Soft-giuon corrections to the drell-yan process

IL NUOVO CIMENTO VoL. 77 A, N. 2 21 Settembre 1983

Soft-Gluon Corrections to the Drell-Yan Process.

) I . I~RIED~AN

Physics Department, 1Vortheastern University - Boston, MA

G. PA~CHER~ a n d Y. :N. SRIVASTAVA

Istituto Nazionale di ~isica Nucleate .~aboratori Nazionali di ~rascati - ~rascati, Italia

(ricevuto il 21 Aprile 1983)

Summary. --The 4-momentum distribution, dd~(K), of the multiple soft- gluon emission is studied using a Bloch-Nordsieck approach. An asymptotic separability is found: d ~ ( K ) _~ d~(K+) d~(K_) d2~(K• where K . = = K o ~- K 3. Their effect in ~ pair production is investigated and explicit expressions for the double differential cross-section are obtained. The slope of the mean transverse momentum with total energy is calculated and compared with data. Our mean scaling curves are also in good agreement with experiment. Further tests are suggested.

PACS. 13.85. - t tadron-induced high- and supcrhigh-encrgy interactions, energy > 10 GeV. PACS. 12.35. - Composite models of particles.

1 . - I n t r o d u c t i o n .

The Dre l l -Y~n (1) (DY) m e c h a n i s m for p r o d u c t i o n of l ep ton pairs in h a d r o n -

h a d r o n sca t t e r ing is b y n o w f i rmly es tabl ished (~,8). Severa l papers h a v e been

(1) S. DRELL and T. ~I. YAN: Phys. l~ev. Lett., 25, 316 (1970); Ann. Phys. (SV. Y.), 66, 578 (1971). (2) L . M . LEDERM~.~: Proceedings of the X I X Internationa~ Con/erence on High Energy Physics, Tokyo, 1978 (Tokyo, 1979), p. 706. (3) G. ~ATTHIA]~: I~iv. NUOVO Cimento, 4, No. 3 (1981).

165

166 ~. FRIED~IAN, G. PANCttERI and Y. N. SRIVASTAVA

devoted to comput ing impor t an t q u a n t u m chromodynamic (QCD) corrections to the basic D u Born ampl i tudes , a l ter ing the absolnte value of the cross- section (~.7) as well as its dependence on the t ransverse m o m e n t u m (8.~o).

I n this pape r we present the effects of summed-up soft gluons on the D Y

spect rum. The formal i sm we use is based on the classic Bloch-Nordsieck me thod of utilizing classical currents to obta in the required summat ion. For this pur-

pose a coherent -s ta te approach was developed (n) and used successfully in a va r i e ty of problems: e+e - annihi lat ion (12), K• of jets (13), deep

inelastic mome n t s (14) and mean scaling (15). A compar ison of this fo rmal i sm with the renormal iza t ion group was also made in ref. (1,).

I n this approach the quan t i ty to be computed is the 4 - m o m e n t u m prob-

abi l i ty distr ibution d*~/d4K, where K is the m o m e n t u m of the QCD radiat ion. We first show tha t there is an a sympto t i c separabi l i ty of this dis t r ibut ion:

d4~(K)~_d~(K+)d~(K_)d2~(K.)~ where K+_-= Ks4-K3. This allows us to obta in ~ ve ry simple expression (eq. (4.3)) for the differential cross-section

da/dXl dx~ which can be used to s tudy scaling violations. Especia l ly for meson- nucleon and ~p initial states~ where the (( sea ~ contr ibut ion is expected to be

small, i t is possible to do a m o m e n t analysis reminiscent of t ha t for deep inelast ic scattering. E v e n if the (( sea )) contr ibut ion is not comple te ly ignored~

large n momen t s probe the infra-red (IR) region and thus our soft-gluon ex-

pressions m a y be useful. As a by-produc t we obtain a compac t formula for (K~,(xt, x~, s)} (eq. (5.3)). Since no (( intrinsic ~; (or (( pr imordia l ~)) K• has been included in this formula, a t bes t we can hope to obta in the (dimensionless) der iva t ive ~(K2~(Xl, x~, s)}/~s. I f we pe r fo rm such an analysis for ~-nucleus

(4) G. PARISI: Phys. Zett. B, 90, 295 (1980). (~) G. ALTARELLI, R. ELLIS and G. ~][ARTINELLI: Nuol. Phys. B, 143, 521 (1978); 146, 544 (1978); 157, 461 (1979). (s) J. KUBAR-ANDRI~ and F. E. PAIG]~: Phys. Rev. D, 19, 221 (1979). (7) G. CURCI and M. GRECO: Phys. Lett. B, 92, 175 (1980). (s) Yr . L. DOXS~ZlTZE~, D. I. D'YAKoNov and S. I. T~OJAN: Phys. Lett. B, 78, 290 (1978); 79, 269 (1978). (9) G. Pa~ISI and R. P]~TRONZIO: Nucl. Phys. B, 154, 427 (1979); K. KAJA~TI~ and J. L1NDFORS: Phys. Lett. B, 74, 384 (1979); J. CLWYMANS and M. KU~ODA: Phys. JLett. B, 80, 385 (1979); J. C. COLLINS: Phys. Rev. ~Lett., 42, 291 (1979). (lO) F. HALZEN and D. M. SCOTT: Phys. Rev. D, 18, 3378 (1978); 19, 216 (1979). (I1) ~[. GRECO, F. PALUMBO, G. PANCHERI-SI~IVASTAVA and u SI~IVASTAVA: Phys. .Lett. B, 77, 282 (1978). (13) G. PANCgERI-SRIvASTAVA and Y. SRIVASTAVA: Phys. Rev. Lett., 43, 11 (1979). (~) G. CmccI, M. GR~co and Y. SRIVASTAVA: Phys. Rev..Lett., 43, 834 (1979); Nud. Phys. B, 159, 451 (1979); PLUTO COLLABORATION: Phys. Lett. B, 100, 351 (1981). (14) G. PANCHERI-SRIVASTAVA, Y. SRIVASTAVA and M. RAM6N-MEDRANO: Phys. Bev. D, 2533 (1981). (15) G. P~NCn~I-S~IVASTAVA and Y. S~IVASTAVA: Phys. tlev. 1), 21, 95 (1980).

