alternate factorization for the drell-yan process

PHYSICAL REVIEW D VOLUME 27, NUMBER 9 1 MAY 1983

Alternate factorization for the Drell-Yan process

Ghanashyam Date* Institute for Theoretical Physics, State University of New York at Stony Brook,

Stony Brook, New York 11 794 (Received 22 October 1982)

An alternate and more general form of factorization for the Drell-Yan process is derived in perturbation theory. Corresponding generalized anomalous dimensions are defined. Sim- ple calculations verify Mueller's conjecture for the recovery of improved parton-model results at very high energies. The approach offers a general framework to analyze the role played by soft partons, and to study their influence on asymptotic behavior.

I. INTRODUCTION

The domain of applicability of QCD perturbation theory has been increased by (a) calculating those quantities which are free of "mass singularities" of the form l o g ~ 2 / m 2 and (b) by using the factorization algorithm proposed by Mueller and ~olitzer. ' This method asserts that for any process that admits a parton-model description, one may (i) calculate the partonic cross section using perturbative QCD, (ii) organize all the mass singularities into a multiplica- tive factor, i.e., express the cross section as a product of factors, one of which contains all the mass singularities while the other is finite, and (iii) absorb the mass-singular factor into distribution and fragmentation functions and treat the mass-finite factor as the "renormalized" partonic cross section, which can be calculated consistently using perturbation theory. Provided that low orders of perturbative calculations are self-consistent and that all the mass singularities can be factorized and absorbed into the D and I: functions, one may apply perturbative QCD with some confidence. The assumptions of universality of distribution and fragmentation functions then correspond to an assumption of universality of mass singularities in perturbation theory, which must be proved. This came to be known as the factorization theorem. Several groups2 gave all-orders arguments for factorization theorems for different processes.

Since then, however, doubts have been raised about the factorization of mass singularities. These were raised due to the examples given by Doria, Frankel, and ~ a ~ l o r ~ ( D m ) and more recently by Bodwin, Brodsky, and ~ e ~ a ~ e ~ (BBL). DFT considered a reaction with two quarks in the initial state and showed that there was a nonleading IR divergence coming from two soft gluons which did not

cancel between real and virtual graphs. Since this divergence is nonleading, it does not contradict the factorization theorem. However, their example does point out that generalization from the case of one quark in the initial state to the case of two quarks in the initial state contains additional features in a non-Abelian theory and the question of cancellation of soft divergences becomes more delicate.

More recently B B L ~ found a leading contribution at the three-loop level, from the so-called "Glauber" region, which does not seem to cancel between real and virtual graphs. This noncancellation occurs due to the mismatch of group factors between virtual graphs. From this they conclude that the usual factorization conjecture fails. This example has received different reactions from different workers. Assuming the noncancellation, ~ u e l l e r ~ observed that Sudakov logarithms (double logarithms) also do not cancel completely. He then argued that the noncanceling contribution found by BBL is suppressed due to a "Sudakov form factor," and thus the usual form of factorization is restored. On the other hand, Collins, Soper, and sterman6 (CSS) maintain that factorization is possible with a somewhat different definition of the D functions. The uncertain- ties regarding the implications of the BBL example show that the issue of cancellation of soft divergences is still unresolved and so is the status of the leg-by-leg factorization theorem for the Drell-Yan (DY) cross section. Following CSS~ one may list some of the open possibilities for the DY cross section.

(1) The factorization conjecture is valid for the DY cross section and the D functions are the same as those measured in leptoproduction.

(2) A factorization is valid in the same form but with D functions different from those measured in leptoproduction.

27 - 2076 (3 1983 The American Physical Society

27 - ALTERNATE FACTORIZATION FOR THE DRELL-YAN PROCESS 2077

(3) A different form of factorization is valid with distribution functions defined for a pair of partons, e.g., the DY cross section may be written as [s] hadron - J ~ X ~ ~ X ~ ~ [ X ~ X ~ - $ ]

where is calculated in perturbation theory and is free of mass singularities.

(4) Factorization just fails. The possibility (1) requires complete cancellation

of soft divergences, (2) requires organization of noncanceling soft divergences in two factors, each associated with a parton in the initial state, (3) just requires factorization of noncanceling (or perhaps some canceling ones) soft divergences into one factor, and (4) of course requires no work in this direction.

In this paper, we consider (3) and prove it to all orders in QCD perturbation theory. Since we do not attempt to show cancellation of soft divergences, the factorization itself is established somewhat straight- forwardly. However, this form of factorization does not decouple completely upon taking the usual moments with respect to T= e2/s , and thus requires a different approach to obtain the asymptotic behavior of du /de2 . Hence a new approach is developed which combines the methods used by Libby-Sterman and ~ u e l l e r . ~

(This alternate form of factorization breaks the universality between deep-inelastic scattering and the DY process, but still implies a universality within the class of hadronic processes initiated by the same hadrons.

The formalism developed below to study the asymptotic behavior leads to a differential equation for the full perturbative cross section without having to absorb the mass singularities into the usual distribution functions. It introduces a generalized anomalous dimension, in terms of which the influence of the mass singularities (those which factorize in the usual way and those which do not, i.e., collinear and soft, respectively) on the asymptotic behavior can be analyzed. By studying the properties of the generalized anomalous dimension one can also see how the usual QCD-renormalized parton model for do/dQ2 may be recovered at very high energies.5

11. DERIVATION OF AN ALTERNATE FACTORIZATION

The idea of the proposed factorization is to express the cross section as a convolution of two func-

tions, one of which has no dependence on the "small" scales p and pP2 - M2. If each of the functions is to be determined in the standard perturbation theory, then the UV renormalized contribution of each individual diagram should be expressible as a sum of a finite number of terms, each of which has a convolution form, with one of the factors independent of the small scales. The present approach to factorization is based upon the work of Refs. 7 and 8. We will begin by reviewing the principal features of this analysis. We will give some com- binatorial definitions and define, for each graph, a certain function Hr, by a forest f o r m ~ l a . ~ We will then prove that Hr so defined is free of mass singularities. Using a slight modification of Zimmer- mann's identity,9 we will derive a convolution form. The additional UV divergences that will arise in the calculations are handled indirectly, and so is the question of gauge invariance. Remarks following Eq. (2.25) are intended to clarify the argument.

Consider a generic Feynman diagram r for the DY process shown in Fig. 1. If one does an approx- imate calculation in the limit s = (p +p2 l2 >> pi2, m '; one encounters logarithms of pi2/s, M2/s. On dimensional grounds, we can write the unrenor- malized (but regularized) contribution of r to a, the cross section as

+ higher orders of [+ ] . Here A is an UV cutoff, M is a common scale introduced such that pi2, m2 are of the order of M ~ . A,@ are some dimensionless functions of their arguments which are all of the order of 1. The A,@ do not have any dependence on either A ~ / S or M2/s.

After renormalization, the scale A is replaced by the renormalization mass p. Since ,u is an arbitrary scale we may take it to be of the same order as A. Thus the ln(,u2/s) factors can be adjusted to be some

FIG. 1. General diagram contributing to the Drell-Yan cross section. I + , [ - denotes the lepton pair.

2078 GHANASHYAM DATE - 27

small numbers. However the 1n(M2/s ) factors diverge as S + w ( M fixed) and make the perturbation expansion unreliable. Formally, the limit s + W , M fixed can be looked upon as the limit s fixed, M-0 (with pi2/M,m 2 / ~ fixed). These divergences which arise as s /M2+ w , p, '/M2, Q '/s fixed, in finite-order perturbation theory, will be re- ferred to as "mass singularities." The analysis of mass singularities begins with the question: Which regions of integration space of Feynman integrals give rise to these mass singularities? This question has been formalized and treated in Ref. 7. A brief summary is given below.

The central observation is that mass singularities can only arise if the denominators in the Feynman integrands vanish. Since Feynman integrals are contour integrals, in many cases the corresponding singular points can be avoided. This is not possible if the contours are "pinched," i.e., cannot be de- formed away from the singularities. However, some of these "pinch singularities" are integrable since the volume element near the singularity vanishes faster than the integrand diverges. The integrability near singular points is studied by the technique of power counting. Thus, the two necessary conditions for mass singularities are (i) Feynman denominators must vanish in such a way as to "pinch" the contours and (ii) near a pinch-singular point, a power counting should indicate the possibility of divergence. Cataloging of the pinch singularities is great- ly facilitated by the physical-picture interpretation given by Coleman and ort ton." Further restriction of this set of pinch-singular points is obtained by the application of the power-counting rules developed in Ref. 7. Since this analysis of mass singularities is done in the neighborhood of a point, the details of UV regularization are not relevant. Also, it is always possible to introduce UV counterterms which have no mass singularities; and, therefore, the details of UV renormalization are also not necessary for this analysis.

