Nuclear Physics B262 (1985) 1-18 © North-Holland Publishing Company

T H E DRELL-YAN P R OC E SS AT 'r --o 1

E.G. DRUKAREV and E.M. LEVIN

Leningard Nuclear Physics Institute, Gatchina, Leningard 188350, USSR

Received 26 October 1983 (Revised 10 December 1984)

The z ~ 1 limit of the Drell-Yan processes is considered in the LLA of QCD. The process is shown to be infrared stable since the large transverse momenta of the spectator quarks dominate the cross section. The total cross section, including the preexponent factor is calculated. The transverse momentum distribution is also obtained.

1. Introduction

In this paper we study the production of the lepton pair in the collision of two hadrons for the case "r ~ 1. As usual, ~" denotes the ratio of the dilepton mass M 2 to the hadron energy s: M 2 = sT; 1 - "r << 1. All the calculations are carried out in the leading logarithmic approximation (LLA) of QCD.

The or thodox (" naive") parton model describes the process as the annihilation of the antiquark from one hadron on the quark of the other hadron [1]. These partons (active partons) are believed to be near their mass shells. Since the parton wave functions are unknown, it is not possible to calculate the cross section of the Drel l -Yan (DY) process ODV. Since the same wave functions are involved in the cross section of deep inelastic scattering (DIS), some predictions can be made on

the ratio ODY/ODI S.

In Q C D the active quarks irradiate gluons with large transverse momentum before converting into the virtual photon. Thus QCD predicts some effects which are incompat ible with the orthodox pat ton model: the violation of scaling, the growth of

cross section do/dqat at large q2 (q is the photon momentum), etc. - see refs. [2, 3] and references therein.

The cross section oDV includes the structure functions of the active quarks. In the case " r - 1 they can be calculated up to an unknown function of q 2 and ~- as the initial state of the active quark has virtuality k 2 - X 2 with ~-1 _ r0 ' the confinement

distance. The cross section OOl s includes the same structure functions. Thus, in the case ~- - I Q C D also makes quantitative predictions only for the ratio aDV/ODI s.

E.G. Drukarev, E.M. Levin / Drell-Yan process at 7 ---, 1

\ /

Fig. 1.

There are some special features of the region 1 - z << 1. They have been consid- ered for DIS in our earlier paper [4], in which, as well as in this paper, we studied mesons as the simplest quark systems. The main properties of the process are valid for any hadrons.

The special choice of gauge reduces the lowest order QCD term for DIS to a single graph, shown in fig. la. The quark c obtains a large part Px of the meson momentum P by exchanging a hard gluon with the other quark. (The Bjorken variable x = - q 2 / 2 P q replaces ~" of the DY process while q is the virtual photon momentum). Here we face some troubles since the corresponding integral over the final state quarks transverse momenta kt, f d 2 k t / k 4 diverges at kt 2 -o 0.

Next comes the problem of the soft gluon corrections. It is more complicated than in the case x - 1 because of the second large logarithm ln(1 - x ) and the possible contributions a , l n2 (1 - x) and a~ ln (1- x), with a~ the strong coupling constant. Note also that the quarks c, c' of fig. la, as well as the hard gluons, obtain large virtuality k2 / (1 - x).

Using the planar gauge, one can reduce all the soft gluon corrections to the three blocks shown in fig. lb. [41. These are the interactions between the valence quarks (block ~p); the irradiations of the struck (block E) and the scattered quarks (block d). The remaining interactions give a small contribution of the order a s.

The product of the blocks E and d (fig. lb) forms the structure function of the active quark [4], while block q0 gives the hadron wave function. All three blocks form the hadron structure function. The interactions of the valence quarks at large distances - r 0 enter block q0 through a constant factor which is connected with the observable amplitudes.

The main contribution to the structure function comes from kt 2 - ) ~ 2 / ( 1 - x)a; a > 0 [4]. This insures the infrared stability of the function. One can also calculate the dependence of the structure function on q2 and x. The calculations of the structure functions, which will be needed during our calculations of ODV are

E.G. Drukarev, E.M. Levin / Drell-Yanprocess at • ~ 1

presented in sect. 2. Our results are valid up to the LLA restrictions:

but

asln( q : / h : )ln(1 - x ) >_ 1,

a~(q2) ln(q2/h2) - 1,

a ~ ( X 2 / ( 1 - x ) ) l n ( 1 - x ) - 1,

as(q2)ln(1 - x ) << 1.

