Summation of large corrections to short-distance hadronic cross sections

Nuclear Physms B281 (1987) 310-364 North-Holland, Amsterdam

S U M M A T I O N OF LARGE CORRECTIONS TO SHORT-DISTANCE HADRONIC C R O S S S E C T I O N S George STERMAN*

Inst~tutefor Advanced Study, Pnnceton, NJ 08540, USA Received I July 1986

The large soft gluon corrections to short-dmtance hadromc cross sectlons are summed to all orders in perturbaUon theory The arguments apply to any cross sectaon whose Born contribution is initiated by quarks, including Drell-Yan and heavy vector boson cross sections. The exponentiation of leading and non-leading In(n) terms in moments of these cross sections is vented. In particular, the In3(n) terms computed by van Neervan at two loops are rederlved on the basxs of one-loop calculations

1. Introduction The aim of this paper is to show how to sum the large perturbative corrections [1,2] that occur in hard hadron-hadron scattering cross sections. Despite initial progress [3, 4], the predictions of Q C D for the normalization of short distance cross sections have not yet been fully understood. This is an important issue, because the interpretation of new particles and interactions depends on an ability to calculate such cross sections [5]. It is also a testing ground for perturbative QCD. In this p a p e r we present techniques and one loop results for the summation of large corrections associated with soft gluons. Numerical evaluations will be the subject of future work. T o illustrate the issues involved, and to be specific, consider the Q C D form for the inclusive Drell-Yan (DY) cross section [6-8]. This is the cross section for the production of a lepton pair of m o m e n t u m Q~ in the scattering of hadrons of m o m e n t u m Pl and P2-

h i ( p 1 ) + h2(P2 ) ---,/t/i(Q ~) + X.

(1.1)

T h e factorized form for this cross section, valid in perturbation theory up to * John Simon Guggenheim Memonal Foundauon Fellow Institute for Theoretical Physics, SUNY, Stony Brook, NY 11794, USA 0550-3213/87/$03 50©Elsexaer ScmncePubhshers B V (North-Holland Physics Publishing Divimon)

G. Sterman / Hadromccrosssecttons corrections of the order

311

1/Q 2, is [6-8]

do = 41re 2 ~.. f,(dxl/x,)(dx2/x2)g dQ 2 9sQ 2 a, b 0

1 a(Xl, Q2)o)(g)ab(,.i./xlx2, Q2 )

X g2,b (X2, Q2),

(1.2)

where "r = a2//s. The prefactor is the Born cross section. The function g,,a(x,, Q2) is interpreted as a parton density, that is, as the probability of finding parton a in ha&on i with a fraction x, of its momentum. The function t%b is dominated by ultraviolet (i.e. of order Q) momenta; it is referred to as the "hard part" below. At zeroth order, o~= 3 ( 1 - ~//xlx2) with this normalization. Also, to(z, Q 2 ) = 0 for z > 1. The parton densities are not calculable in perturbation theory; they must somehow be derived from experiment. The hard part o~(g)ab(z,Q2), however, is calculable in perturbation theory. The fundamental fact which makes this possible is that O)(g)ab is independent of the nature of the external hadrons in eq. (1.2). Thus, (d(g)ab can be calculated with external quarks and/or gluons, as in refs. [1, 2]. The functions derived in this way may then be combined in eq. (1.2) with the parton densities of physical hadrons like protons and pions, to predict the cross sections for these particles. There is another important property of eq. (1.2) which we will exploit in this discussion. The functions g,,a and 09(g)ab are not unique. Two choices of parton density related by convolution with any UV-dominated function are equally acceptable. They simply correspond to different hard parts o)(g)ab. That is, if Cac(f ) is an ultraviolet-dominated function, and if we define

g,.o(x, Q2) =

E f 1(dy/y)Cac(x/y)h,,c(Y,Q2), C

(1.3)

X

then

dG dQ 2

4~ra 2

9sQ 2 ,,E b fol(dYl/Yl)(dY2/Y2)hl,a(

Yl, Q2)t°(h)ab( "r/YlY2, Q2)

Xh2, b(Y2, Q2),

(1.4a)

where the new hard part is related to the old by *o(g),,b(z) = fol(d~l/~l) (d~2/'~2) Ca~(Idl) to(h)~d( zldlld2, Q 2) Cab( td2).

(1.4b)

That is, the change in the parton densities is compensated by a change in the hard part.

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One often takes for the parton density g,, a the particular choice

(1.5)

g,,a(X, Q 2 ) = F ~ , a ( x , Q2),

where /TLa is the contribution of parton a to a deeply inelastic scattering (DIS) structure function [1, 2]. This has the advantage that eq. (1.1) then gives a prediction for the normalization of the Drell-Yan cross section in terms of the observable quantities F,, a. The problem with this choice is that the hard part ~(F) turns out to have large 0 ( % ) corrections [1,2]. At the same time, the experimentally observed normalization of the Drell-Yan cross section is found to be given by eqs. (1.2) and (1.5) only up to a factor of about two for ~(r) at zeroth order in as[9]. To exhibit the large O(a~) corrections, it is useful to observe that the integrals in eq. (1.2) factor multiplicatively under moments with respect to the variable "r = Q2/s,

f01dTTn-l(d~/dQ 2 ) = E gl,a( n , Q2)~d(g)ab(n)gz, a( n, Q2),

(1.6)

a,b where

ga(g),,b(n) =

fo1 d~'~'"-lt~(S)~b(~" ) ,

(1.7)

and similarly for the other two functions. In terms of these moments, the O(as) corrections may be written as [1, 2]

~(F)q~(n,Q=)=as(Q2)CF[21nE(n)-31n(n)+ %r2+~(n)].

(1.8)

The function ~(n) is a bounded function of n. The ln~(n) terms in o~(n) arise from distribution terms in t0(V)(z, Q2) of the form [ln~-l(1 - z)/(1 - z)]+, which are left over when the calculation of ~0(F)(z, Q2) involves the cancellation of real and virtual soft gluons. We shall return to this point below. To understand the origin of these large one-loop corrections, it is useful to write eq. (1.2), with i = q, j = 7:1and g = F, at one-loop as

tdq{(1)(I) =

(1/Oo)(doqrJdO=)0)-

2FqqO) ,

(1.9)

where, following [11, we normalize Fqq(X, Q2) to 5(1 - x ) at zeroth order, and we use the fact that Fq, q = Fva,ra. Eq. (1.9) is illustrated for a set of gluon emission graphs by fig. 1.1. In fig. 1.1 and in succeeding figures, the vertical line separates the graph into contributions to the amplitude and to its complex conjugate. Cut particle lines represent the on-shell particles of the final state. Q(Q') represents the virtual photon for D Y (DIS).

G. Sterman / Hadromc cross secttons

313

!

-2'D Fig. 1.1. Leading logarithm graphs in the Feynman gauge for the DY and DIS cross secUons, an the form of eq (1.9)

If we compute the real and virtual contributions on the right-hand side of eq. (1.9) in dimensional regularization (e = 2 - ½n) we find [1]

°~qq'real°)=(as'/2~r)CF(4~rl~2/Q2)~(l=~e) +2(1+z2)[

e2

3 / e

7 8(l-z)

[ ln(1 - z) 3 ~'2)-]+-[]--~_z]+

~ r(1-e) { 2 3

-6- 4z],

}

t°qra'~t°)=(ad'2~r)CF(4~r#2/Q2) ~(1-~'-~e) e5+-+8+4~'2e 8(l-z). (1.10) t%~ (1) and t%~t(1) are individually divergent, although their sum is finite. The origin of the leading divergences in %ca(x) is easily seen in fig. 1.1. They come from the region of the DIS graphs where the gluon k ~ becomes parallel to the outgoing quark l #. There is no corresponding contribution in the Drell-Yan graph, where there are no outgoing quarks at all. This difference can also be ascribed to the difference in phase space between the DreU-Yan and deeply inelastic scattering cross sections [3]. The point we wish to stress here, however, is that there are singular contributions to deeply inelastic scattering which are not present in Drell-Yan, and that it is precisely these contributions which give the large corrections to to(F) [10].

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Two other important observations have been made concerning these large corrections. The first [3] is that the ln2(n) and part of the ~r2 terms should exponentiate. That is, ~5(F)(n, Q2) = exp{ C r ( a / 2 r r ) [2 ln2(n) + ¢r21}.

(1.11)

The ln2(n) exponentiates because it is associated with soft gluons, while the ~.2 exponentiates because it is associated with the difference between leading behavior in the spacelike and timelike Sudakov form factors. The second is that [11] the ln2(n) terms may be associated with shifts in the argument of an effective coupling. What exponentiates is not just a ln2(n), but an integral of the form ~,F)(n ' Q 2 ) = exp{

Jo[ldzzn-l[[as [(--1 ( l ---z--) z ) Q2] ]+}"

(1.12)

This form has been applied in [4] to the DreU-Yan cross section. In [10] it was pointed out that the freedom to choose different parton densities might be employed to organize the large corrections to the DreU-Yan cross section. It was shown that with a particular choice of the parton density the leading ln2(n) corrections could be absorbed into g,,q and summed in all orders. It is our intention in this paper to build on this observation, although with a different choice of parton density than in [10]. We find that to describe the DY cross section as z ~ 1 it is necessary to use a form of factorization which differs slightly from eq. (1.1). To go beyond the leading logarithm approximation, an extra function must be included which describes the effects of nonleading soft gluons. We also introduce a new parton density, designed to include the effects of phase space restrictions. With these modifications, we derive a form of factorization in which In(n) corrections are separated from the hard part of the Drell-Yan cross section. We then apply analogous reasoning to the DIS structure functions. This allows us to derive an expression for the moments of the hard part oa(F), in which the large-n behavior is explicit. We find that logs of n exponentiate and that the full ~r 4 2 term in eq. (1.8) (not just the Sudakov ~r2) also exponentiates. Of course, it is not possible to infer from our reasoning the behavior of all terms which are n-independent in the large-n limit. The natural exponentiation of the complete large one-loop n-independent component, however, encourages us to suggest that untreated corrections are relatively small. Explicit calculations are carried out with massless quarks. It is also necessary to choose the renormalization prescription and gauge. These choices affect the organization of the ~r2 and related terms in eq. (1.8). We have chosen modified minimal subtraction throughout, and the temporal and axial gauges for DY and DIS, respectively. The criterion for these choices was to simplify, as much as possible, the one-loop hard parts (see eqs. [2.16] and [3.12]). This seems the most natural guide. It

all

all

G Sterman / Hadronw cross secttons

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must be kept in mind, however, that the summation of constant terms is not understood at the same level as for singular terms. Our discussion involves quark scattering cross sections only. Although gluons can also initiate the Drell-Yan and related processes, this requires the emission into the final state of one or more quarks. This, in turn, always leads to a suppression in the z ~ 1 limit. (The one-loop cross section [1, 2] shows this explicitly. It can be proved to all loops by simple power counting arguments [12].) Changing the flavor of the incoming quark also requires the emission of quarks into the final state, and is consequently also nonsingular as z---, 1. Our reasoning thus sums all singular corrections in both singlet and non-singlet Drell-Yan and heavy boson production amplitudes, and in non-singlet cross sections for other final states, such as jets. The method for deriving these results is a judicious use of the freedom to choose parton densities. The following is a simple argument which illustrates the spirit of the approach. Let there be only a single species of parton, so that there is no sum in eq. (1.2). Then suppose we find parton densities ~ with the property that ~(~)(n) has no logarithms of n. Applying eq. (1.6) with g = F and g = + in turn gives

o3(F)(n) = #O/')(n) X

[~(n)/ff(n)] 2.

