Transverse momenta of partons and dimuons in QCD

Nuclear Physics B145 (1978) 24-44 © North-Holland Publishing Company

TRANSVERSE MOMENTA OF PARTONS AND DIMUONS IN QCD

M. GLOCK and E. REYA Institut fiir Physik, Universitfft Mainz, 6500 Mainz, West Germany

Received 3 April 1978

Intrinsic (primordial) transverse momenta of quarks and gluons are calculated as well as those arising from recoil (bremsstrahlung) effects, using only the well-known parton distributions as input. The intrinsic kT'S lie typically in the range of 150-250 MeV. Recent approaches using heuristic integro-differential equations for k T distributions of partons are shown to disagree with the results obtained by rigorous QCD calculations. The transverse momenta of dimuon pairs produced in pp -~ ~+t~- + X at the ISR can be solely explained by dynamical recoil effects, i.e., qq ~ (tz+~-)g and gq --, (~+#-)q, and no significant intrinsic transverse parton momenta are required. These dimuon transverse momenta show a pronounced energy dependence which could be easily tested at the CERN ISR. The only disagreement occurs for the average dimuon <p2 >, but not for (pT), observed in pN collisions. Possibilities to resolve this problem are discussed. Our results are also compared with previous theoretical analyses.

1. Introduction

It is well-known that quarks (partons) within a hadron should possess some intrinsic (primordial) transverse momentum simply because of the finite size of the hadron. The exact shape (e.g., its x dependence) and size of this momentum distribution is quite controversial [1 ], but it is generally accepted that the scale for this momentum is set by the hadron's mass. Furthermore partons acquire dynamically an additional transverse momentum in deep collisions due to recoil off radiated gluons. The scale for these recoil momenta is set by Q2, the characteristic momentum transfer of the process involved. This latter recoil momentum, which depends sensitively on the leptonic or hadronic reaction under consideration, is uniquely calculable in QCD and is much less controversial than the intrinsic 'primordial' transverse momentum. For deep inelastic lepton-nucleon scattering processes, the dynamical recoil transverse momenta of partons can be readily deduced from the familiar expressions [2 -4 ] for the fundamental cross sections (Wilson coefficients) of the relevant subprocesses, as we shall see below.

Similarly, the recoil transverse momenta of dilepton pairs produced in hadronic collisions have been uniquely calculated in QCD [ 5 - 9 ] , whereas there is some controversy on the necessity and size of the intrinsic primordial transverse momenta

24

M. Gliiek, E. Reya / Transverse momenta of partons and dimuons 25

of partons. Even here, however, one finds in the literature confusing heuristic approaches [10-13] which supposedly do describe these recoil phenomena. We will show that these heuristic calculations disagree with the results obtained by the rigorous QCD calculations for primordial as well as recoil momenta, depending only on the well-known longitudinal structure functions.

In sect. 2 we show how information on the primordial transverse momenta of quarks and gluons is already fully contained in the well-known (longitudinal) parton distribution functions. Although the exact expressions are more complicated, it will qualitatively turn out that for a quark distribution x q ( x ) ~ (1 - x) a (for x ~ 0.3), the average intrinsic transverse momentum is given by (k~-(x))~--m2x(1 - x ) / ( a + 1)

2 with m being the nucleo n mass. This implies that the maximal value of (k~_x)) (at x "~ 0.5) of valence quarks (a --~ 3) is about 0.06 GeV 2 , in other words x/(k 2) ~ 240 MeV. For gluons (a ~ 5)_~_d antiquarks (a --~ 7 -10) the primordial momenta are even smaller, namely x/(k÷) "~ 150 MeV. It is clear that these small intrinsic transverse momenta can not account for the observed [14-16] large transverse momenta of the dimuon pair in pp(N) ~ p + p - + X typically (p~)~:-pu- _~ 1-2 GeV 2 , which according to the Drell-Yan picture [17] should be (p~)U*u, ~ (k~-)q + (k~-)~. Rather, these momenta arise entirely from dynamical (gluon) recoil effects described in sect. 3. Here we also compare our QCD predictions at c.m. rapidityy = 0 with PT data, so far available only at y -- 0, and compare our results with other recent calculations and approaches. As we shall see, one should compare the y = 0 data with theoretical predictions at y = 0 and not with theoretical predictions averaged overy as has been done so far. Furthermore, it is of crucial importance to study several PT moments, in order to test the predicted PT distributions, and not just (PT) or (p2). The latter procedure leads to inconsistent and misleading results when compared with experiment. Finally, in sect. 4 we summarize our conclusions and discuss possible applications of these dynamical recoil k T effects of partons in other reactions, like inclusive hadron high PT processes.

