Chapter 3 Thermal Distributions, Saha Equation, Weak Interactions [PDF]

Chapter 3 Thermal Distributions, Saha Equation, Weak Interactions This chapter deals with some background issues important to modeling the BBN as well as stellar reactions: • thermal distributions • the Saha equation • low-energy weak interactions 3.1 Thermal distributions The thermal distributions appropriate for fermions and boson are respectively a) Fermi-Dirac distribution gs ns = (s −µ)/kT e +1 b) Bose-Einstein gs ns = (s −µ)/kT where s − µ is positive e −1 and a classical limit of these distributions that can be applied in many cases is c) Maxwell Boltzmann distribution ns =

gs ( −µ)/kT s e

(1)

(2)

(3)

√ The particle energy s above is m + p~2 + m2 . (Often the rest mass term m is omitted because it can be absorbed into the chemical potential. But one should include it explicitly when nuclear binding energies have to be considered: the Saha equation discussion will illustrate this.) The parameter µ, the chemical potential, is determined for fixed T and particle density: an example will be done below. 3.1.1 Fermi-Dirac Distribution The Fermi-Dirac distribution comes from the continuum limit of indistinguishable particles occupying all possible states, subject to one particle per state, keeping the average particle number and energy fixed. The usual custom is to write µ = F . As kT → 0, (

gs e(s −F )/kT

+1

→

0 if s ≥ F gs if s ≤ F

(4)

Thus F is often called the Fermi level, as it divides the low-energy completely occupied levels from the higher energy completely unoccupied levels. Of course, at finite temperatures, this demarcation is not sharp.

1

ron model

unction

thermally excited to higher energy empty states within kBT in energy kBT

F(E) 1

BT

e

E Fermi energy, EF

22

Figure 1: Fermi distribution occupation probabilities.

2

Maxwell Boltzmann

Bose-Einstein Distribution

Fermi Distribution

Figure 2: A comparison of the Fermi, Maxwell-Boltzmann, and Bose-Einstein distributions.

3

We can integrate over some finite, uniform volume V to count the total number of contained fermions V Z g˜s (5) No = 3 d3 q (− )/kT h e F +1 In this expression (, ~k) is the particle four-momentum and g˜s represents the remaining degeneracy of the quantum level of energy , e.g., perhaps the spin and isospin degeneracy. The degeneracy due to momentum d3 q = 4πq 2 dq where  =

q

q 2 + m2 → m +

q2 if m  q 2m

(6)

is included explicitly in the integral. Setting h ¯ = 1 = k we thus obtain for the number density Z d3 q 1 1 Z q2 n = g˜s = g ˜ (7) dq s (2π)3 e(−F )/T + 1 2π 2 e(−F )/T + 1 This can be converted into an integral over energy by noting that d = qdq √  2 − m2 1 Z n = g˜s 2 d (− )/T 2π e F +1

(8)

In the limit T → 0, n = g˜s

√ 1 Z F 1 2 2 − m2 = g d  ( − m2 )3/2 ⇒  ˜ s 2π 2 m 6π 2 F 

6π 2 n 2 2  F (n, T = 0) = m + g˜s

!2/3 1/2 

(9)

or in the nonrelativistic limit 1 F (n, T = 0) = m + 2m

6π 2 n g˜s

!2/3

(10)

So Eqs. (9) and (10) define F in terms of n at zero temperature. Similarly, one can return to Eq. (5) and invert (numerically) at any T, to determine the full F (n, T ). F is a slowly varying function of kT . A picture of N (), the number of particles of energy , is sketched on the following page. The region around the Fermi surface gradually “softens” as kT is increased, in accordance with the naive expectation that particles with kT of the Fermi surface ought to be occasionally excited above the Fermi surface. BBN applications: In BBN applications for species like neutrinos and electrons/positrons the relativity is important, and in fact the relativistic limit is often appropriate. We also need to specific the chemical potentials appropriate for that application. In standard cosmology one takes µi = 0, i = {p, n, e− , e+ , νx , ν¯x } (11) 4

In the case of the baryons, the nearly complete annihilation of baryons and antibaryons, leading to the very small value of η, guarantees that µ  T and thus is ignorable. Chemical potentials are additively conserved in all reactions. Thus e+ + e− ↔ γ + γ ⇒ µe− + µe+ = 0

(12)

as photons are bosons that can be emitted into any state, so µγ = 0. (This is also an experimental fact, as we have measure the black-body spectrum precisely.) The fact that the universe is seen to have no net (local) charge then requires µe− = µe+ = 0. The case of neutrinos is more interesting. We have not detected the neutrinos, so we do not have an experimental test. Because of reactions at early times such as e− + e+ ↔ νe + ν¯e ⇒ µe− + µe+ = µνe + µν¯e

(13)

The standard assumption that µνe = µν¯e = 0 certainly satisfies this condition, but is not required. There could be an asymmetry in νe and ν¯e . BBN is in fact one of the important constraints on such an asymmetry, and an asymmetry has been invoked as one of many possible “dials” that might be turned to improve the BBN 7 Li discrepancy. In any case, for zero chemical potential and a system in equilibrium we have Z 1 1 3 d3 q d3 q √ = g˜s 2 ζ(3)T 3 ≡ nFU D −→ g ˜ n (µ = 0) = g˜s s R (µ = 0) 3 3 q/T 2 2 (2π) e q +m /T + 1 (2π) e + 1 4π (14) where the results to the right of the arrow are the ultra-relativistic limit we evaluated previously. Similarly the energy density and its ultra-relativistic limit are √ 2 Z Z d3 q q + m2 q 7π 2 4 d3 q FD √ −→ g ˜ = g ˜ T ≡ ρFU D ρ (µ = 0) = g˜s s s R (µ = 0) (2π)3 e q2 +m2 /T + 1 (2π)3 eq/T + 1 240 (15) We can also calculate the pressure. Recall the standard calculation in one dimension, of particles bouncing off a wall. The pressure is the force/unit area, which is given by FD

