Transverse momentum distribution of lepton-pairs in Drell-Yan process in an asymptotically free scalar field theory

ANNALS

OF PHYSICS

153, 205-238

(1984)

Transverse Momentum Distribution of Lepton-Pairs in Drell-Yan Process in an Asymptotically Free Scalar Field Theory* PORTER W. JOHNSON, MIGUEL Department

of Physics,

Illinois

MEDINA-GARCIA,

Institute Received

of Technology. May

AND WU-KI Chicago,

Illinois,

TUNG 60616

4, 1983

The transverse momentum dependence of the parton distribution function and the Drell-Yan cross section is studied in detail in the asymptotically free scalar field theory in six dimensions. In the context of the renormalization group approach of Collins, particular attention is devoted to the transition between the small qr formula (due to multi-quanta effects) and the large (I~ formula (due to single hard-quantum emission). The calculation represents a case study of the application of the renormalization group method and provides a guide to the corresponding study in QCD.

I. 1NTRoDucT10~ The transverse momentum dependence of parton distribution and fragmentation functions is of great interest in the study of the underlying theories of the hadron interaction. From the practical point of view, a quantitative understanding of the transverse momentum distribution can also be of considerable significance in the interpretation of forthcoming data on the production of W and Z bosons in high energy j?p and pp collider experiments [I]. In the conventional approach to perturbative field theory [2], unambiguous firstorder results can be obtained only in the region q: N s, where qT is the transverse momentum variable and s is the overall energy scale of the processin question. Such results are of limited utility because: (i) cross sections become very small in the high qT region, and (ii) the nature of the transition to moderate and low qT region is not understood, so that the range of validity of the first-order result is not known in practice. In the last few years, new resummation methods have been developed [3-51 to obtain predictions also in the “two large massscale” region m2 Q qt Q s, where m is some typical hadron mass. Most of these results are valid only in the “double leading log approximation” (DLLA). The intuitive approach of Parisi and Petronzio [4], in particular, suggeststhe potential importance of this line of study becausethe predicted perturbative effect (the Sudakov form factor) can dominate the contribution

* Work

supported

by National

Science

Foundation

Grant

PHY-81-06908.

205 0003.4916/84

$7.50

Copyright 0 1984 by Academic Press. Inc. All rights of reproduction in any form reserved.

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JOHNSON,

MEDINA-GARCIA,

AND

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of the non-perturbative “intrinsic” transverse momentum distribution, and hence yield meaningful predictions even when detailed knowledge about the latter is lacking. In spite of its great promise, however, the Parisi-Petronzio formalism [4, 6, 71 is still limited in applicability because of the DLLA that is inherent in the approach, and by the uncertainty in transition to the large qT region [ 1, 71. TWO years ago, J. C. Collins [S] investigated the transverse momentum distribution of partons in the asymptotically free nongauge field theory 4:. He applied renormalization group methods to derive results in the region qT comparable to m which are subject only to power-law corrections (i.e., which are valid to all powers of logarithms of large ratios). In addition, he demonstrated theoretically that there is an asymptotic smooth transition from the qT - m formula (due to multi-quanta emission) to the simple first-order perturbative result at qi - s (single hard-quantum emission). Thus, in principle, the entire qT distribution in inclusive lepton-hadron processes can be derived-apart from certain potential non-perturbative effects which can only be treated phenomenologically at present. This approach has since been extended by Collins and Soper [9] to the more interesting case of Quantum Chromodynamics (QCD). Their results theoretically extend the Parisi-Petronzio formalism to all orders in large logarithms and explicitly provide the means to cover the entire qT kinematic range. This paper is the first in a series to investigate the phenomenological consequences of the new theoretical development. Here we explore in detail the case of the asymptotically free (but nongauge) scalar field theory in six dimensions 4:. There are several specific objectives for studying this simplified theoretical model: (i) to investigate, by numerical calculation, whether, and how, the asymptotic matching formula of high and low qT regions obtained by Collins [H] works at practical energies; (ii) to gain a concrete understanding of the X-, qT-, and Qz-dependence, including their correlations, of parton distribution functions in a tractable field theory; (iii) to abstract qualitative features of “lepton-hadron” cross sections which can be contrasted with those of QCD and other candidate theories of hadron interaction and with experimental data; and (iv) to use this simple theory to develop a systematic, efficient, and reliable computational scheme applicable to a variety of physical processes, and which can be extended later to QCD and other theories involving quarks and hadrons. In Section II we first summarize the relevant theoretical formalism and results pertaining to the parton distribution function P(x, k, ; ,u), where x is the longitudinal momentum fraction, k, is the transverse momentum, and y is the momentum scale of the probing process (or renormalization scale)-usually set to be of order Q. This formalism enables the calculation of P(x, k,, Q) provided the usual structure function f(x, Q) is specified at some value Q = Q,. Dependences of P(x, k,, Q) on k, and Q are closely examined, transitions from the short-distance to the long-distance regions are studied in detail in the impact parameter space, and asymptotic matching to firstorder perturbative formulas is demonstrated. In Section III, we apply this formalism to the analogue of the lepton-pair production process (generalized “Drell-Yan” process). The cross section doldx, dx, dqT is calculated both in the qT - m and the

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st N s regions. (Here qT denotes the transverse momentum of the created pair.) The asymptotic smooth transition of the cross section between the two regions of qT is rederived. In Section IV, the numerical calculation is described, and results are presented. Finally, in Section V, the results are discussed, and extension to QCD is commented upon. Most of the theoretical results summarized in Sections II and III are transcribed from Ref. [8] for the purpose of phenomenological application. We refer the reader to the original paper [8] for the theoretical background. Our extensive use of BesselFourier transform’ simplifies many of the original derivations of Ref. [8]. For the convenience of interested readers a number of explicit proofs in this modified approach are presented in the appendices.

