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Identical particles effects on the nucleon structure functions Sucipto, Erwin, Ph.D. The University of Arizona, 1994

U·M·X

300 N. Zccb Rd. Ann Arbor. MI 48106

1

IDENTICAL PARTICLES EFFECTS ON THE NUCLEON STRUCTURE FUNCTIONS

by Erwin Sucipto

A Dissertation Submitted to the Faculty of the

DEPART]'vlENT OF PHYSICS In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF' PHILOSOPHY In the Graduate College

THE UNIVERSITY OF' ARIZONA

1 994

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have read the dissertation prepared by

Erwin Sucipto

------------~-------------------------

entitled

Identical Particles Effects on the Nucleon Structure Functions

and recommend that it be accepted as fulfilling the dissertation

of Philosophy/Physics

requirement

R. L. Thews Date

,

//,/1.-9 /;1 if

T. Bowen

Date

N. D. Scadron

YYl ]),~~~

L. C. McIntyre, Jr.

1J- COYVll ~ ~

J ) Date

{?!t (T 2 operators. The twist

T

= 2 operators

refer to the active

46 parton involving in the hard-scattering, while those of higher twists refer to the parton (spectators) interacting with the active parton. The latter go beyond a simple hand-bag structure, and in general are not calculable.

Figure 2.5: (a) Hand bag diagram for r = 2 and (b) Spectator interference diagram for higher twist.

While the physical region is 0 :::; x :::; 1, the series is only convergent outside of this region. What needed therefore is an appropriate analytic continuation to the complex x-plane, in which the amplitude is analytic except along the cut on the real x-axis from -1 to 1. The discontinuity across the cut is nothing but the structure function W. The n-th term of the expansion can be isolated by taking the (n - 1)-th moment of the amplitude along the contour shown in Figure 2.6,

1 1 . dx x n - 1 T(x,Q 2) = LC~(Q2'fL2) ~nzJc T

0

6~(fL2)

T/2-1

(2.62)

( Q\ )

By shrinking the contour to the physical cut, the left hand side can be expressed in terms of the structure function W,

210' dx

X"-I

W(x, Q')

= ;;: C~( Q', ,,') 6~(J22)

(~,) '/2-1

(2.63)

Generalizing n to assume continuolls values, we can recover the structure function by taking the inverse r.,IIellin transformation of Equation (2.63),

2W(x, Q2)

1

=~ j. _;rz

c+ioo

C-IOO

d.n

Xl-n

L T

_

_"

(

1 )T/2-1

C~( Q2, fL2) O~(W) Q2

(2.M)

47 What we obtain here is the factori'zation of the short distance from the long dis-

~--l--~c

Figure 2.6: The contour for the complex integral of the (n - I)-th moment of the amplitude.

tance effects. The short distance part is implicit in the coefficients

C:: which bear

the near light-cone behavior of the amplitude, and can be calculated within a perturbative framework. The result depends on the specific process being considered, but not on the state of the target; hence, it will be the same whether the target is a proton, a neutron or another hadron. The long distance effect is represented by the constants 6~ which are related to the matrix elements of the composite operators. Contrary to the Wilson coefficients, these constants are process independent; may it be the deep inelastic scattering, e+e- annihilation, or Drell-Van process, the constants are the same for the same hadron, they are solely the property of the hadron. In practice, they are not calculable perturbatively, and must be determined by experiment. The parameter It'2 sets the scale for the separation of the short distance from the long distance effects. Roughly speaking, all parton propagators which are off-shell by less than tt'2 will not take part in the hard scattering and will be included in the non-perturbative part should have written two parameters

fL2

6;, , and

vice versa. To be precise, we

instead of one, ttl and IJ.Jl' The first one,

Ill, refers to the factorization scale that we have just discussed, and the second one,

48

Ph, refers to the renormalization scale that

must be introduced when we perform

perturbative calculations for the singular Wilson coefficients. It is not necessary, but convenient, to choose the two scales to be equal. The interpretation will be more obvious if we consider only the leading twist contribution r

= 2, and

take the Mellin counterparts of the two factors,

(2.65 ) Substituting this definition to Equation (2.63) and rearranging the dummy variables to y and x == yz, we arrive at (2.66) This relation is valid for any n, and so we can infer that (2.67) where we have written :F == 2W for the structure function. In his Parton Model, Feynman took this factorization as the starting point, and interpreted the function

G(y) as the distribution of the partons inside the hadron target. In this view the factorization above takes a simple meaning, that is the hadron structure function is the average of the corresponding function at the parton level weighted by the distribution function of the parton illside the hadroll. We will discuss this subject further in the next chapter. We may see the interpretation from a slightly different perspective if we know the structure function at a certain value of virtuality, say Q2

= Q6.

Solving

for the constant 6~ in Equation (2.63), we have (2.68)

49 Comparing with the Mellin transformation (2.65), and again arguing that they are valid for any n, we may write Equation (2.67) in a different form,

:F( x,Q 2)

=

where the kernel J((y, Q2, Q6)

11 x

d y ,( y,Q 2,Qo2) :F (x -1\. -,Qo2) y y

= F(y, Q2, Jl2)/6~(Q6, JL2)

(2.69)

is calculable perturba-

tively. The equation reads that once we know the structure function at some value of Q6, we can obtain the structure function for any x at some other value of Q2. This is the idea behind the evolution of the structure function. We will clarify this point and its physical meaning in the next chapter.

50

CHAPTER 3

QUARK PARTON MODEL

3.1

Scaling and Impulse Approximation

Feynman proposed the Parton rdodel as a more intuitive approach to explain the phenomenology of the deep inelastic scattering [5]. In this model, a nucleon is pictured as a superposition of states consisting of parallel moving partons q, each of

= y pll, that is a fraction y of the nucleon momentum constrained to 0 S; y S; 1 and I: y = 1, ensuring that all

which carries a momentum kll pll. The fraction y is

partons are moving in the same direction. The number of partons q carrying a momentum fraction between y and y

+ dy

is denoted by Gq(y) dy, which can be

interpreted also (up to a normalization factor) as the probability of finding a type

q parton of that fraction. This distribution function is solely the property of the nucleon, and is independent of the process being considered. The distribution extracted from one process can be fed into another process as an input. Figure 3.1 shows the pictorial presentation of a nucleon in this model.

k:: yp

p

Figure 3.1: Pictorial presentation of a nucleon in the Parton l'vlodel.

51

A scattering from the nucleon occurs entirely at the parton level, regardless of the status of the distribution function. Any nucleon observable is assumed to be the average of this hard process weighted by the parton distribution function,

:F(x, Q2)

=

1

L Jor dy Gq(y) Jor1dz 8(yz _ x)F(z, Q2) q

=

L q

11 dy Gq(Y)F(~, ;r;

y

y

(3.1)

Q2)

On the left hand side is the scattering process viewed in the nucleon frame. A photon of virtuality Q2 enters into the nucleon and is absorbed by a parton of momentum fraction x.

On the right hand side, the same process is viewed in

the parton frame. A parton having a momentum fraction y, when probed by the photon, may be resolved into a group of partons with smaller fractions _. called the secondary partons. Here the photon probes inside the group and scatters with one of the secondary partons carrying a fraction z of the total momentum of the group. Since the scattered pat·tons in the two pictures are the same parton, the fraction z is constrained to zy

= x,

which tells the origin of the delta function in

Equation (3.1) above. Summing the parton process over all possible fractions in

=

I q

J', .."