2. Primordial and recoil transverse momenta of partons

2.1. Theoretical analysis

In order to obtain the relation of intrinsic, i.e., primordial, transverse momenta of partons to their (longitudinal) structure functions, let us repeat the classical variable analysis [18,19] in a covariant parton-model language [20]. Defining

=:; f Cq.X<pj ].(0)1 Jp>

= ~ [ -guy + quqv~ 1 p • q

- - - U ' (1)

26 M. Gliick, E. Reya / Transverse momenta of partons and dimuons

Fig. 1. Deep inelastic scattering in the free parton model.

it is obvious that the contribution of fig. 1 to Wuv is, for massless quarks,

Wuv ~ f d4k f ( ~ - ~ ) 8 ( k 2 ) 6 [ ( k + q)2]e(ko + qo)wt~v

f d3k f ( ~ f f ) f ( 2 k . q (2) = d ' ~ O + q2) w#u

for some Lorentz-invariant wave function f(k" p), in the lab. frame of the hadron, and where

+1 2 - wuv = qukv + qvku + 2kukv ]q g~z, •

To extract W] and W2 from W~u one contracts with guy and PuPu. Then

A -gUVWuu = -314' 1 + (1 + v2/Q 2) W~ , (3)

1 B=m2(1 + vZ/Q2) pUpUWuu = -W 1 + (1 + v2/Q2)W 2 . (4)

i.e.,

Wx - A ) , W2 2(1+v2/Q 2) ' (5) =½(B _ 3 B - A

R = aL --- (1 + v2/Q 2) W2 - W1 _ 2B o T [4] 1 B - A ' (6)

where v = p" q/m and Q2 = _q2. The structure functions FI and F2 are defined by

FI =-- roW1 , F2 - vW2 • (7)

From eqs. (2), (3) and (4) one easily obtains [19,20]

x -A = (1 + 4x2m2/Q2) 1/2 F(~), (8)

n.Z2 X2 1

B = 2 Q2 1 + 4x~-m2/Q 2 f d~' F(~')

1 1

m~4 x3 f d~' : d~" F(~") , (9) + 4 Q4 (1 + 4x2m2/Q2) 3/2 ~ ~,

M. Gliick, E. Reya / Transverse momenta of partons and dimuons 27

where

1

F(~) = (zrm2/2) f d~' f (~ ' ) , x = Q2/2mv,

2x (10) 1 + X/1 + 4x2m2/Q 2"

The key observation in relating (k 2) to the structure function F(~) is that the simple kinematics of fig. 1, where p = (m, ~), q = (v, 0, 0, x / ~ + Q2) and k = (/Co, 0, ko sin 0, ko cos 0), implies

k2 = 1 2(1 + v2/Q 2) (2k2 + 2vk° - ½Q2), (11)

while the term pUpVwuv in the integral leading to B is given by

pUff'wuv = m2(2k 2 + 2vko - ~Q2) .

This immediately leads to

2B 4(k 2 ) R = B --------A = 2(k-~) + Q2" (12)

This relation, depending on the hadron's free parton wave function only, has to be interpreted as the contribution of the primordial (intrinsic) transverse momentum of the partons to R. To leading order in m2/Q 2 , eq. (12) together with eqs. (8) and (9) yields

xm 2 1 (k~(x, Q2))prim --F(x, a2~) f dx' F(x', a 2 ) ,

X

(13)

where we have already indicated that in general the structure function depends also on the momentum transfer at which we probe the hadron. To get an idea on the magnitude of (k2) prim note that neglecting O(Q2/v 2) terms, mF(x) = F 2 ( x ) / x 2,

i.e., the mF for a definite parton distribution xq(x, Q2) obtained in leptoproduc- tion analyses should be identified with xq(x, Q2)/x2 where q = u, d, s, . . . . Note

{a] [b] Fig. 2. Lowest order diagrams for ~,*q ~ gq contributing to the longitudinal fermionic Wilson coefficient C~, i.e., to the recoil transverse momenta of quarks in eq. (16).

28 M. Gliick. E. Reya / Transverse momenta of partons and dimuons

that the above analysis leading to eq. (13) applies to any constituent wave function, and therefore holds also for gluons where now mF(x) should be replaced by xG(x, Q2)/x2 with xG(x, Q2) being the usual gluon distribution in hadrons.

Turning now to transverse momenta produced by recoil effects, these can originate from two different dynamical sources which depend on the quark and gluon content of the hadron. The first process is depicted in fig. 2 whose contribution to the nth moment of the longitudinal Wilson coefficient was shown to be [2,3]

4as(Q 2) C~ - 3rr(n + 1) (14)

Taking the inverse Mellin transform of C~ and convoluting with the quark distributions in the nucleon yields the well-known [2] dynamical quark contribution to R:

4% 1 ~ dx' [x'~2 , , Rqdyn - 31r xq(x, Q2) x "-~ t '~ ) x q (x , Q2). (15)

Here as(,Q 2) is the running strong coupling constant,

O~ (Q2) _ 127r 25 ln(a2/A 2) '

with A--~ 0.5 GeV. Assuming for the moment the validity of relation (12) also for the recoil contribution to R, one obtains the dynamical contribution to the transverse momenta of quarks originating from recoil of the struck quark (fig. 2) to be

a s x 2 Q 2 /~ d x ' , , (k~(x, O2~)dY n - - - - J - - x q ( x , Q2) (16)

~'¢ J q(q) 3rr xq(x, Q2) x x'3

Note, however, that strictly speaking R dyn is not completely due to the recoil kT of the struck partons since in calculating R one also considers, in addition to the recoil graph of fig. 2a, the diagram of fig. 2b which amounts to the recoil of the outgoing parton (jet) as well as the interference term of figs. 2a and 2b. Nevertheless, eq. (16) is as near as one can get to a QCD estimate of (k2) dyn of the struck parton which, as we have seen above, is not a completely meaningful (i.e., gauge-invariant) concept.

