Exclusive Ds Semileptonic Decays [PDF]

Apr 27, 2015 - 1. An example of the nonfactorizable weak annihilation contribution in Ds semi- leptonic decays to an s¯

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CBX 2015-001 Justin Hietala Dan Cronin-Hennessy April 27, 2015

Exclusive Ds Semileptonic Decays Abstract We measure six exclusive Ds semileptonic branching ratios using the data collected at 4170 MeV. We isolate our semileptonic events by reconstructing a tagged Ds to identify a Ds∗ Ds event, then we find an electron and the semileptonic hadron. Dropping the Ds∗ daughter photon gives us additional events and avoids the need to model soft photon backgrounds, at the expense of a clean neutrino missing mass. We obtain B(Ds → φeν) = 2.14 ± 0.17 ± 0.09% and B(Ds → ηeν) = 2.28 ± 0.14 ± 0.20% for the two largest branching ratios.

Contents 1 Introduction 1.1 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Light Meson Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Inclusive Ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Analysis Plan

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3 Data Samples and Monte Carlo

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4 Ds Tagging 4.1 Basic Selection Criteria . . 4.1.1 Track Selection . . 4.1.2 Ks Selection . . . . 4.1.3 Photon Selection . 4.1.4 π 0 and η Selection 4.1.5 η 0 Selection . . . . 4.2 Recoil Mass . . . . . . . . 4.3 Individual Tag Mode Cuts 4.4 Fitting Procedure . . . . . 4.5 Results . . . . . . . . . . . 4.5.1 Monte Carlo . . . . 4.5.2 Data . . . . . . . . 4.5.3 Cross-Checks . . .

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5 Semileptonic Selection Criteria 35 5.1 Electron Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1

6 Measurement of Ds → φeν 6.1 φ Selection . . . . . . . . . 6.2 Ds → φeν Reconstruction 6.2.1 Efficiency . . . . . 6.2.2 Backgrounds . . . . 6.2.3 Fit Procedure . . . 6.3 Results . . . . . . . . . . . 6.3.1 Monte Carlo . . . . 6.3.2 Data . . . . . . . .

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7 Measurement of Ds → (Ks , K ∗ , f0 , η 0 )eν 7.1 General Particle Cuts . . . . . . . . . 7.2 Ds → KS eν . . . . . . . . . . . . . . 7.3 Ds → K ∗ eν . . . . . . . . . . . . . . 7.4 Ds → η 0 eν . . . . . . . . . . . . . . . 7.5 Ds → f0 eν . . . . . . . . . . . . . . .

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8 Measurement of Ds → ηeν 8.1 η Selection . . . . . . . . . 8.2 Ds → ηeν Reconstruction 8.2.1 Efficiency . . . . . 8.2.2 Backgrounds . . . . 8.2.3 Fit Procedure . . . 8.3 Results . . . . . . . . . . . 8.3.1 Monte Carlo . . . . 8.3.2 Data . . . . . . . .

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9 Systematic Uncertainties 9.1 Ds Tagging . . . . . . . . . . . . 9.1.1 Signal Shape Variation . . 9.1.2 Background Functions . . 9.1.3 Multiple Candidate Choice 9.2 Tracking . . . . . . . . . . . . . . 9.2.1 Reconstruction . . . . . . 9.2.2 π and K Particle ID . . . 9.3 Photon reconstruction . . . . . . 9.4 Electron ID . . . . . . . . . . . . 9.4.1 Wrong Sign Electron . . . 9.5 Monte Carlo Consistency . . . . . 9.6 Hadron Efficiencies . . . . . . . . 9.6.1 φ . . . . . . . . . . . . . . 9.6.2 η . . . . . . . . . . . . . . 9.6.3 Ks . . . . . . . . . . . . . 9.6.4 K ∗ , η 0 , and f0 . . . . . . . 9.7 Decays in Flight . . . . . . . . . .

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Splitoff Rate . . . . . . . . Hadronic Branching Ratios Semileptonic Fit Functions Ds Production Efficiencies Final State Radiation . . . Initial State Radiation . . Generating Models . . . . Sum of Systematic Errors

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10 Conclusion A f0 → KK Models A.1 EvtGen Models . . . . . . . . . . . . A.1.1 Default Model (Breit-Wigner) A.1.2 Flatt´e Model . . . . . . . . . A.2 Flatt´e Parametrization . . . . . . . .

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B Alternate Ds → φeν Reconstruction Methods B.1 Particle Selections for Alternate Methods . . . . . . . . . . . . . B.2 Methods 1-4: Cut on Ds Invariant Mass, Fit Ds + γ Recoil Mass B.2.1 Comparison of Methods 1-4 . . . . . . . . . . . . . . . . B.3 Method 5: Cut on Ds + γ Recoil Mass, Fit Ds Invariant Mass . B.4 Ds∗ Daughter Photon Efficiencies . . . . . . . . . . . . . . . . . . B.5 Method 6: No Ds∗ Photon Reconstruction . . . . . . . . . . . . . B.6 Comparison of Alternate Methods . . . . . . . . . . . . . . . . .

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C Ds → ηeν Efficiency Systematic

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D Kaon Tracking and Particle ID Systematic

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E Extra Tables

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F Extra Figures

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List of Figures 1

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An example of the nonfactorizable weak annihilation contribution in Ds semileptonic decays to an s¯ s state (e.g. η, η 0 , φ). A similar contribution can appear in B → Xu lν, which may distort the |Vub | measurement if too large. . . . . . . . 20 Monte Carlo (charm and scaled continuum) simulation of the recoil mass distribution for the Ds tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We reject all Ds with a recoil mass below the cut line. . . . . . . . . . . . . . . . . . 27 Ds invariant mass fits in the data, determining the total number of Ds tags for modes KS K, KKπ, KS Kπ 0 , and KS KS π. The pink line represents our fits’ background component, while the blue line represents the signal component. . . 32

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Ds invariant mass fits in the data, determining the total number of Ds tags for modes KKππ 0 , KS K + ππ, KS K − ππ, and πππ. . . . . . . . . . . . . . . . . . . . Ds invariant mass fits in the data, determining the total number of Ds tags for modes πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. . . . . . . . . Fw/RICH in the 20× MC sample for good tracks that are not used in the tagged Ds . We only plot electrons involved in one of our six semileptonic modes (φeν, ηeν, η 0 eν, f0 eν, KS eν, and K ∗ eν). The electron peak at zero comes primarily from tracks with a momentum below 200 MeV. . . . . . . . . . . . . . . . . . . Extra showers after finding the tagged Ds , the φ, and the electron in φeν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays. We don’t see an improvement in our results from making a cut on any extra shower variable. . Generated and reconstructed momentum spectra from the 20× Monte Carlo. a) φ from Ds → φeν, using our cuts. b) Electron from Ds → φeν. . . . . . . . . . . Hit fraction distributions for kaons from φ → KK in Ds → φeν. a) Percentage of such kaons with a given hit fraction, above and below 300 MeV (roughly the median momentum). b) Relative improvement in our kaon reconstruction efficiency when loosening the hit fraction cut from .5 to .1, by momentum. . . . φ mass distribution when reconstructed from φ → KK in the 20× Monte Carlo. The green lines represent a 10 MeV cut (roughly 2Γφ ), which does not capture the high mass tail. We accept φ masses between the red lines. . . . . . . . . . . Efficiencies for individual semileptonic particles and the overall semileptonic side (φ + electron), by momentum. We include the φ → KK branching ratio in our efficiencies, so εφ and εSL must be less than 49%. The φ “typical cuts” have HF > 0.5 and a 10 MeV mass cut. Section 6.1 gives our looser φ selection. . . Ds → φeν backgrounds with a true Ds tag (peaking background) in the 20× Monte Carlo. Only Ds → f0 eν, f0 → KK presents a sizable peaking background for Ds → φeν. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds → φeν data results after our semileptonic selections. We fit the tagged MDs spectrum with a common branching ratio across all 13 tag modes. The likelihood uses each tag mode’s signal shape on its corresponding masses; the above results show a sum over all tag modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . Fit results in the data after applying Ds → φeν semileptonic cuts for modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization. . . . . . . . . . . . . . . Fit results in the data after applying Ds → φeν semileptonic cuts for modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode receives an independent background normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Missing mass after finding the Ds tag, Ks , and electron in Ds → KS eν, from the 20× Monte Carlo. We keep all events with a M M 2 below the line at 0.4 GeV2 . .

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Ds → KS eν missing mass in the 20× Monte Carlo, by background semileptonic mode. The dominant background comes from Ds → φeν, where φ → KL Ks . Our missing mass cut removes most of this background. . . . . . . . . . . . . . . Our figure of merit for different missing mass cut values in Ds → KS eν. We only consider signal and background events that have a reasonable Ds mass, between 1955 MeV– 1985 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure of merit for different flight significance cuts in Ds → KS eν. We only consider events with a Ds tag mass within 1955 MeV and 1985 MeV, since events outside that region will be dismissed as background in our final fit. . . . . Our Ds → KS eν data fit to the MDs spectrum, after all semileptonic selections. This fit function represents the sum over the fit functions from each Ds tag mode. Ds → K ∗ eν missing mass after finding the Ds tag, the electron, and the K ∗ , 2 from the 20× Monte Carlo. Most of the peaking background has a low M Mνγ , but we remove almost half of the combinatoric background with our cut. . . . . Our figure of merit for different missing mass cut values in Ds → K ∗ eν. We only consider signal and background events that have a reasonable Ds mass, between 1955 MeV– 1985 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K ∗ mass for events that pass Ds , electron, and our other Ds → K ∗ eν cuts in the 20× Monte Carlo. Much of our background has a real K ∗ , so we only obtain moderate background reduction from a tighter mass cut. . . . . . . . . . . . . . Figure of merit for different K ∗ mass cuts in Ds → K ∗ eν. We only consider events that have a Ds tag mass within 1955 MeV and 1985 MeV, since events outside that region will be dismissed as background in our final fit. . . . . . . . Our Ds → K ∗ eν data fit to the MDs spectrum, after all semileptonic selections. This fit function represents the sum over the fit functions from each Ds tag mode. Our Ds → η 0 eν data fit to the MDs spectrum, after all semileptonic selections. This fit function represents the sum over the fit functions from each Ds tag mode. f0 mass for events passing Ds and electron cuts in the 20× Monte Carlo. . . . . Figure of merit for different f0 mass cuts in Ds → f0 eν, considering only events with a Ds tag mass within 1955 MeV and 1985 MeV. Since the f0 width has some uncertainty, a 60 MeV mass cut gives us a good balance between retaining most of the signal while not allowing too much excess background. . . . . . . . . Our Ds → f0 eν data fit to the MDs spectrum, after all semileptonic selections. This fit function represents the sum over the fit functions from each Ds tag mode. The EE259 O.K. cut’s effect on the η pull mass distribution. Top: Reconstructed η spectrum with and without the EE259 O.K. cut. Bottom: Normalized η spectrum with and without the EE259 O.K. cut, showing that the cut doesn’t disproportionally change the pull mass distribution (slightly lower efficiency for large pull masses). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds → ηeν missing mass after reconstructing a Ds tag, an η, and the electron in the 20× Monte Carlo. The solid blue line represents all generated Ds → ηeν events that have a correct Ds tag, while the dotted blue line has the additional requirement that the η gets properly reconstructed from its daughter photons (no splitoff or transition photon fakes). . . . . . . . . . . . . . . . . . . . . . . .

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Our figure of merit for different missing mass cuts in the Ds → ηeν signal region (within 3σ of a correct Ds mass and η pull mass). The black dots correspond to the cuts from this analysis, in which we choose a maximum missing mass cut of 500 × 103 MeV2 to err on the side of high efficiency. We have also tried reconstructing the best Ds∗ → Ds γ transition photon when available and incorporating it into the missing four vector (green dots). However, we don’t see an improvement in our figure of merit within the Monte Carlo by including the transition photon, and using it would expose us to potential problems from the modeling of splitoff showers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Generated and reconstructed momentum spectra from the 20× Monte Carlo. a) Lab frame η momentum from Ds → ηeν. b) Electron momentum in Ds → ηeν. . 85 Efficiencies for η, electron, and the overall semileptonic side (η+electron, with M M 2 cut), by momentum. Our η and semileptonic efficiencies include the η → γγ branching ratio of 39.5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Cause of the shower that leads to a false η combination when we have a correct Ds in an Ds → ηeν event, from our 20× Monte Carlo. These false combinations account for 27% of our counts in Ds → ηeν events with a valid Ds tag. Our systematic addresses possible modeling flaws with the data for the three green slices (π splitoff, K splitoff, and K → µν). . . . . . . . . . . . . . . . . . . . . . 88 Recoil mass against the D0 + K ∗ in a 20× Monte Carlo sample. We keep events between the red lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Recoil mass against the D0 + K ∗ from the 3770 MeV data. . . . . . . . . . . . . 91 Full event missing mass, by D0 tag mode for D0 + K ∗ η in a 20× Monte Carlo sample. We select η combinations between the red lines so that we know we have only true η, then we see if there are any other splitoff η combinations in the event. 92 Full event missing mass, by D0 tag mode for D0 + K ∗ η in the 3770 MeV data. We select η combinations between the red lines to determine true η, then we see if there are any other splitoff η combinations in the event. . . . . . . . . . . . . 93 Ds → ηeν 2D fit projections for the reconstructed Ds mass (top) and η pull mass (bottom) in the 20× Monte Carlo, summing over all tag modes. . . . . . . . . . 99 Ds → ηeν 2D fit projections for the reconstructed Ds mass (top) and η pull mass (bottom) in the data, summed over all tag modes. . . . . . . . . . . . . . . . . . 101 Efficiency (including hadron branching ratios) for each semileptonic mode, by dataset. The solid red line gives the average across the full generic Monte Carlo sample, while the dotted red lines show the 1σ range on this average. . . . . . . 108 Top: φ efficiency in the Monte Carlo, by momentum, before and after correcting the efficiency based on the kaon systematic study in Appendix D. Bottom: Ds → φeν semileptonic efficiency, by φ momentum, before and after correction. . 110 Top: φ mass fit in the data, using the signal and background produced in the Monte Carlo. Bottom: Best φ mass fit in the data after allowing the signal Monte Carlo histogram to shift its peak and convoluting it with variable width gaussians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Normalized Ks momentum distributions in Ds decays to KS eν, KS K, and K ∗ K ∗ (Ks K ∓ π ± π ± ). We use KS K to study Ks reconstruction above 650 MeV and K ∗ K ∗ to study the systematic below 650 MeV. . . . . . . . . . . . . . . . . . . 113 6

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Ds + γ + K recoil mass in data KS K events for “found” and “not found” Ks . . 115 Ds + γ + Kππ recoil mass in data K ∗ K ∗ events for “found” and “not found” Ks . The top row shows only low momentum Ks while the bottom row gives results in our medium Ks momentum region, with pKs determined by the recoil momentum.116 Top: Lab frame electron energy (left) and φ momentum (right) in Ds → φeν for the ISGW2 and pole models. The electron energy has a noticeable increase from ISGW2 to the pole model. Bottom: Lab frame electron energy and η momentum in Ds → ηeν. The decay to a pseudoscalar has a smaller but still positive electron energy shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Difference between the ISGW2 and pole models in the q 2 and electron energy distributions for Ds → φeν (top) and Ds → ηeν (bottom). Empty boxes indicate a surplus in the ISGW2 model, while darkened boxes with an “x” indicate a surplus in the simple pole model. The pole model has higher electron energies in both cases, although it has higher q 2 values for the pseudoscalar η decay and lower q 2 values for the vector φ decay. . . . . . . . . . . . . . . . . . . . . . . . . 125 EvtGen produced lineshape for f0 masses above and below the KK threshold at 987.4 MeV. EvtGen changes its behavior from a non-relativistic Breit-Wigner to a relativistic Breit-Wigner discontinuously as the mass crosses threshold. . . . . 133 CLEO Flatt´e mass lineshape for f0 → KK in the decay Ds → KKπ using the default parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 f0 mass lineshapes for M0 = 980 MeV, ggKπ = 2, and Γ0 = 50 MeV. The dotted red lines indicate our φ mass window. . . . . . . . . . . . . . . . . . . . . . . . . 137 f0 mass lineshapes for the M0 , ggKπ , and Γ0 variations. The sharp peak occurs when M0 > 2mK + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Ds + γ recoil mass vs. Ds invariant mass in the charm Monte Carlo. The lower left (blue) plot shows that the two are fairly independent after mass constraining the Ds four vector. The lines indicate our tag fit’s invariant mass and recoil mass cuts. We scale the total and combinatoric plots by a factor of 1/10th relative to the others to keep those plots less visually congested. . . . . . . . . . . . . . . . 141 Ds + γ recoil mass distribution for the 9 tag modes in Monte Carlo. Our dotted line gives the fit results, with red background and blue signal. Our solid lines give the truth-tagged information: blue shows the signal, yellow shows combinatoric background, red shows true Ds pairing with a non-Ds∗ daughter γ (false γ), and green shows false Ds paired with the Ds∗ daughter γ. . . . . . . . . . . . . . . . 143 Ds + γ recoil mass distribution for the 9 tag modes in the data. Our dotted line gives the fit results, with red background and blue signal. . . . . . . . . . . . . . 144 Ds → φeν event’s missing mass distribution (ν missing mass) given a best photon candidate selection in the Monte Carlo. The red lines indicate our “tight” cut. The green histogram shows true Ds → φeν events that get reconstructed with a false Ds∗ daughter photon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Ds → φeν event’s missing mass distribution (ν missing mass) given a best photon candidate selection in the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Ds → φeν event’s missing mass distribution in the Monte Carlo when we allow multiple photon candidates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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Ds → φeν event’s missing mass distribution in the data when we allow multiple photon candidates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Ds invariant mass for events passing the semileptonic φ and electron cuts in the Monte Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Ds invariant mass for events passing the semileptonic φ and electron cuts in the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 The η momentum distributions in Ds → ηeν and Ds → ηρ both peak near 750 MeV and have comparable widths. This contrasts with the alternate source for a clean η sample, ψ 0 → ηJ/ψ, which creates monoenergetic η with a momentum of 199 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Ds → ηeν efficiency (including the η → γγ branching ratio) by dataset. The solid red line gives the average across the full generic Monte Carlo sample, while the dotted red lines show the 1σ range on this average. . . . . . . . . . . . . . . 158 Top: Ds + γ recoil mass in the Monte Carlo for events with a Ds + γ + ρ recoil mass between 500 MeV and 600 MeV. Bottom: Ds + γ + ρ recoil mass for Monte Carlo events that have a Ds + γ recoil between 1955 MeV and 1990 MeV. . . . . 159 Top: 2D fit projections for the Ds + γ recoil mass in the data from events with a Ds + γ + ρ recoil mass between 500 MeV and 600 MeV. Bottom: Fit projection for the Ds + γ + ρ recoil mass from data events that have a Ds + γ recoil between 1955 MeV and 1990 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Top: Ds +γ+ρ recoil mass in the Monte Carlo for events with −3.5 < ση < 2.5. Bottom: η pull mass for Monte Carlo events that have a Ds +γ +ρ recoil between 500 MeV and 600 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Top: 2D fit projections of the Ds + γ + ρ recoil mass for data events with −3.5 < ση < 2.5. Bottom: Fit projection for the η pull mass from data events that have a Ds + γ + ρ recoil between 500 MeV and 600 MeV. . . . . . . . . . . 162 π ± π ± D∓ recoil mass fits for our φ kaon selections in the Monte Carlo. The left plots show the recoil mass when we find a kaon, while the right plots show the recoil mass when we don’t find the kaon. The top plots contain recoil momenta below 250 MeV, the middle plots have recoil momenta between 250 MeV and 500 MeV, and the bottom plots have recoil momenta above 500 MeV. . . . . . . 164 ππD recoil mass fits for our φ kaon selections in the data. The left plots show the recoil mass when we find a kaon, while the right plots show the recoil mass when we don’t find the kaon. The top plots contain recoil momenta below 250 MeV, the middle plots have recoil momenta between 250 MeV and 500 MeV, and the bottom plots have recoil momenta above 500 MeV. . . . . . . . . . . . . . . . . 165 Monte Carlo kaon efficiency for each set of kaon selections, by momentum. Our φ kaon cuts (hit fraction dropped) show a higher efficiency in each momentum range, with a particular relative advantage in the important low momentum region.166 Kaon efficiencies in the data for each set of kaon selections, by momentum. The error bars on the efficiencies (barely visible) include both a statistical error and the systematic error from fitting. The relative difference between the selection efficiencies roughly matches the Monte Carlo, although the absolute efficiencies for soft kaons all fall below their corresponding Monte Carlo efficiencies. . . . . . 167

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The relative difference between the kaon efficiency in data and the kaon efficiency in Monte Carlo, by selections and momentum range. The high momentum region requires no correction, the middle momentum region requires a slight efficiency correction, and softest kaons require a sizable efficiency correction. The error bars include both statistical and systematic errors from our tracking/PID reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D → Kππ and Ds → φeν daughter kaon momentum distributions, scaled to line up the left momentum bin. Because the kaon momentum distribution for Ds → φeν falls off so sharply in the middle bin relative to Kππ, we perform an additional systematic by splitting the bin into two halves and doing a separate efficiency correction for each. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo (charm and scaled continuum) simulation of the recoil mass distribution for the Ds tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. Again, we reject all Ds with a recoil mass below the cut line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recoil mass distribution for the Ds tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 in the data. We reject Ds with a recoil mass below the cut line. . . Recoil mass distribution for the Ds tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ in the data. We reject Ds with a recoil mass below the cut line. . . . . . . . . . . . . . . . . . . . . . . Monte Carlo truth tagged plots of the invariant mass vs. recoil mass distribution for Ds tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . Note that the recoil mass and the invariant mass are almost entirely uncorrelated. . . . . . . . Monte Carlo truth tagged plots of the invariant mass vs. recoil mass distribution for Ds tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. Again, the recoil mass and the invariant mass show little correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo plots of the invariant mass vs. recoil mass distribution, including properly weighted charm and continuum background, for Ds tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo plots of the invariant mass vs. recoil mass distribution, including properly weighted charm and continuum background, for Ds tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The invariant mass vs. recoil mass distribution in data for Ds tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . The data distribution doesn’t show any unexpected behavior relative to the Monte Carlo expectation. . . . . . . . . . . The invariant mass vs. recoil mass distribution in data for Ds tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fits to the truth-tagged Ds invariant mass from the Monte Carlo. We fix the fit function’s shape parameters (relative normalization, relative width, and crystal ball power law tail) from these results. These plots show the fit results for Ds to KS K, KKπ, KS Kπ 0 , and KS KS π. . . . . . . . . . . . . . . . . . . . . . . . . .

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Fits to the truth-tagged Ds invariant mass from the Monte Carlo. We fix the fit function’s shape parameters (relative normalization, relative width, and crystal ball power law tail) from these results. These plots show the fit results for Ds to KKππ 0 , KS K + ππ, KS K − ππ, and πππ. . . . . . . . . . . . . . . . . . . . . . . Fits to the truth-tagged Ds invariant mass from the Monte Carlo. We fix the fit function’s shape parameters (relative normalization, relative width, and crystal ball power law tail) from these results. These plots show the fit results for Ds to πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. . . . . . . . . . . . . Ds invariant mass fits in the weighted 20× Monte Carlo sample (charm + continuum), determining the total number of Ds tags for modes KS K, KKπ, KS Kπ 0 , and KS KS π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds invariant mass fits in the weighted 20× Monte Carlo sample (charm + continuum), determining the total number of Ds tags for modes KKππ 0 , KS K + ππ, KS K − ππ, and πππ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds invariant mass fits in the weighted 20× Monte Carlo sample (charm + continuum), determining the total number of Ds tags for modes πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. . . . . . . . . . . . . . . . . . Ds invariant mass fits in the data sample using a signal histogram from the truth-tagged Monte Carlo. These plots show our results for Ds to KS K, KKπ, KS Kπ 0 , and KS KS π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds invariant mass fits in the data sample using a signal histogram from the truthtagged Monte Carlo. These plots show our results for Ds to KKππ 0 , KS K + ππ, KS K − ππ, and πππ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds invariant mass fits in the data sample using a signal histogram from the truth-tagged Monte Carlo. These plots show our results for Ds to πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. . . . . . . . . . . . . . . . . . Ds invariant mass fits for a double gaussian/gaussian+crystal ball signal shape compared to fits with a signal histogram for modes KS K, KKπ, KS Kπ 0 , and KS KS π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds invariant mass fits for a double gaussian/gaussian+crystal ball signal shape compared to fits with a signal histogram for modes KKππ 0 , KS K + ππ, KS K − ππ, and πππ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds invariant mass fits for a double gaussian/gaussian+crystal ball signal shape compared to fits with a signal histogram for modes πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. . . . . . . . . . . . . . . . . . . . . . . . . . . Extra showers after finding the tagged Ds , the η, and the electron in ηeν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays. . . . . . . . . Extra showers after finding the tagged Ds , the η 0 , and the electron in η 0 eν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays. . . . . . . . . Extra showers after finding the tagged Ds , the f0 , and the electron in f0 eν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays. . . . . . . . .

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Extra showers after finding the tagged Ds , the Ks , and the electron in KS eν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays. . . . Extra showers after finding the tagged Ds , the K ∗ , and the electron in K ∗ eν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays. . . . Likelihood fit results for the Ds mass spectrum after all Ds → φeν semileptonic cuts in the first four data-sized Monte Carlo samples. The blue histogram represents the Monte Carlo truth-tagged events, while the blue fit line gives the signal part of our fit. The red fit line represents non-peaking background, and the green line shows our peaking background subtraction. . . . . . . . . . . . . . Ds → φeν data-sized Monte Carlo results, second group of datasets. . . . . . . . Ds → φeν data-sized Monte Carlo results, third group of datasets. . . . . . . . . Ds → φeν data-sized Monte Carlo results, fourth group of datasets. . . . . . . . Ds → φeν data-sized Monte Carlo results, fifth group of datasets. . . . . . . . . Ds → KS eν backgrounds with a true Ds tag (peaking background), from the 20× Monte Carlo. These remain after KS eν semileptonic cuts but before any missing mass cut or other, additional background restrictions. φeν with φ → KL Ks dominates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds → KS eν backgrounds with a true Ds tag (peaking background), after all cuts. The other semileptonic modes each give some fake events, while the dominant non-semileptonic contribution comes from Ds tag modes with a kaon faking the electron (e.g. Ds → KKs ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds → KS eν fit results in the data, after all semileptonic cuts, for tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization. . . . . . . . . . . . . . . Ds → KS eν fit results in the data, after all semileptonic cuts, for tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds → K ∗ eν backgrounds with a true Ds tag (peaking background), before our specific K ∗ eν cuts in the 20× Monte Carlo. Our best improvement in peaking background will come from reducing Ds → φeν where one kaon fakes a pion. . . Ds → K ∗ eν backgrounds with a true Ds tag (peaking background), after all cuts. The other semileptonic modes each give some fake events, while the dominant non-semileptonic contribution comes from Ds tag modes where a kaon fakes the electron (e.g. Ds → KKπ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ds → K ∗ eν fit results in the data, after all semileptonic cuts, for tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode receives an independent background normalization. . . . . . . . . . . . . . . . .

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113 Ds → K ∗ eν fit results in the data, after all semileptonic cuts, for tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Ds → η 0 eν backgrounds with a true Ds tag (peaking background) in the 20× Monte Carlo. ηeν with η → ππ(π 0 /γ) produces the most peaking background, while the dominant non-semileptonic contribution comes from Ds tag modes with a kaon faking the electron (e.g. Ds → KKs π 0 ) . . . . . . . . . . . . . . . . . . . 115 Ds → η 0 eν fit results in the data, after all semileptonic cuts, for tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode receives an independent background normalization. . . . . . . . . . . . . . . . . . . . . . 116 Ds → η 0 eν fit results in the data, after all semileptonic cuts, for tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Ds → f0 eν backgrounds with a true Ds tag (peaking background) in the 20× Monte Carlo. η 0 eν with η 0 → ππX provides the plurality contribution, while the dominant non-semileptonic peaking background comes from Ds tag modes where a kaon fakes the electron (e.g. Ds → KKs ). . . . . . . . . . . . . . . . . . 118 Ds → f0 eν fit results in the data, after all semileptonic cuts, for tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode receives an independent background normalization. . . . . . . . . . . . . . . . . . . . . . 119 Ds → f0 eν fit results in the data, after all semileptonic cuts, for tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 χ2 for data fits using various shifts and gaussian smears to the Monte Carlo’s signal Mφ distribution. Smaller shifts and smears tend to be favored, implying a fairly accurate φ mass resolution in the Monte Carlo. . . . . . . . . . . . . . . 121 Our large Mφ cut window means that even φ lineshapes that don’t fit the data ) of particularly well still have a relative efficiency difference from predicted ( ∆ε ε0 less than 0.1%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over the full 20× Monte Carlo when we tag Ds → KS K. The bottom two plots do the same for Ds → KKπ. . . . . . . . . . 123 The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over the full 20× Monte Carlo when we tag Ds → KKππ 0 . The bottom two plots do the same for Ds → KS K + ππ. . . . . .

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124 The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over the full 20× Monte Carlo when we tag Ds → πη. The bottom two plots do the same for Ds → ππ 0 η. . . . . . . . . . . . 125 The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over all Ds tag modes for one data-sized Monte Carlo sample (dataset 0). The bottom two plots give the projections for a different data-sized Monte Carlo sample (dataset 1). . . . . . . . . . . . . . . . 126 The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over all Ds tag modes for another data-sized Monte Carlo sample (dataset 2). The bottom two plots give the projections for a fourth data-sized Monte Carlo sample (dataset 3). . . . . . . . . . . . . . . . . . . . . . 127 Ds + γ + K recoil mass in KS K events for “found” and “not found” Ks , from the Monte Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Ds + γ + Kππ recoil mass in K ∗ K ∗ events for “found” and “not found” Ks , from the Monte Carlo. The top row corresponds to low momentum Ks while the bottom row corresponds to our medium Ks momentum region, as determined by the recoil momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Top: Signal shape fit to the K ∗ mass in K ∗ K. Bottom: K ∗ mass fit after allowing the MK ∗ signal shape to shift left or right and convoluting it with a variable width gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Top: Signal shape fit to the f0 mass in f0 π. Bottom: f0 mass fit after allowing the Mf0 signal shape to shift left or right and convoluting it with a variable width gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Top: Signal shape fit to the η 0 mass in πη 0 , η 0 → ππη. Bottom: η 0 mass fit after allowing the Mη0 signal shape to shift left or right and convoluting it with a variable width gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Top: Lab frame electron energy (left) and η 0 momentum (right) in Ds → η 0 eν for the ISGW2 and pole models. Bottom: Lab frame electron energy and f0 momentum in Ds → f0 eν. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Top: Lab frame electron energy (left) and Ks momentum (right) in Ds → KS eν for the ISGW2 and pole models. Bottom: Lab frame electron energy and K ∗ momentum in Ds → K ∗ eν. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 q 2 distribution under different models for decays with a vector hadron (Ds → φeν) and a pseudoscalar hadron (Ds → ηeν). The difference between the Monte Carlo and ISGW2 for low q 2 in φeν comes from a correction we make to the Monte Carlo’s masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Relationship between the q 2 and electron energy in the Ds rest frame for Ds → φeν. Top: ISGW2 model. Bottom: Pole model. . . . . . . . . . . . . . . . . . . . . . 136 Relationship between the q 2 and electron energy in the Ds rest frame for Ds → ηeν. Top: ISGW2 model. Bottom: Pole model. . . . . . . . . . . . . . . . . . . . . . 137 Ds invariant mass fits after making a Ds + γ recoil mass cut in the Monte Carlo. The dotted blue and pink lines give our signal and background fit functions, while our solid blue and pink lines give the truth-tagged signal and background. 138 Ds invariant mass fits after making a Ds + γ recoil mass cut in the data. The dotted blue and pink lines give our signal and background fit functions. . . . . . 13

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Dataset luminosities determined from Bhabha events (e+ e− → e+ e− ), with statistical and systematic errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 CLEO code release used to generate each background MC sample. . . . . . . . . 22 Kinematically allowed recoil mass and momentum ranges for Ds mesons at 4170 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Recoil mass cut, by Ds tag mode. . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Rejections based on individual tag modes’ daughter particles. . . . . . . . . . . . 28 Ds invariant mass fit functions, by mode. . . . . . . . . . . . . . . . . . . . . . . 29 Tagging results from the 20× Monte Carlo sample, scaled to data size. . . . . . 30 Overall Ds tagging efficiency from signal Monte Carlo, including all branching ratios. Our Ds recoil mass cut creates the efficiency difference between tags from prompt Ds and tags from secondary Ds . . . . . . . . . . . . . . . . . . . . . . . 31 Tagging results from the full data sample (sum of datasets 39, 40, 41, 47, 48). . 35 Signal histogram fit results compared to our standard double gaussian/gaussian+crystal ball fit results in the 20× Monte Carlo sample, scaled to data size. . . . . . . . . 35 Signal histogram fit results compared to our standard double gaussian/gaussian+crystal ball fit results in the full data sample. . . . . . . . . . . . . . . . . . . . . . . . . 36 Fit results from the Monte Carlo’s 20 data-sized samples. The final column in the ”Sum” row gives the total χ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Effect of the electron cuts (track and Fw/RICH ) in the 20× Monte Carlo sample for truth-tagged semileptonic and generic decay modes. These precede any semileptonic hadron cut, but passed Ds tags must fall within the tagging fit window (1900 MeV < MDs < 2030 MeV). . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Effect of an extra track cut on the signal and the background in the peaking Ds mass region after all other selections (e.g. semileptonic hadron cuts) have been S2 , such that ffpost-cut measures our statistical improvement. 38 made. We define f as S+B pre-cut Efficiencies for semileptonic particles in Ds → φeν, with typical φ cuts (HF > 0.5, φ mass within 10 MeV). The efficiencies include the φ → KK branching ratio. . 44 Efficiencies for semileptonic particles in Ds → φeν, with the φ cuts used in this analysis. The efficiencies include the φ → KK branching ratio. . . . . . . . . . . 44 Truth-tagged breakdown for Ds → φeν candidates passing all cuts in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. . . . . . . . . . . . . 46 Absolute branching ratio correction and systematic error for B(Ds → φeν) from peaking background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Test of potential bias in our fitting procedure for Ds → φeν by comparing the number of truth-tagged semileptonic events to the fit result. We allow crossfeed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. 49

14

20

21 22 23 24 25 26 27 28 29 30 31 32 33 34

35 36 37 38 39

Monte Carlo comparison of the measured Ds → φeν branching ratio to its generating branching ratio (2.170%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. . . . . . . . . . . . . . . . . . . . . . . 50 Ds → φeν measurement in the data, including the peaking background correction from Table 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Truth-tagged breakdown for Ds → KS eν candidates in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. . . . . . . . . . . . . . . . . . . . . 61 Absolute branching ratio correction and systematic error for B(Ds → KS eν) from peaking background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Ds → KS eν measurement in the data, including the peaking background correction from Table 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Truth-tagged breakdown for Ds → K ∗ eν candidates in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. . . . . . . . . . . . . . . . . . . . . 69 Absolute branching ratio correction and systematic error for B(Ds → K ∗ eν) from peaking background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Ds → K ∗ eν measurement in the data, including the peaking background correction from Table 26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Truth-tagged breakdown for Ds → η 0 eν candidates in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. . . . . . . . . . . . . . . . . . . . . 72 Absolute branching ratio correction and systematic error for B(Ds → η 0 eν) from peaking background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Ds → η 0 eν measurement in the data, including the peaking background correction from Table 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Truth-tagged breakdown for Ds → f0 eν candidates passing all cuts in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. . . . . . . . . . . . . 77 Absolute branching ratio correction and systematic error for B(Ds → f0 eν, f0 → ππ) from peaking background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Ds → f0 eν, f0 → ππ measurement in the data, including the peaking background correction from Table 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Semileptonic-side efficiencies for φeν, η 0 eν, f0 eν, KS eν, and K ∗ eν, after all cuts. The first column includes the hadron branching ratios into the efficiency, while the second column gives the efficiency considering only hadron decays to the reconstruced decay mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Efficiencies for semileptonic particles in Ds → ηeν, with the η cuts used in this analysis. The η and semileptonic efficiencies include the η → γγ branching ratio. 84 Efficiencies for semileptonic particles in Ds → ηeν, with |ση | < 3.0. The efficiencies include the η → γγ branching ratio. . . . . . . . . . . . . . . . . . . . . . . 84 Truth-tagged breakdown for Ds → ηeν candidates in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. . . . . . . . . . . . . . . . . . . . . 87 D0 tag modes used to estimate splitoff systematic for Ds modes. . . . . . . . . . 89 Rate of additional η formed using splitoff showers, by D0 mode. The data/MC splitoff correction error (extra splitoff factor) includes a small systematic from combinatoric background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

15

40

41 42

43

44 45 46 47 48 49 50 51

52 53 54 55

56 57

58

Rate of additional η formed with splitoff showers after applying an EE259 cut, by D0 mode. The data/MC splitoff correction error (extra splitoff factor) includes a small systematic from combinatoric background. . . . . . . . . . . . . . . . . . 94 Absolute branching ratio correction and systematic error for B(Ds → ηeν) from peaking background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Monte Carlo comparison of the measured Ds → ηeν branching ratio to its generating branching ratio (2.480%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. . . . . . . . . . . . . . . . . . . . . . . 97 Test of potential bias in our fitting procedure for Ds → ηeν by comparing the number of truth-tagged semileptonic events to the fit result. We allow crossfeed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. 98 Ds → ηeν measurement in the data, including the peaking background correction from Table 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Ds → ηeν branching ratio and errors under both η efficiency systematic scenarios.100 D± tag mode used for each Ds mode’s relative normalization and relative width systematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Systematic errors from our Ds tag fits. . . . . . . . . . . . . . . . . . . . . . . . 103 Systematic errors from our Ds tag background shape. . . . . . . . . . . . . . . . 104 Relative systematic error from the multiple candidate efficiency difference between semileptonic and all other Ds decay modes. . . . . . . . . . . . . . . . . . 105 Electron particle identification systematic and efficiency correction, by semileptonic mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Passing events with a good Ds when reconstructing each semileptonic mode using an electron of the wrong charge. Our errors for the reconstructed events in each mode slightly exceed that mode’s counts (all six modes are consistent with zero). 107 Ks efficiency systematic and correction from our found/not found recoil mass fits in each momentum region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Summary of semileptonic hadron systematic errors. Our kaon and η systematics have been included into the K ∗ and η 0 PID columns, respectively. . . . . . . . . 117 Summary of hadron efficiency corrections from particle identification. . . . . . . 117 Systematic for true semileptonic events that pass with incorrect particle identification, mostly due to π or K decays in flight to µ. We take 50% of the effect’s size in Monte Carlo as our systematic. . . . . . . . . . . . . . . . . . . . . . . . 118 Systematic errors and efficiency corrections from uncertain or changed branching ratios in semileptonic daughter hadron decays. . . . . . . . . . . . . . . . . . . . 118 Branching ratio change from a different semileptonic background function. The Ds → ηeν line combines changes to both the pull mass and Ds mass backgrounds. In all cases, the systematic from choosing a different background shape falls well below the statistical or systematic error. . . . . . . . . . . . . . . . . . . . . . . 119 Relative systematic for various Ds production rate uncertainties. This combines the uncertainties from the Ds Ds and Ds∗ Ds cross sections at 4170 MeV with the uncertainty from the Ds∗ branching ratio (the fraction going to Ds γ vs. Ds π 0 ). These combined effects still contribute a negligible systematic. . . . . . . . . . . 120 16