Using the two criteria mentioned above, the following was shown in Ref. 8.

(a) For renormalizable theories, including gauge theories, without superrenormalizable couplings, the mass singularities in a two-particle inelastic scattering cross section are never worse than logarithmic, provided that for gauge theories one works in a physical gauge (e.g., an axial .gauge1') in which the unphysical degrees of freedom do not propagate.

(b) In an axial gauge, only those regions can possibly lead to mass singularities whose physical picture has the form shown in Fig. 2. This figure is the starting point of the present approach to factorization.

In Fig. 2, J1 and J2 (including the lines connected

FIG. 2. Physical picture of a typical region, potentially contributing to mass singularities.

to H ) are jet subdiagrams, S (including the lines at- tached to J1 and Jz) is the "soft subdiagram," and H is the "hard subdiagram," according to the terminology of Ref. 7. With the same terminology, the hard subdiagram consists of one "hard vertex" on either side of the cut and a bunch of jet subdiagrams associated with each observed line in the final state. These jet subdiagrams are called "nonforward jets." All lines are massless. At a divergent pinch-singular point, all the lines in J1 and J2 are on-shell and precisely parallel to p l and pz, respectively; all lines in S have precisely zero momentum; all lines in the nonfonvard jets are on-shell and precisely parallel to their corresponding observed legs, while the hard vertices contain lines which are off-shell.

For power-counting estimation of mass singularities the actual masses of the lines do not really matter however, to be specific, we take all lines to be massless and regulate the mass singularities by keep- ing the external momenta off-shell, pi2=M2. The UV divergences are regularized using dimensional regularization and renormalization is performed at off-shell values of the momenta (the scale is p) . These details are not necessary for the present analysis and will not be exhibited explicitly. The finiteness of H r , defined below, will be proved by showing that for every region that may possibly contribute to a mass singularity, the integrand has a suppression factor which makes it integrable near the pinch-singular point.

We will be considering diagrams, r's, which are generically described by Fig. 3. r is one-particle ir- reducible (1PI) except possibly for the photon line and self-energy insertions on the external lines; and contains the "Born diagram," To described in Fig. 4, as a subdiagram.


Definition: Hard part. A diagram y C T, (y#T) is said to be a hard part of T, if y itself is of the form shown in Fig. 3. In particular y > ro.

Clearly, there is a one-to-one correspondence between classes of divergent singular points, described by Fig. 3, of the integration space of T and the hard parts of T. Also, two hard parts of any given T are either nested or overlapping but never disjoint since every hard part contains To.

Definition: T forest u r . Any collection of non- overlapping (nested) hard parts of T, is said to be a T forest and is denoted by u r . The null set 4 is defined to be a T forest for every T.

For every hard part y C T we introduce a subtraction operator t,, which is defined as follows.

Let the external momenta of y C T be denoted by pip p&, i = 1,2 as shown in Fig. 5. These are linear combinations of the external momenta of T, P l r =pir =PI , p2r =pir =pz; and the loop momenta of T/y. (T /y is a diagram obtained from T by shrinking y to a point.) Loop momenta which flow through the pairs of lines (ply,pZy), (plypi,), (p;,pZy), and (p;,p;,) are called "soft loop momenta," whereas those which flow through the pairs (plYp;,) and (p2,,piy) are called "jet loop momenta." At divergent singular points where all the lines in T/y go on-shell, the soft loop momenta are zero, while the jet loop momenta are collinear with their respective external momenta. We define normal variables of y, as those components of loop momenta from T/y, which vanish (not necessarily at the same rate; see the comment included in Ref. 81, at the pinch-singular point (PSP) corresponding to y. (Note that PSP's are approached only in the limit M=O.) These are (i) all the four components of the soft loops, (ii) the minus component and the sum of

FIG. 4. The "Born diagram" contributing to the Drell-Yan cross section. a, b,c,d, are Dirac indices.

the squares of the transverse components of the jet loops from the ( p , , , ~ ; ~ ) jet, and (iii) the plus component and the sum of the squares of the transverse components of the jet loops from the (p2,p;,) jet.

Let ( k ly, k;,),( k2,, k;,) be the total jet momenta, parallel to p l r and p2r, respectively, flowing through the external lines of y (i.e., jet momenta = total momenta-soft loop momenta). Clearly, kly==k;, and k2,=kiy. We define a set of "off- shell" momenta as

We define subtraction operators t, to act on the integrands of Feynman integrals and/or Hy, functions for y 'C y when the H,,'s are defined.

Definition:

ryf ( P ~ ~ , P ; ~ , P ~ ~ , P ; ~ ) = ~ (phly,phly,ph22y,ph2y) (2.3)

FIG. 3. Generic diagram r considered in the theorem FIG. 5. Topological form of Fig. 2 with a generic of Sec. 11. This is also the topological form of Fig. 1. "hard part" y.

GHANASHYAM DATE

and

iff is independent of pirp/y It is understood that ~f f denotes the Feynman in-

tegrand Ir or H r then the arguments o f f involve only momenta external to T.

Notation: The following algebraic expression will be denoted by F:

where ur is a forest and ( u r ) is the set of all r forests, including 4. For 4,

In a forest, the t's are ordered according to

Definition Cformal): For each r we define a function Hr(plr,p;r,p2r,p;r) formally as

where Ir is the UV-renormalized Feynman integrand associated with r (i.e., the usual Feynman integrand with UV counterterms), lr collectively denotes the loop momenta of T, and Fr is defined in Eq. (2.4). Notice that Hro = Ira.

We now state and prove the main result of this section.

Theorem. Hr(pir,pi;-) is free of mass singularities.

Proof. Instead of directly exhibiting a suppression factor in every possible divergent region in the integration space of a given diagram T, we give an in-

( i v ) ( v )

FIG. 6. Classes of diagram which have only one hard part. The blobs in (i), (ii), (iii), and (iv) are four-particle ir- reducible. Blob in (v) is 1PI. (i) Diagrams with soft and jet lines. (ii) Diagrams with only jet lines. (iii) Diagrams with only uncut soft lines. (iv) Diagrams with only cut soft lines. (v) Diagrams with self-energy insertion on an external line.

direct proof for classes of r's, based upon induction on the number of hard parts of the T's in the same class.

Proof of the theorem for r 's which have only one hard part. Consider all the diagrams r ' s with only one hard part, which has to be ro. T must necessarily be of the form of one of the diagrams shown in Fig. 6. We will prove the theorem for each case.

(i) r with soft and jet lines: Fr = 1 - tro and

Note that we used troIr/ro=Ir/rotro. Explicitly,

where q +PZro =piro +piro is the photon momentum and qo > 0. For the external momenta pir,plr off-shell (pi2#0) there are no PSP's and there is no possibility of a mass-

singular contribution. Hence, away from the mass shell, Hr is certainly free of mass singularities. For pir=Ar,pfr =Er, there is a possibility of a singular contribution from the region where all lines in T / r o go on-shell together with M-0, i.e., piro+Ar0, P~;.,-A'~,.

We note that Iro(Aro,&ro) is well defined and so are the derivatives of Iro with respect topiro,pfr0, evaluat-

27 - ALTERNATE FACTORIZATION FOR THE DRELL-YAN PROCESS

ed at firo,fifr0. Hence, we can Taylor expand Iro about to get

+ (similar term with piro ) + higher-order terms. (2.8)

The differential coefficients are finite and independent of normal variable of To, while the small increments are linear combinations of normal variables of ro.

Since the power-counting degree of is zero (even in the presence of BBL-type regions-see comment included in Ref. 8), and since ( 1 - tro)Iro provides a suppression factor, the integral does not receive any divergent contribution from this region. There are no other regions which could lead to a mass-singular contribution. Hence, Hr(pir,plr) is free of mass singularities.