~IC=:

~ zk

= zy P

Figure 3.2: Description of a deep inelastic nucleon process as a weighted average of the parton counterpart.

the group and summing over all possible groups must give the same result as in the overall nucleon picture. The contribution to the sum can only come from partons having momentum fraction !J larger than the scattered parton x, since the latter

52 must be contained in the former. In this manner, the factorization relation (2.67) obtained from a formal field consideration is recovered in a rather intuitive way. Some discussion on the assumptions used in this model and their validity may be in order at this point, but we would rather reserve this topic for the next section. Until then, we will see how ,this model can explain the scaling phenomenon and other related aspects very easily. in the perturbative approach, the photon-parton interactiolf is presented as a power series of the coupling strength

O:'s,

corresponding to the increasing

number of gluons participating in the process,

At the lowest order, only the Born term contributes and no gluon is involved in

Figure 3.3: Some representative diagrams constituting the series: a. the Born term and b. the one gluon contributions.

the process (the first diagram in Figure 3.3); the corresponding parton tensor can be shown to be

i\HVllv(Born) = e~ 8(2k . q + (/)

~ { (2kll + fJll)(2kv + qv) -

q2 ( -91lv

+ q;;v) }

(3.3)

which, by using the appropriate contraction with the photon polarization tensor,

53

Equations (2.28) and (2.29), then gives

Fdz, Q2)

MvVIl"

=

e; 8(1- z)

=

-gil"

=

4z2 1.IlI."W- = 0 Q2 t\, '0 Il"

(3.4)

where z == Q2/2k . q. By substituting this equation to the master relation (3.1), the corresponding structure functions for the nucleon become

Fdx,Q 2 )

ely x I>; 1 -G (y)8(1--) y y 1

=

q

q

Fd x ,Q 2 )

=

x

=

(3.5)

0

which shows the scaling property. The interpretation is also very straightforward: the structure function is just the superposition of the parton distributions inside the nucleon. When probing the nucleon at the scaling value x, the photon may be absorbed only by a parton having the Tight momentum fraction, and this fraction turns out to match the Bjorken parameter. In this view, the Bjorken parameter x takes a new meaning as the momentum fraction of the parton taking part in the hard scattering. To this lowest order, we can see immediately that not only scaling, but also the Callan-Gross relation is readily satisfied, because FL

= 0,

that is

FJ(x) = F 2 (x) =

2: e; Gq(x) 'I

The sum in Equation (3.5) runs over different parton flavors in the nucleon, and each flavor has its own distribution. The common notations for the parton distributions inside a proton are

u(X) == Gu/p(x)

u(x) == Gu/p(x)

d(x) == Gd/p(x)

d(x) == Gd/p(x)

s(x) == Gs/p(x)

s(x) == Gs/p(x)

c(x) == Gc/p(x)

c(x) == Gc/p(:I:)

(3.6)

The distributions are normalized to match the quantum numbers of the proton. To be specific, the net number of each type of partons must match the number of

54 the corresponding constituent quarks used for proton in the non-relativistic hadron spectroscopy, namely

1 [u(x) - U(X)] dx = 2 , 1 fo [S(X) - S(X)] dx = 0 ,

fo

fo fo

1

[d(x) - d(x)] d.?;

=1

1 [C(X) - c(x)] dx

=0

(3.7)

These distributions are not provided by the model, and must be determined phenomenologically. The experimental observations of the deep inelastic structure functions from the electron, neutrino and anti-neutrino scatterings off a proton and a neutron targets will serve as constraints to the distributions. Numerous relations have been deduced for this purpose by using this model; we list some of them below for illustration,

:F;P(x)

=

~[u(x)+u(x)+c(x)+c(x)]

:r{P(x)

+ d(x) + s(x) + s(x)] = ~ [d(x) + d(x) + c(x) + c(x)] +i [1l{X) + u(x) + s(x) + s(x)] = 2 [d(a:) + u(:r) + s(x) + c(x)] = :2 [1l(X) + d(x) + 8(X) + c(x)] = 2 [u(x) + d(x) + c(x) + s(x)]

:r{n(x)

=

+~ [d(x)

:r;n(x)

:r;P(x) :r;"( x)

2 [d(:r) +u(x)

p.S)

+ c{x) +s(x)]

In writing these relations we have assumed the isospin symmetry between the proton and the nucleon targets, that is Gu/n{x)

Gu/p(x)

= u{x), and

= Gd/p(x) = d(x),

Gd/n(x)

=

all other distributions are the same for both targets. Certain

combinations of the structure functions are constrained by the symmetric property of the target and the probe; these constraints are called the sum rules. Some examples are the Adler sum rule [15],

(3.9)

55

0.2

0.0

0.4

x

0.6

1.0

O.B

Figure 3.4: Parton distributions within a proton.

and the Gottfried sum rule [16],

fa1 dx (F;P(x) -

F;n(x)) =

~

(3.10)

which can be verified directly in the framework of the Parton iVIodel by using the listed structure functions above. Nevertheless, it should be noted that most of the sum rules have been known long before the advent of the Parton Model, showing that the origin of these sum rules is more fundamental than the model itself.. From the construction, it is clear that the expressions for the structure functions in Equation (3.8) reveal only the distributions of the partons which can interact directly with the gauge-boson probe. It is then legitimate to ask whether the distributions so obtained exhaust the nucleon momentum available for the pat'tons. If we define the total fraction carried by the parton of flavor q = u, d, s, etc. as

(3.11) we would expect that Lq(Xq

0.28, Xd

rv

0.15,

+ Xq)

= 1. However, experiments show that Xu '"

.'Cs '" 0.02, and L Xq '" 0.05. Only half of the total proton

56 momentum is carried by the

quark~partons;

in fact, QCD fixes the asymptotic value

of the total momentum that can be carried by the quark-partons to (1 + 16/31)-1, where

I

is the number of quark flavors. The remaining fraction must be carried

by neutral partons which cannot be detected directly by the photon. If we intend to use QCD as the underlying dynamical theory for the partons, we had better attribute these neutral partons to gluons to complete the ingredients for the theory. The behavior of the parton distributions at both ends, i.e. at x

-)0

0,1 ,

is a broad subject of interest by itself; however, for us, a peek at the small-x tail will be sufficient. At large Q2, the limit x energy transfer, v

-)0

00,

-)0

0 corresponds to the region of high

in which the Regge analysis is very useful [17]. In the

Regge picture, the asymptotic behavior of the total scattering cross-section (in the s-channel) is governed by the leading Regge pole in the t-channel.

~ I

I IP, P,w ••• I I I

~ Figure 3.5: The Regge pole dominance.

Correspondingly, the structure functions take the asymptotic form [18]

(3.12) where a is the appropriate Regge intercept: a :::: 1 for Pomeron exchange, a :::: ~ for Reggeon exchange p, w, A, 12, etc. If scaling is to be valid in the Bjorken limit, the Regge residue functions must have a suit.able power of Q2 to compensate the dimension carried by the photon energy, that is (3.13)

57

which yields the following asymptotic scaling form for the structure functions,

(3.14) Experimentally it was observed that :;::1,2 '"

X-

1

as x

-+

0, corresponding

to the Pomeron exchange in the Regge analysis above. Since by definition the Pomeron carries the quantum numbers of vacuum, such as C = +1 and I = 0, it does not recognize flavors but the flavor singlet. Consequently, the structure functions obtained from the nucleon targets (p, n) and the anti-nucleon targets

(]i,71) must give the same result at small x; hence we may conclude that u(x) ~ u(x) ~ d(x) ~ d(x) ~ s(x) ~ s(x).··

x-o 1

--lo

-

(3.15)

X

This part of the parton distribution is known as the sea quark distribution, which arises from the gluon dissociation into a quark-anti quark pair, Figure 3.6. Corrections due to mass asymmetry, m~ > mu,d, are in the order of O(m;/Q 2 ), and hence negligible in the high Q2 limit.

:}

SEA

P

PARTONS

}

VALENCE PARTONS

Figure :3.6: The sea quarks production from the gluon dissociation diagram.

There is another part of the distribution that gives quantum numbers to the nucleon; this part is known as the valence quark distribution. For this separation, it is common to write the parton distributions in the form of the

58

following ansatz,

d(x)

+ uv(x) = ((x) + dv(x)

d(x)

~

u(x) =

u(x)

~

((x)

s(x)

~ ~(x)

(3.16)

...

~

((x)

where ((x) represents the symmetric sea-parton distribution. The leading contribution to the valence quark distributions comes from the a: ~ ~ Reggeon exchange, for which we have uv(x) '" dv(x) ~ x- 1 / 2 • What we learn from this result is that the small-x region of the nucleon is dominated by the sea-partons. The average occupancy of the sea-partons, (nq)

J dx Gq(x)

==

is logarithmically divergent. We see immediately that this cannot be

correct due to the Pauli exclusion principle. Some mechanism to stop the growth must exist. The search for this mechanism and the detailed study of the behavior of the parton distributions at small x are still an active research area. Parallel to this is another active research area devoted to studying the role of gluons in the structure functions, which is the higher order term in the perturbative expansion. The presence of gluons in the hard process breaks the scaling behavior, inducing a Q2-dependence into the structure functions. It is possible to attribute the Q2 dependence to the parton distributions and retain a Born-like interaction for the hard process. For this, the relation (3.5), .1"2 (x, Q2) =

L

e~ {Gq(x, Q2)

+ GTj(x, Q2)}

(3.17)

'i

serves as the new definition for the parton distributions. Note also that the distribution functions are defined throllgh .1"2. A definition using different structure functions is possible, but not necessarily the same as this one, since the CallanGross relation is no longer satisfied.