(1.13)

Now we do not expect either ~ ( n ) or if(n) to be calculable in perturbation theory, since they both involve long-distance effects, but, as we shall see below, their ratio can be a calculable quantity. If we can sum the n-dependence on the r.h.s, of (1.13), we shall have the n-dependence of ~(e), which is what we want. To get a parton distribution with the desired properties, we will use the following guiding principle, discussed in more detail in sect. 2. If the parton density q~ is chosen to include only those collinear and infrared regions which contribute to the original cross section, then real and virtual contributions to the hard part in the factorization formula (1.1) will be separately finite. As we shall see, there will then be no terms in ~(~) of the form [ln~(1- z ) / ( 1 - z)]+. In the terminology of [8], subtractions are chosen so that on a graph-by-graph basis they remove all non-ultraviolet leading contributions, without requiring cancellations between subtractions for different graphs. In summary, then, we have two tasks whose completion will make it possible to compute the large n behavior of ~(r): (i) find a parton density ~k which absorbs the large corrections in 0~tr), and (ii) find a method to sum the ln(n) behavior of densities and cross sections. It is the second task which requires more extensive analysis. Here we shall draw heavily on the work of Collins and Soper [13], Sen [14] and Gatheral [15], which will enable us to sum complete sets of large logarithms. In sect. 2 we discuss factorization near z = 1 for the DY cross section, and its application to moments. Sect. 3 deals with the analogous x ~ 1 limit in DIS. Eq. (3.16) gives an expression for the behavior of ¢o(F)(n, Q2) in terms of a set of functions of n which are studied in sects. 4 through 7. In sect. 8 we summarize our

G. Sterman / Hadromccross,sectwns

316

results and give an explicit expression for the large-n behavior of ~o(~) i n terms of the exponential of functions which are computable order by order in perturbation theory. Our results generalize the exponentiation of ln2(n) and *r2 terms (eq. (1.11)), and the use of rescaled running couplings (eq. (1.12)). Finally, we summarize one-loop calculations, and show how our formalism allows us to predict, on the basis of only one-loop calculations, the correct coefficient of the next-to-leading (ln3n) term at two loops, as calculated in ref. [16]. The basic results for ~(F)(n, Q2) are given in eqs. (8.1), (8.2) and (8.9). Because of the length of the argument, we have summarized the forms and results of all one-loop calculations in an appendix.

2. z --, 1 factorization for the Drell-Yan cross section

In this section we shall see how to write the Drell-Yan cross section in a factorized form in which, following our comments in the introduction, the real and virtual contributions to the hard part are individually finite. We shall find that to do this it is necessary to generalize the factorized form eq. (1.2) somewhat. First, however, let us consider an alternate choice of the density of quarks in quarks defined, with N c colors, as *qq(X, p o / / p , ) = (2'/r X

25/2Nc)-1 f~-oe d Yoexp(-ixPoYo)

× Y'.(p,s, alq+(yo,O)7+q(O)lp, s,a).

(2.1)

Here s labels quark spin, and a quark color, while q is the quark field, l~qq represents the set of graphs shown in fig. 2.1. It is normalized to 3(1 - x) at zeroth order. We shall refer to ~k as the center of mass (c.m.) quark distribution. The quark line created by field q has a fixed energy Xpo; otherwise, its momentum components are integrated over. As defined, ~ is neither gauge invariant nor Lorentz invariant. We shall work specifically in the temporal gauge, A ° = 0, and in

P Fig. 2.1 Graphical contributions to the c m quark dtstnbution tk(x)

317

G. Sterman / Hadromccrosssectwns

the c.m. frame of the DY process. In this frame, the "observed" quark carries a fraction x of the incoming hadron's energy. This means, in particular, that the remaining partons shown in fig. 2.1 can carry no more than a fraction (1 - x ) of that energy. The usefulness of this definition may be seen by studying the partonic phase space of the complete DY process. Consider a specific contribution to process (1.1) with n hadrons, of momenta k,, i = 1,... n, in the final state. The phase space for this contribution differs from ordinary n + 2 particle phase space only in the presence of a delta function which fixes the mass of the DY pair. It is (2.2) Taking z = Q 2/s ~ 1, this becomes 8[s(1-z)+2(s)

1/2

0 0([1__ (,~_.,k,)+

Z]2)] .

(2.3)

Thus, in the limit z--, 1, phase space for the Drell-Yan cross section is defined completely in terms of the energies of the hadrons in the final state. A factorization of the form eq. (1.2), with q~= g, has the potential of including the allowed collinear and infrared regions, and no others. To get a better idea of the relation of ~b to the cross section, we need to identify the correct factorized form of the cross section, applicable in the z --* 1 limit. The arguments which lead to factorization are the same near the edge of phase space as elsewhere [7, 8]. The only difference is that soft gluons do not decouple kinematically from the rest of the graph, since as z---, 1 the energies of real soft gluons may not be negligible compared to ( 1 - z)v~-. Analogous considerations apply to the cross section where the transverse momentum of the Drell-Yan pair is measured [17], and at low x [18]. Also note that when z ~ 1, hard corrections can only appear in the form of virtual lines, since any lines in the final state must carry energy less than ( 1 - z)v~-. We may thus write, to the same level of rigor as in discussions of factorization [7, 8], the z ~ 1 factored form of the cross section as

doqva/dQ 2 = ooIHDy( Q ) [2f o l ( d x l / x l ) ( d x J x 2 ) . rl dw

X~qq(Xl' E1)l~qq(X2' E2) J0 "-I--U(w¢-;) =-w X S ( z - (1 - w)xxx2) + O ( [ 1 - z ] ° ) ,

(2.4a)

or equivalently, in eq. (1.2), to(¢)(z, Q 2 ) = o o l n D v ( a ) l Z u ( 1 - z - [ 1

-xll-

[1 - x 2 l ) .

(2.4b)

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v,/'~

v

Ftg 2 2 Graphical form of z --, 1 factonzation for the DY process,

Eq. (2.4) is illustrated in fig. 2.2. Notice that in eq. (2.4) w, 1 - x~ and I - x b are all less than or equal to z. Thus, to the indicated approximation, z- (1- w)xax b=z-

( 1 - w) + ( 1 - x~) + ( 1 - Xb)

= --[(1--Z)--W--(1--X,,)--(1--Xb)],

(2.5)

is the argument of the delta function in eq. (2.4a), which is therefore equivalent to the phase space delta function eq. (2.3). In contrast to the general form, eq. (1.2), there is no sum in (2.4a) over parton types. ~k is always the distribution of a quark in a quark of the same flavor. Any other choice involves emitting fermions into the final state, and, as discussed in the introduction, can only give non-singular terms in the z ~ 1 limit. Here and below, we shall therefore suppress the quark indices, and use the fact that qJqq= q ~ . In eq. (2.4), q~ and U are functions, with all color sums and averages taken internally. Ia o v ( a ) l 2 is a purely short-distance function. We shall refer to U as the "soft gluon function". It is defined as the sum of "eikonal" graphs shown in the middle of fig. 2.2 [13,14], integrated over all gluon momenta, subject only to the restriction that the total energy of the gluons be equal to wfs-. The eikonal lines are defined as follows. We define light-like vectors v ~ and u~ by v ~' = ~,+,

u ~' = 8~, .

(2.6)

G Sterman / Hadrontc cross secttons

319

The propagators of the v ~ eikonal lines are then given by

i[v. k + re]-1,

(2.7)

where k ~' is defined as the momentum flowing in the direction of the arrow. The eikonal-gluon vertex is simply

- gt av~ ,

(2.8)

with g the usual QCD coupling constant, and t~ a color matrix in the same representation as the quark. Corresponding rules hold for the u~ lines and vertices. The eikonal lines of fig. 2.2 move into the final state. The reasons for this choice of direction are explained in sect. 5 of ref. [8]. It is worthwhile to point out, however, that ref. [8] dealt with cross section with only gauge singlet incoming particles, while here we are discussing incoming quarks. It has been shown, however, that infrared singularities associated with incoming eikonal lines cancel to leading twist [19]. Assuming this result, we may take over the reasoning of ref. [8] to the quark-scattering case at hand. As it stands, eq. (2.4) does not identically factor under moments in the manner of eq. (1.6). This is because the c.m. distribution ~ is an explicit function of the hadronic energy E, = ½s~/2, rather than of the pair momentum Q2. This dependence is not present in the DIS structure functions, which are Lorentz invariant. We observe, however, that q, is not a singular function of E,, so that this extra x-dependence is also not singular. We now derive the consequences of this observation. Consider the moments of eq. (2.4), which is accurate up to corrections which are finite as z --->1. But since for any function f(z).

(2.9)

IfoldZz"-~f(z)l <~( 1 / n ) m a x l f ( z ) l ,

such correction terms are "power suppressed" in n, for large n. We shall therefore consistently drop non-singular contributions to the cross section. F r o m eq. (2.5), we have that w, 1 - x~ and 1 - Xb are all less than 1 - z. So, in eq. (2.4), we can set the variables E a and E b equal to their values when x~ = x b = w = z = 1, since neither H nor ~p is a singular function of the energies. Of course, we keep the dependence of ~b and U on the singular variables w and x. Then, we may rewrite (2.4) as d a / d Q 2-- oolHov(Q 2) 12f(dxl/xl)(dx2/x2)q~(Xl,

Q ) 4 , ( x 2 , Q)

x f(dw/w)U(wQ)8(z-(1-w)x~x2)+O(1-z)

°.

(2.10)

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G. Sterman / Hadromc cross secnons

In this form, the moments factor up to corrections that vanish as n --->oo, so that the noncovariance of the distributions gives negligible corrections in the large-moment limit. We thus find

(2.11) We emphasize that we have taken the singular parts of all the functions in eq. (2.11), and that corrections are O(1/n). It is not difficult to test this procedure at one loop. The one-loop correction to the Drell-Yan cross section, using dimensional regularization, has been calculated in [1]. It may be expressed as

( d o / d Q 2 ) (:)= (do/dQ2)re~a 0) + (do/dQ2)v,~t 0),

(2.12)

where (do/dQ2)re,a (:) is the contribution of the gluon emission graphs, and (do/dQE)wrt O) is the contribution of virtual graphs. If the Born cross section is normalized to 8(1 - z ) , then according to [1], :

(do/dQZ)~al°)----(aJTr)CF(47r/x/Q

r(1-

1

) F-0-~-27)~ Xz*

×[(l-z):-2~-2z(l-z)-'-2q, (do/dQ2)vn_t (1)= -8(1

-- z)(aJrr)CF(4~rl~E/Q

F(1 - e)

2) ~(l ~e )

× [ l i e 2 + 3 / 2 e + 4 - {~r 2] = - 6 ( 1 - z)(as/~r) CF (4~r/x2/Q2)*[l/e 2 + (1/e)(-32 - y)

_~Ir2+½YE_3Y+4]+O(e).

(2.13)

We now compare these terms to the one-loop real and virtual corrections in eq. (2.10). Normalizing the lowest order cross section to 3(1 - z ) , these quantities are given respectively by 2~real(1)+ Ureal(1) and 2+v~t(1)+ Uv,rt (1). The lowest order contributions, real and virtual, to ~ and to U are given in fig. 2.3. After a

321

(7. Sterman / Hadrontc cross secttons

(a)

(b)



u

v

V

~

(c)

(d)

Fig. 2.3 Graphs contributing to one-loop quark distributions (a, b), and to soft gluon ftmctaons (c, d).

straightforward calculation in the A ° = 0 gauge (see the appendix), we find that F(1 - e)

2~real (1) -k- Ureal (1) = -

×

(aJ~r)CF(4rtl.t2/Q 2) ~1 ---~e)

(1i ---2-~e - ~)~ (1 -

1 e

]

z) 1-2~ + 2z(1 - z) -1-z~ +

2 Re f f ~ * ) + U ~ t (1)= - 8 ( 1 - z )( a j a r )CF[ (4rrtt2/Q 2 )~{1/e -~r2+

±Y2--3Y + 2 2

Cv(a/Tr),

z + (1/e)(-32

2-½1n[/~2/Q2]] +O(e).

- y) } (2.14)

From eqs. (2.13) and (2.14) we can see that ( l / e ) singularities cancel individually between the gluon emission contribution to the cross section and the gluon emission contribution of the distributions. Let us define a one-loop non-singular UV function lady (x) by analogy to eq. (1.9) as

I2DV(z)(1) =

(da/dQ2)(1)

_ 2 Re +(1) _ U(1) -

= fnDyl z(x)8(1 -- z) + ~ D Y : e ~ ( Z )

~(I)

.

(2.a5)

This is the hard part of the generalization of eq. (2.4) to include possible real gluon

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G. Sterman / Hadromc cross sectwns

contributions. We find, from eqs. (2.13) and (2.14) that IHt)y] 2(1)= _ (as//2~r)CF(4 + ln[/~Z/a2]),

( 1+z2 ]

~2ov,~O)= (as/rr)C F 1 + -~-S-~-_ z In z .