Further, we note that the suppression factor also implies that the contribution from the neighborhoods of the divergent pinch-singular points vanishes as pir2,p;r2 when ~ ~ ~ : P ; f ~ f i ~ , @ ; ~ . The contribution from the remaining integration space is either constant (independent of pir ,Pir ) or vanishes even faster than p,r2,p[r2. Hence, for pir,pir near &-,phir,

Hr(pir,plr )-[function independent ~ f ~ ~ ~ ~ , ~ , ! ~ ~ ] + ~ ( ~ ~ ~ ~ , ~ i ; . ~ ) . (2.9)

As far as the UV divergences from the subintegrals in lr are concerned, Ir/r, already has the appropriate UV counterterms included in it. The subtracted term ITIT,( -tr,,)Zrn, howecer, contains new UV divergences. The factor ( -tro)Iro does not carry all the components of the loop momenta from lr, e.g., the minus components of the "top-jet" loops, the plus components of the "bottom-jet" loops do not flow through this factor. Because of this, we do not get sufficient suppression when these components become ultraviolet and this results in the new UV divergences.

We will not introduce explicit counterterms for these additional divergences but will keep them regulated so that intermediate manipulations are allowed. In the end we will argue that these divergences must cancel out in the quantities of interest.

(ii) r with only jet lines: Again, T has only one hard part, namely, To and, therefore,

Once again, exactly as in the case of (i), we get a suppression factor in the only possible divergent region. The additional UV divergences are treated in exactly the same manner. Hence, the theorem holds for this case too.

(iii) I' with only uncut soft lines:

In this case, in the mass-singular region, all loops (iv) T with only cut soft lines: This is treated from lr which flow through the external lines of To similarly to the first two cases. are soft. Hence, (v) r with self-energv insertion: This is treated

and hence for Pir =fir

( 1 -fro ) I r o ( ~ i r o , ~ i r o

=Zro(~iro,~i ' ro ) -I=,( hrO,&rO )

-. similarly to case (iii). This case, of course, has no additional UV divergences.

This proves the theorem for all the T's which have only one hard part, viz., To.

Induction hypothesis: Let I?' be any diagram which has m hard parts, where, m < N . Let us assume that

hence, Hr(fir,phir)=O, and clearly is free of mass is free of mass singularities and in particular for singularities. P~T',P[T' +P?:r,,&r,

2082 GHANASHYAM DATE 21

+(function independent of pir'2,p;r 2 ,

Let r be any diagram which has ( N + 1) hard parts. For any such r , one (and only one) of the following is true (for a reduced diagram T/y, the shrunk diagram is counted as a hard part).

(a) There exists a unique r'Cr such that T / r ' has only one hard part and I" has N hard parts.

(b) There exists a unique pair of r ; , r ; C r such that r/ri has only one hard part and I'; has N hard parts. The topology of r/r; is similar to one of (ii), (iii), and (iv) diagrams of Fig. 6.

(c) If r has self-energy insertions on its external lines, then there exists a unique r' such that r/r' has only self-energy blobs [lPI or one-particle reducible (lPR)] and r' has no self-energy insertions on

its external lines. In other words, r' is obtained from r by simply cutting off the self-energies on the external lines. The I?' has m hard parts where m 5 N .

We will consider the theorem case by case. (a) In this case, every hard part of I?, except r ' , is

a hard part of r'. Hence, we can decompose the set of all ur forests into a set of those forests which contain r' and those which do not. Hence,

Consider

By the induction hypothesis dlr.FrcIr. =Hr is free of mass singularities, hence

The integral over lrlr, generates a mass-singular contribution when all lines in Irlr+ go on-shell together. But precisely then (1 -tr, )Hr, provides a suppression factor as described above. Note that by induction hypothesis Hrc -(function independent of pir2,plr2) +0(pir2, pi;.,2), therefore, ( 1 -tr' )Hr, -0 (pir'2,p,!r,2).

Hence Hr is free of mass singularities and Hr (pir,plr )-(function independent of pir2,pi'r ') + ~ ( ~ ~ ~ ~ , ~ j ' ~ 2,

as pir,pi'r -+firfir. This behavior of Hr follows as described in the proof for N = 1. (b) For definiteness, let us consider a r which has 2PI "jet blobs" in the top part and in the bottom part.

See Fig. 7(i). The unique pair of ri and I'; is also shown. Note that l", and Ti overlap. Let r'=r; n I?;. The set of all r forests can be decomposed as

( u r ) = (those forests which do not contain any of r;, r; , r'j U (those which contain r' but not r; , r ; J

U (those which contain r; but not I?',T; J

U (those which contain but not T',T; J

U [those which contain r' and T; but not

U ( those which contain I" and but not r; J . The forests which do not contain any T 1 , r ; , r ; are all the l?' forests. Thus,

Consider


X J dl,P,I,(pir,pi'p;r,,lr )

H r . H r is free of mass singularities by the induction hypothesis. Thus,

This integral has three possible regions that may lead to mass singularities. Again, if pir,p;'- are off-shell, then there is no possibility of a mass-singular contribution. Let us take pirp;= near hrfl.Ir. Introduce two aux- iliary operators S1 and S2 which act on Hr, as follows:

and

Consider the integral

For the specific type of r we are considering, and are the Feynman integrands of the top and the bottom parts, respectively. The integrand and loops of r/r' completely factor.

This integral has the same divergent regions as the integral for H r in Eq. (2.19). If we refer to the region where lines in r; go on-shell together, then we evaluate the H F ) ~ integral. This inte ral is free of mass singu-

!, larities due to a, by now familiar, suppression factor and this serves as a new "HF function which in turn provides suppression for the lPjr integral; i.e.,

Similarly, if lines in r; go on-shell, then we may perform the integration in the reverse order and discover that pr is free of mass singularities. If all lines in r/T' go on-shell together, then we can choose any order of integration and still see that the integral is free of mass singularities. Hence, Rr is free of mass singularities.

Now consider,

IrIr already has the external momenta of r set at the values hr,j?ir; and, therefore, we can take IrIr to the left of all the t operators, i.e.,

Clearly, on Hr , (pir,p;r, )

and since p lr,P i r are already equal to &'-,Pir, tr; =Sf , tr; =S2 , S1S2=S2S1=t'-' .

Substituting these relations in Eq. (2.24) we see that

2084 GHANASHYAM DATE

But this is exactly Hr(fir$;r) which we have shown to be free of mass singularities. Hence, Hr(&r,$:r) is free of mass singularities. Away from these values of momenta, Hr is free of mass singularities by itself. Note that H,#H~ identical- ly, since we cannot take the to the left of the t operators in general.

A similar proof can be given for the remaining types of r in this class (see Fig. 7).

(c) This case is really trivial, since the self-energy blobs are completely factored out from the rest of the integrand and loops. A proof along the lines of that given for case (b) may be constructed.

Thus, in all possible cases we have proved the theorem for r ' s with ( N + 1) hard parts, assuming the induction hypothesis. Hence, by induction, Hr defined by Eq. (2.5) is free of mass singularities for all the diagrams.

Remarks: (1) The prescription to construct H r is very similar to the prescription of "additional subtractions" used by Zimmermann in his proof of Wilson's expansion9 and also used by Mueller in his derivation of the cut vertex expansion.2 The term with ( - t r ) subtraction operators on the right-hand side of Eq. (4.1) of Mueller's paper in Ref. 2 is the analog of Hr. However, in Mueller's formalism, the additionally subtracted contribution vanishes as an inverse power of q2, whereas the Hr constructed above approaches a constant-independent of Pir2,Pir' *--and therefore is unsuppressed. (The "additional divergences" in Mueller's language correspond to the mass singularities in the present language.) This difference in the two approaches arises because in Mueller's approach there is a subtraction for the diagram r itself while in the present approach we do not have a subtraction corresponding to r itself.

(2) We have proved Hr to be free of mass singularities, however, it is full of the new UV divergences introduced due to the subtractions. These, as remarked in the proof of the theorem, are kept regulated but unsubtracted. Eventually we will argue their cancellation.