II. THE PARTON DISTRIBUTION

FUNCTION

(a) Factorization

A central feature in the application of perturbative field theory to high-energy lepton-hadron scattering processes is the factorization of a relevant cross section into a hard scattering amplitude which can be computed in a perturbation series, and a soft factor which can be interpreted in terms of parton distribution (or fragmentation) functions. Both factors depend on a renormalization scale parameter p, which represents the momentum scale at which the system is being probed. The dependences of these functions on ~1 are determined by appropriate renormalization group equations, which contain coefficients that also are computable in perturbation theory. The factorization theorem is most directly proved by examining the asymptotic behavior of a general class of Feynman diagrams.2Y3 In the renormalizable (and asymptotically free) 4: theory, let #a be the renormalized field operator.4 The “current” operator which plays a role in “lepton-hadron” processes will be taken as j, = fZZ,$k, where Z and Z, are the renormalization constants for the wavefunction operator and the mass term in the Lagrangian, respectively. (See Ref. [8].) The factorization of the “deep inelastic scattering” (DIS)

’ We thank J. C. Collins for initially calling our attention to the advantage of using the impact parameter representation. ‘Some physicists favor the use of Wilson’s operator product expansion (OPE). However, the applicability of OPE is relatively limited. In any event, the validity of OPE itself is only established by Feynman diagram analysis in the first place. 3 The validity of the factorization theorem for the pair-production process in QCD has been called into question recently [lo]. The controversy does not apply to the nongauge theory that we are concerned with in this investigation. 4 The precise definition of 4, depends on the renormalization scheme. For definiteness, let us adopt the MS scheme. This is not important for most of our discussions.

208

JOHNSON,

MEDINA-GARCIA,

AND

TUNG

q cl, I 4 I3z EP

/

EP

f

P FIG. section

P

1. The factorization in the Bjorken scaling

of general limit.

P

Feynman

P

diagrams

for

structure function can be represented diagramatically to the formula

the deep

inelastic

scattering

cross

as in Fig. 1. This corresponds

(2.1)

where the hard scattering amplitude (Wilson coeffkient) w is perturbatively computable provided we choose ,u of order Q (so that ln(Q/p) factors, which appear with loop diagrams, do not become large enough to invalidate the series expansion). Similarly, the “pair-production” cross section factorizes in the asymptotic regime s,Q’+co with qTNm, as shown in Fig. 2. Details are given in Section III; also cf. Ref. [8]. Let W(s, xi, x2, xr) N xlxZ dW/dx, dx, dx, where, as usual, x,,~ = (q” f q’)/fi and xf = qt/s, then

where the parton momenta are ki = xi pi + I&, i = 1, 2; iCi, are 4-vectors transverse to pi, P(xi, kiT,p) are the parton distribution functions at scale p, and S(Q/,U,~) is

PI

PI K,

4

s

P I I

K2

K, s

q K2

P

8 P2

FIG. 2. high-energy,

The factorization low transverse

of general Feynman momentum region.

diagrams

P2

for the pair-production

cross

section

in the

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209

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the hard parton annihilation vertex which reduces to 1 if we choose ,u - Q and keep only the leading pertubative term. (b) The Parton Distribution

Functions

P and p

The Feynman diagram analysis (which yields the factorization formula) naturally leads to the following field-theoretic definition of the parton distribution function

W,k,,Pu) 1111: dkP(x, k,, ,u) = k+ j o6

G(k, p, iu)

where k* = (k” + k3)/fl are the light-cone absorptive part of) the Green’s function W,

P, p) = 1 d6y e ik’y(~

components

I T[&.(Y)

(2.3)

of k” and G is (the

&.(O)ll p>.

(2.4)

The Feynman rules for calculating P(x, k,, p) in perturbation theory are given in Appendix B. Because the k,-dependence often occurs in convolution integrals such as Eq. (2.2), it is convenient to define the Fourier-Bessel transform of P and study the dependence on the impact parameter variable b which is conjugate to k7: p(x, b, ,u) = 1 d4k, eik;’ ‘P(x, =-

Working with P(x, b,p) perturbative distribution perturbative component using the inverse of Eq.

47t2 O” k: dk,P(x, b Io

k,,p)J,(k,b).

will also facilitate the convolution of the short-distance, (Subsection (c)) with the longer range confinement, non(Subsection (d)). We will eventually compute P(x, k,, ,u) (2.5)

P(x, k,, ,u) = &I

d4b eik;.‘p(x,

1 =a0 (c) Relation

k,, ,u)

m I

b, ,u)

b2 dbF(x, b, ,u) J,(k, b).

(2.6)

betwen F and f

At the intuitive level the parton distribution should be related to P(x, k,, p) by

of deep inelastic scattering (Eq. (2.1))

j-(x, ,u)= j d4k,P(x,k,, P>= ,';y, F
(2.7)

210

JOHNSON,

MEDINA-GARCIA,

AND

TUNG

However, in scale invariant renormalizable field theories, P(x, k,, ,u) N kr4 at large consequently P(x, b, ,u) diverges logarithmically as b + 0, . In practice, f(x, ,u) has to be defined by a renormalization procedure; say, according to the MS scheme. Then the two functions f(x, cl) and &x, b, ,u) are related by’ k,,

WV

The kernel T(<,pb,p) can be calculated in a leading order perturbation series provided we choose pb N 1. This calculation is described in Appendix C. The result is:

W, ,ub,/$30~:, = S(l - t) - a(~) Ql - Q[ln nb’,u* t y].

(2.9)

Here a@) is the “effective coupling parameter” of the 4: theory. To the leading order in perturbation expansion (see the Appendix) we have (2.10)

where n is a scale parameter characteristic of the theory. Combining Eqs. (2.8) and (2.9), we obtain &,,

(x,b,,u=;)=f(x,p=;)

tPr;(x,b,;)

where c is a constant parameter of order one, f(x,,~) is the “structure function” of Eqs. (2.7) and (2.8), and the second term on the right-hand side arises from the firstorder contribution of Eqs. (2.9), i.e.,

The subscript SD on p (referring to “short distance”) serves as a reminder that these formulas are valid only for small b; say, b G l//i where /i is the scale parameter of Eq. (2.10). The results given above are not yet suitable to be used in Eq. (2.6). First, we need p&x, b, ,u) as a function of two independent variables (u, b), not effectively of one variable b as in Eqs. (2.11) and (2.12). This problem can be solved by using the renormalization group equation satisfied by P(x, k,, cl) and &x, b, p). It governs the ways of ’ One can regard f(x,p) and &x, b,.u) (b + 0) as being the results of two different regularizing the divergent integral in Eq. (2.6). Hence they are related by a finite renormalization kernel T. The moments off and P (in the variables x and v) are related by a finite renormalization constant.

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dependence of these quantities scale”) p:

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on the “renormalization

parameter”

(or “probing

(2.13)

where y = a/6 to first order in a. Detailed definitions of p(d/&) the equation are summarized in Appendix A. The result is:

Lk

and the solution of

b,lu)= B,, (xh;)[-$$“.