(2.16)

As promised, there are no (1/[1 - z ] + ) distributions, and the moments of to(*) are correspondingly well behaved at large n. Also, I2ov,~e~(1) has no x ---, 1 singularity, which is consistent with (2.4). The generalization of this result to all orders is straightforward. By eq. (2.9), 5(*)(n) can only have contributions which are logarithmic or constant as n ~ oo if to(*)(z) is unbounded as z ~ 1. This can happen only if ¢o(g)(z) is defined as a distribution, that is, through terms of the form [ l n ' ( 1 - z ) / ( 1 - z ) ] + . Such a contribution is infrared finite in the sense of a distribution, but it cannot be considered bounded since [ l n ~ ( 1 - z ) / ( 1 - z ) ] + is not defined as a function for z = 1. But such distribution terms come about only from the cancellation between the infrared and collinear singularities in different graphs associated with real and virtual gluons [6, 7, 8]. Thus, if to is defined so that its contribution from every graph is individually finite, there can be no cancellations of singular terms between graphs to produce unbounded distributions in the sum. This is precisely achieved by matching the phase space of the parton density ~ and the function U with the phase space of the full cross section in the z ~ 1 limit. Unlike the DIS density, eq. (1.5), qJ and U have no singular regions of phase space which are not already present in the Drell-Yan cross section as z ~ 1, and which therefore must cancel between virtual and real gluons. Then the singularities of the factorized form eq. (2.10) are in one-to-one correspondence with the singularities of graphical contributions to the Drell-Yan cross section as z ~ 1. Distribution terms of the form [ l n ' ( 1 - z ) / ( 1 - z)]÷ are all absorbed into the functions ~ and U, and the large corrections found in to(F) will be absent in to(~) to all orders in perturbation theory. We have thus achieved the first of our two goals, to find a form of factorization with a hard part to which has no In (n) terms in its moments. We must now turn to the second goal, to compute the large-n behavior of cross sections and parton densities. T o this end, it will be useful first to study the DIS cross section directly. This is the subject of the next section. 3. x --* 1 factorization for the Drell-Yan cross section

We now discuss the deeply inelastic scattering cross section from the same point of view as the DreU-Yan cross section in the preceding section. We shall find appropriate partonic functions (not the same ones as for Drell-Yan) which absorb the singular contributions of real and virtual gluons separately, so that no distribution terms of the form 1/[1 - x]+ are left in the hard part.

G. Sterman / Hadromc cross sectwns

323

The DIS cross section is

h(p) + : ( k ) -o :(/,,) + X(p + / , - / ' 9 ,

(3.1)

with g a lepton and h a hadron. As in the DY case, consider a specific contribution to (3.1) with n hadrons in the final state. The phase space for this process differs from ordinary n + 1 particle phase space only by the presence of a delta function which fixes the momentum transfer q 2 = _ Q 2 = ( k - k ' ) 2 from the lepton to the hadronic system. In a frame where p is lightlike and in the + z direction, this delta function is

,[Q,÷(?where the k, # are the momenta of the final state hadrons. This is a different condition than that given by eq. (2.3) for the DY cross section, and so factorization in the DIS case will require a different choice of parton densities to factorize x ~ 1 singularities, and hence high-moment behavior. To understand how best to factorize the DIS cross section in the limit x ---, 1, we need to know those momentum configurations which give leading contributions in that limit [8,12]. These are shown in fig. 3.1. We work in a frame where p~ is in the + z direction, and in which qT = 0. The subdiagrams L contain virtual lines whose m o m e n t a are nearly parallel to p", and which have plus components of order p÷; in S, the lines are all soft, i.e., have all momentum components much smaller than p ÷; in H all lines are hard, i.e., have all momentum components of order p÷; finally, in J all lines are nearly collinear to the vector ~' = (0, q - , 0 r ) . In the limit x ~ 1, M 2 = ( p + q)2 vanishes as 1 - x, and the final state consists

P Fig. 3 1. Leading reoons for DIS.

G Sterman / Hadromccrosssecnons

324

only of a single jet of particles moving in the direction opposite to p~', and of soft lines. To factor double logarithms associated with the incoming and outgoing directions, we can apply the method of soft subtractions [8,13]. Neglecting the leptonic part, we shall argue that the DI cross section may be written as 2

1

F(x, Q2) = iHBI(Q) I fx (dy/y)ep(y, p+, p. n) × foY-X(dw/[1 - w])V(wp +, wp. n)J[l 2, 1. n] + O(1 - x) ° . (3.3) This factorized form is represented graphically in fig. 3.2. As in the DY case, there is no flavor mixing for terms which are singular in the x ~ 1 limit. The photon momentum q~' and the momentum l~ flowing through the jet J are given by

q~ = ( - x p +, Q2/2xp+,O), t'~ = ([y - x - w]p +, Q2/2xp+,O).

(3.4)

The terms neglected in (3.3) remain finite as x ~ 1; they are associated with contributions which are not singular in the soft or collinear regions, n * is the gauge fixing vector. We will choose the axial gauge n ~ = 8~3

(3.5)

for purposes of explicit calculation in DIS. The function V is analogous to U in (2.4); it is defined as the same sum of eikonal graphs as in fig. 2.2, integrated over

xp

Fig 3 2. Grapbacalform of x ~ 1 factonzauonfor DIS

G. Sterman / Hadromc cross secttons

325

all gluon momenta, but now subject to the restriction that the total plus momentum of the gluons equals wp ÷. The two other functions in (3.3) are defined as follows. We shall refer to ¢ ( y ) as a "light-cone (LC)" quark distribution, ~ ( y , p+) = (2¢r ×

4Nc)-lf~_oodx - exp(-iyp+x -)

× E ( p , s , alq+(O,x-,OT)r+q(O)lp, s,a),

(3.6)

$,a

where q is the quark field. IP, s, a) is a quark state with momentum, spin and color p, s and a, respectively. The distribution ¢ is normalized to 8 ( 1 - x) at zeroth order. Again, we shall not distinguish between quark and antiquark distributions, since for purely partonic calculations they are equal, q~ may be thought of as the distribution of quarks of fixed plus momentum within a quark [20]. The other function will be called the "jet" distribution J, j(12, l. n) =

(1/2~r)4(Q/23/2~r) ×Oiscf d4xexp[-tl.x] • (OIT(q+(x~)q(O))lO).

(3.7)

J is just the two-point function for a quark in the axial gauge. It is normalized at zeroth order to 8(1 - x) by the factor (Q/23/2 rr). We now discuss the factorization properties of eq. (3.3). Compared to the factorized form of the DY process, J is a new function, and the soft gluon function V and the LC quark distribution q~ are like the soft gluon function U and the c.m. distribution ~p respectively, but defined with respect to plus momentum rather than energy. This is because in DIS the phase space gives no restriction comparable to eq. (2.3); the energies of final-state particles can be quite large, even with small plus momentum, since their minus momenta can be as large as q-. Alternately, their transverse momenta can be much larger in DIS than in DY [3]. Energetic particles whose momenta are actually coUinear to fi'~ are included in J, but V and ~ must include particles whose momenta cover the entire "soft" region, where all components of momenta are small and of comparable size. By contrast, in the DY case, only those parts of the soft region consistent with eq. (2.3) were included in U and ft. But there is no sharp division between the soft region and the region where momenta are collinear to ,b'~. This means that V and must include final state particles which are actually collinear to/3'~'. Contributions from such particles, however, will not be leading, because ~ has no collinear enhancements except in the p~' direction, and V has no coUinear enhancements at all [13]. The choice of distributions given above, then, have subtractions for all real and virtual leading regions, as required by our discussion in the sect. 1. Corrections to eq. (3.3) should then be finite, and moments of these corrections bounded.

326

G. Sterman / Hadromc cross secnons

As for Drell-Yan, we can verify that the factorization, eq. (3.3) gives a form in which real and virtual contributions are separately finite at one-loop. Analogously to eqs. (2.15), we define 12DI(1)= F(1) _ (¢(1) + j o ) + U(1)), (3.8) and express this quantity in terms of its real and virtual parts. From [1] we have

I'(1 -e)

F(~)~a = ( aJ2~r )Cv(4~r~Z/Q2 )'Y(1 - 2e) [

(

X 3x+x~(1-x)

FO)'~rt=

11+x2 ~ 1-x

-~

1 --+3-x-~e 21-x

r(1- ~) [

r(1 -

(aJ2vr)CF(4vq~2/Q2)

71}]

3

2 3

2e)

]

8- ~

E

, (3.9)

,(1 - x).

(3.10) The lowest order contributions to ¢ and V are from the same graphs as for ~ and U, respectively, as shown in fig. 2.3. Only the phase space (and gauge) is different. The one-loop graphs which contribute to J are shown in fig. 3.3. The individual values of these functions at one loop are given in the appendix; their total contribution to eq. (3.8) at one-loop is (¢(1) q._j(1) -I- v(X))rea I

~ r ( x - ~)

= ( a s / 2 1 r ) C F (4Vrla2/Q2) F-~-2-ee) ( 1 - x ) - ~ x

l,l+x

/

~

e 1-x

2 1-x

- (1 - x ) h ~ O , V &

2

)] ,

(,0) + jO) + VO))v m = - (aJZ~r)C F 3(1 - x) X 4~r/~Z/Q2)

~5+-(3-2y)

+4+

E

-3V

--

1 2 ln(t~Z/Q2)]

g'a"

(b)

--

L

Fig 3 3. One-loop graphs contributing to the jet dastributaon

(3.11)

G Sterman / Hadromccrosssectwns

327

In the first of eqs. (3.11) we have restricted ourselves for convenience to the singular part of J in the variable (1 - x). Combining eqs. (3.11) and (3.10), and expanding to order e °, we find, as expected, that the real and virtual contributions to the hard part are individually finite,

~Di(1)real

=

(O/s/2q?)C F (1

~'~DI(1)vtrt

=

[HDI[ 2(a) 8(1 - x)

--

x)ln(#2/Q 2) + 2x + 3

l+x2 - 1- l-n xx

] ,

= --(aJ2~r)CF[4 + ln(#2/Q2)] ~(1 - x ) .

(3.12)

Note that, consistent with eq. (3.3), the real contribution to I2D(1) is of order (1 - x ) °.

We are now in a position to discuss moments of the DIS cross section (3.3) from the same point of view as we did the DY cross section, eq. (2.4). Because of the explicit energy dependence in (3.3), moments with respect to x at fixed Q2 do not factorize exactly. On the other hand, if we neglect contributions whose moments vanish as n ~ oo, i.e., finite corrections as x ~ 1, we may replace all p . n dependence in (3.3) by dependence on Q. We then find, analogously to (2.10),

r(x, e 2) = 1i-Io,(o_2) 12£(dy/y ×J[QE(y-x-w)/2x,

0.2 fo'-X(dw/[1 - w ] V(we Q] + O ( 1 - x )

°.

(3.13)

The moments of (3.13) are

F( n, Q Z) = £1 dx x"- 1F( x, Q2 ) =[Hm(Q)12q,(n,QE).V(n).J(n,QE)+o(1/n),

(3.14)

where

v(,,) = fo~(dw/w)(1 - w)"-lU(w), ok(n, Q2) = fot d yy,,-l~(y, Q2), J( n, Q2) = fot d~ ~"- 1j[(1 - ~) Q2/2~, Q].

(3.15)

In deriving (3.14) we again neglect terms that are non-singular as y ~ 1, w ~ 0, and which therefore contribute to corrections of order 1/n.

328

G Sterrnan / Hadromc cross secttons

Finally, using eqs. (1.6), (2.11) and (3.14), and specializing to a single quark flavor, we find the desired expression for t~(F)(n) (compare eq. (1.13)), atF)q~(n)

=

[IHDYl211HDx[4] [ ~ ( n ) lq)( n ) ] 2 ×[U(n)/V2(n)][1/J(n)]2+O(1/n),

(3.16)

where we have suppressed the Q dependence. Thus, the large-n dependence of the moments of the hard part ~(F) is determined by the n-dependence of the functions ¢, 6, U, V, and J. In the following sections we show how their n-dependence may be determined.

4. Behavior of the light-cone distribution In this section, we begin the derivation of the large-n behavior of the various functions entering into eq. (3.16). Here, we shall derive and solve equations which determine the singular behavior of the "light-cone" distribution ¢(x). 4.1 FACTORIZATION OF THE LIGHT CONE DISTRIBUTION

As we saw in sect. 2, the singular part of of the light-cone distribution ~(x) may be thought of as a function of Q and (1 - x). It may be represented schematically as in fig. 4.1, where the diagrams R represent the residues of the on-sheU two-point functions, and I represents all graphs which are one-particle irreducible in the incoming lines p. We begin by showing that the singular (1 - x)Q dependence factorizes from the non-singular Q dependence in q~(x). The reason for this is that the leading (1 - x)Q dependence is associated entirely with the cut lines in fig. 4.1, and that in the limit x - o 1 all these lines must be soft compared to the incoming momentum p. In particular, energetic lines with small plus momentum (lines the minus direction) do

Fig. 4.1 Graphical contributions to the LC quark dtstributton.