Now, using a simple topological identity (a slight variation of one of Zimmermann's identities9), we

I

zr,,= J d4kly J d4k2, J d4i~/yf r /y(Ar ,~ i r ,k i r

I

will rearrange Eq. (2.5). Since the hard parts are either nested or overlap-

ping but are never disjoint, every forest has a unique maximal element, i.e., y,,,>y for all y's belonging to the forest. Hence, the set of all r forests can be partitioned into classes of forests with a given hard part 7, as their common maximal element. 7

enumerates all the hard parts. Hence,

Using this in Hr , we get

or, putting pir,pir =fir,&'r we have

Consider a typical term in the sum, say the one corresponding to T= y. This term is

Recall that we have already identified the jet loop momenta corresponding to any given hard part y C r and kly,k2, are the total jet momenta entering y. kly and k2, belong to the "pl jet" and "p2 jet," respectively. Carrying out a trivial change of variables, we write Eq. (2.29) as

where, Ir<, are the "remaining" loop variables from r/y and we have used &y=fik,P?:r=Er. This may further be wntten as


a,,= J d k ~ + J dk2-~r/y(Rr,ki+,k2-)~l(ki+,kz-) where

Fr/y(fir7ki+,k2- )' J d3kiY J d3k2y J d4Tr/yzr/y(Rpkiprrh)

J d3kl, J d3k2yTr/y(fir,ki,)

and we have used the definition of f i , given in Eq. (2.2).

The external lines of r (and y ) could be quarks and/or gluons, and carry various color, flavor, Lorentz, and spinor indices. Let us denote all the specifications of the external lines by Greek letters a,B, etc. For example, a = l may be quark- antiquark pairs with a certain flavor, color indices, and spinor indices.

Consider all the T'""s specified by an index a. For each I?'"' Eq. (2.28) holds. For each r'"' pick out those T ( ~ ) ' S which have index P. We may sum over all the T(")'s with a fixed T ( ~ ) and generate all possible contributions to T??~. Repeating for every possible dB) and summing over all the ~ ' ~ ' ' s , we can express the sum over all the I"""s as

FIG. 7. Classes of r ' s which have a unique pair of T i and r; such that r/r: has only one hard part. r'=r; n I?;. The broken lines mark the subdiagrams. (i) has jet blobs. (ii) has blobs of uncut soft lines. (iii) has blobs of cut soft lines.

where sum over f i is understood. This is the alternate form of the factorization pro-

posed above. This is illustrated in Fig. 8. Remarks: (1) All the Q2 dependence [Q2 is the

only variable describing the final state for the DY cross section (cross section integrated over q)] of 5 is contained in H ' ~ ) , whereas all the dependence on the variables describing initial state is contained in the function Tap. Since the momentum variables kl,, avearing in HB are parallel to ply, both Tap and H have s dependence.

(2) The form of Eq. (2.32) is very suggestive of a

FIG. 8. The first factor represents FB of Eq. (2.32) whereas the second factor represents HB. The 8 represents the integration over K1+ and K2-. The central parts in the first factor correspond to "bare vertices" while in the second factor, they denote the lepton pair. The arrows indicate momentum flow only. The "cuts" on the momentum lines in the third diagram of the second factor indicates a subtraction term.


convolution form which can be factorized by taking suitable moments (say, moments with respect to 7% Q2/s ). However, because of the s dependence in Tap, the moments with respect to r do not factorize the "convolution" form. If Tap does not have s dependence (unsuppressed logarithms of s ), then the usual moments with respect to T do complete the factorization; however, the s dependence is precisely the signature of uncanceled soft divergences whose presence we have assumed. If this method is ap- plied to the leptoproduction process, then the corresponding Tap will not have dependence on the large-scale -q2 and then there is no difficulty in taking moments with respect to x = -q2/2p.q. Also due to the s dependence of Tap, we cannot derive a renormalization-group equation (RGE) in the usual way to get the asymptotic behavior of the cross section. This question is tackled in the next section.

(3) As mentioned earlier, both TUB and H 6 contain the new UV divergences generated by the subtraction procedure. However, the left-hand side (LHS) of Eq. (2.32) is just the usual renormalized contribution from all the diagrams and knows noth- ing about the new UV divergences in TaB and H ~ . Hence, these divergences from TaB and H B must cancel against each other. Also, we have been using an axial gauge characterized by a vector T ~ . Both

I

TaS and HP individually depend upon rip. (We paid no attention to the gauge invariance of the subtraction procedure.) However, Za is gauge invariant and, therefore, the q,, dependence from Tap and HB must cancel between the two. This alternative form of factorization is therefore full of new UV divergences and is also gauge dependent. Therefore by itself it is not of direct use. We will see in Sec. I11 that we do not use Tap and H~ directly to get physical predictions.

(4) In arriving at Eq. (2.32) we made no approximation to Z and, thus, this equation is really an identity. Approximations will be made in the next section.

( 5 ) The Za is not quite a cross section. In axial gauge, the self-energy diagrams also contain mass singularities (double logarithms per loop). For convenience, we have chosen to include the self-energy insertions on the external lines and then following the correct Feynman rules we have to divide by an appropriate a - t h e wave-function renormalization constant. Since we cannot perform renormalization on the mass-shell due to the infrared divergences, we have an UV-finite Z factor left over after all the UV infinities are canceled. This Z factor contains mass singularities in an axial gauge (due to the nonconvariance of the axial gauge). The cross section is related to Za as

1 a a ( P 1 , ~ , , Q 2 ) = (wave function for each external leg) X an appropriate - for each external leg x Ca . 6

This relation is important in getting a generalized anomalous dimension that will appear in the RGE satisfied by ua.

111. ASYMPTOTIC BEHAVIOR

The usual factorization2 form of cross section also allows one to derive differential equations to analyze the asymptotic behavior. The s dependence of the T~~ of the proceeding sections, however, does not lead to differential equations in a similar manner. We therefore adopt a slightly different approach.

Consider an asymptotic region characterized by two scales, Q and M with M <<Q, i.e., a subset of Lorentz invariants are of the order of Q2 while the remaining ones <a are of the order of M ~ . We i want to consider I' ") in the limit Q/M+ cc . For example, the cross section for the leptoproduction

I

process is a function of p 2 = (hadron massI2, q2= the mass of the virtual photon, and pvq; and we are interested in the limit q 2 - p . q - ~ 2 , p 2 - ~ 2 with Q/M+ CC. Let us define a I?:; in some consistent approximation scheme as

Define d to be the degree of homogeneity of r,,,. Then Fa,, satisfies

along with the renormalization-group equation

a a [p$+RgI: +mn(gl- n y i g l rasy(aQ,M,m,p,g)=O . dg am I (3.3)

P(g), ~ ( g ) , and y(g) are the usual renormalization-group functions. We have written the equations for r,,,


directly and replaced the variables I f p ) by Q and (g,) by M. The homogeneity is in the variables p , m, M, and Q (or a ). Eliminating p( a/ap ) we get

Let us assume that

Then Eq. (3.4) becomes

If the symbolic product @ is the same as the ordinary product then the solution is

rasy(etQ,M,m,g)= r,,,[Q,M,H(t,m,E(t,g)),g(t,g)l

xexp ~ ~ d r ' [ n y ( g ( t l , g ) ) - d -n(et'~,~,~(t'),g(t',g))l .

Remarks: (1) M refers to all the invariants g,, which are of the order of M ~ , and hence

(2) All possible lnetQ/M factors are organized in and the first factor on the RHS of the solution does not have any lnetQ/M though it may have lnQ/M.

(3) E(t ,g( t ,g)) and g(t,g) are the usual effective parameters with t =lna. For a more general symbolic product (typically a convolution of functions) in Eq. (3.6), the exponential in

Eq. (3.7) is generalized (e.g., it may become a "path-ordered exponential"). Nonetheless, we get a solution for r,,, itself and not for one of its "factors," just as in the case of an asymptotic region with a single scale. All the complications of the asymptotic region with two scales are organized into a , which we call a "generalized anomalous dimension," and the asymptotic behavior in such a region can be analyzed by studying a generalized anomalous dimension.

Our strategy is to express M aaa/aM as

with an appropriate definition of e. Following remark (5) of Sec. I1 and using Eq. (2.33) we write

aa'= (wave functions)M- -Za a M Za

I aza a + M - =(wave functions) - -M--" za2 a~ Za aiia a M I =(wave functions)

(wave functions) a aa' = yIRua + M-Sa .

a a M


Z, denotes the product of four v'? factors (of quark or gluon) from the four external lines and

Zi being the (finite) wave-function-renormalization constant of the ith external line. Note that y,, is different from "yuv" (the usual y) which also appears in Eq. (3.28).

We will see below that

Suppressing the wave-function factors which do not affect the argument, Eq. (3.8) can be rewritten as

After explaining the product e we will write the expression for naB. In the above equation we have expressed zB in terms of up, the usual cross section, with the Z ; factors.