59 60 61 62

63 64 65 66 67

Efficiency difference due to final state radiation, by Ds semileptonic mode. . . . Efficiency difference due to initial state radiation, by Ds semileptonic mode. . . Relative systematic from different generating models’ reconstruction efficiency. . Efficiency for each Ds semileptonic mode before and after corrections from systematic biases. These efficiencies include the hadronic branching ratio (taking B(f0 → ππ) = 52% for f0 eν). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total systematic errors (relative) for each Ds semileptonic decay mode. . . . . . Branching ratios for each Ds semileptonic mode before and after our systematic biases and errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of observed signal events for each of our six semileptonic modes. We include the branching ratios with their statistical errors for reference. . . . . . . This analysis’s measured branching ratios for each Ds semileptonic mode. . . . . f0 parameter variations used to determine our f0 → KK correction in Ds → φeν. Our variations correspond to the PDG ranges for the physical mass, Γ0 , and Γππ ππ . In practice, we vary ggKπ instead of directly varying ΓππΓ+Γ since ggKπ Γππ +ΓKK KK as shorthand for has less correlation with the mass and Γ0 . We use f × BBKK ππ +

68 69

70 71 72

73 74 75 76 77 78 79 80 81



121 122 123

126 126 127 128 128

0 →K K ) fwindow × B(f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 B(f0 →π + π − ) Correction and systematic for B(Ds → φeν) from Ds → f0 eν, f0 → KK background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Ds + γ recoil mass tags in the data and Monte Carlo. The crystal ball function tends to undercount the number of tags across all modes, so we adjust the final branching ratio for this systematic effect. . . . . . . . . . . . . . . . . . . . . . . 142 Branching ratios in Monte Carlo for each of the four methods that use Ds + γ tags. Errors are statistical only. . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Branching ratios in the data for each of the four methods that use Ds + γ tags. Errors are statistical only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Ds invariant mass tags in the data and Monte Carlo after cutting on the Ds + γ recoil mass. We only allow each Ds mass to enter once, regardless of the number of Ds + γ combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Ds∗ daughter photon efficiency in data and Monte Carlo. . . . . . . . . . . . . . 152 Rate at which valid Ds without a correctly reconstructed Ds∗ daughter photon will still pass all tagging cuts (including the Ds + γ recoil mass). . . . . . . . . . 153 Ds invariant mass tags in the data and Monte Carlo after a Ds momentum cut. We do not require a pairing with a photon. . . . . . . . . . . . . . . . . . . . . . 154 Branching ratios in Monte Carlo for our different Ds → φeν alternate methodologies. Errors are statistical only. . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Branching ratios in the data for our different Ds → φeν alternate methodologies. Errors are statistical only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Relative kaon systematic and efficiency correction for our loose (φeν) kaon selection.169 Relative kaon systematic and efficiency correction for our medium (K ∗ eν) kaon selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Relative kaon systematic and efficiency correction for our standard kaon selection.169 Number of Ds tags in data and Monte Carlo, by dataset. We fit each dataset independently for this comparison and scale the Monte Carlo to data size. . . . . 171

17

82 83

84

85

86

87

88

89

90

91 92 93 94 95

Number of Ds tags in data and Monte Carlo, by mode. We scale the Monte Carlo to the data luminosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Test of potential bias in our fitting procedure for Ds → KS eν by comparing the number of truth-tagged semileptonic events to the fit result. We allow crossfeed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. 172 Monte Carlo comparison of the measured Ds → KS eν branching ratio to its generating branching ratio (0.090%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. . . . . . . . . . . . . . . . . . . . . . . 173 Test of potential bias in our fitting procedure for Ds → K ∗ eν by comparing the number of truth-tagged semileptonic events to the fit result. We allow crossfeed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. 174 Monte Carlo comparison of the measured Ds → K ∗ eν branching ratio to its generating branching ratio (0.190%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. . . . . . . . . . . . . . . . . . . . . . . 175 Test of potential bias in our fitting procedure for Ds → η 0 eν by comparing the number of truth-tagged semileptonic events to the fit result. We allow crossfeed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. 176 Monte Carlo comparison of the measured Ds → η 0 eν branching ratio to its generating branching ratio (0.860%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. The abnormally high χ2 just reflects low Ds → η 0 eν statistics that distort gaussian error sums (Table 87 gives a more meaningful comparison for this mode). . . . . . . . . . . . . . . . . . . . . . . . 177 Test of potential bias in our fitting procedure for Ds → f0 eν by comparing the number of truth-tagged semileptonic events to the fit result. We allow crossfeed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. 178 Monte Carlo comparison of the measured Ds → f0 eν branching ratio to its generating branching ratio (0.310%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. . . . . . . . . . . . . . . . . . . . . . . 179 Allowed Ds mass range at 3σ, from a gaussian fit. We allow a broader range of masses for the full analysis, but we use this restricted range for systematic checks.180 Summary of various systematic errors for our electron identification. . . . . . . . 181 Relative corrections to the electron identification efficiency for each of our six semileptonic modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Semileptonic-side efficiencies in Ds → η 0 eν, including the η 0 → ππη and η → γγ branching ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Semileptonic-side efficiencies in Ds → f0 eν, including the f0 → ππ branching ratio.182

18

96 97 98 99 100 101 102 103 104 105 106 107 108 109

1

Semileptonic-side efficiencies in Ds → KS eν, including the Ks → ππ branching ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semileptonic-side efficiencies in Ds → K ∗ eν, including the K ∗ → Kπ branching ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . All systematic efficiency corrections (relative) for Ds → φeν. . . . . . . . . . . . All systematic efficiency corrections (relative) for Ds → ηeν. . . . . . . . . . . . All systematic efficiency corrections (relative) for Ds → η 0 eν. . . . . . . . . . . . All systematic efficiency corrections (relative) for Ds → f0 eν. . . . . . . . . . . . All systematic efficiency corrections (relative) for Ds → KS eν. . . . . . . . . . . All systematic efficiency corrections (relative) for Ds → K ∗ eν. . . . . . . . . . . All systematic errors (relative) for Ds → φeν. . . . . . . . . . . . . . . . . . . . All systematic errors (relative) for Ds → ηeν. . . . . . . . . . . . . . . . . . . . All systematic errors (relative) for Ds → η 0 eν. . . . . . . . . . . . . . . . . . . . All systematic errors (relative) for Ds → f0 eν. . . . . . . . . . . . . . . . . . . . All systematic errors (relative) for Ds → KS eν. . . . . . . . . . . . . . . . . . . All systematic errors (relative) for Ds → K ∗ eν. . . . . . . . . . . . . . . . . . .

182 182 183 183 183 183 184 184 185 185 186 186 187 187

Introduction

Ds semileptonic decays have applications both for QCD tests and for light meson spectroscopy. Most notably, exclusive Ds decays to the dominant modes (φeν, η [0] eν) involve no light valence quarks and thus provide an ideal opportunity for comparisons to lattice QCD results. Additionally, since the Ds primarily couples to the final state hadron’s s¯ s component, Ds decay rates can probe the quark content of η/η 0 and of the scalar f0 .

1.1

Lattice QCD

The lattice QCD formalism allows QCD processes to be computed numerically, using a discrete space-time grid with only the bare quark masses and αs as inputs. Lattice QCD has proven very useful in a variety of processes, including the extraction of CKM parameters from semileptonic decays to light mesons [1]. However, the required lattice discretization size depends on the quark masses involved, and smaller lattice sizes require more computing power. While a few lattice analyses have recently begun to get down to the level of the physical light (up and down) quark masses [2], lattice calculations still typically set the light quark masses to a higher than physical value, expressed as a fraction of the strange quark mass [3, 4, 5]. They then use different lattice grid sizes and extrapolate to the continuum limit. These lattice discretization effects tend to dominate lattice calculation errors. Ds semileptonic decays to s¯ s states provide an excellent test of lattice QCD procedures as they allow lattice calculations to use the strange quark mass for the valence quarks rather than extrapolating to light quark masses.

19

1.2

Light Meson Spectroscopy

Semileptonic Ds decays most often result in s¯ s final states, which affords us an opportunity to probe a sector that can otherwise be difficult to access cleanly [6]. In particular, Ds semileptonic decays can potentially probe the s¯ s content of the η and η 0 , and they can shed insight into the f0 (980) quark content and structure. When decaying semileptonically to pseudoscalars, the Ds couples to the s¯ s component of η 0 and η . Conversely, when charged D semileptonic decays result in an η or η 0 , they couple to the mesons’ dd¯ component. Since the decays have related kinematics, a comparison of the four decay widths should determine the strange and nonstrange q q¯ content of the η/η 0 . ¯ molecule, or The f0 (980) has been considered to consist of a q q¯ state, a qq q¯q¯ state, a K K even to have a gluon component [7]. Since the Ds transitions to the f0 particle’s s¯ s component, Ds → f0 eν should provide information on the underlying quark content of the f0 . BaBar may have seen S-wave interference with the φeν, φ → K + K − final state [8], and this mechanism could also lead to a deeper understanding of the f0 substructure.

1.3

Inclusive Ds

Given CLEO’s inclusive Ds measurement [9], the six Ds semileptonic modes considered in this analysis (φeν, ηeν, η 0 eν, f0 eν, K ∗ eν, KS eν) saturate most of the total semileptonic width. Knowing the components of the Ds semileptonic width should improve phenomenological comparisons that use the inclusive Ds spectrum. In the most prominent example, heavy quark symmetry allows a constraint on the weak annihilation (four-quark, Figure 1) component of B → Xu lν semileptonic decays that would otherwise complicate the |Vub | measurement. This constraint comes from comparing the difference of charged and neutral B semileptonic widths to the difference in Ds and D0 semileptonic widths, which should be related up to factors like m3b /m3c [10, 11].

c Ds

ν



W

e s s¯

Xs

Figure 1: An example of the nonfactorizable weak annihilation contribution in Ds semileptonic decays to an s¯ s state (e.g. η, η 0 , φ). A similar contribution can appear in B → Xu lν, which may distort the |Vub | measurement if too large.

20

2

Analysis Plan

We intend to measure branching ratios for six Ds semileptonic decays (Ds → φeν, ηeν, η 0 eν, f0 eν, KS eν, and K ∗ eν). These cover all resonant Ds semileptonic decays up to the singly Cabibbo suppressed level. We use CLEO-c’s 4170 MeV data, where 95% of the Ds sample comes from Ds∗ Ds events [12, 13], and the remainder come from Ds+ Ds− . The Ds∗ decays to Ds γ nearly all the time (94%) [1], with Ds∗ → Ds π 0 making up the difference. Candidate Ds semileptonic events then contain one Ds+ , one Ds− , and either zero, one, or two photons. For all six modes, we reconstruct the nonsemileptonic Ds through one of 13 ”tag” modes. We also reconstruct the semileptonic side’s electron and hadron (φ, η, η 0 , f0 , Ks , or K ∗ ). We do not attempt to reconstruct the photon(s) from a possible Ds∗ decay, which increases our overall efficiency but costs us a clean neutrino missing mass. We use the following 13 Ds tag modes to determine candidate events: KS K; KKπ; KS Kπ 0 ; KS KS π; KKππ 0 ; KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. Of these modes, KS K, KKπ, KKππ 0 , πππ, πη, πη 0 , η 0 → ππη give the most statistical power as they contain over 60% of the tags and have lower relative background than the other 7 modes. The signal modes Ds → φeν and Ds → ηeν should have the largest branching ratios (around a couple percent) due to their s¯ s components, about an order of magnitude higher than the Cabibbo suppressed decays to KS eν, K ∗ eν, and f0 eν. We expect Ds → η 0 eν to have a branching ratio somewhere in between these levels. Ignoring the Ds∗ daughter photons does have the limitation that the branching ratio results dΓ cannot be easily converted into dq 2 . Consequently, we do not determine form factors in this analysis.

3

Data Samples and Monte Carlo

We use the CLEO-c data collected at a 4170 MeV center-of-mass energy (datasets 39, 40, 41, 47, and 48), with a total integrated luminosity of 586 pb−1 (Table 1). At this energy, the e+ e− collisions produce both Ds Ds and Ds∗ Ds . Ds∗ Ds production has the much larger cross section at 4170 MeV, with a σDs∗ Ds of 0.92 nb and a σDs+ Ds− of 0.03 nb [12]. With these cross sections, we expect to have about 1.11×106 Ds mesons in the data sample. Table 1: Dataset luminosities determined from Bhabha events (e+ e− → e+ e− ), with statistical and systematic errors. Dataset 39 40 41 47 48 Total

Integrated lum. (pb−1 ) 55.12 ± 0.03 ± 0.55 123.93 ± 0.05 ± 1.24 119.11 ± 0.05 ± 1.19 109.78 ± 0.05 ± 1.10 178.23 ± 0.06 ± 1.78 586.17 ± 0.11 ± 5.86

21

We use CLEO’s generic ddmix Monte Carlo for each of the 4170 MeV datasets, which generates a weighted mixture of all charm production (DD, D∗ D, D∗ D∗ , Ds Ds , Ds∗ Ds ). Each ddmix sample has 20× the data luminosity, and in all cases we use the final regenerated sample from CLEO’s 20080404 MCGEN 1 release (which includes ISR). The continuum, radiative return, and tau Monte Carlo samples used for backgrounds each simulate 5× the data luminosity, with a release that differs by dataset. Table 2 lists these releases. For all six semileptonic modes, the backgrounds from charm mesons dominate the backgrounds from continuum, radiative return, and tau production. Table 2: CLEO code release used to generate each background MC sample. Dataset 39 40 41 47 48

Release 20060426 MCGEN 20060426 MCGEN 20060426 MCGEN 20080404 MCGEN 20080404 MCGEN

2 2 1 1

EVTGEN [14] generated the charm, continuum, and radiative return samples, while QQ generated the tau samples. The continuum generation used the Lund area law generator. We have also created four signal Monte Carlo samples for each of our six semileptonic modes (φeν, ηeν, η 0 eν, f0 eν, KS eν, and K ∗ eν). The four samples correspond to different Ds production modes at 4170 MeV: Ds Ds ; Ds∗ Ds with Ds∗ → γ(Ds → heν); Ds∗ Ds with Ds∗ → π 0 (Ds → heν); and Ds∗ Ds with the prompt Ds → heν. We generated 250,000 events for each of our φeν and ηeν signal Monte Carlo samples, with 100,000 events for the other semileptonic mode samples. We processed both data and Monte Carlo with the 20060224 FULL A 3 release to maintain consistency with version 2 D skims [15, 16].

4

Ds Tagging

CLEO-c produces nearly all of its Ds sample at a 4170 MeV center-of-mass energy. While this energy gives the most Ds mesons, the total Ds cross section at 4170 MeV still falls slightly below 1.0 nb. That compares to a 9 nb total charm cross section (dominated by D∗ D∗ at 4.7 nb and D∗ D at 2.6 nb) [13] and a 12 nb continuum cross section [1, 12] at 4170 MeV. Since the lost neutrino prevents us from reconstructing the entire Ds semileptonic decay, leaving only the hadron (φ, η, η 0 , f0 , Ks , or K ∗ ) and electron, we need to find some other way to isolate Ds events lest we be smothered by combinatoric background and other decays containing an electron and target hadron. As all Ds events contain two Ds mesons, we can entirely reconstruct, or tag, one of the mesons as an event requirement for Ds semileptonic decay candidates. When measuring branching ratios, Ds tagging yields the further benefit that the measured tags directly provide the branching ratio denominator rather than needing to independently estimate the data sample’s total Ds counts. 22

We use a total of 13 Ds tag modes, chosen for their relative abundancy and their separation from combinatoric background. We have mostly chosen to identify the tag modes by their final state particles rather than their intermediate particles (e.g. KKπ instead of φπ or K ∗ K). This choice maintains consistency with previous CLEO work [17] and avoids the need to worry about overlapping resonances (particularly a concern for KKπ, the most statistically significant mode). We reconstruct the following 13 tag modes: Ds+ → Ks K + , Ds+ → K + K − π + , Ds+ → Ks K + π 0 , Ds+ → Ks Ks π + , Ds+ → K + K − π + π 0 , Ds+ → Ks K + π + π − , Ds+ → Ks K − π + π + , Ds+ → π + π + π − , Ds+ → π + η, Ds+ → π + π 0 η, Ds+ → π + η 0 with η 0 → π + π − η, Ds+ → ππ 0 η 0 with η 0 → π + π − η, and Ds+ → π + η 0 with η 0 → ρ0 γ. Here, and elsewhere, the charge conjugate tag modes are also implied. Once we have our Ds tag candidates, we determine each mode’s tag counts by fitting their invariant mass. Any event with a Ds tag passing our wide mass window gets treated as a semileptonic decay candidate.

4.1

Basic Selection Criteria

We use a common selection criteria for daughter particles in our 13 exclusive tag modes. We have found little gain in deviating from the standard D-tag cuts, so our selection criteria emulates those selections [16, 18]. 4.1.1

Track Selection

Our tag modes include two charged particles that leave tracks: kaons and pions. Our selection for both K ± and π ± have several track quality features in common: • |db | < 5 mm • |z0 | < 5 cm • χ2 < 100, 000 • | cot θ| < 2.53 (equivalent to | cos θ| < 0.93)1 • Hit Fraction > 0.5 For K ± , we further require:2 • 0.125 GeV < pK < 2.0 GeV dE/dx • σK < 3.0 • UsePID true 1

Here, θ represents the angle from the beamline. raised from 0.050 GeV to 0.125 GeV for better dE/dx agreement between data and Monte Carlo [17]

2 min pK

23

We use the standard CLEO parameters for UsePID. Specifically, if we have RICH information with both π and K hypotheses analyzed, p > 0.7 GeV, and |cos(θ)| > 0.8, then we combine the RICH likelihood and σ dE/dx by requiring: 2 • L ≡ (σπ2 − σK ) + (Lπ − LK ) ≥ 0

• At least four RICH photons detected (NγRICH > 3) 2 Otherwise, we just use dE/dx values by requiring (σπ2 − σK ) ≥ 0 [18]. ± Similarly, for π we require:

• 0.050 GeV < pπ < 2.0 GeV dE/dx • σπ < 3.0 • UsePID true The UsePID true here matches that for the kaons, although now we require L ≤ 0. We only apply these track cuts to the tag mode daughter particles. The daughters of the semileptonic hadrons have their own similar, but often looser, selection criteria. 4.1.2

Ks Selection

We make a 1.575σ mass cut on our tag modes’ Ks mesons. This corresponds to a 6.3 MeV nominal mass cut. Our Ks mesons’ π ± daughters don’t have to fulfill the standard π cuts listed in Section 4.1.1 since they don’t necessarily originate from the interaction point. Given CLEO-c’s lower energies than earlier CLEO analyses, we do not use the CleanV0 cuts, nor do we add a flight significance or distance cut. 4.1.3

Photon Selection

Several tag modes include particles like π 0 and η that ultimately decay to photons. Also, Ds+ → π + η 0 , η 0 → ρ0 γ has an explicit photon in the tag mode. These photons share several different selection criteria: • Eγ > 30 MeV • No

E9 E25

cut3

• No splitoff rejection used • Showers with a matched track are disallowed • Showers from hot crystals are disallowed 3

Energy in a shower’s 3 × 3 = 9 central crystals divided by the energy in a shower’s 5 × 5 = 25 central crystals.

24

4.1.4

π 0 and η Selection

In addition to the daughter photon selection, we also consider some additional selection criteria for the π 0 and η mesons used in Ds tags: • The pull mass for both π 0 and η needs to be within 3.0 • We do not reject π 0 that have both showers in the endcap • We do reject η that have both showers in the endcap • Nominal mass less than 1.0 GeV • Max number of σ from expected mass within 1,000 • χ2 ≤ 10, 000 • No additional energy cut on γ in the endcap 4.1.5

η 0 Selection

The tag modes include η 0 reconstructed from its ππη decay mode and from its ρ0 γ decay mode, where the ρ0 decays to π + π − . Each of these decay modes has additional selection criteria. The η 0 → ππη mode involves reconstructing both pions and η mesons. We use the same selection criteria for these as in Section 4.1.1 and Section 4.1.4, respectively. Additionally, we require 947.8 MeV < Mη0 < 967.8 MeV η 0 → ρ0 γ ultimately involves reconstructing two pions and a photon. Again, pions share the same selection criteria as in Section 4.1.1. The photon inherits our standard photon tagging selection. We further require: • 0.5 GeV < Mπ+ π− < 1.0 GeV • 920 MeV < Mη0 < 995 MeV

4.2

Recoil Mass

The tagged Ds mesons are only created in either Ds∗ Ds or Ds Ds events, which constrains their momentum range. The direct Ds momenta depend only on the beam energy (4170 MeV), while the secondary Ds from the Ds∗ decay gain a slight boost. Table 3 gives the kinematic ranges for Ds momenta at our beam energy. We restrict the allowed momentum range by cutting on a directly related variable, the recoil mass, which includes the beam momentum and corresponds physically to the other meson’s mass in the case of prompt Ds decays. We define the recoil mass by r q  2 Ecm − |~pDs |2 + MD2 s − |~pcm − p~Ds |2 , Mrecoil = |pcm − pDs | ≡ where pcm , Ecm , and p~cm correspond to the center-of-mass four vector, energy, and momentum; MDs comes from the PDG [1]; and p~Ds denotes the reconstructed Ds momentum. 25

Table 3: Kinematically allowed recoil mass and momentum ranges for Ds mesons at 4170 MeV. Ds origin Ds Ds Prompt Ds in Ds∗ Ds Ds from Ds∗ → Ds γ in Ds∗ Ds Ds from Ds∗ → Ds π 0 in Ds∗ Ds

Possible Momenta 687 MeV 429 MeV 259 MeV – 542 MeV 351 MeV – 449 MeV

Possible Recoil Mass MDs ≈ 1968 MeV MDs∗ ≈ 2112 MeV 2058 MeV – 2169 MeV 2104 MeV – 2142 MeV

Since we will use both the recoil mass and the Ds invariant mass, we do not use either the beam constrained mass (Mbc ) or ∆E ≡ EDs − Ebeam . Our recoil mass cut varies by tag mode and depends upon the shape and combinatoric background for that mode. Table 4 gives our cut values by tag mode. Table 4: Recoil mass cut, by Ds tag mode. Ds tag modes KS K KKπ πη πη 0 , η 0 → ππη KS Kπ 0 KS KS π KKππ 0 KS K + ππ KS K − ππ ππ 0 η ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ πππ

Minimum Recoil Mass

Maximum pDs

2051 MeV

555 MeV

2099 MeV

459 MeV

2101 MeV

455 MeV

Figure 2 and Figures 75–77 in Appendix F show each tag mode’s recoil mass spectrum and recoil mass cut in the Monte Carlo and data. The background reduction from our recoil mass cut benefits us across the Ds invariant mass spectrum, as the two variables are fairly uncorrelated (shown in Figures 78–83).

4.3

Individual Tag Mode Cuts

Each tag mode has unique backgrounds that we reduce by making a series of additional cuts. These cuts reject D0 or D± mesons, reject unwanted Ks , or remove excess (and often peaking) combinatoric background arising from soft pions. We chose these cuts to maintain consistency with previous CLEO Ds tagging [17] when applicable. Table 5 lists our rejection criteria for

26

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Figure 2: Monte Carlo (charm and scaled continuum) simulation of the recoil mass distribution for the Ds tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We reject all Ds with a recoil mass below the cut line.

each tag mode. We take charges into consideration for our stated invariant mass rejections (e.g. for KS K, we only apply the MKπ rejection when the K and π have opposite signs). After making our previously listed cuts, we choose a best candidate for each tag mode and charge by keeping only the Ds with a recoil mass closest to MDs∗ (2112.3 MeV).

27

Table 5: Rejections based on individual tag modes’ daughter particles. Ds tag mode KS K KKπ KKππ 0

πππ

πη πη 0 , η 0 → ππη KS Kπ 0 KS KS π KS K + ππ KS K − ππ ππ 0 η ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ

Daughter particle cuts (rejections) MKπ ≥ 1830 MeV pπ ≤ 100 MeV 1845 MeV ≤ MKK ≤ 1880 MeV 1860 MeV ≤ MKKπ ≤ 1880 MeV pπ ≤ 100 MeV pπ0 ≤ 100 MeV 475 MeV ≤ Mππ ≤ 520 MeV 1840 MeV ≤ Mππ ≤ 1885 MeV 1845 MeV ≤ MKπ ≤ 1880 MeV, where one of the supposed pions is treated as a kaon. pπ ≤ 100 MeV No additional cuts MKππ0 ≥ 1830 MeV 1860 MeV ≤ MKππ ≤ 1880 MeV pπ ≤ 100 MeV pπ0 ≤ 100 MeV pπ ≤ 100 MeV, where the Ds is rejected if either the Ks meson’s daughter π or the direct π fails the cut. pπ0 ≤ 100 MeV 480 MeV ≤ Mππ ≤ 515 MeV pπ0 ≤ 100 MeV 480 MeV ≤ Mππ ≤ 515 MeV pπ ≤ 100 MeV

28

4.4

Fitting Procedure

Once we make the tag cuts and choose a best candidate for each mode/charge combination in the event, we determine the tag counts for a given mode by fitting its Ds mass spectrum (combining Ds+ and Ds− ). We model our signal mass spectrum by a double gaussian4 for the tag modes reconstructed with only tracks (and for πη 0 , η 0 → ππη). We use the sum of a gaussian and a crystal ball function for modes containing photons or nontrivial FSR (πππ). We take either a linear function or a quadratic function for our background, depending upon the mode and the shape of its combinatoric background. Table 6 lists the particular combination of fit functions for each tag mode. Table 6: Ds invariant mass fit functions, by mode. Ds tag mode KS K KKπ KS KS π KS K + ππ KS K − ππ πη 0 , η 0 → ππη KS Kπ 0 πη ππ 0 η 0 , η 0 → ππη KKππ 0 πππ ππ 0 η πη 0 , η 0 → ργ

Function

Signal: Double Gaussian Background: Linear Polynomial

Signal: Gaussian + Crystal Ball Background: Linear Polynomial Signal: Gaussian + Crystal Ball Background: Quadratic Polynomial

In our signal shape functions, we use a common mean for the two gaussians (or for the gaussian and the gaussian portion of the crystal ball). To reduce the number of free parameters further, we fit our signal shape to a truth-tagged Monte Carlo sample, then we use those results to fix the relative normalization and relative width of the two signal shape component functions. We also fix the two remaining shape parameters in the crystal ball function from the truth-tagged fit, if applicable for the mode. This procedure leaves three free parameters for the signal shape of the reconstructed Ds mass spectrum: an overall normalization, an overall width, and the common mean. Combined with the two or three background parameters for the linear or quadratic polynomial, respectively, we end with five or six free parameters for the reconstructed Ds mass fit.

4.5

Results

In the following sections, we present the results from our fits to the reconstructed Ds invariant mass spectrum in both Monte Carlo and data. We only consider statistical errors on the 4

Sum of two gaussian functions.

29

tag counts here. We do consider systematics associated with our tag counting procedure in Section 9.1.1, but we focus on the branching ratio’s systematic error from tagging rather than on the error for raw tag counts. We typically get smaller tag-related systematics on the branching ratio than on tag counting alone because our procedure involves the Ds tag shape in both the branching ratio’s numerator and denominator. Any comparison with other Ds tagging analyses should keep in mind that our raw tag counts would presumably have a higher systematic error than reflected in just our branching ratio systematics. As mentioned in Section 4.4, we first fit the truth-tagged Monte Carlo MDs distribution to fix all but three parameters for our signal shape function. Figures 84–86 show these fits’ results, by tag mode. The fit functions closely match the truth-tagged histograms, which gives us the freedom to use our functions rather than less flexible signal histograms when fitting the data. 4.5.1

Monte Carlo

Before we fit the data, we first build confidence in our procedure by ensuring that we get the proper tag counts in the generic Monte Carlo sample (charm plus continuum). Figures 87–89 show our fits to the Ds invariant mass for this sample, resulting in the total tag counts displayed in Table 7. While we used a 20× Monte Carlo sample, we have scaled the table’s tag counts down to the data’s luminosity to make direct comparisons with the data counts easier. Table 7: Tagging results from the 20× Monte Carlo sample, scaled to data size. Ds mode KS K KKπ KS Kπ 0 KS KS π KKππ 0 KS K + ππ KS K − ππ πππ πη ππ 0 η πη 0 , η 0 → ππη ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ Sum

Fit counts 5,764.0 ± 100.8 25,242.0 ± 233.9 1,670.5 ± 157.7 1,141.4 ± 69.3 6,693.4 ± 323.6 1,744.1 ± 105.5 3,246.3 ± 92.2 6,081.6 ± 326.3 2,882.3 ± 182.9 6,825.9 ± 700.7 2,132.4 ± 64.3 532.5 ± 84.5 3,904.4 ± 245.2 67,860.7 ± 959.8

Truth tagged counts 5, 693.1 25, 731.6 1, 871.2 1, 081.5 6, 844.5 1, 717.3 3, 200.6 6, 197.6 3, 334.4 6, 560.0 2, 108.4 749.3 4, 079.9 69, 169.5

Nf it −NM C σ

0.70 −2.09 −1.27 0.86 −0.47 0.25 0.50 −0.36 −2.47 0.38 0.37 −2.57 −0.72 −1.36

Fits to the modes ππ 0 η 0 , η 0 → ππη and πη gave the most significant deviations from their truth-tagged counts. In both cases, the background shapes predicted by the Monte Carlo bordered on requiring a non-linear function, like the four other crystal ball modes (KKππ 0 ; πππ; ππ 0 η; and πη 0 , η 0 → ργ). However, we try to avoid such background functions when we have a wide signal shape because the background function can dip inappropriately in the Ds mass region. This lower background leads to an overestimate on the tag counts. The four modes 30

in which we do use a quadratic background have more events than either ππ 0 η 0 , η 0 → ππη or πη, and they tend to have narrower signal shapes (ππ 0 η has a wider shape but more events). These qualities make us less sensitive to the background function when we shift to the data, where we need to be more careful in case the Ds mass reconstruction has a poorer resolution than the Monte Carlo predicts. In addition to procedure cross-checks, we have used the Monte Carlo to determine our tagging efficiency within semileptonic events. As expected, we see essentially the same tagging efficiency independent of the semileptonic mode (Section 9.1.3). However, our recoil mass cut does create a difference in tagging efficiency based on the Ds production method: Ds Ds , Ds∗ Ds with the tagged Ds from the Ds∗ (“secondary”), or Ds∗ Ds where the tagged Ds does not come from the Ds∗ (“prompt”). Table 8 gives the efficiencies for each case. Table 8: Overall Ds tagging efficiency from signal Monte Carlo, including all branching ratios. Our Ds recoil mass cut creates the efficiency difference between tags from prompt Ds and tags from secondary Ds . Ds production mode Ds Ds Ds∗ Ds with prompt Ds → tag Ds∗ Ds with secondary Ds → tag Weighted MC

4.5.2

εtag 0.42% ± 0.01% 7.21% ± 0.03% 5.69% ± 0.03% 6.22% ± 0.02%

Data

Figures 3–5 show our fits to the combined data from datasets 39, 40, 41, 47, and 48. Table 9 summarizes each mode’s tag counts resulting from these fits. Although not directly relevant for this analysis, we find it interesting that we see about 14% more tags in the data than we expected from the 20× Monte Carlo sample. This difference persists across each mode and dataset to within errors, as shown in Tables 81 and 82. 4.5.3

Cross-Checks

In addition to our fitting systematics, described in Section 9.1.1, we have performed two crosschecks for our fitting procedure. In the first cross-check, we use the Monte Carlo truth-tagged histogram for our signal shape instead of the double gaussian or gaussian + crystal ball functions. In the second, we ensure that our chosen procedure consistently fits data-sized samples by breaking the 20× Monte Carlo into 20 equal subsets and fitting each individually. In our signal histogram cross-check, we take the Ds invariant mass spectrum from the truthtagged Monte Carlo as the signal shape instead of a double gaussian or gaussian + crystal ball function. The overall histogram normalization gives us our only free signal parameter. We then add the same background function as in our standard fit (linear or quadratic, by mode). This leaves either 3 or 4 total free parameters, depending on the Ds mode.

31

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We first fit the full Monte Carlo to ensure that the signal histogram fit returns the truthtagged counts. This fit does match both our standard fit results and the true number of tags to within error, as shown in Table 10. We then fit the data with the signal histograms and corresponding background functions. We expect the signal histogram fit to give more or less the same result as our fit function as long as the Ds mass resolution in Monte Carlo accurately represents the true resolution in the data. The fit results, displayed in Figures 90–95 and summarized in Table 11, show consistency between the signal histogram fit and our more flexible double gaussian/gaussian + crystal ball function for all modes, with the exception of ππ 0 η. This discrepancy does not particularly surprise us since ππ 0 η has the worst signal to background ratio, has a wide signal shape, and has a background shape that requires a quadratic function. The combination of these issues allows the signal shape to trade off with the background’s quadratic curvature to some extent. Since the ππ 0 η signal histogram fit clearly doesn’t fit well (both visually and in terms of χ2 ), undershooting the Ds mass distribution’s high side and overshooting the low side, we don’t feel 32

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a need to take an additional systematic based on its results. For our data-sized cross-check, we split the 20× Monte Carlo sample into 20 separate samples to ensure that our fit function will successfully and reliably converge. Table 12 gives our results, where the summation row states the total fit tag counts, the total truth-tagged counts, and the total χ2 across the 20 samples. Our reduced χ2 of 1.90 represents a systematic undercounting of the true number of tags. We believe that the slight, systematic undercounting of about 1% originates from an imperfection in the signal fit shape. The signal function peak falls slightly below the true peak whereas the tails have a small surplus, as seen from our signal Monte Carlo plots in Figures 84–86. When we use this signal shape in a sample with background, the tails of the signal distribution get partially subsumed into a higher background while the slightly low signal shape peak still matches the data peak. The statistical errors for each tag mode exceed the slight Ds tag undercounting effect. However, since the peak region essentially acts like a double gaussian for all modes (as the crystal ball function acts like a gaussian near the peak), we consider the undercounting rate correlated between modes. We would treat the undercounting as a systematic when determining 33

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Figure 5: Ds invariant mass fits in the data, determining the total number of Ds tags for modes πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ.

the branching ratio, but this systematic gets complicated by two factors. The first issue with a proposed undercounting systematic comes from the fact that we ultimately fit the Ds mass spectrum for the number of semileptonic events as well as the number of tags. The semileptonic event counts also have a linear background function, so the slight undercounting can appear in both our branching ratio’s numerator and denominator, significantly shrinking the ultimate systematic. Even if we dismiss the numerator effect, though, we have a second ameliorating factor. Our conventional systematic (Section 9.1.1) involves changing the Ds mass resolution based on results from the D± system. This systematic indicates a wider underlying Ds mass resolution than predicted by the Monte Carlo, which actually makes our signal shape more accurate. Since the conventional systematic involves a wider distribution and gives larger final errors (just due to the systematic’s precision), we consider that study to measure essentially the same signal shape concern, and we don’t add an additional undercounting systematic.

34

Table 9: Tagging results from the full data sample (sum of datasets 39, 40, 41, 47, 48). Ds mode KS K KKπ KS Kπ 0 KS KS π KKππ 0 KS K + ππ KS K − ππ πππ πη ππ 0 η πη 0 , η 0 → ππη ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ Sum

Fit counts 6,226.7 ± 101.2 27,373.5 ± 248.4 2,246.8 ± 209.9 1,125.5 ± 76.5 7,355.5 ± 377.4 1,859.4 ± 120.6 3,377.3 ± 100.0 6,606.3 ± 337.7 3,810.3 ± 190.8 9,476.9 ± 529.0 2,386.6 ± 65.6 1,090.5 ± 118.7 4,272.3 ± 193.3 77,207.5 ± 880.2

Table 10: Signal histogram fit results compared to our standard double gaussian/ gaussian+crystal ball fit results in the 20× Monte Carlo sample, scaled to data size. Ds mode KS K KKπ KS Kπ 0 KS KS π KKππ 0 KS K + ππ KS K − ππ πππ πη ππ 0 η πη 0 , η 0 → ππη ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ Sum

5

Signal histogram 5,813.5 ± 94.0 25,834.7 ± 214.9 1,783.8 ± 123.5 1,161.4 ± 59.5 6,815.9 ± 219.4 1,766.5 ± 87.9 3,326.3 ± 83.1 6,220.3 ± 220.9 3,043.0 ± 138.6 6,615.8 ± 417.8 2,164.2 ± 58.6 624.4 ± 77.1 4,074.1 ± 205.0 69,244.0 ± 656.2

Standard fit 5,764.0 ± 100.8 25,242.0 ± 233.8 1,670.5 ± 159.2 1,141.4 ± 69.2 6,693.4 ± 323.6 1,744.0 ± 105.3 3,246.3 ± 92.2 6,082.3 ± 309.7 2,882.3 ± 182.4 6,825.9 ± 700.7 2,132.4 ± 64.3 532.5 ± 84.0 3,904.4 ± 245.2 67,861.2 ± 954.3

Truth-tagged counts 5, 693.1 25, 731.6 1, 871.2 1, 081.5 6, 844.5 1, 717.3 3, 200.6 6, 197.6 3, 334.4 6, 560.0 2, 108.4 749.3 4, 079.9 69, 169.5

Semileptonic Selection Criteria

While we use the Ds mass spectrum in our fits for both the tag counts and the semileptonic counts, we do make cuts on the other particles to ensure that we have a semileptonic event. In particular, we make an electron cut that gives us the best background rejection in our analysis. 35

Table 11: Signal histogram fit results compared to our standard double gaussian/ gaussian+crystal ball fit results in the full data sample. Ds mode KS K KKπ KS Kπ 0 KS KS π KKππ 0 KS K + ππ KS K − ππ πππ πη ππ 0 η πη 0 , η 0 → ππη ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ Sum

Signal histogram 6,236.9 ± 94.9 27,576.2 ± 225.9 2,039.9 ± 127.7 1,155.0 ± 63.1 7,590.2 ± 256.7 1,854.6 ± 98.9 3,434.5 ± 89.4 6,437.5 ± 233.2 3,527.5 ± 128.7 5,848.1 ± 485.2 2,300.0 ± 58.6 1,069.8 ± 81.6 4,265.9 ± 209.8 73,336.1 ± 724.2

Standard fit 6,226.7 ± 101.2 27,373.5 ± 248.4 2,246.8 ± 209.9 1,125.5 ± 76.5 7,355.5 ± 377.4 1,859.4 ± 120.6 3,377.3 ± 100.0 6,606.3 ± 337.7 3,810.3 ± 190.8 9,476.9 ± 529.0 2,386.6 ± 65.6 1,090.5 ± 118.7 4,272.3 ± 193.3 77,207.5 ± 880.2

We have also studied extra track and extra shower event cuts. While splitoff showers make any extra shower cuts problematic, we do find an extra track cut useful in our final selection.