The second dynamical contribution to the recoil transverse momenta of partons

Fig. 3. Lowest order diagram for ~,*g --* qCt contributing to the longitudinal gluonic Wilson coefficient C~, i.e., to the recoil transverse momentum of quarks emanating from gluons in eq. (19).

M. Gli~ck, E. Reya / Transverse momenta of partons and dimuons 29

depends on the gluon content of the nucleon and originates from quark-pair production off giuons which is shown in fig. 3. The contribution of these diagrams to the nth moment of the longitudinal Wilson coefficient is given by [2,4]

2as (17) C~ = 7r(n + 1)(n + 2)

which, upon Mellin inversion and convolution with the gluon distribution, gives the following total gluonic contribution to R

(~q l d x ' r l x ~ 2 ( 7 ) 3 ] 2% /~dyn ec]) f - (18) = x'G(x', Q2).

" 'g rrF2(x' x ZL -2!

Trying to interpret this as a further contribution to (k2) dyn of the parton q emanating from the gluons in the hadron, one gets, on account of eq. (12),

t.92) ~dyn @s x2Q 2 /~ dx' (k~(x, - j - - ( x ' - x )x 'G(x ' , Q2). (19) ~'~ J'q(g) 2zr xq(x, Q2) x,4

x

However, the interpretation of this contribution to R as due to the recoil transverse momentum of the struck quark is even more questionable than that for the corresponding contributions of fig. 2. Disregarding this minor objection one can now write for the total dynamical recoil <k~)

(k~>dyn =/,/.2 \dyn + zl.2 \dyn (20) q x r'T lq(q) xr'T/q (g) ,

where the two individual contributions are given by eqs. (l 6) and (19). The total average transverse momentum of a patton is then simply given by (k 2 )tot = (k~)qprim

/z.2 \dyn + V~T~q , i.e., using eq. (13)

(k2(x, Q2))tot m 2 x 3 / dx' , , //-2 \dyn = - - x q(x , Q2) + \r'T'q (21) xq(x, Q2) x x'2

Since the diagrams of fig. 3 have to be interpreted as contributing to the (k~) of quarks, it is clear that (k2) dyn in deep leptonic processes is not as straightforward to calculate as (k2)q dyn and will not be undertaken here.

The average transverse momenta can now be uniquely calculated by using just the well-known (longitudinal) parton distributions in hadrons as input. Before doing this, however, let us first briefly consider the transverse patton momenta in the spirit ofrefs. [12] and [13].Defining

h(x, Q2) _- (k~(x, Q2))tot q(x, Q2)

I dx t 1 , as f d x ' = m2x2 f - ~ q(x , Q2) +-~ Q2x q(x', Q 2) J X r2

x x


1 dx ' + ~ a2x f ~7~ (x '-x)G(x' , a 2 ) , (22)

and recalling the standard QCD Q2 dependence of the parton distributions, one can evaluate Q2ah/bQ2 for eq. (22). The resulting expression differs from both corresponding equations of ref. [12] and of ref. [13] obtained by a heuristic consideration of the Q2 dependence of the recoil effects which, contrary to the above rigorous QCD results, leaves us with two free parameters, the transverse momentum cut- off X(x) and the input value of (k~ (x, ,02 ~ \tot ~0J,q at some arbitrary value Q2 = Q2. If Q2 ~ m 2 and for x -~ I, where the gluon distribution in eq. (22) is negligible, the integro-differential equation resulting from eq. (22) agrees with the one obtained in ref. [12] provided one chooses ~.(x) = x/2(1 + x2); the equation of ref. [13] dis- agrees for any choice of X(x). In general, for arbitrary values of x, the application of these heuristic equations [12,13] for calculating the Q2 dependence of (k2), using Feynman's relation R = 4(k~)/Q 2, is wrong. The use of these equations for calculating the transverse momentum of the dimuon pair in pp -+/2+/2- + X is also unjustified, since the fundamental subprocesses responsible for the dynamical recoil effects considered there differ from those of refs. [5,7,8] as is easily seen by inspect- ing the results of refs. [5,7,8] recapitulated in sect. 3.

2. 2. Numerical results

Throughout this paper we shall employ two different sets of parton distributions in nucleons for calculating the QCD predictions quantitatively. One set of distributions we shall use is based on counting rule-like input distributions at Q2 = Q2, which are very similar to those suggested in ref. [21 ]. The x as well as Q2 dependence of the valence distributions is well-known and uncontroversial, while for the SU(3) symmetric sea and gluon distributions we take as input [21] at Q2 = Q~ _~ 1.8 GeV 2

x~(x, Q2) = 0.147(1 - x) 7, xG(x, Q2) = 2.412(1 - x) s . (23)

The full Q2 dependence is then fixed by QCD [21,22]. The other set of parton distributions used correspond to the dynamically [23]

calculated QCD predictions [24]. These are obtained, with the help of well-known renormalization group techniques, by assuming that at low resolution energies, Q2 =/22, hadrons consist of valence quarks only, i.e., x~(x,/22) = xG(x,/22) = 0.

These two different (extreme) sets of counting rule-like and dynamical patton distributions represent roughly the upper and lower bounds of distributions, respectively, compatible with present experiments on deep inelastic lepton-nucleon scattering. The numerical calculations are greatly facilitated by using the simple parametrizations, as given in ref. [25], for the exact x and Q2 dependence of parton distributions as predicted by QCD.