(1)

(2)

Earlier calculations of the meson structure functions [5] have been performed by replacing the lower limit of integration over k 2 by X2. Thus, they gave only double

log terms. The large value of the transverse momentum k 2 >> ~2 enables us also to calculate the preexponent factor which depends on q 2 and x.

In sect. 3 we use these structure functions for the calculation of the Drell-Yan

cross section. They enter bo th the transverse m o m e n t u m distr ibution d o / d q 2 d x F d M 2 and the cross sections d o / d x F d M 2 and d o / d M 2. The distribu-

tion d o / d q 2 d x F d M 2 contains the structure function and the effective form factor T which emerges because of the partial cancellation of the Sudakov form factor by the radiation of soft gluons. We have calculated the latter for the case 1 - T << 1 in sect. 3. In our case T depends on the relative value of ln(1 - ~ ' ) and ln (qZ/q2) . As for the cross sections d o / d x F d M 2, d o / d M 2 they do not include the total hadron

structure functions. Indeed, the latter take into account the lack of cancellation of the final state quark Sudakov form factor by its irradiation of soft gluons. There is no final state quark in the DY process. Hence, only the irradiation of the active quark and its Sudakov form factor enter the cross section. Thus, in the case 1 - ~" << 1 the double log K-factor appears quite naturally in the planar gauge. The 1 - "r limit of the K-factor has been calculated earlier [6] from the analysis of the structure function moments. We give our derivation since it suggests the limpid physical interpretation of the K-factor.

As to the angular distribution of leptons (sect. 4) the general equations obtained in refs. [7] are still valid for the case 1 - T << 1. The longitudinal structure functions which enter the equations are calculated in sect. 2 of our paper. Recently the oscillatory energy dependence of the angular distribution was predicted [8]. We show that there are no such oscillations.

E.G. Drukarev, E.M. Levin / Drell-Yanprocess at r ~ 1

2. The hath'on structure functions

The meson structure function can be presented as

fq2dl~2 2z 2,, f l D ( x ' q 2 ) = J ' - ~ °ts[# )ix dx 'cp2(x"~2)Dq(x /x"qZ ' l ' t2 ) (3)

for 1 - x << 1, see fig. lc. Here #2 = k2/(1 _ x') is the virtuality of the quark c (fig. lb), Dq is its structure

function. The factor/~-2 comes from the propagator of the quark c. In order to obtain the large virtuality /~2>> 22 and the large fraction Px' of the hadron momentum P, the active quark should exchange at least one hard gluon with the spectator quark d. The propagator of this gluon/~-2 is included in the definition of the hadron wave function.

Both functions D and ~0 contain logarithmic terms. In the case 1 - x - 1 the structure function D contains the terms as(qZ)ln(q2/2`2). The function can be presented as D(x, q2) = exp(f (x)~(q2)) [9]. The function f i x ) has been calculated

in ref. [9]. In the case 1 - x << 1 the leading term of f ( x ) is (4C2/f12)ln(1 - x), but the calculation of f ( x ) is not restricted by the leading logarithmic approximation. The way of passing to the limit 1 - x << 1 in the calculations of f i x ) is given in our paper [4]. Thus in the case 1 - x << 1 both the double log terms as(q2)ln(1 - x) × ln(q2/2` 2) and the single log terms as(q2)ln(q2/2` 2) are calculated [4]. In the case 1 - x - 1 they are of the same order [9]. The uncertainty of the value ha gives the terms a s ( q2 ) ln ( 1 - x)<< 1, eq. (2). The function D also contains the double log terms ots(qE)ln2(1 - x) [4, 5].

As to the hadron wave function ~0(x',/~2) with large virtuality ~2 ~> 2`2 of the active quark c and its fraction of hadron momentum x ' close to unity 1 - x ' << 1, one can obtain it by starting from the phenomenological valence quark distribution

(A, Pt) with PA (pt) the relative longitudinal (transverse) momentum of the valence quarks. The exchange of hard gluons between the valence quarks [10] gives the contribution of the order a~(2`2/(1 - x ) ) l n ( 1 - x). These are the leading log terms connected with the coupling constant as(2`2/(1 - x)). Thus, the total wave function is [10] q0(x',/L 2) = ~(x' , /~2)j . Here ~ describes the interaction of the valence quarks separated by a small distance r << 2 ̀ [10] while J is connected with r - 2 ̀

- dA d2pt A j= j~q , ( ,p2), (4)

with pt 2 -2`2. Thus, the valence quark distribution enters the structure function through the factor J, eq. (4), which does not depend on x or q2. It can be expressed through the observable amplitudes, see [10].