G Sterman / Hadromc cross secttons

329

Fig. 4.2 Dlvaslon of hnes in LC dastnbutaon as x ~ 1

not contribute to the leading x--, 1 behavior. The on-shell internal lines of any graph contributing to ~(x) may thus be split into two sets: those which are soft, with momentum components of order ( 1 - x)Q or less, and those which have momenta of order Q, which are therefore nearly parallel to the incoming momentum p. This division is illustrated in fig. 4.2, where the subgraphs S are made up of soft lines, while the subgraphs j are made up of jet-like lines. Clearly, there are different ways to divide any graph G which contributes to q~(x) into jet and soft subdiagrams j and S. Each such division corresponds to a region R& in momentum space, in which the lines of S are soft, while those of j are jet-like. All the singular contributions to 4,(x) must come from such regions in momentum space, since a singular contribution requires some set of internal lines to be on-shell. The x-dependence of ~(x) can be found by factorizing the soft from jet lines in each graph G. This is done by making a series of subtractions according to the tulip-garden prescription of Collins and Soper [13]. This prescription is a variant of the BPHZ subtraction procedure for ultraviolet divergences [21], but applied to infrared behavior. A soft subtraction, written tsG is an approximation to the graph G constructed to be a good approximation to G in the region Rsj; where the subgraph S consists of soft lines. Suppose we write G(x)=

f l-I d4k, J~a" ""(k,")S,,,1, ,,,,,(k,~),

(4.1)

where the set {k, } consists of all the soft lines connecting S with j in fig. 4.2. Assume that the incoming particle p is moving in the plus direction with light-like momentum. Then the soft subtraction associated with this particular division into soft and jet subdiagrams is given by (see, for example, ref. [8], sect. 4),

'sG(x) = f ]-I dak, J +" +(k,~')S

_(k~, ),

(4.2)

330

G Sterman /

H a d r o m c cross sectwns

I

I

I

I

Fig. 4 3 Example of a garden wtth two tuhps, lnchcated by the dashed and unbroken lines.

where the vectors/~," are defined by (4.3)

=

That is, in the soft subtraction the gluons of the soft subdiagram couple only to the current of the jet (in this case, to its "plus" polarization), while the jet is sensitive only to the opposite-moving components of the momenta of the soft lines (in this case, to their minus components). In the terminology of [13], any choice of soft subdiagram S is known as a "tulip". A "garden" K is any set (S } of nested (non-overlapping and not disjoint) sets of tulips S. An example is shown in fig. 4.3. This construction allows one to show [13] that for any graph G,

~.,

1-I (-ts)G

= O(~/Q),

(4.4)

gardens S E K K

where • is the size of the softest momentum within G. (To be precise, the sum in (4.4) is over "inequivalent" gardens. See ref. [13].) Picking out the trivial garden with no tulips from (4.4), we find

G=-

~.,

l-I (-ts)G+h(Q),

(4.5)

non-tnvml S ~ K gardens K

where h (Q) is a function which gets leading contributions only when all the internal gluon lines of G have momenta of order Q. If G is of the form of fig. 4.1, with one or more cut lines, this is never possible. Then, from (4.5) h(Q) is suppressed by the order of (1 - x), since this is the ratio of the momentum components of the cut lines to Q. Neglecting h(Q), we organize the sum in (4.5) according to

Y'.

G=

to 1-I (-ts)G

non-trivial gardens K(G)

= Eto%,(k~ ~) o

S~ K $4, o

E all

gardens L(Jo)

H (-ts)fVI d4k,

S~L

(4.6)

G Sterman / Hadromccrosssecttons

331

where o is the smallest tulip in each garden K, jo is the diagram left over in G when o is removed, and where we have used (4.5) for G, with S = o. Because the tulips are nested within a garden, o is uniquely defined. Note that although the sum is over all gardens of Jo, eq. (4.6) is not suppressed; because Ja has no cut lines, the momenta in Jo may all be of order Q. The next step is to sum over all graphs of the same order in (4.6), and to apply the relevant graphical Ward identities [8,13]. It can then be shown that order-by-order the soft gluons decouple from the jet subdiagrams to give ~ ( x ) = , , ¢ ( a ) x ( [ 1 - x ]Q/21/21~, n ~)/(1 - x ) .

(4.7)

The factor (1 - x ) -1 takes into account the overall scaling. Then X is a function of x only through the dimensionless variable [1 - x]Q/21/2 F. The factor 21/a is purely a matter of convenience. The functions J ( Q ) and X([1 - x]Q/21/2#, n~)/(1 - x) are specified by the graphs shown in fig. 4.4. at(Q) is the residue of the fully subtracted quark self-energy, while X / ( 1 - x) is given by the set of diagrams identical to fig. 4.1, except that the external quark line has been replaced by an "eikonal" line in the same direction as the quark. The factor r E is the residue of the eikonal two-point function and equals either of the eikonal self-energies shown in fig. 4.4. It should be noted that the factorization of eq. (4.7) and fig. 4.4 reflects a different organization of the subtractions than in refs. [8,13] where the largest, rather than the smallest, tulip was isolated in the sum. The proof of factorization via Ward identities is then slightly different, but is easily carried out. Also, we have included subtractions on the self-energy subdiagrams of fig. 4.1, including the residue function R of the quark. This results in the factor rE 1, and m the fact that there is only a single fully subtracted self-energy in fig. 4.4. To see this we note that if j is a quark self-energy, then by (4.5)

j=

~_, non- tnvaal gardens K(G)

t~ I I ( - t s ) j + a C ( Q ) .

(4.8)

S~K

S~o

k

(

-! x%

Fig 4 4 Factorizedform of the LC distnbutaon

332

G. Sterman / Hadromc cross secttons

Now, reorganizing the sum as in (4.6), and using Ward identities, we find the factorization j = (r E - 1 ) J ( Q )

+a¢(Q) = rE,,C(Q ) .

(4.9)

Applying this reasoning to the extra factor R in fig. 4.1 gives the result of fig. 4.4. Starting with eq. (4.7) we can derive an equation which determines the x ~ 1 behavior of the light-cone distribution 4.2. EQUATION FOR THE LIGHT-CONE DISTRIBUTION

To derive an equation for ~(x, Q), we isolate the loop k shown in fig. 4.4. We integrate over all the internal loop momenta of S and over the transverse components of k. X([1 - x]Q) may then be written in the form X([1

- x 1Q/21/2l*, n ~')/(1 =(2~r)-

- x)

2rEJ.d~+dk "r(2k--~k 'k-n-----'= k+n ) 6

~-$--[1-x]Q

),

(4.10)

where p + = Q/21/2. The dimensionless function ~" is the integral over the subdiagram of X which is not reducible by cutting a single eikonal line, and r E takes into account the single leftover self-energy of the external eikonal line. /~ is the renormalization mass. The arguments of ~" in (4.10) are a complete set of independent dimensionless variables which can be formed from the three vectors v ¢, n~ and (k +, k-, 0 r ) subject to the restrictions of (i) boost invariance along the z-axis, 0i) invariance under scalings of n ~ and of v ~ and (iii) v~ in the plus direction. (ii) follows from the fact the propagator in an axial gauge is homogeneous of order zero in n", while the combination of eikonal propagator and vertex is similarly homogeneous of order zero in v r. Defining ~ = 2k+k - and q+ = (1 - x)p +, eq. (4.10) may be rewritten as

x(q+/l~, n")/q +

(2 r)-Z(Zq+)-lrE-

( ~

2(q+)zn-)

,n+

(4.11)

Taking the derivative with respect to In q +, and using the chain rule, we find

0 [x(q+/Iz, O ha q+ [

=

x(q+/l~,n ~) O [x(q+/l~,n¢) ] q+ + 0 ln(n-/n+) 1/2 q+ x(q+/t~, nU)

0 In rl~

q+

o In(n-/.+) x/2"

(4.12)

G. Sterman / Hadromccrosssecttons

333

Fig 4.5. Result of derivative with respect to gauge parameters

The derivatives on the r.h.s, of (4.12) may be carried out using the method of Collins and Soper [13]. The derivative with respect to In(n-/n+) x/2 acts in turn on the gauge terms of each gluon propagator of a graph. In ref. [13] we learn that the resulting terms may be organized as in fig. 4.5, where the square vertex is given by n2

- g t a (o" n ) ( k . n) O~'

(4.13)

with k ~' the momentum flowing out of the vertex along the gluon. The important thing about the vertex (4.13) is that it has no enhancement when the momentum k ~ becomes parallel to v ~. Thus the gluon k ~ cannot become part of the jet subdiagram. A tulip-garden decomposition then gives the result of fig. 4.6, which may be written as [13]

3 [x(q+/#,n~')] OIn(n-/n+) 1/2 [ q+ = fq*d/+ Kx(l +) ×x(q +-l+)/(q +-l+).

(4.14)

~o

K x is defined by the set of graphs at the top of fig. 4.6. It has dimensions of inverse mass. It is easy to verify that the ultraviolet remainder term Gx shown in fig. 4.6 vanishes. This is because the soft approximation on an eikonal line is exact, as is illustrated in fig. 4.7. The application of the above method to the residue function r E alone is shown in fig. 4.8. It generates exactly the negative of the virtual contributions to the function K x ( P +) in eq. (4.14), ~r E

Oln(n-/n +)1/2

-- Kx(~t)rE"

(4.15)

334

G. Sterman / Hadromc cross sections

k

-"

X

~

X

/

Fig. 4.6. Result of soft subtracuons apphed to fig 4.5.

--O |,." Fig. 4.7. Idenmy wtuch shows that the function Gj is zero for X.

Fig. 4.8 Applicatton of soft subtractions to the residue functton.

335

G. Sterman / Hadromc cross secttons

Combining (4.14) and (4.15) with (4.12), we find

O lnq +

[x(q+/l~, n~') ] =

x(q+/l~, n~')

q+

q+

+ [q+dl + Kx(reaa)(l+) × x(q +- l+)/(q +- l+).

(4.16)

~0

As we shall see, Kx~r*~a)(l+)is not generally finite in four dimensions, so we shall use dimensional continuation to define it. Using q + = ( 1 - x)p ÷, redefining Kx(re~a)(l+) - K , ( l +) and using eq. (4.7), we find for the full function q~(x) the equation

9

]

01n(1-x)

+ 1 q>(x) = f x

ldyK~,([x- y ] p + ) ×

~(y)

.

(4.17)

From the nature of its derivation, the corrections to (4.17) are of order (1 - x ) °, compared to its leading behavior of (1 - x) -x. It is this equation which we solve to find the x ~ 1 behavior of ~(x).

43 SOLUTIONAND MOMENTSOF SINGULARBEHAVIOR The solution to eq. (4.17) is

£ 1 --X

X

' fi

("o,

n=0 ~ " ~ 1

(:0"

do~,K~(to,)X8

~

1

(1-x)p

+ "

(4.18)

Here R(Q/tQ is the residue of the quark two-point function. This is the correct constant of integration, since with e = 2 - ~n < 0 (infrared regularization) q~(x = 1) receives contributions only from purely virtual graphs. Both R and K~ are infrared divergent in four dimensions, while ~ ( y ) is infrared finite. The arguments for the infrared finiteness of ~ ( y ) are given in [20]. To verify that (4.18) is a solution to (4.17), we first observe that

din(i--x) + E°~'-~, +1 d l-x-

E o~, --~--0.

(4.19)

Eq. (4.19) allows us to change the derivative with respect to ln(1 - x) into a sum of

336

G. Sterman / Hadronic cross sections

derivatives with respect to the to,. Integration by parts then gives

{f0°°dvK,(,/Q,Q//~)}×

1 "-l'°°dto' n=l

x

(n

-

HJo--to,

1)! ,_~ o

dto;K,(toVQ, Q/,) x8 l - x - ( , , / p ÷) ~,-lto,. (4.20) p+

Changing variables by 1 -y

= 1 - x - (~/f),

(4.21)

we easily derive eq. (4.17). The high-moment behavior of the solution (4.18) is best studied by another change of variables. We define a new set of variables, {z, }, i = 1, 2 .... n by the formulas z--1

~, = 1-1 z, × (1 - z,).

to, = ~,p+,

(4.22)

j=l

If the to, are thought of as momenta being carded by ladders in fig. 4.1, then the z:'s are scaling variables which describe the proportion of momentum left after each ladder exchange. An example with n = 2 is shown in fig. 4.9. The jacobian of the change of variables is just H~',

(1 - z,).

(4.23)

and

T-7

.-o..,-1

l-z,

×{+i P ~_~dy, K~([1-y, lP+) ×8 (i ~ ~ ~--~)). 1-1-I7_:,

ZlZ2P]t

(u2= Zl(I- z2)p

I

~I = (I-Zl)P

,t Fig. 4.9. Ladder with t w o rungs illustrating the relationshp of variables in eq. (4.22).

