Multiple parton scattering in nuclei: Quark–quark scattering

Nuclear Physics A 793 (2007) 128–170

Multiple parton scattering in nuclei:Quark–quark scattering

Andreas Schäfer a, Xin-Nian Wang b,∗, Ben-Wei Zhang c,d,a

a Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germanyb Nuclear Science Division, MS 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAc Cyclotron Institute and Physics Department, Texas A&M University, College Station, TX 77843-3366, USA

d Institute of Particle Physics, Central China Normal University, Wuhan 430079, China

Received 10 May 2007; received in revised form 13 June 2007; accepted 18 June 2007

Available online 28 June 2007

Abstract

Modifications to quark and antiquark fragmentation functions due to quark–quark (antiquark) doublescattering in nuclear medium are studied systematically up to order O(α2

s ) in deeply inelastic scatter-ing (DIS) off nuclear targets. At the order O(α2

s ), twist-four contributions from quark–quark (antiquark)rescattering also exhibit the Landau–Pomeranchuk–Migdal (LPM) interference feature similar to gluonbremsstrahlung induced by multiple parton scattering. Compared to quark–gluon scattering, the modifica-tion, which is dominated by t-channel quark–quark (antiquark) scattering, is only smaller by a factor ofCF /CA = 4/9 times the ratio of quark and gluon distributions in the medium. Such a modification is notnegligible for realistic kinematics and finite medium size. The modifications to quark (antiquark) fragmen-tation functions from quark–antiquark annihilation processes are shown to be determined by the antiquark(quark) distribution density in the medium. The asymmetry in quark and antiquark distributions in nucleiwill lead to different modifications of quark and antiquark fragmentation functions inside a nucleus, whichqualitatively explains the experimentally observed flavor dependence of the leading hadron suppressionin semi-inclusive DIS off nuclear targets. The quark–antiquark annihilation processes also mix quark andgluon fragmentation functions in the large fractional momentum region, leading to a flavor dependence ofjet quenching in heavy-ion collisions.© 2007 Elsevier B.V. All rights reserved.

PACS: 24.85.+p; 12.38.Bx; 13.87.Ce; 13.60.-r

Keywords: Jet quenching; Modified fragmentation; Parton energy loss

* Corresponding author.E-mail address: [email protected] (X.-N. Wang).

0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2007.06.009

A. Schäfer et al. / Nuclear Physics A 793 (2007) 128–170 129

1. Introduction

Multiple parton scattering in a dense medium can be used as a useful tool to study propertiesof both hot and cold nuclear matter. The success of such an approach has been demonstrated bythe discovery of strong jet quenching phenomena in central Au+Au collisions at the RelativisticHeavy-ion Collider (RHIC) [1–3] and their implications on the formation of a strongly coupledquark–gluon plasma at RHIC [4,5]. However, for a convincing phenomenological study of the ex-isting and future experimental data, a unified description of all medium effects in hard processesinvolving nuclei, such as electron–nucleus (e+A), hadron–nucleus (h+A) and nucleus–nucleuscollisions (A+A) has to be developed [6,7]. This must include the physics of transverse momen-tum broadening [8], strong nuclear enhancement in DIS [9] and Drell–Yan production [10,11],nuclear shadowing [12], and parton energy loss due to gluon radiation induced by multiple scat-tering [13–19].

There exist many different frameworks in the literature to describe multiple scattering in anuclear medium [20–22]. Among them the twist expansion approach is based on the generalizedfactorization in perturbative QCD as initially developed by Luo, Qiu and Sterman (LQS) [23].In the LQS formalism, multiple scattering processes generally involve high-twist multiple-partoncorrelations in analogy to the parton distribution operators in leading twist processes. Though thecorresponding higher twist corrections are suppressed by powers of 1/Q2, they are enhanced atleast by a factor of A1/3 due to multiple scattering in a large nucleus. This framework has been ap-plied recently to study medium modification of the fragmentation functions as the leading partonpropagates through the medium [18,19]. Because of the non-Abelian Landau–Pomeranchuk–Migdal interference in the gluon bremsstrahlung induced by multiple parton scattering in nuclei,the higher-twist nuclear modifications to the fragmentation functions are in fact enhanced byA2/3, quadratic in the nuclear size [18,19]. Phenomenological study of parton energy loss andnuclear modification of the fragmentation functions in cold nuclear matter [24] gives a gooddescription of the nuclear modification of the leading hadron spectra in semi-inclusive deeplyinelastic lepton–nucleus scattering observed by the HERMES experiment [25,26]. The sameframework also gives a compelling explanation for the suppression of large transverse momen-tum hadrons discovered at RHIC [27].