SOPT-GLUON CORRECTIONS TO THE DRELL-YAN PROCESS 167

scattering data obtained by the l~A3 group (ms), we find 0.0032 for this deriv- ative, in agreement with the experimental value (16).

An approximate formula for d2~(K• obtained earlier (~5,17) is used to discuss mean scaling. Data from the CFS group (ms) are compared with the theoretical

expression and good agreement is found.

2. - Nota t ion and fac tor iza t ion formula .

In this section we introduce kinematics for the DY process and set up the formalism through which soit gluons will be incorporated in the cross-section. Our approach rests on two conjectures which have been proved valid at least in the leading logarithmic approximation. They are tha t the emission of IR

e gluons is finite to all orders in the QCD coupling constant a, and that it ex- ponentiates as it does in QED, with non-Abelian effects appearing only through the energy dependence of ~, as obtained via the renormalization group equation.

To zeroth order in a,, the cross-section for the DY process, quark -~ anti-

quark -* ~+~- is given by

(2.1) d~~ - - 4 ~ 2 e ~ f f / ) ' - q) d~q 9Q 2

where q is the 4-momentum of the ~ pair, q,q" = Q2, and io' is the momentum

of the qcl system. DY cross-section is obtained from eq. (2.1) by folding in the quark and antiquark probability distributions

(2.2)

where the sum extends over all flavours and s is the square of the total P.m. energy of the incoming hadrons. The D u cross-section does not account for many features of the ~ pair distribution, particularly the Kx-dependenee. To lowest order in ~ , this is obtained through the process q~ -* (~+~-) + gluon for which

(is) j . BADIER, J. BOUCROT, G. BVRGVN, O. C~LOT, PH. Ctr~PENTIER, M. CROZON, D. DECAMP, P. DELPIERRE, A. DIOP, R. DUB~, B. GAND01S, R. HAGELBERG, ~/[. HANS- ROUL, W. KIENZLE, A. LAFONTAINE, P. LE Du, J. LEFRANg0IS , TH. LEI~AY, G. I~IAT- THIAE, A. MICHELINI, PH. MIN]~, H. NGUYEN NGOC, O. RUNOLFSSON, P. SIEGRIST, J. TIMMERMANS, J. VALENTIN, R. VANDERttAGItEN and S. WEISZ: Phys. Lett. B, 89, 145 (1979); 93, 394 (1980); presented at the International Symposium on Lepton and Photon Interactions at High Energies, Fermilab, August 1979. (17) G. PANCHERI-S~IvASTAVA and Y. SRIVASTAVA: Phys. t~ev. D, ]S, 2915 (1977). (18) j . YOH, S. HERB, D. Hol~I, L. LEDERMAN, J. SENS, H. SNYDER, K. UENO, B. BROWN, C. BROWN, W. INNES, R. KEPHAR% T. YAMANOUCHI, R. FISK, A. I~o, H. J(iSTLEIN and D. KArLA~: Phys. t~ev. Lett., 41, 684 (1978).

1 ~ ~ . FRIEDMAN, G. PANC~RI and ~. N. SRIVASTAVA

the differential cross-section reads (~)

(2.3)

where

dQ~dz - 9Q's' q~ s ' / 1 - - z 2' +z'- i-Q~/k~)'

~! - - Q2

s ' ~ / ~ ' ~ 4E ~ ~ k ~ 2

is the gluon c.m. momentum and Z = cos O, where 0 is the scattering angle in the c.m. ~or SUs,oo~o,, C F -- ~.

When this cross-section is folded with the (~ primitive ~) par ton densities~ Q~-dependent par ton densities can be defined (~8), which to first order in ~, agree with those obtained from deep inelastic scattering (DIS). In a previous paper (~4) an expression was obtained for the par ton densities which incorporates the leading soft-gluon corrections to all orders in a,. We would like to show similar momentum-dependent pa t ton densities for the Drell-Yan process as well.

I t is quite useful for what follows to employ the light cone variables k+ ---- k(lq-Z). Equat ion (2.3) m a y then be recast as

(2.~) do" . _ 2~,,e~(7~, 1 (1 -- k + [ ! / ~ ) ~ -~ (1 -- k_/V~) ~

dk+ dk_ 9~' k+k_ 1 - - ~+/v"~- - ~- / 'V~

Equat ion (2.4) allows ns $o form a probabil i ty distribution in k_+:

d ~ 1 do. C,~, 1 0 - - k+lv~) ~ + 0 - - k _ l ~ ) ~ (2.5) dk+dk_:o .o(Q~)dk+dL - zc k+k_ 2 [ 1 - - k + l v f d - - k _ / V ~ 7]

In eq. (2.5), I1~ divergence in both k+ and k variables is exposed. Also, i t is seen tha t the distribution d ~ ( k + , k_) is separablv in the II~ region. This would be quite impor tant to us later in showing the separability of the summed- up gluon spectrum.