In fig. 4 we show the predictions of eq. (13) for the intrinsic (primordial) aver-


0.06

GeV z

0.04

0.02

0.04

0.02

0 0

i

fla 2 'l prim

' I

(b2 ~,prim r ,T ,~

0.5

(k ~ )prim dv

O 2 =3

b 2 ~ prim ~ T ' g

Q2=3

0 0.5 x

Fig. 4. Predictions of eq. (13) for intrinsic (primordial) transverse momenta of partons where u = u v + ~, d = d v + ~ and ff = d = s = s ~ ~. Solid curves correspond to counting rule-like distributions and dashed curves to the dynamically calculated ones. All units are in GeV 2.

age transverse momenta of the various partons in a nucleon. The solid curves refer to counting rule-like distributions, whereas the dashed lines result from using the dynamical distributions predicted by QCD. The predictions at Q2 = 3 GeV 2 correspond to what one would expect from the so called 'naive' (Q2 independent) parton model. To illustrate the rather mild dependence of the intrinsic transverse momenta on Q2 as given by QCD, we also show the resulting (k2) prim at Q: = 100 GeV 2.

Since eq. (13) follows from purely kinematical considerations of the nucleon wave function as described by the covariant parton model, the shaded areas in fig. 4 represent rather general bounds on the allowed intrinsic transverse momenta of partons inside a nucleon. It should be emphasized that these intrinsic transverse momenta lie typically in the range of 150 -250 MeV and are certainly not as large as 5 0 0 - 7 0 0 MeV as commonly advocated in phenomenological analyses of transverse momenta of Drell-Yan dileptons and hadronic high-pT reactions,

The transverse momenta stemming from dynamical recoil effects as given by eq. (20) are shown in fig. 5. For illustration we also show the individual contributions stemming from the two relevant fundamental subprocesses 7*q ~ gq of fig. 2 and eq. (16), and 7*g ~ qCl of fig. 3 and eq. (19). Note that the Q2 dependence for


2

0 G e V 2

0 3

0

2

0

i 2 dyn

( k r ) u v

gq

~ dyn dv

T" uv

oz =~oo .

/I. 2\tot ~r~Tt dv

I 02=100 ~ / - ~ Q2=lO0 "

0.3

0 0.5 0 0.5 x 1.

Fig. 5. Predictions of eq. (20) for dynamical recoil transverse momenta of quarks, where the individual contributions of figs. 2 and 3 (eqs. (16) and (19), respectively) are separately shown for the counting rule-like distributions. The predictions for (k2) t°t axe due to eq. (21). The notation is as in fig. 4.

recoil transverse momenta is much more dramatic than for the intrinsic ones since, according to eqs. (16) and (19), their scale is set by Q2. In addition, we also show in fig. 5 the total transverse momenta of partons as given by eq. (21). In addition to their appearance in R, these transverse recoil momenta should be observable in leptonic deep inelastic semi-inclusive hadron or two-jet production.

Finally, to leading order, the expressions for R(x , Q2) ~ RPrim + Rdyn read, according to eqs. (12);(13) and (20),

= m..~ 2 4 x 3 / dx' , RPrim Q2 F2(x" Q2) x x'--T F2(x ' Q2) , (24)


and R dyn = Fd2Yn/F2 with

1 dx' = 4°t__2s X 2 / - - F2(X' ' Q2) p d y n --2L 37r x '3

x

ldx-- , , +2as x 2 ( ~ e 2 ) / x , 4 ( x ' - x ) x G(x, Q2). (25)

ff q x

These predictions agree of course with those given in refs. [3,4,26] where, in view of the scarce data, a reasonable agreement with experiment has been found, except perhaps at large values of x where the QCD predictions of R lie consistently below the data despite their large error bars. These conclusions, however, would be drastically altered if the more recent preliminary SLAC data [27] do indeed hold up where R is expected to be as large as 30-40%, instead of about 10%. Although only the dominant lowest twist-2 contributions to R have been considered so far, it is not conceivable that higher-twist operators could account for such large values of R, since these contributions are suppressed by additional powers of oqm2o/Q 2, with typically m o = 500 MeV, as has been quantitatively shown in ref. [3].

3. Transverse momenta of dimuon pairs in pp -+ p+ la- + X

The large transverse momenta of dimuon pairs observed in pp and pN collisions [ 14-16] cannot arise from the intrinsic primoridal transverse momenta of the quarks participating in the Drell-Yan fusion mechanism [17]. These can at most yield (p2)U+,- < 0.1 GeV 2, as discussed in sect. 2 and as is evident from the small values of /v2 \ p r im X,~T,q shown in fig. 4, while the observed transverse momenta of dileptons are about 1 -2 GeV 2. Rather, dynamical recoil effects such as q~ -~ (/a+p-)g and gq --> (/.t+bt-)q must account for the observed PT of the pair. The differential cross section for these fundamental processes, as shown in figs. 6 and 7, are given by [8,28]

d2oq~ _8o~2O~s l ( i t~ 2Q2g] dQ2d/ 27Q2 g2 ~+-:-+t [fi / , (26)

d2ogq _ot20ts 1 ( g t~ 2Q2i) (27) dQ2di 9Q2 g2 t~ g ~ '

with x / ~ - M being the mass of the virtual photon (dimuon pair), and g, i and are the usual kinematical invariants for the corresponding subprocess. To obtain the experimentally measured average transverse momentum of the muon pair at c.m. rapidity y = 0,

go_ / =d2pTpjfda OQ 2 dy dZpr y=O / d Q : d y y = o ' (P~)Iy=o (28)


"

p.- , I "1"÷

W Fig. 6. Lowest-order contribution of qq--, (~+u-)g to transverse dimuon momenta.