Z

q2 qx = xn (16) E E where we have noted that half of the particle within any δx of the wall are moving to the left, and that the relativistic velocity is qx /E. In 3D hqx2 i = q 2 /3. Consequently for a system in equilibrium p = (2qx )(n/2)v = qx n

g˜s Z d3 q 1 q2 q g˜s Z d3 q 1 √ √ −→ = ρFU D (µ = 0) 3 3 q/T 2 2 2 2 3 (2π) q + m e q +m /T + 1 3 (2π) e + 1 3 R (17) 3 where again the results on the right are the ultra-relativistic limit. The entropy S(R (t), T ) in a volume R3 (t) is (up to a constant) pF D (µ = 0) =

S(R3 (t), T ) =

R3 (t) (ρ(T ) + p(T )) T 5

(18)

and S is constant if the particles in equilibrium interact only with themselves, so that the energy is conserved. We can the expressions for ρ and p to obtain S

FD

! Z ∞ q 1 g˜s y2 3 2 2 2 √ (R, T ) = y +x + √ 2 (RT ) dy y 2 2 2 2π 3 y + x e y +x2 + 1 0   7π 2 3 FD m (RT ) χ ≡ g˜s 180 T

(19)

where FD

χ

! q 90 Z ∞ y2 1 2 2 2 √ dy y and x ≡ m/T (x) ≡ 4 y +x + √ 2 7π 0 3 y + x2 e y2 +x2 + 1

(20)

In the UR limit, x → 0 and χF D (x) → 1, yielding S F D (R, T ) → g˜s

7π 2 (RT )3 . 180

(21)

3.1.2 Bose-Einstein Distribution In the case of a Bose-Einstein distribution with zero chemical potential and a system in equilibrium one finds Z 1 1 1 d3 q d3 q √ = g˜s 2 ζ(3)T 3 ≡ nBE −→ g ˜ s U R (µ = 0) 3 3 q/T 2 2 (2π) e q +m /T − 1 (2π) e − 1 π (22) Similarly the energy density and its ultra-relativistic limit are √ 2 Z Z q + m2 d3 q π2 4 d3 q q BE √ −→ g ˜ ρ (µ = 0) = g˜s = g ˜ T ≡ ρBE s s U R (µ = 0) (2π)3 e q2 +m2 /T − 1 (2π)3 eq/T − 1 30 (23) The pressure in equilibrium is

nBE (µ = 0) = g˜s

Z

g˜s Z d3 q q2 1 g˜s Z d3 q 1 q √ 2 √ −→ = ρBE (µ = 0) p (µ = 0) = 3 3 q/T 2 2 2 3 (2π) q + m e q +m /T − 1 3 (2π) e − 1 3 UR (24) If the particles in equilibrium interact only with themselves, so that the energy is conserved, then the entropy is conversed. We can the expressions for ρ and p to obtain BE

S

BE

! Z ∞ q 2 g˜s y 1 3 2 √ (R, T ) = y 2 + x2 + √ 2 (RT ) dy y 2 2 2 2π 3 y + x e y +x2 − 1 0   2π 2 m ≡ g˜s (RT )3 χBE 45 T

where BE

χ

! q 45 Z ∞ y2 1 2 2 2 √ (x) ≡ 4 dy y y +x + √ 2 2 4π 0 3 y + x e y2 +x2 − 1

6

(25)

(26)

In the UR limit, x → 0 and χBE (x) → 1, yielding S BE (R, T ) → g˜s

2π 2 (RT )3 . 45

(27)

We used the Bose-Einstein distribution, which describes the distribution of identical bosons, in describing the BBN photon gas. It has additional astrophysics applications in matters such as pion and kaon condensation in dense nuclear matter, etc. With these results we can return to our calculation of the reheating of photons by electronpositron annihilation, but with the time ta now being any time after tb , not necessarily when the annihilation is complete. Chapter (2), Eq. (13), for the neutrino expansion remains as before Chapter 2, Eq.(13) (28) R(tb )Tνb = R(ta )Tνa . But Eq. (14) – equating the entropy before (UR) and after for interacting photons, electrons, and positrons – becomes "

(R(tb )Tνb )3

2 × 2π 2 4 × 7π 2 F D me 2 × 2π 2 4 × 7π 2 + = (R(ta )Tγa )3 + χ 45 180 45 180 Tγ #

"

R(tb )Tνb

=

R(ta )Tγa



4 11

1/3 "

7 me 1 + χF D 4 Tγa

!#

⇒

!#1/3

(29)

Combining the two equations to solve for Tνa /Tγa yields 4 Tν = Tγ 11 

1/3 "

7 me 1 + χF D 4 Tγa

!#1/3

(30)

The resulting Tγ /Tν is plotted as a function of temperatures (units of 109 K) in Fig. 3. 3.1.3 Boltzmann Distribution The Maxwell-Boltzmann distribution describes the behavior of identical, distinguishable particles and can be thought of as the classical limit of Fermi-Dirac statistics, where quantum effects associated with exchange are unimportant. The common situation we will encounter is when the density is low (so that F goes to 0) and the particles are nonrelativistic. Then s /kT is a large number, and the Fermi-Dirac distribution goes over to the Maxwell-Boltzmann distribution. We used this result in the big bang discussion, where these two conditions are met. Some typical uses of the Maxwell-Boltzmann distribution in astrophysics: • Describing the occupation of levels in well-isolated atoms. This is appropriate when quantum effects due to electrons in the plasma and due to other atoms are unimportant. • Describing molecular excitations, such as rotations.

7

1.4

1.3

Tγ/Tν 1.2

1.1

5

10

15

20

Temperature (units of T9)

Figure 3: The calculated ratio of the photon to neutrino temperature, as a function of the photon temperature in units of 109 K. Thus time runs from the right to the left. Note that at T9 ∼ 10, or about 0.86 MeV, the ratio is essentially one. This is well past the time of weak decoupling, so our approximation of treating the neutrinos and photons/electrons/positrons as decoupled at the start of the process is an excellent one. Note that the final ratio of about 1.4 is achieved around T9 ∼ 1 (86 keV), which is near the time of nucleosynthesis. As the photons interact with protons, the photon temperature is the relevant one for BBN.