Also, the collinear hard singularity is rather t ransparent in eq. (2.5). I f we insert a regulator mass m and consider Z - - > • so tha t k• 0~ we find

(2.6) d ~ ~ , ~ ( l n 2 ~ l + (1- -k/~)~ d k - ~ \ -m-/ 2k '

if we retain only the singular par t in m. In the appendix, we consider the n-soft-gluon emission process quark

t (p~) ~ quark (p~)-> (~t+9-) -[- (n soft gluons). We sum over n, color-average

(19) G. ALTARELLI, G. I)At~ISI and R. PETRONZI0: Phys. Zett. B, 76, 351 (1978).

SOFT-GLUON CORRECTIONS TO TI~E DRELL-TAN P R O C E S S 1 6 9

and add virtual contributions to obtain (at the leading order) eqs. (A.12)-(A.14) :

(2.7) - I d, - - d , ~ ( K ) ~o~(q) ~ ' ( P ' - - K - - q)

where %o~=(q)= 4~ro:~/9Q ~ and the 4-momentum distribution of the QCD ra- diation is given by

(2.8) /~(K) ~ - -

with

(2.9)

d4~(K) -:/" d4x d,/i: 3 (2~), exp [ i K . x - - h(x)],

h(x) exp [ - i k ' x ] )

and ds~(k) defined in eq. (A.11). How, as is usual, we inser~ the quark/antiquark probabilities in eq. (2.7) to obtain for the observed hadronic cross-section

- Z dy, y,[],(y,)],(y,) - , \d q/h,hr~+~-+x 9Q ~

In writing eq. (2.10) we have neglected any intrinsic transverse momentum for the partons. We shall return to this problem later.

In the next section, we obtain approximate expressions for .E(K), which we then use to discuss the double differential cross-section as well as the transverse-momentum behavior.

3. - Asymptotic separability and approximate forms for d4~(K).

In this section, we examine the structure of dd~(K) in detail and t ry to exploit the consequences of taking the zero-mass limit, m~-~ 0, in eqs. (2.8) and (2.9). As an inspection of these equations shows, the distribution of dd~(K) exhibits a mass singularity, which can be dealt with by introducing a small regulator (mass) ~ g ~ m ~. One can then show that, in the asymptotic l i m i t / ~ 0, the distribution separates into the product of independent distributions in the light-cone variables, i.e.

(3.1) dd~(K) ~ o d~(K+) d~(K_) d2~(K•

where d~ (K• is finite as ~u-~ 0.

The above separation is obtained by. first considering the function h(x),

12 - I I N u o v o Cimento A .

170 ~ . FRIED~IAN~ G. PANC~II]RI and Y. ~. S~IVASTAV~-

defined in eq. (2.9), in the qcl e.m. frame:

i -~ CP~dk~'- < ~ ~ - k 3 ~ dk3 t , I 1 - - exp [ - -~ (k+t - q- ( 3 . 2 ) = - / " -v7C7~

k_t+)] Jo(k•

where 2E is the c.m. energy and t_+ : t-4-xa, k+ = Ir To s tudy the behavior of h(x) as /z -+ 0, it is convenient to introduce the functions

(3.3)

and

(3.4)

hr(t, xa) =-- h(t, x3; x• = O) =

CF r dk~ 2

v~." + ~", l , -V~'-~'.

hz_(x• =-- h(t = O, xa---- 0; x • ot(k~)[1--Jo(k.Lx•

such tha t h r is logarithmically singular, whereas h• is regular and goes to a constant as # - + 0.

F rom eq. (3.3) we see tha t hr ~ log # as /t --> 0. We now proceed to also determine the behavior of h• and the relationship of h to h~. and h•

F rom eqs. (3.2) and (3.3) we find

(3.5a) [h - - hL] :

r dk3 - Jo(VZx~)] | - j

- - exp [--~ (k'+t_ q- k'--t+)]

! with /r = v/k~ A- # -b ka. As # --> 0 the r.h.s, of eq. (3.5a) -+ (# In #) ](x). Thus (~/~tt)(h- h~)~ log # and is divergent as # -~ 0. We may improve the behavior by considering h - h L - - h z . We find

(3.5b) -- /~ ~-~ [h - - h, . -- h.] =

dk~ [ CF~ ~(~111-- Jo(VYx.)] f ~ / ~ 1 - -

- ~,/~-.

exp

S O F T - G L U O N C O R R E C T I O N S T O T H E D R E L L - Y A N P R O C E S S 171

~ow, as g --> 0, the r.h.s, of eq. (3.5b) ~ f l ( x ) and hence h ~ h~ ~- hz ~- g(x., t+, t ) ~ 0(#), where g(x., t+, t_) is independent o f / 6 and m a y be neglected.