For DY cross sections and p22 are the only "small" momentum invariants. In the c.m. frame with initial motion only in the Z direction, we can write

In the second equality a/apl - and a/dp2+ are taken at fixed p , + and p2-, respectively.

We begin with Eq. (2.32). Acting on it by M a/aM

Since the M a/aM is expressed in terms of derivatives with respect to external momenta, 5 is independent of the loop assignment chosen in the integrals corresponding to TUB. We would like to take the differential operator inside the integral and analyze its action on the integrand TaBH@. This is done conveniently if the limits of integrations for the k I - ,k2+ integrals are independent of p l - and p2+. (The limits on k l + , k2- integrals are independent of p,- ,p2+ .) The routing of the total jet momenta kl and k2, implicit in Eq. (2.32), and shown in Fig. 9(a), is not convenient for this purpose. By choosing the routing shown in Fig. 9(b) we can make the limits of integration independent of pl-,p2+ and take the differential operator inside the integral sign without any extra contribution. Further, the UV counterterms included in Tab for the subtraction of the usual UV divergences can be chosen to be independent of p 1 - and p2+ components and, therefore, will be completely ignored in calculating

on the integrand. Thus in ternis of the momentum routing of Fig. 9(b), we write Eq. (3.11) as

A priori, M(a /aM) also acts on HB, however, tinue to use the same notation for convenience. The knowing the M dependence of HB( K K~ 1, where action of M (a/aM) on TUB can be so organized as Ki =pi -k[ , described in the induction hypothesis of to construct flap. the theorem of Sec. 11, we see that Consider a diagram T, contributing to TUB. The

a a differential operator will act on only those propaga- M - H ~ = M - [ c o ~ s ~ + o ( M ~ ) ] - + o ( M ~ ) .

aM a M tors which carry the momenta p l and/or p2. We will denote the differentiated propagator by putting

We will drop these terms, and therefore, obtain an a prime on it (see Fig. 10). Thus, from a given r, approximation Z fs,' to 5,'. However, we will con- we generate many r 's, each of which has one and

ALTERNATE FACTORIZATION FOR THE DRELL-YAN PROCESS

FIG. 9. (a) shows the momentum routing implicit in Fig. 8. (b) shows the momentum routing suitable for Ma /aM operation.

only one in its propagators differentiated. The topology of the diagram T, contributing to

T " ~ is the same as that of the diagrams contributing to 8"; in particular the regions which may give a mass-singular contribution have the same form. Therefore, we will use the same terminology of hard parts, subtraction operators, forest expression -Pr, etc., used in the previous section. However, unlike

FIG. 10. Example of a diagram with a prime in it. The prime on the 6 1 - k l ) line indicates that this propagator is differentiated.

the previous section we now define r (or r') to be a hard part of itself, while Ao, the "Born diagram" for T (see Fig. 8), is not a hard part of T. All other definitions of Sec. 11 are precisely carried over. In particular, hard parts of T (or T') are either nested or overlapping but never disjoint.

Now consider a particular I". Let ACT' be the largest hard part not containing the differentiation. Clearly for a given I?' there is a unique A. There are diagrams, T"s, for which A = Ao, though A. is not a hard part of T' (by definition). These will be considered at the end of the following analysis.

Define

J dl, R A G - J 2 n ( - t k ) ~ A d , (3.13) ( u A ) h E u A

where U A is a A forest and ( U A ) is the set of all the A forests.

Since the hard parts of A are never disjoint, every A forest has a unique minimal element p; i.e., every hard part h E v A contains p. Hence,

2 = I + 6 2 t - t . (3.14) (u,) P ( V , ( P ) I hEu,(p)

Here u , ( p ) is a forest such that y E v 1 (p) * y >p (y#p). {v l (p) ] is the set of all v l (p) forests and includes the empty forest #. The xp runs over all the hard parts of A. Clearly

Defining


The first term on the right-hand side may be interpreted as a contribution to 2 from p=Ao, with the definitions IAo r fi4 of obvious momenta, and tAoIAo = 1. (Note that A/&= A.) Thus, we rewrite Eq. (3.15) as

where p now runs over all the hard parts of A and includes Ao. Using Eq. (3.18), the contribution of a particular T' to 5"' may be expressed as

Exactly as in the previous section, this can be rewritten as

X @(PI,,& ) (3.21) setting

(3.22) we get

The sum over all I" may be organized as follows. For a fixed A, we sum over all the r' which correspond to this A. (Note that for a given T' we have a unique A and the converse is being used above.) Next we sum over all A's. For the sum over A, we first fix a p and sum over all the A's which satisfy A 2 p ; and then sum over all the p's i.e.,

all A fixed

rt A r' all p fixed A fixed

= [f zp] ? Performing the sum in the order implied above, we

Combining the k ; , k ; integrals with I, integrals, we 'get get all

where, d-- Bpi?; and

(3.26) The expansion of 6"' in terms of diagrams is shown in Fig. 11. The integral in the definitions of fiaS is

FIG. 11. Diagrammatic representation of Eq. (3.26). The first factor denotes I' while the second factor denotes R. The circles denote 4PI blobs (compare Fig. 8). The "cuts" on the lines in the third diagram of the second factor indicates a subtraction term. The o denotes loop integration. The prime inside the smallest circle, in the first factor, indicates that only propagators of the smallest hard parts are differentiated.

27 - ALTERNATE FACTORIZATION FOR THE DRELL-YAN PROCESS 209 1

over all the internal loops of A and of T1/A. This %pression also gives the calculational procedure for nas. The inclusion of those r' for which A = Ao, is straightforward and is represented by the first term in the second factor in Fig. 11. Thus, we have shown that

where @ means the integral expressed in Eq. (3.25). Remarks: (1) As noted earlier, the final answer for

M (a/aM)Za is independent of the momentum routing used in the integrals corresponding to T ~ ~ ; how-

ever, the calculational procedure does depend upon this loop assignment and due attention is to be paid in choosing a particular loop assignment. The position of the differentiation prime will, in general, be different for different assignments.

(2) It may appear that we could have directly obtained

similar to Eq. (3.18) instead of Eq. (2.32). However, now

and we do not get a form we want, to identify a generalized anomalous dimension. Hence, we had to construct the H function (instead of an R function) as an intermediate step.

We note that both aa' and aB in Eq. (3.25) are gauge invariant and free of the "new" UV divergences introduced in the construction of H. There- fore, fias must also be gauge invariant and free of the new UV divergences.

As such, given a a' and as, aas is unique, if it ex-

ists; and, therefore, is independent of any intermediate factorization used in the above manner. The intermediate factorization expressed by Eq. (2.32), however, has been useful in proving the existence of fiaS and has also yielded the calculational procedure for fiaB.

Having defined the generalized anomalous dimension we write the RGE satisfied by on as [compare to Eq. (3.6)]

where ya is the total contribution of the usual UV anomalous dimension and da is the dimension of aa and is given by

3 da =4- ,(external fermion lines) - (external boson lines) -2 (due to the differential cross section) .

flR is defined in Eq. (3.8) while fins is defined in Eq. (3.26). Za(pi,p2) is the product of 1/Z (UV finite) factor for each external line of aa. ZB( k l + ,k2- ) are similar

factors for external lines of 2,. After all the wave functions, initial-state factors, etc., are included da = -4 for dua /de2 , for all a. Now we define the generalized anomalous dimension flap as

so that


Taking into account the kinematics of the Drell-Yan process we write the RGE in full details as

Since we are using M as a regulator of mass singularities, the actual particle masses are irrelevant and, therefore, we drop the ( 1 -q) term by taking m [g(p)]=O. Now setting t =lnh, we write Eq. (3.31) as

where only the relevant functional dependence is exhibited. Note that flap has S dependence.

This is the final form of the RGE that we get. Note that this is an equation for a tensor cross section (i.e., averaging over the initial quantum numbers is not performed). Averaging over the initial quantum numbers does not help, since this averaging removes the index a but retains the index p. Then we do not have the same a on both sides of the above equation. Hence, we will consider the equation for the tensor cross section.

IV. REDUCTION OF THE RENORMALIZATION-GROUP

EQUATION

The indices a and P in (3.32) refer to all the quantum numbers of all the four "external lines" of the cross section and runs over a very large number of values. However, a tremendous simplification is possible because power-counting and invariance arguments can be used to restrict the values of ,O. This is done as follows.