8

As an example, consider a two-level atom, that is, one with a ground state (which we will take to be 1s1/2 ) and an excited state 1p3/2 . The MB weights are, respectively, 2e−gs /kT and 4e−ex /kT

(31)

Thus the population of the excited state is 2e−(ex −gs )/kT 1 + 2e−(ex −gs )/kT

(32)

The result we will use frequently is the Maxwell-Boltzmann velocity distribution law, which comes immediately from N1 (~v1 )d~v1 = N1 (

m1 3/2 −m1 v12 /2kT ) e d~v1 2πkT

(33)

3.2 Saha equation Let’s consider a problem discussed previously n+p↔d+γ

(34)

During the period of interest to us in BBN these nuclear species are nonrelativistic and nondegenerate - a dilute gas that can be accurately described by Maxwell-Boltzmann statistics. The nuclear species n, p, and d are in thermal equilibrium. Previously we studied the detailed balance - primarily to illustrates the role of the high-energy tail of the photon distribution. Here we do things more correctly, using that we know statistical equilibrium holds. We have three nuclear species and a partition function for each one, e.g., Zp ∼

X

e(µp −E(n))/kT

(35)

n

We can write the probability function, noting each species is a set of indistinguishable particles Z Np ZnNn ZdNd S(Np , Nn , Nd ) = p (36) Np ! Nn ! Nd ! The Zs are given by V Z [µp −mp −p2 /(2mp )]/(kT ) 3 gp e dp h3 The integral can be done, yielding Zp =

Zp =

gp V [µp −mp ]/(kT ) e (2πmp kT )3/2 3 h

(37)

(38)

Similarly Zn =

gn V [µn −mn ]/(kT ) e (2πmp kT )3/2 3 h

Zd = 9

gd V [µd −md ]/(kT ) e (2πmd kT )3/2 3 h

(39)

Now we want to find the most probable state, which maximizes S(Np , Nn , Nd ). We note that ln(S) will have the same maximum as S. And ln(n!) ∼ nln(n) − n for large n, by Stirling’s formula. So lnS = Np lnZp + Nn lnZn + Nd lnZd − Np lnNp + Np − Nn lnNn + Nn − Nd lnNd + Nd

(40)

Now let NpT and NnT be the total number of protons and neutrons, regardless of whether they are free or bound. These numbers are constant (integrated over all volume) and Nd = NpT − Np ;

Nd = NnT − Nn ⇒ Nn = NnT − NpT + Np

(41)

That is, we can take Np as our one variable with lnS = Np lnZp + (NnT − NpT + Np )lnZn + (NpT − Np )lnZd − Np lnNp + Np + NnT −(NnT − NpT + Np )ln(NnT − NpT + Np ) − (NpT − Np )ln(NpT − Np ) Thus d(lnS) ∼ lnZp + lnZn − lnZd − lnNp − ln(NnT − NpT + Np ) + ln(NpT − Np ) = 0 d(Np )

(42)

(43)

so that at maximum probability Zp Zn Np Nn Ad 3/2 h3 gd = ⇒ Nd = Np Nn ( ) (2πmN kT )−3/2 e[µd −µn −µp −md +mn +mp ]/(kT ) Zd Nd V gn gp Ap An (44) Here we have taken mp ∼ Ap mN , mn ∼ An mN , and md ∼ Ad mN in the mass ratio, where Ad = 2, Ap = 1, and An = 1 are the atomic numbers of the three species. Converting to number densities (divide by V) nd = h3 np nn

gd Ad 3/2 ( ) (2πmN kT )−3/2 e[µd −md −µp +mp −µn +mn ]/(kT ) gp gn An Ap

(45)

Now at equilibrium µd − µp − µn = 0: the chemical potential is defined as the change in the system energy on adding a particle. Since n + p is in equilibrium with d, the change in energy on adding a neutron and a proton is the change in energy on adding a deuteron. Also the deuteron binding energy – this is defined as a positive quantity – is Bd = mp + mn − md . Finally we define the mass fractions by Xd =

Ad nd nN

XpF =

A p np nN

XnF =

An nn nN

(46)

where the superscript F denotes these are the free p/n mass fractions and where nN = (NpT +NnT )/V is the total number density of nucleons, free or bound. Note Xd +XpF +XnF = 1. It follows Ad 5/2 gd ( ) (2πmN kT )−3/2 eBd /(kT ) Xd = XpF XnF nN h3 gp gn Ap An gd Ad 5/2 3/2 = XpF XnF η ( ) T9 0.337 × 10−5 e25.83/T9 (47) gp gn Ap An 10

where we have used our early result for the photon number density to rewrite this in terms of the baryon/photon ratio η. T9 is the temperature in units of 109 degree Kelvin. (We have left the As and gs in to the end so that this formula can be used for any 1 + 2 ↔ 3 reaction.) Now gd =3 (the deuteron ground state has J=1), gp =2, gn =2 so 3/2

Xd = XpF XnF ηT9 1.43 × 10−5 e25.83/T9

(48)

So let’s solve this for the temperature of deuterium formation. As before, that is defined when half the neutrons are bound. In terms of mass fractions this means the free neutron mass fraction 2XnF = Xd . We also know n/p = 1/7 at freezeout, which means Xd /(XnF + XpF ) = 1/7. And XnF + XpF + Xd = 1. All of this yields XnF =

1 16

XpF =

13 16

Xd =

1 8

(49)