One m a y argue tha t , while the introduction of h A resulted in the improved behuvior exhibited on the r.h.s, of eq. (3.5b), hA is not singular as /x -+ 0 and one should keep g as being potential ly comparable to h A. Howeve 5 we find from eqs. (3.2)-(3.4) tha t g--~ (t+ -~ t )x2 , . Thus it would contribute only

2 when t+, t and x• are ~ll nonzero. In this case the h~ term is also present, which goes as log # and hence dominates the behavior of h as # - + 0. On the other hand~ if we integrate eq. (2.8) over dK+ dK_ in order to get a d~/d~K A distribution we obtain a 8(t+)8(t_) which makes g = 0. Howeve b as can be seen from eq. (3.3), h~ is also zero and, therefore, h = hA -{- 0(/~) for this distribution. Thus g only contributes when h~ is also present, and is thus negligible as/~ -+ 0. In contrast, when looking at d~/d~K• we find h~ is the leading ~erm in h and hence must be retained.

The foregoing discussion leads us to conclude tha t we m a y set h(x)= = k ( t , x~) A- hA(x.) as # -+ 0.

Neglecting terms of order ~ log # in hr(t, xa): we then obtain

(3.6)

,u 0 0

.E

+ + t l - o(kj )l 0

From this, the separation property (3.1) immediately follows with

d K ( d r (3.7) d ~ ( K • •

and

(3.8)

~ 2 ~

r C~ I" dk~ ~ k~ ) I" dk exp[iK• ( ' . Jy(l-exp[-ikt])]

/z 0

d2~(Kz) : d"K~ l d!x• exp [-- iK• x - h . (x . ) ] . J (2~)~

~or K• < 2E, eq. (3.7) can be integrated to give

(3.9)

with

E2

fdk g

172 M. FRIEDMAN, G. PANCHEI~I and Y. N. SRIVASTAVA.

The distributions defined in eqs. (3.7) and (3.8) are all normalized to 1, as is d4~(K), and they all behave as ~-function distributions in the ~ -> O limit. I t is a check upon the approximations which led to eqs. (3.1), (3.7) and (3.8) that , if we integrate d4~(K), as given by those equations, in the momentum variables Ks and K• for Ko < E we obtain the well-known (2o) energy distribution

(3.1o)

with

f d'~(K) d3 K (d~(K+) d~(K_) dKo ~-~ (Ko~ ~-~ d ~ ( g 0 ) = j ~ - =3 ~K~ -- ~ r(~)\E/

~2

2G rdk~

as it should be. A comparison with QED calculations when ~ is a constant and Ce----1 then leads to a more accurate definition # _~ m~/4.

So far, in eq. (3.9), only the soft-gluon contribution has been incorporated. A more complete cMculation must, however, also consider the collinear hard gluons which are also known to exponentiate (7). As suggested in ref. (~4), these corrections can be applied in such a way as to reproduce the first-order result while, at the same time, summing hard collinear ghions to all orders. We propose, for d~(K+) and d~(K_), the following expression:

(3.11)

For fl--> O, this expression approximates to the first-order result, eq. (2.6). I t should be noted that, whereas d4~(K) is a relativistic invariant, the individual probabilities d~(K+) and d~(K_) are not and one has to specify the frame ia which K~ and 2E are calculated. We postpone this to the next section, where explicit expressions for the differential cross-sections are derived and compared with results from deep inelastic scattering.

4. - Differential cross.section and comparison with DIS.

For computing the double differential cross-section da/dxldx2, after the primitive quark densities are folded in, eq. (3.1) is very convenient. The variables xl.2 are defined as

Q2 2q~ x l x 2 : T = --s and x l - - x 2 = zF-~ - ~ ,

(~) E. ]~TIM, G. I)ANCHERI and B. TOUSCItEK: Nuovo Cimento B, 51, 276 (1967).

S O F T - G L U O N C O R R E C T I O N S TO T H E D R E L L - Y A N P R O C E S S 173

where q[ is the longitudinal momentum of the lepton pair in the hadronie e.m. system. I f we integrate over q• af ter a little algebra we find

d 2 o , (4.1) dxl dx~

1 1

�9 d~(u~) d~(u2) ~(y~ - - x~ - - u~) ~ (y~- - x ~ - - u~) ,

where u~,~ : K• Choosing : E ~ : (s/4)yly2, we get (ignoring for the moment the collinear hard term)

(4.2) 1 1

s dxl dx2J~o~ ~ 9xlx~ "~ C ~ ~ J Yl J Y2 xl x2

[ 1 ] 2 / x~\~,.-i / x2~P~.-1

The spectrum fl,. depends on yl and y~. Since the integral is peaked a t y~ ---- x~, we shall evaluate fl~. a t y~ ---- x~. fl~. is then a function of Q~ alone as in DIS (14). We m a y thus write the following expression (which accounts for the eollinear hard gluons as well):

dG (4.3) s

dxl dx2

where

-- 9xlx2eXp -~ C~,as . e~.

�9 {Z , (x l , ~ ) i , ( x ~ , fl~) + L(x~, f l~ ) i , ( x l , fl~)},

(4.4) I , ( x , 8) = - -

I

dy ,

x

Comparison with s tandard approaches leads to the obvious identification of I~(x, fl) as the momentum-dependent quark densities. Also, i t agrees with an earlier paper (14) where the same expression was obtained for (spacelike) DIS.

I f DIS par ton densities are to be inserted in eq. (4.3) for the D u process, the extra factor exp [(z/2) CFas] survives. As has been discussed by various authors (51), i t helps much to alleviate the problem of the absolute value of the cross-section.

As a consistency check, if we let a s -+ 0, so does fl and we obtain from

(~1) G. PARISI: Phys. Lett. B, 90, 295 (1980).