E.G. Drukareo, E.M. Levin / Drell-Yan process at T --* 1

One can see that although ~/p2 _ 1//~4 ~ (1 - xh2/k 4 the main contribution to . 1 / t ~

(3) comes from finite kt 2 - X 2 / ( 1 - x) a and there is no infrared divergence. This happens because of the Sudakov suppression for the function Dq(X'/X, q2,/~2) at small/z 2. Detailed calculations are given in ref. [4].

For the calculations of integral of eq. (3) we introduce the usual QCD notation

1 k 2 ~ ( k 2 ) = ~ - ~ l n l n ~ - , r2 = ~ - ~ n f , C2= 4,

with n / t h e number of flavours. The integrand of integral (3) reaches its maximum value at the point (#2, x')

determined by the set of equations

1 - x ' 1

1 - x = 4C2[~(q2)-~( l~2(x ' - x))] '

4C2~(/~ 2) -4C2~(/~2(x ' - x)) - 2 = O, (5)

which can be obtained by using the explicit presentation of Dq [91 and q~ [10]. The solution of eqs. (5) (/x 2, Xo) is, up to the terms - as(q2)iasX2/(1 - x)

x0__x 1 - x ~1

*i = 4 C 2 [ ~ ( q 9 ) - ~ ( h : / ( X - x))] + 2,

= exp( - fl2/2C2),

= 0.034, for n i = 3.

(6)

(7)

Expression (6) for/~2 is identical to

~2

k 2 = (1 - x ~ rla' a = 1 -----e- (8)

Thus, the main contribution to (3) comes from k t 2 >> X 2. It insures the infrared stability of the process.

6 E. (7. Drukarev, E.M. Levin / Drell- Yan process at "r --} 1

For the transverse structure function of the meson with spin S we obtain

5/ exp(5-12 E ,2) D ( x , qZ) = 1 6 . ~ V C2 8 4C 2

X ~1-4C2/fl2 (1 - x ) ~ )

F(~I + 2 -(4C2/f12)ln(1 - e)) ~+(nc:/a~)O+~(1-~))(

y

\ l - x

l.(q2/X2) ] - 4vE)c

2,*1 2 + ½f12~ ~ a~ ~ f J s (1 - e) 1 +(3-4yE)/~2+(2C2/[~Z)3sl

(9)

Here

,= ) Y = + 16C2(1 - e) ' f0 = 1 , f l = ~ In r/ ,

while Js and e were defined by eqs. (4) and (7). The double log terms of eq. (9) coincide with those obtained earlier, see ref. [10]

and references therein. Since the single log contribution of the functions Dq and cp are known [9,10] and finite kt z dominate the process [4], eq. (9) also gives the single log terms and the preexponent factor. All effects due to large distances are contained in the factor Js 2 which can be expressed through the observables. Note that the power of 1 - x differs from that given in [5] because large kt 2 - )k2/(1- x) a dominate the process.

Note also that at 1 - x - 1 the meson structure function

D(x, q2)= fq~2(x',~2)D(x'/x, q2,~2)d~2dx '

(there is no need to extract one gluon as in the case 1 - x - 1) contains the unknown hadron wave function qffx', #2) at #2 _ h2. Thus, the structure function contains the unknown preexponent factor which depends on x and q2.