5.1

Electron Selection

We select electrons using the same general track quality requirements applied to Ds tracks in Section 4.1.1. Further, we use the Fw/RICH variable [19], a likelihood ratio that tests the electron hypothesis under a weighted combination of E/p, dE/dx, and RICH information. Fw/RICH runs from 0 to 1, with 1 being electron-like, as shown in Figure 6. We require Fw/RICH ≥ 0.8. We also add the requirement that the hit fraction falls below 1.2, although this cut has no real impact beyond consistency with previous systematic work. We do not attempt to reconstruct electrons with a momentum below 200 MeV. These soft electrons cost some efficiency, but as Table 13 shows, we get substantial background reduction from our combined electron cuts.

5.2

Event Selection

We look for Ds semileptonic decays in Ds∗ Ds and Ds Ds events, where the Ds∗ decays to a Ds with some photons (either directly to a γ or via a π 0 ). Consequently, we should not have any tracks other than those from the tagged Ds , the electron, or the semileptonic hadron. We reject any event with an extra track, which cuts out events with an e+ e− pair that would otherwise pass our electron selection, semileptonic events with the wrong hadron mode (e.g. Ds → η 0 eν faking Ds → ηeν), and some semileptonic events with misreconstructed tags. Not surprisingly, we rarely throw out signal events with our extra track cut, as shown in Table 14. We have considered a rejection on extra showers above various energy thresholds (25 MeV, 36

Table 12: Fit results from the Monte Carlo’s 20 data-sized samples. The final column in the ”Sum” row gives the total χ2 . Datasize sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sum

Fit counts 67,932.6 ± 1,022.9 71,422.1 ± 1,214.3 71,579.7 ± 1,386.5 71,409.4 ± 919.6 69,774.6 ± 1,046.3 69,795.4 ± 983.4 71,017.8 ± 1,221.3 72,092.7 ± 1,275.1 67,852.4 ± 1,133.5 70,128.0 ± 809.5 69,108.9 ± 926.7 70,547.9 ± 1,182.5 71,312.2 ± 1,125.7 71,769.1 ± 1,390.2 69,466.9 ± 840.6 69,036.8 ± 992.4 71,385.3 ± 1,250.5 71,574.6 ± 1,352.9 70,312.3 ± 1,266.5 69,738.3 ± 1,027.0 1,407,257.0 ± 5,060.6

Truth tagged counts 70, 585 71, 265 71, 112 71, 119 71, 552 71, 326 71, 007 71, 154 71, 084 71, 044 71, 265 71, 208 71, 260 71, 292 70, 906 71, 235 71, 528 70, 952 71, 439 70, 842 1, 423, 175

Nf it −NM C σ

−2.59 0.13 0.34 0.32 −1.70 −1.56 0.01 0.74 −2.85 −1.13 −2.33 −0.56 0.05 0.34 −1.71 −2.21 −0.11 0.46 −0.89 −1.07 38.04

Table 13: Effect of the electron cuts (track and Fw/RICH ) in the 20× Monte Carlo sample for truth-tagged semileptonic and generic decay modes. These precede any semileptonic hadron cut, but passed Ds tags must fall within the tagging fit window (1900 MeV < MDs < 2030 MeV). Ds semileptonic mode φeν ηeν η 0 eν KS eν K ∗ eν f0 eν All other modes

# Passed Ds tags 55, 399 50, 775 19, 282 1, 022 4, 148 7, 132 6, 741, 304

# Passed electron cuts 31, 864 35, 772 12, 038 641 2, 582 4, 411 204, 944

Cut efficiency 58% 70% 62% 63% 62% 62% 3%

100 MeV, 300 MeV, and 500 MeV), but we did not find them useful (Figures 7 and 96–100). Our

37

Fraction of tracks

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Figure 6: Fw/RICH in the 20× MC sample for good tracks that are not used in the tagged Ds . We only plot electrons involved in one of our six semileptonic modes (φeν, ηeν, η 0 eν, f0 eν, KS eν, and K ∗ eν). The electron peak at zero comes primarily from tracks with a momentum below 200 MeV.

Table 14: Effect of an extra track cut on the signal and the background in the peaking Ds mass region after all other selections (e.g. semileptonic hadron cuts) have been made. We define f S2 , such that ffpost-cut measures our statistical improvement. as S+B pre-cut Ds semileptonic mode φeν ηeν η 0 eν KS eν K ∗ eν f0 eν

εsignal 99% 97% 97% 98% 98% 99%

εbackground 61% 42% 52% 69% 64% 59%

fpost-cut fpre-cut

1.01 1.31 1.06 1.38 1.40 1.23

Ds and electron selections keep the background levels low, so even the handful of signal events improperly rejected by an extra shower cut will cause a reduction in our statistical significance. 38

Several different effects can lead to improper signal rejection under an extra energy cut: modes containing kaons can have decays in flight, leaving a shower without a properly matched track; the Ds∗ can decay to a π 0 , yielding one extra shower; legitimate single showers sometimes get misreconstructed as two or more showers; and splitoff showers may not get identified as such. Although we can reduce the impact of each effect through various selections, we have not found that such improvements save the extra shower cut. Individual energies of extra showers

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Figure 7: Extra showers after finding the tagged Ds , the φ, and the electron in φeν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays. We don’t see an improvement in our results from making a cut on any extra shower variable.

39

Measurement of Ds → φeν

6

While we ultimately reconstruct six different Ds semileptonic modes, we use a similar reconstruction and fitting procedure for five of them (φeν, η 0 eν, f0 eν, KS eν, and K ∗ eν). To illustrate the procedure, we first describe how the process works for Ds → φeν. Then, in section 7, we show how the procedure differs for each of the other four semileptonic modes that have a similar reconstruction but fewer events. We determine the number of Ds → φeν events by fitting the tagged Ds mass spectrum after making the electron selections in Section 5.1, the φ cuts in Section 6.1, and the event cuts in Section 5.2. We do not directly use the electron or φ kinematic information beyond requiring that they pass our particle cuts, although our φ reconstruction gets used indirectly in our f0 → KK background subtraction.

6.1

φ Selection

We reconstruct the φ meson in Ds → φeν via the φ → KK decay mode, which makes up roughly half of all φ decays. Ds → φeν presents a challenge in its φ reconstruction as the semileptonic φ tends to be fairly soft (Figure 8a). The soft φ and low Q value in the φ → KK decay (about 32 MeV) leads to soft daughter kaons, which decreases our detection efficiency significantly relative to higher energy φ decays (e.g. Ds → φπ). As an additional challenge, the long tail of the φ Breit-Wigner forces us to use a wide φ mass window relative to its decay width of Γφ ≈ 4.26 MeV. The combination of low Ds → φeν background and soft kaons encourage loose kaon selection criteria. In particular, we have essentially dropped any hit fraction cut to accommodate the higher likelihood that a soft kaon will decay in flight (Figure 9). We considered loosening other typical track cuts, but we did not find the slight efficiency improvement to be worth deviating from established systematic studies [20]. We ultimately require the following cuts on the φ meson’s daughter kaons: • |db | < 5 mm • |z0 | < 5 cm • χ2 < 100, 000 dE/dx < 3.0 • σK • Hit Fraction > 0.1

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Figure 8: Generated and reconstructed momentum spectra from the 20× Monte Carlo. a) φ from Ds → φeν, using our cuts. b) Electron from Ds → φeν.

41

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42

With the low combinatoric background in Ds → φeν and the long φ Breit-Wigner tail, we extend the high side of our φ mass window as far as we reasonably can. On the low side of the φ mass window, we run into contamination from Ds → f0 eν, where f0 → KK. While we do measure Ds → f0 eν via f0 → ππ and use it to guide our background subtraction, the uncertainty in f0 parameters leads us to restrict the low φ mass range somewhat to reduce the impact of these systematic uncertainties on our φeν result. For our final φ mass cut, shown in Figure 10, we use: • −15 MeV < Mφreconstructed − MφPDG < 30 MeV We do not attempt to reconstruct the φ through modes other than KK (e.g. πππ 0 ).

Reconstructed φ → KK mass, truth-tagged Counts / 2 MeV

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×10 120 100 80 60 40 20 0 970

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Figure 10: φ mass distribution when reconstructed from φ → KK in the 20× Monte Carlo. The green lines represent a 10 MeV cut (roughly 2Γφ ), which does not capture the high mass tail. We accept φ masses between the red lines.

43

Ds → φeν Reconstruction

6.2 6.2.1

Efficiency

To determine our efficiencies, we generated signal Monte Carlo samples with one Ds decaying to φeν and the other decaying generically. We made four samples with 250,000 events, one for each different type of semileptonic Ds production at 4170 MeV: one sample for Ds+ Ds− and three for Ds∗ Ds , where the semileptonic Ds can either be “prompt” (not from the Ds∗ side) or “secondary” in one of two ways (Ds∗ → Ds γ or Ds∗ → Ds π 0 ). We present the efficiencies for Ds → φeν in Table 15 for typical φ cuts and Table 16 for this analysis’s looser φ cuts. We determine all our semileptonic efficiencies after successfully reconstructing a Ds tag within the fit window (1900 MeV ≤ MDs ≤ 2030 MeV). The efficiency for both the φ and the electron increases with higher momenta, as shown in Figure 11. This causes the overall semileptonic efficiency to be slightly lower than the simple product of hadron and electron efficiencies, since high momentum electrons are correlated with low momentum φ and vice versa. Table 15: Efficiencies for semileptonic particles in Ds → φeν, with typical φ cuts (HF > 0.5, φ mass within 10 MeV). The efficiencies include the φ → KK branching ratio. Ds production mode Ds Ds ∗ Ds Ds with Ds∗ → (Ds → φeν) γ Ds∗ Ds with Ds∗ → (Ds → φeν) π 0 Ds∗ Ds with prompt Ds → φeν Weighted MC

70.4% 70.2% 70.3% 70.7% 70.5%

εe ± ± ± ± ±

2.6% 0.6% 0.6% 0.7% 0.5%

12.9% 15.2% 15.1% 14.1% 14.9%

εφ ± ± ± ± ±

1.1% 0.3% 0.3% 0.3% 0.2%

εSL 9.4% ± 1.0% 10.2% ± 0.2% 10.1% ± 0.2% 9.5% ± 0.3% 10.1% ± 0.2%

Table 16: Efficiencies for semileptonic particles in Ds → φeν, with the φ cuts used in this analysis. The efficiencies include the φ → KK branching ratio. Ds production mode Ds Ds Ds∗ Ds with Ds∗ → (Ds → φeν) γ Ds∗ Ds with Ds∗ → (Ds → φeν) π 0 Ds∗ Ds with prompt Ds → φeν Weighted MC

εe 70.4% ± 70.2% ± 70.3% ± 70.7% ± 70.5% ±

44

2.6% 0.6% 0.6% 0.7% 0.5%

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45

6.2.2

Backgrounds

Our predicted Ds → φeν background from the 20× Monte Carlo primarily consists of events that don’t peak within our Ds mass fit region (Table 17). We use a linear background function in our data fit for such events. However, we do have to take special account of events that fake the semileptonic side (electron or φ → KK) while having a true Ds tag, as these events will look like signal events in our fit to the Ds mass spectrum. Table 17: Truth-tagged breakdown for Ds → φeν candidates passing all cuts in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. Event truth True Ds tag, true Ds → φeν True Ds tag, not Ds → φeν False Ds tag, true Ds → φeν False Ds tag, not Ds → φeν

Passing candidates 210.90 4.10 46.40 16.55

Most events with a true Ds tag that are incorrectly reconstructed as Ds → φeν come from Ds → f0 eν, where f0 → KK (Figure 12). The large decay width of the f0 means that a nontrivial fraction of f0 → KK decays have an invariant mass within our φ → KK mass window. At the same time, the low overall number of Ds → f0 eν, f0 → KK decays relative to other backgrounds (particularly combinatoric background near KK threshold with false Ds tags) prevents us from simply fitting the MKK spectrum to determine the number of f0 eν background events. We instead use our Ds → f0 eν (f0 → ππ) measurement from Section 7.5, Γ values from the Particle Data Book [1], and a model for the f0 → KK a range of Γff0 →KK 0 →ππ lineshape to estimate the amount of Ds → f0 eν, f0 → KK background that appears within our Ds → φeν sample, as described in Appendix A. We give our correction for the Ds → φeν branching ratio from the Ds → f0 eν, f0 → KK peaking background and its associated systematic in Table 18. We also include a correction for events with a correct Ds tag that fake Ds → φeν from sources other than Ds → f0 eν, primarily φρ± . We use Monte Carlo estimates for the latter correction since the decay kinematics are well understood (relative to their statistical significance). Table 18: Absolute branching ratio correction and systematic error for B(Ds → φeν) from peaking background. Background mode Ds → f0 eν Non-semileptonic Total Ds → φeν correction

46

BR correction (0.0080 ± 0.0115)% (0.0041 ± 0.0014)% (0.0120 ± 0.0116)%

MC true D+s tags for successful φ+e+ν

f0 e + ν e (90.1%)

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mode

(8.6%)

f 0 e+ νe φ a+ 1

Ds, tag mode Figure 12: Ds → φeν backgrounds with a true Ds tag (peaking background) in the 20× Monte Carlo. Only Ds → f0 eν, f0 → KK presents a sizable peaking background for Ds → φeν.

6.2.3

Fit Procedure

In most tagged neutrino analyses (e.g. [21, 22]), the event’s missing energy and momentum will be reconstructed and used in some form (e.g. missing mass, ∆E and Mbc ) to determine the number of signal events. Since we’ve dropped the Ds∗ meson’s daughter γ in this analysis, our missing four momentum contains both the neutrino and the photon. The missing variables (mass, energy, momentum) are then distorted and do not give clear separation between signal and all backgrounds (although some background may still be rejected, as described for KS eν in Section 7.2). When searching for an alternate fit parameter, we have noticed that a major systematic error in our φeν and ηeν modes comes from the lineshape uncertainty associated with the Ds tag. To minimize this systematic, we have chosen to fit the Ds mass spectrum for our branching ratio’s numerator as well as its denominator, which eliminates most of the impact from our reconstructed Ds lineshape uncertainty. As described previously, we do not directly involve the φ or electron kinematic information in our fit, using that particle information only to establish that we have a legitimate Ds → φeν event. Even in our relatively high statistics Ds → φeν semileptonic mode, we only expect about 47

200 events spread across our 13 tag modes, with some tag modes likely to contain only one or two events. Our low statistics semileptonic modes (η 0 eν, f0 eν, KS eν, and K ∗ eν) have even fewer events spread across the same number of tag modes. To deal with the errors associated with our low statistics, we use an unbinned, extended maximum likelihood fit to our Ds mass spectrum. Further, since the small number of events per tag mode will distort a weighted sum combining individual results,5 we instead use a common branching ratio parameter across all 13 tag modes. We fix our signal Ds mass lineshape from each mode’s tag fit results. This leaves only the normalization floating (via the common branching ratio parameter), giving us just one signal parameter in our fit. We determine a linear background for each tag mode from the 20× Monte Carlo, then we allow the overall normalization of this shape to float independently for each tag mode. If we have fewer than 20 background events for a tag mode in the 20× Monte Carlo sample (less than one expected background event in the data), we choose a constant background function instead of a linear function as our background shape for that tag mode. Our fit function then has 14 floating parameters in our Ds mass fit – one signal parameter (the branching ratio) and one background normalization for each of our 13 Ds tag modes. We have a log likelihood function to minimize given by: ~ BG ) ≡ − ln L(B, N ~ BG ) F(B, N " 13 # " 13 # X X i i = B · εSL · Ntag + NBG −

X

i=1 h i=1 i [j] [j] [j] [j] ln B · εSL · Ntag · fsig (mj ) + NBG · fBG (mj ) ,

mj

~ BG is the background normalization (one per tag mode), [j] where B is the branching ratio, N i i (m) are the (m) and fBG refers to the tag mode associated with reconstructed Ds mass mj , fsig normalized mass distributions of the signal and background for the given tag mode, respectively, i and Ntag is the number of Ds tags for mode i. The first two terms in our function just represent the overall signal and background normalizations, while the third term corresponds to the sum of each event’s log likelihood given our signal and background shapes.

6.3

Results

The fit results presented in the following subsections only involve the statistical error. We determine our systematic errors in Section 9 and give our full errors with the final, efficiencycorrected result in Section 10. 6.3.1

Monte Carlo

To ensure that our procedure properly measures the branching ratio on the couple hundred events expected in data, we have split the 20× Monte Carlo into twenty data-sized subsamples. These subsamples allow us an in-vs.-out test in which we accurately measure the number of 5

Bienaym´e formula for adding errors in quadrature need not apply.

48

truth-tagged semileptonic events, as seen from the lack of fit bias in Table 19. We also obtain the proper branching ratio in our Monte Carlo test after correcting for peaking background, as demonstrated in Table 20 with our branching ratios’ χ2 of 21.6 over the twenty subsamples. Table 19: Test of potential bias in our fitting procedure for Ds → φeν by comparing the number of truth-tagged semileptonic events to the fit result. We allow cross-feed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sum

truth-tagged NSL+tag 234 224 199 226 208 191 240 222 191 208 194 201 218 219 235 216 212 220 210 213 4281

fit NSL+tag 231.74 ± 16.58 227.52 ± 14.79 191.93 ± 15.43 227.01 ± 16.54 202.15 ± 15.74 191.44 ± 13.94 248.97 ± 17.05 229.31 ± 16.71 194.93 ± 15.13 208.35 ± 14.50 201.72 ± 15.31 197.51 ± 14.91 217.17 ± 16.13 226.68 ± 16.23 242.62 ± 16.59 212.87 ± 15.74 216.87 ± 15.85 212.31 ± 15.52 212.19 ± 15.63 201.87 ± 15.55 4295.17 ± 70.28

Difference (# σ) −0.137 0.238 −0.458 0.061 −0.371 0.032 0.526 0.438 0.260 0.024 0.504 −0.234 −0.051 0.473 0.460 −0.199 0.307 −0.496 0.140 −0.716 0.202

For reference, we present our fits to these twenty data-sized subsamples as Figures 101–105 in Appendix F.

49

Table 20: Monte Carlo comparison of the measured Ds → φeν branching ratio to its generating branching ratio (2.170%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Weighted averages/χ2

Raw fit BR (2.439 ± 0.175)% (2.306 ± 0.150)% (1.917 ± 0.154)% (2.289 ± 0.167)% (2.076 ± 0.162)% (1.972 ± 0.144)% (2.520 ± 0.173)% (2.300 ± 0.168)% (2.014 ± 0.156)% (2.101 ± 0.146)% (2.094 ± 0.159)% (2.013 ± 0.152)% (2.178 ± 0.162)% (2.236 ± 0.160)% (2.504 ± 0.171)% (2.217 ± 0.164)% (2.196 ± 0.160)% (2.138 ± 0.156)% (2.111 ± 0.156)% (2.058 ± 0.158)% (2.171 ± 0.036)%

50

Corrected BR (2.405 ± 0.175)% (2.271 ± 0.150)% (1.883 ± 0.154)% (2.255 ± 0.167)% (2.041 ± 0.162)% (1.937 ± 0.144)% (2.485 ± 0.173)% (2.266 ± 0.168)% (1.979 ± 0.156)% (2.066 ± 0.146)% (2.059 ± 0.159)% (1.978 ± 0.152)% (2.143 ± 0.162)% (2.201 ± 0.160)% (2.470 ± 0.171)% (2.182 ± 0.164)% (2.161 ± 0.161)% (2.104 ± 0.156)% (2.076 ± 0.156)% (2.023 ± 0.158)% (2.137 ± 0.036)%

#σ 1.34 0.67 −1.86 0.51 −0.80 −1.62 1.83 0.57 −1.22 −0.71 −0.70 −1.26 −0.17 0.19 1.75 0.07 −0.06 −0.42 −0.60 −0.93 21.54

6.3.2

Data

We give our measured branching ratio and number of signal events for Ds → φeν in Table 21, which includes our correction from background with a peaking Ds mass. The branching ratio here assumes the Monte Carlo’s efficiency; our systematics section (Section 9) discusses some corrections to this efficiency (Table 98) that appear in our final result (Table 66). Table 21: Ds → φeν measurement in the data, including the peaking background correction from Table 18. Measurement Raw fit result Peaking BG correction B(Ds → φeν)

Branching Ratio (1.935 ± 0.152)% (0.012 ± 0.012)% (1.923 ± 0.153)%

# Events 208.0 ± 16.6 1.3 ± 1.2 206.7 ± 16.7

Figure 13 shows the result of our likelihood fit on the Ds mass spectrum after our φ, electron, and event selections. The plot shows the sum over all Ds masses and fit functions for simplicity, even though the underlying mass distribution varies by tag mode. We give our individual functions and mass plots for each tag mode in Figures 14 and 15. These individual plots better represent how the likelihood fit operates, although the common branching ratio does connect each of the tag modes’ signal normalizations to one another.

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Figure 13: Ds → φeν data results after our semileptonic selections. We fit the tagged MDs spectrum with a common branching ratio across all 13 tag modes. The likelihood uses each tag mode’s signal shape on its corresponding masses; the above results show a sum over all tag modes.

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Figure 15: Fit results in the data after applying Ds → φeν semileptonic cuts for modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode receives an independent background normalization.

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Measurement of Ds → (Ks, K ∗, f0, η 0)eν

7

Our procedure for the Ds semileptonic modes KS eν, K ∗ eν, f0 eν, and η 0 eν closely follows that for Ds → φeν. We use the same Ds tags and electron selections for each of the additional four semileptonic modes, but each mode has its own cuts for the semileptonic-side’s hadron. In some cases, we also make additional background rejection cuts and event missing mass cuts, as described in Sections 7.2 and 7.3.

7.1

General Particle Cuts

The semileptonic modes K ∗ eν, f0 eν, and η 0 eν each have pions in their final states. CLEO has made substantial effort to understand pion behavior in the detector over a range of momenta, so we follow the standard pion selections [20] for these modes: dE/dx < 3.0 • σπ • | cos(θ)| < 0.93 • |db | < 5 mm • |z0 | < 5 cm • χ2 < 10, 000 • Hit Fraction > 0.5 • If pπ > 550 MeV and we have info from the RICH, we also require: – 2 or more photons in the RICH (pion hypothesis) 2 ) < 0, – Particle ID: (LLHπ − LLHK ) + (σπ2 − σK where LLH refers to the RICH log likelihood and σ comes from the dE/dx measurement.

K ∗ eν also has a kaon in its final state. While φeν required two kaons to make a φ meson, the kaon in a K ∗ reconstruction gets paired with a pion. This forces us to be more aggressive in our kaon selections to avoid excess combinatoric pairings. Relative to φeν, we have increased the hit fraction requirement (although only to 0.3), we have added a particle ID cut, and we have slightly decreased the χ2 requirement. For kaons in K ∗ eν, we require dE/dx < 3.0 • σK • | cos(θ)| < 0.93 • |db | < 5 mm • |z0 | < 5 cm • χ2 < 10, 000

55

• Hit Fraction > 0.3 • If pK > 550 MeV and we have info from the RICH, we also require: – 1 or more photons in the RICH (kaon hypothesis) 2 – Particle ID: (LLHπ − LLHK ) + (σπ2 − σK ) > 0, where LLH refers to the RICH log likelihood and σ comes from the dE/dx measurement.

• If pK < 550 MeV and we have info from the RICH, we require 4 or fewer photons (pion hypothesis) in the RICH.

7.2

Ds → KS eν

Unlike Ds → φeν, where our signal events dominated our background even with fairly loose cuts, Ds → KS eν has a large background component. While some of this background comes from combinatoric effects, by far the dominant contribution comes from other Ds semileptonic modes that have both a valid Ds tag and a valid electron. φ → KL Ks in φeν produces the most problematic background as it also has a true Ks , so tightening our K-short cuts will not help our background rejection (K ∗ → Ks π 0 gives a similar but smaller problem). To deal with the problem of Ds → KS eν background coming from other semileptonic modes, we have reintroduced the notion of an event missing mass cut. In this case, the missing four vector consists of both the neutrino and the Ds∗ meson’s daughter photon. Even though the missing mass doesn’t peak at zero as it would with only a missing neutrino, we still get good separation between KS eν and φeν since the soft photon doesn’t push the missing mass as far as the extra KL (or, to a lesser extent, as far as the extra π 0 in K ∗ eν). We show this missing mass separation in Figures 16 and 17. We’ve optimized our missing mass cut using a simple  S2 , as seen in Figure 18. figure of merit S+B In addition to our peaking background, the missing mass cut rejects the majority of our combinatoric background. However, we still retain more total combinatoric events than signal events. As these combinatoric events often lack a true Ks , we have also considered adding a flight significance cut to our Ks selection. We again use a simple figure of merit for events in our signal region to evaluate our potential flight significance cut. We present this figure of merit for various flight significance cuts in Figure 19. While the Monte Carlo analysis favors a very large minimum flight significance, we have elected to make the cut at 4.0 to capture most of the benefit while avoiding any potential systematics from data/Monte Carlo deviations at higher flight significance values.

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Figure 17: Ds → KS eν missing mass in the 20× Monte Carlo, by background semileptonic mode. The dominant background comes from Ds → φeν, where φ → KL Ks . Our missing mass cut removes most of this background.

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We make a few additional rejections on potential KS eν events to avoid specific backgrounds. In particular, we have small but identifiable backgrounds from Ds → KKs and Ds → KKs π 0 where the kaon fakes an electron. To deal with these, we treat the electron track like a kaon and reject the event if it combines with the Ks (or Ks + π 0 ) to have a Ds mass within 10 MeV of the PDG value. We also see some background with real electrons, notably in Ds → τ ν where a fake Ks gets formed far out in the detector. We eliminate most of these events by requiring that each π track forming the Ks comes from within 20 cm of the origin. This cut also removes a small, similar background from Ds → ηeν. We give our full list of Ds → KS eν semileptonic-side requirements below: • |MKrecon − MKPDG | < 6.3 MeV s s • Ks flight significance > 4.0 2 < 400, 000 MeV2 (.40 GeV2 ) • M Mγν recon • Reject if |MKK − MDPDG | < 10 MeV when the electron is treated as a kaon s s recon PDG • If any π 0 are found, reject if |MKK | < 10 MeV when the electron is treated 0 − MDs sπ as a kaon p • ρπ0 ≡ d2b + z02 < 20 cm for the π from Ks

We obtain a much improved signal relative to background, particularly peaking background, by making these cuts as seen in Table 22. In this table, our “before cuts” column only uses the Ks mass cut, while the “after cuts” column contains the events after making the other listed cuts. Figures 106 (before cuts) and 107 (after cuts) in Appendix F further breakdown the modes that produce a peaking background. Table 22: Truth-tagged breakdown for Ds → KS eν candidates in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. Event truth True Ds tag, true Ds → KS eν True Ds tag, not Ds → KS eν False Ds tag, true Ds → KS eν False Ds tag, not Ds → KS eν

Candidates before cuts 24.95 162.30 4.35 292.30

Candidates after cuts 23.00 3.05 2.55 62.30

After all cuts, we get an efficiency for Ds → KS eν of 30.9% (including the Ks branching ratio). For ease of comparison, we have placed our efficiencies for this and the other non-η branching ratios in Table 34, at the end of the section. Our stated efficiencies come from signal Monte Carlo with different Ds production modes; we cover the systematic uncertainty from this weighting for all modes in Section 9.11. Ds → KS eν still has some remaining peaking background (true Ds ) events from each of the other semileptonic modes we measure, as well as peaking background from events with a fake 61

electron. We correct our Ds → KS eν branching ratio for the semileptonic background sources by using their faking efficiencies and our measured branching ratios. Since the non-semileptonic fakes come from more precisely measured Ds decay modes (primarily through some of the same decay modes that we use for tags), we simply use a Monte Carlo estimate in their branching ratio correction. We give the individual components and the overall sum of these corrections in Table 23. Table 23: Absolute branching ratio correction and systematic error for B(Ds → KS eν) from peaking background. Background mode Ds → ηeν Ds → f0 eν Ds → φeν Ds → K ∗ eν Ds → η 0 eν Non-semileptonic Total Ds → KS eν correction

BR correction (0.0010 ± 0.0005)% (0.0001 ± 0.0001)% (0.0014 ± 0.0005)% (0.0044 ± 0.0011)% (0.0002 ± 0.0002)% (0.0052 ± 0.0011)% (0.0123 ± 0.0017)%

As in Ds → φeν, we have used our twenty data-sized Monte Carlo samples for an in-vs.-out test to ensure that our fitting procedure accurately measures the input branching ratio. Our χ2 of 13.3 over the twenty samples and overall fit-to-truth difference of 0.5σ gives us confidence that our underlying procedure works on this sample size and background rate (full results in Tables 83 and 84 from Appendix E). We show our Ds → KS eν data fit results in Figure 20 for the sum across all Ds tag modes, with the component fits from each tag mode in Figures 108 and 109 (Appendix F). We present both the raw branching ratio and our branching ratio after correcting for peaking background (but before correcting the Monte Carlo efficiency for systematic biases) in Table 24.

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Figure 20: Our Ds → KS eν data fit to the MDs spectrum, after all semileptonic selections. This fit function represents the sum over the fit functions from each Ds tag mode.

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Table 24: Ds → KS eν measurement in the data, including the peaking background correction from Table 23. Measurement Raw fit result Peaking BG correction B(Ds → KS eν)

7.3

Branching Ratio (0.186 ± 0.035)% (0.012 ± 0.002)% (0.173 ± 0.035)%

# Events 44.5 ± 8.4 2.9 ± 0.4 41.5 ± 8.4

Ds → K ∗ eν

Ds → K ∗ eν has a small branching ratio relative to the other Ds semileptonic modes considered in this analysis, but it also has a relatively low number of background events. Our only problematic backgrounds come from Ds → KKπ (e.g. K ∗ K) when a kaon fakes an electron and from Ds → φeν when one of the kaons fakes the K ∗ pion. We can deal with most of the kaon-faking-electron background by simply treating the electron as a kaon and rejecting the event if it combines with the K ∗ to form a Ds . We similarly deal with the φeν background by treating the K ∗ daughter pion as a kaon and rejecting the event if it pairs with the other kaon to form a φ. A missing mass cut on the event rejects much of the combinatoric background (Figures 21 and 22), which would otherwise present the largest remaining challenge to our measurement. As in Ds → KS eν, our missing 4-vector consists of both the neutrino and the unobserved Ds∗ daughter photon, shifting the missing mass away from zero.

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Since our prior cuts remove most of the background, we only need a loose selection on  S2 ∗ the K invariant mass, as shown in Figures 23 and 24. Our figure of merit S+B for the MK ∗ cut window plateaus around 100 MeV. We choose to keep all K ∗ within 106 MeV, which corresponds to a 5σ mass cut (about 2ΓK ∗ ).

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Figure 24: Figure of merit for different K ∗ mass cuts in Ds → K ∗ eν. We only consider events that have a Ds tag mass within 1955 MeV and 1985 MeV, since events outside that region will be dismissed as background in our final fit.

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We list our final Ds → K ∗ eν cuts below: • The K ∗ daughter kaon’s charge must be opposite to the Ds tag charge • |MKrecon − MKPDG | < 106 MeV ∗ ∗ 2 • M Mγν < 450, 000 MeV2 (.45 GeV2 ) recon | < 20 MeV when the electron is treated as a kaon • Reject if |MKKπ − MDPDG s

• Reject if MKK < 1060 MeV when the K ∗ daughter pion is treated as a kaon These cuts remove over half of the peaking background and just under half of the combinatoric background, as seen in Table 25. The “before cuts” column only includes the K ∗ charge requirement and a wide K ∗ mass cut of 150 MeV. The “after cuts” column includes all our listed cuts. Figure 110 gives the breakdown of our peaking background before cuts, while Figure 111 shows the peaking background components after all our cuts. We get a K ∗ eν semileptonic-side efficiency after all cuts of 24.1% (including the K ∗ branching ratio). Table 25: Truth-tagged breakdown for Ds → K ∗ eν candidates in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. Event truth True Ds tag, true Ds → K ∗ eν True Ds tag, not Ds → K ∗ eν False Ds tag, true Ds → K ∗ eν False Ds tag, not Ds → K ∗ eν

Candidates before cuts 33.05 5.35 10.45 126.90

Candidates after cuts 30.15 2.10 7.50 58.85

We correct our raw branching ratio result for peaking background from events with a true Ds . These events come from other semileptonic modes, τ ν, and Ds decays where a kaon fakes the electron. As in Ds → KS eν, we use our measured branching ratio for the semileptonic correction while using the Monte Carlo rates for the non-semileptonic correction. We show the components of this correction and their sum in Table 26.

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Table 26: Absolute branching ratio correction and systematic error for B(Ds → K ∗ eν) from peaking background. Background mode Ds → KS eν Ds → f0 eν Ds → φeν Ds → η 0 eν Non-semileptonic Total Ds → K ∗ eν correction

BR correction (0.0001 ± 0.0001)% (0.0002 ± 0.0001)% (0.0055 ± 0.0012)% (0.0017 ± 0.0007)% (0.0032 ± 0.0010)% (0.0107 ± 0.0017)%

Like our other semileptonic measurements, we have used our twenty data-sized Monte Carlo samples for an in-vs.-out test to ensure that our fitting procedure works with the signal and background levels in Ds → K ∗ eν. We placed the results from this test in Tables 85 and 86 from Appendix E. Figure 25 contains the result of our data fits, summed over all Ds tag modes. Figures 112 and 113 in Appendix F show the fits by individual tag modes. We present our Ds → K ∗ eν branching ratio measurement before and after peaking background corrections in Table 27. Table 27: Ds → K ∗ eν measurement in the data, including the peaking background correction from Table 26. Measurement Raw fit result Peaking BG correction B(Ds → K ∗ eν)

Branching Ratio (0.180 ± 0.040)% (0.011 ± 0.002)% (0.170 ± 0.040)%

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7.4

Ds → η 0 eν

Unlike our other semileptonic modes, where we observe the semileptonic side’s hadron through its direct decay to two final state particles, we reconstruct the η 0 in Ds → η 0 eν through its decay to ππη with η → γγ. This gives us two mass constraints, reducing both our combinatoric and peaking background to very low levels (with signal-to-background comparable to the much higher statistics φeν mode). The extra constraint allows us to use minimal rejections on the semileptonic side and maintain a high efficiency. Although we could use the invariant mass of the η as a constraint, we instead choose to use the η pull mass, which takes the different errors on each daughter photon measurement into account. Adding on a comfortable η 0 mass cut of 10 MeV, we then just have the Ds → η 0 eν cuts • |ση | < 3.0 • |Mηrecon − MηPDG | < 10 MeV 0 0 As seen in Table 28, these cuts give a marginal improvement over our extremely loose “before cuts,” with a 5.0 pull mass on the η and a 30 MeV mass cut on the η 0 . We see a semileptonic-side efficiency for η 0 eν of 4.0% after cuts, including all branching ratios. Table 28: Truth-tagged breakdown for Ds → η 0 eν candidates in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. Event truth True Ds tag, true Ds → η 0 eν True Ds tag, not Ds → η 0 eν False Ds tag, true Ds → η 0 eν False Ds tag, not Ds → η 0 eν

Candidates before cuts 24.60 0.60 3.75 12.40

Candidates after cuts 22.75 0.30 3.10 6.25

What little peaking background we do see (Figure 114) tends to come from Ds → ηeν, where the η → πππ 0 . A photon from the π 0 then combines with another shower (like the Ds∗ daughter photon) to make a fake η. This ηeν peaking background contains few enough events relative to signal that we reject too many true events when we try a direct π 0 reconstruction. We instead just do a peaking background subtraction based on our ηeν measurement. We’ve given this correction and our smaller peaking background corrections in Table 29. Our in-vs.-out Monte Carlo test results for Ds → η 0 eν can be seen in Section E, Tables 87 and 88. Figure 26 shows our data fits for the sum over all Ds tag modes. We have placed our individual tag mode fits in Appendix F, Figures 115 and 116. Table 30 gives our raw Ds → η 0 eν measurement and the result after correcting for peaking background.

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Table 29: Absolute branching ratio correction and systematic error for B(Ds → η 0 eν) from peaking background. Background mode Ds → ηeν Ds → φeν Non-semileptonic Total Ds → η 0 eν correction

BR correction (0.0065 ± 0.0033)% (0.0002 ± 0.0002)% (0.0035 ± 0.0024)% (0.0102 ± 0.0041)%

Table 30: Ds → η 0 eν measurement in the data, including the peaking background correction from Table 29. Measurement Raw fit result Peaking BG correction B(Ds → η 0 eν)

Branching Ratio (0.646 ± 0.140)% (0.010 ± 0.004)% (0.636 ± 0.140)%

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Figure 26: Our Ds → η 0 eν data fit to the MDs spectrum, after all semileptonic selections. This fit function represents the sum over the fit functions from each Ds tag mode.