174

eq. (4.4)

(4.5)

M. I~RIEDMAN, G. PANCHERI and Y. N. SRIVASTAVA

I ,(x, 8) ~ /,(x) .

Thus, the DY result is recovered in this limit. Before proceeding further, we want to stress tha t the above expressions

are valid for small energy losses, i.e. for x _~ 1 and small q• For large 1 -- x values and large q~, the hard, wide bremsstrahlung contributions m a y become dominant and must be added to this calculation. I t now appears possible to do a moment analysis similar to tha t employed in DIS. I t is especially suited for meson-nucleon or p~-initiated lepton pair processes from which the <~ sea >> contribution can either by completely eliminated (by suitable linear combinations) or at least minimized.

Consider, for example, the difference

[ da~-, dan+, ] (4.6) L(-'(xl, x~, 8) -~ s [dxldx~ ~ , j ~-/~-'(x~, fl)G(-'(x,, 8)

from which the sea contribution cancels out. The N- th moment in the variable x, (for fixed x,) then gives

(4.7)

where

and

(4.s)

I

1" dxl ~ ( ) K(N, x,, 8) = J ~ xl F~- (x~, fl) G(-'(x2, fl) ~-- U(N) <xf(8) ) (/(-'(x2, 8) ,

0

1

0

N(N -}- 1) ] exp [18]r(lv) 1 + <xf(8)) = 27~r(N + 8) (N + 8)(~ + 8 + l i "

Here x~ is a redundant variable and it m a y be useful to integrate over some fixed range in x2 to increase the statistics. In fact, let

(4.9) K(.Y, fl) = f dx2K(N, x2, fl) = C(N) O(fl)(xf(fl)> . x~o

S O F T - G L U O N C O R R E C T I O N S TO T I t ~ D R E L L - Y A N P R O C E S S 175

~ow consider the ratio of 2 different moments:

(4.1o) fdx~ x~ fdx2fs (da~-'/dx~ dx2) -- s (da'~+'/dx~ dx2)} 0 ~2o

fdx~ x~ fdx~{s(da~-,/dxl dx2) -- s (da~+'/dx, axe)} 0 ~ o

K(iV, ~) ~(iV) <x~(~)> -- X(M,,~) ~'(M) <x~(/~)>

Variations in in R(IV, Mfl) vs. fl are given by our model, independent of the ~ primitive )> quark densities. Larger values of iV, M sample larger x and thus the Ii~ region for which our formulae should be reliable.

5. - E n e r g y d e p e n d e n c e o f < q ~ ) .

(5.~)

where

In this section we derive an expression for the mean of the squared transverse momentum, (q~), of the lepton pair. Let us first assume that the partons inside the hadron carry no intrinsic transverse momentum. Then, eq. (2.10) gives

2 ~(yl, Y2) = ~_, e~{/dyl) ],(y2) -[- (yl +-~ y2)} . i

I t can be shown that the probability distribution d4~(K) satisfies the following general relation:

(5.2)

where

f K 2 d 2 K ~ d '~ (K) [~ 3 H ( k ) d ~ ( K 0 - k, K 3 - ka) k 2 d4K J~" dKo dKa

[~2 K d4~(K) d2~(Ko, ga) = j a z ~ �9

Inserting eq. (5.2) in (5.1), we obtain

da(s, q) <q~(xl, x,, s)) dxldx~ --

47ra ~ f ? - - ~-~ | d y l [ d y 2 ~(Yl

J J , Y~)fd3~(k) k ~, d ~ ( P ~ - - qo--k , -P'

1 ~ M. F R I E D M A N , G. P A N C H ~ R I and ~ . N. SRIVAST~V~k

For x~, x~ > �89 one can exchange the limits of integration and obtain the sum rule

(5.3) a~(~, q) f d,n(k)kl d~(s, q § k) (q~(xl, x,, s)) dx~ dx, dx~ dx,

Equation (5.3) is quite general. The only assumptions made in deriving eq. (5.3) are that i) the collinear hard- and soft-gluon contributions can be fac- torized from the (~ hard ~) cross-section and that ii) the parton densities have no (intrinsic) transverse momentum. Thus, given the experimental differential cross-section alone, eq. (5.3) allows one to test the above faetorization hypothesis quite clearly. I f there is an energy-independent, intrinsic transverse momentum, eq. (5.3) is no longer correct. However, it can be used to compute the energy-dependent terms in (q~,~, i.e. to predict the slope (8/Ss)(q2,~. A detailed numerical test based on the above is in preparation and will be presented elsewhere. Here we present a simple application to the ~-nucleus data.

In the soft limit

d3~(k) k~_ _~ CF~, dk+ dk_. 7g

Then eqs. (4.3) and (5.3), for x~, x~>�89 give

(5.4)

1 1

4~, ~ e~(~dy~ I,(y~, fl~) ~dy~i~(y~, ilL) -4- (x~*--~x~)} t Xl ~S

t

where ~, has been assumed to be a constant, for simplicity. For ~-nucleus case, valence quarks alone seem to work very well, at least

for x1,~>0.2 (a.~e). In this approximation then we have

(5.5)

1 1

. Sdyl I,~(yl) Sdy~ I~(y~) <q.(xl, x~, s)> ___ - - s

3z I,~(x~)I~(x~)

Using the parameterization presented in ref. (16), we obtain for the slope at x~ = x~ = ~ ~ 0.5

(5.6) ~(q~') ~ , _ 0 . 5 = o .ola . 8S

We require that the model also provide a consistent value for the absolute cross-section. The factor K, which is experimentally known (le) to be _~ 2, in the model is given by K ~ exp [ w By requiring K ~ 2, we obtain

SOFT-GLUON CORRECTIONS TO THE D R E L L - Y A N PROCESS 1 7 7

~ ~ 0.33 and

(5.7) ~<q~->~s ~=~=o.5 -~ 0.0033.