E.G, Drukarev, E.M. Levin / Drell-Yan process at • ~ 1 7

The series expansion of eq. (9) in powers of e leads to a simpler equation:

D ( x ' q e ) = 1 6 " - 9 v C 2 8 4C 2 r ( n + 2 )

x ( l _ x ) n + , c 2 / a ~ ( 71 ]~ ( 2 C 2 1 n z ( 1 - x ) ) ~--2Sj exp ¢~ hi(qVX~)

~ ( /~2 ) 4c2 . - - 1 + ~= 1-~ 16G(:-~) ---~-ln(:-~). (lO)

Now we calculate the longitudinal meson structure function. It is given by eq. (3) where Dq denotes the longitudinal structure function of the quark. One knows [9] that in order to calculate the latter one sacrifices the logarithm in the last cell of the ladder which describes the structure function in the planar gauge [11]. In this gauge the gluon propagator is

d.~(t, c) D~,~(I ,c)= 12 + i e

l.c. + c G d~,o(l, C ) = g~,o l . C

C = P + q ' ( q ' = q + P x ; q'2 = 0), (II)

while ! is the gluon momentum. Note that the irradiation of the real non-logarithmic gluon (fig. 2a) leads to the

appearance of the extra factor (1 - x). Indeed the phase space of the real gluon is limited:

f : - x d a ~ - l - x ,

(Sudakov variables: l = a eP + fl~eq' + It are used), while for the virtual gluon (fig. 2b) f d a e - 1.

Since in DIS q2< 0, we choose the frame with q0 = 0. Thus, the longitudinal y-matrices can be presented as

bx + O' bx - O'

E.G. Drukarev, E.M. Levin / DrelI-Yan process at ~ -~ 1

\ \ ~ / /

P p

,, ot /

p

Fig . 2.

The sum over longitudinal polarizations gives for any operator A

] t L A ] t L "~- - - (2/q2)(pAO'+ O'AP),

while for the vector f

(12a)

yefYL = --2 L . (12b)

Thus, the structure function is given by the graph of fig. 2b with the shown directions of the longitudinal matrices:

r dx' , /~2)im dF(_~f 2 ) DLq(X, q2,1~2)=j___~_Eq(X ,q2, f

. d41 1 ^ ^ - l)yffO dt,.(l) x J ~ i 4 ~ r a s C E ~ s S P P T . ( P "l)~'(f - ^ ~^,

1 X (13)

i2(p _/)2(f_ 1)2,

where f = P ( x ' - x ) + q' is the momentum of the scattered quark; the function E(x', q2) describes the radiation of the active quark. The latter is connected with the transverse structure function by the planar gauge equation

fEq(X , ,q2 , ,2 ) imdF(qZ(x ' - - x ) ) d x ' l (14) Dq(x, q2, ~2) = x ' - x x' 9r

E.G. Drukarev, E.M. Levin / DrelL Yan process at r--, 1 9

with d F the quark Green function in the planar gauge [11]. The LLA equation for Eq is thus

Dq(x, q2,/x2 ) = eq( x, q2,/~2 )dF( q2(1 _ x )) (15)

with the leading contribution coming from x' - x << 1 - x. It is the term P(x' - x) of momentum f which gives the leading contribution to

eq. (13). Thus,

DLq(X, q2)___ asC22rr 2 J,,[x dx,~q(X,,q2)ln(x, x)lmdF(q2(x, x)) . (16)

Using the explicit expression for d v [11]

dF(q2(x'--x))=exp( 2a~(q2~)lna(x'-3~r x) ) . (17)

(Here and later we omit the second argument of the function dF(KZ,(K • C)//C2); K" C/C 2 = q2). With the help of another LLA formula

ImdF(q2(x'--x)) = -C2as(qZ)ln(x '-X)dF(qZ(x'-x)) (18)

we obtain

2C2 ( I n ( l - x) ) 2 1 - x DLq(X'q2'lz2)='-~- 2 ln(q2/h2) Dq(x'q2'lz2) ~ j ( q 2 ) _ ~ ( ~ 2 ( l _ x ) ,

(19)

with Dq the transverse structure function. As to fig. 2a, its contribution is of the order Dq(x, qE)fda~a~- (1 - x)2Dq(x, q2)

since eq. (12b) leads to the factor 12 = -a~#,S in the numerator. Thus, in the region

ln(1 - x) 1 - x _< ln(q2/X2 ) (20)

The longitudinal structure of the quark is given by the diagram of fig. 2b and eq. (19).

