How to impose initial conditions for QCD evolution of double parton distributions?

# How to impose initial conditions for QCD evolution of double parton distributions?

## Abstract

Double parton distribution functions are used in the QCD description of double parton scattering. The double parton distributions evolve with hard scales through QCD evolution equations which obey nontrivial momentum and valence quark number sum rules. We describe an attempt to construct initial conditions for the evolution equations which exactly fulfill these sum rules and discuss its shortcomings. We also discuss the factorization of the double parton distributions into a product of two single parton distribution functions at small values of the parton momentum fractions.

quantum chromodynamics, parton distributions, evolution equations, sum rules

## I Introduction

In high-energy hadron scattering, the final state particles could be produced from two hard interactions in one collision. This process, called the double parton scattering (DPS), is viewed as two hard interactions of two pairs of partons from the scattering hadrons. The DPS is the simplest process in the analysis of multiparton interactions, studied for many years from both the theoretical Kirschner:1979im (); Shelest:1982dg (); Zinovev:1982be (); Ellis:1982cd (); Bukhvostov:1985rn (); Snigirev:2003cq (); Korotkikh:2004bz (); Gaunt:2009re (); Blok:2010ge (); Ceccopieri:2010kg (); Diehl:2011tt (); Gaunt:2011xd (); Ryskin:2011kk (); Bartels:2011qi (); Blok:2011bu (); Diehl:2011yj (); Manohar:2012jr (); Ryskin:2012qx (); Gaunt:2012dd (); Snigirev:2014eua () and phenomenological sides DelFabbro:1999tf (); Kulesza:1999zh (); DelFabbro:2002pw (); Cattaruzza:2005nu (); Berger:2009cm (); Gaunt:2010pi (); Kom:2011bd (); Berger:2011ep (); Kom:2011nu (); Bartalini:2011jp (); d’Enterria:2012qx (); d’Enterria:2013ck (). The experimental evidence of the DPS has been presented in Refs. Akesson:1986iv (); Abe:1997bp (); Abe:1997xk (); Abazov:2009gc (); Aad:2013bjm (); Chatrchyan:2013xxa (); Aad:2014rua ().

The DPS processes allow one to gain information on parton correlations by measuring the DPS cross section in high energy scattering of two hadrons, and . In the collinear approximation, the inclusive DPS cross section is given in the form Diehl:2011yj (); Ryskin:2011kk ():

 σh1h2AB = N2∑f1f2f′1f′2∫dx1dx2dx′1dx′2d2q(2π)2 (1) × Dh1f1f2(x1,x2,Q1,Q2,q)^σAf1f′1(x1,x′1,Q1) × ^σBf2f′2(x2,x′2,Q2)Dh2f′1f′2(x′1,x′2,Q1,Q2,−q),

where and denote the two hard parton processes and is a symmetry factor, equal to 1 for and 2 otherwise.

In the above, are double parton distribution functions (DPDFs) of hadrons , which depend on the two parton flavors , parton momentum fractions , two hard scales involved in the DPS and an additional transverse momentum . The presence of the latter momentum is related to the loop structure of the exchanged four partons in the forward scattering amplitude which ultimately enters into the definition of the DPDFs Diehl:2011yj (). The importance of this variable for the DPS cross section computations has been discussed at length in Ref. Ryskin:2011kk (). The longitudinal momentum fractions obey the condition

 0

which says that the sum of parton longitudinal momenta cannot exceed the total proton momentum (taken for definiteness from now on). This is the basic parton correlation which has to be taken into account. For more advanced aspects of parton correlations, see Ref. Manohar:2012jr ().

The analysis of the DPS is crucial for a better understanding of background for many important processes measured at the experiments at Tevatron and the LHC, e.g. for the Higgs boson production Krasny:2013aca (), as well as for a better description of multiparton interactions, needed for example for modeling of the underlying event; see Ref. Bartalini:2011jp () for a comprehensive review of these issues. Thus, it is very important to use a rigorous approach based on QCD evolution equations for the DPDFs. These equations are known in the leading logarithmic approximation Kirschner:1979im (); Shelest:1982dg (); Zinovev:1982be (); Snigirev:2003cq (); Korotkikh:2004bz (). They conserve new sum rules Gaunt:2009re () which relate the double and single parton distribution functions at any evolution scale. In this presentation we address a problem of specifying initial conditions for the evolution equations of the DPDFs which exactly obey the new sum rules.