Any given aa is a tensor with indices corresponding to space-time, Dirac matrices, color, and flavor. Let ( TF, i = 1, . . . , qa) denote a set of linearly independent tensors with the collective index a. Let

4i's are invariant functions of Lorentz scalars and

group scalars, and carry no indices at all. The dots in the arguments of TF denote any numerical (invariant) tensors (e.g., 6i,g,,), which can be used to construct T and those in 4 refer to g(p),ncolor,nflavor. aa is gauge invariant and, therefore, there is no q, (axial gauge vector) dependence.

Substituting Eq. (4.1) into Eq. (3.33, we get

where

The tensors TY can be absorbed into aaS and flaS~,! itself can be expanded in the same basis. Let

Substituting in Eq. (4.2) gives

'la qa "P 1 x T;(DI$~)= 2 T F x 2 dxa#'(x)I$,(r/x) . i=l i=1 fl ] = I

(4.4)

Using linear independence of Tia, we get

Thus, using a tensor-basis expansion, we have translated the equation for ua into an equation for "scalar" quantities #Ji. The matrix dimension of w is (2, nu) and is still very large.

We are interested in the mass singularities present in flap (and hence in mi,), i.e., the lnp12,1np22 dependence of wij. To determine this, we reapply power- counting arguments to the diagrams from which flap and, hence, wij are calculated. Not all the tensors T,! absorbed into flap lead to lnp 2, and we drop these T ~ S . Effectively, not all j's in the 2p2x1 contribute. Suppose only j = 1 . . N << ( x p q p ) contribute. Then, Eq. (4.5) decouples as

and

Equation (4.6a) has the same form as the original equation (3.32), but with much smaller matrix dimension. Equation (4.6b), however, is of a different


form. If 4j's are obtained by solving Eq. (4.6a), then the right-hand side of Eq. (4.6b) is completely known and is independent of +izr. Thus, Eq. (4.6b) does not have the form of the renormalization-group equation (3.21, but is similar, in form, to the Callan- ~ ~ m a n z i k " equations.

We proceed to obtain the asymptotic behavior of the tensor cross section by solving Eq. (4.6a) and then Eq. (4.6b). Once the tensor cross section is known, various averages over initial quantum numbers can be performed.

In order to determine the tensors T! that may give twist-two logarithms in flap, we use power- counting and invariance properties. For simplicity, we consider only a single flavor for the quarks. We take these quarks to be in the fundamental representation the color group SU(3).

We group the tensor cross section (or tensors) into three classes according to the type of lines they refer to (see Fig. 12).

(i) Quark-quark. All four lines are fermionic.

= Dirac = Group

index index

(ii) Quark-gluon. Two are fermions and two are gluons.

(iii) Gluon-gluon. All four are gluons. For class (ii), any two lines could be fermions.

The fermion arrows may go in all possible directions.

First, consider class (i). A priori, we consider the most general contribution to the cross section with four fermion lines, i.e., the two fermion arrows may be followed in all three possible ways. They are as follows. Ma) Each arrow stays exclusively in a jet IJ1,J2). (i)(b) Each arrow passes through both jets ,, without crossing the cut. (i)(c) Each arrow passes through both jets and crosses the cut.

Consider Ma). From power-counting arguments, we see that from the fermion line passing through jet J1 , we get a y+ while from the one passing through J2 , we get a y-.

For both classes (i)(b) and (i)(c), power counting allows only yl,, for each line and effectively we get only one tensor, namely, (yl )(yl)+(yz)( y2).

Now consider class (ii). We first rule out the possibility that the fermion line may pass through both jets. The general tensor of this class has two Lorentz indices of the gluons and Dirac matrices, yp, due to the fermion line. The Dirac matricesA yp, ~ u s t contract with a momentum vector, either k or k,. If the fermion line passes through both jets, then power counting alloys onlyJransverse y matrix structure. But neither k l nor kz has a transverse component and, hence, the contribution from the above possibility is suppressed. Thus, the fermion line must pass through either J1 or J2. If it passes through J 1 , it gives y+ and if it passes through J,, it gives y-. The indices of the gluons must be transverse [as in class (iii)] and the Lorentz tensor must be gii .

Consider class (iii) (see Fig. 12)

--I

(iii )

FIG. 12. The q-q, q-g, and g-g sectors. In the q-g sector there is one more diagram which is not shown.

Aijkr is a group tensor and which is a Lorentz scalar and TpmP is a momentum-space tensor which is a group scalar. a is gauge invariant and, therefore, does not depend upon the axial gaugz v ~ t o r r] .

The most general form of TpY8( k kz ) consistent with Lorentz and parity invanances, has 43 basis tensors. Gauge invariance and power counting reduces this large number to only three, namely, g,,g,p, gpagvp, g,g,p, with only transverse components.

Thus, to summarize: if we write o a = ~ (PUP)T (space-time) and classify T into classes (i), (ii), and (iii), then the expansion of T in' terms of contributing tensors alone, is (see Fig. 12)


T? 'AB,~=( y- )ABg,B~!i' , g,vgag~yil +g,agvg~!ii)

+g,ggva~:"il . Here, W s are Lorentz scalars (and also group sca-

Jars) and p,v,a,P can only be 1 and 2. The position of the fermion indices refers to jets through which the fermion arrows are followed.

Now we want to analyze the tensor A(groUp'. Clearly, power counting is irrelevant. However, the momentum-space analysis restricts the way fermion arrows can pass through the jets [e.g., the class (ii) tensor analysis], and this simplifies the analysis of group factors. Once again, we classify A into classes (i), (ii)(l), (ii)(2), and (iii), with the restrictions due to power counting. The procedure to get these is described below.

Let us write the tensor A schematically as A =A:! where AB now refers to the grorp indices of the top two lines while CD refers to those of the bottom two lines. This we write as

A:: = 2 dB@ E j c ~ wij , (4.9) i,j

where J is a basis in the space of tensors with top indices and (E, J is a basis in the space of tensors with the bottom indices. The @ denotes a tensor product of the two bases and Wij are "expansion coefficient," which carry no group indices. It is clear that the contribution of any diagram contributing to aa can be written in the above form with unique Wij .

A pair of the top (bottom) lines can be a pair of gluons or a quark-antiquark pair. For the classes we consider, we never have a quark-gluon pair and, as usual, we assume that we never have a quark-quark (antiquark-antiquark) pair. This is one of the reasons to group the top (bottom) two lines together rather than the left (right) two lines. We may call this an "expansion in the t channel."

Now we note that the QCD Hamiltonian is invariant under global SU(3) transformations and, therefore, the S matrix is diagonal in the subspace of states which transform irreducibly under SU(3) transformations. Hence, choosing the bases

1, { E ~ 1 suitably, we may find that Wij =O for many combinations of i and j; and for some classes of diagrams, it may be proportional to aij. This will reduce the number of terms in the xi, considerably. For example, if we consider class (i) with quark-

antiquark top and bottom pairs, then these pairs could be in a singlet or in an octet state. Therefore, a priori, we have 2 >< 2 = 4 terms in the sum. Howev- er, Wijaalj and, thus, only two terms survive. We can carry out a similar analysis for the gluon pair also.

In this way, all contributing tensors are determined. To obtain WIJ, these tensors are to be absorbed into flap. This is done by introducing "bare vertices" for each of the contributing tensors and as- signing the tensors to them as the corresponding "bare vertex factors." After flaS is calculated using these new graph elements, we project out the contribution to the ith tensor to get wlJ. The wlJ appear- ing in (4.6a) are to be calculated using the following Feynman rules.

The bare vertices of Fig. 13 are to be taken as new graph elements. The vertex factors to be associated with these elements have a momentum-space part and a group-space part. The momentum-space factors are given explicitly while the group factors are given in terms of the projection operators P. All the

FIG. 13. Bare vertices. The P s represent projection operators in the group space of the t channel.


bare vertices have momenta K1 (top) and K2 (bot- pret the formal solution? tom) going into them on the left side of the cut. If (2) Regardless of the interpretation, what are the kl ,k2 are the corresponding "jet momenta" [see Eq. general, qualitative features of the solution? (2.211, then all the bare vertices have (3) Given some answers to (1) and (2), what are

2 the physical implications of the results?

8 (C l+ )8(C2- )e(2khl+C2- -Q 1 . These questions have been explored to some ex-

The remaining momentum-space factors are the ten- tent and a discussion of each is presented below.

sors with Lorentz, spinor, and group indices. For (1) Internretation of the solution - .

the vertices with ferkions, other directions for arrows are also possible. Corrections to these bare vertices are drawn by using the usual Feynman rules. For cut diagrams, replace the denominator of each cut line by ( - 2 ~ i ) G + ( ~ ~ - r n ~ ) ; where 6+ has a 8 function included in it which ensures positive energy flow across the cut from the left to the right.