The emphasis of recent studies of medium modification of fragmentation functions has beenon radiative parton energy loss induced by multiple scattering with gluons. Such processes in-deed are dominant relative to multiple scattering with quarks because of the abundance of softgluons in either cold nuclei or hot dense matter produced in heavy-ion collisions. Since gluonbremsstrahlung induced by scattering with medium gluons is the same for quarks and antiquarks,one also expects the energy loss and fragmentation modification to be identical for quarks andantiquarks. However, in a medium with finite baryon density such as cold nuclei and the for-ward region of heavy-ion collisions, the difference between quark and antiquark distributions inthe medium should lead to different energy loss and modified fragmentation functions for quarksand antiquarks through quark–antiquark annihilation processes. To study such an asymmetry, onemust consider systematically all possible quark–quark and quark–antiquark scattering processes,which will be the focus of this paper.

In this study we will calculate the modifications of quark and antiquark fragmentation func-tions (FF) due to quark–quark (antiquark) double scattering in a nuclear medium, working withinthe LQS framework for generalized factorization in perturbative QCD. For a complete descrip-tion of nuclear modification of the single inclusive hadron spectra, one still have to considermedium modification of gluon fragmentation functions in addition to modified quark fragmenta-

130 A. Schäfer et al. / Nuclear Physics A 793 (2007) 128–170

tion function due to quark–gluon scattering [18]. The theoretical results presented in this paperwill be a second step toward a complete description of medium modified fragmentation func-tions. However, one can already find that quark–quark (antiquark) double scattering will givedifferent corrections to quark and antiquark FF, depending on antiquark and quark density ofthe medium, respectively. This difference between modified quark and antiquark FF may shedlight on the interesting observation by the HERMES experiment [25,26] of a large differencebetween nuclear suppression of the leading proton and antiproton spectra in semi-inclusive DISoff large nuclei. Such a picture of quark–quark (antiquark) scattering can provide a competingmechanism for the experimentally observed phenomenon in addition to possible absorption offinal state hadrons inside nuclear matter [28,29].

The paper is organized as follows. In Section 2 we will present the general formalism of ourcalculation including the generalized factorization of twist-4 processes. In Section 3 we will il-lustrate the procedure of calculating the hard partonic parts of quark–quark double scattering innuclei. In Section 4 we will discuss the modifications to quark and antiquark fragmentation func-tions due to quark–quark (antiquark) double scattering in nuclei. In Section 5, we will focus onthe flavor dependent part of the medium modification to the quark FF’s due to quark–antiquarkannihilation and we will discuss the implications for the flavor dependence of the leading hadronspectra in both DIS off a nucleus and heavy-ion collisions. We will summarize our work in Sec-tion 6. In Appendix A.1, we collect the complete results for the hard partonic parts for differentcut diagrams of quark–quark (antiquark) double rescattering in nuclei. We also provide an alter-native calculation of the hard parts of the central-cut diagrams in Appendix A.3 through elasticquark–quark scattering or quark–antiquark annihilation as a cross check.