(2.14)

The second factor on the right-hand side is the renormalization group (RNG) evolution factor which allows the evaluation of ps,,(x, b, p) for all values of p for which a@) < 1. The combined use of Eqs. (2.12)-(2.14) enables us to compute the short distance transverse momentum parton distribution function ps,,(x, b, cl) provided the inclusive longitudinal momentum parton distribution function f(x, p) is given. This connection appears to be a quite remarkable result. Its root lies in the intuitive relation, Eq. (2.7). Its quantitative realization represents an elegant application of the renormalization group equation applicable to renormalizable and asymptotically free theories. These results alone, however, are still not sufficient for the calculation of P(x,k,,p) (Eq. (2.6)) and W(s, xi, x2, xT) (Eq. (2.2)). We need to know the behavior of P(x, b, p) in the intermediate and long-distance region. Before passing to those regions, it is instructive to examine the behavior of psr,(x, b,p) as a function of b for fixed x and P. For extreme short distances, b < p-’ < II -I, it follows from Eqs. (2.11)-(2.14) that

This function increases without limit as b decreases to zero for all x. The proof is given in Appendix D. Fs’,,(x, b, p) continues to decrease as b increases in the range bp - 1. This can be seen by rewriting Eqs. (2.1 l), (2.12) as k,(x,

byP) bu--l.f(X,~) -

aCu)Pnnb2p2+ ~1g(x,c1)

(2.16a)

where (for frequent later use) we have defined (2.16b)

For fixed P, the dependence on b comes from the second term, which is decreasing. In Fig. 3, we show a typical curve of ps,,(x, b, p) as a function of b in the short-distance

212

JOHNSON, MEDINA-GARCIA, 0

AND TUNG

P”(x,b,Q) t

x=045 A=OlO(A) Q= 45(A)

025-

0.20

-

015

-

O‘OOo5 b(A-')

FIG. 3. The short-distance parton distribution function P,, in the impact parameter space.

function P,, , and the long-distance distribution

region (along with an anticipated long-distance curve to be discussedbelow). Details of the numerical calculational procedure will be discussedin Section IV. (d) The Parton Distribution at Large and Intermediate Impact Parameters The large-distance behavior of &x, 6, p) cannot be determined by the perturbative methods. On the other hand, existing non-perturbative methods do not yet provide reliable quantitative predictions. Therefore, we can only rely on reasonable phenomenological prescriptions. “Confinement effects,” as reflected in typical hadron size and the width of transverse momentum distributions, suggest a cutoff factor characterized by a scale parameter the order of 0.5-0.9 GeV. One of the simplest possible prescriptions is therefore to have h,

b, P) latgeb

P&x,

b) = A e-A2b2’2.

(2.17)

Here the subscript LD denotes “long distance.” It is an important issue as to how to match the short-distance formula, Eq. (2.14), with the long-distance formula, Eq. (2.17). Clearly we need to examine the behavior of p,,(b) as b decreasesfrom the region bA << 1, and of ps,,(b) as b increasesfrom the region b/i < 1. A moment’s reflection reveals that, whereasp,,(b) is well behaved for all b (and hence poses no problem), FsD(b) has an artificial singularity as b/i -+ O(1). (N.B. a(c/b) = S/3 ln(c/bA).] Th is means that we cannot use ps,,(x, b, ,u) as given in Eq. (2.14), beyond its region of validity (i.e., b/i < 1) in a simple-minded

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prescription-such as defining P(x, b, ,L) as the product of Fs,,(x, b, ,D) and pLD(x, b) over the full range of b. In order to ensure a reasonable transition from the short-distance to the longdistance regions, we need to regulate ps,, in such a way that: (i) the artificial singularity mentioned above is removed, and (ii) the behavior of the regulated function in the “intermediate region” (d -i < b < /i - ‘) permits a smooth passage to pLD(x, b). Clearly, there is no unique way to do this. Therefore, we shall have to investigate the sensitivity of our results to the prescription adopted for this purpose. (This problem is encountered in practically all papers studying the transverse momentum distribution using the perturbative method. However, this prescriptiondependence as such is seldom discussed explicitly in the literature.) A commonly used prescription for avoiding the singularity of a@) as b -+ l/A is to make the substitution, a@) -+ a*&)

= a(~*)

where

(Qo > A>;

cl*=drn

(2.18)

so that a*@) is well defined for all ,u, and a*(p) N a($) for large ,u (i.e., ,U9 Q,). As a consequence,

0 C

“b

A

const =a large

b

hence

%
Q; I, P).

(2.20)

This behavior has two drawbacks: (i) it does not naturally represent the trend of ps,,(x, b,p) from its region of applicability at short distance (as depicted in Fig. 3); and (ii) it does not match well with the expected long-distance behavior as represented by Eq. (2.17). These observations are illustrated in Fig. 4 in which the extrapolated curve is represented as the dashed line labelled PSI, [a(l/b*)]. Applied to actual calculations, these features could be the source of unphysical results for P(x, k,, ,LL)and W(s, xi, x,, xT). A kink in b-space results in artificial oscillations in k,-space through the Fourier-Bessel transform, cf. Eqs. (2.6) and (2.2)) An alternative procedure which is designed to alleviate these problems is to make the substitution, (2.21) where /I is a (positive) parameter. It is easily seen that for large p (or short distance), (2.22)

214

JOHNSON, MEDINA-GARCIA,

-.‘;- ------

AND TUNG

P” so[a(+)]

.\

0.20

',\,FSD

[a*+]

\

0.15

I

o.lo IL-&&-& 0

0.5 b(A-'1

1.0

FIG. 4. Two prescriptions for extrapolating the short-distance formula into the long-distance region discussed in Section II.

as required. On the other hand, at long distance (i.e., small p), we have:

or a*

1

(

-b

1

Gf

W)‘W4.