59

3.2

Validity of the Impulse Approximation

In the master relation (3.1), the sum is taken over the transition probability instead of over the transition amplitude, under the assumption that the subprocesses involving different pm'tons are incoherent. The reason behind this assumption is not very transparent, especially when we remember that in quantum mechanics there is always a possibility of interference from coherent amplitudes. Therefore, it is necessary to seek the condition for which the assumption is valid. Previous experience from the inelastic scattering of a non-relativistic electron by the Coulomb field of a nucleus may shed a light on our situation. The transition rate for this process is

(3,18) Representing the nucleus as a superposition of the nucleon constituents, IX~) =

L INi ),

we can write the transition rate in the Born approximation as

ITid 2

=

411" 0:2

I(n' 411"0'2

=

Ir7P

N,) +t(N,INj)} {t(Nd .=, ',1'/· {Z + Z(Z - 1) 1(1j({n+1})

+ 1)-transition probability

=

(n+1)lP1 P2 "'PnPn+1

=

(n

+ 1) Pi->j( {n})

Pn+1

(4.11)

which also follows from (4.8) by induction. By using the theorem for conditional probability, we can write the transition probability for the (n making a transition to the final state

IJ),

+ l)-th

particle

knowing that the other n particles have

already been there, which is

Pi->j(n+ 11{n})

=

Pi->j( {n + I}) ({ }) Pi->j n

= (1 +n) Pn+1

(4.12)

which is just the Bose enhanced probability. Here also no reference to any specific distribution has been made, and hence the result should be valid in general. For this reason, any transition occuring in the background of bosons and fermions must be modified by the Bose Enhancement (BE) and the Fermi Blocking (FE) factors as given by Equations (4.12) and (4.4). In particular, with this modification, the transition rate (2.6) becomes

where the momentum volume CICIl1Cllts in square brackcts stand for the Bosc Enhanced and the Fermi Blockcd volumc clemcnts, (4.14) As in the Parton Nlodel, the single photon approximation transition rate for electron-nucleon scattering in the parton framework can be written as Wfi

= (4.15)

86 The incoherent sum is still assumed, as shown by the sum over the initial quark distribution,

J d3 k 1)(k) nF(k),

while the statistical factor for the final state is

hidden in the parton tensor,

~vt;: = _1_/ 41rJl1

2: JII [(Pk'] 1)(/:;') [d G] 1)(6) 3

k',G

(21f)" 84 (2:(k'

k',G

+ G) - k + q) ~ 2: (kl J

1l

Ik', G)(k', GI

JJ Ik)

( 4.16)

3,S'

Comparing this transition rate with the rate in the nucleon frame of view given in Equation (2.18), we can extract the expression for the nucleon tensor WI~ in terms of the parton tensor ~i!Il!l' (4.17) where the sum covers the spin, flavor and color degrees of freedom. This is the generalization of the master equation (3.1), relating a nucleon observable to the parton sub-process counterpart. By taking the appropriate contraction with the polarization tensor of the photon, we can express this equation into the more familiar form for the structure function,

PN 2 ')E F(x, Q2 ) = ~ ~ Cq ~

N

q

Jd

Pk ~ 2 k 1)(1.) np(k) ')E F(z, Q )

3-

~

(4.18)

k

Although the form is almost identical to the old relation employed in the Parton fvlodel, this new master equation has a capability to accomodate the effects of identical particles in the background. At this point, this relation is still valid for any parton distribution, not limited to the unidirectional distribution of the Parton f\IIodei or the equilibrium distribution of the thermal case.

4.2

The Role of Direction

As a first step toward this generalization, let us concentrate on the effect of direction to the quark distributions and the evolution equation obtained in the previous

87 chapter. For the moment we will ignore the statistical correlation factors in the final state, so that the perturbative calculation for the parton sub-process will proceed exactly as in the Parton Model, including the sum of the leading logarithmic terms. The only difference here is in expressing the Q2-dependence of the parton distribution in terms of the long distance and short distance factors; instead of a one dimensional convolution (3.1), now we have a three dimensional integral,

= J(2rr)32k llNs(k):;::2

3~

3~

d k ~ 2 (2rr)32k nNs(k, Q ) 8(1 - z)

J

d k

-0

~

-

LL

1

2

~ (z, Q )

(4.19)

Again, for simplicity we have worked only with the Non Singlet part of the distribution. On the left hand side we have shifted the Q2 dependence to the distribution function and assumed the interaction to be Born-like. On the right hand side, the function .i,fLA(z, Q2) is the same as the leading logarithmic sum in Equation (3.80). Assuming azimuthal symmetry around the photon axis, we can perform the azimuthal integral immediately. For the rest, it is convenient to introduce a new set of variables (z, w) to represent the parton moment urn,

(4.20)

w ==

As before, the variable z is Loundcd by 0 :::; z :::; 1, while the variable u.i is limited by w ~

= Hq3 - qo). We will use the same notation for the distribution

Wmin

function in the new set of variables, 1lNs(k, Q2)

== llNs(k, cos 0, Q2)

~ 1lNS(Z, W, Q2).

The integral over the variable z on the left hand side can now be performed with the help of the delta function, and the remaining integral becomes

1

00

Wmin

2

dw nNs(l,w,Q ) =

1

00

wmin

rt

dz -0 - LLA 2 dw Jo Z3 HNS (1,w):;::2 (Z,Q)

Since this relation must be valid for any value of changing qo ==

II

while keeping

Q2

Wrnin

(4.21)

(which can be varied by

fixed), we may argue that the equality holds for

the integrand as well. lIenee, we can write

(4.22)

88 In principle, the variable w carries the dependence on the polar angle. The same wvalue appearing on both sides of the extraction above indicates that only partons moving in the same direction will contribute to the distribution.

This can be

understood since in the Leading Logarithmic Approximation the most dominant contribution arises when the gluons are emitted practically parallel to the quark, and hence do not change the quark direction. This interpretation can be seen more clearly if we re-express the relation in terms of the regular momentum and angle variables,

1 Tk dk 00

W

2

( TINS W,

cos 0, Q2 ) =

w

2 -0 7/.

Ns (k, cos 0)

k' Q2)

- LLA(W

:;:2

(4.23)

The similarity of this relation to its one dimensional counterpart in Equation (3.81) is more striking if we recall that P TINs(k,cosO,Q2)

= dNjdk dn

is the particle

number density per unit length of momenbm per unit solid angle, congruent with the parton distribution per unit momentum fraction in the Parton l'vlodel. Note also that the momentum and direction on the left hand side are not independent, but related by Equation (4.20). To probe the distribution at any value of momentum and direction, we have to vary the photon energy v while keeping Q2 fixed, just like in the Parton Model when we want to obtain the distribution function at any value of .1:. The evolution equation can be obtained by taking the derivat.ive of ('1.23) with respect to Q2, or equivalently with respect to fe,

d

df\,W

1 dkyk 00

2

l1NS

(

2 w,cosO,Q)=

w

2

W 2) P,/q(k)TINS(k,cosO,Q

This is the equivalent of the Altarclli-Parisi equation, governing the evolution of the quark distribution per unit length of momentum in a certain direction. The interpretation is very much the same as before: The change in the quark distribution at momentum w in the direction cos

°

comes from all quarks moving

in that direction with momelltum /.: 2:: w which split illto a gluon and another quark having a fractional momentulll z of the original, where z = wj k. The probability

89

of the splitting is given by the splitting function Pqq(z). What we learn from this result is that different directions contribute independently to the evolution of the distribution, a condition which is automatically satisfied in the Parton Model.

It is appropriate to ask whether we can revisit the Parton Model from this result with some suitable conditions. In that model, the parton momentum is bounded by the nucleon momentum P, and the lower limit w is taken to be the fraction x of the upper-bound, w

= xP.