2. General formalism

In order to study quark and antiquark FF’s in semi-inclusive deeply inelastic lepton–nucleusscattering, we consider the following processes,

e(L1) + A(p) → e(L2) + h(�h) + X,

where L1 and L2 are the four momenta of the incoming and outgoing leptons, and �h is theobserved hadron momentum. The differential cross section for the semi-inclusive process can beexpressed as

EL2E�h

dσhDIS

d3L2d3�h

= α2EM

2πs

1

Q4LμνE�h

dWμν

d3�h

, (1)

where p = [p+,0,0⊥] is the momentum per nucleon in the nucleus, q = L2 − L1 =[−Q2/2q−, q−,0⊥] the momentum transfer carried by the virtual photon, s = (p + L1)

2 thelepton-nucleon center-of-mass energy and αEM is the electromagnetic (EM) coupling constant.The leptonic tensor is given by Lμν = 1/2 Tr(γ ·L1γμγ ·L2γν) while the semi-inclusive hadronictensor is defined as,

E�h

dWμν

d3�h

= 1

2

∑X

〈A|Jμ(0)|X,h〉〈X,h|Jν(0)|A〉2πδ4(q + p − pX − �h), (2)

where∑

X runs over all possible final states and Jμ = ∑q eqψqγμψq is the hadronic EM current.

Assuming collinear factorization in the parton model, the leading-twist contribution to thesemi-inclusive cross section can be factorized into a product of parton distributions, parton frag-mentation functions and the partonic cross section. Including all leading log radiative corrections,


Fig. 1. Lowest order and leading-twist contribution to semi-inclusive DIS.

the lowest order contribution [O(α0s )] from a single hard γ ∗ +q scattering, as illustrated in Fig. 1,

can be written as

dWSμν

dzh

=∑q

∫dx f A

q

(x,μ2

I

)H(0)

μν (x,p, q)Dq→h

(zh,μ

2), (3)

H(0)μν (x,p, q) = e2

q

2Tr

(γ · pγμγ · (q + xp)γν

) 2π

2p · q δ(x − xB), (4)

where the momentum fraction carried by the hadron is defined as zh = �−h /q−, xB = Q2/2p+q−

is the Bjorken scaling variable, μ2I and μ2 are the factorization scales for the initial quark dis-

tributions f Aq (x,μ2

I ) in a nucleus and the fragmentation functions in vacuum Dq→h(zh,μ2),

respectively. The renormalized quark fragmentation function Dq→h(zh,μ2) satisfies the DGLAP

QCD evolution equations [30]:

∂Dq→h(zh,μ2)

∂ lnμ2

= αs(μ2)

2π

1∫zh

dz

z

[γq→qg(z)Dq→h

(zh/z,μ

2) + γq→gq(z)Dg→h

(zh/z,μ

2)]; (5)

∂Dg→h(zh,μ2)

∂ lnμ2

= αs(μ2)

2π

1∫zh

dz

z

[ 2nf∑q=1

γg→qq (z)Dq→h

(zh/z,μ

2) + γg→gg(z)Dg→h

(zh/z,μ

2)], (6)

where γa→bc(z) denotes the splitting functions of the corresponding radiative processes [31,32].In DIS off a nuclear target, the propagating quark will experience additional scatterings with

other partons from the nucleus. The rescatterings may induce additional parton (quark or gluon)radiation and cause the leading quark to lose energy. Such induced radiation will effectivelygive rise to additional terms in the DGLAP evolution equation leading to a modification of the


Fig. 2. A typical diagram for quark–gluon double scattering with three possible cuts [central (C), left (L) and right (R)].

fragmentation functions in a medium. These are power-suppressed higher-twist corrections andthey involve higher-twist parton matrix elements. We will only consider those contributions thatinvolve two-parton correlations from two different nucleons inside the nucleus. They are propor-tional to the thickness of the nucleus [18,23,33] and thus are enhanced by a nuclear factor A1/3

as compared to two-parton correlations in a nucleon. As in previous studies [18,19], we willlimit our study to such double scattering processes in a nuclear medium. These are twist-fourprocesses and give leading contributions to the nuclear effects. The contributions of higher twistprocesses or contributions not enhanced by the nuclear medium will be neglected for the timebeing.