Since F&x, b, ,u) is inversely proportional to a(c/b) (cf. Eq. (2.15) and Appendix D), the substitution of a* for a in the calculation of ps,, will result in a power-law falloff of Fs,, as a function of b. This naturally extends the decreasing trend of Fs’,, from the region of small impact parameter b (Fig. 3) and, at the same time, makes the transition to the long-distance exponential falloff at large b a smooth one. An extrapolation curve based on this prescription is shown as the dash-dotted line in Fig. 4 labelled P”,,[a*(l/b)]. Using a reasonable regularization procedure, such as that described above, we can now present the Ansatz for the parton distribution function in impact parameter space over the entire range of 6, (2.24)

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The superscript * on the first two factors on the right-hand side indicates that a regularization procedure beyond the region b/i < 1 is applied. We recall that the first factor is given (to first order in a) by Eqs. (2.1 l), (2.12), and the second factor arises from the evolution from the scale c/b to ,u. We shall call the latter the renormalization group factor. (In QCD, this factor is a part of the generalized “Sudakov Factor” in the Collins-Soper formulation.) Equation (2.24) forms the basis for the calculation of the transverse momentum parton distribution function P(x, k,, Q) (Eq. (2.6)) and the pair-production cross section W(s, xi, x2, x7) (Eq. (2.2), and Section III below). The numerical results of these calculations are presented in Section IV. (e) Asymptotic Formulas for P(x, k,, Q) It is instructive to examine the asymptotic behavior of P(x, k,, Q) at large k, for three reasons: (i) to exhibit the internal consistency of the theory, (ii) to serve as the precursor of a similar result for the “cross-section” formulas which play a key role in the next section, and (iii) to provide the basis for checking the accuracy of numerical calculations. We shall derive two asymptotic formulas for P(x, k,, ,u) which have slightly different regions of validity, but which should agree with each other in the overlapping region. First, let us consider the behavior of P(x, k,, P) in the region where k, and p are both large and of the same order, i.e., k, - 001) $ d. According to familiar properties of the Fourier Transform, we expect the Fourier-Bessel integral, Eq. (2.6), to be dominated by small L-of order ,K’. The behavior of the integrand &x, b,,u) in this region is given by Eq. (2.16). Substituting Eq. (2.16) into Eq. (2.6) we obtain (2.25) The kq4-dependence is characteristic for this scale invariant renormalizable field theory. The a(,~) factor indicates that the large k, limit is associated with the emission of a single hard quantum with effective coupling a(k, -,u). The above formula can be “renormalization-group-improved” by the observation that, because of the evolution equation (2.13), P(x, k,, p) can be written as (cf. Eq. (2.14)) 119

P(x, k, , p) = P(x, k,, ak,)

* [ 4W

1

’

(2.26)

This equation is valid for k, and ,u “large” but not necessarily of the same order. In this formula a is an arbitrary number of order unity. Invoking Eq. (2.25) for the first factor on the right-hand side of Eq. (2.26), we obtain (2.27)

216

JOHNSON,MEDINA-GARCIA,AND

TUNG

In the section on numerical calculations, we always compare the function P(x, k,, p), computed from the Fourier-Bessel integral, with the asymptotic formulas (2.25) and (2.27) to check for consistency. III.

INCLUSIVE

PAIR PRODUCTION

AT NON-ZERO

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(a) General Considerations Let us turn to the inclusive pair-production

process depicted in Fig. 5,

!+4(PI> + $*(P*) + “Yv7) +x -

4IUI> + M4)*

(3.1)

The cross section, after summing over X and integrating over the center-of-mass variables of the (I,, I,) pair, is proportional to the W-function introduced in Eq. (2.2),

WSl PI, P2)= $w*x:s3~

a62e’““%v* lAz)Ao)l PI P*>

(3.2)

where s = -(p, + pZ)*, and j(z) is the (renormalized) current operator defined in Section II. The function W is depicted in Fig. 6. For convenience, W(s, x, , x2, x,) is chosen to be dimensionless. We summarize some of the kinematic relations which are useful for subsequent discussions. We always assume s, Q* % m*, where m represents a typical hadronic mass (of “physical” states as well as partons).

0) (ii) (iii) (iv> (v)

s = -(P, + P2YT

Q’ = -(I, + 12)2; p1.qT=P2.qT=@

qU=x,p’;+x2pt+q$ t=Q*/s=x,x,-x;; xF=xl -X2, XT = dfi; boundary of allowed physical region:

(3.3)

“7” time-like -+ xi x2 > xt , “X’ time-like + (1 - x1)( 1 - x2) > x:.

FIG.

5.

The general

structure

of inclusive

pair-production

amplitude.

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217

4

4

FIG. 6. The general structure of diagrams contributing to the cross section W for pair production.

In the center-of-mass frame of the initial particles, it is convenient to use the “lightcone components” of vectors: Vu = (I’+, VT,, V-), where V* = (T/O f V3)/d2, V3 = longitudinal component along the beam axis. Thus, PY = (l/w,

0’3O),

P; = (0, 0’7m), and 4” = (XI \/s/2,

t&,x2

&q.

It is then obvious that (3.4a) and the rapidity variable y is given by q” + q3 y = In qo-q3 =q.

(3.4b)

Two sets of independent variables will be used in the numerical calculations: (s, xi, x2, qT) and (s, 7, y, qT). When the second set is used, it is convenient to define an alternative set of longitudinal momentum fraction variables: xA = fi xB=fieey.

eiy (3.4c)

(b) W in the Low qT Region As mentioned in the previous seckion, the function W factorizes in the limit s, Q2 -+ co and qT - m is accordance with Fig. 2 and Eq. (2.2). The transverse momentum of the produced pair results from those of the annihilating partons which in turn arise from the cumulative effect of almost collinear parton emission (real and virtual). Transforming to impact parameter space according to the definition

218

JOHNSON,

Eq. (2.5), and invoking rewrite Eq. (2.2) as

MEDINA-GARCIA,

the convolution

W(S,X,,X~,~~)=~~~X~S~

AND

TUNG

theorem for Fourier transforms, we can

I

d4be-+b

.

.&q,b,W2

%2

7

b,~1.

(3.5)

According to the discussion in Section II, P”(x, b,,u) can be represented as a product of two factors, one due to perturbative effects (for small b) and the other due to nonperturbative effects (for large b) as shown in Eq. (2.24). We obtain: w,(s,

x,, x2, qT) = 4n4x;s2

I

d4b e-i~T”~Azb2

]S2($Pj

0) 219 I a*(clb) J (3.6)

I

where PSI, is explicitly given by Eq. (2.11). The subscript L on W, reminds us that this is the formula for the low qT region. The vertex function S (Fig. 2) is calculable in a perturbative series if we choose p = c’Q with c’ of order 1. To first order in a, we have (cf. Ref. [B]) S’(Q/p,p)

= 1 t a(Q)[ln 4~”

t 3 - Y].