Setting the lower bound like this is equiv-

alent to fixing the direction, which explains why the Parton Model requires that all partons move in the same direction. In addition, the distribution function in the Parton Model is defined per unit length of momentum, and so the correspondence should be G~;rLon(x, Q2) --; w2 nNS(w, cos 0, Q2). Substituting all these ingredients into the evolution equation (4.24) above, and rescaling the integration variable to k

= yP, we recover the AltareUi-Parisi equation immediately, _d GP>rLO" (X. , Q2) _NS d /\,

11dy _ P X

Y

qq

x GP"'LO"(y (?) (_) NS , 1 Y

Solution of the evolution equation is usually done by taking the moment of the equation with respect to the variable in the equation, also known as the r.,ilellin transform. A similar approach applied to Equation (4.24) fails because the integral over w mixes the contribution from different directions. However, if the distribution does not depend on the direction angle explicitly, but depends only on the magnitude of the momentum, we can still have the moment equivalent of Equation (3.86). Defining the n-th moment of the distribution as ( 4.25) where nNS(w,Q2) == HNS(W,coSO,Q2) is independent of direction angle, we can reexpress (4.24) as the evol u tion

CC[ uation

~M (Q2) d/\' 11

=

for the moments, p(n) IH (Q2) '1'1 • 11

90 The n-th moment for the splitting function is the same as given in Equation (3.87), and the solution to the moment equation is the same as in the Equation (3.88). Another case for which t.he moment method is also applicable is when all the partons move in the same direction, for example in the Parton Model. In this case the moment method works because no directional mixing can take place. As a calculational example, let us consider the case where at Q2 = Q6 the quark distribution is given by the Fermi-Dirac distribution. The Non Singlet distribution at this scale is (4.27) which is independent of angle explicitly. Its n-th moment can be calculated from the definition (4.25) with the result

{t

i=O

(n -: I)! (1 _ 2i-n-1) ((n + 2 ~ j) Il i J! (3n+2- J

+

1/11+2

_t'"_

n+2

(x Il)n+l } + (1 _ (_l)n-l) 11-' dx ~-~0

e~+l

(4.28)

Unless n is odd, we cannot express the general moment in an analytical form. For

n

= 1, the moment is related

to the baryon number of the system, (4.29)

and this quantity is conserved because ~1~) = O. In case of a proton quantum numbers, there are two valence ll-quarks and one valence d-quark, so that the ratio of their moments is

MI'/Mil

= 2.

At low-temperature, the first term of the

moment can be neglected, and we recover the well known result

ltd

= 2-1/3Itu,

showing that the formation of u-quarks at low temperatures is more 'difficult' than that for d-quarks. Qualitatively, this conclusion can be expected due to the Pauli exclusion principle; with more ll-than d-quarks in the background, the Fermi blocking factor for'll will be more effective than for d.

91 For n

= 2,

the moment is related to the excess of energy carried by the

quarks over that by the antiquarks,

M 2(Q 02) = (Jl" 4 At high temperature,

e-{31l

rr2 Jl2 (32

+ '~) -f

21 ((4) (3"

+ '~)

12

~ (-l)i -i{3ll)

+ (34 j=l L..J

.4 e J

(4.30)

1, the last two terms cancel, so that the energy

difference at high temperature behaves as (3-2, in contrast to the total energy (density)

2:: q (€q +cq)

=

{(Jl" / 4) + (rr2p2 /2(32) + (7rr" /60(3")} /(2rr2), which behaves

as (3-4 at high temperature. However, as we increase Q2 at fixed temperature, the second moment decays exponentially to zero, because

PJi l = -16/9

is negative.

At very high Q2, the energy carried by the quarks is the same as the energy carried by the antiquarks.

4.3

Nucleon Structure Functions in the Thermal Model

Some previous attempts to include statistical correlation factors have been done in the context of the Thermal Bag Model, in which hadrons are pictured as a collection of almost free quarks and gluons in thermal equilibrium, confined in a volume \~v of hadronic size by the vacuum or bag pressure [24J. This model has a slightly different philosophy from the Parton lVlodel; here the idea of factorization is still maintained, but the quark and gluon distributions are known from the very beginning, usually taken to be the equilibrium Fermi-Dirac and Bose-Einstein distributions. Because of this, no evolution equation is usually sought in this approach. So far, a great deal of effort has been devoted to studying the IR divergences arising in the calculation. Since all distributions are known, there will be no room for renormalization, and strict cancellation must be required. Normal methods of cancellation no longer hold due to the emergence of new types of singularities when one modifies the theory to a temperature-dependent theory [10, 25J. Upgrading to the temperature-dependent theory is usually done by taking the thermal average of the zero-temperature counterpart [26J. In doing so, many

92

rules have to be changed accordingly. As an illustration, let us consider the thermal average of a scalar-field propagator iD(x - V), with Xo < Yo for simplicity,

(iD(x - V)) Ixo

-

Tr {e- f3li T[¢( xO)¢(Yo)]}

=

Tr {e- f3fl ¢(yo)e- f3li ef3il ¢(xo)}

=

Tr {¢(yo)e- f3f1 ¢(xo - i(3)}

=

Tr

=

(iD(x - y))l x o-if3

{e- f3f1 T[ ¢( Xo -

(4.31)

i(3)¢(yo) }

In all these expressions we have suppressed the space coordinates which remain unchanged during the calculation. To get to the third line we have used the time translation operation,

= i{3.

extended to an imaginary time displacement Xo

The convention for the time

ordering with imaginary time is that the one nearest to the end point -i(3 is the latest. This simple calculation shows that the propagator is periodic along the imaginary time axis, and consequently its Fourier representation in momentum space has a discrete energy component [27],

J~)3 3-

iD(x - y) = _1_. '\"" e-iwn(xo-Yo) _

• 1':1 ~

lfJ

11

(') ~7r

eik.(x- y ) iD(w

\

F) 'I)

"

(4.33)

where iD(wn' k) is the momentum representation of the propagator with discrete . /(3 , . . - ')_7rZ11. an d IInagmary energy, Wn = .

-

l

ID{w n , I.:) = ----=_-w~ - k2 - m 2 Analytic continuation of this free propagator from imaginary time to real time was first established by Dolan and Jackiw [28], which in momentum space corresponds to analytic continuation from discrete imaginary energy,

Wn,

to continuous real

energy, 1.'0. The result is

(4.35)

93

== (exp(,Blkol) -: 1)-1 is the usual Bose-Einstein distribution, and

where nB(k) P =

k5 - k2 •

The first term is the usual zero-temperature propagator, and the

second term is the contribution of identical particles in the medium in which the particle propagates.

The interpretation is more obvious when we decom-

pose the propagator into the positive and negative energy propagators, iD(x)

B(xo)iD+'(x)

==

+ B( -xo)iD-(x), whose Fourier transforms are (4.36)

The emission of the (on-shell) scalar particle is enhanced by the Bose-factor from the background. One soon runs into trouble when applying this thermal propagator. For example, the on-shell part of the propagators labeled by

e in the

process in Fig-

ure 4.1 gives a factor [8(£2 - m 2)J2 in the amplitude, which is not a well-defined quantity.

Figure 4.1: An example of a graph containing an ill-defined factor [8(£2 - m 2 )F.