(4.24)

G.Sterman/ Hadromccrosssectwns

337

Because of the delta function, we find n

I-I z,= x,

(4.25)

so that every z, > x. In the limit x --, 1, then ~,= (1 - z,) + O([1 - x]2).

(4.26)

Eqs. (4.25) and (4.26) make the moments of (4.18) particularly easy to evaluate in the x ---, 1 limit. We find a simple exponentiation in the form

fol dxx"-l•( x) = R( Q/# )

×exp[/' dz [J0 1 - z

z._X[ldy, p+K,([l_Y, lp+)]+O(1/n), J:

(4.27)

where the corrections come from terms which are smooth as x ~ 1. We emphasize that since K , involves only real graphs, it cannot be infrared finite in four dimensions. These divergences must, and do, cancel in the final result, given in sect. 8.

4 4. ONE-LOOP CONTRIBUTION The only one-loop graph which contributes to K , is shown in fig. 4.10. Evaluated using dimensional regularization (see appendix), it gives K , ([ 1 - x ] QI21/2) = (21/2/Q)(2 a s l y ) C v ( 4 ~ # 2 / 0 2 )~

r(2+e) (4.28) (1 - x ) ~+2"'

where we have used the fact that 2po = Q, with Po the energy of the incoming quark.

+C.C. v

Fig 4.10 One-loop contribution to the function K,(y)

338

G. Sterman / Hadromccrosssecttons 5. Center of mass distribution

Most of the discussion for the center of mass distribution ~(x) parallels that for the light-cone distribution. By exactly the same factorization reasoning as led to eq. (4.7), we have for the energy distribution q~(x) = J ( Q ) ~ ( [ 1 - x]Q/2/~, n~')/(1

-

x),

(5.1)

where J ( Q ) , which is made up of subtracted virtual graphs, is the same as in (4.7). There is, however, a difference when we derive the equation satisfied by ~k(x). In place of X([1 - x]Q/21/2~) we now have ~([1 - x]Q/2/~, n~), which contains all the information about cut gluons. It may be expressed as ,~([1 -

-

xlQ/2~, n")/(1

- x)

dk'dk (2k k (2~)-2r~f 2k+k - "r

#2

)

,k-n + 8

21/2po

)

[ 1 - x I , (5.2)

where 2p0 = Q. The only difference between X and ~ is the phase space delta function. As before, we derive an equation for the x-dependence of ~, and the form of this equation will reflect the nature of the phase space integral in (5.2). As with X, we derive the equation by relating the derivative with respect to x to a derivative with respect to the gauge parameters. This turns out to be slightly more complicated in this case, since, after using the phase space delta function, we have, instead of (4.11~.

i(w/~, ,.)/w = (2~r)-2(2)-lrgf(dTl/Tl[ge'2-~]l/2)r(

~,ea[n-/n

+

]),

(5.3)

where

.=~

w + ( w 2 - n)1/2 w-

(5.4)

( w 2 - n) x :

and where

w - - (1 - X)po.

(5.5)

Here the W, or alternately x, dependence appears in the factor [W 2 17]1/2, as well as in the argument of T. The derivative with respect to ln(W) cannot be immediately changed into a derivative with respect to l n ( n - / n +), as in eq. (4.12). To deal with this complication, it is useful to decompose ~([1- x]Q/2/~) into two terms. We perform the decomposition in such a way that both terms can be treated -

339

G. Sterman / Hadromc cross secnons

in the same way as X ( [ 1 - x]Q/2VElz), at least after yet another factorization procedure has been applied to one of them. Another way of writing (5.3) is

I~(W/#, n~')/W = o~x(W/Iz,n~')/W+ ~2(W/Iz, n~')/W,

(5.6)

where the functions to1 and o~2 are defined by

o~l(W/#,n~)/W=(2~r)-2(2W)-lrEf(d,/~),(-~,e"[n-/n+]),

(5.7)

~2( W/l~, nl')/W = (2~)-:(2W)-IrEf(d,/n[wE-nll/:)(W

- [W:- nl 1/:)

0.

(,.8)

~o1 is of the general same form as eq. (4.11); we need to study o~2 a little more. It can be rewritten in the form

~:(W/#, n~)W

X~

(-+21/2p0

[l--x]

) .

(5.9)

The factor k - is the sum of the minus momenta of all the cut lines of any graph, k-=Eq;

.

(5.10)

a

We now imagine organizing each graph into "ladder rung" subdiagrams ~, which are two-particle irreducible in the vertical channel, as in fig. 5.1. Each cut line qa will appear in one of these subdiagrams h,. We write 6)2 as

.,2(w/#, n~)/w = ~

(1/W)fd41:(2~r)-4u,,_,(l:)fd41(27r)-'~kzh,(l,,l;),

(5.11)

340

G. Sterman

/

Hadromc cross sectwns

=X

h(1)

IIii

X~ u(n-O

Fig. 5 1 Generalized ladder graphs in the function qJ(y).

where

l,-- E k,, y=l

1,--

k,

(5.12)

j~t+l

In eq. (5.11), h,(l,, I;) includes the ladders up to X,, and from h, + 1 to h , . Schematically, l

h,(t,,l;) = FI x,,

u,_,(17)

includes those

n

u._,(t;) = I-I x,,

j=l

(5.13)

j~t+l

where the eikonal lines between ladders are associated with the rung below. Ladder graphs of this type were studied in the original discussions of factorization [6]. It was shown that (i) no infrared or coUinear divergences can come from any inner ladder unless all outer ladders are divergent and (ii) all divergences are logarithmic in physical gauges, so that a single extra numerator factor of momentum is enough to spoil both infrared and coUinear divergences within any ladder. This means that, because of the extra factor q~, h,(l, l') contains neither collinear nor infrared internal logarithms (although its overall scaling remains the same). We can now factor the ladders contained in u . . . . which contain divergences, and therefore logarithms of (1 - x), from the ladders in h,, which do not. This factorization is carried out by defining a new subtraction operator, t-m,which acts on the subgraph L m = 1"-Ira_iX,, by the relation

imL.(tm, t;~) = Lm(t., lm), ~'

(5.14)

fm~=21/2(lm)oS~+.

(5.15)

where

G. Sterman / Hadrontc cross secttons

341

That is, 1~~' is the momentum which has the same energy as l,~, but is always lightlike and in the plus direction. With this definition, the quantity

(1 -

i,)h,(l,, l[)

(5.16)

is suppressed whenever lff is either soft compared to w or is in the plus direction. This is enough to eliminate the possibility of soft or collinear divergences in the next rung out, )~,+1- Let ~2(n) be some n-ladder contribution to w2- Then, from (5.14) we have

in,4 ") = , 4 ~) .

(5.17)

We may write %(')

= ~ %('"),

(5.18)

1=1

where ~2(''') is the contribution to ~2(~) from terms in eq. (5.10) where qa is in the ith ladder. Then, from (5.17),

f i (1- [k)w2'"')(W)/W=O,

(5.19a)

t~l k~t or

o~2(")(W)/W= ~_, ~,[kki--[1(1--i,)o~2("")(W)/W.

(5.19b)

In the last form, we have isolated the outermost subtraction in each term. This allows us to write ¢o2t') in terms of the subgraphs inside and outside the outermost subtraction. According to eqs. (5.14) and (5.15), these subgraphs are linked only by an energy integral, so we have

.,2(')(wl/w=Zfd k,o[~°-~(~,o)/t~,o]×U~(w-tk,o), II

I I

!

!

(5.20)

where ~,_~( l'k,o)/l ' k,o is an n - k ladder contribution to the complete function ~([1 - x ] Q / 2 # , n ") of eq. (4.1). n

~.-AtLo)/l~,o=fd3z'(2~) -'

FI x,,

(5.21)

j~k+l

Hk(W--

[1/,]o) contains no logarithms by construction. k

k-1

k

Hk(P~,0) = t-k E FI (1 - i,) FI X~ho. a~l j=a

b=a+l

(5.22)

342

G. Sterman / Hadromc cross sections

where we have suppressed momentum integrals. Because Hk is evaluated at fixed total energy, it has dimensions of an inverse mass. Summing eq. (5.22) over all graphs, we find finally

to2(W)/W= (1/W) foWdPo[~(lo)/1o] X H(W- lo),

(5.23)

where H(w- lo) is the sum over all the Hk'S of eq. (5.22) from all graphs which contribute to o~2. Substituting (5.23) back into (5.6), we find the recursive formula

I~(W)/W= tol(W)/W+ foWdlo[~(lo)/lo] X H(W- lo).

(5.24)

From this form we will be able to find an equation satisfied by ~(W)/W. Analogously to eq. (4.12), we begin by taking the derivative with respect to In w,

(8/8 lnW)t;(W)/W= (8/8 lnW)[o~l(W)/W ]

+ foWdlo[~(to)/to] x

(0/0 h i W ) H ( W - to).

(5.25)

Next, we write out the first term on the r.h.s., using (5.7), to exhibit its W-dependence.

(8/0 InW)[,~x(W)/W ] = -~,,(W)/W x (2rr)-zO/2w)f(dn/n)

( aa/a In W)l.(a/aa)

+ n+l)r -- - ~ ( w ) / w

+ [ o/o In( , - / n +)1/211~(w)/wl

-[~(W)/W][OInr~/Oln(n-/n+)~/2], where we have used (5.3). The analysis of the gauge derivative of just as for x(q+)/q ÷,and we find (compare eq. [4.141)

(5.26)

~(W)/W proceeds

0 I n ( n - / n + ) 1/2

= [Wdt o r~(w- t0)x ~(to)/to, .'13

(5.27)

343

G. Sterman / Hadromc crass sectwns

while 0"trE/Oq 111( n

+/n -):/2

(5.28)

= gt/(vart.)rE -----Kx<~.~)rE.

The function K~(W-lo) is given by exactly the same set of graphs in fig. 4.6 as Kx(q+). The only difference is that the phase space is now defined at fixed energy rather than plus momentum. Setting

(5.29)

w = (1 - X)po.

and combining (5.26)-(5.28) with (5.24), we find the equation which determines the x dependence of ~(W/I.t, n~'), and hence (by [5.1]), ~b(x),

aha(1-x) +1 ~(x)=f~

dyK,~([x-y]po)×~(y ).

(5.30)

where now

K,~(flQ) = K{'e~a)(flQ) + [ ( 0 / 0 InQ)H(flQ) + H(flQ)] .

(5.31)

The solution to (5.30) is obviously of the same form as the solution to (4.17). It is

~ ( Y ) --

1 -~

x

, . 0 ,,. ,=1

d,o:r~(,o:)x8

,o,

1

(a-x)p°]

'

where again corrections are of order ( 1 - x) °, and R is the residue of the quark two-point function. The moments are

fol dXxn-lq,(x) = R (Q/l~) x exp

I:o

1 1d-z z z "-~

z'

dypoK,([1-ylpo)

] +O(1,/n).

(5.33)

As with K,, K~ is infrared sensitive. As we shall see, this dependence cancels in eq. (3.16). At one loop, referring to eq. (5.31), K~(re~)(flQ/#) is given by fig. 4.10, but now with fixed energy, while H(flQ) is given by fig. 5.2 with an extra factor of

G. Sterman / Hadromccrosssecttons

344

Fig 5.2. Contnbutlon to H at one-loop

(21/2q-/q °) in the phase space integral. Direct calculation gives • r(1 -

Kffreal)([1-y]Q/21~)

~)

= (2/Q)(2as/cr)CF(4Crl~2/Q 2) F-~ - 2-ee)(1 _y)-X-2e, r(2 - e)

H([1 - y]Q/2)

y)-1-2.

= (2/Q)(Eeq/cr)cF(4Crl~2/Q2)~-~'S'2-~)

(1 (5.34)

where we have used P0 = ½Q" Now using eq. (5.31), we get

r ( 2 - ~)

K~([1

- y]Q/21,) = (2/Q)(2t~J~r)CF(4W12/Q2)~-2_--27)

-1-2. (1 - y ) (5.35)

6. Jet distribution

The jet distribution, eq. (3.7), is handled in a slightly different manner from the c.m. and light-cone distributions. To leading power in (1 - x), it is of the form

(12)-l(p,y~,) ×j[12/~t2, l.n/(tt(n2)l/2),g(iQ].

(6.1)

The leading behavior in the moments of J comes from the limit 12 ~ 0, I. n = O(Q). In accordance with eq. (3.14), we parametrize l ~ as

t~ = ([1 - ~]p+, Q2/2t;P+,OT) = ( 1+, 1-, l,r).