So that 3/2

1.72 × 105 = ηT9 e25.83/T9

(50)

which relates η and Td . We find for η = 10−9 that Td = 0.779 and for η = 10−10 Td = 0.726, in units of 109 Kelvin. We employed this result without derivation previously. The Saha equation also be used to determine the time that atoms formed. There is a slight complication: there are multiple bound states. An interesting exercise is to consider two bound states, after first considering only one, to see the pattern. What is the temperature of recombination, defined as the time when half of the electrons are bound in neutral hydrogen? 3.3 Low-energy weak interactions In nuclear beta decay a parent nucleus decays to a daughter with the same atomic mass, but an atomic number changed by one unit, with the missing charge carried off by an electron or positron ( (A, Z + 1) + e− + ν¯e (51) (A, Z) → (A, Z − 1) + e+ + νe The best known example is the decay of a free neutron into a proton and electron, with a half life of about 10 minutes. Another is the decay of a bound neutron in tritium to produce an electron and 3 He with a half life of 12.26 years: the effects of the nuclear binding in changing the energy released in the decay is responsible for the great increase in the half life. At the end of the 1920’s the existence of such beta emitters was well established, and it was obsereved the spectrum of the emitted electrons was continuous, rather than monochromatic. This led Pauli to propose that a new unobserved spin-1/2 neutral particle, the neutrino, was co-produced with the outgoing electron/positron. In this way the decay energy, Q ' M (A, Z) − M (A, Z ± 1) would be shared between the two outgoing leptons.

11

(52)

e

e GF

1

2

a)

1

2

b)

Figure 4: Fermi’s four-fermion interaction description of weak interactions (b), built in analogy with electromagnetism (a) apart from the replacement of the electric field in the latter by a point-like interaction. Pauli’s proposal was made in 1930 (though not published or discussed publicly for several more years), and in 1932 Chadwick discovered the neutron. Fermi was present at a number of Pauli’s presentations and discussed the neutrino with him on these occasions. In 1934 he published his insightful model for the beta decay process, and indeed for weak interactions in general. He described beta decay in analogy with Dirac’s successful model of the electromagnetic interaction, wherein two charged particles interact via the exchange of a (virtual) photon that is produced and then absorbed by the electromagnetic currents associated with the particles (see Fig. 4). Fermi represented the weak interaction in terms of the product of weak “currents,” one connecting the initial and final nucleon and the other connecting the final state electron/positron and neutrino. In electromagnetism the virtual photon connects the two currents at distinct points in space-time: indeed the masslessness of the photon is the reason for the long-range Coulomb force. In his weak interaction theory, however, Fermi connected the currents at the space-time point, in effect assuming that the weak interaction is very short ranged. The strength of the interaction was determined by an overall coupling strength GF GF (53) Hw = √ ψp† jµ ψn ψe† j µ ψν 2 One weak current is associated with the conversion of a neutron into a proton, and the other with the production out of the vacuum of an electron and antineutrino, which carry off almost all of the released energy. The electron spectrum predicted by this weak Hamiltonian can be readily estimated dimensionally by using Fermi’s golden rule for the differential decay rate, yielding in the no-nuclear-recoil approximation dΓ ∼

d3 pe d3 pν 2πδ(Q − Ee − Eν )|Hw |2 (2π)3 (2π)3 12

∼

q G2 dE E p (Q − Ee )2 − m2ν (Q − Ee )|Mw |2 e e e 2π 3

(54)

If one evaluates this at low energies and Taking mν = 0 and assuming |Mw |2 ∼ constant one finds dΓ ∼ pe Ee (Q − Ee )2 (55) dEe where pe and Ee are the momentum and energy of the electron. Fermi’s interaction accounted for the continuous electron spectra observed in experiment. From the 886 second lifetime of the neutron one finds that Γn = h ¯ /τn ∼ 7.4 × 10−28 GeV, while Fermi’s interaction gives Γn

G2F Z mn −mp ∼ dEe Ee pe (mn − mp − Ee )2 |Mw |2 2π 3 me ' 1.34 × 10−18 GeV5 G2F |Mw |2

(56)

2 As |M GF ∼ 10−5 GeV−2 . The subsequent standard model identifies √ w | ∼ 5.7, one finds GF / 2 = (1/8)(gw /MW )2 with MW ∼ 80 GeV and gw ∼ 0.66. The propagator mass corresponds to a distance scale ∼ 0.002 f, pointlike on the scale of either the nucleon size (∼ f) or the momentum scale of typical beta decays (q −1 ∼ 100f ).

The field operator ψq in the four-fermion interaction contains an annihilation operator for generic particle q as well as a creation operator for the corresponding antiparticle q¯. Thus the semi-leptonic interaction of Eq. (53) describes, in addition to neutron β decay and its generalization to nuclei, i) electron capture, where an atomic electron orbiting the nucleus interacts with one of the nuclear protons, converting it to a neutron and producing an outgoing neutrino, e− + p → n + ν. (An analogous process for positrons, e+ + n → p + ν¯, is important in the hot plasmas encountered in the big bang and in explosive stellar environments.) ii) The charged current neutrino reactions ν + n → p + e− and ν¯ + p → n + e+ (which are the inverses of the reactions in i), but are often referred to as “inverse beta decay”). iii) The exotic resonant reactions ν¯ + e− + p → n and ν + e+ + n → p, the true inverse reactions of beta decay. The first can occur in an atom; both can take place in astrophysical plasmas. Reaction ii) is the one relevant for neutrino detection, e.g., with detection of the produced electron in a water Cerenkov detector. The cross section (again ignoring nuclear recoil) is G2F GF Z d3 pe 2 2πδ(m + E − m − E )|M | ∼ pe Ee |Mw |2 σ ∼ | √ |2 p ν n e w (2π)3 2π 2

(57)