A remedy to this problem has been discovered relatively recently, where the real time Green functions are rederived directly from first principles without having to perform any analytic continuation. At least two independent approaches have been adopted, the path-integral method [29] and the operator method of the Thermo-Field Dynamics [30], that is quantum field theory at finite temperature,

94

both giving the same result. In both approaches, the number of field degrees of freedom is doubled to,p == (,pI, ,p2)j the type-l field is the physical one, and the type2 field is the unphysical one (ghost) . Corresponding to this doubling, the types of vertices are also doubled to type-1 (-ig) and type-2 (+ig) vertices. External lines can only be of type-1 fields connected to the other part of the diagram through type-1 vertices. All propagators become (2 x 2) matrices, the (11)-component propagates between two type-1 vertices, the (12)-component propagates from type1 to type-2 vertex, and so on. The explicit form for a fermion propagator in this theory is

is(k, m) = (JC + m)iS(k, m)

(4.37)

where is(k, m) is the following matrix,

(4.38)

where

Xk

==

Ik· UI- jlf.(k. U),

with f.(z) ==

z/Izl

the sign function, and 6.(k, m) ==

i(P - m 2 + if.)-I . For a massless gauge boson (e.g. gluon), the propagator is (4.39) where iD(k, m) is the propagator for a thermal scalar field,

iD(k, m) ==

(

6.(k, m)

0

o

6.-(/':, m)

)

+ 27r8(1~2 _ In'2) e{JXk + 1

(

_cf3

1 _ef3 x d

z

Xk /

2

)

1 ( 4.'10)

In this two-component approach, the ill-defined singularities of the type of products of delta functions cancel among themselves [31]. Doubling the degrees of freedom does not change the topological and combinatorial structures of the Feynman diagrams, but it does increase the number of diagrams involved. However,

95 the number of diagrams to be evaluated may be reduced significantly by using the Cutkosky rules for finite temperature [32]. 'With this new formalism, many calculations have been done for processes at finite temperature. Related to our work is the deep inelastic structure function in a Statistical Model for the nucleon which has been calculated in detail up to the first order. Thews and Cleymans initiated the calculation at the zeroth order [9]. At this level, the statistical correlation factors do not playa significant role; the OE-factor is absent because no gluon appears in the process, while the FO-factor, which has the form Fo-factor = 1 -

1 e{3(k-+11-1' )

+1

despite being present, has a value of practically one, because in the deep inelastic limit v is always large. As can be expected, to the lowest order, the Callan-Gross relation is still satisfied,

:;:1

=

:;:2, and explicit calculation shows that

Since the quark constituents are confined in a finite size nucleon, their momentum is bounded below by approximately half of the size of the nucleon; hand, from the

1301'11

011

the other

term delta function we find that the lower limi L for the

momentum integral is krnin =

tM x.

Combining the two, we find that the region

of validity of this model is roughly x ~ 0.4 ; this is reflected in the result of data fitting below, Figure 4.2. The best fit for large x falls within a small range of values for the parameters T and Il, they are around T!'oJ 40 fvIeV and f.L !'oJ 200 f'l'1eV. At small x, instead of going to a finite value, the structure function goes to zero due to the nature of the Fermi statistics which in this model produces a finite number of constituents. Since this part is dominated by sea quarks, higher order processes may be expected to fill ill the deficiency in this region. Calculations to the first order of as have been performed by many and in many different ways [9, 25, 33] ..Just with one gluon ill the process, the length

96

_

T=40 MeV;

T

0.4

02

,,= 200 MeV

=50 MeV; ,,= 170 M oV

10

0.6

Figure 4.2: Data fitting for the Born structure function in the Thermal Model [9J.

and difficulty of the calculation increases dramatically such that analytical results cannot be obtained without making approximations. In the case of no background, one can perform perturbative calculations in a covariant way or by specializing to a certain frame and still obtain the same result. However, with the background, a new parameter, the plasma velocity U~, enters the formulation, and we have to take into account the transformation of the background when we transform to another frame.

As an illustration for typical calculations in this model, let us

consider the one gluon emission process. The parton subprocess in a general frame can be written as

- J

Ff.,L =

[dLipsJ

( + 1) ( 1

e{JG.u _

1 + 1)-" IA/tI;,L

1 - e{J(k"U-~)

1

(4.'12)

where [dLipsJ is the final particle phase space factor as given by Equation (3.74), and IA11~,L is the invariant matrix for the process. For the total sum, the invariant matrix is the same as in the non-thermal case,

-1"/12 _ {it.-:- Tl.s-=- JV c ~ - -01'0 'I:"

811

2

.)(m - Q2){ _ __ US

+ 1/1'!Q2

(1s- + -:--:;n-1)} -::;-

97

while for the longitudinal one is much more complicated,

16Fo

us

Q2

Q2

+ (q. U)2

- (G. U)( q . U)

{_ (G. U)2 (Q2

+ m2~) U

(u + Q2 +

2

m (1

+ ~))

- 2(G. U)((k + q). U) (Q2 - m 2 (1

+ %))

+ (q. U)((k + q). U) (U -:5 + 2Q2 + m 2 (s ~Su)2)

+ 4(q. U)2 ((:5 -

m 2))

_ 4((k + q). U)2 (2Q 2 + m 2 (s

+

(4.44)

~SU)2)

~ (.5 2 + U2 + 2Q2(U +:5) + 2Q" + m2Q2(S ~SU)2)}

The on-shell part of the propagator does not contribute in this case, because with the mass regularization we have avoided the pole in the propagator.

It is interesting to note that when we take the deep inelastic limit at zero temperature from this expression we do not immediately recover the corresponding result previously calculated in the Parton Model. The total sum is still the same as in the Equation (3.33), but the longitudinal contribution becomes

il:::;; ~O:sCF { 2.:~ 1 -""

- 38(1- z)

+ O(~)}

(4.45)

v

which is different from Equation (3.36). We have written this expression in the plasma rest frame; however, if in addition to this we are also allowed to use the Ctll -center of mass- momentum in (4.'14), the expression returns to the old result. In the Parton l'vIodel, this choice is allowed because the parton distributions do not depend on the velocity of the 'plasma'. Not so in the Thermal Moclel, since the velocity of the plasma fixes the frame [or the calculation; choosing the rest frame for the plasma prevents us rrolll IIsillg the Cill-rrame. Analytical calculation or the struct me function by usi ng ('l.t13)

illld

('1.4'1),

along with other processes of the same order, has been attempted, but in the

98 end must rely on numerical calculations [9J. One of the many important results from this effort is the cancellation of the IR singularities, even before the final integrations are performed. This is very crucial to the model because by fixing the distributions from the beginning, no renormalization scheme can be applied.

It turns out that this cancellation can only take place if we take into account all processes of the first order including those neglected in the Parton Model, such as the parton recombination processes [10J. Furthermore, the distribution functions for these processes must satisfy the detailed balance relations, in the sense that the statistical factors for a process must be balanced by those for the time reversed process, for example the statistical factors for the process q that for q+ G

~

~

q+

c: must equal

q, (4.46)

Thermal equilibrium distributions certainly satisfy this requirement.

No other

distributions are known to satisfy this condition; it seems that for non-equilibrium distribution the IR-singularities persist [34J. The inclusion of order O( as) expands the validity region of this model to x ~ 0.21, with best fitting to the observed data occuring at T '" 670 MeV and J.L '"

133 lVIeV. However, with these values of the parameters, fitted values of the

ratio R

= (J"L/ (J"T

fall about a factor of six above the experimental data. This

inability to reproduce the nucleon structure indicates a shortcoming of the model, and further improvement is necessary. Since the thermal equilibrium condition is not a definitive assumption, relaxing this coudition may give a more realistic feature to the model as well as more flexibility to manuveur the experimental data. However, less work has been done in this area due to the difficulty caused by the indefiniteness of the distributions. No concensus has been reached so far regarding the status of the pinch singularities associated with products of delta functions; some claim they clo not cancel [35], and others claim they do not even appear [36J.

99 4.4

Splitting Functions in the Statistical Model

Although the statistical correlation factors have been formally incorporated into the formulation, their real effects are still difficult to be evaluated due to lack of analytical results, simply for mathematical reasons. A numerical approach sheds some light, but cannot replace the transparency of an analytic expression. Central to the discussion is the structure function which in the absence of statistical factors can be factorized into the hard-process and the non-perturbative parton distribution. The dynamics of the hard-process is represented by the splitting functions, which subsequently governs the evolution of the distribution itself. When the statistical factors are brought in, the factorized non-perturbative part is reconnected to the perturbative hard-process. It is still a question how this reconnection affects the splitting functions and the evolution equation. To answer this question, let us consider again the gluon emission process in the background of quark and gluon distributions. Explicit calculation with thermal equilibrium background indicates that the Fermi Blocking effect is negligible in the deep inelastic limit; so to simplify the exposition, we will discard this factor from the setup. The quark structure function as usual is given by

- J {+

:F~ =

dLips 1

1

e{3G.u _

1

}-2 liv11~

(4.47)

where we have included the Bose factor from the background in the expression. The phase space factor is the same as the one defined in Equation (:3.7,1) and the invariant matrix in a general frame is provided in (4.43). As we have noted earlier, the presence of the plasma velocity does not allow us to use center of momentum variables in the rest frame of the plasma; calculation with arbitrary momentum variables in this frame produces a very complicated region of integration [10]. A simpler region of integration can be obtained if we work in the center of momentum frame, retaining the plasma velocity until the end of the calculation. After casting the result into an invariant form, then we can translate the final expression into the

100

rest frame of the plasma. The price that we have to pay in simplifying the boundary condition is that we have to use rather complicated variables of integration. Below we will outline some of the relevant steps which will lead us to the final result. Four out of six dimensional integrals in the phase space factor can be performed with the help of the momentum conserving delta function; the last two integrals are usually expressed in terms of the center of momentum angles, dLips

-t

s- m2 _2 •

327r

J

dflcM

S

However, the exponential part in the Bose factor contains one of these angles, and we have to change the variables further more to simplify the integration of this factor. Here we use

x - -G.O/Go Y =

(4AS)

-lUI (G· k/Gol.·o)

'With these variables, the phase space integrals become (4.49)

where IJI is the Jacobian of the transformation, IJI a

_

_

2~XY _

y2} -1/2

== -k· 0/1.'0

b = c

{(b 2 _ ( 2)c2 _ X 2

=

(4.50)

101 IGI/Go

Note that all the vectors are evaluated in the center of momentum frame. The region of integration for these variables is the inner side of the ellipse (4.51)

which is sketched in Figure 4.3. The Bose factor is now independent of the Y variable, and the integration ovcr }" can be done explicitly.