(6.2)

We will be interested in the singular behavior as ~ ~ 1. As usual we take the quarks as massless. The function J satisfies three equations: (i) a homogeneity equation,

[~,2(a/a~ ~) + (l. ,,)2( a/a[t.,,l ~) + 12( a/al2)] ×J[

12/, 2, I. n/{ l~( n2)1/2}, g(/t)] = O;

(6.3)

G. Sterman / Hadromccrosssectwns

345

(ii) a renormalization group equation, [ # ( a / a # ) + fl(g)( O/ag) + 2"yq]J[12/# 2, l. n / { #(nZ)t/2}, g(#)] = 0 ,

(6.4)

where 7q is the quark anomalous dimension in the axial gauge [22],

Vq = - CF3a,/4rr ;

(6.5)

and (iii) a gauge-dependence equation [13] of the same type obeyed by the fight-cone and energy distributions above,

[ o/o

.-/.+

)1j2]s[ ;2/.2, ;../( .(. 2)1,2}, g(#)] #(n2)t/2}, g(#)l

= [0/0 ln(l.n/{n2}l/2)]j[12/#2, l . n / ( ~- Gj( I" "//#, g(#))j[12//#2,

l" n//(#(n2)1/2), g(#)]

+ foO-X)P+(d~o)Kj(o~v. ?l/#(n2) 1/2,

g(#))

XJ [ ( 1 - 600)2/# 2, l" n / { # ( n2 )X/2} , g ( # ) ] ,

(6.6)

where the vector v ~ is a fight-like vector in the plus direction, v ~= (v +, v-, VT) = (1,0,0).

(6.7)

The functions Ks(~ov. n/#(n2) l/z, g(#)) and Gj(l. n/#, g(#)) are defined by the graphs shown in fig. 6.1, with the square vertex given by eq. (4.13). The graphs are evaluated with a phase-space delta function 8(Eq, + - ¢0), which fixes the total plus momentum, Eq,+, of the cut lines of K s to be ~0. The functions Gj and K s are infrared finite order-by-order by the same reasoning as for the corresponding functions in ref. [13]. The finiteness of K s involves the cancellation of real and virtual contributions, and it is convenient to think of K s as a distribution in terms of the dimensionless variable

y = 1 - o~/p +.

(6.8)

We then write Ks([1 - y ] p +) = ( l i p +)[k s, + ([1 - y ] p +)/(1 - y )] + 8(1 - y )

+/~J,a p+

+ks,f([1-y]p+).

(6.9)

346

G Sterman / Hadrontc cross secttons

+

C.C.

J

4- C.C.

Fig. 6 1. Graptucal form of terms m eq. (6.6)

k j, 8 is an infrared finite number, kj, + an infrared finite function for y * 1, and kj, f is a finite, non-singular function for all y. The plus distribution is defined to act, for any functions f(z) and g(z), as

£1 dzg(z)[f+(z)/(1 -

z)] + - £ 1 d z [ g ( z ) -

g(1)l[/+(z)/(1

-

z)]

-g(1) j ; t f +( z )/(1- z )] .

(6.10)

In particular,

fo:[i+(.)/(1- z)]+=0.

(6.11)

Graphical contributions to K j with real gluons require no new UV subtractions, while the reasoning of ref. [13] shows that the combination Gj + Kj,,~r t is unrenorrealized. We thus have

[(~a/a~,) + #(g)( O/Og)] (Gj + k j, 8) =

O,

[(~a/a~) + ,8(g)( a/ag)] K ] ( y --1=I) =

O,

(6.:2)

so that

kj,8[g(M)] + Gj[Q/M,

§(M)] = kj,8[g(#)] +

Gj[Q/Iz,

g(#)].

(6.13)

G. Sterman / Hadromccrosssections

347

Similarly, exhibiting dependence on the renormalization mass, we have

[kj, + [(1 - y ) p + / M ,

g(M)]/(1 - y ) ] + + k:,/[(1 - y ) p + / M , g(M)]

= [ky,+[(1-y)p+/l~,g(#)]/(1-y)]++

ks,/[(1-y)p+/l~g(#)].

(6.14)

We will find these observations useful below. Here and elsewhere g ( M ) refers to the solution to the equation O~(t)/Ot = fl(~) evaluated at t = i n ( M / # ) with boundary condition g(t = 0) = g(#). To one-loop, then,

g2( ) g E ( M ) = 1 - gE(#)blln (M2/1~2) '

(6.15)

where fl(g) = big 3 + .. • . Since we are interested in the singular behavior of J as l 2 --~ O, we can make the replacement

(6.16) Corrections will be of order (1 - ~)o. We can then use eqs. (6.3), (6.4) and (6.6) to derive the equation satisfied by the singular parts of J,

[( O/O tnIl - ~]) - ~fl(g)( O/Og) + 1] X (J[(1-

~)Q2/#2, Q(n+/n-)l:/#, g ( # ) ] / ( 1 - x ) } 1

=-½fldyp+[-~_yks,+([1-y]p+/#,g[#])]+

-~[1 k :,n(g[t~])+Gs(Q/#,g[#]) x(JI(1-f)Q2/#2,

2Vq]

Q(n+/n-)l/2/#,g(l~)]/(1-x)},

(6.17)

where we have divided J by (1 - x) for convenience, so that the solution for J will be of the same form as for q, and tk. Here and below, we have dropped kj, p which does not contribute to the singular behavior. We solve eq. (6.17) to find the singular behavior of J as x ~ 1.

348

G. Sterman / Hadromc cross sections

To express the solution to eq. (6.17) it is convenient to suppress the dependence on gauge parameters and to define

S(Q/I.t,g) =

kj,~(g)

+

Gj(Q/I-t,g).

(6.18)

The solution can then be expressed as J I(1 - x)Q2//~2,

Q/l~,g(#)]

oo

= E - - n! n~0

,r=l

2fo dw, T(w,/l~,Q/Iz,g(Iz))8 j ~= l (w,/Q)-(1-x) (6.19a)

where the function T is given by

r(w,l , OIt,,

= [f~_,~,/Qdy[kj, +([1-y]p+/Iz, ~[{w,/(1-y)Q }1/2/x])/(1-y)]

+]

Wt

S(Q/i.I,, g [ ( w d / Q } 1/2/.tl) - 2 y q ( g [ ( w , a

+

}1/2]) 1

]

+ ½#(g[(w,/Q}a/~])F'(g[{w,/Q}l:t*)

+

Wt

+

(6.19b)

S~-w,sedYlkJ,÷([1 - y]S21/2,gl ( w ~ ( 1 - y ) Q

}1/2])/(1-y)]+

Wz

+

÷

S(1,~[{w,Q}I/2]) - 2yq(g[(w,Q}X/2])] + ½fl(~[{ w,Q} x/2])F'(~[{ w,Q} t/2]) Wt

-8(w,)F(g[Q]).

+

(6.19c)

In (6.19c) we have set/~ = Q and used eq. (6.14). F(g) is an polynomial in g, which must be determined order-by-order by direct computation.

G. Sterman / Hadromc cross sections

349

It is not difficult to verify that (6.19) satisfies eq. (6.17). Since T in (6.19) is linear in the functions Ks, ÷.... F, we can study the role of each function separately. Consider, for example, S = ks, 8 + G. We denote by L s all terms proportional to S on the 1.h.s. of (6.17), when J is given by (6.19a, b). As usual, the derivative with respect to ln(1 - x) is converted to a sum of derivatives with respect to the w,, using eq. (4.19). After integration by parts, we find L s = _ l f o ~ ( d w / w ) [ w [ O/Owl - ½B(g)[ O/Og]]

× J ( 1 - x - w / p +) - I S ( Q/Iz, ~[0]) J(1 - x ) ,

(6.20)

where the last term, which must cancel in the final answer, is the end-point contribution from w = 0. Next, we observe that

[w(0/0w) -

d/og)] g[( w/Q )1/2] = 0,

(6.21)

so that only the 8 ( w ) term contributes to the w integral in (6.20). Using (6.21) in (6.20) gives

= S(Q/I~

g[#]) J(1 - x ),

(6.22)

as required by (6.17). The other terms in (6.19b) are treated similarly. In (6.19) the function F plays the role of a constant of integration. The exponentiation of the two terms involving F is forced by the requirement that (6.19) satisfy both (6.17) and the renormalization group equation. In particular, multiplication of J by the arbitrary function of as[Q] is not allowed by (6.17). Of course, the exponentiation of the lowest order term in F(g[Q]) should not be taken too literally, since F has arbitrary higher order corrections. The term 8(oo,)F(g[Q]) must be a function of the effective coupling at #2 = Q2 rather than (1 - x ) Q 2 = l 2 because the 8(1 - x ) term in the distribution J must be finite order-by-order in perturbation theory [12]. Note finally that, unlike K, and K~, T is infrared finite, since it includes both real and virtual contributions. The arguments for this result are exactly the same as for the corresponding function K(b) given in ref. [13]. Thus, unlike K, or K~, K s and S may be evaluated directly as distributions in four dimensions.

350

G. Sterman / Hadromc cross sections

+ c.c.

+ C.C.

'

(b)

Fig. 6 2. (a) One-loop contributions to Kj. (b) One-loop contnbutmn to Gj.

The moments of the solution, eq. (6.20), are found in just the same way as for the energy and light-cone distribution. The result is

fo~d~~"-ls [ ( a

-

~)Q~/~, Q/w,g(~)]

=exp[foldZz"-XQT([1-zlQ/l~,Q/t~,g(~))]+O(1/n

).

(6.23)

At one-loop, the graphs in fig. 6.2 contribute to K s and S. We find (see appendix)

kj 8 ,~ + Gj= (as/~r)Cv(4~r~Z/Q2) ~1"(2 ' '

E)F(1

~. (1 -

+ 2e)

2~)

'

/%o~, [(1 - ~)Q/zx/2,.J = (21/2/Q)(2%/~r)CF(4~r#2/QZ)~(1 - e)F(1 + e) x (1 - ~)-1-z~. (6.24) From these expressions, using eq. (6.9) and the distribution identity (l-x)

-l-a= -(1/a)3(1-x)

+ (1/[1-x])++

we find

[ks,+([1-YlP+'as)/[1-y]]°+)=(zaj~r)Cv

-.. ,

[1]

S(1, a S)(1) = (2av/, r) CF,

~

(6.25)

+' (6.26)

G. Sterman / Hadromc cross sections

351

which are finite distributions, as expected. Finally, by substituting (6.26) into (6.19) at order as, and comparing the result to the one-loop calculation of J in the axial gauge given in the appendix, we find

F(g[Q]) = - (av/2~')Cv(3) .

(6.27)

7. Soft gluon Itmetions We now turn to the soft gluon functions U and V. Because they describe the interaction of eikonal lines, they require a different treatment from the c.m. and light-cone quark densities ~ and 4, and the jet distribution J. To be specific, we consider the function U(w). The arguments for V are exactly the same. We are considering, then, the sum of eikonal graphs shown in the middle of fig. 2.2. The energy crossing the cut is fixed to be Wpo; otherwise all momenta are integrated over. Gatheral [15] has shown that all such graphs exponentiate "before phase space is integrated over". That is, we may write U (or V) as

u= fexp

ns

(7.1)

,

where the integral is over all allowed momenta, where s labels the order, and where the ~s are of the form fis = )', C(W)F(W).

(7.2)

W

The C(W) are color factors, while the F(W) are integrands, with color factors omitted, for graphs which are not reducible by cutting two eikonal lines except at the vertices where the opposite-moving eikonal lines join. Each such graph is referred to in ref. [15] as a "web". The generic form of a web is illustrated in fig. 7.1. The graph F(W) is two-line irreducible, except for the cuts a and b. For w 4: 0, the sum is over all cuts c of F(W) except the two-line reducible cuts a and b. For w = 0, there is a delta function contribution to which only cuts a and b contribute. The webs W in eq. (7.1) are still linked by the phase space integrals appropriate to the cross section being calculated. For us, however, these are simple, involving only a fixed total energy crossing the cut. So, the formal equation (7.1) may be written out more explicitly as V(Wpo) =

(a/nO n~0

~,(o~) = Efi~(o~). $

Wpo-

,o, , t=l

(7.3)

352

G. Sterman / Hadromc cross secttons

Ib Fig. 7 1. General form of "web" contributingto eq. (7.2). a and b are the only two-linereducablecuts of the graph.

Now to take the moments of U, we must change to scaled variables, defined by analogy to eq. (4.22)

[l-[~__z] -1

~ , = |[=~X' • (1 - X , ) p 0 ,

(7.4)

so that

,~'~o,= 1 -

hj Po.