The paper is organized as follows. In Secs. II and III we briefly describe evolution equations for parton distributions. In Sec. IV the new sum rules are presented, while in Sec. V the most popular initial conditions are described. In Sec. VI we discuss a problem with them and describe an attempt to solve it. In Sec. VII factorization of the DPDFs into a product of two single parton distribution functions (SPDFs) is discussed.

## Ii Evolution equations for SPDFs

To set the notation, let us recapitulate the QCD evolution equations in the collinear approximation for SPDFs, , which are used in the description of the single parton scattering. The general form of these equations is given by

 ∂tDf(x,t)=∑f′∫10duKff′(x,u,t)Df′(u,t), (3)

where the evolution parameter and the parton momentum fraction obey the condition . The integral kernels, , describe the real and virtual parton emissions

 Kff′(x,u,t)=KRff′(x,u,t)−δ(u−x)δff′KVf(x,t). (4)

The real emission kernel corresponds to the parton transition , where the momentum fraction , and is given by

 KRff′(x,u,t)=1uPff′(xu,t)θ(u−x). (5)

The virtual part, , can be computed from the imposed momentum sum rule

 ∑f∫10dxxDf(x,t)=1 (6)

where the normalization to unity means that partons carry the whole nucleon momentum. Thus we find

 xKVf(x,t)=∑f′1∫0duuKRf′f(u,x,t). (7)

The functions in Eq. (5) are splitting functions computed perturbatively in QCD in powers of the strong coupling constant:

 Pff′(z,t)=αs(t)2πP(0)ff′(z)+α2s(t)(2π)2P(1)ff′(z)+.... (8)

The first term on the rhs corresponds to the leading logarithmic approximation while the higher terms are computed in the next-to-leading approximations. In this way, the well known DGLAP evolution equations for SPDFs are obtained

 ∂tDf(x,t) = ∑f′1∫xdzzPff′(z,t)Df′(xz,t) (9) − Df(x,t)∑f′1∫0dzzPf′f(z,t).

Note that the diagonal in flavors splitting functions, , have a simple pole singularity at which is removed by the virtual term [so called prescription].

## Iii Evolution equations for DPDFs

The evolution equations for the DPDFs are only known for in the leading logarithmic approximation Kirschner:1979im (); Shelest:1982dg (); Zinovev:1982be (); Snigirev:2003cq (); Korotkikh:2004bz (); Gaunt:2009re (). The first discussion of the next-to-leading corrections can be found in Ceccopieri:2010kg (). We start from considering two equal hard scales, , and introduce the following notation for the DPDFs in such a case

 Df1f2(x1,x2,t)=Df1f2(x1,x2,Q,Q,q=0). (10)

A phenomenological discussion of the case in the context of evolution equations, discussed below, can be found in Ryskin:2011kk (); Golec-Biernat:2014nsa ().

The QCD evolution equations take general form

 ∂t Df1f2(x1,x2,t) (11) = ∑f′∫1−x20duKf1f′(x1,u,t)Df′f2(u,x2,t) + ∑f′∫1−x10duKf2f′(x2,u,t)Df1f′(x1,u,t) + ∑f′KRf′→f1f2(x1,x2,t)Df′(x1+x2,t),

where the integral kernels are given by Eq.(4) with the real part (5) in the leading logarithmic approximation and the virtual part found from Eq. (7). The two integrals in the above describe the DGLAP evolution of a single parton with the second parton treated as a spectator. This gives the upper integration limits resulting from condition (2).

The third term needs special attention. It describes the real emission splitting of a single parton into two partons which undergo two independent hard scatterings. This is why the SPDFs appear here and the evolution equations (3) and (11) form a coupled set of equations which has to be solved simultaneously. In the leading logarithmic approximation, there is only one parton flavor, , which leads to two parton flavors, and . Thus, we have the following splittings: , , , and . In such a case

 KRf′→f1f2(x1,x2,t)=αs(t)2π1x1+x2P(0)f′f1(x1x1+x2)

where are splitting functions in the leading logarithmic approximation. It can easily be checked that the splitting functions can also be used in this case. Thus the rhs. of the evolution equations (11) is invariant with respect to the parton interchange, . If the initial conditions for them, specified at some initial scale , are parton exchange symmetric,

 Df1f2(x1,x2,t0)=Df2f1(x2,x1,t0), (13)

the evolution will preserve this symmetry for any value of .