Recalling that in

101 v

,

/

------:{ /1

/1

.

I

1 1

x

-bel

I 1 I I / v..:. _____ _

/ /

Figure 4.3: Region of integration in the eM-frame.

terms of X and Y, the Mandelstam variable it

= (k -

G)2

= m2 -

2G oko(Y + b)/b,

we can write the following list containing various dependence on it found in the integrand of the structure function

j dY

PI

=

j dY Plii =

7r

7r (

m

2- 2Goko(1 - :2 X))

-b 1 2Goko jR b(b2d - aX) 1 7r = jdYPI;2 2G oko (VR)3 j dYIJI

~

=

(4.52)

7r---

where

d

-

R f2

=

m2

1--2G oko

(X - ad)2 + f2 4m 2 sQ2(s + Q2 _ m2) (b2 _ ( 2 ) (5 + m2)2(5 + Q2)2

( 4.53)

Up to this point, no approximation has been made, and all results are still exact. The function R-n/2 is a sharply peaked function around X

=

ad when m 2

-t

O. We can use this fact to simplify the integration over X with the following

102 identifications,

~ F(ad)jbe d~ = F(ad) In 4{(bc)2f~ (ad)2}

j-bebe dX F(X) VR

-be V R be dX _F-=(X~) ~ F( ad) jbe dX j -be (VRP -be (VR)3

=

2F( ad)

(4.54)

f2

We can verify the reliability of this approach by performing the calculation for the 'one' part in the Bose factor, and indeed we recover the result of the Parton Model as expected. For the thermal part the calculation is now very straightforward, but the algebra is tediously long, and so is the resultj there is no point of writing it here. Most of the terms will drop out when m 2

~

O. The more useful form will be

obtained if we go to the deep inelastic limit, and neglect terms of order O(1/Q2)j in this case the result can be written as 2o: s 37l'

{

3 1 1 + Z2 1 - Z Q2 5 ----+(1+z)+--log---+-o(l-z) 2 21 - z 1- Z Z2 m 4

1 ( e/3(l-z)k

-

1

) ( -1-" ( -)

1 + Z2

1-

Z

Q2 ) }

-2 + --10 1 - z g -Z2- m

(4.55)

The statistical corrections to the longitudinal structure function can be calculated in the same way, with the invariant matrix given in Equation (4A4). The 'one' part does not contain any Q2 dependence, just like in the Parton r..lodel, but the explicit form is differentj we have discussed the reason when we were discussing Equation (4,45). The terms in the thermal part are all in the order of O(1/Q 2 ), and hence will be neglected. The total contribution to the structure function

Pi,

including the Born

term, can be written in the form -r'/(~. . . ' .r2

= w here

Q2 , (31') =.r~ - -r'l t.

{

the function

0(1- z)

0: Q2 + 2;B'I,/({1,z}j{3k)logm + O:sf:j(z,{3k) } 2

.11 (z, f3 k)

0: .

+ '2JL 3-r'l (4.56)

does not depend on Q2. The Born part does not

involve any gluon, and hence is the same with or without the gluon statistical

103 factor. The modified splitting function is defined as (4.57) The ordered variables a and b, with 0 ::; a ::; b ::; 1, are the momentum fractions of the quark parton after and before emitting the gluon, respectively, and the function 'Pqq(z) is the same splitting function defined in Equation (3.68) of the previous chapter. All these fractions are taken relative to the momentum k of the incoming quark taking part in the hard process. Obviously, (b-a) is the momentum fraction carried by the emitted gluon, and ajb is the final fractional momentum of the splitting quark with respect to its momentum before the emission. from this point on the parameter (31.. in the argument will be understood and will be omitted from the list.

Figure 4.4: The enhanced splitting process.

The interpretation is very similar to the regular splitting. A quark of momentum bk splits into another quark with momentum (ajb)(bk)

= ak and a gluon

with momentum (b-a)k. The probability of splitting is given by P'Iq(a/h), and the presence of gluons in the background stimulates the splitting by the Bose factor {I

+ nB((b -

a)k)}. The enhanced splitting probability is just the product of the

two factors as represented in Equation (4.57). Interestingly, for the one step emission under consideration, the gluon momentum in the enhancement factor takes

104 the value of (1- z )k, although we did not assume anything when we performed the final phase space integrals. This indicates that the leading logarithmic contribution comes from an almost parallel gluon emitted by the quark. Here also we cannot truncate the expansion at the first order because of the inverse logarithmic dependence of the coupling strength,

Ct s (

Q2) log Q2 '" 0(1).

In analogy to the multiple gluon emission in the Parton model, we make an ansatz that the leading logarithmic contribution still comes from the cascade diagram shown in Figure 4.5 , and that this enhanced splitting function also holds for each emission of gluon.

..1------------

Figure 4.5: T!Je cascade diagram for multiple gluon emission in the background.

To avoid long writing of multiple integrals, we introduce a new convolution notation for functions with ordered variables like the enhanced splitting function,

{A*B}({a,b}) ==

(b d.,.

Ja

z-A({a,z}) B({z,b})

From the definition, it is straightforward to verify that this

*

(4.58) -operation is not

commutative, but still associative,

{A*D}({a,b}) f. {B*A}({a,b}) {(A*B)*C}({a,b}) =

{A*(B*C)}({a,b})

(4.59)

105 With this notation, we can express the one gluon emission as a convolution of the modified splitting function and the Born term (up to the function

CtsrJ.

which is

independent of Q2),

(4.60) where

/l,

is the same as the one defined in Equation (3.76) for the Parton Model,

and the Born function wi th ordered variables is defined as

Ff ({ y, z})

= 5(1 - y / z )

This function is the same as the usual Born term. By keeping track of the quark and gluon momenta in the cascade diagram, the n-gluon emission can be expressed as a multiple convolution of the enhanced splitting function and the Born function,

(4.61) and the sum over the leading logarithmic contributions becomes

(4.62) We still have to calculate the process with a gluon in the initial state, and the nucleon structure function can be obtained by inserting the two results together into the master equation.