(7.5)

In the w ~ 0 limit, all the X's must go to unity, since the 8-function in (7.3) becomes (l/P0)8

w- 1 +

X, ,

(7.6)

and so, changing variables (as in [4.24]) from { ~, } to { X, },

(7.7) But then

foldW(1-w)nU(wpo)=exp(fold~npov[(1-h)po])+O(1/n),

(7.8)

so that the moments of U exponentiate. Clearly, the same reasoning applies to V, so that we may also write

foldw(1-w)"V(wp+)=exp(foldXh"p+#[(l-X)p+])+O(1/n), (7.9)

where # differs from v only in the phase space over which it is integrated.

353

G. Sterman / Hadromc cross secttons

An important point needs to be made here. For this exponentiation to be useful, p and p should behave at worst as (1 - ~)-1, with no logarithms, as 2~~ 1. It is not difficult to demonstrate this result. Consider first a contribution to F(W) in fig. 7.1 with no ultraviolet divergent subgraphs. When summed over all cuts and integrated over phase space, any graph of the form of fig. 7.1 is finite [12,13]. All infrared divergences from the purely virtual cuts a and b, which contribute only via 8(1 - X) must cancel the sum of the real cuts. But the virtual cuts can have at most a simple infrared pole at e = 2 - ½n = 0, since F(W) is a "web", and so has no nestings of subdiagrams which are individually infrared divergent. As a result, the sum over all real cuts can also give at worst a singularity of the form (1 - X)-I. Note this is not true for individual cuts. An example is given in fig. 7.2, which contains a term of the form - (I/e)(1

- X)-'-"

(1/e2)8(1

- ~ ) - (1/e)(1/[1

- X])+

e--~0

+ (ln[1 - ~.]/[1 - X])+,

(7.10)

since either gluon momentum can vanish while the other gluon carries energy (1 - ~ ) P 0 .

Now we can drop the restriction that F(W) have no renormalization parts (2-, 3or 4-point subdiagrams). The presence of such subdiagrams introduces in general extra factors of ln([1 - } ' ] P o / # ) . We can organize these factors by using the fact that U (and V) are multiplicatively renormalizable [13], since the only divergences in U and V not taken care of by the usual renormalization prescription are corrections to the vertex where the two opposite-moving eikonal lines meet. Thus, U obeys [(0/0 ln#) +

fl(g)(O/Og)

+ Yu] U = O,

(7.11)

with Vu the same anomalous dimension as in [13]. On the other hand, except for contributions with no gluons in the final state (cuts a and b in fig. 7.1), the function J,(wQ/#, g[/t]) is ultraviolet finite. By analogy to

Fig. 7.2. Example of a cut web.

354

G. Sterman / Hadromc cross secttons

eq. (6.9), we write ~,(Wpo ) = ~,~(wpo ) + (1/po) [~,+ (wpo)/w] +.

(7.12)

Renorma!ization dependence is entirely in the vn term, which obeys [( O/O In/~) + ,8(g)( O/ag)] v~ + 7v = O.

(7.13)

7'+(wQ/l~, g[/~]) = I,+(wQ/M, g [ M ] ) .

(7.14)

while 1,+(w) obeys

Thus, choosing/~ = Q in eq. (7.8),

(7.15) and similarly for V. One-loop calculations give, for the real and virtual contributions to U and V,

r(1-e)

Ureal([1 - ~,] Q/2) fx) = (2/Q)(EotJ~r)CF(4rrF2/Q2) ~~ ( 2 S - ~ ) (1 - ~ ) - 1 - 2 ' ,

U~-t ([1 - X]Q/2) (1) = ( 2 / Q ) ( 2 a J ~ r ) C F ( 1 / e ) 8 ( 1

- ~)(1 + e[ln4~r - "t]),

~eal ([1 - ~k]Q/21/2) (1)= (21/2/Q)(2as/~r)CF(4~r#2/Q2)~F(1 + e)(1 - h ) - l - 2 ~ , v ~ t ([1 - X]Q/2X/2) fl)= (21/2/Q)(2as/~r)CF(1/e)8(1 - X)(1 + e[ln4~r - 3']). (7.16) F r o m (7.16) and (7.3) we find v +(wpo) °) = p +(wp +)(1) = ( 2 a j a r ) CF, V~(I) :

p8 (I)

= (as//rr)ln(/~2/Q2).

(7.17)

8. Summary We now apply the results of the previous sections to organize the complete high moment behavior of the DY cross section when factorized in terms of DIS structure

355

G. Sterman / Hadrontccrosssections

functions. We then summarize one-loop results, and discuss next-to-leading terms at two loops. We can get a form for moments of the hard part to(r) simply by substituting the results (4.27), (5.33), (6.23) and (7.15) into (3.16). Choosing/~ = Q, this result may be expressed as

o~(V)(n,Q2) =lHm(g[O])[-ZxlZ(g[a])[ 2 ×exp[fol dzz,-X( P+(1, g[ (1- Z}Po]) - 20+(l' g[ {1- z }P+]) } 1-z + +Ps(g[Q]) - 2os(g[Q])] [ el d z z n _ l f l d y

× exp[2/o 1 - z

~

[pOK'~((1-y)P°' - p +K,((1

g[Q])

-y)p+,

g[Q])]

×exp[2foXdzz"-lQ × r(1- z, X, g[ Q])] , where from (6.19c), defining z = 1 -

(8.1)

w/Q,

Q × T(1 - z,1, g[Q])

[1

fa

1

s(1,g[{1-z}l/ZQ])-27q(g[{1-z}1/2Q]) ] + ½fl(g[ (1 - z }'/2QI)F' (g[(a ~ }1/2Q])

i-7 4-

-8(1

-z)F(g[Q]).

In (8.1) the function

(8.2)

L(Q, g(Q)) is R(Q, n) ×/-/Dr L(Q, g(Q)) = R(Q, n') × Hm

(8.3)

from eqs. (4.24) and (5.32), and n ~ and n '~ are the gauge vectors chosen for quark densities ~b(x) and ~(x), respectively. Another way of writing L is

R(Q, n) x HDVX U,~_t L(Q, g(Q)) = R(Q, n') × HDI× Vv~.t '

(8.4)

G. Sterman / Hadromc cross sections

356

where we have used the fact that the purely virtual parts of the soft gluon functions U and V must be the same in dimensional renormalization, since they are pure counterterms, having no scale. Then if we apply the reasoning of sections 2 and 3 only to the virtual graphs which occur in DY and DIS respectively, we see that L is precisely equal to the ratio of timelike to spacelike Sudakov form factors F3 [14, 23], independent of the gauge chosen in each case,

L(Q) = Ir3(p, p')/F3( p, - P ' ) 1-

(8.5)

Thus the ratio of the time-like to space-like Sudakov form factors [3] occurs naturally in this analysis. We shall make the usual assumption [3, 4] that, following the leading behavior [14, 23], the ~r2 term exponentiates,

Relnl"3(Q2)=lnI"3(-Q2)+(~s(QZ)/2~r)CF(½~r2).

(8.6)

The importance of large non-singular terms of this type for practical evaluations of the K-factor [9] has been emphasized in [4, 24]. As noted above, there are no logarithms of 1 - y in v+ or p +. Similarly, although K , and K+ individually contain a ln(1 - y ) term, it comes from a region where cut gluons are both soft and nearly collinear to the momentum p~. This part of phase space is included in the definition of both functions, and, since the two functions differ only in terms of the phase space in their definitions, the ln(1 - y ) contribution must cancel in the difference of the two. The leading logarithm terms, of the form ln(1 - y ) / ( 1 - y ) , giving In2 (n) in the moments, are thus contained completely in the last exponential, associated with the final-state jet in the DIS factorization formula. Now we outline the straightforward substitution of one loop results into eqs. (8.1) and (8.2). IHDd 2(1) is given by eq. (3.12). Note that only the virtual part contributes to leading order in 1/n. ILl 2 is given by eqs. (8.5) and (8.7), and v+ and p+ are specified by eq. (7.17). The combination K, (1) - K , (1) is evaluated by combining eqs. (4.28) and (5.35). As expected, all singular terms cancel. The result, using the distribution identity (6.25), is exp [2fol ldZ- z z~-I f01 dy (K~ ( ( 1 - Y ) P o , g[Q])C1) _ K , ( ( 1 - y ) p + ,

=exp[(aJ2~r)CF(}~r2)] + O(e).

g [ Q ])o))] (8.7)

Finally, Kj and S are given in eq. (6.26), F is given by eq. (6.27) and 7q =

G. Sterman / Hadromc cross secttons

357

- C v ( 3 a J 4 r r ). Substituting all this into eqs. (8.1) and (8.2) we find

9.

(2a~tQl/~r)]exp[(aJ2,r)CF(%r2- 3)] [ 1 Z n-l[z dy ×exp[--(2CF/~r)f ° dz ~ - - z l j o ~_yat{1-z)X/2(1-y}'/~Q]

= [1 +

[

×exp (7CF/2rr)

fol d z z"-l-la[(1-z}X/2Q]] 1------7 n--1

Xexp[-(2CF/~r)foldzZ~--zaa[(a-z )Q/21/2]].

(8.8)

Note that the full term proportional to ~r2 in the one-loop correction, eq. (1.8) has exponentiated. Eq. (8,7) summarizes what we can learn about the high-moment corrections to the DreU-Yan cross section from one-loop calculations supplemented by the reasoning of sects. 2-7. Although (8.7) includes only one-loop calculations, it can be checked by a two-loop calculation already in the literature. A contribution to the next-toleading In(n) term at two loops is found by expanding the running coupling using the one-loop beta function. ¢X

ff(t,a) =

1 - 4*rblaIn(t)

= a ( 1 + 4~rbxain(t)+ .-. ).

(8.9)

In particular, applied to the kj, + term in (8.8), this gives

1

1

_y)lj2e])] +

= (as[Q.l/2*r)CF(4in(1 - z) - 6a~[Q](4rrbl)in2(1-

z) + - - . }. (8.10)

Substituting this result into (8.8), and fixing the remaining running couplings, we derive a form which gives the leading and next-to-leading in(n) behavior to all

358

G. Sterman / Hadromccrosssectwns

orders, as well as the correct one-loop d(1 - x) terms, ,o(e)(n, Q 2 ) = [1 + 2as[Ql/~r] x exp[(a~[Ql/2~r)(~*r 2 - 3)]

x exp[ CF( ast Q l/~r )ln2( n ) - CF(3ast Q l/2~" )ln( n ) -Cr,(as2[Q]/~r)(4.bl)ln3(n) + . . - ] ,

(8.11)

which agrees with the result of van Neervan [16]. The author thanks the Institute for Advanced Study for its hospitality. It is also a pleasure to thank John Collins and Davison Soper for conversations which were crucial in the development of the project, and Paul Mackenzie for many useful discussions. This work was supported in part by US Department of Energy Contract No. DE-AC0276ER02220 and by National Science Foundation grant PHY-81-09110.

Appendix In this appendix we summarize the form and results of the one-loop calculations used in the paper. Everywhere we have used dimensional regularization and modified minimal subtraction (MS), defining e=2-½n.

(A.1)

with n the number of dimensions. All calculations are carded out in physical gauges where the gluon propagator is Dr, (q) = ( - i / [ q2 + ie]) × N~,(q, n),

N~,,(q,n) -- g~,,

n~q, + qj,n,, n2q~,q, + -(n.q)2" n.q

(A.2)

The denominators of N~, are defined as principle values. In all calculations of distributions with incoming quark momentum p~ we take pO= {Q corresponding to the c.m. system of the Drell-Yan process in the z --* 1 limit. Finally, we always work with massless quarks. A.1 CENTER OF MASSQUARKDISTRIBUTION At lowest order the c.m. distribution is given by the contribution of the two graphs, figs. 2.3a and 2.3b to eq. (3.6). For calculations in the DY case we choose

359

G. Sterman / Hadromc cross sectmns

the A ° = 0 gauge. Eq. (3.6) normalizes ~k(Y) at zero loops to ~b(y) t°) = 8(1 - y ) .