When considering double scattering with nuclear enhancement, a very important process isquark–gluon double scattering as illustrated in Fig. 2. Such processes give the dominant contri-bution to the leading quark energy loss and have been studied in detailed in Refs. [18,19]. Themodification to the vacuum quark fragmentation function from quark–gluon scattering is,

�Dqg→qgq→h (zh) = α2

s CA

Nc

∫d�2

T

�4T

1∫zh

dz

z

{Dq→h(zh/z)

[1 + z2

(1 − z)+T A

qg(x, xL)

f Aq (x)

+ δ(z − 1)

× �T Aqg(x, �2

T )

f Aq (x)

]+ Dg→h(zh/z)

[1 + (1 − z)2

z

T Aqg(x, xL)

f Aq (x)

]}, (7)

where the +function is defined as

1∫0

dzF (z)

(1 − z)+≡

1∫0

dzF (z) − F(1)

1 − z(8)

for any F(z) that is sufficiently smooth at z = 1 and the twist-four quark–gluon correlationfunction,

T Aqg(x, xL) =

∫dy−

2πdy−

1 dy−2 ei(x+xL)p+y−(

1 − e−ixLp+y−2)(

1 − e−ixLp+(y−−y−1 )

)× 〈A|ψq(0)

γ +F +

σ (y−2 )F+σ (y−

1 )ψq(y−)|A〉θ(−y−2 )θ(y− − y−

1 ), (9)
2


Fig. 3. Diagram for leading order quark–antiquark annihilation with three possible cuts [central (C), left (L) andright (R)].

has explicit interference included. The matrix element in the virtual correction [the term withδ(z − 1)] is defined as

�T Aqg

(x, �2

T

) ≡1∫

0

dz1

1 − z

[2T A

qg(x, xL)|z=1 − (1 + z2)T A

qg(x, xL)]. (10)

Since T Aqg(x, xL)/f A

q (x) is proportional to gluon distribution and independent of the flavor ofthe leading quark, the suppression of the hadron spectrum caused by quark–gluon or antiquark–gluon scattering should be proportional to the gluon density of the medium and is identical forquark and antiquark fragmentation. It was shown in Ref. [24] that such modification of partonfragmentation functions by quark–gluon double scattering and gluon bremsstrahlung in a nuclearmedium describes very well the recent HERMES data [25] on semi-inclusive DIS off nucleartargets.

In this paper, we will consider quark–quark (antiquark) double scattering such as the processshown in Fig. 3 and its radiative corrections at order O(α2

s ) in Fig. 4. The contributions of quark–quark double scattering is proportional to the quark density in a nucleon, while the contributionof quark–gluon double scattering is proportional to the gluon density in a nucleon; and the gluondensity is generally larger than the quark density in a nucleon at small momentum fraction.However, as pointed out in earlier works [18], quark–quark double scattering mixes quark andgluon fragmentation functions and therefore gives rise to new nuclear effects. The annihilationprocesses as shown in Figs. 3 and 4 will lead to different modifications of quark and antiquarkfragmentation functions in a medium with finite baryon density (or valence quarks). Such differ-ences will in turn lead to flavor dependence of the nuclear modification of leading hadron spectraas observed in HERMES experiment [25,26].

Quark–quark double scattering as well as quark–gluon double scattering are twist-4 processes.We will apply the same generalized factorization procedure for twist-4 processes as developedby LQS [23] for semi-inclusive processes in DIS. In general, the twist-four contributions can beexpressed as the convolution of partonic hard parts and two-parton correlation matrix elements.In this framework, contributions from double quark–quark scattering in any order of αs , e.g.,the quark–antiquark annihilation process as illustrated in Fig. 4, can be written in the followingform,


Fig. 4. A typical diagram for next-to-leading order correction to quark–antiquark annihilation with three possible cuts[central (C), left (L) and right (R)].

dWDμν

dzh

=∑q

∫p+ dy−

2πdy−

1 dy−2 HD

μν(y−, y−

1 , y−2 ,p, q, zh)