This value is in good agreement with the slope 0.0029 measured in ~-nucleus scattering at x~ -- x~ -- 0.275. In consideration of the gross approximations made in obtaining eq. (5.7), this result is quite encouraging, ~s it is of the correct order of magnitude. A more refined analysis is in progress.

6. - Mean sealing and the transverse-momentum distribution.

In this section, we shall present a phenomenological analysis of the transverse-momentum distribution of ~ pairs produced in lop collisions. We shall show that the data exhibit scaling in the mean and that they can all be fitted by the same universal curve derived from the soft-gluon emission formula. We analyze the data by K~eTd~ et aL (22), at 400 GeV/c incident proton energy, for two different values of the invariant mass, including the T region. The remarkable fact is that, although the mean transverse-momentum squared <q~,> changes in ~he resonance region, the normalized transverse momentum distribution for the two different mass regions is fitted by the same dimensionless function of the dimensionless variable z• ~ q•177 This follows from the separability of the cross-section into a (( soft ,> and a (~ hard )> part and from the asymptotic factorization of the BN function dd~(K) intod ~+ d~_ d~(K•

As shown in sect. 2 and 3, one can write for the normalized ~ pair distribution the expression

(6.1) 4 ~ ( K • - - [ d~x• d2K• j (2~) 2 exp [-- i K • h(x• E)],

where q• ~--- K~ is the transverse momentum of the ~ pair and

(6.2) E) exp [ + x j ) .

An approximate closed-form solution of eq. (6.1), discussed in ref. (~5), is given by

(6.3) d ~ ( K • (2~)-1 fl• [ K• ~-l+~j2jf [ Kz asK• -- 2 2A r (1 +

(2~) D. M. KAPLAN~ R. FISK, A. ITO, H. JOSTLEIN, J. APPEL, B. BROWN, C. BROWN, W. INNES, R. KEPHART, K. UENO, T. NIrAMANOUCHI, S. HERE, D. HOM, L. LEDERMAN, J. SENS, H. SNYDER and J. YOH: Phys. 2ev. •ett., 40, 435 (1978).

178 ~. FRIEDMAN, G. PANCttERI and v. N. SI~IVASTAVA.

where 9if, is a modified Bessel function and where fl• and E V ~ are defined through the large- and small-x~ behavior of h(x., E), i.e.

fl• In iEx•

h(~• ~),., ~xlfd~(k) ~ ~ I~• ,

.Ex• --~ c ~

.Ex• --> 0,

and we have used the leading logarithmic approximation to define the func- t ion fl• which contains the (integrated) dependence upon ~s. ~ot iee tha t fl• is the t ransverse-momentum analogue of fiT. In the above approximation schem% one does not really need to know the details of the single soft-gluon distribution d3~(k), excepting tha t i t has a logarithmic singularity as k -+ 0.

The appearance in eq. (6.3) of the quant i ty E%/_4 is what ensures t h a t d ~ ( K • exhibits sealing in the mean. In fact, f rom eq. (6.1), one obtains

(6.4) <K~_> =fd3~(k) k ~, = 2E~Afl•

Both fl• and A can be separately calculated in QED (17), where the soft-photon spectrum is known. In QCD, there are some yet unsolved questions which make a computat ion of A and fl• problematic, i.e. the behavior of gs in the deep infra-red region and the folding of parton scattering processes with intrinsic momentum dependences inside the hadrons.

The inclusion of an intrinsic transverse momentum in our picture can be phenomenologically accomplished by considering fl• and fl• as empiric param- eters to be determined from the data. Thus we use experimental values in expressions obtained from eq. (6.3), as in the following dispersion formula:

(6.5) <q• _ 1

A(fl• ~ v/<q2.>__<q• -- W ~ F ( f l • ~ 1)~2__1

iF(Z• + �89

In fig. 1 we show the renormalized transverse-momentum distribution as obtained from the data of ref. (28), plot ted in terms of the variables z• We have taken <qz> ~ 1.2 GeV for the mass variable M ~ ~ (5--6)GeV. For the T region, we use <q• -~ 1.35 GeV (18). The two sets of da ta indeed fall on the same curve and they can all be fitted by the function (1/z•177 with

d @ V ~ I ' ( f l • + � 8 9 ~ /8_ �9 �9 dz• = ~ f l • 8 z~z :Z (~.ds,_,tZT) ,

where

.V"Y_P(/7~/2 + - z , = r ( / 7 . / 2 ) 21) z ~ .

80FT-GLUON CORRECTIONS TO THE DRELL-YAN PROCESS 179

lo ~ t . ~

~% 10-I

10-2~__-

10 ̀3 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 z•

Fig. 1. - A p lo t of t he normal ized t r a n s v e r s e - m o m e n t u m d i s t r ibu t ion in t e rms of t he va r i ab le zj=q~_/(q• for t he reac t ion p p - - ~ + ~ - + X at 400GeV/c : o M ~ = ( 5 - - 6 ) G c V , a M~v = ( 9 - - 1 0 ) G e V . The da t a were ob ta ined f rom rcf. (~*). W e h a v e chosen ( q . } = 1.2 GeV for t he mass va r i ab le M ~ ---- (5--6) GeV. I n t he T region, we choose (q~} = 1.35 GeV.

