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Double parton correlations versus factorized d i s t ri b u t i o n s
V.L. Korotkikh, A.M. Snigirev M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics 119992, Vorobievy Gory, Moscow, Russia E-mail: vlk@lav01.sinp.msu.ru, snigirev@lav01.sinp.msu.ru

arXiv:hep-ph/0404155 v1 19 Apr 2004

Abstract Using the generalized Lipatov-Altarelli-Parisi-Dokshitzer equations for the two-parton distribution functions we show numerically that the dynamical correlations contribute to these functions quite a lot in comparison with the factorized comp onents. At the scale of CDF hard process ( 5 GeV) this contribution to the double gluon-gluon distribution is nearly 10% and increases right up to 30% at the LHC scale ( 100 GeV) for the longitudinal momentum fractions x 0.1 accessible to these measurements. For the finite longitudinal momentum fractions x 0.2 ? 0.4 the correlations are large right up to 90% in accordance with the predicted QCD asymptotic b ehaviour.

P AC S :

12.38.нt

K ey w or ds: many parton distributions, leading logarithm approximation


Recent CDF measurements [1] of the inclusive cross section for double parton scattering have provided new and complementary information on the structure of the proton and possible parton-parton correlations. Both the absolute rate for the double parton pro cess and any dynamics that correlations may intro duce are therefore of interest. The possibility of observing two separate hard collisions has been proposed since long [2, 3], and from that has also developed in a number of works [4, 5, 6, 7, 8, 9, 10, 11]. The Tevatron and specially LHC allow us to obtain huge data samples of these multiple interactions and to answer to many challenging questions of yet po orly-understo o d aspects of QCD. A brief review of the current situation and some progress in the mo deling account of correlated flavour, colour, longitudinal and transverse momentum distributions can be found in ref. [12]. Multiple interactions require an ansatz for the structure of the incoming beams, i.e. correlations between the constituent partons. As a simple ansatz, usually, the two-parton distributions are supposed to be the pro duct of two single-parton distributions times a momentum conserving phase space factor. In recent paper [13] it has been shown that this hypothesis is in some contradiction with the leading logarithm approximation of perturbative QCD (in the framework of which a parton mo del, as a matter of fact, was established in the quantum field theories [14, 15, 16]). Namely, the two-parton distribution functions being the pro duct of two single distributions at some reference scale become to be dynamically correlated at any different scale of a hard pro cess. The value of these correlations in comparison with the factorized components is the main purpose of this Letter. In order to be clear and to intro duce the denotations let us recall that, for instance, the differential cross section for the four-jet pro cess (due to the simultaneous interaction of two parton pairs) is given by [6, 7] d =
q /g

d12 d34 Dp (x1 , x3 ) Dp (x2 , x4 ), п ef f
ff

(1 ) in the denominator

where dij stands for the two-jet cross section. The dimensional factor e e

represents the total inelastic cross section which is an estimate of the size of the hadron,
ff 2 2 rp (the factor 2 is intro duced due to the identity of the two parton pro cesses). C DF

With the effective cross section measured by CDF, (ef f ) can estimate the transverse size r of (ef f )
p

= (14.5 ‘ 1.7

+1.7 -2.3

) mb [1], one

0.5 fm, which is to o small in comparison with the

proton radius Rp extracted from ep elastic scattering experiments. The relatively small value
C DF 2 with respect to the naive expectation 2 Rp was, in fact, considered [9, 10] as

evidence of nontrivial correlation effects in transverse space. But, apart from these correlations, the longitudinal momentum correlations can also exist and they were investigated in ref. [13]. The factorization ansatz is just applied to the two-parton distributions incoming in eq. (1): Dp (xi , xj ) = Dp (xi , Q2 ) Dp (xj , Q2 ) (1 - xi - xj ), 1 (2 )


where Dp (xi , Q2 ) are the single quark/gluon momentum distributions at the scale Q2 (determined by a hard pro cess). However many parton distribution functions satisfy the generalized Lipatov-Altarelli-ParisiDokshitzer evolution equations derived for the first time in refs [17, 18] as well as single parton distributions satisfy more known and cited Altarelli-Parisi equations [15, 16, 19]. Under certain initial conditions these generalized equations lead to solutions, which are identical with the jet calculus rules proposed originally for multiparton fragmentation functions by Konishi-UkawaVeneziano [20] and are in some contradiction with the factorization hypothesis (2). Here one should note that at the parton level this is the strict assertion within the leading logarithm approximation. After intro ducing the natural dimensionless variable g 2 (І 2 ) Q2 1 ln 1 + b ln t= 2 b 4 І2 ln( 1 = ln 2 b ln(
Q
2 2 QC D І2 2 QC D