In the case when the two hard scales are significantly different, e.g. , the large logarithms appear. They have to be resummed which leads to the DGLAP evolution equation with respect to the second parton

 ∂t2 D f1f2(x1,x2,t1,t2) (14) = ∑f′1−x1∫0duKf2f′(x2,u,t2)Df1f′(x1,u,t1,t2),

where . Thus the evolution has two steps, from equal initial scales to the equal final scales , according to Eq. (11), and then to the scales , according to Eq. (14). However, we do not discuss such a case in our analysis, concentrating only on the first step of the evolution. We also refrain from discussing the impact parameter representation of the DPDFs and corresponding evolution equations, sending the reader to Ref. Diehl:2011yj ().

## Iv Sum rules for DPDFs

The DGLAP evolution equations (3) obey the momentum sum rule (6), while the evolution equations (11) preserve a new momentum sum rule:

 ∑f1∫1−x20dx1x1Df1f2(x1,x2,t)Df2(x2,t)=1−x2. (15)

This relation can be understood by treating the ratio of the parton distributions under the integral as the conditional probability to find parton with the momentum fraction , while the second parton characteristics, and , are fixed. In such a the total momentum fraction carried by partons equals . In this way, the momentum sum rule (15) relates the double and single parton distribution functions for any value of :

 ∑f1∫1−x20dx1x1Df1f2(x1,x2,t)=(1−x2)Df2(x2,t). (16)

The valence quark number sum rule for the SPDFs has the well-known form

 ∫10dx{Dqi(x,t)−D¯qi(x,t)}=Ni, (17)

where is the number of valence quarks . For the DPDFs, the analogous sum rule depends on the flavor of the second parton (see Refs. Gaunt:2009re (); Gaunt:thesis () for more details):

 ∫1−x20 dx1{Dqif2(x1,x2,t)−D¯qif2(x1,x2,t)} (18) = ⎧⎪ ⎪⎨⎪ ⎪⎩NiDf2(x2,t)\rm~{}~% {}~{}~{}for f2≠qi,¯qi(Ni−1)Df2(x2,t)\rm~{}~{}~{}~{}for f2=qi(Ni+1)Df2(x2,t)\rm~{}~{}~{}~{}for f2=¯qi.

It is important to emphasize that the momentum and valence quark number sum rules are conserved by the evolution equations (3) and (11) once they are imposed at an initial value . If not true, the sum rules will not be exactly satisfied during evolution.

The sum rules (16) and (18) are written with respect to the first parton. Assuming the parton exchange symmetry (13) to be valid for any value of , the sum rules could also be written with respect to the second parton. In this case, the integration is performed over up to with the first parton flavor and the momentum fraction fixed.

## V Symmetric initial conditions

To solve Eqs. (3) and (11) we need to specify initial conditions for both the SPDFs and DPDFs. The initial SPDFs can be taken from well-established parameterizations, e.g. from the leading-order (LO) MSTW parametrization Martin:2009iq (), which we use in the forthcoming analysis. However, the specification of the initial DPDFs needs assumptions since the experimental knowledge on the DPDFs is very limited. For practical reasons, their form is built out of the existing SPDFs. For example, in Refs. Korotkikh:2004bz (); Gaunt:2009re () the symmetric initial conditions with respect to the parton interchange (13) were proposed,

 Df1f2(x1,x2,t0) = Df1(x1,t0)Df2(x2,t0) (19) × (1−x1−x2)2(1−x1)2+n1(1−x2)2+n2,

where in the correlation factor, for sea quarks and for valence quarks. These distributions are also positive definite provided that the SPDFs are positive.

In Fig.1 we show how ansatz (19) fulfills the momentum and valence quark number sum rules by plotting the ratio of the rhs to lhs for Eqs. (16) and (18), respectively. The sum rules are fulfilled if the ratios equal one. In the valence number sum rule and (for simplicity, we also set ). We see that the momentum sum rule is quite well satisfied while the valence quark number sum rule is significantly violated. The limiting values, for and for , correspond to the values for , which are obtained from Eq. (19) computed for . The case with for valence quarks leads to similar results.

## Vi Attempt to satisfy sum rules

Is it possible to construct initial distributions which exactly fulfill the discussed sum rules? The answer is in the affirmative if we concentrate on the first parton, treating the second one as a spectator (or vice versa).