For incident neutrino energies large compared to the electron rest mass this becomes σ ∼ G2F Eν2 |Mw |2 /2π ∼ 3.7 × 10−42 cm2 for Eν ∼ 10 MeV. Fred Reines and Clyde Cowan were 13

the first to detect the neutrino, some 25 years after Pauli’s suggestion, using the intense ν¯e flux from the Savannah River reactor, in South Carolina, of about 1013 /cm2 s. Their detector consisted of two plastic tanks each filled with 200 liters of water, in which was dissolved cadmium chloride. The protons in the water provided the target for the reaction ν¯ + p → n + e+ , while the cadmium has a large cross section for neutron capture. The tanks were sandwiched between three scintillation detectors. The group looked for a signal consisting of gamma rays from the annihilation of the emitted positron on an electron followed closely (within a few microseconds) by gammas from the deexcitation of the cadium nucleus that had captured the neutron. The observed signal was correlated with the reactor being in operation. They announced in 1956 that the neutrino had been detected. In 1996 Reines was awarded the Nobel Prize for this discovery. (Clyde Cowan had died years earlier.) A second Nobel Prize was awarded to Lederman, Schwartz, and Steinberger for demonstrating the existence of the νµ . The muon The muon decays by a purely leptonic process with a mean lifetime of ∼ 2.2 × 10−6 s, µ− → e− + νµ + ν¯e .

(58)

This decay conserves charge, total lepton number (family number), and electron/muon flavor, under the standard additive conservation laws with the assignments particle e− e+ νe ν¯ µ− µ+ νµ ν¯µ

` `e `µ +1 +1 0 −1 −1 0 +1 +1 0 −1 −1 0 +1 0 +1 −1 0 −1 +1 0 +1 −1 0 −1

Dimensionally one can estimate the lifetime 1 GF Z d3 pe d3 pν d3 pν¯ (2π)4 δ 4 (pµ − pe − pν − pν¯ ) ∼ | √ |2 τµ (2π)3 (2π)3 (2π)3 2 G2F m5µ ∼ i.e. τµ ∼ 2.5 × 10−5 sec 3 1636π

Γµ =

(59)

Muon decay fits easily into Fermi’s interaction provided the lepton current is generalized to Jµlep = ψe† jµ ψνe + ψµ† jµ ψνµ

(60)

Then the product of the lepton current with its hermitian conjugate yields an interaction responsible for muon decay. This generalization thus yields a lepton current-lepton current 14

interaction to our previous lepton-hadron current interaction for β decay, and also extends the interaction to two flavors. It is assumed that the overall coupling GF is universal. More correctly, the universality holds at the quark level, and only after insertion of the appropriate quark mixing matrix. The neutron is then described as a composite of a pair of d quarks and a u quark, while the proton consists of a pair of u quarks and a d quark. The weak current connecting the proton and neutron can then be replaced by a corresponding current connecting a u quark with a d quark, Jµhad = ψn† jµ ψp → ψd† jµ ψu

(61)

As the field operator ψd† can both create a d quark and destroy the corresponding antiparticle ¯ this same current describes the quark component of the process where a du ¯ system (i.e. (d), + a π meson) decays to a muon and a neutrino π + → µ+ + ν

(62)

which is the dominant decay mode of the charged pion. Experimenters collided neutrinos from such decays with neutrons in an attempt to produce electrons and muons, as predicted by the current of Eq. 60. But they found only muons, not electrons. The explanation for this result is that neutrinos come in two distinct species, an electron type νe and a muon type νµ , with the weak current coupling electrons only to νe and muons only to νµ as in Eq. (60). The neutrino produced in pion decay thus must be a νµ and of the wrong type, or “flavor,” to produce an electron. The third charged lepton, the τ , was discovered in 1977, adding another term to this equation—the coupling of the τ to its neutrino, the ντ . Measurements of the decay width of the neutral Z-boson and astrophysical arguments based on the helium abundance in the universe suggest that this may exhaust the set of lepton-neutrino pairs: there appear to be no more light neutrinos beyond the ντ . The modern picture of the weak interaction consists not only of three doublets of charged lepton-neutrino pairs but also of three doublets (often called “generations”) of charge 2/3, charge -1/3 quarks—(u,d),(c,s),(t,b). The charged weak current then can be written as the sum of six separate currents connecting such quark and lepton doublets 



Jµ = Jµhad + Jµlep = ψd† ψs† ψb†









ψu ψνe       UKM jµ  ψc  + ψe† ψµ† ψτ† jµ  ψνµ  ψt ψντ

(63)

Low-energy weak interactions are then described by a effective current-current interaction with a single overall coupling GF GF Hw = √ Jµ† J µ (64) 2 Such a contact interaction is a good approximation at low energies to a more complete theory described in terms of the exchange of a heavy charged W-boson, as was mentioned earlier. 15

Here UKM is a general unitary 3×3 matrix, which is not needed in the case of the lepton current due to the assumption in the standard model that the three neutrinos are degenerate. Consequently Fermi’s weak interaction, in its modern guise, contains an enormous range of physical processes. In addition in 1972 an additional neutral current (diagonal in quark and lepton identities) was discovered. An effective low-energy Hamiltonian for neutral-current weak processes is obtained by coupling this current to its hermitian conjugate. 



Jµ = Jµhad + Jµlep = ψe† ψµ† ψτ† 

+



ψu† ψc† ψt†











ψνe ψe     0  † † 0  † jµ  ψµ  + ψνe ψνµ ψντ jµ  ψνµ  ψντ ψτ







ψu ψd     † † † 0  jµ0   ψc  + ψd ψs ψb jµ  ψs  ψt ψb

(65)

with

GF Hw ∼ √ Jµ† J µ (66) 2 In this case the interaction arises from the exchange of a heavy neutral particle—the Z-boson with mass mZ ∼ 91 GeV—and thus can again be taken to be of contact form for low-energy reactions. Thus the inventory of interactions are leptonic, semi-leptonic, and hadronic weak interactions mediated by both charge and neutral currents. Fermi’s model for β decay, when made covariant (adding a current to the charge operator he introduced), was the correct effective charge-changing weak interaction at low energies with one exception, handedness. As both quarks and leptons are spin-1/2 objects, they can be described by four-component Dirac fields and their currents can be expressed as bilinear forms of such fields connected by Dirac matrices. Experimentally – this was first deduced by Gamow and Teller in 1936 – the proper combination is an equal mixture of polar and axial vector structures ψa† jµ ψb ≡ ψa† γ0 γµ (1 − γ5 )ψb (67) (Gamow and Teller deduced the need for comparable vector and axial vector operators, but remarkably did not consider the required amplitudes to accomplish this, or they might have discovered parity violation 20 years before Lee and Yang.) The Dirac chirality operator 1 − γ5 =

1 −1 −1 1

!