The data for (q~} and (q*,} in the two mass regions suggest ft . ~ 13. Wi th this choice, one obtains the solid curve in excellent agreement with data.

In a previous publicat ion (15), i t was shown tha t distr ibution (6.3), expressed as a funct ion of z~, fits the t ransverse mom en tu m of pions in

c+e- -+ = -~X and pp ~ = - b X .

IS0 M. FRIEDMAN, G. PANCHERI and Y. N. SRIVASTAVA

I t was also shown that eq. (6.3) was consistent with an experimental fit to ~-pair production in a wide x, range. Recently, new evidence has been presented for mean scaling in the processes (23)

o

lop --* K,~ pp ~ = - ~ X , ~p --* r c ~ X .

Thus mean scaling appears to be characteristic of most inclusive hadronie and semi-hadronic processes. Our analysis strongly suggests it to be a consequence of a factorizable collective effect like soft-gluon emission. This hypothesis is further confirmed by the fact that mean scaling is known experimentally to occur for almost all (34) one-dimensional distributions, and not just for the transverse momentum. Such an occurrence is indeed a natural consequence of the asymptotic faetorization of d4~(K), eq. (3.1).

7 . - C o n c l u s i o n .

We have presented here a detailed account of soft-gluon corrections to the DY processes. The formalism is borrowed from the Bloch-lgordsieck approach in QED. This approach is direct, intuitive and physically appealing. Based as it is in terms of probability distributions, it meshes nicely with the

underlying probabilistic parton picture of the DY process. As far as we are aware, for the first time the complete soft-gluon distribu-

tion d4~(K) has been discussed. The asymptotic separability in terms of the light-cone variables K• and K• is a new result. I t allowed us to obtain an expression for (q~,) as an integral over the measured energy and longitudinal distributions alone. This remarkable property can be used to test factorization itself. A partial test was at tempted here with positive results.

The details of qz-spectrum were examined using an approximation to

d2~(Kz) which showed mean scaling. Even though (q~) changes appreciably in the T-resonance region, we found that the same mean scaling curves work

(2s) B. BATYUNYA, I. BOGUSLAVSKY, I. GRAMENITSKY, R. LEDINCKY, L. TIKIIONOVA, .~.. VALVAROVA, V. VRBA, D. JERMILOVA, V. PHILIPPOYA, V. SAMOJLOV, T. TEMIRALIEV, S. DUMBRAJ8, J. ERVANNE, E. HANNULA, 1:~. VILLANEN, ~. DEMENTIEV, I. KHORZA- VINA, E. LEIKIN, A. PAVLOVA, V. RUD, I. HERYNEK, V. KOL, J. ~IDK~, V. ~IM~K, L. ROB, I~. SUE, J. PATO~KA, G. KURATASlIVILI, A. HUDZHADZE, V. TSINTSADZE, T. TOPURIYA and S. LEVONJAN: Scaling in the mean and associative multiplicities ]or the inclusive ~eaetions ~p--*K~ and ~p--*AX aS 22.4GeV/c, JINR preprint, Dubna E1-80-316 (1980). (3~) The one exception is the stange-baryon distribution. W. Dv~woonIv,: private communication.

SOFT-GLUON CORRECTIONS TO TttE DRELL-YAN PROCESS 181

within and wi thout this QS range. This at tests ~o the basic soundness of the approach.

More work needs to be done to include the var ia t ion of a and some calculable (nonsingular) hard corrections.

APPENDIX

In this appendix we wish to exhibi t the origins of the exponent ia l form for the soft-gluon emission. Le t us consider a quark of m o m e n t u m 21 and an an t iqua rk of m o m e n t u m p~. We are in teres ted in the process

(A.1) q(Pl) + ~t(P~) -~ ~t+~ - -[- ng

where ng means n gluons. T h e q and ~t come f rom two different hadrons. One calculates the t rans i t ion ampl i tude for the process, squares and sums over final states (of color) and averages over the spin and color of the incident quarks. One can show tha t the leading logari thmic terms are then the same as would have been obta ined f rom an Abelian theory with a classical current j~,(k), where k is the m o m e n t u m of the emi t t ed gluon. This is so because the non- Abelian graphs wi th three and four gluons are nonleading and thus do not ap- pear in the soft l imit . Thus the cross-section for the process of eq. (A.1) m a y be wr i t t en as

(A.2) dab . . . . ---- da(qq ~ i~+~t - + n soft gluons) =

( �9 dSq ( 2 ~ ) ' 6 ' ( P ' - - ~ k , - - q ) (2~)3~

I n eq. (A.2) s' is the to ta l energy squared in the center of mass of the two quarks, Mo(P'-- ~ k,) is the( ( elastic ~) ampl i tude with its a rgument shif ted to

i allow for the m o m e n t u m carr ied off by the gluons, q is the to ta l m o m e n t u m of the ~ pair .

To get the cross-section for the emission of an y number of g h 0 n s , we wish to in tegra te over each of the variables k, and then sum from n ~ 0 to oo I t is convenient to do this by performing the in tegrat ion subject to the condit ion t ha t the to ta l gluon 4 -momentum is K and then performing ~ .