) ) , b=

33 - 2nf in QCD, 1 2

where g (І2 ) is the running coupling constant at the reference scale І2 , nf is the number of active flavours, 19]
j dDi (x, t) = dt 1 j


QC D

is the dimensional QCD parameter, the Altarelli-Parisi equations read [15, 16, dx j D (x , t)P x i x . x

j j

(3 )

x

j They describe the scaling violation of the parton distributions Di (x, t) inside a dressed quark

or gluon (i, j = q /g ). We will not write the kernels P explicitly and derive the generalized equations for twoparton distributions D
j1 j i
2

(x1 , x2 , t), representing the probability that in a dressed constituent i

one finds two bare partons of types j1 and j2 with the given longitudinal momentum fractions x1 and x2 (referring to [13, 15, 16, 17, 18, 19] for details), we give only their solutions via the convolution of single distributions [17, 18] D
t j j1 j
2

j1 j i

2

(x1 , x2 , t) =
2

(4 )
z1 z 1 +z
2

dt

1-x x
1

2

0

dz1 z1

1-z x
2

1

dz2 z2

j Di (z1 + z2 , t )



1 z 1 +z

P

j j1 j

2



j1 j2 1 2 Dj1 ( x1 , t - t )Dj2 ( x2 , t - t ). z z

This convolution coincides with the jet calculus rules [20] as mentioned above and is the generalization of the well-known Gribov-Lipatov relation installed for single functions [14, 16] (the distribution of bare partons inside a dressed constituent is identical to the distribution of dressed constituents in the fragmentation of a bare parton in the leading logarithm approximation). The solution (4) shows that the distribution of partons is correlated in the leading logarithm

2


approximation: D
j1 j i
2

j j (x1 , x2 , t) = Di 1 (x1 , t)Di 2 (x2 , t).

(5 )

Of course, it is interesting to find out the phenomenological issue of this parton level consideration. This can be done within the well-known factorization of soft and hard stages (physics of short and long distances) [21]. As a result the equations (3) describe the evolution of parton distributions in a hadron with t (Q2 ), if one replaces the index i by index h only. However, the initial conditions for new equations at t = 0 (Q2 = І2 ) are unknown a priori and must be intro duced phenomenologically or must be extracted from experiments or some
j mo dels dealing with physics of long distances [at the parton level: Di (x, t = 0) = ij (x - 1);

D

j1 j i

2

(x1 , x2 , t = 0) = 0]. Nevertheless the solution of the generalized Lipatov-Altarelli-Parisi-

Dokshitzer evolution equations with the given initial condition may be written as before via the convolution of single distributions [13, 18] D
1-x j1 j
2 2

j1 j h

2

j (x1 , x2 , t) = Dh1 j2C D) (x1 , x2 , t) + (Q
1

(6 )

x

1

dz1 z1

1-z x
2

dz2 z2

D

j1 j h

2



j2 j1 2 1 (z1 , z2 , 0)Dj1 ( x1 , t)Dj2 ( x2 , t) , z z

where D
t j j1 j
2

j1 j2 h(QC D )

(x1 , x2 , t) =
2

(7 )
z1 z 1 +z
2

dt

1-x x
1

2

0

dz1 z1

1-z x
2

1

dz2 z2

j Dh (z1 + z2 , t )



1 z 1 +z

P

j j1 j

2



j1 j2 1 2 Dj1 ( x1 , t - t )Dj2 ( x2 , t - t ) z z

are the dynamically correlated distributions given by perturbative QCD (compare (4) with (7)). The reckoning for the unsolved confinement problem (physics of long distances) is the unknown nonperturbative two-parton correlation function D mentum conserving factor at this scale І2 : D Then D
j1 j h
2

j1 j h

2



(z1 , z2 , 0) at some scale І2 . One

can suppose that this function is the pro duct of two single-parton distributions times a mo-

j1 j h

2

j j (z1 , z2 , 0) = Dh1 (z1 , 0)Dh2 (z2 , 0)(1 - z1 - z2 ).

(8 )

(x1 , x2 , t) = D
1 dz1 z1 1 x
2

j1 j2 h(QC D )

j j (x1 , x2 , t) + (1 - x1 - x2 ) Dh1 (x1 , t)Dh2 (x2 , t) +


(9 )

j1 j2 x

1

dz2 z2

j1 j j j2 1 2 Dh1 (z1 , 0)Dh2 (z2 , 0)Dj1 ( x1 , t)Dj2 ( x2 , t)[(1 - z1 - z2 ) - 1] , z z



(1 0 ) where D (x, t) =
j


1 j h

x

dz j jx Dh (z , 0) Dj ( , t) z z 3

(1 1 )


j is the solution of eq. (3) with the given initial condition Dh (x, 0) for parton distributions inside

a hadron expressed via distributions at the parton level. This result (9) shows that if the two-parton distributions are factorized at some scale І2 , then the evolution violates this factorization inevitably at any different scale (Q2 = І2 ), apart from the violation due to the kinematic correlations induced by the momentum conservation (given by functions)1 . For a practical employment it is interesting to know the degree of this violation. Partly this problem was investigated theoretically in refs. [18, 23] and for the two-particle correlations of fragmentation functions in ref. [24]. That technique is based on the Mellin transformation of distribution functions as
1 j Mh

(n, t) =
0

j dx xn Dh (x, t).