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7.5

Ds → f0 eν

Ds decays often contain kaon final states, which makes the f0 → ππ reconstruction mode for Ds → f0 eν fairly clean. Although f0 eν has some peaking background from other semileptonic modes (like Ds → η 0 eν, η 0 → ππγ), most of our peaking background comes from events with a kaon faking the electron. These events also generally involve another kaon faking a pion, as the large Vcs ensures that kaons in Ds decays tend to come in pairs. Due to this double-fake rarity, none of the individual modes with electron faking occur with significant frequency. Since we can’t simply reconstruct all such modes to reject the event without hitting our signal through the combinatorics, we instead just apply a correction using the Monte Carlo expected rates for such fakes. As shown in Figures 27 and 28, we have a fairly low background in Ds → f0 eν, so we use a relatively broad mass cut. Other than our standard pion and electron cuts, our f0 eν reconstruction only contains the f0 mass cut: • |Mfrecon − MfPDG | < 60 MeV 0 0 We take 980 MeV to be the f0 PDG mass.

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Figure 27: f0 mass for events passing Ds and electron cuts in the 20× Monte Carlo.

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Figure 28: Figure of merit for different f0 mass cuts in Ds → f0 eν, considering only events with a Ds tag mass within 1955 MeV and 1985 MeV. Since the f0 width has some uncertainty, a 60 MeV mass cut gives us a good balance between retaining most of the signal while not allowing too much excess background.

Table 31 shows our peaking and combinatoric background levels compared to our signal events. Figure 117 in Appendix F shows the breakdown of our true Ds tag background, while Table 32 has our branching ratio corrections for this peaking background. Given the uncertainty in B(f0 → ππ), we’ve chosen to quote a result for Ds → f0 eν, f0 → ππ rather than assuming any particular branching ratio.

76

Table 31: Truth-tagged breakdown for Ds → f0 eν candidates passing all cuts in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. Event truth True Ds tag, true Ds → f0 eν True Ds tag, not Ds → f0 eν False Ds tag, true Ds → f0 eν False Ds tag, not Ds → f0 eν

Passing candidates 55.10 3.75 6.70 171.80

Table 32: Absolute branching ratio correction and systematic error for B(Ds → f0 eν, f0 → ππ) from peaking background. Background mode Ds → φeν Ds → K ∗ eν Ds → η 0 eν Non-semileptonic Total Ds → f0 eν, f0 → ππ correction

77

BR correction (0.0002 ± 0.0001)% (0.0001 ± 0.0001)% (0.0014 ± 0.0005)% (0.0037 ± 0.0008)% (0.0054 ± 0.0010)%

As with our other modes, we have done an in-vs.-out test for Ds → f0 eν using the twenty Monte Carlo data-sized samples. Tables 89 and 90 in Appendix E contain the results of this comparison. Figure 29 has the summed results of our data fits across all Ds tag modes. Figures 118 and 119 in Appendix F contain the individual tag mode fits. We give our raw measurement and background corrected branching ratio for Ds → f0 eν, f0 → ππ in Table 33.

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Figure 29: Our Ds → f0 eν data fit to the MDs spectrum, after all semileptonic selections. This fit function represents the sum over the fit functions from each Ds tag mode.

78

Table 33: Ds → f0 eν, f0 → ππ measurement in the data, including the peaking background correction from Table 32. Measurement Raw fit result Peaking BG correction B(Ds → f0 eν, f0 → ππ)

Branching Ratio (0.135 ± 0.024)% (0.005 ± 0.001)% (0.130 ± 0.024)%

# Events 43.6 ± 7.8 1.7 ± 0.3 41.9 ± 7.8

For comparison, we summarize all our Monte Carlo efficiencies, by semileptonic mode, in Table 34. We will later correct these efficiencies through the various effects in Section 9 (Tables 100–103), leading to the final branching ratios in Table 66. Table 34: Semileptonic-side efficiencies for φeν, η 0 eν, f0 eν, KS eν, and K ∗ eν, after all cuts. The first column includes the hadron branching ratios into the efficiency, while the second column gives the efficiency considering only hadron decays to the reconstruced decay mode. Semileptonic Mode Ds → φeν Ds → KS eν Ds → K ∗ eν Ds → η 0 eν Ds → f0 eν

εSL (13.91 ± 0.18)% (30.93 ± 0.53)% (24.13 ± 0.35)% (4.02 ± 0.16)% (21.69 ± 0.34)%

79

BR εexcl SL (28.33 ± 0.38)% (45.09 ± 0.77)% (36.25 ± 0.53)% (23.46 ± 0.96)% (41.72 ± 0.66)%

Measurement of Ds → ηeν

8

We find Ds → ηeν events in much the same way as Ds → φeν events: we reconstruct the Ds tag, the electron, and the semileptonic-side hadron, while ignoring the Ds∗ → Ds γ transition photon or other extra showers in the event. We originally made the choice to ignore the transition photon in Ds → φeν because of the photon’s low efficiency, generally low backgrounds in Ds semileptonics, and complications from splitoff showers. We still benefit from leaving the transition photon out of our reconstruction for Ds → ηeν, but the η → γγ reconstruction has higher backgrounds from both combinatoric background and splitoff showers that contribute to fake η. The rise in combinatoric background doesn’t present a huge problem, as we still have a potential sideband subtraction and fewer background than signal events in the signal region. Splitoff showers used in fake η do cause difficulties, however, since true ηeν events can have improperly reconstructed η with a rate not always well modeled by the Monte Carlo. With Ds → φeν, we reconstructed the φ but only used it as a consistency check rather than directly involving it in the fit since false φ didn’t present much of an issue. For Ds → ηeν, however, false η in background modes become a problem, so we need to use the η kinematic information directly in our fit. Specifically, we do a two-dimensional fit to the η pull mass and the Ds mass to determine the number of Ds → ηeν events. We retain the electron selections and event cuts from Sections 5.1 and 5.2, respectively.

8.1

η Selection

We considered a cut on EE259 for the η daughter photons, but we found that the reduction in background did not compensate for the relative 6% efficiency loss (Figure 30). In general, we have found that our 2D fit to the η pull mass and Ds invariant mass separates signal from background well enough that we can use a fairly loose selection on the η. We make the following selections on the η daughter photons: • No track matches the shower location • No showers may come from hot crystals • The shower must come from the barrel or the endcap of the calorimeter, not the transition region in between • Eγ > 30 MeV Beyond the individual photon cuts, we also make a simple η selection. We use the η pull mass instead of the nominal mass to take advantage of the two showers’ uncertainty information. Our loose pull mass cut of 5.0σ ensures that we have a sideband region for false η in our eventual 2D fit. As in Section 7.2 with Ds → KS eν, we have made a cut on the mass of the missing four vector. This missing mass includes both the neutrino and the Ds∗ → Ds γ transition photon. The soft transition photon’s low energy ensures that the missing mass for properly reconstructed events stays closer to the neutrino’s missing mass (zero) than incorrectly reconstructed events. This allows us to cut out most of the combinatoric background, the majority of fakes from other Ds semileptonic modes (e.g. η 0 eν, η 0 → π 0 π 0 η), and a significant portion of misreconstructed η from true events, as seen in Figures 31 and 32. 80

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Figure 30: The EE259 O.K. cut’s effect on the η pull mass distribution. Top: Reconstructed η spectrum with and without the EE259 O.K. cut. Bottom: Normalized η spectrum with and without the EE259 O.K. cut, showing that the cut doesn’t disproportionally change the pull mass distribution (slightly lower efficiency for large pull masses).

81

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Figure 31: Ds → ηeν missing mass after reconstructing a Ds tag, an η, and the electron in the 20× Monte Carlo. The solid blue line represents all generated Ds → ηeν events that have a correct Ds tag, while the dotted blue line has the additional requirement that the η gets properly reconstructed from its daughter photons (no splitoff or transition photon fakes).

82

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Figure 32: Our figure of merit for different missing mass cuts in the Ds → ηeν signal region (within 3σ of a correct Ds mass and η pull mass). The black dots correspond to the cuts from this analysis, in which we choose a maximum missing mass cut of 500 × 103 MeV2 to err on the side of high efficiency. We have also tried reconstructing the best Ds∗ → Ds γ transition photon when available and incorporating it into the missing four vector (green dots). However, we don’t see an improvement in our figure of merit within the Monte Carlo by including the transition photon, and using it would expose us to potential problems from the modeling of splitoff showers.

83

Overall, we require the following cuts on the η beyond those for its daughter photons: • |ση | < 5.0 2 < 500, 000 MeV2 (.50 GeV2 ) • M Mγν

8.2 8.2.1

Ds → ηeν Reconstruction Efficiency

As with Ds → φeν (Section 6.2.1), we generated four signal Monte Carlo samples of 250,000 events for each Ds production mode (Ds+ Ds− , Ds∗ Ds with prompt Ds → ηeν, Ds∗ Ds with Ds∗ → Ds γ or Ds∗ → Ds π 0 ). We give our efficiencies for our wide two-dimensional fit region (1900 MeV ≤ MDs ≤ 2030 MeV and ση < 5.0) in Table 35. For comparison, we also provide the efficiency with a tighter η selection ( ση < 3.0) in Table 36. The semileptonic efficiencies include our event missing mass cut, so they’re slightly smaller than the simple product of the electron and η efficiencies. Table 35: Efficiencies for semileptonic particles in Ds → ηeν, with the η cuts used in this analysis. The η and semileptonic efficiencies include the η → γγ branching ratio. Ds production mode Ds Ds ∗ Ds Ds with Ds∗ → (Ds → ηeν) γ Ds∗ Ds with Ds∗ → (Ds → ηeν) π 0 Ds∗ Ds with prompt Ds → φeν Weighted signal MC Generic MC

e 81.7% ± 79.9% ± 80.7% ± 80.4% ± 80.2% ± 80.3% ±

2.7% 0.7% 0.7% 0.7% 0.5% 0.5%

η 26.8% ± 26.4% ± 26.5% ± 26.7% ± 26.6% ± 26.5% ±

1.6% 0.4% 0.4% 0.5% 0.3% 0.3%

SL 20.7% ± 20.7% ± 20.7% ± 20.5% ± 20.6% ± 20.4% ±

1.4% 0.4% 0.4% 0.4% 0.3% 0.2%

Table 36: Efficiencies for semileptonic particles in Ds → ηeν, with |ση | < 3.0. The efficiencies include the η → γγ branching ratio. Ds production mode Ds Ds ∗ Ds Ds with Ds∗ → (Ds → ηeν) γ Ds∗ Ds with Ds∗ → (Ds → ηeν) π 0 Ds∗ Ds with prompt Ds → φeν Weighted signal MC Generic MC

e 81.7% ± 79.9% ± 80.7% ± 80.4% ± 80.2% ± 80.3% ±

2.7% 0.7% 0.7% 0.7% 0.5% 0.5%

η 25.4% ± 24.5% ± 24.6% ± 24.9% ± 24.7% ± 24.7% ±

1.5% 0.4% 0.4% 0.4% 0.3% 0.3%

SL 19.8% ± 19.3% ± 19.3% ± 19.2% ± 19.3% ± 19.1% ±

1.4% 0.3% 0.3% 0.4% 0.3% 0.2%

We have a higher efficiency for electron detection in ηeν than we do in φeν or most of our other semileptonic modes. This improvement comes from the lower mass of the η, leading to a higher Q value in Ds → ηeν and fewer of the low efficiency, slow electrons (Figure 33). We 84

also find that the η → γγ efficiency does not have a strong dependence on the η momentum (Figure 34), in contrast to our other semileptonic modes where slower hadrons have difficultto-reconstruct charged tracks.

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Figure 33: Generated and reconstructed momentum spectra from the 20× Monte Carlo. a) Lab frame η momentum from Ds → ηeν. b) Electron momentum in Ds → ηeν.

Previous CLEO η studies [23] suggest a correction to the relative η efficiency of -5.6% with a relative systematic of 5.9%. We have done our own systematic (discussed in Section 9.6.2 and Appendix C) that does not show a needed correction, although we get a large, relative systematic error of 7.9%. We use this systematic in our final results, as it comes from a run environment that more closely matches our own. For completeness, however, we also include the Ds → ηeν branching ratio when using the corrected efficiency and smaller η efficiency systematic (final results in Table 45).

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Figure 34: Efficiencies for η, electron, and the overall semileptonic side (η+electron, with M M 2 cut), by momentum. Our η and semileptonic efficiencies include the η → γγ branching ratio of 39.5%.

86

8.2.2

Backgrounds

Our event missing mass cut eliminates most of our background combinations from events other than Ds → ηeν or with incorrect Ds tags (Table 37). We can fit the remaining background combinations fairly well with either a linear background function in the η pull mass, a linear background in the Ds mass, or both, as described in Section 8.2.3. Table 37: Truth-tagged breakdown for Ds → ηeν candidates in the 20× ddmix and 5× continuum Monte Carlo, scaled to data size. Event truth True Ds tag, true Ds → ηeν True Ds tag, not Ds → ηeν False Ds tag, true Ds → ηeν False Ds tag, not Ds → ηeν

Candidates before cuts 604.55 109.60 90.80 787.75

Candidates after cuts 499.30 22.20 59.05 198.70

We still have a problem with combinations from true Ds → ηeν events where the tagged Ds and electron get properly constructed, but where we have an improper η reconstruction (“volunteer” events). The η can be formed by using a splitoff shower from one of the tagged Ds tracks or by using the Ds∗ daughter transition photon (Figure 35). These misreconstructed η combinations make up 27% of all true Ds tag, true Ds → ηeν combinations in the Monte Carlo, which can be seen in the difference between the solid and dotted Ds + ηeν lines in Figure 31. We attempted a best candidate selection on the η, such that each true event only gets counted once whether it has a volunteer combination or not. However, this shapes our other false η backgrounds away from a simple linear distribution, and it still requires us to estimate how many volunteer-only events we have for our efficiency systematic. While we expect the Monte Carlo to model the volunteer η combinations from a real η photon paired with the transition photon fairly well, we have found instances (e.g. Ds → φeν) where the Monte Carlo underestimated the number of splitoff showers. Simply rejecting all splitoff showers costs us too many true signal events, so we ultimately ran a separate systematic correction to account for false η pairings with splitoff events. For our splitoff systematic, we take advantage of CLEO’s large sample of D0 and D± events from ψ(2S) → DD events at 3770 MeV. In particular, we use D0 → K ∗ η decays, which gives us a fairly pure η sample after we do the full event reconstruction. From this sample, we see how often we get an extra η combination in events with a correct reconstruction, which tells us the false η rate from splitoff showers (or similar causes, like K → µν decays from the tag side). We then scale the number of anticipated splitoff showers from each Ds mode by the rate of extra splitoffs that we observed in the data from similar D0 modes. We use four different D0 tags (Kπ, Kππ 0 , Ks ππ, Kπππ), which we link with our 13 Ds tag modes (Table 38) to model the possible splitoff opportunities in Ds → ηeν events. We cut fairly harshly on the D0 to get as clean a sample as possible, requiring the beam constrained mass to be within 5 MeV of the D0 mass and the ∆E within 20 MeV of zero. We then reconstruct K ∗ → Kπ from the other tracks, requiring consistent charges with the D0 tag, good tracks (within 5 cm of z0 and 5 mm of db ), and that particle ID matches the K or π track 87

Origin of non-η shower in false η combinations

Transition γ (45.6 %)

Two splitoff showers (0.7 %) Other (4.6 %) K → µ ν (3.7 %)

π0 daughter (9.7 %)

π splitoff (13.9 %)

K splitoff (21.7 %)

Figure 35: Cause of the shower that leads to a false η combination when we have a correct Ds in an Ds → ηeν event, from our 20× Monte Carlo. These false combinations account for 27% of our counts in Ds → ηeν events with a valid Ds tag. Our systematic addresses possible modeling flaws with the data for the three green slices (π splitoff, K splitoff, and K → µν).

2 (L = (σπ2 − σK ) + (Lπ − LK ) less than zero for π, greater than zero for K). We also require that the reconstructed K ∗ mass be within 35 MeV of its PDG mass. Once we have a reconstructed D0 tag and a K ∗ , we ensure that we have a D0 → K ∗ η event by requiring that the recoil of the event fall near the η mass (525 MeV to 600 MeV), as seen in Figures 36 and 37. We then reconstruct η → γγ with a 5σ pull mass cut and ensure that we have only properly reconstructed η by cutting tightly (±10 MeV) on the event’s missing mass with the η included (Figures 38 and 39).

88

Table 38: D0 tag modes used to estimate splitoff systematic for Ds modes. D0 mode Kπ

Kππ 0

Ks ππ Kπππ

Corresponding Ds modes KS K KKπ KS Kπ 0 KKππ 0 πη ππ 0 η πη 0 , η 0 → ππη ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ Cross-check only KS KS π KS K + ππ KS K − ππ πππ

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Figure 36: Recoil mass against the D0 + K ∗ in a 20× Monte Carlo sample. We keep events between the red lines.

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Figure 37: Recoil mass against the D0 + K ∗ from the 3770 MeV data.

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Figure 39: Full event missing mass, by D0 tag mode for D0 + K ∗ η in the 3770 MeV data. We select η combinations between the red lines to determine true η, then we see if there are any other splitoff η combinations in the event.

93

When we have identified events with a correct η reconstruction, we see how many proper events have additional η from splitoff showers after dropping the missing mass requirement. By comparing the extra η from splitoff in Monte Carlo and in data, we can determine the difference in splitoff rate (Table 39). We don’t see a large needed splitoff correction for η reconstruction with our cuts, mostly because the photons involved have larger energies where the Monte Carlo models splitoff better. Since our Ds → ηeν fits still depend somewhat on the Monte Carlo’s splitoff rate, we incorporate these results in a systematic, as described in Section 9.8. Table 39: Rate of additional η formed using splitoff showers, by D0 mode. The data/MC splitoff correction error (extra splitoff factor) includes a small systematic from combinatoric background. Mode

Recon Data %

Kπ Kππ 0 Ks ππ Kπππ Integrated

(25.9 (27.4 (20.9 (39.2 (30.1

± ± ± ± ±

3.6)% 2.5)% 6.2)% 3.4)% 1.7)%

Recon MC % (27.0 (29.9 (34.7 (38.4 (32.2

± ± ± ± ±

0.8)% 0.5)% 1.6)% 0.7)% 0.4)%

Truth-tagged MC % 27.1% 30.0% 34.6% 38.7% 32.3%

Extra splitoff factor 0.959 ± 0.137 0.915 ± 0.085 0.603 ± 0.181 1.020 ± 0.089 0.936 ± 0.054

Foregoing any cuts on EE259 increases our signal at the expense of some extra splitoff background. However, the Monte Carlo models the extra rate of background about as well with no E9 cut as when we do apply an EE259 O.K. cut, shown in Table 40. For our full analysis, we E25 simply drop EE259 to maximize our signal and use our splitoff systematic results to correct for the difference in splitoffs between the data and the Monte Carlo. Table 40: Rate of additional η formed with splitoff showers after applying an EE259 cut, by D0 mode. The data/MC splitoff correction error (extra splitoff factor) includes a small systematic from combinatoric background. Mode

Recon Data %

Kπ Kππ 0 Ks ππ Kπππ Integrated

(10.9 (13.3 (16.7 (20.2 (15.1

± ± ± ± ±

2.8)% 2.1)% 6.2)% 3.0)% 1.4)%

Recon MC % (10.6 (12.5 (13.5 (16.4 (13.4

± ± ± ± ±

0.6)% 0.4)% 1.3)% 0.5)% 0.3)%

Truth-tagged MC % 10.7% 12.5% 13.8% 16.5% 13.4%

Extra splitoff factor 1.033 ± 0.266 1.064 ± 0.169 1.239 ± 0.476 1.236 ± 0.186 1.127 ± 0.110

We also have a small, peaking background from events with a true Ds and a correct η that aren’t Ds → ηeν events. These events come from Ds → φeν where φ → ηγ, Ds → η 0 eν where the η 0 decays to a state with no tracks and an η, and events where a kaon fakes the electron. Most such peaking background get rejected by our event missing mass cut, but we use our 94

measured Ds → φeν and Ds → η 0 eν branching factions to correct the remainder. We give the resultant correction to our Ds → ηeν branching ratio from these corrections in Table 41. Table 41: Absolute branching ratio correction and systematic error for B(Ds → ηeν) from peaking background. Background mode Ds → φeν Ds → η 0 eν Non-semileptonic Total Ds → ηeν correction

8.2.3

BR correction (0.0037 ± 0.0011)% (0.0104 ± 0.0029)% (0.0017 ± 0.0008)% (0.0158 ± 0.0032)%

Fit Procedure

In our other semileptonic modes, we had low background on the semileptonic side. This let us fit the Ds invariant mass for both our tags and semileptonic events without worrying about the specific reconstruction of the electron or hadron. Ds → ηeν has somewhat more background on the semileptonic side due to the relative ease of making an η. We solve this problem by directly incorporating the η into our fit, doing a 2D fit to the η pull mass and the tagged Ds mass after getting a good electron. We find no correlation between the η pull mass and the Ds mass (as expected), so we can use a simple product of the two distributions for our fit functions. This allows us to reapply the Ds mass signal lineshape that we determined previously from our tag fits for each Ds mode. We then generate a truth tagged η lineshape with our cuts in the Monte Carlo, which we take to be the η signal distribution. We use a linear background for combinations with a false η or a false Ds . We determine a normalized slope for the η background and for the background in each of the Ds modes by fitting their 1D background projections in the Monte Carlo. As in Ds → φeν, we do an unbinned, extended maximum likelihood fit over each Ds tag mode, with the fits linked by a common branching ratio parameter. However, instead of one background normalization for each tag mode due to false Ds , we now have the possibility of a false η, a false Ds , or both. We don’t require an extra parameter for combinations with a false η and true Ds as we determine their rate by one of two methods. The larger component of false η, true Ds combinations comes from volunteer (true) events, where the reconstructed η contains a splitoff shower or the Ds∗ transition photon. We use our splitoff study to estimate the splitoff combination rate, and we use the Monte Carlo for the transition photon combination rate (which only involves kinematics). We then tie both to the rate for correctly reconstructed, true events (the branching ratio). The smaller component of false η, true Ds combinations come from events that do not have a Ds → ηeν. The majority of these combinations arise from other Ds semileptonic modes, where either splitoff showers combine with a real η shower or where the semileptonic hadron decays to multiple π 0 . We determine the rate of such combinations from the Monte Carlo and 95

correct that rate using our measured branching ratio for each of those Ds semileptonic modes. We have a small remaining component (half an event) expected from all other sources; we use the Monte Carlo rate for such combinations. This leaves 13 parameters (one per tag mode) for false Ds combinations with a true η. Similarly, we have another 13 parameters for combinatoric background (false Ds , false η), one per tag mode. We have a total of 1 signal parameter (the common branching ratio) and 26 background parameters in our fit. Our log likelihood function to be minimized is given by ~ BG , K ~ BG ) ≡ − ln L(B, N ~ BG , K ~ BG ) F(B, N " 13 # X i i = B · εSL · Ntag · (1 + r ) " + !ηeν ·

i=1 13 X

# [j] Ntag

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" +

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X

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ln F (mj , ση[j] ),

mj

where [j]

[j]

F (mj , ση[j] ) ≡ B · εSL · Ntag · fsig (mj ) · gsig (ση[j] ) [j]

[j]

+ B · εSL · Ntag · fsig (mj ) · gBG (ση[j] ) · r[j] [j]

[j]

+ !ηeν · Ntag · fsig (mj ) · gBG (ση[j] ) [j]

[j]

[j]

[j]

+ NBG · fBG (mj ) · gsig (ση[j] ) + KBG · fBG (mj ) · gBG (ση[j] ). Here, B is the Ds → ηeν branching ratio, εSL is our ηeν efficiency, ri is the total rate of false η from splitoff and transition photon combinations for Ds tag mode i, !ηeν is the total ~ BG is the background normalization for rate for false η combinations from all non-ηeν sources, N ~ false Ds /true η events (one per tag mode), and KBG is the combinatoric background. [j] refers i to the tag mode associated with the Ds mass, mj . Our distributions are given by fsig for the i normalized signal mass distribution of true Ds with tag mode i, fBG for the normalized, linear background function from false Ds of tag mode i, gsig for the normalized pull mass distribution from true η, and gBG for the normalized, linear background function from false η. In F, the first term corresponds to the number of signal Ds → ηeν events. The second term (with ri ) gives the extra false η combinations formed from true events, while the third term gives false η combinations from other sources. The fourth and fifth terms combine to form the total background from false Ds tags. The last, log term gives the sum over the distributions for each of the different signal and background sources, explicitly stated in the definition of [j] F (mj , ση ). 96

8.3

Results

Unlike the other semileptonic modes where our statistical errors dominate, our Ds → ηeν measurement has comparable levels of statistical and systematic error. The difficulty of obtaining a clean and comparable sample for the η efficiency drives the systematic error, so we have included that systematic (described with more detail in Section 9.6.2) in the following results. Section 10 contains the final result with our additional, less dominant systematic errors from Section 9. 8.3.1

Monte Carlo

We first break our 20× Monte Carlo into twenty data-sized samples to test our analysis technique with a limited statistics data set. Our comparison across these data-sized samples, given in Tables 42 and 43, show that our analysis reproduces the generating Ds → ηeν branching ratio and number of signal events to within statistical error (χ2 of 26.9 over 20 samples). Table 42: Monte Carlo comparison of the measured Ds → ηeν branching ratio to its generating branching ratio (2.480%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Weighted averages/χ2

Raw fit BR (2.712 ± 0.132)% (2.423 ± 0.124)% (2.251 ± 0.119)% (2.602 ± 0.128)% (2.313 ± 0.122)% (2.588 ± 0.125)% (2.485 ± 0.122)% (2.492 ± 0.124)% (2.321 ± 0.113)% (2.263 ± 0.121)% (2.530 ± 0.125)% (2.384 ± 0.123)% (2.511 ± 0.123)% (2.480 ± 0.124)% (2.365 ± 0.122)% (2.758 ± 0.131)% (2.539 ± 0.123)% (2.354 ± 0.121)% (2.343 ± 0.120)% (2.569 ± 0.125)% (2.457 ± 0.028)%

97

Corrected BR (2.692 ± 0.132)% (2.404 ± 0.124)% (2.231 ± 0.119)% (2.583 ± 0.128)% (2.293 ± 0.122)% (2.568 ± 0.125)% (2.465 ± 0.122)% (2.473 ± 0.124)% (2.301 ± 0.113)% (2.243 ± 0.121)% (2.511 ± 0.126)% (2.364 ± 0.123)% (2.491 ± 0.123)% (2.460 ± 0.124)% (2.346 ± 0.122)% (2.738 ± 0.131)% (2.519 ± 0.123)% (2.335 ± 0.121)% (2.323 ± 0.120)% (2.549 ± 0.125)% (2.437 ± 0.028)%

#σ 1.60 −0.62 −2.09 0.80 −1.53 0.70 −0.12 −0.06 −1.58 −1.97 0.25 −0.94 0.09 −0.16 −1.10 1.98 0.32 −1.20 −1.31 0.55 26.87

Table 43: Test of potential bias in our fitting procedure for Ds → ηeν by comparing the number of truth-tagged semileptonic events to the fit result. We allow cross-feed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sum

truth-tagged NSL+tag 369 369 350 397 343 373 367 348 324 369 369 348 374 380 361 370 390 350 348 377 7276

fit NSL+tag 382.00 ± 22.30 354.60 ± 21.61 334.04 ± 21.13 382.58 ± 22.38 334.00 ± 21.03 372.55 ± 21.62 364.06 ± 21.34 368.40 ± 21.82 333.11 ± 19.36 332.66 ± 21.24 361.51 ± 21.47 346.87 ± 21.35 371.25 ± 21.68 372.84 ± 22.19 339.79 ± 20.99 392.65 ± 22.23 371.83 ± 21.54 346.62 ± 21.28 349.24 ± 21.32 373.66 ± 21.68 7184.27 ± 96.09

Difference (# σ) 0.583 −0.666 −0.755 −0.644 −0.428 −0.021 −0.138 0.935 0.471 −1.711 −0.349 −0.053 −0.127 −0.323 −1.011 1.019 −0.844 −0.159 0.058 −0.154 −0.955

Figure 40 shows the 1D projections of our 2D fit to the Ds invariant mass and η pull mass in the Monte Carlo, after summing over all twenty datasets and each tag mode. We have also added the 1D projections of our Monte Carlo results for the four highest statistics tag modes (summed over all twenty datasets) and for the first four datasets (summed over all tag modes) in Figures 122–126 from Appendix F.

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Figure 40: Ds → ηeν 2D fit projections for the reconstructed Ds mass (top) and η pull mass (bottom) in the 20× Monte Carlo, summing over all tag modes.

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8.3.2

Data

Table 44 contains our Ds → ηeν branching ratio with statistical error after we correct for true Ds , true η feedthrough from other semileptonic modes. Figure 41 shows the projections of our reconstruction and fit along the Ds invariant mass and the η pull mass. Table 44: Ds → ηeν measurement in the data, including the peaking background correction from Table 41. Measurement Raw fit result Peaking BG correction B(Ds → ηeν)

Branching Ratio (2.265 ± 0.136)% (0.016 ± 0.003)% (2.249 ± 0.136)%

# Events 360.7 ± 21.9 2.5 ± 0.5 358.2 ± 21.9

As the η efficiency systematic (Section 9.6.2) dominates our error for Ds → ηeν (a relative 7.9% systematic versus a relative 6.0% statistical error), we show the branching ratio with just that systematic error added in Table 45. For comparison, we’ve also included the Ds → ηeν result when using the standard CLEO η efficiency systematic [23]. That analysis uses different η selections and has a cleaner environment (ψ 0 → ηJ/ψ), extrapolating the systematic on their monoenergetic η from a π 0 study. They saw a relative systematic error of 5.9% with a relative efficiency correction of -5.6%. Although we feel that the standard systematic provides a viable alternative, we have chosen to use our own systematic for the final result, believing that we gain improved accuracy at the expense of finer precision. Table 45: Ds → ηeν branching ratio and errors under both η efficiency systematic scenarios. η efficiency systematic This analysis Alternate systematic

Ds → ηeν branching ratio (2.249 ± 0.136 ± 0.179)% (2.375 ± 0.143 ± 0.134)%

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Figure 41: Ds → ηeν 2D fit projections for the reconstructed Ds mass (top) and η pull mass (bottom) in the data, summed over all tag modes.

101

9

Systematic Uncertainties

Our main limit on measurement precision in Ds semileptonic decays with our sample comes from the amount of available data. However, all six of our semileptonic modes have non-trivial errors of the “how well do we know what we know” variety, which we have determined and enumerated to the best of our abilities in this chapter.

9.1

Ds Tagging

We have a variety of possible systematic effects that could cause a mismatch between the Monte Carlo efficiency and the true efficiency for reconstructing Ds tags, like our recoil mass resolution or resonant substructure (e.g. KKπ can come from φπ, K ∗ K, or be nonresonant). However, by using the Ds invariant mass for both the numerator and denominator of our branching ratios, the effect of any such errors in our tagging efficiency cancels out. We only have to worry about systematic effects that create different biases in our tag counting than in our semileptonic fits. 9.1.1

Signal Shape Variation

Our Ds tag fits have either a double gaussian or a gaussian plus crystal ball for their signal components, as mentioned in Section 4.4 (Table 6). We use this same shape for both the number of tags (our branching ratio denominator) and the number of semileptonic counts (our numerator), so we don’t expect our result to depend strongly on minor errors with our lineshape. However, we do see far more background relative to signal for our tag counts than for our semileptonic counts, so we could have some bias due to a wider than expected signal looking like background in our tag fits. We allow our Ds mass signal shape’s overall normalization, mean, and overall width to vary when we do the tag counts, so we should expect no bias between the data and Monte Carlo from those parameters. However, we use the sum of two shapes for our signal functions, and we fixed the relative normalization and relative width between them using our predicted (Monte Carlo) histograms. While leaving the overall width to float deals with most MDs resolution issues, we can tell a story about the poorer quality tracks having a worse than expected resolution while the high quality tracks match well, or one where we have more poor quality tracks than the Monte Carlo expects. The relative width or normalization, respectively, would then need to adjust to properly match the true signal shape in data. Our tagged Ds backgrounds make it impractical to simply allow the relative width or normalization to float, so we need to look elsewhere to study any potential biases between the Monte Carlo and data. Since the D± has similar decay modes to the Ds (often with just a K to π conversion), we look at that system to study our tag signal shapes. While probably overkill, we wanted to keep the procedure as close to our Ds tagging as possible, so we use DD∗ events at 4170 MeV instead of moving to the high data running at 3770 MeV. This costs us some precision (and generates much more work), but it allows us to use a similar choice of a best recoil mass to protect us from the (unlikely) possibility that a best choice somehow biases the track quality in a way not predicted from the Monte Carlo. We use seven different D± tag modes, each of which corresponds to one or more Ds tag modes (Table 46). We reconstruct the D± tags with the same daughter particle cuts as listed

102

in Section 4.1 for the Ds . Our best recoil mass selection for each charge now takes the D± with a recoil mass closest to the D∗+ mass instead of the recoil mass closest to a Ds∗ . Table 46: D± tag mode used for each Ds mode’s relative normalization and relative width systematics. D± mode Kππ Ks ππ 0 Kπππ 0 Ks π Ks K KKπ ππ 0 π 0

Corresponding Ds modes πππ KS Kπ 0 πη 0 , η 0 → ππη KKππ 0 πη KS KS π KS K KS K + ππ KS K − ππ KKπ ππ 0 η ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ

We fit each MD± spectrum with our analogous Ds signal function (double gaussian or gaussian plus crystal ball), fixing the relative normalization and relative width to the Monte Carlo signal. Then, we fit the MD± spectrum in data with the relative normalization or relative width allowed to float. We treat the ratio between the data’s relative normalization (or width) and the Monte Carlo’s relative normalization (or width) as our 1σ systematic variation. Once we have the systematic variation on the relative normalizations or widths, we redo the Ds semileptonic analyses with our rescaled values. We show the systematic on our semileptonic branching ratios for each mode in Table 47. Table 47: Systematic errors from our Ds tag fits. Semileptonic mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

Tag statistics 1.0% 1.0% 1.3% 1.0% 0.9% 1.2%

Fit shape, 1.5% 1.9% 1.6% 1.2% 1.3% 0.5%

N2 N1

Fit shape, 0.3% 0.7% 0.3% 0.3% 1.5% 2.7%

σ2 σ1

Total systematic 1.8% 2.2% 2.1% 1.6% 2.2% 3.0%

Table 47 also gives the branching ratio systematic from the statistical uncertainty in our Ds tag count measurements. Since the tag counts for each mode provide an explicit weighting in 103

our linked semileptonic fits, we can’t simply add their errors in quadrature with the numerator errors. Instead, we follow each statistical variation through the entire process and add the final results from each tag mode’s fluctuations in quadrature. Whether this error should be considered statistical in nature because it arises from underlying statistics on the tagging or systematic in nature because it creates a bias in our numerator fits’ weights is a matter of philosophy; we include it here as it has more in common with our tagging systematics than with our semileptonic measurements. 9.1.2

Background Functions

We approximate the combinatoric backgrounds on our Ds tag fits with either a linear or quadratic background function, depending on the tag mode (Table 6). This gives us some flexibility in case the data has slightly different combinatorics than predicted by the Monte Carlo, but it may give the background too much freedom to add or steal counts from our signal (particularly the quadratic backgrounds). To estimate our systematic error from our choice of background function, we also fit using a one parameter background histogram with our signal function. Our background histogram includes both a charm and a continuum component, which we simply fix to the Monte Carlo expectation so that we only have one free background parameter. The histogram background fits give us roughly the same tagging results as our normal procedure, with a relative difference in total Ds counts of less than 1%. However, since each mode’s tag counts also act as a weighing function for our semileptonic fits, we follow the changes through the entire procedure to our branching ratio. After adding the branching ratio variation from each tag mode’s background histogram fit in quadrature, we obtain the systematic errors shown in Table 48. Table 48: Systematic errors from our Ds tag background shape. Semileptonic mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

9.1.3

Relative systematic error 1.3% 0.9% 1.3% 0.8% 0.9% 2.1%

Multiple Candidate Choice

In our Ds tag selection, we make a best candidate choice based on the Ds recoil mass. When multiple candidates exist in the event, this selection can occasionally cause us to throw out the proper tag and instead choose the extra candidate (with a non-peaking Ds mass). The multiple candidate rate when the other side Ds decays semileptonically differs slightly from when it decays generically, creating a slight bias between tags in semileptonic events and tags 104

without a semileptonic event. The Ds tags’ multiple candidate efficiency comes primarily from kinematics and differences in charged/neutral daughter hadron decay rates, both of which should be well modeled by the Monte Carlo. We determine the systematic shown in Table 49 by combining the tag multiple candidate efficiency, each semileptonic mode’s branching ratio, and the small difference in multiple candidate efficiency for the semileptonic mode compared to the overall multiple candidate efficiency. Table 49: Relative systematic error from the multiple candidate efficiency difference between semileptonic and all other Ds decay modes. Semileptonic mode Ds → φeν Ds → ηeν Ds → η 0 eν Ds → f0 eν Ds → KS eν Ds → K ∗ eν

9.2 9.2.1

Relative systematic 0.11% 1.67% 0.21% 2.20% 3.05% 0.28%

Tracking Reconstruction

Our semileptonic hadron reconstruction efficiencies in Section 9.6 depend in part upon the reconstruction efficiency for their daughter pion and kaon tracks. CLEO has performed tracking systematics for π and K reconstruction using the baseline track selections that we also adopt for our Ds tagging [24]. They see no correction needed between the Monte Carlo and data track reconstruction efficiencies, with a 0.3% systematic on π tracks and a 0.6% systematic on kaon tracks. The standard CLEO systematic analysis works for our typical π reconstruction, used in f0 eν, K ∗ eν, and η 0 eν. However, the K systematic does not carry over to our analysis for a couple reasons. The 0.6% kaon reconstruction systematic depends on a momentum distribution peaking near 500 MeV, while we have a much softer momentum distribution with a peak closer to 250 MeV in Ds → φeν. This alone might be dealt with using CLEO’s kaon systematic broken into momentum bins, but we also loosen the kaon cuts for both φ and K ∗ reconstruction to increase the efficiency. To this end, we have performed our own kaon systematic with the track selections used in this analysis and a momentum binning that better follows the kaon momentum distribution in our events. Since the standard CLEO particle ID cuts also differ from our own, we have combined both the kaon track reconstruction and kaon particle ID into one study, given in Appendix D. We find that the Monte Carlo does a good job of modeling higher momentum kaon tracks but that it overestimates the efficiency for low momentum tracks. CLEO’s particle ID study [25] also found a large correction required at low kaon momenta, with our work matching those results. 105

We incorporate the π tracking systematic and the kaon systematics from Appendix D into our overall hadron efficiency systematic in Section 9.6. 9.2.2

π and K Particle ID

CLEO has performed a systematic study for the data/Monte Carlo efficiency difference in its standard π and K particle ID selections [25] using D0 and D+ decay modes in the 3770 MeV data. While we have a customized kaon systematic for our looser cuts (Appendix D), we do follow the standard cuts for pions in K ∗ , η 0 , and f0 . The standard study sees a systematic error of 0.02% for pions with an efficiency correction that has a strong momentum dependence. We correct each mode’s particle ID efficiency based on that mode’s pion momenta. Our f0 and K ∗ pions have an efficiency correction close to the average from the original study (−0.49% per pion), but our slower η 0 pions give us a slightly larger correction. We’ve summarized the integrated corrections and systematics for these three modes from all constituent particles in Section 9.6.4.