The function given by eq. (19) can now be used for the calculation of the meson longitudinal structure function. The latter is given by eq. (3) where DLq from eq. (19) stands for Dq. Since DLq obtains extra powers of (x' - x), the position of the saddle

10

point is shifted:

E.G. Drukarev, E .M. Levin / Drel l -Yan process at • ~ 1

1 - x o = 1 t 1/( l -e) . 1------~-- ~ +------1 ' #~= h2( 7/+\I--xl ] (21)

Thus, the meson longitudinal structure function is

1 )1-4cj,2. 8C~ as(q 2) 12 1 _ _ _ ~ / 1 + ~ DL(X'q:)=D(x'q:)-fff2 ( l - x ) as (X2-~ Z x ] ) ] , + 2 \

(22)

with D(x, q2) the transverse structure function of eqs. (9), (10); for V and 3' see eq. (9). If ~(q2) >> 1 (but ((q2) < In 1/(1 - x)), eq. (22) takes the form

2 2C2 { as(q2) ):~(q2)'l- x DL(x, qZ)=D(x,q )-~22 / c q ( X z - ~ Z x ] ) _ (23)

Note that eqs. (22), (23) are valid until eq. (2) is true. For very small I - x - ~ 2 / q 2 D L

becomes larger than D [5].

3. The transverse m o m e n t u m distribution: the total cross sect ion

We introduce the usual Drell-Yan notation [2, 3]. The hadron A (B) momentum is denoted by PA(B)- The process is considered in the hadron c.m. frame; PA + PB = 0. The dilepton mass M 2 = s~-; 1 - z << 1. The active quark of hadron A (B) (fig. 3a) carries the fraction XA(B) of the momentum PA(B). Thus,

X A X B ~ T,

while the longitudinal momentum of the dilepton is

We introduce also [2]

1 q = ~[XA-- XBI~.

x r = x a - x B << 1, x v= Iql/{v~.

The cross sections do/dM2dq2tdxv; do/dM2dxF; d o / d M 2 are determined within the framework of LLA by the impulse approximation. This can be proved in the same way as it has been done for the case of DIS [4].

E.G. Drukarev, E.M. Levin / Drell-Yanprocess at r ~ 1 11

3.1. THE TOTAL CROSS SECTION

We begin with the calculation of the cross sections integrated over the transverse momentum d o / d x F d M 2 ; d o / d M 2. Note that they do not include the total structure functions of the active quarks Dq(X, q2/~2) but only their parts which describe the irradiation of the active quarks Eq(x, q2,/xz) given by eqs. (14), (15). The unification of the quark function Eq and the hadron wave function gives the hadron function

I a ( x , q2) = D(x , q 2 ) d F l ( q , ( 1 - x ) ) . (24)

Thus, we obtain the equation

d , , / d M 2 dxF = (4 2/9S eA(XA' B(XB, qq, = ~ ( 2 5 ) ot 137

with the functions L calculated with the help of eqs. (9), (10), (24). The cross section integrated over x F is

do 4~ra 2 { 1 + • 2 ~ [ 1 + "r ) dM 2 ~ OAk---f--,q )19B~---2--,q2 ( 1 - ~ ) d ~ 2 ( q 2 ( 1 - ~ )

× 22"- 1B (~1, ~1). (26)

For ~/ and D see eqs. (6), (9). Thus, for the K factor (the ratio of the DY cross section to that of DIS) we obtain

K( , , q2 ) = dF:( q2(1 - "r ) ). (27)

Eqs. (17) and (27) give the formula for the K factor which has been obtained earlier [6] from analysis of the moments of the structure functions. We have presented eq. (27) since it suggests a simple physical interpretation of the origin of the K factor: the lack of total cancellation between real and virtual irradiation of the final state quark in DIS. Note also that in our calculations the question of the argument of the running coupling constant a S did not emerge.

Let us discuss the accuracy of the equations obtained in this section. The leading corrections to eq. (11) come from the vertex graphs. They are of the order as(q2). During the calculation of the integral (14) we have neglected the region x ' - x - 1 - x ' which gives a contribution as(q2) ln(1- x). The functions D and d v in integral (14) are known up to these terms. Thus the next step to improve the accuracy of the K factor calculations should be more accurate calculation of the functions D and d F.

As to eq. (25), the leading corrections as(q 2) come from the interaction between the active quarks of the hadrons A and B in the upper cells of their structure functions - compare with the analysis given in [12].

12 E.G. Drukarev, E.M. Leoin / Drell-Yan process at "1" ~ 1

The other interactions between hadrons A and B give the contribution of the order Ots(q2)~2/q 2 at 1 - - r - 1 [13a]. For 1 - r<< 1 they give the corrections Ots(qE)()~2//q2)(1 - 'r) 2 [13b]. Thus, the impulse approximation (fig. 2a) is valid up to the terms - 1 / q 2, i.e. with this accuracy the interaction between hadrons is reduced to the interaction between the active quarks.