(3.7)

If we only keep the leading terms (in a) for S and P,, in Eq. (3.6), we obtain: W,(s, x,, x2, qT) = 47c4x;s2

I

d4b e-i’+6-A2b2

What is the practical region of validity of the asymptotic formulas Eq. (3.6) and Eq. (3.8)? Existing theory cannot answer this question in a definitive way. However, a good clue is offered by examining the limit of Eq. (3.6) as qr becomes of order Q and comparing the result with the “high qT” formula to be discussed in the next subsection. Let W(s, XI, x2, x-r) = 4Tlii(o) W,(s, x,, X2,&).

(3.9)

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Using arguments which are parallel to those given in Subsection II(c), we obtain

WCs,x, >~2,x,) = (27+(Q) x; *b-(x,, +

terms of order a’or m"/Q'

Q>g(xz,

Q>g(xll Q>l

Q>+.0x2,

(3.10)

'

Since the basic formula for IV, (Eq. (3.5)) is valid only for “low” qT, the above expression for w does not really represent the behavior of the physical cross section at large qT. Its use will be restricted to providing a better understanding of the transition from the low qT to the high qT regions as will be discussed in Subsection (c) below. (c) W in the High qT Region In the qT- O(G) k inematic region, the cross section factorizes in a slightly different fashion, as indicated by Fig. 7. The parton momenta are collinear with respect to the corresponding parent particle, and the central “hard scattering” amplitude COis given in leading order perturbation theory by the diagrams of Fig. 8. In this instance, the origin of the (large) transverse momentum for the produced pair comes from recoil after emission of a hard parton (the intermediate state summed in Fig. 7). We obtain W, = (27+(Q)

*f(rlyQ)

FIG. 7. The factorization represents a “hard scattering”

595/153/Z-2

x1x2x;

[(r,-xlfx2+x;

of the W-function amplitude arising

+

(~*-x2;xl+x:

’ fG2 I

3

Q>.

in the high-energy, high transverse momentum from diagrams such as those of Fig. 8.

(3.11)

region.

w

220

JOHNSON, MEDINA-GARCIA,

AND TUNG

FIG. 8. The lowest-order Feynman diagrams contributing Fig. 7 and Eq. (3.11).

to the hard scattering amplitude w of

This can be simplified to

where the subscript H on W, indicates that these formulas apply in the high qT kinematic region and the variable 5 is related to rl and & by

&-x,

=x,e’

(3.13)

<2-x2=xTe-5.

The limits of integration in Eq. (3.9) are determined from the constraints ci < 1 to be [, = In -+J$

2

(3.14)

I& = In +.

It is well known that the first-order “hard scattering” formula diverges when it is extrapolated to the low qT region. It is not supposedto be valid there. For what value of qT does Eq. (3.12) cease to be applicable? This is another question of practical importance for which perturbative theory has no good answer. As mentioned earlier, however, a clue can be obtained by comparing W, with W,. To this end, we isolate the most singular term in W, as xT + 0 (i.e., the low qT limit). We observe that in the xT -+ 0 limit, the b-function in Eq. (3.11) may, in effect, be replaced by 1 r2

--x2

2

f3 r,L

(

X,ft2yX2

)I

+(1-Z)

(3.15)

TRANSVERSE MOMENTUM

DISTRIBUTION

221

where the argument of the d-function in the 1st (2nd) term is interpreted as a function of
W,(hx1,x2,x4z

W +W ln-4)

IV=P70"4Q>~,'V~~,~ Q>g(xz,Q>+f(xz3Q,g(x,,Q>l.

(3.16) (3.17)

Comparing Eq. (3.17) with Eq. (3. lo), we see that the two limiting functions are identical in form. This does not, however, imply that the low qr formula W, and the high qT formula W, necessarily join smoothly onto each other, because, in principle, the two limits applied (Eq. (3.16) and Eq. (3.9)) do not overlap. (d) Complete Formula for W(s, x, x, q)

We need an Ansatz for W(s, xi, x2, qT) which reduces to W, of Subsection (b) for “small qT,” coincides with W, of Subsection (c) for “large qT,” and interpolates smoothly in between the two regions. The equality of the large limit of W, and the leading singular term of the small qT limit of W, discussed above led Collins to the following Ansatz [ 81: w= w,-

w+ w,

(3.18)

where the common expression I@ (Eq. (3.10) and Eq. (3.17)) is subtracted in order to compensate for the unwanted term W, or W, in the low or high q.,. region, respectively, and to provide a smooth transition in the intermediate range. Whether this Ansatz is a good one depends on two considerations: (i) where it is true that at large qT (W, - w> is of higher order in (r than W,, is this difference smaller than W, (which is supposed to represent the true value of W in this region)? (ii) Similarly, although (W, - w) is rid of the XT’ singularity at small qT, is this difference smaller than W, (the correct answer for W at low qT)? We shall demonstrate that the answer to both of these questions turns out to be “no” near the ends of the kinematic boundary. Nevertheless, it will also become clear that: (i) Eq. (3.18) does provide a good interpolating formula for the intermediate region, and (ii) the failure at both the high and low ends in qT can be easily handled by inspection of the results.

IV. NUMERICAL

CALCULATIONS

AND RESULTS

The formalism developed in the last two sections transverse momentum parton distribution function production cross section W(s, x,, x2, qT) from an distribution function f(x, Q). Since f(x, Q) itself renormalization-group) equation in Q, it can in turn be

allows us to calculate the P(x, k,, Q) and the pairinput longitudinal parton satisfies an evolution (or determined from initial data

222

JOHNSON,

MEDINA+ARCIA,

AND

at Q = Q,. In the present model calculation, function used is xf(x, Q,) = x’.‘(l

TUNG

the input longitudinal

distribution

- x)*.?

The evolution in Q is computed numerically using the method of Furmanski Petronzio [ 121. Figure 9 shows the function f(x, Q) for three values of Q. (a) The Parton Transverse

Momentum

Distribution

(4.1)

and

Function

We calculate P(x, k,, Q) by (numerically) taking the Fourier-Bessel transform of P(x, b, Q) given by Eq. (2.24). For definiteness, the size parameter A is assumed 1.0

0.8

FIG. 9. The input parton longitucinal momentum distribution as a function of x, for three values of Q.