(68)

projects out, for massless particles, only “left-handed” state components—i.e., components whose spins are aligned opposite to their momenta. The Dirac equation for a free particle of mass m is (i 6 ∂ − m)ψ(x) = 0 (69) 16

where 6 ∂ = ∂0 γ0 − ∂~ · ~γ . Using the standard representation for the 4×4 Dirac matrices[?] 1 0 0 −1

γ0 =

!

0 ~σ −~σ 0

~γ =

!

(70)

and the positive energy plane wave solutions of Eq. 70, s

ψ(x) =

E+m 2E

!

χ ~ σ ·~ p χ E+m

exp(−ip · x)

(71)

where χ is a two-component Pauli spinor, then in the limit as m → 0 and E→ |~p| this becomes ! 1 χ exp(−ip · x) (72) ψ(x) −→ ψ0 (x) = √ 2 ~σ · pˆχ m→0 so that

(

(1 − γ5 )ψ0 (x) =

ψ0 (x) ~σ · pˆχ = −χ 0 ~σ · pˆχ = χ

(73)

as claimed. Thus, to the extent that neutrino masses are much smaller than the momenta at which they are typically probed, only left-hand νs and right-handed ν¯s interact through the weak interaction. This was proven in the extraordinarily elegant experiment of Goldhaber, Grodzins, and Sunyar in 1957, who exploited the reaction 152

Eu(0− ) + e− → 152 Sm∗ (1− ) + νe → 152 Sm(0+ ) + γ + νe

(74)

The clever idea behind this scheme is that one can select those γ’s from the decay of the Sm excited state which travel oppositely to the direction of the electron-capture νe ’s (i.e. in the direction of the nuclear recoil) by having them resonantly scatter from a Sm target. By angular momentum conservation the helicity of the downward-going γ is the same as that of the upward-traveling νe . The results of the experiment strongly confirmed the twocomponent hypothesis. It is interesting to note that the chirality structure of the weak current also explains why the decay of the charged pion proceeds predominantly via π + → µ+ + νµ rather than by the mode π + → e+ + νe which is strongly favored by phase space. The point is that if the positron were massless, it too would be described by a two-component theory and any such particle coupled to the weak interaction would have to be purely right-handed. Then, as diagrammed in Figure 3, the decay of a pion into a positron and a neutrino must be forbidden because angular momentum conservation prohibits the coupling of a right-handed positron and left-handed neutrino to a spinless system. Of course, in the real world the positron is light but not massless. Thus the positron decay of the charged pion is not completely forbidden but rather highly suppressed compared to its muonic counterpart— Γ(π + → e+ + νe ) = Re ≡ Γ(π + → µ+ + νµ )

me mµ

!2

17

m2π − m2e m2π − m2µ

!2

= 1.23 × 10−4

(75)

Se+ e

S

e

+

right-handed

e

left-handed

Figure 5: Schematic representation of a pion at rest decaying to a massless positron and a neutrino. Such a decay is forbidden by angular momentum conservation. which is confirmed by experiment, Reexp = (1.230 ± 0.004) × 10−4 .

One additional point should be made about neutrinos and helicity. Throughout the discussion the νe produced when a proton beta decays in a nucleus has been distinguished from the ν¯e produced in neutron beta decay. The concept of a distinct antiparticle is certainly clear for charged leptons like the electron, as its antiparticle—the positron—carries the opposite charge. More generally, particle-antiparticle conjugation reverses the signs of all of a particle’s additively conserved quantum numbers. The neutrino is immediately seen to be quite interesting then, as it lacks a charge, magnetic moment, or other measured quantum number that would necessarily reverse under such an operation—it is unique among the leptons and quarks in that the existence of a distinct antiparticle is an open question. Early on, before the handedness of the weak interaction was discovered, there appeared to be a simple test of the particle-antiparticle properties of the neutrino. If one defines the νe as the neutrino produced when a proton decays in a β + source, then one finds that νe ’s produce electrons by the reaction νe + n → p + e−

(76)

but not positrons in the analogous reaction νe + p 6→ n + e+

(77)

Similarly if we define the ν¯e as the particle produced in the β − decay of the neutron decay, then ν¯e ’s produce positrons by the reaction ν¯e + p → n + e+

(78)

ν¯e + n 6→ p + e−

(79)

but not electrons by the reaction Thus it would appear that the νe and the ν¯e are operationally distinct. In fact, the absence of the reactions in Eqs. 77,79 became apparent around 1950 from an experiment done by 18

nature, a form of natural radioactivity known as double beta decay. If, for example, the reaction in Eq. 79 were allowed, certain nuclei could undergo the second-order weak decay (A, Z) → (A, Z + 1) + e− + ν¯e → (A, Z + 2) + 2e−