× 〈A|ψq(0)γ +

2ψq(y−)ψq(y−

1 )γ +

2ψq(y−

2 )|A〉. (11)

Here we have neglected transverse momenta of all quarks in the hard partonic part. Transversemomentum dependent contributions are higher twist and are suppressed by 〈k2⊥〉/Q2. Therefore,all quarks’ momenta are assumed collinear, k2 = x2p and k3 = x3p. HD

μν(y−, y−

1 , y−2 ,p, q, zh)

is the Fourier transform of the partonic hard part Hμν(x, x1, x2,p, q, zh) in momentum space,

HDμν(y

−, y−1 , y−

2 ,p, q, zh)

=∫

dxdx1

2π

dx2

2πeix1p

+y−+ix2p+y−

1 +i(x−x1−x2)p+y−

2 HDμν(x, x1, x2,p, q, zh)

=∫

dx H(0)μν (x,p, q)HD(y−, y−

1 , y−2 , x,p, q, zh), (12)

where, in collinear approximation, the hard partonic part H(0)μν (x,p, q) [Eq. (4)] in the lead-

ing twist without multiple parton scattering can be factorized out of the high-twist hard partHD

μν(x, x1, x2,p, q, zh). The momentum fractions x, x1, and x2 are fixed by δ-functions of theon-shell conditions of the final state partons and poles of parton propagators in the partonic hardpart. The phase factors in HD

μν(y−, y−

1 , y−2 ,p, q, zh) can then be factored out, which in turn will

be combined with the partonic fields in Eq. (11) to give twist-four partonic matrix elements ortwo-parton correlations. The quark–quark double scattering corrections in Eq. (11) can then befactorized as the convolution of fragmentation functions, twist-four partonic matrix elements andthe partonic hard scattering cross sections. For scatterings (versus the annihilation) with quarks(antiquarks), a summation over the flavor of the secondary quarks (antiquarks) should be in-cluded in two-quark correlation matrix elements and both t , u channels and their interferencesshould be considered for scattering of identical quarks in the hard partonic parts.

After factorization, we then define the twist-four correction to the leading twist quark frag-mentation function in the same form [Eq. (3)],

dWDμν

dzh

≡∑∫

dx f Aq (x)H(0)

μν (x,p, q)�Dq→h(zh). (13)
q


3. Quark–quark double scattering processes

In this section we will discuss the calculation of the hard part of quark–quark double scatteringin detail. The lowest order process of quark–quark (antiquark) double scattering in nuclei isquark–antiquark annihilation (or quark–gluon conversion) as shown in Fig. 3. The hard partonicparts from the three cut diagrams in this figure are [18]:

HD0,C(y−, y−

1 , y−2 , x,p, q, zh)

= Dg→h(zh)2παs

Nc

2CF

xB

Q2eixp+y−

θ(−y−2 )θ(y− − y−

1 ), (14)

HD0,L(y−, y−

1 , y−2 , x,p, q, zh)

= Dq→h(zh)2παs

Nc

2CF

xB

Q2eixp+y−

θ(y−1 − y−

2 )θ(y− − y−1 ), (15)

HD0,R(y−, y−

1 , y−2 , x,p, q, zh)

= Dq→h(zh)2παs

Nc

2CF

xB

Q2eixp+y−

θ(−y−2 )θ(y−

2 − y−1 ). (16)

The main focus of this paper is about contributions from the next-leading order correctionsto the above lowest order process. There is a total of 12 diagrams for real corrections at one-loop level as illustrated in Fig. 5 to Fig. 16 in Appendix A.1, each having up to three differentcuts. In this section, we demonstrate the calculation of the hard parts from the quark–antiquarkannihilation in Fig. 4 in detail as an example. We will list the complete results of all diagrams inAppendix A.