(1 2 )

After that the integro differential equations (3) become systems of ordinary linear-differential equations of first order with constant co efficients and can be solved explicitly [18, 23]. In order to obtain the distributions in x representation an inverse Mellin transformation must be performed
j Dh (x, t) =

d n -n j x Mh (n, t), 2 i

(1 3 )

where the integration runs along the imaginary axis to the right from all n singularities. This can be done numerically. However the asymptotic behaviour can be estimated. Namely, with the growth of t (Q2 ) the first term in eq. (6) becomes dominant 2 for finite x1 and x2 [23]. Thus the two-parton distribution functions "forget" the initial conditions unknown a priori and the correlations perturbatively calculated appear. The asymptotic prediction "teaches" us a tendency only and tells nothing about the values of x1 , x2 , t(Q2 ) beginning from which the correlations are significant (the more so since the asymptotic behaviour takes place over the double logarithm dimensionless variable t as a function of Q2 ). Naturally numerical estimations can give an answer to this specific question. We do it using the CTEQ fit [22] for single distributions as an input in eq. (7). The nonperturbaj tive initial conditions Dh (x, 0) are specified in a parametrized form at a fixed low-energy scale

Q0 = І = 1.3 GeV. The particular function forms and the value of Q0 are not crucial for the CTEQ global analysis at the flexible enough parametrization, which reads [25]
j xDp (x, 0) = Aj xA1 (1 - x)A2 eA3 x (1 + eA4 x)A5 . 0
1 2
j j j j j

(1 4 )

This is the analogue of the momentum conserving phase space factor in eq. (2) Such domination is the mathematical consequence of the relation between the maximum eigenvalues (n) in the moments representation (after Mellin transformation): (n1 + n2 ) > (n1 ) + (n2 ) in QCD at the large n1 , n2 (finite x1 , x2 ), because (n) - ln(n), n 1.

4


The independent parameters Aj , Aj , Aj , Aj , Aj , Aj for parton flavour combinations uv 0 1 2 3 4 5 п, g and u + d are given in Appendix A of ref. [25]. To distinguish the u and d п п u - u , dv d - d п п п пп distributions the ratio d/u is parametrized as a sum of two terms:
d u п Dp (x, 0)/Dp (x, 0) = A0 xA1 (1 - x) п A
2

+ (1 + A3 x)(1 - x)

A

4

(1 5 )

with the co efficients A0 , A1 , A2 , A3 , A4 again from ref. [25]. The initial conditions for strange quarks are assumed:
п s п d s u Dp (x, 0) = Dp (x, 0) = 0.2 Dp (x, 0) + Dp (x, 0) . j The parton distribution functions Dp (x, t) at all higher Q(t) are determined from the input j initial conditions Dp (x, 0) by the Altarelli-Parisi evolution equations. The CTEQ Evolution п

package [26] was used and adapted by us in order to obtain numerically single distributions
j Di (x, t) at all t and at the parton level also. We fixed the fundamental parameter of per-

turbative QCD,

QC D

= 0.281 GeV, that is in accordance with the strong coupling constant,

s (MZ ) 0.2, at the Z resonance in one-lo op approximation. Only the light quarks u, d, s (nf = 3) are taken into account in the evolution equations and are treated as massless, as usual. After that the triple integral (7) was calculated numerically for three values of Q = 5, 100, 250 GeV as a function of x = x1 = x2 . To be specific we considered the double gluon-gluon distribution function in the proton. In this case only the kernel P
g g g

can be taken into account as

giving the main contribution to the perturbative double gluon-gluon distribution. The remnant terms of sum in eq. (7) are relatively small and can only increase the effect under consideration because they are positive. The results of numerical calculations are presented on fig. 1 for the ratio: R(x, t) = D
gg p(QC D ) g g (x1 , x2 , t) Dp (x1 , t)Dp (x2 , t)(1 - x1 - x2 ) 2 x 1 =x 2 =x

.