9.3

Photon reconstruction

We only reconstruct photons for use in our Ds tags and in our η modes (ηeν, η 0 eν). In the case of the Ds tags, our reconstruction efficiency doesn’t impact our branching ratio measurements, as mentioned in Section 9.1. We also don’t need to make an explicit photon reconstruction systematic for the η since we roll both daughter photon efficiencies into our overall η systematic (described later in Section 9.6.2 and in Appendix C).

9.4

Electron ID

Electron systematic errors have a strong momentum dependence due both to direct particle identification and to changes in the electron identification efficiency when in the presence of other tracks and showers [26]. While the systematic from direct electron identification dominates, we have also included the non-trivial environmental effects (following the D+ → Xe+ ν procedure). Each semileptonic mode has a distinct electron momentum distribution, causing each mode to have its own electron systematic. Table 50 gives our final electron identification systematics and efficiency corrections after integrating over momentum and combining each systematic effect. We’ve included the different components for these systematics and corrections in Tables 92 and 93 (Appendix E). 9.4.1

Wrong Sign Electron

Any peaking background in our six semileptonic modes requires that we have a true Ds and that some track passes the electron cut. We explicitly correct each semileptonic mode for peaking background due to other semileptonic modes, which leaves a small peaking component from events with no direct electrons. Some of our modes (KS eν, K ∗ eν) have problems with kaon tracks faking electron tracks, which we deal with by applying a missing mass cut. After correcting for events with a direct electron and those where another track fakes an electron, we only have to worry about real electrons that get produced indirectly, like through photon interactions in the detector. 106

Table 50: Electron particle identification systematic and efficiency correction, by semileptonic mode. Semileptonic mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

Electron PID systematic 0.68% 0.37% 0.59% 0.59% 0.38% 0.60%

Electron PID correction -1.91% -1.24% -1.71% -1.69% -1.24% -1.69%

Since these indirect electrons don’t have a preferred sign, we can test the impact of such events by reconstructing the entire event with the required electron sign flipped (matching the Ds tag charge instead of the semileptonic Ds charge). We compare our expected numbers of wrong sign electron events from the Monte Carlo to our reconstructed wrong sign events in the data to see if we have an inconsistency. We find very few wrong sign electron events in the data, with fit errors higher than the number of reconstructed events in each case. As Table 51 shows, this consistency with zero events matches our prediction from the Monte Carlo. Only Ds → f0 eν even had a single measured wrong sign event in the data, and this mode also (not coincidentally) had the highest background from false Ds . Table 51: Passing events with a good Ds when reconstructing each semileptonic mode using an electron of the wrong charge. Our errors for the reconstructed events in each mode slightly exceed that mode’s counts (all six modes are consistent with zero). Semileptonic mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

Expected data events (from MC) 0.1 0.2 0.0 0.2 0.1 0.1

Actual data events 0.0 0.7 0.0 1.1 0.1 0.1

The predicted indirect electron events already have a statistical error from our Monte Carlo determined correction. We take no additional systematic for the Monte Carlo’s modeling of such events.

9.5

Monte Carlo Consistency

CLEO collected the 4170 MeV data over five data sets, which correspond to roughly two calendar time periods. The Monte Carlo generation reflects this separation in time, as datasets 107

η e ν efficiency

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39-41 use a different code release from datasets 47 and 48. While we expect no noticeable impact from the different generating time periods on our efficiencies, CLEO regularly tweaked its code to improve the accuracy between data and Monte Carlo. We have thus checked the generic Monte Carlo efficiencies for our six semileptonic modes across each dataset, as well as checking our signal Monte Carlo against the generic Monte Carlo (Figure 42).

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Figure 42: Efficiency (including hadron branching ratios) for each semileptonic mode, by dataset. The solid red line gives the average across the full generic Monte Carlo sample, while the dotted red lines show the 1σ range on this average. The fluctuation across datasets and signal Monte Carlo for Ds → ηeν pushed the limits of random variation. This, combined with a different run environment and η momentum range than in the standard CLEO η efficiency systematic, led to our performing a custom η systematic that we discuss in Appendix C. We also find a discrepancy in the Ds → KS eν efficiency between our signal Monte Carlo and the generic Monte Carlo. We traced this to a random 3σ fluctuation between the two in B(Ks → π + π − ). When we change our denominator to only consider events where the Ks decays to charged pions, this efficiency discrepancy disappears. We do not take any systematic to our efficiency from internal Monte Carlo inconsistency for the other semileptonic modes, as the efficiencies in the signal Monte Carlo and across the 108

five datasets falls within a reasonable distribution around the average.

9.6 9.6.1

Hadron Efficiencies φ

We have two sources of possible systematic error to consider for our Monte Carlo determined φ efficiency. The largest concern comes from the kaon reconstruction and particle ID efficiencies, discussed in Appendix D, which requires both a systematic error and an efficiency correction. The other possible systematic arises from the possibility that the combined kaon tracks in the data could have a broader invariant mass distribution (poorer resolution) than predicted by the Monte Carlo, extending part of the φ distribution beyond our mass window. Our kaon reconstruction and identification efficiency measurement shows that we tend to properly reconstruct soft kaons in the Monte Carlo more often than we actually do in the data. Since the two daughter kaons from the φ have correlated momenta, we need to correct our φ efficiencies based on each kaon momentum pair. Figure 43 shows the result of our φ efficiency correction, by momentum. For the predicted φ momentum distribution from Ds → φeν in the Monte Carlo (ISGW2 model), this results in a φeν semileptonic efficiency change of -8.2% (relative). We additionally obtain two systematic errors from our kaon reconstruction study in Appendix D. The first comes about directly from our measurement limitations on the kaon efficiency in each momentum bin. We treat the individual kaon systematics as correlated and obtain a relative systematic error of 1.7% for φeν given our φ momentum distribution. The second systematic results from the process of splitting the kaon momenta into bins in the first place, given that each bin may not have a constant efficiency. Considering different efficiency distributions across the bins gives us a relative systematic error of 0.5%. In addition to finding and correctly identifying the kaons, they also have to combine to form a φ mass that falls within our cut window (−15 MeV < Mφrecon − MφPDG < 30 MeV). Since our mass window already captures most of the φ spectrum, we don’t expect any resolution difference to significantly affect our efficiency. We explicitly test this by assuming that the data φ could peak in a slightly different location (a shift) and by taking the data resolution to have a gaussian smear convoluted with the Monte Carlo resolution. We don’t have enough data to test the φ resolution explicitly in our tagged Ds → φeν analysis, so we instead use an inclusive approach by plotting the KK spectrum when we find an electron in the event (no Ds tagging). We use our standard Ds → φeν kaon cuts but to avoid electron-only events (e.g. ee → eeee, where two electrons fake kaons), we also require that the kaons not pass electron cuts and that the tracks not be too close to the beamline (| cos(θ)| < 0.8). We have redone the systematic relaxing these additional kaon restrictions with a similar result, but we get less precision due to the extra background. We use the Monte Carlo signal and background functions to fit our data spectrum, allowing the signal to shift or have a poorer resolution from a gaussian smear. Figure 44 shows our best fit to the data (a peak shift of −0.05 MeV and smear with a σ of 0.1 MeV) alongside the fit with no shift or smear allowed. Not only do we obtain a small relative systematic of 0.025% from the φ resolution over our large window, but we find that the Monte Carlo matches the data well enough that we’d see a small systematic even with a tighter mass window. We also

109

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Figure 43: Top: φ efficiency in the Monte Carlo, by momentum, before and after correcting the efficiency based on the kaon systematic study in Appendix D. Bottom: Ds → φeν semileptonic efficiency, by φ momentum, before and after correction.

show more detailed fit results for our different shifts and smears with Figures 120 and 121 in Appendix F.

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Figure 44: Top: φ mass fit in the data, using the signal and background produced in the Monte Carlo. Bottom: Best φ mass fit in the data after allowing the signal Monte Carlo histogram to shift its peak and convoluting it with variable width gaussians.

111

9.6.2

η

The best existing study of η efficiency systematics at CLEO [23] uses a 3σ pull mass cut in its η selection, compared to the 5σ cut we use in this analysis. That study also uses monoenergetic η from ψ 0 → ηJ/ψ with a relatively soft η momentum of 199 MeV. Ds → ηeν involves a much wider η momentum spectrum, as seen in Figure 33, which gives us a much wider range of resultant photon energies. Furthermore, ψ 0 → ηJ/ψ with J/ψ → µ+ µ− produces a cleaner environment (fewer photons and poorly tracked hadrons) than the environment we see in Ds∗ Ds events. For these reasons, we have decided to do our own η efficiency systematic study. This results in worse precision on our systematic than the high statistics ψ 0 study, but we feel that it more accurately represents the η selections and environment in our analysis. Our η efficiency systematic uses the same Ds tag modes as our standard analysis, but it takes advantage of the relatively large Ds± → ρ± η branching ratio to get a clean η sample. We have described our full efficiency systematic technique and results in Appendix C. Unlike the previous study of η efficiency systematics at CLEO, we do not find that we need a Monte Carlo correction to match the data. That study suggested a relative correction of -5.6% to the η efficiency for η without an EE259 O.K. cut (-6.5% for those with an EE259 O.K. cut). In our study, with a broader pull mass cut and wider η momentum range, we find that the data and Monte Carlo agree to within 1.2%. This falls well within the precision of our study, so we take no efficiency correction. Our systematic procedure has the downside of requiring a large systematic error given the statistical error in our ηeν measurement. While the previous η study had a relative 5.9% efficiency systematic after extrapolating to a wider momentum region, our η systematic procedure yields a 7.9% relative systematic. This makes it a limiting error in our overall Ds → ηeν measurement. 9.6.3

Ks

The standard CLEO systematic study on Ks reconstruction [27] shows no efficiency difference between Monte Carlo and data up to 0.8%, as long as both daughter π tracks have been found. However, our backgrounds in Ds → KS eν lead us to make tighter selections on the Ks than the prior study. While they used a 12 MeV mass cut with no flight significance selection for the Ks , we use a 6.3 MeV mass cut and a flight significance greater than 4.0. Furthermore, we require that the Ks daughter tracks fall within 20 cm of the origin to avoid τ ν backgrounds, which the generic Ks study did not have to concern itself with. Since our selections lead to a significantly different efficiency than the loose cuts from the standard study (a relative difference of about 30%), we have run our own systematic for Ks reconstruction. The low statistics in our Ds → KS eν measurement mean that we can have a fairly forgiving precision from our Ks systematic without impacting our overall error. We consequently try to keep our systematic study’s environment as close to Ds → KS eν as possible by using tagged Ds∗ Ds decays (4170 MeV data) in our systematic measurement rather than the higher statistics 3770 MeV data. We compared the Ks momentum spectrum in Ds → KS eν to that from several other Ks modes, and we ultimately chose Ds → KS K for our systematic above 650 MeV and Ds → K ∗ K ∗ (Ds± → Ks K ∓ π ± π ± ) for our lower momentum systematic (Figure 45). We use a procedure 112

similar to that for our kaon systematic (Appendix D) by reconstructing all particles other than the Ks , then fitting the recoil mass both when we successfully reconstruct a Ks (“found” events) and when we don’t find a Ks (“not found” events). Rather than trust that the general π tracking systematic applies for Ks daughter tracks, we combine our track and Ks reconstruction into one systematic (i.e. we don’t require two extra tracks before looking for candidate Ks events).

Comparison of p Fraction of events (%)

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Figure 45: Normalized Ks momentum distributions in Ds decays to KS eν, KS K, and K ∗ K ∗ (Ks K ∓ π ± π ± ). We use KS K to study Ks reconstruction above 650 MeV and K ∗ K ∗ to study the systematic below 650 MeV. In reconstructing all other particles in the event, we need to find a Ds tag, the Ds∗ daughter photon, and either a single kaon (for KS K) or a kaon and two pions (for K ∗ K ∗ ). Our Ds tag involves the same 13 tag modes as our full analysis, with selections given in Section 4. For a cleaner sample, we also add a mass cut to the Ds , with each tag mode’s cut listed in Table 91 (Appendix E). We use the selections from Section 8.1 for our Ds∗ daughter photons, excepting the minimum energy cut (irrelevant here). After ensuring that the Ds + γ recoil mass matches a Ds (Mrecoil between 1950 MeV and 1990 MeV), we improve the later resolution slightly by rescaling our 113

photon energy to match the Ds∗ Ds kinematic requirement. Our kaons and non-Ks pions must have the proper charges relative to the tagged Ds and pass the selections from Section 7.1. We also require pπ > 100 MeV to eliminate soft pion swaps. After reconstructing all other particles, we separate candidate events into three systematic regions based on their recoil momentum: a high momentum Ks region (650+ MeV), a medium pKs region (400 MeV– 650 MeV), and a low pKs region (200 MeV– 400 MeV). We then attempt recon to reconstruct a Ks using the selections from our Ds → Kp − S eν analysis in Section 7.2 (|MKs π PDG 2 2 MKs | < 6.3 MeV, Ks flight significance > 4.0, and ρ0 = db + z0 < 20 cm for the π from Ks ). We make both a “found” and “not found” plot for the recoil mass against the Ks candidate in each momentum region. We fit each momentum region’s recoil mass plots with a double gaussian for the signal shape and either a linear background function, a scaled histogram background, or both depending on the characteristics of each mode (e.g. the KS K “not found” recoil mass fit requires a Kη background shape; the K ∗ K ∗ requires an extra shape for softly peaking false Ds∗ daughter photons). Figure 46 contains the “found” and “not found” fits for our KS K data, while Figure 47 has the “found” and “not found” data fit results for the two K ∗ K ∗ momentum regions. For completeness, we have also included the fits from Monte Carlo in Figures 127 and 128 as part of our extra figures section (Appendix F). Table 52 gives the efficiency results from our various momentum region fits. We had decided before the study to take a correction to our Ks efficiency if we found a 2σ or larger difference between data and Monte Carlo, and the high momentum region just reached this threshold. The lower Ks efficiency in this region comes from the daughter tracks themselves not being reconstructed properly. When we repeat the analysis requiring that the event has two candidate Ks tracks (with invariant mass between 300 MeV–700 MeV and a combined momentum within 60 MeV of the Ks , following [27]), the difference between data and Monte Carlo disappears (with both about 80% efficient). We get a final correction to our overall Ks efficiency of -11.1% after weighting by the Ks momentum distribution in Ds → KS eν. Table 52: Ks efficiency systematic and correction from our found/not found recoil mass fits in each momentum region. pKs region 200 MeV–400 MeV 400 MeV–650 MeV 650+ MeV Integrated

Syst. mode K ∗K ∗ K ∗K ∗ KS K Combined

εtrue MC 18.5% 19.2% 23.7% N/A

εdata (17.7 ± 3.7)% (20.6 ± 3.1)% (20.0 ± 1.7)% N/A

Correction — — -15.2% -11.1%

Systematic 21.1% 15.5% 8.6% 7.3%

We base our systematic on the combination of our data precision and the Monte Carlo in-vs.-out precision. Since each Ks momentum region has the same efficiency to within error, we do not apply an additional systematic to account for using finite-sized Ks momentum bins. Like our Ks efficiency correction, we have weighted each momentum region’s systematic error based on the KS eν momentum spectrum to get an integrated Ds → KS eν systematic of 7.3%.

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Figure 46: Ds + γ + K recoil mass in data KS K events for “found” and “not found” Ks .

9.6.4

K ∗ , η 0 , and f0

Our K ∗ eν, η 0 eν, and f0 eν modes all have hadrons with relatively broad mass distributions that decay to a final state involving pions (with one kaon for K ∗ and one η for η 0 ). Since we do make a (broad) cut on each resonant particle’s mass distribution to avoid allowing in too much background, we need to ensure that the mass resolution in data matches the Monte Carlo. Additionally, each mode has a distinct momentum spectrum for its final particles, which we need to incorporate into our momentum dependent tracking and particle identification systematics. In all three semileptonic modes, we determine the mass resolution by reconstructing the candidate hadron (K ∗ , η 0 , or f0 ) in a fairly clean Ds mode (Ds → K ∗ K, Ds → πη 0 , η 0 → ππη, or Ds → f0 π, respectively). We fit the candidate hadron’s mass in the data by using the Monte Carlo signal shape. However, we allow the signal mass distribution to shift either direction, and we convolute the signal shape with a gaussian to model potentially poorer data resolution. We take our systematic to be the relative change in events passing our mass cut window for the smeared and unsmeared distributions. In all three cases, the systematic fell well within the precision of our measurement, so we find no need for a correction to our efficiency. To get a clean sample for our mass resolution, we reconstruct a Ds tag, a Ds∗ daughter 115

χ 2 / ndf Nsig µ

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Figure 47: Ds + γ + Kππ recoil mass in data K ∗ K ∗ events for “found” and “not found” Ks . The top row shows only low momentum Ks while the bottom row gives results in our medium Ks momentum region, with pKs determined by the recoil momentum.

photon, and a kaon (K ∗ ) or pion (η 0 , f0 ), following the same procedures and particle selections as for our Ks systematic (Section 9.6.3). We reject any events with extra tracks after the candidate hadron reconstruction, and we cut on the momentum or missing mass of the event to further clean up our sample, as follows: Ds rest • 650 MeV < pin < 750 MeV for the kaon in K ∗ K K Ds rest • 650 MeV < pin < 800 MeV for the pion in f0 π π Ds +γ+π • 900 MeV < Mrecoil < 1000 MeV in πη 0 , η 0 → ππη Ds +γ+π+η0 • Mrecoil < 75 MeV in πη 0 , η 0 → ππη

We have placed the resulting K ∗ , η 0 , and f0 mass resolution plots in Appendix F (Figures 129 – 131). We have put our final mass resolution systematic in Table 53 (along with our tracking and particle identification systematics, described below). 116

We’ve taken a correlated 0.3% track reconstruction systematic for each π and electron track [24]. Since our failures in kaon track reconstruction come primarily from decays in flight, we consider this uncorrelated to the other tracking systematics and have instead incorporated it into our momentum-dependent kaon particle ID systematic. For simplicity, we’ve also absorbed our rather large η reconstruction efficiency systematic into the η 0 particle identification systematic. Only pion and kaon particle identification require a correction to our efficiency, which we have summarized in Table 54. Table 53: Summary of semileptonic hadron systematic errors. Our kaon and η systematics have been included into the K ∗ and η 0 PID columns, respectively. Semileptonic hadron η0 f0 K∗

Track reconstruction 0.90% 0.90% 0.60%

PID 7.90% 0.04% 1.21%

Mass resolution 3.15% 2.63% 2.59%

Table 54: Summary of hadron efficiency corrections from particle identification. Semileptonic hadron η0 f0 K∗

9.7

Relative ε correction -2.94% -0.50% -2.88%

Decays in Flight

For all modes other than Ds → ηeν, we consider any passing event containing the semileptonic mode in question as a true event, regardless of whether or not we correctly reconstructed the semileptonic side. Normally, we don’t have any ambiguity in this procedure because we actually did reconstruct the semileptonic side correctly according to the Monte Carlo. However, we do see a few cases in each mode where a true semileptonic event passes our cuts with an incorrectly reconstructed semileptonic-side hadron. The vast majority of these cases come from either a kaon or pion on the semileptonic side decaying in flight, nearly always to a muon. The Monte Carlo should model the decay-in-flight kinematics without difficulty. However, the “kinked” track could cause problems with tracking reconstruction and the resultant track momentum, used indirectly (and sometimes directly) in our event reconstruction. Given the small effect, we simply take 50% of the efficiency for true but incorrectly reconstructed events as our systematic error to model any possible data/Monte Carlo differences. Table 55 gives the efficiency for all such true but incorrectly reconstructed events and our ultimate systematic (expressed as a relative error). This systematic includes both decay-inflight events and all other events, although the efficiency from other events always falls well below the threshold at which we include systematic errors (a relative 0.3% error). 117

Table 55: Systematic for true semileptonic events that pass with incorrect particle identification, mostly due to π or K decays in flight to µ. We take 50% of the effect’s size in Monte Carlo as our systematic. Semileptonic mode Ds → φeν Ds → η 0 eν Ds → f0 eν Ds → KS eν Ds → K ∗ eν

9.8

εSL pass, wrong MC tag 0.023% 0.040% 0.226% 0.390% 0.343%

Relative systematic 0.08% 0.49% 0.52% 0.63% 0.71%

Splitoff Rate

Our Ds → ηeν fits need to correct for “volunteer” combinations, which predominantly come from either the Ds∗ daughter photon or a splitoff shower combining with a true η daughter photon to make an extra η candidate. Section 8.2.2 discusses this effect with a procedure using D0 → K ∗ η that lets us correct the splitoff rate from the Monte Carlo. We obtained a splitoff correction consistent with the Monte Carlo rate (1σ difference), so we don’t take a bias correction from splitoff. However, our splitoff rate procedure has an associated uncertainty, which we take as the systematic error for the Monte Carlo’s splitoff model. Varying the splitoff rate across its 1σ range gives us a B(Ds → ηeν) relative systematic of 1.16%.

9.9

Hadronic Branching Ratios

We measure each semileptonic hadron decay through a particular decay mode (e.g. η → γγ). The semileptonic modes’ efficiencies depend upon the branching ratio for these hadronic decays. However, updates to the known hadronic branching ratios from more recent measurements require an efficiency correction, while uncertainties in those branching ratios contribute a systematic error. Table 56 gives these corrections and systematics using the most recent branching ratios from the Particle Data Group [1]. The η 0 decay includes both the direct uncertainty in B(η 0 → ππη) and the uncertainty from B(η → γγ). Table 56: Systematic errors and efficiency corrections from uncertain or changed branching ratios in semileptonic daughter hadron decays. Hadron decay φ → KK η → γγ η 0 → ππη Ks → ππ

BMC 49.1% 39.5% 43.7% 68.6%

BPDG (48.9 ± 0.5)% (39.4 ± 0.2)% (42.9 ± 0.7)% (69.2 ± 0.1)%

118

Systematic 1.0% 0.5% 1.7% 0.1%

Correction — — −1.8% 0.9%

9.10

Semileptonic Fit Functions

After making our semileptonic cuts, we fit each mode’s Ds mass spectrum using a linear background function and a signal shape from the tagging fits. The Ds mass signal shape shouldn’t generate an additional systematic beyond that discussed in Section 9.1.1 because we use the same shape for our branching ratio’s numerator and denominator. However, we made the choice of a linear background function empirically, with parameters from a fit to the Monte Carlo’s predicted background. To investigate a potential systematic from our choice of background function, we have replaced our linear background function with a constant function and compared the branching ratio results. A constant function generally goes beyond the 1σ variation on our linear fit to background, but we take this as a worst case scenario on the Monte Carlo’s effective background model. The results from Table 57 show that we get a negligible systematic even for this worst case. The Ds → ηeν mode also includes an explicit fit to the η pull mass spectrum, using a signal histogram shape and a linear background. For this mode, we independently take a constant background on the η pull mass and each Ds mode. Table 57 contains the results after combining all ηeν background systematics in quadrature, which still yields a negligible systematic. Table 57: Branching ratio change from a different semileptonic background function. The Ds → ηeν line combines changes to both the pull mass and Ds mass backgrounds. In all cases, the systematic from choosing a different background shape falls well below the statistical or systematic error. Semileptonic mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

δB/B 0.50% 0.39% 1.04% −0.86% 0.63% −0.35%

σBG syst /σstat 0.063 0.075 0.048 0.048 0.034 0.016

We took the η signal shape for our Ds → ηeν fit from the Monte Carlo. We’ve used two techniques to obtain a systematic on this signal shape. In the first technique, we extract an η pull mass spectrum from a clean Ds → πη sample, we fit the data’s pull mass spectrum with a crystal ball shape, and then we use the resultant crystal ball function as our η signal shape in the branching ratio fit. For our second technique, we convolute the Monte Carlo’s η pull mass spectrum with a gaussian of varying widths and compare the best fit branching ratio to our standard branching ratio (without a gaussian smear). Both techniques come to similar relative systematic estimations (1.0% vs. 0.8%), but we have chosen the Ds → πη method as most closely representing the uncertainty in signal shape. To get our clean η pull mass spectrum from Ds → πη, we reconstruct a tagged Ds + γ, do a basic kinematic fit on the Ds∗ daughter photon, then find the other side π and η. We use the same Ds tag modes and cuts as in our normal analysis (Section 4), dropping the four tag 119

modes with η daughters to avoid any possible complications. We cut on the Ds mass based on its tag mode (Table 91 in Appendix E), on the Ds + γ recoil mass (1950 MeV–1990 MeV), and on the π momentum in the Ds rest frame (within 20 MeV of the ideal 902 MeV). Once we have a reconstructed η, we also require that the event’s missing mass fall within 100 MeV of zero. Events passing all our Ds → πη cuts have nearly no background, giving us a very pure η sample. We fit the data’s pull mass spectrum to a crystal ball function and use that (slightly wider than Monte Carlo) shape in our branching ratio fit. Our branching ratio changes by a relative 1.0%, so we take that to be our η signal shape’s systematic.

9.11

Ds Production Efficiencies

Tagged and semileptonic Ds get created through multiple modes at 4170 MeV. The e+ e− collision can directly produce Ds+ Ds− (σDs Ds = 0.034 nb), or it can produce Ds∗ Ds (σDs∗ Ds = 0.916 nb) [13]. Further, the Ds∗ may decay to either Ds γ (94%) or to Ds π 0 (6%) [1]. Each of these Ds production mechanisms have associated uncertainties, while the Monte Carlo that we use to determine our semileptonic efficiency simply takes each production mode’s most likely value. To incorporate the Ds production mode uncertainty into our overall error, we have determined each process’s tag and semileptonic efficiencies (Table 8 for tags; Tables 16, 35, and 94–97 for semileptonics). We then vary each Ds production uncertainty by 1σ and take the change in our average efficiency as a systematic. In practice, Ds Ds∗ , Ds∗ → Ds γ production dominates our efficiency. This dominance and the fact that each production mode has similar semileptonic efficiencies makes the Ds production systematic negligible (Table 58). Table 58: Relative systematic for various Ds production rate uncertainties. This combines the uncertainties from the Ds Ds and Ds∗ Ds cross sections at 4170 MeV with the uncertainty from the Ds∗ branching ratio (the fraction going to Ds γ vs. Ds π 0 ). These combined effects still contribute a negligible systematic. Semileptonic mode Ds → φeν Ds → ηeν Ds → η 0 eν Ds → f0 eν Ds → KS eν Ds → K ∗ eν

9.12

syst σD s Ds

syst σD ∗ s Ds

B(Ds∗ → Ds π 0 )syst

0.0033% -0.0001% -0.0091% -0.0035% -0.0019% 0.0029%

-0.0018% 0.0001% 0.0050% 0.0019% 0.0011% -0.0016%

-0.0028% -0.0000% -0.0192% 0.0072% 0.0316% 0.0113%

Total relative systematic 0.0047% 0.0001% 0.0218% 0.0082% 0.0317% 0.0118%

Final State Radiation

When the Ds decays to charged particles, the decay can also include photons emitted via an electromagnetic interaction with the final state charged particle. This final state radiation (FSR) doesn’t cause us a problem in tagged Ds , as any tag efficiency drop will be reflected 120

proportionally in our branching ratio’s numerator and denominator. However, the quarks that make up the semileptonic hadron and particularly the electron produced in the semileptonic decay may have FSR that distorts the semileptonic efficiency. We use the PHOTOS 2.0 package to estimate FSR in our Monte Carlo. Since FSR emission from charged particles mostly results in soft photons and our particle efficiencies stay fairly flat outside the extreme regions, our decays’ efficiencies don’t change much with the inclusion of FSR. Only about 2% of semileptonic decays (varying slightly by mode) have FSR that alters daughter particle momenta enough to push the combined heν momenta outside its allowed kinematic range. Of those decays, 90% still have relative efficiencies within 5% of the non-FSR efficiency. Roughly 0.2% of semileptonic decays see a significant efficiency drop, mostly due to the electron momentum falling below threshold. Table 59 gives the efficiency difference with and without FSR for each semileptonic mode. Past work [17] has taken 30% of this difference as a systematic, but none of our efficiency variations affect the overall systematic error even if we take the entire drop as our systematic. We include the results here for reference, but we otherwise dismiss FSR as a systematic effect. Table 59: Efficiency difference due to final state radiation, by Ds semileptonic mode. Semileptonic mode Ds → φeν Ds → ηeν Ds → η 0 eν Ds → f0 eν Ds → KS eν Ds → K ∗ eν

9.13

Relative systematic 0.19% 0.06% 0.06% 0.28% 0.19% 0.28%

Initial State Radiation

In the initial e+ e− collision, one of the two charged particles may emit a soft photon. This initial state radiation (ISR) lowers the collision’s center-of-mass energy. Since CLEO-c ran at 4170 MeV, just above the Ds∗ Ds threshold of 4081 MeV, the Ds momenta for events with even moderate ISR can vary significantly. Fortunately, the Monte Carlo provides a good model for ISR, with Ds single tags at 4170 MeV matching the Monte Carlo’s ISR prediction to within 0.6% [17]. Nonetheless, the Monte Carlo predicts that just over 10% of events in our sample will have a center of mass below 4160 MeV, so we have checked the semileptonic efficiency difference for events produced at lower center-of-mass energies. Table 60 gives the efficiency difference between events produced without ISR and events that include ISR. Not surprisingly, we find very little difference between the two given the fairly flat efficiency across Ds momenta and the fact that most events don’t have significant ISR. Even if the Monte Carlo had too little ISR by 30% (well above the precision extrapolated from the single tag study mentioned previously), we could ignore this systematic. We thus take no additional systematic from ISR effects. 121

Table 60: Efficiency difference due to initial state radiation, by Ds semileptonic mode. Semileptonic mode Ds → φeν Ds → ηeν Ds → η 0 eν Ds → f0 eν Ds → KS eν Ds → K ∗ eν

9.14

∆ε/ε 0.12% 0.79% 0.09% 0.09% 0.77% 0.27%

Generating Models

Our semileptonic efficiency primarily depends on the electron and hadron momenta in the detector (the lab frame). These momentum distributions and their correlations are determined in part by the form factors’ q 2 dependence, which isn’t easily calculable from first principles. Our Monte Carlo uses the ISGW2 [?] quark model when generating the semileptonic Ds decay, but various pole dominance models [?] offer alternate form factor dependencies and consequent momentum distributions. While we believe the ISGW2 model best represents the underlying physics given the relatively heavy c quark and the wide q 2 range relative to the number of close resonances in the charm system, the pole model has been used most often in the literature and gives us a simple alternative to estimate our efficiencies’ form factor model dependence. We have used the pole model’s simplest form as our point of comparison, in which a single 2 resonance dominates the form factors. In this case, each form factor has a (1− Mq 2 )−1 dependence on q 2 , where M is the mass of the nearest meson resonance with appropriate quantum numbers. We use a Ds∗ pole mass for our vector form factors and a Ds1 (2460) pole mass for our axial form factors, matching prior work [?]. For Ds decays to vector hadrons (φ, K ∗ ), we have three form factors and also need the relative normalizations between them; we use rv = 1.81 and 2 (0) and r2 = A are the relative normalizations at q 2 = 0 for the r2 = 0.82, where rv = AV1(0) (0) A1 (0) vector/axial and axial/axial form factors, respectively. We generate our Ds decays using both our baseline model (ISGW2) and the simple pole model, then we treat the difference between the two as a 1σ systematic arising from the generating model. The default CLEO Monte Carlo had some minor coding errors in the masses for its ISGW2 implementation; we corrected those in our own implementation but found that the final systematic didn’t change. For a further comparison point between models, we also include the original, less sophisticated ISGW model (no relativistic corrections, exponential form factor dependence). We did not use the ISGW model in our systematic estimate, although it would have had only a minor effect in any case. Table 61 contains our systematic for each semileptonic decay mode. In all cases, the pole model created events with a higher efficiency than the ISGW2 model. The slightly more energetic electron spectrum in pole model events dominated this efficiency increase with fewer electrons below our 200 MeV minimum pe cut (Figure 48). While decays to pseudoscalar and scalar hadrons have similar q 2 /Ee correlations between models and thus don’t have a significant efficiency change beyond the higher electron efficiency, decays to vector hadrons see a further

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efficiency increase in the pole model from a more beneficial q 2 /Ee correlation, pairing lower q 2 (higher Ehadron ) with more energetic electrons (Figure 49). Table 61: Relative systematic from different generating models’ reconstruction efficiency. Ds Ds Ds Ds Ds Ds Ds

mode → φeν → ηeν → η 0 eν → f0 eν → KS eν → K ∗ eν

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In our extra figures section, we have included the lab frame hadron and electron momenta for each of the other four semileptonic modes (Figures 132 and Figures 133). We’ve also included the q 2 and q 2 vs. Ee distributions for the different models in Ds → φeν and Ds → ηeν decays (Figures 134–136).

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9.15

Sum of Systematic Errors

Each Ds semileptonic decay mode has its own collection of systematic biases and errors, as described earlier in this section. Our extra tables section contains a full systematic error and bias listing for each mode. The systematic bias corrections can be found in Tables 98-103, while the relative systematic errors can be found in Tables 104-109. We’ve included a summary of our total systematics here, with Table 62 giving the efficiency corrections from biasing effects and Table 63 giving each mode’s relative systematic errors. Table 62: Efficiency for each Ds semileptonic mode before and after corrections from systematic biases. These efficiencies include the hadronic branching ratio (taking B(f0 → ππ) = 52% for f0 eν). Semileptonic mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

εMC 13.9% 20.6% 4.0% 21.7% 30.9% 24.1%

εcorrected 12.5% 20.4% 3.8% 21.2% 27.4% 23.0%

Table 63: Total systematic errors (relative) for each Ds semileptonic decay mode. Semileptonic mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

Relative systematic error 4.46% 8.70% 10.11% 4.91% 8.56% 7.13%

In five of our six modes, the statistics available limits our branching ratio measurement rather than our systematic errors. Even the one exception, Ds → ηeν, effectively gets restricted by statistics because that’s the limiting factor in the dominant Ds → ρ + η systematic. Our largest required efficiency bias corrections come in Ds → φeν, driven by soft kaon track reconstruction (true either with our custom kaon selections or with the default CLEO cuts), and in Ds → KS eν, driven by high momentum Ks reconstruction. Incorporating our efficiency corrections and systematic errors into our measurements gives us the final branching ratio results shown in Table 64.

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Table 64: Branching ratios for each Ds semileptonic mode before and after our systematic biases and errors. Decay mode Ds → φeν Ds → ηeν Ds → η 0 eν Ds → f0 eν, f0 → ππ Ds → KS eν Ds → K ∗ eν

Raw B (1.92 ± 0.15)% (2.25 ± 0.14)% (0.64 ± 0.14)% (0.13 ± 0.02)% (0.17 ± 0.03)% (0.17 ± 0.04)%

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Corrected B (2.14 ± 0.17 ± 0.09)% (2.28 ± 0.14 ± 0.20)% (0.68 ± 0.15 ± 0.06)% (0.13 ± 0.02 ± 0.01)% (0.20 ± 0.04 ± 0.01)% (0.18 ± 0.04 ± 0.01)%

10

Conclusion

We’ve measured Ds semileptonic branching ratios for six modes: Ds → φeν, Ds → ηeν, Ds → η 0 eν, Ds → f0 eν, Ds → KS eν, and Ds → K ∗ eν. We primarily used CLEO-c’s 586 pb−1 Ds∗ Ds sample, where the Ds∗ generally decays to a Ds via a soft photon emission. One Ds gave us a tag to identify the event, while the other became a candidate for our target semileptonic decays. Table 65: Number of observed signal events for each of our six semileptonic modes. We include the branching ratios with their statistical errors for reference. Semileptonic mode Ds → φeν Ds → ηeν Ds → η 0 eν Ds → f0 eν, f0 → ππ Ds → KS eν Ds → K ∗ eν

B (2.14 ± 0.17)% (2.28 ± 0.14)% (0.68 ± 0.15)% (0.13 ± 0.03)% (0.20 ± 0.04)% (0.18 ± 0.04)%

Signal Events 206.7 ± 16.4 358.2 ± 21.6 20.1 ± 4.4 41.9 ± 7.8 41.5 ± 8.3 31.6 ± 7.5

As the soft Ds∗ photon had a low reconstruction efficiency and dubious background predictions from our Monte Carlo, we sacrificed the ability to reconstruct a zero neutrino missing mass in exchange for additional events by dropping the photon reconstruction. Since each mode showed fairly low background even without the photon, we could safely loosen our other particle cuts in the key φeν and ηeν modes to gain further signal events. These looser cuts required new analysis for these particles’ reconstruction efficiencies, but the atypically slow kaons and relatively unexplored η momentum range warranted such study in any case. Table 65 summarizes the number of signal events we obtained, while Table 66 states our final results for all six Ds semileptonic modes, including all statistical and systematic errors. Table 66: This analysis’s measured branching ratios for each Ds semileptonic mode.