One can write down the hard partonic part of the central-cut diagram of Fig. 4 (Fig. 5 inAppendix A.1) according to the conventional Feynman rule,

HDCμν(y

−, y−1 , y−

2 ,p, q, zh)

=1∫

zh

dz

zDg→h

(zh

z

)∫dx

dx1

2π

dx2

2πeix1p

+y−+ix2p+y−

1

× ei(x−x1−x2)p+y−

2

∫d4�

(2π)4Tr

[/p

2γμHγν

]2πδ+

(�2)2πδ+

(�2g

)δ

(1 − z − �−

q−

),

H = C2F

Nc

g4 γ · (q + x1p)

(q + x1p)2 − iεγα

γ · (q + x1p − �)

(q + x1p − �)2 − iεγβ

/p

2γσ

× γ · (q + xp − �)

(q + xp − �)2 + iεγρ

γ · (q + xp)

(q + xp)2 + iεεαρ(�)εβσ (�g), (17)

where δ+ is a Dirac δ-function with only the positive solution in its functional variable, εαρ(�) =−gαρ + (nα�ρ + nρ�α)/n · � is the polarization tensor of a gluon propagator in an axial gauge(n ·A = 0) with n = [1,0−, �0⊥], � and �g = q + (x1 +x2)p − � are the 4-momenta carried by thetwo final gluons respectively. The fragmenting gluon carries a fraction, z = �−

g /q−, of the initialquark’s longitudinal momentum (the large minus component).

To simplify the calculation in the case of small transverse momentum �T q−,p+, we canapply the collinear approximation to complete the trace of the product of γ -matrices,

H ≈ γ · �q

1

4�− Tr[γ · �qH ]. (18)
g


According to the convention in Eqs. (11) and (12), contributions from quark–quark double scat-tering in the nuclear medium to the semi-inclusive hadronic tensor in DIS off a nucleus can beexpressed in the general factorized form:

dWDqq,μν

dzh

=∑q

∫dx H(0)

μν (x,p, q)

∫p+dy−

2πdy−

1 dy−2 HD(y−, y−

1 , y−2 , x,p, q, zh)

× 〈A|ψq(0)γ +

2ψq(y−)ψq(y−

1 )γ +

2ψq(y−

2 )|A〉. (19)

After carrying out the momentum integration in x, x1, x2 and �± in Eq. (17) with the help ofcontour integration and δ-functions, one obtains the hard partonic part, HD , of the rescatteringfor the central-cut diagram in Fig. 4 (Fig. 5) as

HD5,C(y−, y−

1 , y−2 , x,p, q, zh)

= α2s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zDg→h

(zh

z

)C2

F

Nc

2(1 + z2)

z(1 − z)I5,C(y−, y−

1 , y−2 , x, xL,p), (20)

I5,C(y−, y−1 , y−

2 , x, xL,p)

= ei(x+xL)p+y−θ(−y−

2 )θ(y− − y−1 )

(1 − e−ixLp+y−

2)(

1 − e−ixLp+(y−−y−1 )

), (21)

where the momentum fractions xL is defined as

xL = �2T

2p+q−z(1 − z). (22)

Note that the function I5,C(y−, y−1 , y−

2 , x, xL,p) contains only phase factors. One can combinethese phase factors with the matrix elements of the quark fields to define a special two-quarkcorrelation function

TA(5,C)qq (x, xL) =

∫p+ dy−

2πdy−

1 dy−2 〈A|ψq(0)

γ +

2ψq(y−)ψq(y−

1 )γ +

2ψq(y−

2 )|A〉× I5,C(y−, y−

1 , y−2 , x, xL,p). (23)

The contribution from quark–antiquark annihilation in the central-cut diagram in Fig. 4 to thehadronic tensor can then be expressed as

dWDqq,μν

dzh

=∑q

∫dx H(0)

μν (x,p, q)α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zDg→h

(zh

z

)

× C2F

Nc

2(1 + z2)

z(1 − z)T

A(5,C)qq (x, xL). (24)

Contributions from all quark–quark (antiquark) double scattering processes can be cast in theabove factorized form.