÷

÷

Fig. 7. Lowest-order contribution of gq ~ (t~+/~-)q to transverse dimuon momenta.

one has to convolute with the patton distributions in the hadrons h 1 (beam) and h2(target ). The constraint y = 0 implies

(T +XlX2 12 x~ - - 4 p 2 = 4 - - 4 r , (29) $ \ X 1 +X 2 /

where x/s is the total c.m. energy of the colliding hadrons, r = Q2/s, and Xl and x 2 are the fractional momenta of the partons participating in the fundamental subprocesses. The kinematical invariants of these subprocesses are determined by g=XlX2S and f o r y = 0

= ls(2r - XI.~T) , U = ls(2r - X2.~T) , (30)

with 22 = x~ + at . Performing the PT integration in eq. (28), using p2 = Et/g, one obtains for a given reaction h l h 2 ~/a+/l - + X

( a d2° : 1 PT) d - d a ~ y=o = ~ r dx 1 : dX2(1N/~XT)a &'~T

X 1 +X 2

d2oq~ d2o_

X ( x p x 2 ) + ( l ~ 2 ) , (31)

with the standard expression

d2a - 4~ra2 [/-/~q~h2(x/r, x/r, Q2) + (1 ~ 2)] dQ 2 dy 9Q 4

y=O

and where

(32)

hqhlh2." ~1 i Xl,X2, Q2)= XlX2 ~ e2qhl(xl , Q2)~h2(x2, Q2), flavors


H~glh2(xl,X2, Q2)=XlX2 ~ e2qGhl(xl,Q2)[qh2(x2, Q2)+~h2(x2, Q2)]. (33) flavor$

Note that in eq. (31) the substitution 1 ~ 2 has to be performed also in the expression for the cross section Ogq since Ogq(Xl, x2) = agq(i, a)4: Ogq(a, t') - Oqg(X 2, xl). The treatment of this crossing problem in ref. [9] is incorrect. The lower limit X/r for x 1 and x 2 in the integrals of eq. (31) is dictated by x 2/> 0. This also satisfies automatically the requirement xlx 2 >>- r.

3.1. Comparison with experiment

The cleanest test of the above predictions is clearly achieved if one compares with hadronic scattering experiments off single nucleon targets where no possible collective nuclear enhancement effects exist, in contrast to hadron-nuclei scattering. In fig. 8 we compare our predictions of eq. (31) with the recent pp ISR dimuon production data [15] at x/s = 52 GeV. Although few events in these data correspond to x/s = 28 and 62 GeV, the agreement with the theoretical transverse momenta originating from dynamical recoil effects (figs. 6 and 7) is good. To show the dependence of our results on different parametrizations of parton distributions, we have again used the two (extreme) sets of distributions as discussed in subsect. 2.2, namely the counting rule-like (solid curves) and the dynamical ones (dashed curves). These different results are partly due to the different choice of Q2 for the counting rule-like input parametrizations, i.e., Q2 = 1.8 GeV 2, in contrast to Q2 = 3 GeV 2 for the dynamically calculated distributions. It is clear from fig. 8 that the ISR pp data require no intrinsic primordial transverse momenta of partons whatsoever, in agreement with our general result of sect. 2 that primordial momenta can contribute to (p2) at most 0.1 GeV 2.

. . . . . . giilt > ~ = 52 GeV

% 2

e ~

0

>" ~ gq~' f q | CL

v 2 6 10 1~ 18 22

M (GeV)

Fig. 8. Comparing the ISR pp data [ 15] with the predictions of eq. (3 l) , where the individual contr ibut ions o f figs. 6 and 7 are separately shown. Solid curves correspond to count ing rule- like par ton distributions, dashed curves to the dynamical ly calculated ones.


As PT "-~ 0 the cross sections in eqs. (26) and (27) diverge (i.e., parallel emission) preventing us from directly comparing the PT distributions d2o/dMd2p T with experiment in an unambiguous way. Since only PT moments [5] are well behaved, we calculated (PT) as well as (p2) in order to compare indirectly the theoretical PT distributions with the experimentally observed ones. The agreement as shown in fig. 8 is indeed satisfactory. Furthermore we also have plotted the two individual contributions stemming from the diagrams in figs. 6 and 7. The q~ subprocess of fig. 6 dominates the average transverse momenta over a large region of the dilepton mass, in contrast to the rather small gluon-quark (gq ~ 7* q) scattering subprocess. Naively one might expect the q~ component to be much smaller than gq, simply because the nucleonic sea distribution x~(x, Q2) in eq. (31) is about 10 times smaller than xG(x, Q2). However, in taking PT moments the weight coming from daqfi in eq. (26) is on the average 20 times as large as the contribution from dogq in eq. (27). The q~ recoil process of fig. 6 plays therefore an essential role in explaining the transverse momenta of dimuons!.