Ds Ds Ds Ds Ds Ds

Decay Mode → φeν → ηeν → η 0 eν → f0 eν, f0 → ππ → KS eν → K ∗ eν

Branching ratio (2.139 ± 0.170 ± 0.086)% (2.277 ± 0.137 ± 0.196)% (0.680 ± 0.150 ± 0.064)% (0.133 ± 0.025 ± 0.006)% (0.196 ± 0.039 ± 0.015)% (0.178 ± 0.042 ± 0.012)%

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f0 → KK Models

A

f0 → KK decays present a unique problem for the Ds → φeν analysis as the f0 resonance lies very near (and likely below) the KK threshold, yet the f0 ’s wide decay width extends its mass spectrum well into the φ mass region. Thus, Ds → f0 eν events where the f0 → KK invariant mass falls within the φ mass region become effectively indistinguishable from Ds → φeν events and need to be deducted from our φeν signal. To remove these f0 eν events, we use our Ds → f0 eν measurement from the f0 → ππ mode via B(Ds → φeν)correction = B(Ds → f0 eν, f0 → ππ) ∗

B(f0 → K + K − ) εf0 →KK ∗ ∗ fwindow , B(f0 → π + π − ) εφ→KK

(1)

where fwindow represents the fraction of f0 → KK decays with an invariant mass in the φ mass cut window (−15 MeV < Mφrecon − MφPDG < 30 MeV) and εf0 →KK is the reconstruction efficiency for f0 within the mass window. The fraction of f0 → KK decays that fall within our φ mass window (fwindow ) depends upon the f0 → KK mass lineshape. Regardless of the underlying model, this lineshape will necessarily depend upon parameters such as the f0 resonant mass and width, which have large uncertainties from previous measurements [1]. Unfortunately, EvtGen does not describe this lineshape in a consistent manner for resonant masses across the KK mass threshold, as described below, so we have used our own model based on a Flatt´e parametrization [31, 32]. The Particle Data Book estimates the f0 physical mass at 980 MeV ± 10 MeV [1], which extends both above and below the K + K − threshold near 987.4 MeV. We chose to use the central mass (980 MeV) as the input value for our B(Ds → φeν) correction from Ds → f0 eν, f0 → KK, then we estimate one systematic error on the correction by varying the physical mass throughout the 1σ mass range (10 MeV). We use a similar method to vary the total and partial f0 widths. However, since the Particle Data Book does not give a central value for either, we chose 50 MeV as a reasonable value for the total width (with 40 MeV to 100 MeV as our 1σ ππ (with 0.52 to 0.82 for our 1σ systematic variation) and 0.80 as the central value for ΓππΓ+Γ KK systematic variation). We vary each of these values independently in our systematic. This may not be entirely appropriate since the three different f0 parameter values contributed by each experiment are correlated, but we find it prohibitively time-consuming and of marginal benefit to disentangle each experiment’s correlations (if even possible without delving into unpublished results). In section A.1, we discuss the models available in EvtGen that we have chosen not to use, as the information may prove useful to others using EvtGen or similar software. We discuss the Flatt´e model that we instead use, along with its results, in section A.2.

A.1 A.1.1

EvtGen Models Default Model (Breit-Wigner)

In the CLEO Monte Carlo, the mass and width of a particle can be altered by changing its values from the evt.pdl file and passing the modified pdl file to EvtGen. However, EvtGen switches the generating model used when the f0 resonant mass sits below the KK mass threshold from its model for f0 above the KK mass threshold. Specifically, EvtGen uses a non-relativistic 131

Breit-Wigner (Equation 2) for the f0 → KK mass lineshape when the f0 resonant mass lies below threshold, and it uses a relativistic Breit-Wigner (Equation 3) for the lineshape when the resonant mass lies above KK threshold: 2 dΓ NR B-W 1 ∝ (2) , Γ 0 m − m0 + i 2 dm   2 p Rel B-W Γ 0 p0 dΓ   , (3) ∝  dm (m2 − m20 ) + im0 Γ0 mm0 pp 0

where Γ0 is the f0 width, m0 is the resonant mass, m is the invariant KK mass, p is the daughter kaon momentum in the rest frame of m, and p0 is the daughter kaon momentum in the rest frame of m0 . The different lineshapes and their dependence on different f0 masses can be seen in Figure 50. The discontinuous change in lineshape models as the f0 resonant mass crosses the KK threshold produces the dominant effect in our systematic when we try these models, with fwindow between the relativistic and non-relativistic Breit-Wigners differing by a factor of two or more. Ultimately, we do not believe this systematic represents true variation across the threshold, nor do we think that a Breit-Wigner properly models the f0 near threshold in any case, so we instead use a Flatt´e model to describe the lineshape. A.1.2

Flatt´ e Model

The f0 mass lies very near (likely just below) 2mK + , which leads to substantial threshold effects in f0 → K + K − decays that a simple Breit-Wigner does not model well. One can fix the biggest issue by changing the constant width, Γ, to a momentum-dependent width. However, the f0 requires still more work, as the opening of the KK decay mode also alters the ππ mass lineshape below threshold due to analyticity, with non-trivial effects for both modes. The Flatt´e model gives a form for the lineshape that preserves unitarity and analyticity in the threshold region [33], making it appropriate for analysis of the f0 . EvtGen does have a Flatt´e model available, and while CLEO doesn’t use it for all f0 decays (as shown in the previous section), it does use the model for one of six resonances in the Ds → KKπ Dalitz decay. EvtGen’s Flatt´e model for this mode uses the formula: 2 1 dΓKK CLEO Flatt´e ρ3 , ∝ 2 (4) 2 2 ρ ) dm m − m20 + i (gKK ρK + gππ π s  2 2mπ ρπ (m) = 1− , m  q 2  K 1 − 2m above KK threshold, m q ρK (m) =  2mK 2  i − 1 below KK threshold, m where m0 here is the bare mass of the f0 and ρ3 (m) is the three-body phase space factor (relevant in Ds → KKπ but not our semileptonic decay): 132

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v u  2  2 ! u m m π , ρ3 (m) = mρK tλ 1, , mDs mDs λ(a, b, c) = a2 + b2 + c2 − 2ab − 2bc − 2ac. Figure 51 shows the f0 → KK mass lineshape generated from this model with the default parameters m0 = 965 MeV, gKK = 800 MeV, and gππ = 406 MeV. While this Flatt´e model uses reasonable parameters and gives a more sensible f0 mass lineshape than the basic Breit-Wigner, we have chosen to use our own model for a couple reasons. The EvtGen parameters are essentially hard-wired, such that we have to recompile EvtGen each time we want to try a new parameter set. Further, the Flatt´e model only incorporates the 0 K + K − and π + π − decay modes of the f0 , while the f0 can also decay to K 0 K (and π 0 π 0 ). The 133

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K 0 K mode becomes relevant when we do our systematic variation across the f0 mass range listed in the Particle Data Book, as the range extends to 990 MeV— higher than 2mK + but below 2mK 0 , leading the K-coupling to split into both real and imaginary parts.

A.2

Flatt´ e Parametrization

Our Flatt´e model follows the notation of the original paper [31] with the dimensionless coupling constants gK and gπ : 2 √ dΓKK Flatt´e Γ0 ΓK , ∝ 2 2 dm m − mr + imr (ΓK + ΓK 0 + Γπ + Γπ0 ) where

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with mr as the resonant (bare) mass of the f0 and Γ0 as the f0 width at the physical mass. Γ and ΓK 0 follow the same equations as Γπ and ΓK , respectively, with the appropriate mass differences and Γπ0 = 12 gπ pπ0 due to isospin.6 To get the Ds → f0 eν correction on our Ds → φeν measurement from equation 1, we need to determine both the relative amounts of π + π − to K + K − and the fraction of K + K − that falls within our φ mass window. Specifically, we need to use our Flatt´e model to get the product B(f0 →K + K − ) × fwindow for our parameters’ central values and for their 1σ variations. B(f0 →π + π − ) In our formula, we have three parameters we can vary: the bare mass; Γ0 ; and the ratio of couplings, ggKπ . As stated previously, the Particle Data Book gives experimental ranges for three related f0 parameters: the physical mass, Γ (which we take as the width at the physical ππ . We can convert the physical mass, M0 , to the bare mass, mr , using mass, Γ0 ), and ΓππΓ+Γ KK the quadratic given by π0

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below K + K − threshold, above K + K − threshold.

ππ We vary ΓππΓ+Γ by running across a range of different ggKπ values, then integrating the KK ππ resulting lineshapes for Γππ and ΓKK . Our default value of 0.80 stated previously for ΓππΓ+Γ KK gK corresponds to a default gπ coupling ratio of about 2, with a 1σ range from about 1.8 to 9.2. This range also covers most of the results given from experiments in the literature [34, 35], although not quite all [36]. + − 0 →K K ) × fwindow , We find that ggKπ , Γ0 , and M0 are weakly correlated in their effect on B(f B(f0 →π + π − ) so we get our final systematic by adding the results of each variation in quadrature. We give + − 0 →K K ) the default value of B(f × fwindow and the extreme values for each parameter variation B(f0 →π + π − ) in Table 67. The mass lineshapes for each f0 mode are shown with the same parameter values in Figure 52 and Figure 53. + − 0 →K K ) × fwindow correspond to the extreme values of our systemThe extreme values of B(f B(f0 →π + π − ) +



0 →K K ) ππ atic range for the physical mass and for ΓππΓ+Γ . However, the maximum B(f × fwindow B(f0 →π + π − ) KK occurs in the middle of our Γ0 range (79 MeV). This maxima remains even if we extend our possible Γ0 up to 200 MeV, to match some values found in the literature [37, 38, 39]. We present our final correction to the Ds → φeν branching ratio after combining the variations in M0 , ggKπ , and Γ0 with the uncertainty in the Ds → f0 eν branching ratio in Table 68.

6

Or due to the fundamental behavior of states with identical bosons in quantum mechanics. Whatever explanation strikes your fancy.

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Table 67: f0 parameter variations used to determine our f0 → KK correction in Ds → φeν. ππ Our variations correspond to the PDG ranges for the physical mass, Γ0 , and ΓππΓ+Γ . In KK gK gK Γππ practice, we vary gπ instead of directly varying Γππ +ΓKK since gπ has less correlation with the mass and Γ0 . We use f × Physical mass (MeV) 980 990 970 980 980 980 980

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B

Alternate Ds → φeν Reconstruction Methods

In our various Ds semileptonic measurements, we elected not to reconstruct the photon from Ds∗ → Ds γ decays. We made this choice for multiple reasons, notably the efficiency hit for the soft photon, potential systematic issues arising from extra candidate showers in the event (from splitoffs or decay-in-flight tracking failures), and the otherwise low background in Ds semileptonic measurements thanks to the electron and Ds tag. However, we are aware that analyses of this type (e.g. [29]) typically use full event reconstruction to identify the neutrino by its missing mass. Further, other analyses on the CLEO data sample [21, 40] have arrived at a somewhat different B(Ds → φeν) using the missing mass and the Ds∗ meson’s daughter photon in their tagging procedure. To accommodate any misgivings over the approach that we have taken in our primary analysis, we have also measured B(Ds → φeν) using six variant procedures that approximate the CLEO analyses and other potential approaches. In four of our six alternate approaches, we reconstruct the Ds∗ meson’s daughter photon and estimate the number of tags from the Ds + γ recoil mass spectrum. We measure the number of semileptonic decays by using the event’s missing mass; we get four slightly different combinations by toggling the missing mass fit range and whether or not we make a best candidate choice for the photon. In another alternate approach, we reconstruct the Ds∗ daughter photon and require that the Ds + γ recoil mass falls in a reasonable range, but we fit the Ds invariant mass spectrum without the photon for the number of tags. This allows true semileptonic events containing false photon combinations to pass our selection, and it gives us a data estimate for the rate of such false photon combinations when combined with our Ds + γ recoil mass tags. Finally, we do an intermediate approach where we use the same Ds tag modes as the other approaches, but we do not reconstruct the Ds∗ daughter photon. In this case, we do a simple fit to the Ds invariant mass after all cuts to determine the number of semileptonic events, much like our standard analysis. We have also tested an approach that uses a two dimensional tagging fit to the Ds invariant mass and the Ds +γ recoil mass. We found that this method gave us essentially the same results as when we cut on the Ds mass and fit the Ds + γ recoil spectrum. However, our fits to the two dimensional spectrum become sensitive to initial parameters, possibly due to a small remaining correlation between the two variables. Since our fit shape systematic dominates the tagging errors, we have chosen to drop this method and focus on the one dimensional fits instead.

B.1

Particle Selections for Alternate Methods

Given the significant differences in B(Ds → φeν) between our primary analysis and another analysis on the CLEO data sample [40], we have attempted to eliminate any comparison complications by using that analysis’s particle selections for all six of our alternate measurement approaches. We restrict ourselves to 9 of our 13 tag modes: KS K, KKπ, KKππ 0 , KS K − ππ, πππ, πη, 0 ππ η, πη 0 , η 0 → ππη, and πη 0 , η 0 → ργ. We drop our normal Ds momentum cut (in the form of a recoil mass cut that varies by mode) since we will instead be using the Ds + γ recoil mass for our fits and selections. Also, we add a 150 MeV ρ cut in Ds → ππ 0 η. We otherwise retain the

139

individual tag mode cuts listed in Table 5.7 We use the same electron cuts as in our primary analysis, with the sole exception that we adopt a slightly more conservative | cos θe | < 0.90 angle cut instead of | cos(θe )| < 0.93. Although we loosen the overall φ mass requirement to 60 MeV for all of our alternate methods, we otherwise adopt tighter kaon cuts. Specifically, we institute a minimum hit fraction cut of 0.5 rather than the token hit fraction cut of 0.1 from our primary analysis. In addition dE/dx to our standard dE/dx consistency cut of σK < 3.0, we also add an additional rejection from a basic particle ID. If we have |pK > 700 MeV|, then we use both σ dE/dx and the RICH 2 ) + (Lπ − LK ) ≥ 0; otherwise, we drop the RICH likelihood likelihood by requiring (σπ2 − σK 2 2 and simply require (σπ − σK ) ≥ 0. We require that candidate showers for the Ds∗ meson’s daughter photon do not come from hot channels in the calorimeter, they can’t have an associated track, and they need to pass CLEO’s E9 O.K. cut.8 Showers must have an energy above 50 MeV if in the endcap or 30 MeV in the E25 barrel, although the kinematic range of the Ds∗ photon limits these extremes in any case. We also reject any event with an unused shower that meets the above criteria but has an energy above 300 MeV.

B.2

Methods 1-4: Cut on Ds Invariant Mass, Fit Ds + γ Recoil Mass

Our first four alternate methods all use the Ds + γ recoil mass to tag candidate events. We first | < 17.5 MeV. We then allow each − MDPDG restrict the Ds invariant mass to the range |MDrecon s s passing Ds to pair with any valid shower to form a Ds + γ tag candidate. We determine the number of Ds + γ tags by fitting the recoil mass spectrum, where the recoil four momentum is constrained + pγ ). By using the mass constrained Ds four vector for given by precoil = pbeam − (pmass Ds our Ds +γ recoil, we make the recoil mass fairly independent of the reconstructed invariant mass (Fig. 54). Aside from conceptual simplicity, this also reduces the remaining recoil background after our basic invariant mass cut. We fit the recoil mass from each Ds tag mode separately, using a crystal ball function and 4th degree polynomial background function. This gives us a total of five signal parameters and five background parameters. We use the Monte Carlo to fix the crystal ball function’s mean, its α (the number of σ at which the gaussian turns into a polynomial), and its n (polynomial power). We allow the signal normalization, the signal width (σ), and all of the background parameters to float. We only count Ds + γ tags that have a recoil mass within 3.782 GeV2 < 2 Mrecoil < 4.0 GeV2 , in accordance with [40]; not only does this match our recoil mass cut for the full event reconstruction, but it prevents our tagging counts from being unduly influenced by the long crystal ball tail. We give our Ds + γ tag fit results in Table 69, with Figure 55 (Monte Carlo) and Figure 56 (data) showing the actual fits. For two of our four methods, we use all Ds + γ combinations with a valid recoil mass when reconstructing a semileptonic event. In the other two methods, we choose a best γ candidate from among those that pass the recoil mass window. We choose this best γ by determining For our intermediate method (Method 6), we do not find the Ds∗ meson’s daughter photon. While we still restrict ourselves to the 9 tag modes, we otherwise maintain the Ds tag cuts from our primary analysis. 8 9 9 The energy dependent “ EE25 O.K.” cut requires a minimum EE25 value for the shower. For low energy photons, the central nine crystals must contain around 80% of the total shower energy. This minimum smoothly scales to requiring roughly 90% of the energy in the central crystals for higher energy photons. 7

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what its lab energy should have been given the Ds + γ recoil mass and the shower’s position in the calorimeter, then choosing the photon whose energy lies closest to its ideal energy. This procedure provides a slight improvement toward choosing the correct photon in Monte Carlo over simply taking the candidate with the best recoil mass, and it produces less shaping of the final missing mass spectrum. We call the former two methods our “multiple candidate” methods and the latter two our “best candidate” methods, in both cases referring to candidate photons allowed to pair with our Ds . Once we have our Ds + γ tags, we look for a passing φ meson and a passing electron from the semileptonic Ds decay. We then calculate the event four vector. The φ width prevents us from improving its resolution with a kinematic fit, but we are able to improve the Ds∗ daughter photon resolution. For all four of our methods, we adjust the photon energy to its ideal energy, essentially doing a one variable kinematic fit by using the fact that we know the photon location much better than its energy. We then calculate the missing mass from the sum of all four vectors, the beam energy, and the beam momentum: pmissing mass = pbeam − constrained (pmass + pcorrected + pφ + pe ). γ Ds 141

Table 69: Ds + γ recoil mass tags in the data and Monte Carlo. The crystal ball function tends to undercount the number of tags across all modes, so we adjust the final branching ratio for this systematic effect. Ds mode KS K KKπ KKππ 0 KS K − ππ πππ πη ππ 0 η πη 0 , η 0 → ππη πη 0 , η 0 → ργ Sum

Data tags 3,772 ± 90 16,069 ± 218 5,254 ± 251 2,273 ± 109 4,618 ± 189 1,863 ± 78 3,168 ± 152 1,419 ± 56 3,351 ± 144 41,788 ± 468

MC, fit tags 72,405 ± 370 326,072 ± 887 97,446 ± 934 50,753 ± 436 89,019 ± 511 36,095 ± 265 74,488 ± 516 26,888 ± 226 62,202 ± 444 835,368 ± 1,683

MC, truth tags 74, 138 341, 134 118, 900 53, 110 96, 594 37, 198 74, 635 26, 681 63, 918 886, 308

We determine the number of signal counts from the missing mass plot, with two signal region possibilities. For our “tight” missing mass range, we consider a signal region with |M M 2 | < 40, 000 MeV2 . Our “wide” missing mass range extends over |M M 2 | < 400, 000 MeV2 . The two M M 2 ranges deal differently with events that have a true Ds → φeν decay but whose Ds tag has been paired with an incorrect Ds∗ daughter photon. Such false γ events do not peak in our Ds + γ recoil mass spectrum and consequently do not get counted as tags. We then need to cut such events out of our branching ratio’s numerator with a background subtraction (the “tight” methods), or we need to trust the Monte Carlo to correct our efficiency properly for such surplus events (the “wide” methods). Both the “tight” and “wide” cut regions have their own difficulties. The “wide” missing mass range covers nearly all semileptonic events (true Ds∗ daughter photon or otherwise). We have doubts about using the Monte Carlo to get the false γ rate correct for this method, as we discuss in Section B.4.9 Combinatoric background also enters into the signal region, which we estimate with the Monte Carlo.10 The “tight” background subtraction gets complicated because false γ events peak softly in the |M M 2 | distribution. We need to rely on the Monte Carlo to determine the soft peak’s shape, as a flat background non-trivially overestimates the branching ratio. Every method also has the same peaking background from Ds → f0 eν, f0 → KK that we see in our standard analysis. For this comparison, we just use the Monte Carlo to correct the f0 eν rate. Overall, we get four slightly different methods of determining a branching ratio, by taking 9

We can also try estimating the false γ rate from a Ds + γ recoil sideband. However, this gets conflated with combinatoric background in the sideband, and it requires us to trust the Monte Carlo to correctly extrapolate the false γ distribution from the sideband to signal region in any case. 10 We could estimate the background with a Ds mass sideband, but we’d still be relying on the Monte Carlo to estimate how that sideband propagates through the recoil mass distribution with its false γ combinations. We find it best to make Monte Carlo dependence explicit, especially with a small effect like this.

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either a tight/wide missing mass range and by allowing either the best candidate/multiple candidates for the Ds∗ daughter photon. We give our missing mass plots from the Monte Carlo and the data for each type of candidate selection in Figures 57–60.

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B.2.1

Comparison of Methods 1-4

Each method’s photon candidate multiplicity and |M M 2 | cut range gives it particular advantages and drawbacks, which we briefly discuss below. As in the previous section, we refer to Ds +γ tags with a γ that did not come from a Ds∗ daughter photon as “false γ” combinations. In particular, we address the case with more false γ in the data than predicted by the Monte Carlo (although the same problems would occur in reverse if the data has fewer false γ candidates than the Monte Carlo predicts). Our best tight method (best photon candidate, tight |M M 2 | range) has the potential flaw that when we get more false γ tag combinations than expected from the Monte Carlo, the false γ contribution to the M M 2 spectrum becomes more peaked than predicted (a consequence of choosing the best γ). This effect causes us to slightly under subtract false γ background from the signal region, leading to an overestimate of the branching ratio. On the other hand, we choose the correct γ less often than predicted with our efficiency (since more false photons exist to potentially make a best candidate), giving us a slight underestimate of the branching ratio. The former effect dominates, as false γ events get pushed into the signal region with the extra combinations more frequently than events with a correct γ get thrown out. The best wide method allows both false and true γ combinations into our |M M 2 | range, so we don’t have to worry about whether we chose the correct γ combination or not. However, since the Ds + γ tag spectrum only peaks with true Ds∗ daughter photons, the efficiency determined by the Monte Carlo implicitly assumes a certain rate of extra events from false γ. When we have more false γ than expected, we get more events in the |M M 2 | range than we took credit for given our tags and efficiency, causing us to overestimate the branching ratio. We also need to trust the Monte Carlo to determine the number of combinatoric and false Ds background, since our |M M 2 | range extends far enough that a sideband subtraction isn’t reasonable; this estimate could be systematically high or low, sending the branching ratio either way. Our multiple tight method solves both flaws of the best tight method since we don’t have a best candidate choice to shape the false γ background, nor do we have an efficiency issue when choosing false γ in place of true γ. We do have the statistical drawback that we aren’t making use of true Ds → φeν events that happen to have only false Ds∗ daughter photons, but that’s also true of the best tight method and doesn’t drive our error in any case. Although different from the best candidate methods used in prior analyses [40], we consider this the most accurate of our four methods that use Ds + γ recoil mass tags. We include a multiple wide method for completeness, although this suffers from a larger efficiency sensitivity to false γ than the other methods. This occurs because we can frequently get false γ combinations even when the correct γ was found and had the best combination, with no sideband available to estimate such combinations. We expect this method to overestimate the branching ratio due to the extra false γ events (or due to the low Monte Carlo efficiency, depending on your perspective). Keeping these potential systematic biases in mind, we compare the results from each of our four methods in Table 70 for the Monte Carlo and in Table 71 for the data sample. We used a Monte Carlo input Ds → φeν branching ratio of 2.170%. We get largely similar branching ratios from each of our four methods that use Ds + γ tags. At worst, our relative systematic error based on the |M M 2 | cut window and choice of best candidate comes to around 4%, which can be ignored given our 13% relative statistical error

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Table 70: Branching ratios in Monte Carlo for each of the four methods that use Ds + γ tags. Errors are statistical only. Allowed candidates Best candidate Best candidate Multiple candidate Multiple candidate

M M 2 range Narrow Wide Narrow Wide

εSL 9.63% 13.24% 10.89% 18.85%

NSL+Ds +γ 1836 ± 51 2505 ± 52 2080 ± 59 3561 ± 64

B(Ds → φeν) (2.150 ± 0.059)% (2.134 ± 0.045)% (2.156 ± 0.061)% (2.132 ± 0.038)%

Table 71: Branching ratios in the data for each of the four methods that use Ds +γ tags. Errors are statistical only. Allowed candidates Best candidate Best candidate Multiple candidate Multiple candidate

M M 2 range Narrow Wide Narrow Wide

εSL 9.63% 13.24% 10.89% 18.85%

NSL+Ds +γ 90.1 ± 11.1 118.5 ± 11.4 98.3 ± 12.9 173.7 ± 14.0

B(Ds → φeν) (2.110 ± 0.260)% (2.019 ± 0.195)% (2.036 ± 0.269)% (2.079 ± 0.170)%

and our 10% relative systematic error from the tag fits.

B.3

Method 5: Cut on Ds + γ Recoil Mass, Fit Ds Invariant Mass

Fitting the Ds + γ recoil mass spectrum suffers from the problem of non-linear background, which gets exacerbated by the signal’s long power law tail from photon reconstruction. In contrast, the Ds invariant mass spectrum has a fairly flat background, and a smaller power law tail even in photon tag modes since we have one fewer photon to reconstruct. To take advantage of the cleaner fitting while retaining the Ds∗ daughter photon reconstruction, we have tried one method using Ds invariant mass tags after cutting on the Ds + γ recoil mass. We allow a Ds tag to enter our invariant mass plot once if it pairs with one or more photons 2 to create a Ds + γ recoil mass within 3.782 GeV2 < Mrecoil < 4.0 GeV2 . We then fit the Ds invariant mass spectrum for each tag mode with a linear background function and the sum of a gaussian and crystal ball to represent the signal. We fix the relative normalization and relative width of the gaussian and crystal ball in the Monte Carlo, as well as the crystal ball function’s α and n parameters. We allow all other parameters to float in our fit, including both linear background parameters and the common mean for the gaussian and crystal ball. We count tags within |MDrecon − MDPDG | < 17.5 MeV, since we only allow Ds within this mass s s range to later combine with an electron and φ meson for our full semileptonic event. We give our Ds invariant mass tagging results from both the Monte Carlo and the data in Table 72, with the fits shown in Figure 137 and Figure 138 (Appendix F). Since we use the same Ds invariant mass and Ds + γ recoil mass ranges as our previous four methods, we have exactly the same missing mass reconstruction (Figures 57–60). Unlike the Ds + γ recoil mass tagging, our Ds invariant mass tags peak whether the Ds∗ daughter 150

Table 72: Ds invariant mass tags in the data and Monte Carlo after cutting on the Ds + γ recoil mass. We only allow each Ds mass to enter once, regardless of the number of Ds + γ combinations. Ds mode KS K KKπ KKππ 0 KS K − ππ πππ πη ππ 0 η πη 0 , η 0 → ππη πη 0 , η 0 → ργ Sum

Data tags 4,726 ± 76 21,603 ± 181 8,498 ± 241 3,887 ± 96 6,490 ± 186 2,341 ± 80 3,329 ± 143 1,754 ± 47 3,388 ± 135 56,017 ± 434

MC, fit tags 87,061 ± 317 402,906 ± 742 149,721 ± 920 66,833 ± 348 109,144 ± 460 42,248 ± 229 90,113 ± 505 31,320 ± 191 71,189 ± 425 1,050,530 ± 1,534

MC, truth tags 86, 050 405, 485 145, 063 63, 501 111, 756 42, 876 86, 663 30, 930 74, 086 1, 046, 410

photon is true or false. Rather than trust the Monte Carlo to estimate how many tags come from true photon combinations, we use a best candidate selection over the “wide” range for the |M M 2 |, giving one potential event for each tag and keeping both true and false photons. This procedure means that we only have to use our Monte Carlo to provide the efficiency for the φ and electron, independent of the false photon rate. Aside from the possibility that the Monte Carlo underestimates the combinatoric background (less than a 4% effect), we do have a slight complication with our Ds tags in this method. Generic Ds decays may have photons as final decay products, while real Ds → φeν events do not (at least not for our reconstructed mode of φ → KK). This means that generic Ds decays have a slightly higher rate of photon candidates available to create a passing Ds + γ recoil mass than Ds → φeν decays, allowing relatively more Ds tags with fake Ds∗ daughter photons in generic decays than we see with real φeν events (about a 3% correction). If the Monte Carlo underestimates the number of generic Ds decay photons, we will also slightly underestimate our branching ratio. Overall, we expect a relative systematic from this method of 6%, which is partially independent of the systematic from our recoil mass tag methods. We state our branching ratio results for this method in our summary section, with Tables 76 and 77 giving the results in Monte Carlo and data, respectively.

B.4

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Some of our alternate Ds → φeν methods depend upon the Monte Carlo predicted rates for reconstructing the Ds∗ daughter photon. Fortunately, with the various tagging methods that we’ve performed (and an additional fit for the number of Ds before combining with photons), we can get an estimate for the actual photon efficiency in data. We can also get an estimate for the rate at which events without a true Ds∗ daughter photon still pass the Ds + γ recoil mass cut by pairing with another shower (the “false γ” rate). This cross-check indicates both

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that the Ds∗ daughter photon gets successfully reconstructed at a lower rate than expected from the Monte Carlo and that false showers allow the Ds to pass our Ds + γ cut more often than expected. To determine our true efficiency for the Ds∗ daughter photon reconstruction, we perform an additional fit to the Ds invariant mass before it combines with a photon. This gives us our efficiency denominator, which we can combine with our number of Ds + γ tags to get the efficiency. We use the same fitting methodology for this MDs fit that we used when determining invariant mass tags (method 5), although here we don’t cut on the Ds + γ recoil mass before getting our invariant mass distribution. As seen in Table 73, we obtain a lower photon efficiency in data by about 8% (relative) than we get in the Monte Carlo. We don’t list the errors on the efficiency, as the systematics from tagging dominate the small (sub-percent) level statistical errors (we haven’t done a thorough systematic evaluation, but a relative 3%-4% seems likely given our previous work). Table 73: Ds∗ daughter photon efficiency in data and Monte Carlo. Sample type MC, truth-tagged MC, fit Data, fit

NDrecoil s +γ 886,308 886,309 ± 1,785 44,336 ± 497

NDpre-recoil s 1,267,860 1,268,470 ± 1,821 68,999 ± 561

εγ 69.9% 69.9% 64.3%

The Ds tag can combine with showers from sources other than the Ds∗ daughter photon (false γ) to form a passing Ds + γ recoil mass. These showers can come from real photons on the other side (untagged) Ds decay, or they can arise when the Ds decays to soft kaons. The kaons have a high rate of unmatched showers due to splitoff and decays in flight that may not be well determined in the Monte Carlo. Although we don’t use this result directly, we can get a feel for the data/Monte Carlo difference by looking at how often a tag passes the Ds + γ recoil by pairing with a false γ when the real photon was not found. We use the number of tags from our recoil mass fit (methods 1-4), our invariant mass fit tags after a Ds + γ recoil cut (method 5), and the previously determined photon efficiency to determine the rate of fake γ. As seen in Table 74, the data has a 10% higher likelihood of finding such a fake shower than the Monte Carlo predicts. This gives us the sense that either generic Ds decays have more photons than the Monte Carlo, or we have more non-photon extra showers from Ds decay products than we’d expect. The former effect will distort our efficiency estimate for the invariant mass tag procedure (method 5), while the latter particularly affects Ds → φeν and can distort all of our recoil mass results other than the multiple tight method. We can also get a more direct estimate for the rate of false Ds∗ daughter photons in Ds → φeν decays by comparing our multiple tight method’s background estimates in data and Monte Carlo. Background makes up about 45.4% of events in the Monte Carlo compared to 47.3% for the data. Since combinatoric background only makes up some 5% of the total events (according to the Monte Carlo), it’s likely that these surplus data background primarily come from false photon combinations. If we interpret the extra background in data as entirely false photon combinations, we estimate a relative 7% higher rate of false γ combinations for Ds → φeν events over what the Monte Carlo predicts. 152

Table 74: Rate at which valid Ds without a correctly reconstructed Ds∗ daughter photon will still pass all tagging cuts (including the Ds + γ recoil mass). Sample type MC, truth-tagged MC, fit Data, fit

B.5

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NDpost-recoil s 1,046,410 1,050,530 ± 1,534 56,017 ± 434

εγ 69.9% 69.9% 64.3%

ffake γ 42.0% 43.0% 47.4%

Method 6: No Ds∗ Photon Reconstruction

For ease of comparison, we also include a simplified version of our primary analysis that uses cuts and fits similar to the other five alternate methods. As in our primary analysis, we do not attempt to reconstruct the Ds∗ daughter photon, instead using the Ds invariant mass spectrum for both tags and the number of semileptonic events. We reconstruct the same 9 Ds tag modes listed in B.1, leaving out four of the higher background modes from our primary analysis (KS Kπ 0 ; KS KS π; KS K + ππ; and ππ 0 η 0 , η 0 → ππη). Since we do not reconstruct a potential Ds∗ daughter photon, we don’t have a Ds + γ recoil mass available for our selection. Instead, we use our primary analysis’s Ds momentum cut in the form of the Ds recoil mass range from Table 4. Once we’ve selected our Ds candidates, we determine the number of Ds tags by fitting the invariant mass spectrum. As in our previous Ds → φeν reconstruction method using the Ds invariant mass for tags (method 5), we fit each tag mode to the sum of a gaussian and crystal ball for the signal with a linear background function (using fixed signal shape parameters from the Monte Carlo). This fit choice differs from our primary analysis, where we sometimes use a double gaussian or a quadratic background, based on the tag mode. While our choice of fit function for this alternate method may not be as accurate as in our primary analysis, it does reduce the likelihood that any difference in results between the alternate methods came from a fit systematic on the tags. Overall, we see an 8% difference in tags from our primary analysis over these modes, although ππ 0 η drives nearly the entire difference with its non-linear background shape. Our tag results for this method are given in Table 75, with our data fits shown in Figure 139. After making the same electron and φ cuts as for our other alternate methods (rather than the φ/e cuts from our primary analysis), we again plot the Ds invariant mass. We fit each tag mode with the signal shape determined from our tag fits (only the overall normalization floats), plus a linear background function. Unlike our primary analysis, we refrain from using a common branching ratio to ensure that this method remains both simple and as similar as possible to the other alternate methods. We thus fit each tag mode independently. Figures 61 and 62 show our Monte Carlo and data plots for the Ds mass after making our semileptonic selections. We include the results in our summary section with Tables 76 and 77.

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Table 75: Ds invariant mass tags in the data and Monte Carlo after a Ds momentum cut. We do not require a pairing with a photon. Ds mode KS K KKπ KKππ 0 KS K − ππ πππ πη ππ 0 η πη 0 , η 0 → ππη πη 0 , η 0 → ργ Sum

MC, fit tags 115,958 ± 380 510,195 ± 888 119,558 ± 750 65,749 ± 351 118,575 ± 521 64,179 ± 317 108,063 ± 800 42,550 ± 232 72,327 ± 453 1,217,150 ± 1,700

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B.6

Comparison of Alternate Methods

Our best Ds → φeν measurement using Ds + γ recoil mass tags comes from the multiple tight method, so we compare it, the Ds invariant mass tag method, and our simplified no Ds∗ photon analysis in Tables 76 and 77 for the Monte Carlo and data, respectively. The multiple tight method and the simplified no photon method give us the biggest branching ratio range, with a relative difference of about 10%. We expected our non-correlated systematic errors between methods to also be at about the 10% relative level, so this relative difference seems reasonable. Note that while the statistical errors also cover this range, they don’t completely explain the difference since they have a correlation with each other (all methods run over the same data sample). Table 76: Branching ratios in Monte Carlo for our different Ds → φeν alternate methodologies. Errors are statistical only. → φeν method tags, multiple tight MDs tags, best wide MDs tags & signal, no γ

Ds MDrecoil s +γ

raw Ntags 835,368 ± 1,683 1,050,530 ± 1,534 1,216,450 ± 1,698

εSL 10.89% 11.48% 13.97%

NSL+Ds +γ 2080 ± 59 2505 ± 52 3160 ± 61

B(Ds → φeν) (2.156 ± 0.061)% (2.150 ± 0.045)% (2.150 ± 0.042)%

Table 77: Branching ratios in the data for our different Ds → φeν alternate methodologies. Errors are statistical only. Ds → φeν method recoil MDs +γ tags, multiple tight MDs tags, best wide MDs tags & signal, no γ

raw Ntags 41,787.7 ± 468.1 56,017.1 ± 433.6 65,476.6 ± 551.6

εSL 10.89% 11.48% 13.97%

NSL+Ds +γ 98.3 ± 12.9 118.5 ± 11.4 144.8 ± 13.3

B(Ds → φeν) (2.036 ± 0.269)% (1.908 ± 0.183)% (1.831 ± 0.169)%

Our primary analysis yields a branching ratio roughly in the middle range of our alternate methods, with the full range covered by its statistical error. We take this to mean that our primary Ds → φeν result is fairly robust to the changes in fit function, particle cuts, and Ds∗ photon reconstruction considered in this appendix. Our multiple tight method and our Ds invariant mass tag method both involve an f0 → KK correction based on our Monte Carlo. Given the discussion in Appendix A, the Monte Carlo likely has more f0 → KK than the data, so this could lead to an underestimate of the branching ratio by a relative 2%-3%. We do not attempt an f0 → KK correction in our simplified no Ds∗ photon analysis; this could lead to an overestimate of our branching ratio by up to a relative 2%. However, the systematic error from our (different) f0 correction methodologies comes out to roughly the size of the correction itself, so this doesn’t alter the consistency between our different methods.

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C

Ds → ηeν Efficiency Systematic

As described in Section 9.6.2, we have elected to create our own η efficiency systematic rather than adopting a preexisting one. This gives us an η reconstruction efficiency with our exact η selections, a comparable η lab momentum range (Figure 63), and a run environment similar to that in the Ds → ηeν analysis.