(80)

where the neutrino produced in the first decay is reabsorbed by the nucleus, producing a final state with two electrons and no neutrinos. The absence of such “neutrinoless double beta decay,” which has a distinctive experimental signal because the entire energy release is carried off by the electrons, thus seemed to show that the νe and ν¯e were indeed distinct particles. This prompted the introduction of a distinguishing quantum number, lepton number. The νe and electron were assigned le = +1, the ν¯e and positron le = -1. The assumption of an additively conserved lepton number in weak interactions then allows the reactions in Eqs. 76,78, but explains the absence of the reactions in Eqs. 77,79,80. A neutrino with a distinct antineutrino is a Dirac neutrino. However the discovery of the apparent exact handedness of the weak interaction invalidates this simple conclusion. All of the results are also explained by the assignments νe → νeLH

and ν¯e → νeRH

(81)

and a weak interaction that violates parity maximally. Here RH denotes a right-handed particle and LH a left-handed one. Thus the possibility that the neutrino is its own antiparticle — a so-called Majorana neutrino — is still open. In this case a reaction like that of Eq. 80 is not forbidden by an exact additive conservation law, but rather by helicity. Therefore if a Majorana neutrino had a small mass, neutrinoless double beta decay would occur, but the decay rate would by suppressed by the small quantity 

mν Eν

2

(82)

where Eν ∼ 50 MeV is an energy characteristic of the virtual neutrino emitted and reabsorbed in the decay. Modern searches for neutrinoless double beta have established limits on half lives of ∼ 1025 y, corresponding to a Majorana neutrino mass below 1 eV. Given that the familiar charged leptons have only Dirac masses, it is natural to ask why neutrinos, which can have two kinds of masses, would then be the only massless leptons in the standard model. The absence of Dirac neutrino masses in the standard model follows from the need to have both left-hand and right-handed fields in order to construct such masses. We have noted that neutrinos interact only weakly and that weak interactions involve only left-handed components of the fields. The standard model, being very economical, has no right-handed neutrino fields and thus no Dirac neutrino masses. However, the absence of Majorana masses has a more subtle explanation. One can construct a left-handed Majorana mass with the available standard model neutrino fields, but it turns out this term is not “renormalizable,” i.e., it generates infinities in the theory. Our point-like Fermi β decay 19

theory is another example of a nonrenormalizable theory, though it works quite well in the domain of low-energy weak interactions. If we were to relegate the standard electroweak model to a similar status — that of an effective theory — Majorana mass terms could then be introduced. In effect, most extensions of the standard model do precisely that, and also generally introduce new fields such as those creating right-handed neutrinos. Thus almost all theorists, believing the standard model is incomplete and must be extended in some such ways, also believe that neutrinos have masses. Indeed, the puzzle is rather to explain why these masses are so much smaller than those of charged particles. With this background out of the way, we now move on to discussions of the specific weak interactions needed for BBN (and later for stellar burning). 3.4 Nucleon/nuclear β decay Now we want to repeat what we did above for β decay but in detail, while also adding in various effects such as blocking, Coulomb effects, etc. With the conventions I use (Bjorken and Drell) the nuclear β decay rate is dω = |M |2

d3 pN f Mn d3 pe me d3 pν mν (2π)4 δ 4 (pN i − pN f − pe − pν ) (2π)3 En (2π)3 Ee (2π)3 Eν

(83)

The invariant amplitude M is, as we discussed in the big bang section, effectively a contact interaction, because the momentum transferred between leptons and nucleons is so much smaller than the mass of the W boson. Thus it can be written GF M = cos θc √ U¯ (n)γ µ (1 − gA γ5 )U (p)U¯ (ν)γµ (1 − γ5 )V (e) 2

(84)

This amplitude must be averaged over the initial neutron spin – that inserts a factor of 1/2Ji + 1 = 1/2 – and summed over all final spins (proton, electron, neutrino). The spinors are normalized as follows X ±s

Uα (p, s)U¯β (p, s) =

p/ + m 2m

! X ±s

αβ

Vα (p, s)V¯β (p, s) =

p/ − m 2m

!

(85) αβ

The axial coupling in Eq. (84), gA , accounts for the renormalization of the axial coupling from its bare quark value of 1 by various strong interaction effects: the nucleon is a composite particle. The experimental value is 1.274. The vector coupling is not modified because the total electric charge is conserved. But the axial-vector coupling has a nontrivial relation to the underlying quark couplings. Also note that a factor correcting for the weak mixing we discussed earlier - the Cabibbo angle – has been inserted: this accounts for the fact that in β decay, u → cos θc d + sin θc s. That is, flavor changing charged-current reactions are observed. This flavor mixing is represented as a rotation: weak universality is maintained.

20

While the neutron β decay rate can now be evaluated in the usual way, with the spin sums yielding standard traces, we instead will treat the decay nonrelativistically, as this will allow us to easily generalize to β decay within a nucleus. Nuclear β decay involves momenta that are typically small compared to the inverse size of the nucleus, (1.2A1/3 f )−1 ∼ 165 MeV A−1/3 . Thus the momentum transfer to the nucleus – the effects of which are ~ encoded in the operator eik·~x – can be neglected. In this “allowed” approximation the operators in our amplitude become γµ µ=0 1 p ~ µ = 1, 2, 3 M ∼

~v c

γµ γ5 ~σ · Mp~ ∼ ~σ

v c

(86)

The space-like part of the vector operator and the time-like part of the axial-vector operator are suppressed by v/c. As bound nucleon have v/c ∼ 1/10, and as these operators require a parity change, they are also ignored in the allowed approximation. Thus it is the time-like part of the vector current – the Fermi matrix element – and the space-like part of the axialvector current – the Gamow-Teller operator – that survive in the nonrelativistic limit. It follows that our β decay invariant amplitude can be approximated by  GF  cos θc √ Φ† (n)Φ(p)U¯ (ν)γ 0 (1 − γ5 )V (e) − Φ† (n)gA~σ Φ(p) · U¯ (ν)~γ (1 − γ5 )V (e) 2

(87)

where the Φ are now two-component Pauli spinors for the nucleons: we have done a standard nonrelativistic reduction of the four-spinors. The above result is written for the β decay p → n. It is convenient to generalize it for n ↔ p by introducing the isospin operators τ± where τ+ | ni = | pi and τ− | pi = | ni, with all other matrix elements being zero. With this generalization, we can now finish the calculation. We square the invariant amplitude, integrate over the outgoing electron, neutrino, and final nuclear three-momenta, average over initial nucleon spin, and sum over final nucleon spin, electron spin, and neutrino spin. The result is ω=

G2F

 √ 1 1 ZW (W − )2  2 − m2 d |hf ||τ± ||ii|2 + gA2 |hf ||στ± ||ii|2 cos θc 3 2π m 2 2