As can be seen from fig. 9 the predicted energy dependence of (p-~) is very pronounced, which could be easily tested at CERN-ISR. Here, and for the rest of this paper, we have used the Q2 dependent counting rule-like parton distributions since they appear to be in better agreement with the data of fig. 8.

In fig. 10 we compare our predictions with the experimental results obtained from p-nucleus scattering at y = 0 [14,16]. The agreement for (PT)ty=0 is certainly illusory in view of the by far too low predictions for (p2)iy= o . It would also be misleading to suggest that large intrinsic transverse momenta of partons could account for the large value of (p2)ly= o, since (PT)1 y = o allows only for small contributions from the intrinsic momenta. Because of the agreement reached for the ISR pp data in fig. 8; it appears to be impossible to conclude that the PT distributions themselves, as predicted by the recoil diagrams of figs. 6 and 7, are too steep as compared with experi-

~ 3 ®

~ 2 o

B 1

~ 2 L~

, i , , , , , i , r ,

pp~R." ~" ÷ X

(GeV)

62

2 6 10 14 18 22 M (GeV)

Fig. 9. Predictions of eq. (31) for the energy dependence of the dimuon's transverse momenta, using the counting rule-like patton distributions.


3

2 o

O . v

0 2

o

0

Fig. 10. Comparing the pN data parton distributions.

, i r , , , , , r i ,

pN ~ J * p - * X

= 27.4 GeV

**++ +~

q ~ v • g

,..Y 2 6 10 1/, 18 22 M (GeV)

[14] with the predictions of eq. (31) with counting rule-like

ment. Rather, as we shall argue in subsect. 3.2, there are qualitative indications that collective nuclear enhancement effects might be responsible for the large value of (p~). This receives further support from the fact that perturbative higher-order effects in as are unlikely to account for this discrepancy as we shall discuss in a moment.

The situation is very similar for the total (longitudinal) cross section as shown in fig. 11. The agreement with the ISR pp data is acceptable although it should be noted that our predictions (solid curves) could be easily increased by 30-50% by just changing the QCD input momentum Q2 to, say, 3 or 4 GeV 2 and/or decreasing the renormalization mass scale A to, say, 0.3 GeV thus weaking the effect of scaling violations. For illustration we also have included in fig. 1 l the pN data which lie, by a factor 2 -3 , consistently higher than the QCD predictions. The predictions of the Q2 independent, naive parton model (dotted curves) agree approximately in magnitude with the data, but this is unsatisfactory not only because of the wrongM-dependence. Again, nuclear enhancement effects appear to be the only feasible mechanism to.account for this discrepancy. This is because, on the one hand, higher-order contributions in ~ to eq. (32) are unlikely to account for a factor of two in the cross section. On the other hand, it is entirely illusory [4,11 ] to 'explain' the pN data by adopting a SU(3) symmetric sea as large as x~(x, Q2) ~ 0.3(1 - x) 6 , i.e., f l dxx~(x, Q2)~ 0.04, in disagreement with deep inelastic lepton-nucleon data: the experimentally observed ratio o~/o" = 0.38 -+ 0.02 at low neutrino energies, for example, tightly constrains the sea component to

1

: dx x~(x, 02) < 0.02. (34) 0

A further possibility to enhance the pN Drell-Yan cross section relative to the pp


10 "31 , , , , ,

~ ~ 0 > ® 10 10-32 -33 ~ I _ "'~._.....,

"' . .?,>/t. e

~r

1U'--35 pp -~'~- • x ~ _ _ - "-... e ~

"~ I ~ = 28 GeV L

• ~ 52 GeV =

• ~ : 62 GeV

1 0 - 3 6 pN ~ - • X

4- 4s : 27 4 GeV

- - -J~ I I I - - J ~

0.1 0.2 0.3 0./-, 0.5 M / 4-g"

Fig. 11. Comparison of the total cross section with the Drell-Yan formula, eq. (32), where the solid curves refer to the Q2 dependent counting rule-like parton distributions. The pp data are from ref. [15] and the pN data are taken from refs. [14,16].

predictions shown in fig. 11 would be to assume a broken SU(2) sea [29], i.e., if(x) < fir(x), still satisfying the constraint equation (34) with ~ = I - ~(fi + ~r~. This being so because of the very different elm. couplings of up and down quarks. However, this SU(2) broken sea can clearly not account for the large discrepancy of predictions for (P~)ly=o with the pN data shown in fig. 10.

3.2. Comparison with other calculations for rapidity averaged transverse momenta o fd imuons

For the sake of clarity we will compare our results with recent at tempts [7,8] to calculate average transverse momenta of dimuons. In ref. [8], (p~) was averaged over all rapidities y , and compared with the y = 0 pN data. However, as we shall see y-averaged PT'S are smaller than those evaluated at y = 0. Moreover, in order to derive conclusive results one has to consider more PT moments, (PT) and (p~) for


example, to check the PT dependence of d2o/dM d2pT as discussed in sect. 2. For comparison let us recall the y-averaged (p~) of ref. [8]

f _ _ . 3f.hlh do a2as (Ix 1 dx2 (1 - -12J /*'q~ (Xl,X2 ' Q2) (p2) dQ2 = 27Q 2 Xl X2

r r/x 1

-16 × 16rt2 -7 (1 -- TI2)2J +/-/~glh2 (XA'X2' Q2)