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In addition to these benefits, our Ds → ηeν Monte Carlo datasets show an ηeν efficiency variation that borders on the edge of allowed random variation (Figure 64). Of course, with six semileptonic modes, it shouldn’t be shocking if one mode’s efficiency variations have a one in six chance of being consistent with a random distribution. Even so, creating our own systematic for the efficiency from the same datasets gives us more confidence that our analysis has a sound foundation. We obtain an η sample by taking advantage of the large (8.9%) Ds± → ρ± η branching ratio [41]. We determine the presence of an η by finding the recoil mass after reconstructing a Ds tag, the Ds∗ daughter photon, and a ρ. Then, we explicitly reconstruct the η → γγ with 157

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Figure 64: Ds → ηeν efficiency (including the η → γγ branching ratio) by dataset. The solid red line gives the average across the full generic Monte Carlo sample, while the dotted red lines show the 1σ range on this average.

our analysis’s η selection to get an η efficiency. As this can be done in both data and Monte Carlo, we can compare the efficiencies for our overall systematic. For our Ds tag, we use the same 13 tag modes as our full analysis, described in Section 4. Since we don’t need to fit the Ds mass spectrum, we make the additional mode-dependent Ds mass cuts shown in Table 91 (Appendix E). We follow the selections from Section 8.1 for our Ds∗ daughter photon candidates, with the exception of the irrelevant minimum energy cut. For the ρ± , we reconstruct a π ± and a π 0 . Our charged π follow the same selection criteria as in Section 7.1, although we make minor adjustments by lowering the χ2 threshold to 1,000, dE/dx < 3.0 cut when we have both we don’t allow hit fractions above 1.2, and we drop the σπ RICH information and pπ > 550 MeV. The π and Ds tag must have opposite charges, and the event can have no other charged tracks. We require a 3.0 pull mass cut on the π 0 , and its showers must meet the requirements from Section 8.1 (although we drop the distinction between barrel and endcap showers). The background to Ds → ρη mostly consists of events with soft pions. We can eliminate much of this background by adding cuts on the ρ. Specifically, we require 600 MeV < Mρ < 960 MeV and 500 MeV < pρ < 1000 MeV. We also eliminate particular backgrounds by 158

rejecting π ± that have a rest frame momentum within 5 MeV of 712 MeV, 743 MeV, or 902 MeV for Ds to πφ, πη 0 , and πη, respectively. To determine our number of Ds → ρη events, we perform a 2D fit to the Ds + γ recoil mass and the Ds + γ + ρ recoil mass. We use the Monte Carlo to get four lineshapes for the fit corresponding to events with true or false η and true or false Ds + γ + ρ. The Monte Carlo accurately reproduces the widths of these lineshapes in data, but the peak locations have a slight shift. We allow the distributions to shift in each dimension and take the best χ2 . In the data, this shifts our Ds + γ recoil mass fit function by 1.3 MeV and our Ds + γ + ρ recoil fit function by 4.5 MeV. We give our projections for each fit dimension in Figures 65 (Monte Carlo) and 66 (data). recoil

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Carlo, keeping the same 4.5 MeV shift to the Ds + γ + ρ recoil mass that we obtained from our previous fit. We show our distributions from this fit for the Monte Carlo in Figure 67 and for the data in Figure 68. We expected to see a lower η efficiency from the data than the Monte Carlo, in accordance with a previous CLEO analysis using ψ 0 → ηJ/ψ that saw a relative 5.6% correction to the η efficiency [23]. However, with our η environment and selections, we only see a relative 1.2% lower efficiency in the data compared to the Monte Carlo (32.7% to 33.1%), well within our error. Consequently, we do not take a correction to our η efficiency. Our systematic error on the efficiency comes almost equally from our error on the numerator (2D pull mass/Ds +γ +ρ recoil fit) and our error on the denominator (2D Ds +γ +ρ recoil/Ds +γ recoil fit). We use a binomial error for the our efficiency, although we have to adjust it upward by a factor of 1.2 to account for fit backgrounds. Ultimately, we obtain an η efficiency in the data of (32.7± 2.6)%, giving us a relative η efficiency systematic of 7.9%.

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D

Kaon Tracking and Particle ID Systematic

While all track reconstructions depend on the particle’s momentum to some extent, kaon reconstruction and identification have a particularly strong dependence on momentum given the possibility of the kaon decaying within the detector. Kaons in Ds semileptonic decays tend to have lower momenta than kaons in D0 /D+ decays, so we use loose cuts to gain efficiency for those otherwise low yield kaons. However, that means that our cuts don’t match the prior CLEO systematics, requiring us to perform our own systematic study for kaons. We choose an approach that combines the reconstruction and particle ID efficiencies, giving a single, momentum-dependent systematic error for kaons. We initially looked in the Ds system for a kaon systematic, given the relatively large numbers of kaons produced thanks to the Ds meson’s strange component. However, our systematic errors from these studies typically came out to about 3% per kaon, or about 6% for Ds → φeν as a whole given the correlation between the systematics on the two kaons. The higher statistics from D+ → K − π + π + (and its charge conjugate) during CLEO’s 3770 MeV running allow us to achieve a lower systematic on the kaons, at the expense of not having an exactly identical run environment. While we don’t use it directly, our original Ds → KKπ study yielded roughly the same needed kaon efficiency correction (with higher errors) that we will see from our D+ study below. To isolate D → Kππ decays, we first reconstruct a D± tag. We use five modes in our D± tag reconstruction: Kππ, KKπ, Ks π, Ks ππ 0 , and Kπππ 0 . We ensure that we have a D+ D− event by requiring that the tag’s beam constrained mass falls within 5 MeV of the D+ mass and that the ∆E falls within 20 MeV of zero. We then choose each charge’s best tag by D± invariant mass. Once we have a D+ tag candidate, we rescale its total momentum to match D+ D− production, improving the resolution of our later recoil masses. After finding a tag, we look for two additional tracks with proper charges passing pion particle ID. We reject any event with a total extra energy above 250 MeV or with an extra track passing simple electron cuts, avoiding backgrounds from π 0 modes and semileptonic modes, respectively. We can then identify D → Kππ events by checking that the recoil mass against the D and two π matches a kaon. From here, we have two ways of calculating the kaon efficiency. We can take all events with a recoil mass near the kaon mass as the denominator, then obtain our numerator by explicitly reconstructing the kaon and finding a zero missing mass for the event. Alternately, we can try to find the kaon, plotting the “found” recoil mass when we reconstruct it and the “not found giving the total kaon efficiency. found” recoil mass when we don’t, with εK = NfoundN+N not found As it turns out, the found/not found approach makes it slightly easier in practice to get good precision from our fitting because nearly all the background comes from events without a found kaon, allowing us to focus on those events as the source of any nonstatistical error. To get separate efficiencies for each kaon momentum region, we split our sample into three bins based on the D+ + ππ recoil momentum: one for kaon momenta below 250 MeV, one for kaons between 250 MeV and 500 MeV, and one for kaon momenta above 500 MeV. We’ve chosen these momentum regions so that Ds → φeν kaons split roughly evenly between the lowest and middle bins. We fit to the “found” and “not found” plots in each of these regions using histogram shapes from the Monte Carlo, with a fit systematic error determined by doing a simple cut and count to the same plots. 163

When fitting, we discovered that the recoil kaon mass in the data tends to fall slightly below the recoil mass in the Monte Carlo. To account for this, we allow our signal shape to shift by small amounts to the left, and we take the best χ2 from all such shifts (which results in a recoil shift of about 0.8 MeV). We show the final Monte Carlo and data plots from each momentum region for our φ meson’s kaon selections in Figures 69 and 70, respectively. recoil

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In our analysis, we use three different sets of selections for kaons: one for φ (Section 6.1); one for K ∗ (Section 7.1); and one for Ds tags (Section 4.1.1), where our kaon selections for the Ds tags follow the standard CLEO kaon cuts. We have repeated our systematic for each of these kaon cuts, with the final efficiencies across our kaon selections and momentum regions shown in Figure 71 for the Monte Carlo and Figure 72 for the data reconstruction. In all cases, we find that the data efficiency deviates from the Monte Carlo efficiency for soft kaons and requires an efficiency correction, as shown in Figure 73 and from our final results in Tables 78–80.

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Table 78: Relative kaon systematic and efficiency correction for our loose (φeν) kaon selection. Momentum region pK < 250 MeV 250 MeV < pK < 500 MeV 500 MeV < pK

Statistical error 1.59% 0.43% 0.18%

Fit systematic 1.69% 0.08% 0.59%

Total systematic 2.32% 0.44% 0.62%

εK correction -9.47% -1.35% N/A

Table 79: Relative kaon systematic and efficiency correction for our medium (K ∗ eν) kaon selection. Momentum region pK < 250 MeV 250 MeV < pK < 500 MeV 500 MeV < pK

Statistical error 1.78% 0.60% 1.37%

Fit systematic 1.69% 0.08% 0.59%

Total systematic 2.45% 0.60% 1.49%

εK correction -9.08% -1.50% N/A

Table 80: Relative kaon systematic and efficiency correction for our standard kaon selection. Momentum region pK < 250 MeV 250 MeV < pK < 500 MeV 500 MeV < pK

Statistical error 1.67% 0.76% 0.51%

Fit systematic 1.69% 0.08% 0.59%

Total systematic 2.38% 0.76% 0.78%

εK correction -7.21% N/A N/A

Our large momentum bins mean that the kaon momentum distribution within each bin (e.g. 250 MeV < pK < 500 MeV) can differ between D → Kππ and the semileptonic mode, shown for Ds → φeν in Figure 74. We add an additional systematic to the efficiency error for this effect by splitting the momentum bin into two halves and allowing each half of the bin to have a different efficiency correction, constrained by the bin’s total kaon efficiency and the adjacent bins’ efficiency corrections. This procedure results in a relative 0.5% systematic error for Ds → φeν. Since the φ meson’s daughter kaons tend to have strongly correlated momenta, we correct our φeν efficiency based on the daughter kaon momentum pairs rather than on the individual φeν kaon momentum distribution.

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E

Extra Tables

Table 81: Number of Ds tags in data and Monte Carlo, by dataset. We fit each dataset independently for this comparison and scale the Monte Carlo to data size. Dataset 39 40 41 47 48 Sum

Data fit counts 7,246.8 ± 255.3 15,609.7 ± 414.2 16,308.6 ± 443.9 14,686.3 ± 408.9 23,823.5 ± 752.0 77,674.9 ± 2,274.3

MC fit counts 6,482.8 ± 308.6 14,278.5 ± 422.0 13,886.4 ± 438.2 12,940.5 ± 396.6 20,283.1 ± 580.2 67,871.3 ± 2,145.6

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Table 82: Number of Ds tags in data and Monte Carlo, by mode. We scale the Monte Carlo to the data luminosity. Ds mode KS K KKπ KS Kπ 0 KS KS π KKππ 0 KS K + ππ KS K − ππ πππ πη ππ 0 η 0 0 πη , η → ππη ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ Sum

Data fit counts 6,226.7 ± 101.2 27,373.5 ± 248.4 2,246.8 ± 209.9 1,125.5 ± 76.5 7,355.5 ± 377.4 1,859.4 ± 120.6 3,377.3 ± 100.0 6,606.3 ± 337.7 3,810.3 ± 190.8 9,476.9 ± 529.0 2,386.6 ± 65.6 1,090.5 ± 118.7 4,272.3 ± 193.3 77,207.5 ± 880.2

MC fit 5,764.0 25,242.0 1,670.5 1,141.4 6,693.4 1,744.1 3,246.3 6,081.6 2,882.3 6,825.9 2,132.4 532.5 3,904.4 67,860.7

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counts ± 100.8 ± 233.9 ± 157.7 ± 69.3 ± 323.6 ± 105.5 ± 92.2 ± 326.3 ± 182.9 ± 700.7 ± 64.3 ± 84.5 ± 245.2 ± 959.8

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2.6)% 1.4)% 17.9)% 9.0)% 7.7)% 9.5)% 4.3)% 8.0)% 10.7)% 16.2)% 4.6)% 39.4)% 8.5)% 2.1)%

Table 83: Test of potential bias in our fitting procedure for Ds → KS eν by comparing the number of truth-tagged semileptonic events to the fit result. We allow cross-feed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sum

truth-tagged NSL+tag 27 25 37 35 23 23 18 29 26 29 20 22 22 22 37 28 25 26 20 27 521

fit NSL+tag 23.95 ± 5.72 26.86 ± 6.08 39.67 ± 7.79 28.34 ± 6.15 25.55 ± 5.94 19.51 ± 5.79 20.18 ± 5.91 23.20 ± 5.90 21.46 ± 6.59 26.82 ± 6.19 22.74 ± 5.78 23.99 ± 5.88 27.44 ± 6.13 18.36 ± 5.09 31.85 ± 6.52 32.67 ± 6.59 17.67 ± 5.99 26.27 ± 6.26 23.32 ± 5.61 23.80 ± 5.83 503.66 ± 27.31

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Difference (# σ) −0.533 0.306 0.343 −1.083 0.429 −0.603 0.368 −0.982 −0.689 −0.353 0.474 0.339 0.888 −0.715 −0.790 0.710 −1.224 0.043 0.592 −0.549 −0.635

Table 84: Monte Carlo comparison of the measured Ds → KS eν branching ratio to its generating branching ratio (0.090%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Weighted averages/χ2

Raw fit BR (0.113 ± 0.027)% (0.122 ± 0.028)% (0.178 ± 0.035)% (0.129 ± 0.028)% (0.118 ± 0.027)% (0.090 ± 0.027)% (0.092 ± 0.027)% (0.105 ± 0.027)% (0.100 ± 0.031)% (0.122 ± 0.028)% (0.106 ± 0.027)% (0.110 ± 0.027)% (0.124 ± 0.028)% (0.081 ± 0.023)% (0.148 ± 0.030)% (0.153 ± 0.031)% (0.080 ± 0.027)% (0.119 ± 0.028)% (0.104 ± 0.025)% (0.109 ± 0.027)% (0.112 ± 0.006)%

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Corrected BR (0.100 ± 0.027)% (0.109 ± 0.028)% (0.165 ± 0.035)% (0.115 ± 0.028)% (0.104 ± 0.027)% (0.077 ± 0.027)% (0.078 ± 0.027)% (0.091 ± 0.027)% (0.086 ± 0.031)% (0.108 ± 0.028)% (0.093 ± 0.027)% (0.096 ± 0.027)% (0.110 ± 0.028)% (0.068 ± 0.023)% (0.134 ± 0.030)% (0.139 ± 0.031)% (0.067 ± 0.027)% (0.105 ± 0.028)% (0.091 ± 0.025)% (0.096 ± 0.027)% (0.099 ± 0.006)%

#σ 0.36 0.68 2.13 0.89 0.52 −0.49 −0.44 0.04 −0.13 0.64 0.09 0.24 0.73 −0.98 1.46 1.60 −0.85 0.54 0.03 0.21 14.38

Table 85: Test of potential bias in our fitting procedure for Ds → K ∗ eν by comparing the number of truth-tagged semileptonic events to the fit result. We allow cross-feed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sum

truth-tagged NSL+tag 28 37 35 28 33 34 33 35 27 38 36 33 37 28 38 29 31 24 29 30 643

fit NSL+tag 27.80 ± 6.33 33.39 ± 5.24 35.63 ± 7.18 29.57 ± 6.48 34.40 ± 6.73 29.57 ± 6.72 26.81 ± 6.75 35.68 ± 6.96 26.01 ± 6.83 40.91 ± 7.10 30.72 ± 6.98 38.05 ± 7.58 42.45 ± 7.28 30.47 ± 6.74 31.65 ± 6.73 24.58 ± 6.64 34.54 ± 6.63 25.21 ± 5.65 29.34 ± 6.52 24.11 ± 5.97 630.89 ± 29.84

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Difference (# σ) −0.032 −0.689 0.087 0.242 0.208 −0.660 −0.916 0.098 −0.144 0.411 −0.756 0.667 0.749 0.367 −0.945 −0.665 0.534 0.214 0.052 −0.987 −0.406

Table 86: Monte Carlo comparison of the measured Ds → K ∗ eν branching ratio to its generating branching ratio (0.190%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Weighted averages/χ2

Raw fit BR (0.169 ± 0.038)% (0.195 ± 0.031)% (0.205 ± 0.041)% (0.172 ± 0.038)% (0.204 ± 0.040)% (0.176 ± 0.040)% (0.156 ± 0.039)% (0.206 ± 0.040)% (0.155 ± 0.041)% (0.238 ± 0.041)% (0.184 ± 0.042)% (0.224 ± 0.045)% (0.245 ± 0.042)% (0.173 ± 0.038)% (0.188 ± 0.040)% (0.148 ± 0.040)% (0.202 ± 0.039)% (0.146 ± 0.033)% (0.168 ± 0.037)% (0.142 ± 0.035)% (0.183 ± 0.009)%

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Corrected BR (0.157 ± 0.038)% (0.183 ± 0.031)% (0.193 ± 0.041)% (0.160 ± 0.038)% (0.192 ± 0.040)% (0.164 ± 0.040)% (0.145 ± 0.039)% (0.194 ± 0.040)% (0.143 ± 0.041)% (0.226 ± 0.041)% (0.172 ± 0.042)% (0.212 ± 0.045)% (0.234 ± 0.042)% (0.161 ± 0.038)% (0.176 ± 0.040)% (0.136 ± 0.040)% (0.190 ± 0.039)% (0.134 ± 0.033)% (0.156 ± 0.037)% (0.130 ± 0.035)% (0.171 ± 0.009)%

#σ −0.86 −0.22 0.08 −0.79 0.04 −0.66 −1.15 0.11 −1.15 0.87 −0.43 0.49 1.03 −0.75 −0.34 −1.36 −0.01 −1.69 −0.90 −1.72 15.92

Table 87: Test of potential bias in our fitting procedure for Ds → η 0 eν by comparing the number of truth-tagged semileptonic events to the fit result. We allow cross-feed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sum

truth-tagged NSL+tag 32 10 27 24 15 19 32 23 23 16 27 16 22 26 18 17 19 23 27 31 447

fit NSL+tag 29.34 ± 5.77 8.33 ± 4.01 21.86 ± 5.02 25.28 ± 5.52 13.87 ± 4.29 17.85 ± 4.34 33.64 ± 5.46 21.90 ± 4.88 22.66 ± 5.42 11.48 ± 4.08 27.47 ± 5.37 15.18 ± 4.70 21.65 ± 4.84 24.09 ± 4.96 15.75 ± 4.38 20.49 ± 4.74 20.03 ± 4.15 20.65 ± 4.94 28.88 ± 5.35 22.83 ± 5.41 423.20 ± 21.96

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Difference (# σ) −0.461 −0.417 −1.023 0.231 −0.264 −0.265 0.300 −0.226 −0.062 −1.109 0.087 −0.176 −0.073 −0.386 −0.513 0.736 0.249 −0.476 0.351 −1.509 −1.084

Table 88: Monte Carlo comparison of the measured Ds → η 0 eν branching ratio to its generating branching ratio (0.860%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. The abnormally high χ2 just reflects low Ds → η 0 eν statistics that distort gaussian error sums (Table 87 gives a more meaningful comparison for this mode). Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Weighted averages/χ2

Raw fit BR (1.068 ± 0.210)% (0.292 ± 0.141)% (0.755 ± 0.173)% (0.881 ± 0.193)% (0.492 ± 0.152)% (0.636 ± 0.154)% (1.177 ± 0.191)% (0.760 ± 0.169)% (0.809 ± 0.194)% (0.400 ± 0.142)% (0.986 ± 0.193)% (0.535 ± 0.165)% (0.751 ± 0.168)% (0.821 ± 0.169)% (0.562 ± 0.156)% (0.738 ± 0.171)% (0.701 ± 0.145)% (0.719 ± 0.172)% (0.993 ± 0.184)% (0.805 ± 0.191)% (0.702 ± 0.038)%

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Corrected BR (1.057 ± 0.210)% (0.281 ± 0.141)% (0.744 ± 0.173)% (0.871 ± 0.193)% (0.481 ± 0.152)% (0.625 ± 0.155)% (1.166 ± 0.191)% (0.749 ± 0.169)% (0.799 ± 0.194)% (0.389 ± 0.142)% (0.975 ± 0.193)% (0.524 ± 0.165)% (0.740 ± 0.168)% (0.810 ± 0.169)% (0.551 ± 0.157)% (0.727 ± 0.171)% (0.690 ± 0.145)% (0.708 ± 0.172)% (0.982 ± 0.184)% (0.794 ± 0.191)% (0.691 ± 0.038)%

#σ 0.94 −4.12 −0.67 0.05 −2.48 −1.52 1.60 −0.66 −0.32 −3.31 0.60 −2.03 −0.72 −0.29 −1.97 −0.78 −1.17 −0.88 0.66 −0.35 53.13

Table 89: Test of potential bias in our fitting procedure for Ds → f0 eν by comparing the number of truth-tagged semileptonic events to the fit result. We allow cross-feed from other semileptonic modes for this fitting comparison, as those events produce real peaking background that we deal with outside the fitting apparatus. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sum

truth-tagged NSL+tag 56 52 49 51 57 51 52 49 56 46 70 42 45 59 53 53 63 56 48 46 1054

fit NSL+tag 62.93 ± 9.28 45.11 ± 8.09 45.96 ± 7.14 42.88 ± 8.17 48.99 ± 7.61 53.61 ± 8.27 48.57 ± 8.07 44.73 ± 8.02 57.19 ± 8.96 40.18 ± 7.58 77.54 ± 9.43 43.89 ± 7.42 43.39 ± 8.09 65.11 ± 9.08 52.42 ± 7.20 63.73 ± 8.71 61.51 ± 9.16 53.97 ± 8.40 53.70 ± 8.33 41.28 ± 7.31 1046.68 ± 36.87

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Difference (# σ) 0.747 −0.851 −0.425 −0.994 −1.053 0.316 −0.425 −0.533 0.133 −0.769 0.799 0.255 −0.199 0.674 −0.080 1.232 −0.163 −0.242 0.684 −0.647 −0.198

Table 90: Monte Carlo comparison of the measured Ds → f0 eν branching ratio to its generating branching ratio (0.310%), in data-sized samples. The weighted average line contains the χ2 across the 20 samples rather than the number of σ between the measured/generated branching ratios. Dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Weighted averages/χ2

Raw fit BR (0.425 ± 0.063)% (0.293 ± 0.053)% (0.294 ± 0.046)% (0.277 ± 0.053)% (0.323 ± 0.050)% (0.354 ± 0.055)% (0.315 ± 0.052)% (0.288 ± 0.052)% (0.379 ± 0.059)% (0.260 ± 0.049)% (0.516 ± 0.063)% (0.287 ± 0.048)% (0.279 ± 0.052)% (0.412 ± 0.057)% (0.347 ± 0.048)% (0.426 ± 0.058)% (0.399 ± 0.059)% (0.349 ± 0.054)% (0.343 ± 0.053)% (0.270 ± 0.048)% (0.333 ± 0.012)%

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Corrected BR (0.413 ± 0.063)% (0.282 ± 0.053)% (0.283 ± 0.046)% (0.266 ± 0.053)% (0.311 ± 0.050)% (0.343 ± 0.055)% (0.304 ± 0.052)% (0.276 ± 0.052)% (0.368 ± 0.059)% (0.248 ± 0.049)% (0.505 ± 0.063)% (0.276 ± 0.049)% (0.268 ± 0.052)% (0.401 ± 0.057)% (0.336 ± 0.048)% (0.414 ± 0.058)% (0.388 ± 0.059)% (0.337 ± 0.054)% (0.331 ± 0.053)% (0.258 ± 0.048)% (0.321 ± 0.012)%

#σ 1.65 −0.54 −0.59 −0.83 0.03 0.60 −0.12 −0.65 0.97 −1.25 3.10 −0.71 −0.81 1.58 0.54 1.79 1.31 0.50 0.40 −1.08 27.42

Table 91: Allowed Ds mass range at 3σ, from a gaussian fit. We allow a broader range of masses for the full analysis, but we use this restricted range for systematic checks. Ds tag mode KS K KKπ KS Kπ 0 KS KS π KKππ 0 KS K + ππ KS K − ππ πππ πη ππ 0 η πη 0 , η 0 → ππη ππ 0 η 0 , η 0 → ππη πη 0 , η 0 → ργ

Minimum Ds mass (MeV) 1,949.69 1,952.93 1,941.32 1,951.94 1,944.48 1,953.76 1,953.60 1,948.80 1,934.89 1,930.60 1,945.81 1,939.39 1,938.02

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Maximum Ds mass (MeV) 1,987.31 1,984.07 1,995.68 1,985.06 1,992.52 1,983.24 1,983.40 1,988.20 2,002.11 2,006.40 1,991.19 1,997.61 1,998.98

Table 92: Summary of various systematic errors for our electron identification.

SL mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

Base eID systematic 0.23% 0.10% 0.16% 0.17% 0.13% 0.18%

Base eID correction uncert. 0.03% 0.03% 0.03% 0.03% 0.03% 0.03%

Base eID correction adjust. 0.39% 0.28% 0.35% 0.34% 0.28% 0.34%

Event environ. correct. 0.48% 0.20% 0.42% 0.42% 0.19% 0.44%

Event environ. uncert. 0.16% 0.10% 0.14% 0.14% 0.11% 0.14%

Total systematic 0.68% 0.37% 0.59% 0.59% 0.38% 0.60%

Table 93: Relative corrections to the electron identification efficiency for each of our six semileptonic modes. Semileptonic mode φeν ηeν η 0 eν f0 eν KS eν K ∗ eν

Base electron ID -1.55% -1.10% -1.38% -1.37% -1.12% -1.36%

Event environment -0.36% -0.14% -0.32% -0.32% -0.13% -0.33%

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Total correction -1.91% -1.24% -1.71% -1.69% -1.24% -1.69%

Table 94: Semileptonic-side efficiencies in Ds → η 0 eν, including the η 0 → ππη and η → γγ branching ratios. Ds production mode Ds Ds Ds∗ Ds with Ds∗ → (Ds → η 0 eν) γ Ds∗ Ds with Ds∗ → (Ds → η 0 eν) π 0 Ds∗ Ds with prompt Ds → η 0 eν

εe 75.5% ± 74.9% ± 75.5% ± 74.0% ±

4.2% 1.0% 1.1% 1.1%

εη 0 7.1% ± 1.3% 5.5% ± 0.3% 5.0% ± 0.3% 6.2% ± 0.3%

εSL 5.5% ± 1.1% 3.8% ± 0.2% 3.6% ± 0.2% 4.4% ± 0.3%

Table 95: Semileptonic-side efficiencies in Ds → f0 eν, including the f0 → ππ branching ratio. Ds production mode Ds Ds ∗ ∗ Ds Ds with Ds → (Ds → f0 eν) γ Ds∗ Ds with Ds∗ → (Ds → f0 eν) π 0 Ds∗ Ds with prompt Ds → f0 eν

εe 78.7% ± 72.6% ± 73.2% ± 72.7% ±

4.3% 1.0% 1.0% 1.1%

εf 0 32.7% ± 30.1% ± 30.5% ± 29.7% ±

2.8% 0.6% 0.7% 0.7%

εSL 24.6% ± 2.4% 21.7% ± 0.6% 22.1% ± 0.6% 21.6% ± 0.6%

Table 96: Semileptonic-side efficiencies in Ds → KS eν, including the Ks → ππ branching ratio. Ds production mode Ds Ds ∗ ∗ Ds Ds with Ds → (Ds → KS eν) γ Ds∗ Ds with Ds∗ → (Ds → KS eν) π 0 Ds∗ Ds with prompt Ds → KS eν

εe 81.7% ± 80.5% ± 81.8% ± 81.0% ±

6.3% 1.5% 1.5% 1.6%

εKs 44.2% ± 41.8% ± 44.0% ± 43.3% ±

4.6% 1.1% 1.1% 1.2%

εSL 33.2% ± 4.0% 30.6% ± 0.9% 33.1% ± 1.0% 31.0% ± 1.0%

Table 97: Semileptonic-side efficiencies in Ds → K ∗ eν, including the K ∗ → Kπ branching ratio. Ds production mode Ds Ds ∗ ∗ Ds Ds with Ds → (Ds → K ∗ eν) γ Ds∗ Ds with Ds∗ → (Ds → K ∗ eν) π 0 Ds∗ Ds with prompt Ds → K ∗ eν

65.1% 71.8% 71.6% 71.6%

εe ± ± ± ±

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εK ∗ 32.9% ± 2.9% 34.8% ± 0.7% 35.7% ± 0.7% 35.1% ± 0.8%

εSL 21.3% ± 2.3% 24.1% ± 0.6% 24.8% ± 0.6% 24.0% ± 0.7%

Table 98: All systematic efficiency corrections (relative) for Ds → φeν. Systematic Kaon track efficiency Electron ID Total

Relative ε correction −8.17% −1.91% −10.08%

Table 99: All systematic efficiency corrections (relative) for Ds → ηeν. Systematic Electron ID Total

Relative ε correction −1.24% −1.24%

Table 100: All systematic efficiency corrections (relative) for Ds → η 0 eν. Systematic Electron ID π (and K) ID Semileptonic hadron B Total

Relative ε correction −1.71% −2.94% −1.83% −6.48%

Table 101: All systematic efficiency corrections (relative) for Ds → f0 eν. Systematic Electron ID π (and K) ID Total

Relative ε correction −1.69% −0.50% −2.19%

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Table 102: All systematic efficiency corrections (relative) for Ds → KS eν. Systematic Ks efficiency Electron ID Semileptonic hadron B Total

Relative ε correction −11.08% −1.24% 0.86% −11.46%

Table 103: All systematic efficiency corrections (relative) for Ds → K ∗ eν. Systematic Electron ID π (and K) ID Total

Relative ε correction −1.69% −2.88% −4.57%

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Table 104: All systematic errors (relative) for Ds → φeν. Systematic Distribution within pK bin Kaon track reconstruction Ds tag signal fitting Ds tag BG shape Electron ID Form factor model Multiple candidate choice Semileptonic hadron B Track reconstruction Efficiency statistics Total

Relative systematic error 0.54% 1.71% 1.80% 1.27% 0.68% 2.91% 0.11% 1.02% 0.30% 1.33% 4.46%

Table 105: All systematic errors (relative) for Ds → ηeν. Systematic Ds tag signal fitting Ds tag BG shape Electron ID Form factor model Signal shape Multiple candidate choice Semileptonic hadron B Particle ID Track reconstruction Efficiency statistics Splitoff rate Total

Relative systematic error 2.23% 0.92% 0.37% 0.73% 1.04% 1.67% 0.51% 7.90% 0.30% 1.07% 1.16% 8.70%

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Table 106: All systematic errors (relative) for Ds → η 0 eν. Systematic Ds tag signal fitting Ds tag BG shape Particle ID Track reconstruction Mass resolution Electron ID Form factor model Multiple candidate choice Decay in flight Semileptonic hadron B Efficiency statistics Total

Relative systematic error 2.07% 1.31% 7.90% 0.90% 3.15% 0.59% 1.64% 0.21% 0.49% 1.71% 4.09% 10.11%

Table 107: All systematic errors (relative) for Ds → f0 eν. Systematic Ds tag signal fitting Ds tag BG shape Particle ID Track reconstruction Mass resolution Electron ID Form factor model Multiple candidate choice Decay in flight Efficiency statistics Total

Relative systematic error 1.60% 0.80% 0.04% 0.90% 2.63% 0.59% 2.29% 2.20% 0.52% 1.57% 4.91%

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Table 108: All systematic errors (relative) for Ds → KS eν. Systematic Ds tag signal fitting Ds tag BG shape Ks efficiency Electron ID Form factor model Multiple candidate choice Decay in flight Semileptonic hadron B Track reconstruction Efficiency statistics Total

Relative systematic error 2.20% 0.86% 7.28% 0.38% 1.35% 3.05% 0.63% 0.07% 0.30% 1.72% 8.56%

Table 109: All systematic errors (relative) for Ds → K ∗ eν. Systematic Ds tag signal fitting Ds tag BG shape Particle ID Track reconstruction Mass resolution Electron ID Form factor model Multiple candidate choice Decay in flight Efficiency statistics Total

Relative systematic error 2.97% 2.08% 1.21% 0.60% 2.59% 0.60% 5.10% 0.28% 0.71% 1.47% 7.13%

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Invariant mass vs. Recoil mass, π η

Ds Invariant Mass (MeV)

Invariant mass vs. Recoil mass, π π π

2060

Ds Invariant Mass (MeV)

2060

2060

2080

2100

2120

2140 2160 2180 Ds Recoil Mass (MeV)

1900

2060

2080

2100

2120

2140 2160 2180 Ds Recoil Mass (MeV)

Figure 83: The invariant mass vs. recoil mass distribution in data for Ds tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ.

196

Counts / 1.30 MeV

Ngauss1

1348 / 95

χ2 / ndf Best invariant mass for mode K K π (MC True)

1.359e+04 / 95

1.358e+05 ± 5.288e+02

Ngauss1

6.137e+05 ± 9.965e+02

µgauss

1968 ± 0.0

σ gauss1

10000

5.357 ± 0.019

Ngauss2/N gauss1

0.08981 ± 0.00275

σ gauss2/ σ gauss1

3.303 ± 0.035

8000

Counts / 1.30 MeV

χ2 / ndf Best invariant mass for mode Ks K (MC True)

µgauss

60000

1968 ± 0.0

σ gauss1

50000

0.09027 ± 0.00090

σ gauss2/ σ gauss1

4.61 ± 0.02

40000

6000

30000

4000

20000

2000

0 1900

10000

1920

1940

1960

1980

2000

0 1900

2020 MD (MeV)

1920

1940

1960

1980

2000

s

2

2400 2200 2000 1800

2

335 / 93

Ngauss µgauss

2.991e+04 ± 9.378e+02

σ gauss

7.158 ± 0.105

1968 ± 0.1

Ncryst/N gauss

0.0175 ± 0.0017

σ cryst/ σ gauss

1.767 ± 0.025

α cryst

0.8374 ± 0.0200

ncryst

130 ± 86.6

1600

2020 MD (MeV) s

χ / ndf Best invariant mass for mode K s Ks π (MC True) Ngauss1

Counts / 1.30 MeV

χ / ndf Best invariant mass for mode K s K π0 (MC True)

Counts / 1.30 MeV

4.521 ± 0.007

Ngauss2/N gauss1

1400

396.4 / 95 2.597e+04 ± 2.131e+02

µgauss

2500

1969 ± 0.0

σ gauss1

4.474 ± 0.035

Ngauss2/N gauss1

0.08293 ± 0.00497

σ gauss2/ σ gauss1

3.833 ± 0.091

2000

1500

1200 1000

1000

800 600 500

400 200 0 1900

1920

1940

1960

1980

2000

0 1900

2020 MD (MeV) s

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 84: Fits to the truth-tagged Ds invariant mass from the Monte Carlo. We fix the fit function’s shape parameters (relative normalization, relative width, and crystal ball power law tail) from these results. These plots show the fit results for Ds to KS K, KKπ, KS Kπ 0 , and KS KS π.

197

χ / ndf Best invariant mass for mode K K π π0 (MC True)

1968 ± 0.0

σ gauss

9000 8000 7000

2

χ / ndf Best invariant mass for mode K s K+ π π (MC True)

1350 / 93

Ngauss1

1.242e+05 ± 1.340e+03

6.63 ± 0.04

Ncryst/N gauss

0.01189 ± 0.00052

σ cryst/ σ gauss

1.943 ± 0.015

α cryst

0.834 ± 0.011

ncryst

130 ± 13.3

6000

Counts / 1.30 MeV

Counts / 1.30 MeV

2

Ngauss µgauss

4000

2000

3000

1500

2000

1000

1000

500 1960

1980

2000

0 1900

2020 MD (MeV)

0.07613 ± 0.00294

σ gauss2/ σ gauss1

4.967 ± 0.093

3000 2500

1940

1920

1940

1960

1980

2000

s

2

Counts / 1.30 MeV

Ngauss1

9000

µgauss

8000 7000

χ2 / ndf Best invariant mass for mode π π π (MC True)

2257 / 95 7.683e+04 ± 3.405e+02 1968 ± 0.0

σ gauss1

3.887 ± 0.016

Ngauss2/N gauss1

0.08326 ± 0.00216

σ gauss2/ σ gauss1

5.311± 0.070

6000

2020 MD (MeV) s

Counts / 1.30 MeV

-

χ / ndf Best invariant mass for mode K s K π π (MC True)

3.974 ± 0.022

Ngauss2/N gauss1

3500

4000

1920

1969 ± 0.0

σ gauss1

4500

5000

0 1900

938.8 / 95 4.149e+04 ± 2.510e+02

µgauss

10000

5000

964.7 / 93

Ngauss µgauss

1.327e+05 ± 2.313e+02

σ gauss

5.552 ± 0.010

Ncryst/N gauss

1968 ± 0.0

0.00515 ± 0.00013

σ cryst/ σ gauss

8000

2.623 ± 0.030

α cryst

0.9693 ± 0.0870

ncryst

3 ± 26.5

6000

4000 4000

3000 2000

2000

1000 0 1900

1920

1940

1960

1980

2000

0 1900

2020 MD (MeV) s

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 85: Fits to the truth-tagged Ds invariant mass from the Monte Carlo. We fix the fit function’s shape parameters (relative normalization, relative width, and crystal ball power law tail) from these results. These plots show the fit results for Ds to KKππ 0 , KS K + ππ, KS K − ππ, and πππ.