To obey the momentum sum rule (16), it is enough to postulate the following form (we skip in the notation):

 Df1f2(x1,x2)=11−x2Df1(x11−x2)Df2(x2). (20)

However, the valence number sum rule (18) needs corrections for identical quark flavors or antiflavors which do not spoil the already fulfilled momentum sum rule. The form below does the job

 Dfifi(x1,x2) = 11−x2{Df1(x11−x2)−12}Df1(x2) Dfi¯fi(x1,x2) = 11−x2{Df1(x11−x2)+12}D¯fi(x2) (21)

where are quark flavor or antiflavors. Unfortunately, there is a price to pay. The DPDFs for identical flavors or anti-lavors are not positive definite. For

 Dfifi(x1,x2)≈{Df1(x1)−1/2}Df1(x2) (22)

and for all the existing parametrizations for bigger than some . Thus, is negative in this range.

This is shown in Fig. 2 where the initial distributions and (multiplied by ) are plotted for fixed . They are constructed from the LO MSTW parametrization of SPDFs at , using ansatze (19) (sym) and (21) (our). We see that from ansatz (21) is negative for . Using a numerical program which we constructed to solve Eqs.  (3) and (11), we show in Fig. 3 that this effect does not change when the distributions are evolved up to .

The proposed form of initial conditions is not symmetric with respect to parton interchange described by Eq. (13). A simple symmetrization,

 Df1f2(x1,x2)→Df1f2(x1,x2)+Df2f1(x2,x1), (23)

does not solve the problem since the discussed sum rules are violated in such a case. For example, the integration over as in the sum rule (16) gives the correct result with the first term in Eq. (23), which is spoiled by a nonzero contribution from the second term. Unfortunately, we could not find a better symmetrization prescription which conserves the sum rules.

To summarize our attempt, it seems that in the construction with SPDFs we cannot find the initial distributions which fulfill the sum rules in both the variables and . Therefore, the symmetric parametrization (19), discussed at length in ref. Gaunt:2009re (), is still the best proposition for applications.

## Vii Factorization at small x

The sum rule problem for the initial distribution concerns the large behavior of the DPDFs. If parton momentum fractions are small, , both parametrizations of the initial distributions, given by Eqs. (19), (20), and (21), tend to the factorized form

 Df1f2(x1,x2,t0)≈Df1(x1,t0)Df2(x2,t0) (24)

where denote quark flavors, antiflavors, or a gluon.

Does the approximate factorization hold during the evolution? The inspection of Eq. (11) reveals that in general the third, splitting term violates the factorization during the evolution. However, the scale of the violation depends on the numerical value of the splitting term in comparison to the values of the first two terms in Eq. (11).

The analysis with our numerical program shows that the violation is significant only for the splitting , due to a large value of the single gluon distribution at small . This is illustrated in Fig. 4, where and evolved to from the symmetric input are compared to the corresponding products of and at the same scale. The effect of the violation of factorization for small values of is only seen for , while for the distributions, and (not shown here), relation (24) holds very well. The same conclusions are valid for the other quark flavors. To avoid a possible confusion, let us stress that factorization (24) is not expected to be true at large values of parton momentum fractions (i.e. for ).

## Viii Summary

The specification of initial conditions for the QCD evolution equations of the DPDFs (written in the leading logarithmic approximation) is not a simple task, mainly because of the new sum rules which they should obey.

In the presented attempt we tried to build the initial DPDFs out of the existing SPDFs treating one of the two partons as a spectator. The form which we found obeys the momentum and valence quark number sum rule with respect to the longitudinal momentum fraction of the active parton. The difficulty with symmetrization of the proposed form is the reason why the sum rules are not symmetric with respect to the interchange of partons. Therefore, the approach Korotkikh:2004bz (); Gaunt:2009re () in which parameters the symmetric form of initial DPDFs are optimized to approximately fulfill the sum rules is still the best one can achieve.

We also discussed the factorization of the DPDFs into a product of two SPDFs at small values of parton momentum fractions . We showed that such a factorization is to a good approximation conserved by the QCD evolution except for the distribution for which the splitting contribution in the evolution equations (11) is quite important due to a large gluon distribution at small . For large values of , the factorization is not expected.

###### Acknowledgements.
This work was supported by the Polish NCN Grants No. DEC-2011/01/B/ST2/03915 and No. DEC-2012/05/N/ST2/02678 as well as by the Center for Innovation and Transfer of Natural Sciences and Engineering Knowledge in Rzeszów.

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