TRANSVERSE MOMENTUM

DISTRIBUTION

223

independent of x and chosen to be equal to unity. This amounts to using d as the momentum scale in terms of which all the other parameters and variables are measured. The “short-distance” distribution function P”s,, is obtained from Eq. (2.14), with the first factor given by Eq. (2.11). The inclusion of the first-order correction term in Eq. (2.11) is optional, as not all terms of this order are kept in the crosssection formulas. Figure 10 shows, in solid line, a typical k,-distribution curve. For comparison, also shown on this graph are: (i) a Gaussian distribution (dash-dotted) corresponding to the input “intrinsic” distribution of Eq. (2.17); (ii) asymptotic (perturbative) k,distribution according to Eq. (2.25) (long- and short-dashed); and (iii) asymptotic k,distributions according to Eq. (2.27) with u = l/3 (long- and two short-dashed), and a = 0.9 (dashed), respectively. We see that the renormalization group approach used P(x, k,,Q)

x * 0.45 Q = 45(A) A = 0.25(A)

6 kT(A) FIG. 10. distance” asymptotic

6

A typical parton transverse momentum distribution is shown in solid line. The input “longGaussian distribution is shown in dash-dotted line. Three “short-distance” perturbative distributions are shown in dashed lines.

224

JOHNSON, MEDINA-GARCIA,

AND TUNG

in the present calculation reproduces at large k, (short distance) the first-order perturbative results, both in shape and magnitude; it deviates from the latter as k, decreases, and follows the non-perturbative (Gaussian) distribution into the small k, (long distance) region. We show here three asymptotic curves in order to illustrate the arbitrariness inherent in first-order perturbative predictions-they agree with each other within corrections of the next order. We see that the “renormalization-groupimproved” asymptotic curves agree with the full result represented by the solid line better than the “straight first-order” curve (long- and short-dashed) does. In the scalar field theory model, the low transverse momentum distribution is purely phenomenological in nature: the Gaussian shape parameter A is put in by hand, and the normalization at k, = 0 also depends on the “regularization” prescription in bspace (since it is equal to J”d4b& b, Q)). In QCD, the expectation is that the Sudakov form factor, due to soft gluon effects, would be the determining factor for long-distance behavior, hence obviating the reliance on phenomenological input. >

Ptx, k,, Q) .

I-

Q -45(A) A = 0.25 (A)

I-

is

FIG. 11. The parton transverse momentum distribution for four values of x with all other variables fixed.

TRANSVERSE MOMENTUM

225

DISTRIBUTION

The dependence of P(x, k,, Q) on the (probing-) scale parameter Q is quite simple: it is proportional to [a(Q)]“‘, as is evident from Eq. (2.14). Thus P decreases slowly with increasing Q, uniformly in the other variables (x, kT). We shall not show the Qdependence explicitly. Does the k,-distribution depend non-trivially on x? The answer is not hard to find: at large k,, the asymptotic formula Eq. (2.25) indicates that P(x, k,, Q) is firoportional to g(x, Q)--a convolution integral of f(x, Q)-over x, Eq. (2.16b); at small k,, P(x, k,, Q) (i.e., J‘ d4b&x, b, Q)) is determined by the integral of f(x, c/b) over a range of b. Hence there is no direct relationship between the x-dependence of the transverse momentum distribution in the two k, regions. Figure 11 shows the kdistribution for four values of x. We can also ask how P(x, k,, Q) depends on the coupling scale parameter A. Figure 12 shows the comparison of the k,-distribution curves corresponding to three

I 0

I

I

I

I

I

I

2

4

6

8

IO

12

kT(A)

FIG. 12. The parton transverse momentum distribution for three values of A with all other variables fixed.

226

JOHNSON, MEDINA-GARCIA, a

AND TUNG

IC S = IO’

GeV2

5 = 0.05 y = 0.45 = 0.35 x B = 0.143 ‘A

A - 0.15 GeV A - 1.0 GeV

q,

LOG

SCALE

FIG. 13. (a) A representative transverse momentum distribution for pair production in a log-log format; (b) The combined transverse momentum distribution over the full range in a semi-log format.

different values of A for the same (x, Q) and input f(x, Q). The large k, perturbative part is sensitive to A, but the small k, part remains the same. (b) Pair-Production Cross Section W(s, t, y, qT) Using the formalism of Section III, we calculate the low qT cross section W, given by Eq. (3.6), the high qT cross section W, given by Eq. (3.12), and the common limiting from w given by Eq. (3.10) and Eq. (3.17). We then examine how these results relate to each other in various kinematic regions and addressthe problem of determining a reliable transverse momentum distribution for the full range of qT. Finally, we discuss the dependenceof the q,-distribution on the other variables and parameters. In order to simplify the calculation of the Fourier-Bessel integral for W,, we shall

TRANSVERSE

b

MOMENTUM

227

DISTRIBUTION

IO

(SAME

I

I

PARAMETERS)

1

5

0 4,

LINEAR FIG.

I

I

I

I

I

1

IO

SCALE 13-Continued.

use as independent kinematical variables the set (s, r, y, qT) and substitute the longitudinal momentum fraction variables (xl, x2) in the W, and I@ formulas by (xA , x,) (cf. Eqs. (3.4a), b, c)). The advantage is that, for given s, (x, , x,, Q’) can be regarded as independent of qT, whereas (xi, x2, Q’) cannot. In the small qT region where IV, represents the main contribution to the cross section, the difference between the two sets of variables (xi, x2) and (xa, x,) is of order qt//s, which is negligible. The difference is, of course, significant in the large qT region where W, represents the correct answer. We show in Fig. 13a the three IV’s as functions of qT in a typical case which displays most of the common features of our model calculation. On this log-log plot, the limiting function W appears as a straight line (being proportional to qf*). We notice that, on the one hand, the W, curve approaches this straight line for large qT;

228

JOHNSON,

MEDINA-GARCIA,

AND

TUNG

on the other hand, the W, curve approaches it for small qT. This is just the expected behavior discussed in Section III. This general feature suggests the possibility of making the smooth transition between the two extreme regions, thus predicting a reliable qT distribution for the full kinematic range without an arbitrary cutoff on the region of applicability for each of the theoretical expressions W, and W, . In order to do this quantitatively, we need to examine the results in greater detail. In particular, we shall investigate the applicability of the “reasonable” Collins prescription, Eq. (3.18). For low qT, as seen in Fig. 13a, W, diverges and w represents its most divergent part (-x;‘). Although the percentage difference (W, - r)/w becomes quite small, IO4

IO3 s =lO”

r = 0.05

IO2

Y=o

IO

I

16’

IO2

IO3

ItI4

qltvlAX

lb

I

I FIG.

14.

2

Behavior

5 qT

IO LOG

of the three

20

50

100

zuv

SCALE W-functions

at a very

high energy

value.