By definition, the Q2-dependent quark distribution

can be extracted from this structure function by shifting the Q2 dependence to the distribution and assuming the interaction to be Born-like. However, to avoid dealing with the initial gluon process, we will again consider only the Non Singlet distribution. The relation for extracting this Non Singlet part is the same as in Equation (4.19), (4.63) As before, the integration over the azimuthal angle can be performed by symmetry, and that over the variable::

011

the left hand side by introducing the variables (z,w)

106 defined in Equation (4.20). The argument applied to equate the integrands on both sides of Equation (4.21) still holds, and so we can claim a similar result for our case, nNS ( 1,w,

-LLA( z, Q2 ) Q2 ) = la1dz.,..,..o 3" nNs(l,w) :;:2

(4.64)

o z or in terms of the regular momentum (k, cos 0), W

2

-1

n NS ( W, cos 0, Q 2) -

00

w

dk yk.2.,..,..0. nNs(k, cos 0)

- LLA(W :;:2

k' Q 2)

( 4.65)

Up to this point, everything is apparently the same as without the statistical factors. The usual interpretation, saying that different directions contribute independently to the distribution and that only those quarks with momentum 10

2:

W

contribute after emitting gluons of total momentum (10 - w), is still valid. However, this is as far as we can go; the distribution on the right hand side is still unknown, and we cannot express this relation in terms of the measured distribution at a certain value of Q2. Taking the derivative of (4.65) with respect to bring one factor of

Eqq

t\,

will

down from the exponent, but its dependence on the quark

distribution prevents us from separating this function from the distribution in the integral. As a consequence, we can no longer have the usual integra-differential evolution equation. For the same reason, the moment method will not help in this situation. There is a recent idea to introduce the statistical correlation effects into the scene by modifying the AltareHi-Parisi equation [37]. The idea is based on the observation that this evolution equation has a similar structure to the Boltzmann transport equation if the left hand side were treated as a time evolution and the right hand side as a collision factor. The final state correlation effects are brought in through the collision factor. However, as we have seen in our derivation, the modification of the splitting function prevents us from having the aforementioned equation. Starting from the Altarelli-Parisi equation to include the background effects in this way is unjustifiable. In closing this chapter, we would like to highlight one plausible reason why

107 this approach fails to produce the conventional evolution equation. This equation is built on the premise of the factorization theorem, where the perturbative hard-process is completely free from the influence of the non-perturbative effects. Correlating the perturbative calculation with the background through statistical factors violates this basic premise, and hence the resulting evolution equation does not factorize as in the conventional approach.

108

CHAPTER 5

STATISTICAL CORRELATIONS IN THE PARTON MODEL

From the previous chapter we learn that the splitting process is not affected by the direction of the momentum of the splitting parton but by its magnitude only. If we constrain the distribution to one direction, we recover a Parton lVlodel with the capability to accomodate the statistical correlation factors, following the general rule that any emission of quarks or gluons parallel to the background will be blocked by the Fermi factor or enhanced by the Bose factor, respectively.

5.1

General Picture

Before we plunge into the calculation, it will be helpful if we review the picture of a nucleon in the Parton Model, and point out any necessary modification when we introduce the final statistical factors into the formulation. In the one dimensional model, all partons move in the same direction carrying a fraction y of the nucleon momentum P. The probability of finding a particular parton within the range of y and y + dy is denoted by the parton distribution function GO(y). This distribution function contains the structure of the nucleon, and is probed by the photon during the nucleon deep inelastic scattering. In lowest order, where no gluons are involved in the process, the photon is absorbed only by quarks having the correct momentum fraction, that is the fraction matching the Bjorken parameter x

=

Q2/2P . q.

The scattered quark takes the momentum of the photon and moves in a different direction from those in the background, eluding blocking by the Fermi factor. In this model, the blocking factor is absellt completely from the lowest order, in

109 contrast to the Thermal Model where the blocking factor is present but its effect is negligible [9]. Since the partons are not free, but interacting, they emit and absorb other partons continuously, and hence display different characteristics of their distributions when probed at different resolution powers. A primary parton previously perceived as having a momentum k

= yP, by a finer

resolution is perceived as a

cloud of secondary partons each carrying a fraction z of the primary momentum.

In the cloud frame, the photon is absorbed by one of the secondaries having the right momentum fraction z

= Q2/2k . q of the primary, transfering its momentum

to this secondary and leaving the rest undisturbed (Figure 5.1). This argument was used when we derived the master equation (3.1) in the previous chapter. The statistical correlations may enter the formulation at this point, that is when the state of the secondaries matches that of the other primaries, the emission of secondary gluons will be enhanced by other gluons in the primary, while the emission of secondary quarks will be blocked by other quarks in the primary.

SECONDARY

k

PARTONS

\ P

V

PRIMA RY

zk

PARTO NS

Figure 5.1: Statistical correlations for secondary pat·tons

A t first order, order O( as), a primary parton is resolved in to two secondary pat'tons, and for correlation with the background, they must be in the same direction as that of the original parton. However, only one of them can correlate with

110 the background at one time, since the other one must absorb the photon momentum and move away from the primary direction. To interact with the photon, this scattered parton must have the right momentum zk = xP, so that the remaining momentum carried by the unscattered one has to be (1 - z)k = (1 - z)xP/z. By fixing the direction of one of the outgoing secondaries, all the momenta are fixed and no more final momentum integration is to be performed. Since the log Q2 dependence can only arise when we perform an integration over some final state, no such term can be expected from the correlation factors. Furthermore, since the correlations occur with the parton spectators in the background, we may expect that the correlation effects will be of the order of higher twist 0(1/Q 2 ).

5.2

One-Dimensional Model - Parton Model Revisited

We know already that the 3D-parton distribution is not identical with the 1Dparton distribution of the Parton Model. By suppressing the occupation of the transverse modes such that the distribution is sh;;trply peaked around the longitudinal direction, we expect to mimic the condition for the Parton Model. For this purpose, we write the distribution as

(5.1 ) where 1\2 is the normalizatioll factor which will be chosen later such that G~,dkz) coincide with the quark and gluon distributions of the Parton l'vlodel. The function

h

is a sharply peaked function around 1::1. = 0, with width fl

the factorization of the transverse mode is the assumption that

-1

O. Implicit in

Ikl.l « Ikzl; if the

longitudinal momentum is too small, the transverse distribution will be more dominant, and the assumption breaks down [38]. In the Parton Model this assumption is satisfied by going to an infinite momentum frame (see Equation (3.27)). Assuming that we are working in this region, we can set the width of the transverse distribution to zero and associate the transverse distribution with a delta function,

111 so that the distribution functions (5.1) simplify to (5.2)

In the original 3D-distribution, the momentum is not bounded and covers all directions; hence, suppressing the transverse momentum will lead to a onedimensional model with an unbounded momentum, which not quite the unidirectional distribution of the Parton fvlodel. To recover the Parton Model completely, we have to set an upper-bound for the momentum and require all the partons move in the same direction. We will save these last two requirements for later, and see for the moment what results from this unbounded one-dimensional model. Substituting the definition (5.2) to the master equation (4.18), we have (5.3) For the Born approximation, the parton sub-process is given by 1

J (27r(Pk')321.:' (27r)'1 8 (k + q - k') 4(k· 1.:') 4

=

-

=

Jd'lk' O(I.:~) 8(1.:'2) 8 (1.: +

=

0("'0

47r

4

q - k') 2k .1.:'

(5.4)

+ qo) 8([1.: + q]2) 2(k + q). I.:

and to obtain the final result, we substitute this expression to (5.3).

We will

use a similar infinite-momentum frame to the one used in the Parton fvlodel (see Equation (3.24)), that is the frame where the photon is mostly transverse, with small longitudinal component q_ from

-00

to

+00;

= Q2/2xP

ql = Q2,

. The integral over kz ranges

however, the theta function limits the contribution only to the

positive range of k z • The final integral is then

(5.5)

112 The last line is mathematically identical to the result of the Parton NIodel if we set the normalization to (5.6) Even without requiring the partons to move in the forward direction, only the forward moving partons will take part in the scattering, and even more, since the value of x is set kinematically to 0

~

x ~ 1 regardless of the model, only the

partons with momentum 0 ~ k" ~ P can be probed by the photon. This shows that only a portion of the entire parton distribution can be probed by the photon at Born level in this one dimensional model. If we agree that probing by the photon should be able to cover the par tons in the nucleon entirely, then we have to accept the accessible domain of momentum as a criterion for the model. This is what was done in the Parton Model. From this point on we will adopt the convention of the Parton Model, where all the pat·tons are moving in the same direction and their momenta are bounded above by the nucleon momentum P. Equation (5.2) for the distribution should be restricted to kz 2: 0, and can be written in general as

(5.7) where A2

IS

given by (5.6).

With this definition, the structure function (5.3)

becomes

(5.8) The upper limit of the integral k:: = P is set by the model, while the lower limit k min = xP, with x

== Q2/2P . q is set by the requirement that

s ==

(k

+ q)2

2: O.

The last line is nothing but the master equation for the Parton l\'Iodel. For a more familiar form, we can express (5.S) ill terms of invariant variables, since the

113

structure function itself is a scalar quantity,

F 2(x,Q2)

= I:e~ q

ely - x G~(y)F2(-,Q2) 1y y 1

(5.9)

x

The recipe to incorporate the statistical factors into the Parton Model is now obvious, that is we have to employ the definition (5.7) in the general expression for the transition rate in a medium (4.13). Note that every further appearance of the distribution function from the background will always be accompanied by the normalization factor A2 • By dimensional consideration, we know that we have to compensate its dimension by Q2. This supports our assertion that the background corrections will be of the order of higher twist 0(A2/Q 2).