(A.3)

The one-loop real graph is then given by

e/(Y)~'am=(CF/2S/2)g2#E~f ( d~))"(2p-1 q) 2N~(q' n) X tr[ #3'~'(# - q)3'+(# - ~) 3'~] X 2¢r8+ (q 2) X 8(q 0 - [1 - Y ] P o )

= (aJcr)Cv(4Cr#2/Q2) ~x

(a I + a 2 + a3) ,

(g.4)

where a, is the contribution of the ith gauge term in eq. (A.2), a 1 = - ½(1 _ y ) 1 - 2 ,

r ( 2 - e) 1 -e r ( 2 - 20 x - - ,

a2= _(1_y)_1_2, F(1-e) 1 --=---r(2-0 F ( 1 - 2 e ) X-+(1-y)-2'F(2--2e)e a3= _(l_y)_,_2,

X 1

e'

F(1-e)

r ( 2 - 2e)"

(A.5)

The virtual contribution is [/', /,~5/2~,v2.2e /" d~q

~(Y)v~rt(1)=t'~V/"-

]~e~ j(2~r)n

1

1

(2p.q+q2+te) (q2+ie)N~,,(q, n)

× tr[ ~T~'( ~ - ~) y ~ - x y +]8([1 - Y] Po) •

(A.6)

The first and third gauge terms of the gluon propagator are "pure counterterms" in dimensional regularization, because all mass scales except/~ cancel in their integrals. Only the second gauge term gives a non-trivial result. It is important to treat the gauge denominators as principle values in this calculation. The complete answer is ~k( y ) ~ t 0 ) = - 8 ( 1 - y ) ( a J T r ) C F [ ( 4 ~ r l x 2 / Q 2 ) ' ½ ( [ s g n ( n 2 ) e - 2 " ] ' +

r 0 - e)F(1 + 2e) X

2e2(1 - 2e)

] X (1 - e) + (3/4e)(1 + e[ln4cr - 3'])J • (A.7)

360

G. Sterman / Hadronw cross sectwns

The rather strange factor involving n 2 comes from the principle value prescription. The last term is the modified minimal subtraction counterterm. In the A ° = 0 gauge, where n 2 = 1, the principle value term contributes a factor (1 - e2~r2), which is included in eq. (2.14). A.2. LIGHT-CONE QUARK DISTRIBUTION

One-loop contributions to the LC distribution are given by the same graphs, figs. 2.3a and 2.3b. The virtual contribution is exactly the same for the light cone and the center of mass distribution, ~ (Y)wt (1) = tk (Y)~art (1).

(A.8)

Eq. (A.8) does not mean that we must use the same gauge to evaluate the two functions, only that the formulae for the two are the same. In the A 3 = 0 gauge, where n 2 = - 1, the principle value term gives a factor (1 - ½e2~r2) which is included in eq. (3.11). The real distribution requires recomputation. Its form is the same as eq. (A.4a), except for the normalization and phase space specified by eq. (3.6), 1/-,

2 2~

~(Y)real(1) = 4t"Fg ~

/" d"q 1 J (2~r)'~" ( 2 p .

x tr[~'f'(g-

4)v+(g-

q)2Nt,.(q,

n)

4 ) v q x 2~r3+(q 2) x

= (as/~r)CF(4~rl~2/Q2) ~X (b I +

8(q+-tl-ylp

b E + b3).

+) (A.9a)

where again the b,'s correspond to the different gauge terms in (A.2), and where 1 bl = -

- x) x -,

E

1

b== -x(1-x)-~-2"r(1 + ~)x 7(1- }e2~r2), b 3 = - (1

-y)-l-2T(1 + ~).

(A.9b)

The factor (1 - ½e2~rz) comes from the gauge denominators, which can vanish for physical momenta in the A 3 = 0 gauge. A 3. JET DISTRIBUTION

We need to calculate the jet distribution only up to terms which are nonsingular in the x --, 1 limit, since other terms must be infrared finite and cannot be summed by our methods anyway. The reason for calculating the LC and c.m. distributions

361

G. Sterman / Hadromc cross secttons

exactly above is to check that all ( l / e ) singularities cancel in the "hard" parts, eqs. (2.16) and (3.12), even those which are not proportional to (1 - x) -1. The one-loop contributions to J are given by fig. 3.3a and 3.3b. The virtual contribution to J(1 - y ) is the same as for q~(Y),~-t at one-loop,

(A.10)

J ( 1 -- y )vxrt (1) = ~ (y)vtrt (1),

where here we must use the same (A 3-- 0) gauge in both cases for the factorized form of the DIS cross section, eq. (3.13). From eq. (3.7) the real contribution, fig. 3.3a is 1

g2/~2'CF

4,41-,) x

xN~,p(q, n) x

f d~q

x .I (2~r).(2~)8(q2)(2~')8([l-ql2)

tr[In'q~,"(l-4)~,.l~'] r(2-

= (a#'~r)CF(4~r#2/Q2)*

~)

F-"~-~'2-7) ( 1 / 4 - l/e)(1 -

x)

-1-,

1 + F(1 + e ) ( 1 / e - 1)(1 - ½e2~rz)(1 - x ) - l - E ' J .

(A.11)

Again, the factor (1 - ½e2~r2) comes from the gauge denominators. A 4. SOFT GLUON DISTRIBUTIONS

The soft gluon distributions U and V are computed from figs. 2.3c and 2.3d, plus their complex conjugates. The computation is simplified considerably by the observation that the first two gauge terms in the gluon propagator, exI. (A.2) cancel in these graphs, leaving only the q~'qP term to contribute. The real contribution to U is given by

=

f

d"

1 2=8(q2)8(Wpo - qo)_77_-qq N+-(q)

= (2/Q)(2aJrr)CF(4~r#E/Q=)

. r ( 1 - e)

~(2-- 27) w-1-2e"

(a.12)

The real contribution to V is similar, an

= (2x/2/Q)(2aJrr)Cv(4~r#2/Q2)'F(1

+ e)w -1-2~

(A.13)

G. Sterman / Hadromccrosssections

362

Finally, the virtual contributions to U and V are pure counterterms in modified minimal subtraction, p ° t : ( Wpo),~ ~ = p + v ( wp + ),~?~

= ( a # q r ) C r ( 1 / e ) ( 1 + e[In4rr - 3']).

(A.14)

Using the above results, we easily verify eqs. (2.16) and (3.12) on the one-loop hard parts.

A.5 K, AND K~ The function K , ( [ 1 - y ] p + ) is given by the real gluon graph, fig. 4.10 and its complex conjugate, with the square vertex specified by (4.13). It is d~ K,([1 - y 1Q/21:)(1)= 2Cvg~t~z~f(2~q). 2rr~(qZ)~(q+

×

_ [1 - ylp+)

n2o,,pl~N"O( q) (v.n)(n.q)(v.q) ,

r(2 +

~)

=(21/2/Q)(2aJrr)(4rtg2/Q 2) (1Zy51-72 ' . (A J5) The function K, a) is found from the eq. (5.31) to be the sum of fig. 4.10 and its complex conjugate, K~(real)f1), and the quantity (0/0 ln Q)Ha)(flQ)/fl, which is given by fig. 5.2. The first contribution analogous to eq. (A.15), is

K ~ ( [ 1 - y ] p o ) ( t ) = 2CFg2g 2e X

2~rS(q2)8(qo - [1 --ylP0)

n2o,paN~'a( q ) (v.n)(n.q)(v.q) r(1

=(2/Q)(2aJ~r)(~q~2/tp'nl2)

-

e)

F - ~ - ~ e ) (1-Y)-I-2'"

(A.16)

On the other hand, fig. 5.2 is given by ( 0/O In Q)(H([1 - y]Q/2)l[1 - y])

=(o/o

ln^X.. dnq r 1 -q3/qo] ~e),..~g2~2~t" j-~gj-~t

× (2~r)8(q2)8(qo - [1 - y ] Q / 2 ) ×

v,,poU~'#( q ) (o.q) ~

= -2e(2/O)(aJrt)CF(4rq~/O

: , r ( 1 - e)

) ~-~-._7~7e) (1 -

y)-1-2~

. (A.17)

G. Sterman / Hadromccross sectwns

363

From (A.16), (A.17) and (5.31) we get 2

2 ~ r(2-e)

K¢([1-YlP°)(s)=(2/QI(2aJ~r)CF(4~rI~ /Q ) F-~---~e) ( 1 -

y)-X-2~.

(A.18)

A.6 Kj AND Gj

At one-loop, the functions K s and Gj are computed from fig. 6.2. It is convenient first to compute the combination gj, vlrt(1)+ aj (1). The two terms can then be disentangled by using the facts that Kj,~t is a pure counterterm and that Gs is infrared finite [13]. We have

[[ gj,vxrt(1)-b aj (1)]

d~q ['Y~(_____[-~I 5 n2v.Na#(q)

=2ig2.2~ f P u f f (2~r)"

q2(p_q)2 (v.n)(q.n)

= (a#qr)CF(4~r#2/Q2),r(2

- e)r(1 + 2e) ~(1 - 2~)

X ½([sgn(n2)e-2'"] "+ [sgn(n2)] ~).

(A.19)

The real one-loop contribution to K s is given by the real gluon graph in fig. 6.2. It is

Kj,~¢~I([1_y]p+)(x)

= 2CFgZl~2~f(2~q)"2~rS(q2)3(q+--[l--y]P÷)(~;:n-~-q-~.q) = (21/2/a)(2as/~r)(a~rp2/a2)

~ (1 - e ) / ' ( 1 + e)

(1 _y)~+2~

(A.20)

References [1] G Altarelfi, R.K. Ellis and G. Martinelli, Nucl. Phys B143 (1978) 521, (E) 146 (1978) 544, B157 (1979) 461 [2] J. Kubar-Andre and F E. Paige, Phys Rev., D19 (1979) 221, K Harada, T Kaneko and N. Sakai, Nucl. Phys B155 (1979) 169; (E) B165 (1980) 545; J Abad and B. Humpert, Phys. Lett. 80B (1979) 286; B Humpert and W.L. van Neervan, Phys. Lett 84B (1979) 327 [3] G. Parisi, Phys. Lett. 90B (1980) 295; G. Curci and M. Greco, Phys. Lett. 92B (1980) 175

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G. Sterman / Hadromc cross sectzons

[4] P. Chiapetta, T Grandou, M. Le BeUac and J L. Meumer, Nucl Phys. B207 (1982) 251 [5] E. F_.ichten,I. Hinchliffe, K Lane and C. Qmgg, Rev. Mod. Phys 56 (1984) 579 [6] D Amati, R. Petronzto and G. Venezaano, Nucl. Phys. B140 (1979) 54, B146 (1978) 29, S. Lthby and G. Sterman, Phys. Rev. D18 (1978) 3252; A.H. MueUer, Phys. Rev. D18 (1978) 3705; R.K. Ellis, H Geor~, M. Machacek, H.D. Politzer and G.G. Ross, Nucl. Phys. B152 (1979) 285; A.V. Efremov and A.V. Radyushkin, Theo. Math Plays 44 (1981) 664, 774 [7] G. Bodwin, Phys. Rev D31 (1985) 2616 [8] J.C. Collins, D E. Soper and G. Sterman, Nud Phys B261 (1985) 104 [9] I R. Kenyon, Rep. Prog. Phys. 45 (1982) 1261 [10] A Ramalho and G Sterman, Plays Rev. D29 (1984) 2517 [11] D. Amatt, A Basseto, M Ciafalom, G. Marchesmi and G. Venezaano, Nucl. Phys. B173 (1980) 429; M Ciafaloni and G. Curd, Phys. Lett. 102B (1981) 352 [12] G Sterman, Phys Rev. D17 (1977) 2773 [13] J C. Colhns and D E. Soper, Nucl. Phys B193 (1981) 381 [14] A. Sen, Phys Rev D24 (1981) 3281 [15] J.G.M. Gatheral, Phys. Lett. 133B (1983) 90 [16] W L. van Neervan, Phys. Lett 147B (1984) 175 [17] G. Parisi and R. Petronmo, Nucl. Phys. B154 (1979) 427; G Curci, M. Greco and Y. Srivastava, Nucl Phys. B159 (1979) 451; G. Altarelli, R.K. Elhs, M. Greco and G. Martinelh, Nucl. Phys. B246 (1984) 12; J.C Collins, D E Soper and G Sterman, Nucl Phys. B250 (1985) 199 [18] J Milana, Phys. Rev. D34 (1986) 761 [19] J Frenkel, J G.M. Gatheral and J.C. Taylor, Nucl. Phys. B233 (1984) 307 [20] G. Curca, W. Furmanski and R Petronzto, Nucl. Phys B175 (1980) 27; J.C. Collins and D.E. Soper, Nucl Plays B194 (1982) 445 [21] W. Zimmermann, m Lectures on Elementary particles and quantum field theory, eds S Deser, M. Crnsaru and H. Pendleton (MIT Press, 1970) [22] D.E. Soper, Nucl Phys. B163 (1980) 93; B175 (1980) 465 [23] A.H. Mueller, Phys. Rev D20 (1979) 2037; J C. Collins, Phys. Rev. D22 (1980) 1478 [24] F. Khalafi and W J. Stirhng, Z Phys. C18 (1983) 315