(88)

where f and i are the final and initial nucleon states, and where m is the electron mass, W is the energy release in the decay, and  is the outgoing electron energy. The τ+ operator corresponds to β − decay and the τ− to β + decay. The matrix elements have been reduced in angular momentum, as all spins have been summed over Jf −Mf

hJf Mf |OJM |Ji Mi i ≡ (−1)

Jf J Ji −Mf M Mi

!

hJf ||OJ ||Ji i

(89)

This result easily generalizes to nuclear decay, under similar approximations, including now the neglect of two-body currents arising from, for example, the decay of charged mesons 21

being exchanged between nucleons. The operators are replaced by τ ± → ΣA i=1 τ± (i) στ± → ΣA i=1 σ(i)τ± (i)

(90)

The factor of 21 before the square of the nuclear matrix elements is replaced by 2Ji1+1 where Ji is the initial nuclear spin. Finally, the Coulomb effects on the outgoing electron or positron can be approximately accounted for by including the Coulomb factor F (Z, ) = |F0 (Z, )|2 =

2πη Zf Ze α where η = −1 β

e2πη

(91)

where β is the electron/positron velocity and F0 (Z, ) is the s-wave Coulomb wave function in the field of the daughter nucleus of charge Zf , evaluated at the nuclear origin. This corrects for the use of outgoing plane waves, by inserting the correct s-wave point-nucleus nonrelativistic Coulomb wave function probability. If Z becomes large, more care is needed, as a numerical solution of the Dirac equation for an extended nucleus is then required. Thus we obtain finally ω = G2F cos2 θc

√ 1 ZW F (Z, )(W − )2  2 − m2 d 3 2π m

A A X X 1 στ± ||ii|2 τ± ||ii|2 + gA2 |hf || × |hf || 2Ji + 1 i=1 i=1

!

(92)

The spin-independent and spin-dependent operators appearing above are known as the Fermi and Gamow-Teller operators. The Fermi operator is the isospin raising/lowering operator: in the limit of good isopsin, which typically is good to 5% or better in the description of low-lying nuclear states, it can only connect states in the same isospin multiplet. That is, it is capable of exciting only one state, the state identical to the initial state in terms of space and spin, but with (T, MT ) = (Ti , MT i ± 1) for β − and β + decay, respectively. This state is called the isospin analog state. The Gamow-Teller operator can excite final states with Jf = Ji − 1, Ji , Ji + 1. 3.5 Neutron lifetime in BBN 1. For the neutron, the Wigner-Eckart theorem quickly gives √ h1/2; 1/2, 1/2||τ+ ||1/2; 1/2, −1/2i = 2 √ h1/2; 1/2, 1/2||στ+ ||1/2; 1/2, −1/2i = 6

(93)

so that A A X X 1 τ± ||ii|2 + gA2 |hf || στ± ||ii|2 = 1 + 3gA2 ∼ 5.87 |hf || 2Ji + 1 i=1 i=1

!

22

(94)

2. We will ignore Coulomb corrections, as they are small for Z=1. 3. We correct for the fact that our decays are not occurring in vacuum, but in a plasma where some final states are partially occupied. Thus decays into possible final states in vacuum must be reduce by the probability that the state is already occupied. That factor is immediately obtained from Fermi-Dirac statistics 1−

1 eE/T

+1

=

1 1 + e−E/T

(95)

Thus we find a rate for neutron destruction via β decay in the BBN plasma of ω(n → p + e− + ν¯e ) = q 1 ZW 1 1 G2F cos2 θc (1 + 3gA2 ) 3 de (W − e )2 e 2e − m2 − /T −(W −e )/T 2π m 1+e e 1+e

(96)

where the outgoing neutrino energy is W −e where W = 1.293 MeV is the difference between the neutron and proton mass. The RHS can also be written as an integral over neutrino energy q 1 1 1 Z W −m 2 d  (W − ν )2 − m2 (W −  ) cos θc (1 + ν ν ν 3 −(W − )/T ν 2π 0 1+e 1 + e−ν /T (97) where the outgoing electron energy is W − ν .

G2F

2

3gA2 )

There are three other ways to destroy a neutron that naively appear consistent with our basic lepton conservation laws n + νe → p + e−

n + e+ → p + ν¯e

n + e+ + νe → p

(98)

but the third doesn’t work kinematically, since one is putting lepton energy into a hadron target to convert a neutron to a heavy but lower mass final state. Thus we need consider only the first two. Because the nucleon is heavy, we can consider it to be at rest. (This will contrast with the case of nuclear cross sections, where both projectile and target are heavy.) For the first reaction, the number of reactions per target neutron – and thus the rate for destroying neutrons - is (see Bjorken and Drell, page 113, for the general formula for non-colinear beams) Z ∞ 0

1 mν Mp d3 pp me d3 pe 1 d3 pν 2 4 4 |M | (2π) δ (p + p − p − p ) n ν p e 3 p /T 3 3 ν (2π) e + 1 pν Ep (2π) Ee (2π) 1 + e−e /T

(99)

which yields ω(n + νe → p + e− ) = q 1 Z 1 1 G2F cos2 θc (1 + 3gA2 ) 3 dν 2ν (W + ν ) (W + ν )2 − m2 (100)  /T −(W +ν )/T 2π 1+eν 1+e 23

Finally, the rate for n + e+ → p + ν¯e is Z ∞ 0

d3 pe 1 me Mp d3 pp mν d3 pν 1 2 4 4 |M | (2π) δ (p + p − p − p ) n e p ν 3  /T 3 3 e (2π) e + 1 e Ep (2π) Eν (2π) 1 + e−ν /T

(101)

which yields ω(n + e+ → p + ν¯e ) = G2F cos2 θc (1 + 3gA2 )

q 1 1 Z∞ 1 2 2 2 d  (102) e e ee − me (W + e ) 3  /T −(W +e )/T 2π me 1+ee 1+e

24