5.3

Statistical Factors in Order 0(0:.)

Actually, what we have discussed so far is the effect of the statistical corellations on the asymptotic states of the pm'tons.

For interacting partons, there should

be some effect also on their dynamics, for example, in the Thermal rvlodel, the effect appears on the on-shell part of the propagators and is handled by doubling the field degrees of freedom. Unfortunately, at this moment, we do not have a suitable field theory in an arbitrary background distribution which can be applied to the Parton ivlodel. Adopting the thermal propagators to fit our case is not very straightforward since there is no equivalent counterpart to the temperature parameter. To first order, for tree diagrams such as the gluon emission and the initial gluon processes, the on-shell part of the propagators will be avoided by regularization, and only the vacuum part will participate, so that for these processes we can apply the usual QeD theory in the calculation. The only place of order 0(0:.) where the type-2 vertex can appear is in the self-energy diagrams. We expect this part will not contribute significantly to the observable structure functions other than for renormalization as in the vacuum case, so that we can proceed to use

114

the usual vacuum QeD theory for our calculations. The only new feature in our approach is the appearance of the statistical correlation factors in the phase space of the outgoing partons.

5.3.1

Initial Gluon Process

Experience from the ordinary Parton Model shows that the initial gluon process, /* +G ~

q+ij does not suffer from

III

divergences, except in the logarithmic term.

I-loping this simplicity to continue, even under the influence of the background, we will begin our study with this process. The relevant phase space element, including the statistical correlation factors, for the outgoing quark-antiquark pair is given by [dLips]' =

=

[dLips]{l - nq(k)}{l - nq(k')} [dLips]{l - nq(k) - nq(k')

+ nq(k') nq{k')}

= [dLips]{l - i\2G~(kz)8(kJ.) - i\2G~(k~)8(k~)

(5.10)

_1\4 G~(kJ8(kl.) G~{k~)8(k~)} where [dLips] is the usual phase space without the background effect, defined for k and k' following the definition in Equation (3.74). In the last expression, we have replaced the 3D-distribution functions by the unidirectional distribution functions given in Equation (5.7). The six-dimensional integrals can be performed with the help of the six available delta fllnctions, four [rom the momentum conserving delta function in the d[Lips] and two from the transverse momentum delta function. The last term, containing two distributions and eight delta functions, will not contribute due to the conOict in the argument of the delta functions. Physically, this term cannot contribute because one of the final particles must absorb the photon momentum and move away from the background. For explicit calculation we will use the same infinite-momentum frame as the one used previously in the Born term.

115 Following the correlation terms in the phase space factor (5.10, the structure function can be written in three parts, (5.11) The first term comes from the 'one' part, the second term from the blocking effect by the quark background, and the last term from the blocking effect by the antiquark. For the invariant matrix, we can still use the old expression from the Parton Model given in Equation (3.57). To be consistent with the regularization in the 'one' part, here also the gluon must be slightly off··shell and spacelike, GI,GI' _m 2 =

_EQ2.

=

Following the usual procedure, by integrating the invariant matrix

over the final phase space, and substituting the result to the master relation, we obtain the initial gluon contribution to the structure function. The 'one' part is identical to the previous result of the Parton Model without the background, so we will not pursue this part any further. Also we need to calculate only the parallel quark case, since the parallel anti-quark will yield the same result. The calculation is somewhat simplified if we include the gluon mass in the definition of the parameter z,

z == 2q. G _

m2

(5.12)

In the limit m 2 ~ 0, this definition coincides with the one used in the 'one' part. In terms of this parameter, the invariant matrix for a parallel quark can be written as (5.13) For the moment, our interest will be just on the total sum structure function, because the longitudinal one will only shift the finite part. After integrating over the final momentum and the transverse momentum of the quark, the integration left is

(5.H)

116 The delta function fixes the magnitude of the outgoing quark to kz = (1- z)G + /2, where the gluon momentum is given by G+ == Go+G z = 2xP(1-ez)/z. Substituting the result of (5.14) into the master equation, we obtain the structure functions for the blocking effect,

(5.15) The lower limit of the integral is

Zmin

= x/(l + ex).

Apart from the 2/e term,

no other singularity appears in the hard process, and so we can drop all other appearances of e from the expressions, simplifying the expression to (5.16) This result shows explicitly ",Iwt we Illentioned earlier in the general consideration. The blocking effect is indeed in the order of higher twist, and is provided by the unscattered secondary parton (the quark) having momentum kz

= (1- z)G z. The

other secondary (the anti-quark) with momentum k~ = zG z absorbs the transverse momentum of the photon and scatters away. As we can see in this result, the soft-gluon is excluded from the integration region, because the initial gluon must have enough momentum to produce the final quark-antiquark pair. Therefore, the appearance of any divergence should be attributed to the collinear singularity. In contrast to the logarithmic singularity log 1\3/m 2 in the main part, the singularity arising from the statistical correction factor has the form 2/ e, or more precisely 2A 2 /1'n2. In addition to having the same origin, both singularities are independent of Q2, and we expect to treat them simultaneously on equal footing. We will deal with the singularity later after we discuss the gluon emission process. One other thing to note from this result is that, unlike in the Thermal ~dodcl, the splitting probability docs not resemble any sort of enhanced or blocked splitting function; this is because the main part and the correlation pnrt involve differcllt ordcrs of twist.

117 5.3.2

Gluon Emission Process

The final phase space factor for the gluon emission process "1*

+q

-jo

q + G is

similar to Equation (5.10), except now the gluon enhancement factor replaces the anti-quark blocking factor. Here also the structure function can be written in three parts, (5.17) The term with double distributions does not contribute for the same reason as in the initial gluon process. The 'one' part is again the old Parton rvIodel result, and will be skipped in the subsequent discussion. The last t,wo terms contain the statistical factors, the blocking effect from the quark background, and the enhancement effect from the gluon background, respectively. Here too, we will use the same massive gluon method for the regularization, and we can take over the invariant matrix from the ordinary Parton Model as given in Equation {3.38 for the structure function, In case of a parallel quark, one of the Mandelstam variables vanishes,

I == (k - k')2

= 0, and

the invariant matrix simplifies to

--2

IMIE = (5.18)

In this expression, the variable z is defined with respect to the initial quark momentum

Z

== Q2/2h. q, and the parameter

t

== m 2/Q2 is the uSlla.l gillon mass

regulator. The invariant matrix from the longitudinal photon polarization is propOl·tional to t, and hence will not contribute. After integrating over the gluon momentum G and over the final quark transverse momentum

_A2

=

J(~~~ G~(k~) 8([k +

q - k']2 - m 2)lk

~: G~(ql -

{{l ~

-2a s C p

Z - tZ])

=0

L

z)2

kl.,

we are left with

1M I~

+ (1 _Zc:;)2 }

(5.19)

118

= (1 -

The last delta function fixes the final quark momentum to k~

z-

f.Z )kz -+

(1 - z)kz as expected. Inserting this relation to the master equation, we have

:FJ'l(x, Q2) '-'

=

J

~ 2 dkz L..J eq kz GO(k q'z ) ( GOq(k[l _

= -

A2

E2e;a C s

F

Q2

2as C) A2 F Q2

z]) {(1 -

Z

z mu

dz

1

-

X

Z

x

z

GO(~ _ .)) Tq

Z

Z)2

x

G~(-)

Zmax

= 1/(1

+ f.).

(5.20)

z

{1 +(1 (1_ - Z)2}

We have omitted all terms that will not contribute when limit for the integral is

+ z}

Z)2

f. -+

O. The upper

The emitted gluon is knocked off the

primary direction, and so no collinear term 2/ f. appears. However, another type of singularity appears at the end point, corresponding to the soft momentum of the outgoing quark. A similar treatment can be given to the enhancement part; in this case, the outgoing gluon is moving parallel in the forward direction, experiencing the enhancement from the background. Integration over the final quark momentum and the transverse gluon momentum can be performed without any problem by using the delta functions. The remaining integrals are

dk G O( Gz )-2 Fr,