Thus eq. (A.2) becomes

(A.3) dO.brems i [j/~(~i) [2 ] = -

4 (2~r)32c% '

182 M. FI~,IEDMA.N~ G. PANCttEI~,I a n d Y. N. SRIVASTAV~

(A.4)

and

1 /" d'x _ 1 [ d3k ]" dgb .... -- 2~,J(-2~)4 d 'K exp [iK.x] ~ . . L(2=)'2k Ii,(k)p exp [-- ik.x] �9

�9 I T r I M o ( P ' - - K ) J 2 d3q (2J r ) ' 5 ' (P ' - -K- -q )

(A.5)

where

d 1 ~ ddx ab .... = ~ s ' J ~ ddK exp [iK.x] exp [-- hB(x)].

.1_ Tr ]Mo(P'-- K)I ~ d3q 4 (2~)32~%

- - - - (2~)'a'(P'-- K - - q) ,

(A.6) ~" d~k

h~(x) = - j ( ~ ) ~ Ij.(k)l ~ e~p [ - ~k. x].

Equation (A.3) is the product of two parts. One describes the scattering without soft-gluon emission, while the other describes the soft-gluon emission. In fact, the latter can be used to define the probability that there will be an emission of gluons with 4-momentum K. This distribution is given by

(A.7) d '~ (K) C k, , d 'K - - (27~)a2k i [1.(]r K -

whexe C will be determined below. we must require

This gives

(A.8)

For this to be ~ probability distribution

fd d~(K) ddK = 1.

The factor C is missing in eq. (A.3) and one has to multiply by it. I ts origins are, of course, due to the virtual soft gluons, which when included will just have the effect of multiplying eq. (A.3) by C. They are thus contained in the scattering term and one has

(A.9) mr ]M(P ' - - K)I2 --~ exp [-- h,]~ mr I M e ( P ' - - K ) ] 2 .

Thus in eq. (A.5) we must replace h~(x) by h(x) -= h~-~ h~. We then have

(A.10) f d3 k h(x) = (2~)~2k IJ"(k)l~ [1--exp [--ik'x3].

SOFT-GLUON CORRECTIONS TO THE DRELL-YAN PROCESS 183

I t will be convenient to define d3~(k), where

dak (a . l~) d3~(k) - - (2~)s2 ~ Ii,(k)[2.

~(k) has the physical significance of the average number of soft gluons of m o m e n t u m k.

We note t ha t h(x) of eq. (A.9) has no II~ divergence, the v i r tual soft-gluon exchange te rm h~ of eq. (A.8) providing the cancellation which is missing in hs (x ) of eq. (A.6). Thus we have, including both vir tual and real soft-gluon emission, for the cross-section

d4~ = f d 4 ~ ( K ) a~o,n(q) ~4(p, (A.12) ddq

4 ~ z 2 (A.13) a~o,.= 9Q ~ e~,

--K--q),

where ea is the charge of the quark in units of the proton charge, and

(A.14) d ' ~ ( K ) _ f d ' x d ' K J (2~)4 exp [ i K . x - - h(x)].

�9 RIASSUNT0 (*)

Si studia la distribuzione del quadrimpulso dd~(K) dell'emissione multipla di gluoni morbidi usando un approccio di Bloch-Nordsieck. Si trova una separabilit~ asintotica: d4~(K) ~ d~(K+) d~(K_) d2~(K• dove K• = Ko 5= K3. Si studia il loro effetto nella produzione di coppie di 9 e si ottengono espressioni esplicite per la sezione d'urto dop- pio differenziale. Si caleola la pendenza dell'impulso medio trasversale con l'energi~ totale e si confronta con i dati. Le curve medie di scala sono in buon accordo con l'espe- rimento. Si suggeriscono altri controlli.

(*) Traduzione a cura della Redazione.

rIOltpaBgH, CBH3aHHble C HCHycKa~HeM MEWKHX UJIIOOHOB, K n p o ~ e c c y ~ p e ~ a - ~ l a H a .

PeaIoMe (*). - - Hcnoab3ya no~xo~ Bnoxa-Hop~ca~a, accae;ayeTca pacnpe~e~eHne ~eTbIpex- m~nyar, ca, d4~(K), ~nz MHOgeCTBem~oro ncnycimm~a Marr, nx rnrooi~oB. O6~apyz<eHa acrtMrtroTrI~ecKan pa3~esiaeMOCTb: da~(K) ~ d~(K+) d~(K_) d2~(K.), r~e K~. ~K0 5=Ks. I/ICC$le~yeTot Bm~gaae HcrIyc~r Mgrrmx rsllOOrlOB Ha po~c~eHrie MIOOHHblX nap. Hony- aamTcn TO~a~Ie B~ipa~eHn~ Rzul )XBO~HOrO ~rlqbqbeperttmanbHoro nonepe~Horo ce~eH~z. B ~ c z ~ e T c a nar~o~ cpe~laero noaepe~Horo nMny~ca c noanofl ~neprHe~. lTosry~en- H~U] pe3ya~rar cpaBmmaerc~ c 3~cnepnMeaTam, rmn~l~ ~lanH~n~m Ham~ cpe~xRne ~pnB~ie c~e~m~Hra Taro~e xopomo corzacymTc~ c ~crIepnMeHTOM. rlpe~naramrc~ ~ononHH- TeYlbHbIe rlcc~/e~oBarltI~/.

(*) flepeeeOeno peOaKttue~.

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Soft-giuon corrections to the drell-yan process