The structure of the phase factors in I5,C(y−, y−1 , y−

2 , x, xL,p) is exactly the same as forgluon bremsstrahlung induced by quark–gluon scattering as studied in Refs. [18,19]. It resemblesthe cross section of dipole scattering and represents contributions from two different processesand their interferences. It contains essentially four terms,


I5,C(y−, y−1 , y−

2 , x, xL,p)

= θ(−y−2 )θ(y− − y−

1 )ei(x+xL)p+y−

× [1 + e−ixLp+(y−+y−

2 −y−1 ) − e−ixLp+y−

2 − e−ixLp+(y−−y−1 )

]. (25)

The first term corresponds to the so-called hard–soft processes where the gluon emission is in-duced by the hard scattering between the virtual photon γ ∗ and the initial quark with momentum(x + xL)p. The quark then becomes on-shell before it annihilates with a soft antiquark fromthe nucleus that carries zero momentum and converts into a real gluon in the final state. Thesecond term corresponds to a process in which the initial quark with momentum xp is on-shellafter the first hard γ ∗-quark scattering. It then annihilates with another antiquark and producestwo final gluons in the final state. In this process, the antiquark carries finite (hard) momentumxLp. Therefore one often refers to this process as double-hard scattering as compared to the firstprocess in which the antiquark carries zero momentum. Set aside the change of flavors in theinitial and final states, the double-hard scattering corresponds essentially to two-parton elasticscattering with finite momentum and energy transfer. This is in contrast to the hard–soft scatter-ing which is essentially the final state radiation of the γ ∗-quark scattering and the total energyand momentum of the two final state gluons all come from the initial quark. The correspond-ing matrix elements of the two-quark correlation functions from these first two terms are called‘diagonal’ elements.

The third and fourth terms with negative signs in I5,C(y−, y−1 , y−

2 , x, xL,p) are interferencesbetween hard–soft and double hard processes. The corresponding matrix elements are called ‘off-diagonal’. The cancellation between the two diagonal and off-diagonal terms essentially givesrise to the destructive interference which is very similar to the Landau–Pomeranchuk–Migdal(LPM) interference in gluon bremsstrahlung induced by quark–gluon double scattering [18,19].One can similarly define the formation time of the parton (quark or gluon) emission as

τf ≡ 1

xLp+ . (26)

In the limit of collinear emission (xL → 0) or when the formation time of the parton emission,τf , is much larger than the nuclear size, the effective matrix element vanishes because

I5,C(y−, y−1 , y−

2 , x, xL,p)|xL=0 → 0, (27)

when the hard–soft and double hard processes have complete destructive interference.We should note that in the central-cut diagram of Fig. 4, the final state partons are two gluons.

Therefore, in Eq. (20) the gluon fragmentation function in vacuum Dg→h(zh/z) enters. If theother gluon (close to the γ ∗-quark interaction) fragments, the contribution to the semi-inclusivehadronic tensor is similar except that the corresponding effective “splitting function” should bereplaced by

1 + z2

z(1 − z)→ 1 + (1 − z)2

z(1 − z). (28)

As we will show in Appendix A.1, the two gluons in the quark–antiquark annihilation processes(central-cut diagrams) are symmetric when contributions from all possible annihilation processesand their interferences are summed. Therefore, one can simply multiply the final results by afactor of 2 to take into account the hadronization of the second final-state gluon.

In addition to the central-cut diagram, one should also take into account the asymmetrical-cutdiagrams in Fig. 4, which represent interference between gluon emission from single and triple


scattering. The hard partonic parts are mainly the same as for the central-cut diagram. The onlydifferences are in the phase factors and the fragmentation functions since the fragmenting partoncan be the final-state quark or gluon. These hard parts can be calculated following a similarprocedure and one gets,

HD5,L(R)(y

−, y−1 , y−

2 , x,p, q, zh)

= α2s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zDq→h

(zh

z

)C2

F

Nc

2(1 + z2)

z(1 − z)I5,L(R)(y

−, y−1 , y−

2 , x, xL,p), (29)

I5,L(y−, y−1 , y−

2 , x, xL,p)