(1 6 )

Here one should note that the momentum conserving phase space factor (1 - x1 - x2 )2 is intro duced in eq. (16) instead of (1 - x1 - x2 ) usually used. The reason is simple: this factor was intro duced in eq. (2), generally speaking, "by hand" in order to "save" the momentum conservation law, i.e. in order to make the pro duct of two single distributions is equal to zero smo othly at x1 + x2 = 1. However the generalized QCD evolution equations demand higher power of (1 - x1 - x2 ) at x1 + x2 1: only the phase space integrals in eqs. (6) and (7) give
1-x
2

1-z

1

dz1
x
1

dz2 = (1 - x1 - x2 )2 /2.
x
2

In fact this power must depend on t increasing with its growth as this takes place for single distributions at x 1 [16, 27]. The asymptotic behaviour of two-particle fragmentation functions at x1 + x2 1 was investigated, for instance, in ref. [28] with the similar result. Our 5


numerical calculations support this assertion also: the power of (1 - x1 - x2 ) for the perturbative QCD gluon-gluon correlations is higher than 2 and increases with t(Q) as one can see from fig. 1 However the intro duced factor (1 - x1 - x2 )2 has not an influence practically on the ratio under consideration in the region of small x1 , x2 . And namely this region, in which multiple interactions can contribute to the cross section visibly, is interesting from experimental point of view. Fig. 1 shows that at the scale of CDF hard pro cess ( 5 GeV) the ratio (16) is nearly 10% and increases right up to 30% at the LHC scale ( 100 GeV) for the longitudinal momentum fractions x 0.1 accessible to these measurements. For the finite longitudinal momentum fractions x 0.2 ? 0.4 the correlations are large right up to 90% . They become important in more and more x region with the growth of t in accordance with the predicted QCD asymptotic behaviour. The correlation effect is strengthened insignificantly (up to 2%) for the longitudinal momentum fractions x 0.1 when starting from the slightly lower value Q0 = 1 GeV (early used by CTEQ Collaboration). We conclude also that R(x, t) const at x 0 most likely, calculating this ratio ( 0.1) at xmin = 10-4 . Seemingly the correction to the double gluon-gluon distributions at the CDF scale can be smo othly absorbed by uncertainties in the e
ff

increasing the transverse effective size rp by a

such way. But this augmentation is still not enough to solve a problem of the relatively small value of rp with respect to the proton radius without nontrivial correlation effects in transverse space [9, 10]. In summary, the numerical estimations show that the leading logarithm perturbative QCD correlations are quite comparable with the factorized distributions. With increasing a number of observable multiple collisions (statistic) the more precise calculations of their cross section (beyond the factorization hypothesis) will be needed also. In order to obtain the more delicate their characteristics (distributions over various kinematic variables) it is desirable to implement the QCD evolution of two-parton distribution functions in some Monte Carlo event generator as this was done for single distributions within, for instance, PYTHIA [29]. Acknow ledgements It is pleasure to thank V.A. Ilyin drawing the attention to the problem of double parton scattering. Discussions with E.E. Bo os, M.N. Dubinin, I.P. Lokhtin, S.V. Molo dtsov, L.I. Sarycheva, V.I. Savrin, T. Sjostrand, D. Treleani and G.M. Zinovjev are gratefully acknowledged. Authors are specially thankful to A.S. Proskuryakov for the valuable help in adapting CTEQ Evolution package. This work is partly supported by grant N 04-02-16333 of Russian Foundation for Basic Research.

6


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[21] J.C. Collins, D.E. Soper and G. Sterman, in: Perturbative QCD, edited by A. Mueller (World Scientific, Singapore, 1989). [22] CTEQ Collab., H.L. Lai et al., Eur. Phys. J. C 12 (2000) 375. [23] V.P. Shelest, A.M. Snigirev and G.M. Zinovjev, JINR Communication, E2-82-194, Dubna (1982); Preprint ITP-83-46E, Kiev (1983), unpublished. [24] M.J. Puhala, Phys. Rev. D 22 (1980) 1087. [25] J. Pumplin et. al., JHEP (2002) 0207:012. [26] http://www.phys.psu.educteq/ [27] Yu.L. Dokshitzer, D.I. Dyakonov and S.I. Troyan, Phys. Rep. 58 (1980) 269. [28] I. Vendramin, Nuovo Cimento 62 A (1981) 21. [29] T. Sjostrand, Comp. Phys. Com. 135 (2001) 238.

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0.9 0.8 0.7 0.6

250 GeV

R(x)

0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

5 GeV 100 GeV

x

Figure 1: The ratio of perturbative QCD correlations to the factorized component for the double gluon-gluon distribution in the proton as a function of x = x1 = x2 for three values of Q = 5(solid), 100(dashed), 250(dash-dotted) GeV.

9