We close this section by exhibiting a formal solution to Eq. (4.6a). Let

where 1

T.(u,t,r,g)= s o u d u ' S dxo(t-u1,x,g(u' ,g))

and

is an arbitrary function of T and g. Then $ defined by Eq. (4.10) satisfies Eq. (4.6a) with the initial con- dition

The proof by induction is straightforward. Note that every 4o generates a solutio~i.

V. DISCUSSION

The main outcome of the present analysis is the framework of the renormalization-group equation, Eq. (3.32), in which the asymptotic behavior of the perturbative cross section can be analyzed without having to cancel IR divergences. In the course of the analysis, we defined the "generalized anomalous dimension," did tensor analysis to simplify the calculational procedure for it and, subsequently, obtained a formal solution to the RGE.

Three main categories of questions pose themselves:

(1) In view of the fact that the perturbative cross section (with quarks and gluons) is not the observed cross section (with hadrons), how should one inter-

As emphasized above, the analysis has been done entirely within the perturbative framework; and, in particular, the tensor cross section, aa , satisfying '

Eq. (3.32) is a perturbative cross section. With the intuition of the parton picture, it seems natural to identify the perturbative cross section, a~ol,rl,,,, with the partonic cross section and convolute it with some "distribution" and "fragmentation" functions, D' and 3''. However, unlike the usual QCD- improved parton model in which the distribution functions acquire a Q2 dependence due to the ab- sorption of the mass singularities from the perturbative cross section, the D"s above, will not acquire any Q2 dependence since no pieces of the perturbative cross section are absorbed into them. The formal solution itself gives all the Q2 (or s) dependence of the full, hadronic cross section. In this picture, the D' and F' functions play a somewhat passive role in linking the hadronic and perturbative cross sections.

We also note that given naS (or wIJ), every choice of a$(r,g( t,g) ) [or qO( r,g( t,g) )] generates a formal solution. The framework by itself does not provide a clue as to what to choose for F g ( ~ , g ( t , ~ ) ) . One could choose it also, as a perturbative cross section in which case as t- CO, Fg-G( 1 -x), i.e., is the "Born" contribution. One might also take it to be some nonperturbative function which, as t + cc , looses all its dependence on the perturbative effective coupling, g(t,g), and approaches some unknown constant (independent of t) value which has to be ex- tracted from the experiments. Presently this ambi- guity is not resolved and it remains an open question.

As a side remark, we note that the formal solution given above may also be used for the leptoproduction (DIS) process, where we know that the convolution form can be factorized by taking the usual moments. One then does not have to invert the moments to get the x ( - -q2/2p .q) dependence. This application of the present method has not been studied in detail.

(2) Qualitative features of the solution

The study of qualitative features of the solution begins with the study of some lowest (nontrivial) or-


der properties of (wrJ). In the Appendix, the leading contributions from 0 (g2) correction to the fermion- fermion sector (i.e., bare vertices and the external lines both belong to the fermion-fermion class) are calculated. For the leading contributions, the integral in Eq. (4.6a) reduces to a simple product, and it is possible to integrate (4.6a) directly, rather than to use the formal solution. We note the following: (a) The anomalous dimension wrJ itself may not have an s dependence, but its x dependence can introduce a logarithmic s dependence from the "convolution" integral. The net result is that the "effective" anomalous dimension gets a t dependence. (b) To 0 (g2 lm), in the fermion-fermion sector, there is no mixing of the (color) singlet and octet cross sections (or 4's). For the singlet cross section, the t dependence cancels and we once again get the usual anomalous dimension (independent of t) which gives the usual asymptotic behavior of the singlet cross section. For the octet cross section, the t dependence does not cancel but leads to a Sudakov-type suppression of the octet cross section [see Eq. (A13)I. Note that the subleading contributions such as those coming from a soft-gluon exchange between the p , and p 2 lines do mix the singlet and the octet cross sections, but are s independent. Effects of these contributions are, presumably, small as t-+ oo . (c) The BBL graphs [O(g6)] show the possibility of mixing of the singlet and octet cross sections for the leading twist contribution. If taken by themselves, they lead to a t-independent matrix (in the color space), the anomalous dimension of which is different from the usual anomalous dimension (diagonal in the color space), implying a deviation from the usual QCD-improved parton model. However, as pointed out by M ~ e l l e r , ~ we see the Sudakov-type suppression of these contributions due to the noncancellation of the leading t dependence at 0 (g2).

(3) Consequences

From these examples we tentatively consider the following scenario for the recovery of the standard results. Consider Eq. (4.6a):

Let us assume that WIJ is diagonalizable in the color space. If so, we define 4'= U4 and diagonal- ize the equation. This will lead us to N decoupled equations of the form

Suppose that for J corresponding to the singlet

bare vertices all the explicit t dependence of UIJ cancels for all I = 1, . . . , N. (This has also been suggested by ~ u e 1 l e r . l ~ ) In that case, the t-dependent entries, ~ I J , have the form

J Singlet Nonsinglet I

Singlet 0 Nonzero Nonsinglet 0 Nonzero

Clearly, as far as t dependence of w , ~ is concerned, the singlet and nonsinglet blocks completely decouple, i.e., the singlet eigenvalues hJ in Eq. (5.2) have no t dependence and are completely independent of the entries, wlJ, corresponding to the nonsinglet blocks. The asymptotic behavior of 4Hinglet is then governed by the usual anomalous dimension. The nonsinglet eigenvalues, hl, should all have large negative coefficients for the leading S dependence so that they are suppressed in a manner similar to the Sudakov-type suppression explained in the Appen- dix. If this is true, then for large enough t all the nonsinglet contributions will vanish and we will be left with the singlet contributions which have the usual behavior. This is a possible mechanism for the recovery of the asymptotic behavior implied by the usual factorization conjecture as suggested by ~ u e l l e r . ~

Apart from the study of the asymptotic behavior, the present framework offers a limited universality of normalization of cross section. Assuming that a moderate generalization of this method is done for cross sections with more complicated final states, e.g., jet cross section, semi-inclusive cross sections, etc., the formalism predicts the same anomalous dimension for different final states produced by the same initial state; and thus, leads to the same asymptotic behavior for these cross sections. With the appropriate D1,F' functions and a suitable interpretation of the solution, the formalism suggests the same normalization for all the hadronic cross sections initiated by the same pair of hadrons. Thus, we predict the same K factor for these various cross sections. However, we cannot relate the normalization of cross sections for, say, a-p and p-p scattering, in the present approach.

ACKNOWLEDGMENT

I wish to thank George Sterman for suggesting this topic and for helpful discussions. This paper is supported in part by the National Science Founda- tion Grant No. PHY8 1-09 1 10.

ALTERNATE FACTORIZATION FOR THE DRELL-YAN PROCESS 2097

APPENDIX suppression of the "octet cross section." We also include a calculation of the Glauber region contribu-

In this appendix, we calculate the leading graphs tion to point out a mixing in the group space, contributing to the generalized anomalous dimen- We recall the definition of f laB and subsequently, sion to illustrate some of its features in the O(g2) calculation. We use these to show a Sudakov-type of ~ I J

naB= product of the two factors shown in Fig. 11, and

Consider the contribution coming from the first term in the first factor (there are many graphs in this term), and the first term ("Born" term) of the second factor (Fig. 11). In the first factor the differentiation prime always appears in the smallest hard part. The first term in the first factor has only one hard part and therefore we can calculate it by operating by

on the diagrams (with one hard part only) without a prime.

We calculate the graphs shown in Fig. 14. From graphs (a) and (b) of Fig. 14, we will obtain the coefficient of the leading logarithm (double logarithm), whereas, from (c) and (d) we extract the contributions from the "Glauber" region. These exhibit the t dependence ( t =lns/so in the RGE) of the generalized anomalous dimension, and the role played by group factors associated with the bare vertices, in the (non)cancellation of the t dependence, as well as in the mixing in the group space. The contribution from graph (a) is

The integrand of Eq. (A2), together with the factors outside the integral, is the leading contribution to oaB (in the momentum space). This itself has no s dependence; and, hence, no t dependence. Its x dependence is of the form x2/1-x where x =(p l - k)+/p l+ , and from Eq. (A2) we see that this x dependence leads to a logarithmic dependence on s from the region near x -- 1 (k + - 0).