3.2. THE TRANSVERSE MOMENTUM DISTRIBUTION

Now we calculate the distribution d a / d M 2 d q2t d x. Note that the LLA of QCD is valid while qt 2 is not too small:

as(q2)ln(q2/q2t) << 1. (28)

There is the upper limit of q2 as well. Denote the active quarks from the upper cells of their structure functions which directly annihilate into a virtual photon as

PeA, PcB" Thus, the condition

leads to

(Pea + PcB); - S = S ( r - 1), (29)

q2 ~< S(1 - r ) . (30)

There is a still stronger restriction within our model (the single photon annihila- tion). Indeed, the photon transverse momentum qt should be compensated either by that of the radiated gluon or of the quark spectator. In the first case

(Pc^ + ecB - 1) ~ = s T ;

hence

s ( 1 - = s [ ( 1 - x A ) + (1 - + •,].

Since for the real gluon a , , f l , > 0 we obtain a , , fl,~< 1 - ~-. As qt 2 = l 2 = -a~e f l ,S we get the constraint

qt 2 ~< S(1 - ~.)2. (31)

In the second case, since the mass of the quark-spectator should be positive, /30(1- ~')S + qt 2 > 0. Eq. (29) gives us fld ~< 1 - - z and we come to constraint (31) again.

In order to calculate the distribution we follow the method developed in ref. [11] for the case 1 - z - 1 . We introduce the function M ( x , flq2, q2t); the product

f lq , qt ) = d F ( f l q 2 ) M ( x, flq2, q2t ) is the probability of finding the quark carry- S ( x , 2 2

ing the fraction x of the momentum of the hadron with virtuality up to flq2 provided its transverse momentum does not exceed the value qt 2. The latter is

E.G. Drukareo, E.M. Let)in / Drell- Yan process at ,r ~ 1

assumed to be limited by constraint (31). One can see that

M(x, q2/(1 - x), q]) -- D(x, q2/(1 - x)).

Using one more LLA equation for the E function

dF(q2(1 -- x ) ) D ( x , q2(1 - x)) = fill(X, q2),

13

(32)

(33)

with

T(x, q2,q 2)

4(72-1-x dfl do = e x p ----'~2 Yq2/q2(l_x) ~ fq~Sp-t-~2S

4Qfx dfl Bs

The DDT equation [11] in our case is

do 4~'ot 2 0 q2 DB XB, . dq2t dMEdxv = 9S4q 20lnq 2 DA XA' 1 --'r 1--'r ]

(38)

(36)

(37)

o r

X(x, q2(1 - x ) , q 2) = D(x, q2/(1 - x))T(x, q2, q2t ),

one can write

d F ( q 2 / ( 1 - x ) ) M ( x , q 2 / ( 1 - x ) ) = ~ ( x , q2t/(1-x)2 ). (34)

The LLA equation for the function X is

qt/q ( l --x) ~O£ ~ /fl.eS Ot£"l- ]J£

X X(x - a£, fl£q2, q2). (35)

We are interested in fl = 1 - x. The solution of eq. (35) is

l-- l ( l 1 - - x ] d F \ I - - x ]

[4C2 f l - x dfl£ [ddpas(p)] x e x p [ fl-"'~,l q2/q2(l_x ) fl£ JX 2 ,13 -t- fl~'---""~ '

14 E.G. Drukarev, E.M. Levin / DreU-Yan process at "r ~ 1

A - - ? - -

Fig. 3.