X 1 - r 1 2 + 1 ( 1 - r 1 2 ) - 2r12 +(1 ~2 ) (35)

with r12 -= r/XlX2, and where the coefficients of the Drell-Yan kernels are simply obtained by calculating the p2 moments from eqs. (26) and (27). The usual Drell- Yan cross section is given by

do" _47rot2 j~ dXIFrrhlh2/ T_ ) )1 dQ2 9Q4 r xl L nq~ ~XI,xl ' Q2 + (51 ~ 52 . (36)

Let us also record the final result for the y-averaged linear transverse momentum (PT) of dimuons in h lh 2 -+ p+p- + X

do O~2~s~ j~ dx1 j ~ dx2 N~12(1 _ T12) (PT) OQ2 - 18Qa x~- x--T

r T/X 1

zzhlh2. 32 T12 1 X inq~ [ X l ' X 2 ' Q 2 ) I 4(1 -7"12)+ 3 1--T12A

+/-/~gqlh2 (xl ,x 2, Q2)[1 + -~(1 - TI2) 2 -- -~r12(1 - r12)] + (1 ~ 2)/ , (37) J

where the coefficients of the Drell-Yan kernels result from multiplying eqs. (26) and (27) by PT = ~ a n d integrating over dL The gq piece of eq. (37) was already considered in ref. [7].

Note that since we use throughout Q2 dependent patton distributions, we did not include the higher order q~ -+ ~+p- )g and gq ~ ~+p- )q contributions to the total cross sections of eqs. (32) and (36). The leading In Q2 parts of these higher- order contributions are already included in the Q2 dependence of the distributions [5,30] entering the lower order Drell-Yan formulae in eqs. (32) and (36), while a consistent treatment of the non-leading terms necessitates [26] the inclusion of higher-order effects (two-loop contributions to the 7's and to/3) in the Q2 dependence of the patton distributions. It is believed [26] that these partly cancel the


contribution of the non-leading terms in the q~ ~ (/a+/2-)g and gq ~ ~+/~- )q total cross sections.

The calculations of refs. [7] and [8] were done using naive (i.e., Q2 independent) parton distributions and, assuming the q~ -+ (/a+/a-)g component to be negligible on account of the small fraction of antiquarks in the nucleon, the suitably regularized contribution of gq-+ ~+/ . t -)q was therefore added to the lowest-order Drell-Yan expression in evaluating the total cross sections. In this way one has introduced an additional free parameter, the infrared regularization mass. In ref. [7], the leading as well as non-leading higher-order terms were added, while in ref. [8] only the leading in Q2 term was taken into account.

In fig. 12 we compare the y-averaged predictions [7,8] of eqs. (35) and (37) with the data at y = 0. Working this way [7,8] the theoretical predictions are about 20% below the predictions a t y = 0 as can be seen from fig. 10, where both theory and experiment are consistently compared at y = 0. Looking only at (pT), one deduces from fig. 12 that recoil effects (dominated by the qq graphs of fig. 6) do adequately describe the data. Perhaps an intrinsic (k T) per parton of about 0.1-0.2 GeV might be needed [7], in agreement with our results of sect. 2. However, considering only (p~) the disagreement of the recoil effects (solid curve) with the data suggest [8] an intrinsic (k~-) per parton of at least 0.5 GeV 2 ! Again, the situation is similar to the one encountered in fig. 10: the agreement of the QCD predictions with the ISR pp data (fig. 8) prevents us from drawing the conclusion that the predicted PT distribution d2o/dM d2px is too steep and that the agreement for (PT) is purely accidental.

3

o 2

A

V

0 2

>

o I

~ 0

i , i , i i , , , , ,

p N~Ij*IJ- ÷ X

q¥ = 27.t, GeV

~ * + +-.

...... i; ................. .".-".!.t'...,.~:,oo

...... ~ - ~ ' " ' . ~ na~ve par- 1

2 6 10 I~ 18 22

M (GeV)

Fig. 12. Comparison of the pN data at y = 0 [ 14] with the y-averaged predictions for <pT ) and <p.]-> of eqs. (37) and (35), respectively. The solid curves refer to the Q2 dependent counting rule-like parton distributions.

M. Gli~ck, E. Reya / Transverse momenta of partons and dimuons 41

Rather, if one does not question the so far successful parton model and QCD as a whole, one needs a mechanism which enhances (p~-) and d 2 a/dM dy, but not (PT), for dimuon production off nuclear targets and which leaves the successful predictions for proton targets (figs. 8 and 11) unaltered. Since, as discussed above, higher- order effects in as are unlikely to account for such large effects, the only feasible possibility left appears to be collective nuclear enhancement effects. That these effects can, at least qualitatively, remedy the above inconsistent results for nuclear targets can be seen in the following way. In the extreme case [31 ] collective nuclear effects cumulatively enhance the c.m. energy squared s = 2mPlab of a hadron-nucleon collision to sn = ns for a hadron-nucleus collision, where n is the number of nucleons which collectively interact with the incoming hadron. Thus, on the average, the effective rn'S will be smaller than T = Q2/s for a pure pp collision making d2o/dQZdy in eq. (32) larger [32], because antiquarks are probed at smaller values of x. This would enhance the theoretical curves as required by the pN data in fig. 11. In addition, a smaller value of r will substantially increase (p~) in eq. (35) because of the strong threshold suppression factor (1 - r12) a, whereas its effect on (PT) in eq. (37) will only be marginal because of the weak suppression factor (1 - r l2) 1 . Any serious quantitative calculation of nuclear enhancement effects has of course also to take into account the Q2 dependences of collective parton distributions predicted by QCD. In order to check this conjecture, it would be highly desirable to measure transverse dimuon momenta in pp reactions at x/s ~ 28 GeV and to see whether (p~) lies significantly below the pN Fermilab data. Just to illustrate the collective nuclear enhancement effects more quantitatively, note that the Fermilab data [14] have been obtained mainly from Pt targets. Taking [31] n "~ A 1/3, the effective energy X/~n of this experiment corresponds to about 52 GeV. From fig. 9 it is obvious that the PT moments at this effective value ofx/s follow the data in fig. 10.