To get the right-hand side of the RGE, Eq. (4.6a), we multiply the integrand of Eq. (A2) by $(r/( 1 - k + /p + 1) where $ is the scalar coefficient corresponding to the bare vertex (y+)AB(Y- )CD; (different 4's for singlet and octet group factors). Thus, the contributions from graph (a) to the right-hand side of the RGE is

(A3) #(T/x)-4(r) gives a finite contribution as x - t 1,

whereas $(r) isolates the divergence from x -+ 1 region. The divergent part is

which can be expressed as

+nonleading terms . (A4)

The group factors can be evaluated very simply, and the singlet (octet) bare vertex leads to a singlet (octet) tensor structure for the contribution of graph (a) and, thus, there is no mixing in the group space.

Now consider the self-energy graph (b), of Fig. 14. In axial gauge it is known to have a double logarithm. The calculation is straightforward. The final answer in momentum space is

The group factors are the same for both singlet and octet bare vertices. The effect of the SR terms is to cancel the contribution from one of the self-energy insertions. (There are two such insertions, one on thepl leg on each side of the cut.)

Identical contributions are given by the graphs of types (a) and (b) involving the p2 legs. There are all the graphs which give leading contributions from the O(g2) corrections to the fermion-fermion bare

2098 GHANASHYAM DATE 27 -

vertex. (Also, the external lines are fermlons.) For these contributions, the singlet and octet functions, 40 and 48 satisfy a diagonal RGE.

For the octet bare vertex, all the l n f i /M dependence does not cancel, due to the mismatch of the group factors. It gives the right-hand side of the RGE, Eq. (4.6a), as

where N =n2- 1 and n =3 for SU(3). The RGE we get for this contribution is

The "anomalous dimension" has t dependence unlike the usual anomalous dimension (we have neglected it since it is independent of t).

Introducing g( t,g) defined by

we write the solution of Eq. (A71 as

=~,(o,g(O,gi)exp sotdtla(tl ,g(t ' ,g)) , (A9)

where

Replacing g by g( -t,g) everywhere in Eq. (A9) and using

g(tl ,g(t2,g))=g(tl +t2,g) , we get

48(t,g)=48(O,g( -t,g))

xexp ~otdt la( t l ,g(r ' - t ,g)) ( ~ 1 0 )

or, setting t'-t = -t" in the integral we get

48(t,g)=48(0,g( -t,g))

xexp c d t f l a ( t -t l ' ,g-t",g)) . ( ~ 1 1 )

If P(F) = - bg3, then

Hence, for b >O (i.e., for asymptotically free theories), g2( -t,g)-+O as t--+ C C . The integral in the exponent gives

FIG. 14. Graphs calculated in the Appendix.

which for t" - t --+ cc gives

t -t" -n ( t lnt -t)

2n2

-n -- - ( t lnt) . 4n2b

(A13)

Thus, we see that the "octet cross section" $8 is suppressed like a Sudakov-type form factor.13

Next we consider graphs (c) and (d) of Fig 2. In these sixth-order graphs, Bodwin et aL4 found a contribution from the so-called "Glauber region" (described below) which is associated with a subleading logarithm (single logarithm from a three-loop diagram) of the leading twist type (i.e., it is unsuppressed by an inverse power of s). This contribution does not cancel between the "real" [graph (c)] and the "virtual" [graph (d)] graphs, due to the


mismatch of group factors. In an Abelian theory, The "Glauber region" corresponds to on-shell these contributions do cancel. In the present frame- propagations of lines 1, 2, 3, and 4 [Fig. 14(c)]. work, we will see that these graphs "mix the repre- Furthermore, the momenta kl-and k2 have trans- sentations," i.e., the singlet bare vertex leads to both verse components of the order k, but thcr plus and singlet and octet contributions and so does the octet minus components are of the order of ( k )2. Keep- bare vertex. For the singlet bare vertex, the contri- ing only the leading terms from the numerator, we butions from real and virtual graphs do cancel. get

Symmetrizing in k l and k2, this simplifies to

N r - - 3 2 p l + 2 k - 2 [ ( ~ + c l ) . ( ~ + g 2 ) ] ( y + ) ( y - )

Now, evaluating the contour integrals kl? ,k2& to pick up the poles in the lines 1,2, 3, and 4, we get

where we have put p 1 2 = p 2 2 = ~ 2 and x =k- /p2-. The transverse components can be integrated us-

ing polar coordinates. The final answer is

where A1,2 are the ratios of the uppe; limits_to the layer lim_its for the integrals over I k l / / I k I and I k2 I / I k I , respectively. These are independent of

x and correspond to the "boundary" of the "Glauber region." A is a UV cutoff. Taking -Ma/aMi and incorporating +(r/l-x), we write the right-hand side of the RGE as

This is the s-independent but x-dependent contribution from the Glauber region of the real graph.

Proceeding in exactly the same manner, we extract the corresponding contribution from the virtual graphs. For these graphs, the Glauber region corresponds to picking up the poles from lines 1, 2, 3, and 4 shown in Fig. 14(d). The momentum-space integrals become identical to those for the real graph, after a shift in the k integral and a symmetri- zation with regard to k l and k2 momenta is carried out. The factors of i from the Feynman rules, give a relative minus sign between the contribution of the real graph and that of the sum of the two virtual graphs. For singlet bare vertex, the group factors for real and virtual graphs are the same. Hence, for the singlet bare vertex, these contributions cancel between the real and the virtual graphs. For the octet bare vertex, the group factors are different and a noncancellation results.

'Current address: Weizmann Institute of Science, Physics Department, Rehovot, Israel.

'A. H. Muller, Phys. Rev. D 9, 963 (1974); H. D. Politzer, Phys. Lett. m, 430 (1977); Nucl. Phys. u, 301 (1977).

2D. Amati, R. Petronzio, and G. Veneziano, Nucl. Phys. m, 29 (1978); S. Libby and G. Sterman, Phys. Rev. D l8, 3252 (1978); A. H. Mueller, ibid. 18, 3705 (1978); S. Gupta and A. H. Mueller, ibid. 20, 1 1 8 (1979); R. K.

Ellis, H. Georgi, M. Machacek, H. D. Politzer, and G. G. Ross, Phys. Lett. m, 281 (1978); Nucl. Phys. u, 285 (1979).

3R. Doria, J. Frenkel, and J. C . Taylor, Nucl. Phys. m, 93 (1980).

4G. T. Bodwin, S. J. Brodsky, and G. P. Lepage, Phys. Rev. Lett. a, 1799 (1981).

5A. H. Mueller, Phys. Lett. m, 355 (1982). 6J. C. Collins, D. E. Soper, and G. Sterman, Phys. Lett.


m, 388 (1982). 'G. Sterman, Phys. Rev. D u, 2773 (1978). %. Libby and G. Sterman, Phys. Rev. D 18, 4737 (1978).

In obtaining their results Libby and Sterman took the ''normal variables" defined in their paper to vanish at the same rate. However, the example given by Bodwin et al. in Ref. 4, points out that there are additional divergent regions for which the normal variables do not vanish at the same rate. In particular, corresponding to the Glauber region, the transverse components of the soft loops scale as 1Th while the plus and minus components scale as h. These regions, which were not considered by Libby and Sterman, however, do not upset their conclusions. In particular, the physical picture of the divergent PSP's is still as shown in Fig. 2; and the

divergences are still no worse than logarithmic. 9W. Zimmerman, in Lectures on Elementary Particles and

Quantum Field Theory, edited by S. Deser, M. Grisaru, and Pendleton (MIT, Cambridge, Mass., 1971).

1°S. Coleman and R. E. Norton, Nuovo Cimento 38, 438 (1965).

I1R. I. Arnowitt and S. I. Fickler, Phys. Rev. 127, 1821 (1962); W. Konetschny and W. Kummer, Nucl. Phys. m, 106 (1975); J. Frenkel, Phys. Rev. D 13, 2335 (1976).

12C. Callan, Phys. Rev. D 2, 1541 (1972); K. Symanzik, Commun. Math. Phys. 18,227 (1970).

13A. Sen, Phys. Rev. D 3, 3281 (1981). 14A. H. Mueller, Phys. Rep. 73, 349 (1981).

alternate factorization for the drell-yan process

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