As shown in ref. [14], the form factor T is described by the ladder graphs (fig. 3b) in a rather broad region where the product (asln(q2/q2t))nln(q2/q2t) can become much larger than unity. One can see that this is true for the case 1 - r << 1 as well. Thus, T can be presented as

{ 2C2[ ' cts(q2), q2 q2 l n 2 ( l _ ~ . ) ] } T ( ' r ' q 2 ' q 2 ) = e x p - - - ~ 2 [ m a ~ - f f ~ m - ~ - l n ~ - 2 q t ln(q2/)t2 ) . (39)

Note that T(~-, q2, q2(1 _ r)2) = dFl(q2(1 _ T)2). For ln(1 - z ) - 1 eq. (39) coincides with the results obtained in ref. [14]. In the case asln2(q2/q2t)- 1 we obtain

osln "'Jl 40) T( ' r ' q2 'q2 )=exp - -~2 qt

For In(1 - ~') - 1 eq. (40) is identical to those given in ref. [151. Unlike the case 1 - z - 1 one should take into account the derivatives of the D

functions in eq. (35) as well those of T. The relative contribution is

@ D / lnq 2 = ( 4C2/ flz ) Da s ( q2/(1 - x ))ln(1 - x ) ,

OT/ln q2 = _ (2C2/fl 2) Ta s (qE)ln(q2/q2) . (41)

Thus

do 4~raa4c2as(q2t)'m q2(1 - r ) 2 - ~ D A (x A, qt2 ) dq 2 dM2 dXF 9 •2 S4q2t qt 1 ~ ,r

The LLA should calculate the upper non-logarithmic cell of the structure function ladder

x D B ( xB' 1 q2t-~']]T2('r'qz'q~)" (42)

is valid until ln(q2(1 - "r)2)/q 2 >> 1. If ln(q2(1 - ~')2)/qt2 - 1 one

E.G. Drukareo, E.M. Leoin / Drell-Yanprocess at r ~ 1 15

accurately. The calculations show that eq. (39) is true in this region as well. Since T(r, q2, q2(1 _ ~.)2)= dFl(q2(1 _ r)),

d r

dq2dM2dxF 4~r42 4C2 2 1 q2(1 - r) 2

9 -~2 ct~(q )S--~t21n q2 DA(XA'q2(1--r))

×Da(xB,q2(1 -- r))d~2(q2(1 - r ) ) , (43)

for q 2 q2(1 _~.)2. We conclude the section by stating the mean transverse momentum

( q2} = 4C2/flEas( qE )q2(1 _ ~.)2. (44)

4. The lepton angular distribution: no oscillatory energy dependence

The angular distribution of the leptons is given by the equations obtained in ref. [7]. The distribution, integrated over the azimuthal angle is [7]

dODy dM2dxFd(CosO ) ~ DT(1 + cos20) + DE(1 -- COS28). (45)

The functions DT, L a r e given by eqs. (9), (10), (20), (21). Thus, in the region as (q2) ln (1 - x)<< 1, D L << D T and the role of the second term of eq. (43) di- minishes while 1 - r becomes smaller. But for smaller 1 - r it increases, since D L >_ D T at 1 - "r - ~k2//q 2 [5].

The oscillatory energy dependence of the distribution of do~d9 was predicted in ref. [8]. Oscillations were expected to emerge in the terms which describe the interference between the amplitudes of the transverse and longitudinal virtual photon production. Since these amplitudes are complex, a phase difference appears. We show that the latter is small and does not lead to oscillations. This is true for the case 1 - r - 1 as well.

The vertices which describe the radiation of the transverse (longitudinal) virtual photons Its,) =/"t~)e i~,~*~(q2). Thus, the interference term W a - / ~ t / ~ + F , F * - Ft° F~cos( St( q 2) - 8 e(q2)). If

Im F t Im F~e

Re F t Re E2' (46)

we have ~t = ~£" Note first of all that the imaginary parts emerge only if the interaction between

the active quarks C A and C B is taken into account. Indeed the virtualities of the active quarks are negative. Thus, the self-energy corrections have no imaginary part. Hence I m G ( P 2) = ~rP~(p2)dF(P2). The extra term which is proportional to ~ [17]

16 E.G. Drukarev, E.M. Levin / Drell-Yan process at r--* 1

Ca

c/

9

/ \ CA

Fig. 4.

does not have a pole Pc 2 = 0 and does not contribute the imaginary part. The irradiation of real gluon also can not move the quark to the mass shell, but increases the absolute "value of its virtuality: p2 = t i c s , tic < 0; (Pc - l ) 2 - ( t ic - f l e ) S < 0

since fl~e > 0. Thus, one should include gluon exchange between quarks C A, C a (fig. 4a).