For a purely illustrative purpose we also present in fig. 12 (dotted curves) the results obtained by (inadequately) using naive, Q2 independent parton distributions. However, the large difference in the normalization of both predictions will be reduced by adopting a larger value of 2 2 Qo, e.g., Qo = 4 GeV 2.

In closing let us mention that besides the PT moment analysis presented here and in refs. [7,8], PT differential cross sections a ty = 0 were studied in refs. [9] and [33 ] with an incorrect prescription for the ~ ~ fi crossing of the quark-gluon scattering contribution and in ref. [34] with the correct one. In these studies naive (Q2 independent) parton distributions are used and, except for ref. [33], the divergence at PT ~ 0 was not regularized. The regularization in ref. [33] affords a primordial transverse momentum of ~1 GeV. Note further that the dominance of the Compton gq scattering as emphasized in ref. [34] refers only to large values of PT in d2a/dMdp~. This implies that gq scattering dominates only higher PT moments, i.e., (p~-) for n > 2.


4. Conclusions

Using only the well-known parton distributions as input, we have calculated intrinsic (primordial) transverse momenta of quarks and gluons as well as those arising from purely dynamical recoil (bremsstrahlung) effects. The intrinsic (k2)'s lie typically in the range of 150-250 MeV and are therefore expected to play only a minor role in explaining transverse momenta of hadrons or jets produced in deep inelastic lepton-nucleon scattering, of dimuon production in hadronic collisions, or in purely hadronic high-pT processes. In addition, there are (dominant) purely dynamical contributions to the transverse momenta of partons which originate from recoil effects, such as ~,*q ~ gq and 7*g ~ qq as shown in figs. 2 and 3. Apart from their appearance in R = alia T, these transverse recoil momenta should be observable in leptonic deep inelastic semi-inclusive hadron or two-jet production experiments. Previous approaches using heuristic integro-differential equations [12,13] for the parton's transverse momenta turn out to disagree with these rigorous QCD calculations.

Since intrinsic (primordial) transverse momenta of partons are almost negligible, the experimentally observed large transverse momenta of dimuons produced in hadronic collisions have to arise almost entirely from the dynamical (giuon) recoil effects such as q7:1 ~ ~+/~-)g and gq ~ 0a+/~ - ) q depicted in figs. 6 and 7. Indeed the predictions are in good agreement with ISR pp data for (PT) and (p2) a ty = 0, indicating that even the PT distributions d2o/dMd2pT themselves are in fair agreement with experiment. The subprocess q~l -~ (,u+/a-) g plays the dominant role in explaining the PT data. In order to reach conclusive results, it is of essential importance to compare several PT moments with experiment, e.g., (PT) and (p~-), to make sure that d2o/dMd2pT is indeed correctly reproduced. This is because the infrared divergent (parallel emission) cross sections of the various subprocesses pre- vent us from directly comparing d2a/dMd2pT with experiment in an unambiguous way. QCD predicts also a pronounced energy dependence for (p~-) which could be easily tested at CERN-ISR.

When comparing the QCD recoil momenta with dimuon data obtained from proton scattering off nuclear targets, the predictions for (p~) are far too small, whereas the predicted (PT) is in good agreement with experiment. Because of the agreement obtained with the ISR pp data, it is unreasonable to conclude that the predicted PT distribution d 2 a/dMd2pT is too steep. Having discussed alternatives to remedy this inconsistency, we came to the conclusion that only collective nuclear enhancement effects, of one sort or another, could be capable of increasing (p~) as well as the total cross section d 2 o/dMdy, bu t leaving (PT) almost unaltered. Exper- imentally this could be most easily tested by measuring the <p~) of dimuons produced in proton-proton collisions at x/s "" 28 GeV, which should yield a result significantly lower than the pN Fermilab data.

Since, as we have seen, intrinsic transverse momenta of partons cannot significantly exceed 250 MeV, (k T) as large as 500-700 MeV used to enhance theoretical


high PT hadronic cross sections appear to be highly overestimated. The relevant x and Q2 range entering present high PT analyses imply an even lower value for the corresponding intrinsic (kT), as is evident from fig. 4. It would be clearly interesting to study further implications of dynamical recoil effects in other deep inelastic pro-

cesses as for example semi-inclusive hadron or jet(s) production.

We would like to thank P. Minkowski for some clarifying discussions and F. Halzen and D,M. Scott for emphasizing the crossing properties of eq. (3 t).

References

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44 M. Gliick, E. Reya / Transverse momenta of patrons and dimuons

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COO-881-21.

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Transverse momenta of partons and dimuons in QCD