198

Counts / 1.30 MeV

3000 2500

χ2 / ndf Ngauss µgauss σ gauss Ncryst/N gauss σ cryst/ σ gauss α cryst ncryst

0

345.4 / 93 5.648e+04 ± 1.821e+03 1966 ± 0.1 10.15 ± 0.12 0.01146 ± 0.00127 1.678 ± 0.020 0.8969 ± 0.0270 130 ± 85.1

Best invariant mass for mode π π η (MC True) Counts / 1.30 MeV

χ2 / ndf Ngauss µgauss σ gauss Ncryst/N gauss σ cryst/ σ gauss α cryst ncryst

Best invariant mass for mode π η (MC True)

2000

5000

4000

183.6 / 93 2.004e+04 ± 2.702e+03 1966 ± 0.1 23.32 ± 0.93 0.2221 ± 0.0325 0.5205 ± 0.0164 0.9953 ± 0.0219 6.457 ± 0.815

3000

1500 2000 1000 1000

500

1920

1940

1960

1980

χ2 / ndf Ngauss1 µgauss σ gauss1 Ngauss2/N gauss1 σ gauss2/ σ gauss1

Best invariant mass for mode π η’ (MC True) Counts / 1.30 MeV

2000

3000 2500

0 1900

2020 MDs (MeV)

1920

1940

1960

1980

0

1028 / 95 4.72e+04 ± 4.08e+02 1967 ± 0.0 6.868 ± 0.051 0.1615 ± 0.0083 2.802 ± 0.040

2000

2000

χ2 / ndf Ngauss µgauss σ gauss Ncryst/N gauss σ cryst/ σ gauss α cryst ncryst

Best invariant mass for mode π π η’ (MC True) Counts / 1.30 MeV

0 1900

800 700 600

2020 MDs (MeV) 203 / 93 1.066e+04 ± 7.029e+02 1967 ± 0.1 7.854 ± 0.219 0.02237 ± 0.00380 1.682 ± 0.041 0.8645 ± 0.0335 130 ± 120.7

500 400

1500

300 1000 200 500 0 1900

100 1920

1940

1960

1980

2000

0 1900

2020 MDs (MeV)

χ2 / ndf Ngauss µgauss σ gauss Ncryst/N gauss σ cryst/ σ gauss α cryst

Best invariant mass for mode π η’ -> ρ γ (MC True) Counts / 1.30 MeV

1920

4500 4000 3500

1940

1960

1980

2000

2020 MDs (MeV)

339.7 / 94 1600 ± 344.0 1968 ± 0.0 30 ± 0.2 2.843 ± 0.612 0.2979 ± 0.0016 1.477 ± 0.024

3000 2500 2000 1500 1000 500 0 1900

1920

1940

1960

1980

2000

2020 MDs (MeV)

Figure 86: Fits to the truth-tagged Ds invariant mass from the Monte Carlo. We fix the fit function’s shape parameters (relative normalization, relative width, and crystal ball power law tail) from these results. These plots show the fit results for Ds to πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ.

199

700 600

χ 2 / ndf Ngauss1 µ gauss σgauss1 constlin slopelin

1671 / 95 6876 ± 120.3 1968 ± 0.1 5.226 ± 0.098 134.3 ± 2.3 -0.3551 ± 0.0283

500

χ2 / ndf 8204 / 95 Ngauss1 3.01e+04 ± 2.79e+02 µ gauss 1968 ± 0.0 σ gauss1 4.246 ± 0.044 constlin 1070 ± 6.6 slope -1.608 ± 0.083 lin

Best invariant mass for mode K K π Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode K s K

400

4000 3500 3000 2500 2000

300 1500 200 1000 100 0 1900

500 1920

1940

1960

1980

2000

0 1900

2020 MD (MeV)

1920

1940

1960

1980

2000

s

500

400

χ2 / ndf Ngauss µ gauss σ gauss const lin slope lin

820.5 / 95 1382 ± 105.7 1968 ± 0.7 6.512 ± 0.585 447.7 ± 4.4 -1.265 ± 0.051

300

s

Best invariant mass for mode K s Ks π Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode Ks K π 0

2020 MD (MeV)

250

χ 2 / ndf Ngauss1 µ gauss σgauss1 constlin slopelin

718.5 / 95 1370 ± 83.2 1969 ± 0.3 4.279 ± 0.291 161.2 ± 2.5 -0.3494 ± 0.0313

200

150 200 100 100

0 1900

50

1920

1940

1960

1980

2000

0 1900

2020 MD (MeV) s

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 87: Ds invariant mass fits in the weighted 20× Monte Carlo sample (charm + continuum), determining the total number of Ds tags for modes KS K, KKπ, KS Kπ 0 , and KS KS π.

200

1400 1200

χ2 / ndf Ngauss µ gauss σ gauss constquad slope quad x2quad

1007 / 94 6112 ± 244.9 1968 ± 0.3 6.477 ± 0.276 1058 ± 8.5 -2.311 ± 0.338 0.01333 ± 0.00256

1000

2 Best invariant mass for mode Ks K+ π π χ / ndf

Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode K K π π0

600 500 400

800

300

600 400

200

200

100

0 1900

971.6 / 95 Ngauss1 2107 ± 127.5 µ gauss 1969 ± 0.2 σ gauss1 3.767 ± 0.258 const lin 389.3 ± 4.1 slope 0.3867 ± 0.0544 lin

1920

1940

1960

1980

2000

0 1900

2020 MD (MeV)

1920

1940

1960

1980

2000

s

-

s

2

600 500

χ 2 / ndf Ngauss µ

Best invariant mass for mode π π π Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode K s K π πχ / ndf

1925 / 95 Ngauss1 3896 ± 110.7 µ gauss 1969 ± 0.1 σ gauss1 3.656 ± 0.118 const lin 203.6 ± 3.0 slope 0.1818 ± 0.0394 lin

2020 MD (MeV)

400

479.4 / 94 6548 ± 315.4 1969 ± 0.2

gauss

2000

σgauss

1800

const quad slope

1600

x 2quad

quad

5.351 ± 0.271 1844 ± 11.8 -5.428 ± 0.457 0.007412 ± 0.003390

1400 1200 1000

300

800 200

600 400

100 200 0 1900

1920

1940

1960

1980

2000

0 1900

2020 MD (MeV) s

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 88: Ds invariant mass fits in the weighted 20× Monte Carlo sample (charm + continuum), determining the total number of Ds tags for modes KKππ 0 , KS K + ππ, KS K − ππ, and πππ.

201

χ2 / ndf Ngauss µgauss σ gauss const lin slope

400 350

lin

χ2 / ndf Ngauss µgauss σ gauss const quad slope quad 2 xquad

Best invariant mass for mode π π 0 η

568.1 / 95 2500 ± 131.4 1967 ± 0.6 9.423 ± 0.545 348.8 ± 4.2 -1.025 ± 0.046

Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode π η

300 250

2000 1800 1600 1400

570.3 / 94 1024 ± 64.8 1967 ± 0.8 23.81 ± 2.15 1912 ± 11.3 -9.38 ± 0.42 0.02815 ± 0.00327

1200 1000

200

800 150

600

100

400

50

200 1920

1940

1960

1980

χ2 / ndf Ngauss1 µgauss σ gauss1 const lin slope

Best invariant mass for mode π η’ Counts / 1.30 MeV

2000

180 160

0 1900

2020 MDs (MeV)

lin

1920

1940

1960

1980

Best invariant mass for mode π π 0 η’

1029 / 95 2387 ± 72.0 1968 ± 0.2 6.669 ± 0.223 39.05 ± 1.29 -0.1266 ± 0.0150

Counts / 1.30 MeV

0 1900

140 120

2000

χ2 / ndf Ngauss µgauss σ gauss const lin slope

160 140

lin

2020 MDs (MeV) 968.6 / 95 418.6 ± 54.7 1967 ± 1.1 6.165 ± 0.883 153.9 ± 2.6 -0.4664 ± 0.0297

120 100

100 80 80 60

60

40

40

20

20 0 1900

1920

1940

1960

1980

2000

0 1900

2020 MDs (MeV)

χ2 / ndf Ngauss µgauss σ gauss const quad slope quad x2quad

Best invariant mass for mode π η’ -> ρ γ Counts / 1.30 MeV

1920

900 800 700

1940

1960

1980

2000

2020 MDs (MeV)

473.8 / 94 80.39 ± 3.89 1968 ± 0.4 28.54 ± 1.59 856.8 ± 7.2 -3.878 ± 0.262 0.01016 ± 0.00195

600 500 400 300 200 100 0 1900

1920

1940

1960

1980

2000

2020 MDs (MeV)

Figure 89: Ds invariant mass fits in the weighted 20× Monte Carlo sample (charm + continuum), determining the total number of Ds tags for modes πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ.

202

700

χ 2 / ndf Nsignal constlin slopelin

92.69 / 97 0.1082 ± 0.0016 120.3 ± 2.2 -0.3653 ± 0.0262

600 500

χ2 / ndf 253.1 / 97 Nsignal 0.1072 ± 0.0009 const lin 1254 ± 7.1 slope -2.123 ± 0.089 lin

Best invariant mass for mode K K π Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode K s K

4500 4000 3500 3000 2500

400

2000

300

1500 200 1000 100 0 1900

500 1920

1940

1960

1980

2000

0 1900

2020 MD (MeV)

1920

1940

1960

1980

2000

s

s

500

400

Best invariant mass for mode K s Ks π Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode K s K π0 χ2 / ndf

90.59 / 97 Nsignal 0.1082 ± 0.0068 const lin 463 ± 4.3 slope -1.289 ± 0.052 lin

2020 MD (MeV)

300

χ 2 / ndf Nsignal constlin slopelin

77.5 / 97 0.1055 ± 0.0058 201.6 ± 2.8 -0.5096 ± 0.0343

250 200

300 150 200 100 100

0 1900

50

1920

1940

1960

1980

2000

0 1900

2020 MD (MeV) s

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 90: Ds invariant mass fits in the data sample using a signal histogram from the truthtagged Monte Carlo. These plots show our results for Ds to KS K, KKπ, KS Kπ 0 , and KS KS π.

203

1800

χ2 / ndf Nsignal constlin slope

lin

quad

1600

lin

Best invariant mass for mode Ks K+ π π χ2 / ndf

95.92 / 96 0.111± 0.004 1285 ± 11.0 -2.079 ± 0.446 0.009896 ± 0.003339

1400 1200

Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode K K π π0

800 700 600 500

1000

400

800 600

300

400

200

200

100

0 1900

98.02 / 97 Nsignal 0.1073 ± 0.0057 const lin 541.7 ± 4.8 slope 0.2765 ± 0.0632 lin

1920

1940

1960

1980

2000

0 1900

2020 MD (MeV)

1920

1940

1960

1980

2000

s

-

s

800 700 600 500

χ 2 / ndf 86.28 / 96 Nsignal 0.1039 ± 0.0038 constlin 1914 ± 12.9 slopelin -6.695 ± 0.482 quadlin 0.01055 ± 0.00353

Best invariant mass for mode π π π Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode K s K π π χ2 / ndf

102.7 / 97 Nsignal 0.1077 ± 0.0028 const lin 278.9 ± 3.4 slope 0.081 ± 0.045 lin

2020 MD (MeV)

2200 2000 1800 1600 1400 1200

400

1000

300

800

200

600 400

100 0 1900

200 1920

1940

1960

1980

2000

0 1900

2020 MD (MeV) s

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 91: Ds invariant mass fits in the data sample using a signal histogram from the truthtagged Monte Carlo. These plots show our results for Ds to KKππ 0 , KS K + ππ, KS K − ππ, and πππ.

204

χ 2 / ndf Nsignal const lin slope

400

lin

350

Best invariant mass for mode π π 0 η

134.1 / 97 0.1058 ± 0.0039 264.8 ± 3.4 -0.7751 ± 0.0395

Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode π η

300 250

χ2 / ndf Nsignal const lin slope quad lin

1800

lin

1600

155.6 / 96 0.08934 ± 0.00741 1771 ± 12.5 -6.26 ± 0.57 0.005223 ± 0.004306

1400 1200 1000

200

800 150 600 100

400

50 0 1900

200 1920

1940

1960

1980

2000

0 1900

2020 MD (MeV)

1920

1940

1960

1980

2000

s

χ 2 / ndf Nsignal const lin slope

200

lin

180

s

Best invariant mass for mode π π η’

160 140 120

200

lin

84.03 / 97 0.1403 ± 0.0107 155.9 ± 2.5 -0.5135 ± 0.0295

180 160 140 120

100

100

80

80

60

60

40

40

20 0 1900

χ 2 / ndf Nsignal const lin slope

0

97.71 / 97 0.1095 ± 0.0028 30.35 ± 1.14 -0.1006 ± 0.0135

Counts / 1.30 MeV

Counts / 1.30 MeV

Best invariant mass for mode π η’

2020 MD (MeV)

20 1920

1940

1960

1980

2000

0 1900

2020 MD (MeV)

1920

1940

s

1980

2000

2020 MD (MeV) s

χ2 / ndf Nsignal const lin slope lin quad

Best invariant mass for mode π η’ -> ρ γ Counts / 1.30 MeV

1960

1000

lin

105.5 / 96 0.1048 ± 0.0052 919.1 ± 8.9 -4.514 ± 0.354 0.01145± 0.00260

800

600

400

200

0 1900

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 92: Ds invariant mass fits in the data sample using a signal histogram from the truthtagged Monte Carlo. These plots show our results for Ds to πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ.

205

Ds invariant mass fit comparison for mode K K π Counts / 1.30 MeV

Counts / 1.30 MeV

Ds invariant mass fit comparison for mode K s K

500

400

3000

2500

2000

Standard fit

Standard fit

300

1500 BG + signal histogram

200

BG + signal histogram

1000

100

0

500

1920

1940

1960

1980

2000

0

2020 MD (MeV)

1920

1940

1960

1980

2000

s

s

Ds invariant mass fit comparison for mode K K π0

Ds invariant mass fit comparison for mode K s Ks π

s

120

Counts / 1.30 MeV

Counts / 1.30 MeV

2020 MD (MeV)

100

80

120 100 80

Standard fit

Standard fit

60 60 BG + signal histogram

40

BG + signal histogram

40

20

0

20

1920

1940

1960

1980

2000

0

2020 MD (MeV) s

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 93: Ds invariant mass fits for a double gaussian/gaussian+crystal ball signal shape compared to fits with a signal histogram for modes KS K, KKπ, KS Kπ 0 , and KS KS π.

206

Ds invariant mass fit comparison for mode K s K+ π π Counts / 1.30 MeV

Counts / 1.30 MeV

Ds invariant mass fit comparison for mode K K π π0 500

400

300

Standard fit

240 220 200 180 160 Standard fit

140 120 100

200

BG + signal histogram

BG + signal histogram

80 60

100

40 20

0

1920

1940

1960

1980

2000

0

2020 MD (MeV)

1920

1940

1960

1980

2000

s

s

-

Ds invariant mass fit comparison for mode π π π Counts / 1.30 MeV

Counts / 1.30 MeV

Ds invariant mass fit comparison for mode K s K π π 450 400 350 300

500

400

Standard fit

250

2020 MD (MeV)

Standard fit

300

200 BG + signal histogram

150

BG + signal histogram

200

100

100

50 0

1920

1940

1960

1980

2000

0

2020 MD (MeV) s

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 94: Ds invariant mass fits for a double gaussian/gaussian+crystal ball signal shape compared to fits with a signal histogram for modes KKππ 0 , KS K + ππ, KS K − ππ, and πππ.

207

Ds invariant mass fit comparison for mode π π0 η Counts / 1.30 MeV

Counts / 1.30 MeV

Ds invariant mass fit comparison for mode π η

140 120 100

Standard fit

250

200

Standard fit 150

80 60

BG + signal histogram

BG + signal histogram

100

40 50 20 0

1920

1940

1960

1980

2000

0

2020 MD (MeV)

1920

1940

1960

1980

2000

s

s

Ds invariant mass fit comparison for mode π π0 η’ Counts / 1.30 MeV

Ds invariant mass fit comparison for mode π η’ Counts / 1.30 MeV

2020 MD (MeV)

140 120 100

Standard fit

50

40

Standard fit 30

80 60

BG + signal histogram

BG + signal histogram

20

40 10 20 0

1920

1940

1960

1980

2000

0

2020 MD (MeV)

1920

1940

s

1960

1980

2000

2020 MD (MeV) s

Counts / 1.30 MeV

Ds invariant mass fit comparison for mode π η’ -> ρ γ 250

200

150

Standard fit

100

BG + signal histogram

50

0

1920

1940

1960

1980

2000

2020 MD (MeV) s

Figure 95: Ds invariant mass fits for a double gaussian/gaussian+crystal ball signal shape compared to fits with a signal histogram for modes πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ.

208

Individual energies of extra showers

Total extra shower energy, showers > 25 MeV

Total extra shower energy, showers > 100 MeV

5000

2500

6000

4000

2000

3000

1500

2000

1000

5000

4000

3000

2000 1000

500

0 0

100

200

300

400

500 600 Eγ (MeV)

# extra showers > 300 MeV

1000

0 0

100

200

300 400 500 600 700 Total shower energy (MeV)

0 0

100

200

300 400 500 600 700 Total shower energy (MeV)

# extra showers > 500 MeV 12000

All recon Ds + ηeν

10000 10000 8000 8000 6000

!Ds + ηeν

6000

4000

Ds + !ηeν

4000

2000

2000

0

0

0

1

2

3

4 5 6 7 8 Number of extra showers

!Ds + !ηeν 0

1

2

3

4 5 6 7 8 Number of extra showers

Figure 96: Extra showers after finding the tagged Ds , the η, and the electron in ηeν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays.

209

Individual energies of extra showers

Total extra shower energy, showers > 25 MeV

160

90

90 140

80

80

120

70

100

60

70 60 50

50

80

Total extra shower energy, showers > 100 MeV

40

40

30

30

20

20

10

10

60 40 20 0 0

100

200

300

400

500 600 Eγ (MeV)

# extra showers > 300 MeV

0 0

100

200

300 400 500 600 700 Total shower energy (MeV)

0

100

200

300

400 500 600 700 Total shower energy (MeV)

# extra showers > 500 MeV 800

700

All recon Ds + η’eν

700

600

600

500

500

400

400

300

300

200

200

100

100

0

0

0

1

2

3

4 5 6 7 8 Number of extra showers

!Ds + η’eν Ds + !η’eν !Ds + !η’eν 0

1

2

3

4 5 6 7 8 Number of extra showers

Figure 97: Extra showers after finding the tagged Ds , the η 0 , and the electron in η 0 eν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays.

210

Individual energies of extra showers

Total extra shower energy, showers > 25 MeV

Total extra shower energy, showers > 100 MeV

450

700

250

400 600 350

200

500

300

400

250

150

200

300

100

150 200 100

50

100

50

0 0

100

200

300

400

500 600 Eγ (MeV)

# extra showers > 300 MeV

0 0

100

200

300 400 500 600 700 Total shower energy (MeV)

0

100

200

300

400 500 600 700 Total shower energy (MeV)

# extra showers > 500 MeV 3000

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Figure 98: Extra showers after finding the tagged Ds , the f0 , and the electron in f0 eν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays.

211

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# extra showers > 500 MeV

1600

1000

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1

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Figure 99: Extra showers after finding the tagged Ds , the Ks , and the electron in KS eν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays.

212

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!Ds + !K*eν 0

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Figure 100: Extra showers after finding the tagged Ds , the K ∗ , and the electron in K ∗ eν (20× MC sample). Our shower quality selections include both EE259 O.K. and a splitoff rejection. The peak near 140 MeV is due to the γ from Ds∗ decays.

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10 0 1900

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Figure 101: Likelihood fit results for the Ds mass spectrum after all Ds → φeν semileptonic cuts in the first four data-sized Monte Carlo samples. The blue histogram represents the Monte Carlo truth-tagged events, while the blue fit line gives the signal part of our fit. The red fit line represents non-peaking background, and the green line shows our peaking background subtraction.

214

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Figure 102: Ds → φeν data-sized Monte Carlo results, second group of datasets.

215

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Figure 103: Ds → φeν data-sized Monte Carlo results, third group of datasets.

216

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Figure 104: Ds → φeν data-sized Monte Carlo results, fourth group of datasets.

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Figure 105: Ds → φeν data-sized Monte Carlo results, fifth group of datasets.

218

249 1964 17.8

+

MC true Ds tags for successful Ks+e+ν

φ e+ ν

e

(87.8%

)

φ e+ νe f 0 e+ νe K*0 e+ νe 0 K e+ νe η e+ νe η’ e+ νe τ+ ντ φ a+ 1 Ds, tag mode Ds hadronic, non-tag

Figure 106: Ds → KS eν backgrounds with a true Ds tag (peaking background), from the 20× Monte Carlo. These remain after KS eν semileptonic cuts but before any missing mass cut or other, additional background restrictions. φeν with φ → KL Ks dominates.

219

+

MC true Ds tags for successful Ks+e+ν )

K*

0

e+

, Ds

νe (28.8%)

+

0

K

e

νe

(1

+ 10.2%) η e νe (

)

2% 0.

g ta

φe+ νe

m

(8.

e od

% .2 2 3

(

5%

)

φ e+ νe f 0 e+ νe K*0 e+ νe K0 e+ νe η e+ νe η’ e+ νe τ+ ντ Ds, tag mode Ds hadronic, non-tag Figure 107: Ds → KS eν backgrounds with a true Ds tag (peaking background), after all cuts. The other semileptonic modes each give some fake events, while the dominant non-semileptonic contribution comes from Ds tag modes with a kaon faking the electron (e.g. Ds → KKs )

220

Events / 6.5 MeV

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s

s

4.5 4

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7 6 5 4

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s

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MD after Ks +e cuts, K K

2020 MDs (MeV)

Figure 108: Ds → KS eν fit results in the data, after all semileptonic cuts, for tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization.

221

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Figure 109: Ds → KS eν fit results in the data, after all semileptonic cuts, for tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization.

222

*

MC true D+s tags for successful K +e+ν

+

φe

) 6% 56.

νe(

νe

Ds hadronic, non-tag (12.3%)

+

e η’

) .3% 2 (1

D

s,

tag

mo d

e( 10

.4%

)

φ e+ νe f 0 e+ νe K0 e+ νe η e+ νe η’ e+ νe Ds, tag mode Ds hadronic, non-tag

Figure 110: Ds → K ∗ eν backgrounds with a true Ds tag (peaking background), before our specific K ∗ eν cuts in the 20× Monte Carlo. Our best improvement in peaking background will come from reducing Ds → φeν where one kaon fakes a pion.

223

*

MC true D+s tags for successful K +e+ν

+

φ

Ds ,

)

tag m

ode

νe

(9.8

%)

ha

dr

on ic

) .8%

(9

-ta g

(3

+

,n on

ηe

(1 4. 6%

)

η’ e+ νe (26.8%)

e

νe

1% 4.

D

s

φ e+ νe f 0 e+ νe K0 e+ νe η e+ νe η’ e+ νe Ds, tag mode Ds hadronic, non-tag

Figure 111: Ds → K ∗ eν backgrounds with a true Ds tag (peaking background), after all cuts. The other semileptonic modes each give some fake events, while the dominant non-semileptonic contribution comes from Ds tag modes where a kaon fakes the electron (e.g. Ds → KKπ).

224

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s

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1

0 1900

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s

s

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0 1900

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Figure 112: Ds → K ∗ eν fit results in the data, after all semileptonic cuts, for tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode receives an independent background normalization.

225

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Figure 113: Ds → K ∗ eν fit results in the data, after all semileptonic cuts, for tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization.

226

MC true D+s tags for successful η’+e+ν

+

ηe

νe tag

mo

de (

20.0

%)

φe+ ν

e

(20

.0%

)

) .0% (60 Ds ,

φ e+ νe η e+ νe Ds, tag mode Figure 114: Ds → η 0 eν backgrounds with a true Ds tag (peaking background) in the 20× Monte Carlo. ηeν with η → ππ(π 0 /γ) produces the most peaking background, while the dominant non-semileptonic contribution comes from Ds tag modes with a kaon faking the electron (e.g. Ds → KKs π 0 )

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6

0 1900

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Figure 115: Ds → η 0 eν fit results in the data, after all semileptonic cuts, for tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode receives an independent background normalization.

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1940

1960

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Figure 116: Ds → η 0 eν fit results in the data, after all semileptonic cuts, for tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization.

229

+

MC true Ds tags for successful f +e+ν

η’ e +

νe (4 4.9% )

0

)

D s,

tag

de mo

(3

% 2.7

Ds hadr onic, n φe + on-tag (10.2%) ν e ( 12 .2 % )

φ e+ νe η’ e+ νe Ds, tag mode Ds hadronic, non-tag Figure 117: Ds → f0 eν backgrounds with a true Ds tag (peaking background) in the 20× Monte Carlo. η 0 eν with η 0 → ππX provides the plurality contribution, while the dominant non-semileptonic peaking background comes from Ds tag modes where a kaon fakes the electron (e.g. Ds → KKs ).

230

Events / 6.5 MeV

mass_plot_0_0 Entries 6 Mean 1969 RMS 7.064

s

s

3.5 3

MD after f 0 +e cuts, K K π

mass_plot_0_1 Entries 48 Mean 1958 RMS 29.75

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MD after f 0 +e cuts, K K

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mass_plot_0_4 Entries 41 Mean 1960 RMS 28.75

7

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Total f 0+e+ν signal Background

6

0 1900

1940

s

s

2.5

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1920

MD after f 0 +e cuts, K Ks π Events / 6.5 MeV

0 1900

2020 MDs (MeV)

Figure 118: Ds → f0 eν fit results in the data, after all semileptonic cuts, for tag modes KS K, KKπ, KS Kπ 0 , KS KS π, and KKππ 0 . We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode receives an independent background normalization.

231

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2 1937 31.48

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Figure 119: Ds → f0 eν fit results in the data, after all semileptonic cuts, for tag modes KS K + ππ; KS K − ππ; πππ; πη; ππ 0 η; πη 0 , η 0 → ππη; ππ 0 η 0 , η 0 → ππη; and πη 0 , η 0 → ργ. We fit the tagged MDs with a common signal normalization (branching ratio) for all 13 tag modes. Each mode does receive an independent background normalization.

232

Χ2

Χ2 vs. σ 140 Mass shift, -0.10 MeV Mass shift, -0.05 MeV Mass shift, 0.00 MeV Mass shift, 0.05 MeV Mass shift, 0.10 MeV

138 136 134 132 130 128 126 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 0.4 0.45 σgauss smear (MeV)

Figure 120: χ2 for data fits using various shifts and gaussian smears to the Monte Carlo’s signal Mφ distribution. Smaller shifts and smears tend to be favored, implying a fairly accurate φ mass resolution in the Monte Carlo.

233

Χ 2 vs. ε, -15/+30 MeV φ cut Χ2

140 Mass shift, -0.10 MeV

138

Mass shift, -0.05 MeV Mass shift, 0.00 MeV

136

Mass shift, 0.05 MeV Mass shift, 0.10 MeV

134 132 130 128 126 99.9

99.92 99.94 99.96 99.98

100 100.02 100.04 100.06 ε 100.08 ε0 (%)

Figure 121: Our large Mφ cut window means that even φ lineshapes that don’t fit the data particularly well still have a relative efficiency difference from predicted ( ∆ε ) of less than 0.1%. ε0

234

ση for full MC, mode Ks K Counts / 0.25 σ

Counts / 1.3 MeV

MDs for full MC, mode Ks K All recon

70

Ds + !ηeν

60

!Ds + ηeν Ds + ηeν

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150

200 100

150 100

50 50 0 1900

1920

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0

2020 MD (MeV)

-4

-2

0

2

4 ση

s

Figure 122: The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over the full 20× Monte Carlo when we tag Ds → KS K. The bottom two plots do the same for Ds → KKπ.

235

ση for full MC, mode K K π π0 Counts / 0.25 σ

Counts / 1.3 MeV

MD s for full MC, mode K K π π0 100

120

All recon Ds + !ηeν

80

100

!Ds + ηeν Ds + ηeν

60

80

!Ds + !ηeν 60 40 40 20 20

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Counts / 1.3 MeV

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25

20

30 25

15

20 10

15 10

5 5 0 1900

1920

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0

2020 MD (MeV)

-4

-2

0

2

4 ση

s

Figure 123: The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over the full 20× Monte Carlo when we tag Ds → KKππ 0 . The bottom two plots do the same for Ds → KS K + ππ.

236

ση for full MC, mode π η Counts / 0.25 σ

Counts / 1.3 MeV

MDs for full MC, mode π η 30

All recon Ds + !ηeν

25

!Ds + ηeν

30 25

Ds + ηeν

20

35

!Ds + !ηeν

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ση for full MC, mode π π0 η Counts / 0.25 σ

Counts / 1.3 MeV

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90 80 70

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10

20

5

10

0 1900

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0

2020 MD (MeV)

-4

-2

0

2

4 ση

s

Figure 124: The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over the full 20× Monte Carlo when we tag Ds → πη. The bottom two plots do the same for Ds → ππ 0 η.

237

60

ση for all modes, dataset 0 Counts / 0.25 σ

Counts / 1.3 MeV

MDs for all modes, dataset 0 All recon Ds + !ηeν

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25 30 20 15

20

10 10 5 0 1900

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0

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-4

-2

0

2

4 ση

s

Figure 125: The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over all Ds tag modes for one data-sized Monte Carlo sample (dataset 0). The bottom two plots give the projections for a different data-sized Monte Carlo sample (dataset 1).

238

ση for all modes, dataset 2 Counts / 0.25 σ

Counts / 1.3 MeV

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All recon Ds + !ηeν

40

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30

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40 30

20

20 10 10 0 1900

1920

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2000

0

2020 MD (MeV)

-4

-2

0

2

4 ση

s

Figure 126: The top two plots show the projections from our 2D ηeν fit to Ds invariant mass (left) and η pull mass (right) over all Ds tag modes for another data-sized Monte Carlo sample (dataset 2). The bottom two plots give the projections for a fourth data-sized Monte Carlo sample (dataset 3).

239

Counts / 4.0 MeV

K s found, p

recoil

χ2 / ndf Nsig µ σ NBG

> 650 MeV

400 350

79.41 / 58 1.295e+04 ± 2.198e+02 497.7 ± 0.3 17.47 ± 0.30 0.2604 ± 0.0195

300 250 200 150 100 50 0300

350

400

450

500

550

600

650

D s+ γ +K

Mrecoil

Counts / 4.0 MeV

K s not found, p

recoil

χ2 / ndf Nsig µ σ y

> 650 MeV

1400

70 / 56 4.383e+04 ± 4.968e+02 497.7 ± 0.2 17.64 ± 0.23 66.5 ± 4.2 0.5501 ± 0.0403 0.7544 ± 0.1125

400

1200

700 (MeV)

m NK η

1000 800 600 400 200 0300

350

400

450

500

550

600

650

D s+ γ +K

Mrecoil

700 (MeV)

Figure 127: Ds + γ + K recoil mass in KS K events for “found” and “not found” Ks , from the Monte Carlo.

240

180

m

Counts / 4.0 MeV

240 220

400

52.91 / 54 783.9 ± 29.6 497.9 ± 0.3 15.28 ± 0.66 0.2216 ± 0.0564 -0.2059 ± 0.6255 0.01804 ± 0.00309

160

χ 2 / ndf K s not found, precoil ∈ (200 MeV, 400 MeV) Counts / 4.0 MeV

200

χ 2 / ndf Nsig µ σ N!γ y

K s found, precoil ∈ (200 MeV, 400 MeV)

Nsig µ σ N!γ y

1000

400

m

68.56 / 56 3095 ± 86.0 497.8 ± 0.1 16.29 ± 0.50 0.5255 ± 0.0619 49.91 ± 4.56 0.2699 ± 0.0183

800

140 600

120 100

400

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200

40 20 350

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550

Counts / 4.0 MeV

K s found, precoil ∈ (400 MeV, 650 MeV) 250

600

χ 2 / ndf Nsig µ σ N!γ y 400

m

200

0 300

650 700 D s+ γ +K π π (MeV) Mrecoil

37.16 / 56 1281 ± 42.3 497.2 ± 0.3 21.03 ± 0.73 0.5995 ± 0.0954 0.5662 ± 1.0040 0.01645 ± 0.00460

350

400

450

500

550

600

χ 2 / ndf K s not found, precoil ∈ (400 MeV, 650 MeV) Counts / 4.0 MeV

0 300

Nsig µ σ N!γ y

1200

400

1000

m

650 700 D s+ γ +K π π (MeV) Mrecoil

75.7 / 56 5207 ± 127.1 497.2 ± 0.2 23.7 ± 0.6 0.3169 ± 0.0824 114.5 ± 5.9 0.1633 ± 0.0218

800

150 600 100 400 50

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600

0 300

650 700 D s+ γ +K π π Mrecoil (MeV)

350

400

450

500

550

600

650 700 D s+ γ +K π π Mrecoil (MeV)

Figure 128: Ds +γ +Kππ recoil mass in K ∗ K ∗ events for “found” and “not found” Ks , from the Monte Carlo. The top row corresponds to low momentum Ks while the bottom row corresponds to our medium Ks momentum region, as determined by the recoil momentum.

241

χ 2 / ndf Nsig Ntagged BG Ncomb BG

Counts / 4.0 MeV

MK*, No signal smear or shift 40

82.64 / 50 0.04375 ± 0.00261 0.0101 ± 1.2661 0.0841 ± 0.0920

35 30 25 20 15 10 5 0 700

750

800

850

900

950

χ 2 / ndf Nsig Ntagged BG Ncomb BG

MK*, Best fit (5.6 MeV smear, -0.8 MeV shift) Counts / 4.0 MeV

1000

40

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1100 MK* (MeV)

75.6 / 50 0.04501 ± 0.00248 2.771e-12 ± 2.807e-01 0.06375 ± 0.07198

35 30 25 20 15 10 5 0 700

750

800

850

900

950

1000

1050

1100 MK* (MeV)

Figure 129: Top: Signal shape fit to the K ∗ mass in K ∗ K. Bottom: K ∗ mass fit after allowing the MK ∗ signal shape to shift left or right and convoluting it with a variable width gaussian.

242

χ 2 / ndf Nsig Ntagged BG Ncomb BG

Mf , No signal smear or shift Counts / 6.5 MeV

0

60

135.3 / 66 0.05117 ± 0.00531 0.07774 ± 0.01532 0.01811 ± 0.02162

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1300 Mf (MeV) 0

χ 2 / ndf Nsig Ntagged BG Ncomb BG

Mf , Best fit (3.9 MeV smear, -6.5 MeV shift) Counts / 6.5 MeV

0

60

130.4 / 66 0.05286 ± 0.00534 0.08203 ± 0.01510 0.01109 ± 0.02132

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1300 Mf (MeV) 0

Figure 130: Top: Signal shape fit to the f0 mass in f0 π. Bottom: f0 mass fit after allowing the Mf0 signal shape to shift left or right and convoluting it with a variable width gaussian.

243

χ2 / ndf Nsig

Counts / 7.0 MeV

Mη’, No signal smear or shift 22

15.1 / 11 0.04659 ± 0.00577

20 18 16 14 12 10 8 6 4 2 0

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χ2 / ndf Nsig

Mη’, Best fit (8.4 MeV smear, +2.8 MeV shift) Counts / 7.0 MeV

1000

22

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11.52 / 11 0.04983 ± 0.00601

20 18 16 14 12 10 8 6 4 2 0

900

920

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1000

1020 Mη ’ (MeV)

Figure 131: Top: Signal shape fit to the η 0 mass in πη 0 , η 0 → ππη. Bottom: η 0 mass fit after allowing the Mη0 signal shape to shift left or right and convoluting it with a variable width gaussian.

244

plab

lab

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η’

Fraction / 15 MeV

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Figure 132: Top: Lab frame electron energy (left) and η 0 momentum (right) in Ds → η 0 eν for the ISGW2 and pole models. Bottom: Lab frame electron energy and f0 momentum in Ds → f0 eν.

245

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lab

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ISGW2 ISGW1 Simple pole MC

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Figure 133: Top: Lab frame electron energy (left) and Ks momentum (right) in Ds → KS eν for the ISGW2 and pole models. Bottom: Lab frame electron energy and K ∗ momentum in Ds → K ∗ eν.

246

Fraction / 12100 MeV

q2, Ds → φ e ν ISGW2 ISGW1 Simple pole MC

0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002

3

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0

× 10 0

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800

1000

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1400

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Figure 134: q 2 distribution under different models for decays with a vector hadron (Ds → φeν) and a pseudoscalar hadron (Ds → ηeν). The difference between the Monte Carlo and ISGW2 for low q 2 in φeν comes from a correction we make to the Monte Carlo’s masses.

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q2 vs. Ee, Ds → φeν , ISGW2 3 1200

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q2 vs. Ee, Ds → φeν , Pole 3 1200

× 10

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800

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400

200

0

0

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Figure 135: Relationship between the q 2 and electron energy in the Ds rest frame for Ds → φeν. Top: ISGW2 model. Bottom: Pole model.

248

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q2 vs. Ee, Ds → ηeν , Pole 3 × 10

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Figure 136: Relationship between the q 2 and electron energy in the Ds rest frame for Ds → ηeν. Top: ISGW2 model. Bottom: Pole model.

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200 0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

5000 4000 3000

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 M Ds (MeV)

MD , π η’, η’ -> ρ γ

s

s

Counts / 1.0 MeV

Counts / 1.0 MeV

6000

MD , π η’, η’ -> π π η Counts / 1.0 MeV

0

MD s , π π η

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

2000 1800 1600 1400

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1200 3000

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200 0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 M Ds (MeV)

Figure 137: Ds invariant mass fits after making a Ds + γ recoil mass cut in the Monte Carlo. The dotted blue and pink lines give our signal and background fit functions, while our solid blue and pink lines give the truth-tagged signal and background.

250

MD , K K π

350 300

2000

1500

250

0

s

2500

Counts / 1.0 MeV

400

MD , K K π π

s

Counts / 1.0 MeV

Counts / 1.0 MeV

MD s , Ks K

2000 1800 1600 1400 1200 1000

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50

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0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

s

500 400

MD , π η

MD s , π π π Counts / 1.0 MeV

Counts / 1.0 MeV

s

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 M Ds (MeV)

s

Counts / 1.0 MeV

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MD , K K π π

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

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200 180 160 140 120

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100 80 60 40 20

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

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MD , π η’, η’ -> π π η

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 M Ds (MeV)

MD , π η’, η’ -> ρ γ

s

s

Counts / 1.0 MeV

Counts / 1.0 MeV

600

Counts / 1.0 MeV

0

MD s , π π η

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

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0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

300

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 MDs (MeV)

0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 M Ds (MeV)

Figure 138: Ds invariant mass fits after making a Ds + γ recoil mass cut in the data. The dotted blue and pink lines give our signal and background fit functions.

251

MD , K K π

MD s , K K π π0

s

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4500

Counts / 1.3 MeV

Counts / 1.3 MeV

Counts / 1.3 MeV

MD s , Ks K

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MD , K K π π

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2020 MDs (MeV)

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2020 MDs (MeV)

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2020 MDs (MeV)

MD , π η

MD s , π π π

s

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Counts / 1.3 MeV

Counts / 1.3 MeV

s

0 1900

2020 MDs (MeV)

s

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Counts / 1.3 MeV

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MD , π η’, η’ -> ρ γ

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2020 MDs (MeV)

MD , π η’, η’ -> π π η Counts / 1.3 MeV

Counts / 1.3 MeV

0 1900

2020 MDs (MeV)

MD s , π π0 η

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Counts / 1.3 MeV

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Figure 139: Ds invariant mass fits using a gaussian+crystal ball signal shape and a linear background fit function.

252

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