TRANSVERSE

MOMENTUM

229

DISTRIBUTION

the difference (W, - I@ itself (which enters Eq. (3.18)) still diverges as (ln x,), and becomes large as compared to the expected main contribution W, at some value of qT (around qT = 1 in the example). Therefore, Eq. (3.18) fails as it stands in this region. However, in view of the fact that the fractional difference is very small over a large intermediate range of qT (l-8 in the example) where the difference is also small compared to W,, and the fact that W, has no significance in the very small qT region, it clearly makes sense to disregard (W, - w> in Eq. (3.18) below the intermediate region of overlap. The above discussion suggests that this region starts where x+ (In x,) becomes small. For high qT, a similar situation occurs. The expected main contribution W, falls considerably below the other two functions W, and w. Even though the difference (W, - w) is formally of higher order in a(q,), it eventually becomes larger than W, when the latter decreases rapidly to zero approaching the kinematical boundary.

\ I I

I 2

, 3

I IO

45

\ c

20

30

%

FIG. 15. Behavior of the three W-functions at a low value W, and @ are outside the kinematical boundary.

of s; the dashed

parts

of the curves

for

230

JOHNSON,

MEDINA-GARCIA,

AND

TUNG

Neither I+‘, nor E, being relevant physical expressions only at low qT, is constrained by the high qr boundary condition. Again, to be sensible, the difference (W, - w> must be removed from Eq. (3.18) above the intermediate region of qT where it becomes small as compared to W,. As noted earlier, generally speaking, this occurs at values of q,. where a(qT/A) first becomes small. We are led therefore to the conclusion that the Ansatz given by Eq. (3.18) should only apply to the region between the two domains discussed above. (In the case represented by Fig. 13a, this region occurs around q m 8.) Its application provides a smooth interpolation between the low and high qT formulas needed to obtain the ‘A’( 6, T, y, q$ ( A2/s)

r = 0.05 y-0 A = 0.25

FIG. 16. The transverse the other variables fixed.

momentum

distribution

function

W (scaled

by s) for four values

of s with

TRANSVERSE MOMENTUM

DISTRIBUTION

231

complete transverse momentum distribution of the cross section. In Fig. 13b we draw the predicted IV@, t, y, qr) as a function of qT in the more familiar semi-log format. It should also become obvious from this study that it does not make sense to assign a fixed region of applicability for W,, W,, or Ansatz (3.18) as is often done. The proper range of applicability in qT for these functions (if they apply at all) depends also on the value of the other kinematic variables. Of particular importance is s which sets the overall energy scale and XTmax which is equal to Min{x,x,, (1 - x,)(1 - x2)}. Thus, in addition to the situation displayed in Fig. 13a, we can envision the following two types of general behavior: (i) At very high values of s, there is a considerable overlap region within the kinematic boundaries where both xk (lnx,) and a(q,/A) are small. The three functions W,, W,, and (W, - W + W,) closely track each other in this extended “intermediate range.”

s = IO4 y=o T

0.005 0.05 0.10 0.15

-.------

---h - 0.25

9, FIG. 17. The transverse momentum distribution for pair production for four values of r with the other variables fixed.

232

JOHNSON,

MEDINA-GARCIA,

AND

TUNG

There is no problem specifying the expected cross section over the entire range of qT. Figure 14 represents such a case. (ii) At relatively low s, or for extreme values of (t, y) close to the kinematic boundary (even for large s), the available qT range is so restricted that the three functions are quite different, as depicted in Fig. 15. No clear prediction can be made in case like this. On the one hand, W, is inapplicable because of the small qT involved; on the other hand, W, can not be trusted at all because it does not respect the required kinematical boundary. One possible way to remedy this situation is to modify the prescription for calculating W, in order to enforce the kinematic boundary constraints by hand [6]. We did not delve into this approach because it would introduce another source of ad hoc input into the calculation. We now exhibit the dependence of the transverse momentum distribution on the other kinematic variables. For given (r, y, qT), the cross section is explicitly proporId

s = IO4

r = 0.05 Y 0

IO

-.-.------

0.15 0.45

I

IO'

FIG. 18. The transverse other variables fixed.

momentum

distribution

for pair

production

for three

values

of y with the

TRANSVERSE

MOMENTUM

233

DISTRIBUTION

tional to (s/d 2), and it also depends on s implicitly through scaling variable Q’ = rs. We show in Fig. 16 the qT distribution divided by the factor (s/d2) for four different values of s at fixed (t, u). The two top curves, corresponding to s = lo6 and lo’, respectively, almost coincide. The other two curves, corresponding to lower values of s, also agree at smaller qT, but deviate and vanish faster than the large s curve as qr increases. This is due to the closing-in of the kinematic boundary for lower values of S.

For fixed (s, y, qT), varying r leads to simultaneous changes in the longitudinal momentum fractions (x,.,, xB) and Q’. An example of this r-dependence of the cross section is shown in Fig. 17. We see that the distribution becomes progressively sharper as r increases. We have no simple explanation of this behavior. IC

IC

I

IO

I

I

I

0

I

,

I

,

I

IO

I

1

I

1

I

I

20

4, FIG. 19. The transverse other variables fixed.

momentum

distribution

for pair production

for four

values

of ,4 with

the

234

JOHNSON,MEDINA-GARCIA,ANDTUNG

For given (s, r, qT), varying y leads to changes in the longitudinal momentum fractions (xA,xs). The resulting variation in the transverse momentum distribution reflects mostly the x-dependence of f(x, Q), which enters the functions W, and W,. Figure 18 shows three curves corresponding to different values of y with the other variables fixed. We can also test how sensitive the transverse momentum distribution is to the value of /i, the effective coupling scale parameter. Figure 19 shows four such distribution curves for four values of/i. At small qT, the cross section decreases with increasing /1; but at large qT, the relation is reversed. Hence, the transverse momentum distribution is broader for larger effective strong coupling. This is intuitively reasonable.