This interaction leads to the appearance of the phase 8 t ( ~ ( q 2 ) a r c t g ( ~ / p 2 t ) >__ 1 [8] with pt 2 the transverse momentum transferred in interaction between quarks c A and c B. The active quarks obtain momenta (PeA + P) 2 and (PcB -- p)2 after rescattering. One can see that p 2 should be small enough to let the radiation of the real gluons compensate the Sudakov damping of the quark Green function d F

Thus,

X2 (47) p2~ 1 - ~ "

~t(~) - arctg ~ ( 1~--~_2 T ) •

Since only the upper cells of the structure functions are sensitive to the photon polarization, we obtain Eta,)= Eta)(1 + i~(~2//[1- x])) and no phase difference, according to eq. (46). In order to obtain the phase shift one should include the interaction between the active quarks once more. This is the first QCD correction to

E.G. Drukarev, E.M. Leoin / DrelI-Yan process at "r ~ 1 17

the vertex function, which is of the order as(q 2) in the planar gauge [11] (fig. 4b).

Thus, the phase difference

3 t ( q 2 ) - 3 ~ ( q 2 ) - a s ( q 2 ) ~ ~ - x <<1 (48)

does not lead to oscillations.

5. Conclusion

We have calculated the main characteristics of the Drell-Yan process at I - z << 1. The special feature of the case 1 - ~- << 1 is that not only the power of exponent, but also the preexponent factor can be obtained for all the calculated functions. Indeed, all the distributions include the hadron structure functions. These can be calculated with the declared accuracy since the large values of the spectator quark transverse momen ta k 2 >> X 2 dominate the process - eq. (8). Thus, the functions are infrared

stable. The meson structure functions are calculated in sect. 2, eqs. (9), (10), (17), (21). The earlier calculations of the transverse structure functions have been per- formed by replacing the lower limit of integration over k 2 by k2 [5]. Thus, they gave only the double log terms. We have also obtained the single log terms and the preexponent factor. Note that the shift of kt 2 into the region k 2 >> X 2 leads to a modificat ion (though, rather small) of a power of 1 - x.

The obtained structure functions are used in the calculations of the DY cross sections d a / d M 2, d a / d M 2 d x F , eqs. (25), (26), with the same accuracy. The ratio

of the D Y and DIS cross sections coincides with that, obtained in ref. [6]. Note, meanwhile, that eq. (27) gives a simple physical interpretation to this ratio.

The transverse momentum distribution d o / d q 2 dx F d M 2 is also obtained in sect. 3. We calculated the effective form factor T for the case 1 - r << 1. In our case it depends on the relative value of ln(q2/q2t) and In(1 - I-); eqs. (40)-(43).

The lepton angular distribution involves the longitudinal structure functions, calculated in sect. 2. In sect. 4 we also dispute the recent predictions of the oscillatory energy dependence of the angular distribution.

We are indebted to A.H. Mueller, R. Petronzio and M.G. Ryskin for talks on the subject; also to G. Stepanova and L. Valyamova for typing the manuscript.

References

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(1983) 1550; Phys. Lett. 130B (1983) 223

18 E.G. Drukarev, E.M. Levin / Drell-Yan process at r ---* 1

[5] A.H. Mueller, Phys. Reports 73 (1981) 237 [6] P. Chiappetta, T. Grandow, M. Le Bellac and J.L. Meunier, Nucl. Phys. B207 (1982) 251 [7] C. Lam and W. Tang, Phys. Rev. D21 (1972) 2712;

J.C. Collins and D.E. Soper, Phys. Rev. D16 (1977) 2219 [8] J.R. Ralston and B. Pire, preprint ANL-HEP-CP-82-67 [9] Yu.L. Dokshitzer, ZhETF 73 (1977) 1216

[10] S.J. Brodsky and C.P. Lepage, Phys. Lett. 87B (1979) 359; Phys. Rev. D22 (1980) 2157 [11] Yu.L. Dokshitzer, D.I. Dyakonov and S.I. Troyan, Phys. Reports 58 (1979) 269 [12] S.I. Troyan, Ph. Thesis, LNPI, Leningrad (1981) [13] (a) A. Khalafi, preprint DAMPT 82/89;

(b) E.G. Drukarev and E.M. Levin, preprint LNPI-873 (1983) [14] M.G. Ryskin and S.I. Troyan, Sov. J. Nucl. Phys. 35 (1982) 756 [15] G. Parisi and R. Petronzio, Nucl. Phys. B154 (1979) 427