V. COMMENTS

AND CONCLUSION

The asymptotically free scalar field theory model provides a simple, self-consistent testing ground for many theoretical ideas and phenomenological techniques. The detailed study of this model presented here clears the way for a systematic investigation of the predictions of QCD on the transverse momentum distribution of partons in hadrons, as well as for physical particles in semi-inclusive lepton-hadron processes, at high energies. We have shown in detail how the renormalization group formalism is applied in this class of important phenomenological problems, and demonstrated by numerical calculation that already at reasonable energies the computed distribution functions for distinct qT regions satisfy the theoretical (asymptotic) relations which make it possible to determine the cross section over the complete transverse momentum range. The scalar field theory model is, of course, highly unphysical. The absence of massless vector mesons, in particular, means the absence of the most interesting Sudakov factor. As a consequence, all the calculated distributions for low transverse momentum are dominated by the phenomenological input “intrinsic” (Gaussian) factor. The predictions are, therefore, not very interesting per se. The most important issues to emerge from the present exercise are: (i) how to handle the transition from the low transverse momentum region to the high one, (ii) how to set up the numerical calculational scheme, and (iii) what are some of the relevant phenomenological problems to bear in mind for the corresponding QCD calculation. Of course, even the “relatively dull” results can be of much value, as concrete contrasting examples, in the comparison of QCD predictions with experimental findings. APPENDIX

A. Summary of Renormalization Group Equations for P(x, k, ,u) and f (x, Q)

Let g be the usual renormalized coupling constant. We define the effective coupling parameter, a(p) = g2(u)/64n3 = 314 ln(,u*/A ‘). (A-1)

TRANSVERSE

The renormalization

MOMENTUM

group differential

235

DISTRIBUTION

operator is (A.21

where, to leading order in perturbation

expansion, /? = -3ga/4

G4.3)

Y, = Wi and mR is the renormalized mass. The renormalization group (or evolution)

w?) equation for P(x, k,, p) is

[Q+y]P(x,k,,~u)=O, where y is the “anomalous

dimension”

of the #-field operator; y = a/6.

(‘4.6)

The solution to Eq. (A.5) is of the familiar form P(x, k,, p) = P(x, k,, ak,) &~~?dg~g)‘4(g) = W,

k,, ~k,)[aCu)la(~k,)l

Similar formulas hold for P”(x, b,,~). The renormalization group (or evolution)

equation for f(x,

“9-

G4.7)

Q) is

(A.81 where, to first order in a, p[(, g(u)] = a@)[-@(1

- 4 + 2(t - <‘)I.

C4.9)

Equation (A.8) cannot be solved in explicit form. But, if we take the moments (in x) on both sides, we get (A. IO) where y(n) = -f

+

2 (a + l>(n + 2) ’

(A.ll)

and the moments are defined as (A.12)

595/153/2-3

236

JOHNSON,

MEDINA-GARCIA,

AND

TUNG

Equation (A.10) can be solved. The result is

f(n,p)=f(n,pJ[+]cz’3”‘“’

(A.13)

These results will be used later. B. Feynman Rules for Calculating

P(x, k,, p) in Perturbation

Theory

The Feynman rules for calculating G(1, p), Eq. (2.4), are standard. We can rewrite the definition of P(x, k,,,u) in terms of G(1, p), Eq. (2.3), as:

f’Ww)=$$ k+ 6(l+

- k+) S”(& - /?T) G(Z, P).

Hence the rules for calculating P(x, k,, ,u) in perturbation theory can be easily formulated in terms of those for G(1, p). These rules can be read off from the diagram of Fig. 20, where the double line stands for the factor k+ 6(Z+ -k+)d4(&&)

and where 1 is treated as a loop momentum C. Calculation

(B.2)

variable.

of the Kernel T(<, b,p)

Keeping in mind the general relation, Eq. (2.7),

we first calculate both f(x, ,u) and P(x, b., ,u) in leading order perturbation theory, and hence obtain the expression for T(<,pb,p). We then apply the result to the actual distribution function f(x, ,u) (determined, say, from the “deep inelastic scattering” process) to evaluate the physical distribution P(x, k,, p). The details are given below. k

k \

FIG. general

J

20. The effective Feynman diagram for calculating the parton distribution function diagrams for the Green’s function G(I, p). The rules are explained in the text.

in terms

of

TRANSVERSE

MOMENTUM

231

DISTRIBUTION

To first order in a+), using the rules of (A), one can easily write down [8]: ~pert(X~ 4 9Pu)= z 6(x - 1) 64Gr) ati) +

Integrating

712

x(1 -x) [k; + 4 -x + -4]2

over & and applying dimensional

f,,,, (+,p)=Zd

regularization,

(T-l)-o&):(1-+)/ln

+

one obtains [8]

(lPtZ+:2@’

On the other hand, taking the Fourier-Bessel

cc.21

o(a’).

+y/.

(C.3)

transform of Eq. (C.2) one gets

P”pert(r, 6, Pu>= z 4r - 1) + a@) tll - r) 2Wb)

(C.4)

where l2 = m2( 1 - x + x’) and K, is the modified Bessel function of the 2nd kind (or Kelvin function) of order zero. The perturbative expansion of r(& pb, cl) involves powers of a@) In pb, and for the latter to be small we require a@) < 1 and ,ub - O(1). We are, therefore, only interested in pPert, Eq. (C.4), when p2 9 /i * and b-* $ n 2. Let us use the expansion of K, for small arguments: K,(lb) N In f

- y + O(12b2).

We obtain Lt(X,

Z&x-

byPu)w

l)+a@‘)x(l

4eC2y

-x)ln

b2m2(1 -x+x*)

+ terms of order a2 or m2b2. After substituting

(C.3) and (C.5) into Eq. (C.l), we conclude that T(t,,&p)

D. Asymptotic

(C.5)

Behavior

= 46 of&x,

1) -a@)

((1 - O[ln nb2iu2 + rl.

P5)

b, p) at Short Distance

We would like to show that &, zero. The starting point is Eq. (2.15),

b,,u) increases without limit

k,(x, b,pu) =f (x,;) [31

1’9.

as b decreases to

P-1)

Taking moments (cf. Eq. A. 12)) on both sides, we obtain

&,(Owu)=f (n,;)[&]“‘.

CD.21

238 Substituting

JOHNSON,

MEDINA-GARCIA,

AND

TUNG

(A. 13) on the right-hand side, one obtains F,,(n, b, p) =f(n,p)[a(u)/a(c/b)]

1’9+(2’3)Ffn)

=f(n, p)[cf(u)/a(C/b)]*‘((“+ =f(n,~)[ln(c/b~/lnOl/~)]*““+

1)(n+2)) IJCnf2)).

(D.3)

We see that all moments of Fs,, (n > 0) grow as some positive power of ln(c/b/l). Hence p&x, b,~) also grows without limit as b * 0 for fixed x and P.

ACKNOWLEDGMENTS The authors would like to thank J. C. Collins for explaining the theoretical formalism present paper is based and for frequent discussions during the course of this work.

on which

the

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