= −ei(x+xL)p+y−(1 − e−ixLp+(y−−y−

1 ))θ(y−

1 − y−2 )θ(y− − y−

1 ), (30)

I5,R(y−, y−1 , y−

2 , x, xL,p)

= −ei(x+xL)p+y−(1 − e−ixLp+y−

2)θ(−y−

2 )θ(y−2 − y−

1 ). (31)

In the asymmetrical cut diagrams, the above contributions come from the fragmentation of thefinal-state quark. Therefore, quark fragmentation function Dq→h(zh/z) enters this contribution.For gluon fragmentation into the observed hadron in this asymmetrical-cut diagrams, the con-tribution can be obtained by simply replacing the quark fragmentation function by the gluonfragmentation function Dg→h(zh/z) and replacing z by 1 − z. Summing the contributions fromthree different cut diagrams of Fig. 4, we can observe further examples of mixing (or conversion)of quark and gluon fragmentation functions. This medium-induced mixing was first observed byWang and Guo [18] and is a unique feature of quark–quark (antiquark) double scattering amongall multiple parton scattering processes.

With the same procedure we can calculate contributions from all other cut diagrams of quark–quark (antiquark) double scattering at order O(α2

s ), which are listed in Appendix A.1. There arethree types of processes: two annihilation processes, qq → gg (central-cut diagrams in Figs. 5, 6,7, 8 and 9), qq → qi qi (central-cut diagram in Fig. 10) and quark–quark (antiquark) scattering,qqi(qi) → qqi(qi) (central-cut diagram in Fig. 11). One also has to consider the interferenceof s and t-channel amplitude for annihilation into an identical quark pair, qq → qq (central-cutdiagrams in Figs. 12 and 13) and the interference between t and u channels of identical quarkscattering qq → qq (central-cut diagram in Fig. 14).

Contributions from left and right-cut diagrams correspond to interference between the am-plitude of gluon radiation from single γ ∗-quark scattering and triple quark scattering. Theamplitudes of gluon radiation via triple quark scattering essentially come from radiative cor-rections to the left and right-cut diagrams of the lowest-order quark–antiquark annihilation inFig. 3 (as shown in left and right-cut diagrams in Figs. 5, 6, 8, 9, 12, 13, 15 and 16). Two othertriple quark scatterings with gluon radiation, shown as the left and right-cut diagrams in Figs. 11and 14, correspond to the case where one of the final state quarks, after quark–quark scattering,annihilates with another antiquark and converts into a final state gluon.

4. Modified fragmentation functions

In order to simplify the contributions from quark–quark (antiquark) scattering (annihila-tion), one can first organize the results of the hard parts in terms of contributions from central,left or right-cut diagrams, which are associated by contour integrals with specific products ofθ -functions,


HD = HDC θ(−y−

2 )θ(y− − y−1 ) − HD

L θ(y−1 − y−

2 )θ(y− − y−1 )

− HDR θ(−y−

2 )θ(y−2 − y−

1 ). (32)

These θ -functions provide a space–time ordering of the parton correlation and will restrict theintegration range along the light-cone. For contributions from central, left and right-cut diagramsthat have identical hard partonic parts, H

D(c)C = H

D(c)L = H

D(c)R , they will have a common com-

bination of θ -functions that produces a path-ordered integral,

y−∫0

dy−1

y−1∫

0

dy−2 = −

∫dy−

1 dy−2

[θ(−y−

2 )θ(y− − y−1 ) − θ(−y−

2 )θ(y−2 − y−

1 )

− θ(y− − y−1 )θ(y−

1 − y−2 )

](33)

that is limited only by the spatial-spread y− of the first parton along the light-cone coordinate.For a high-energy parton that carries momentum fraction xp+, y− ∼ 1/xp+ should be verysmall. Those contributions that are proportional to the above path-ordered integral are referred toas contact contributions (or contact interactions).

Similarly, y−1 − y−

2 is the spatial spread of the second parton and can only be limited by thespatial size of its host nucleon even for small value of momentum fraction. The spatial positionof its host nucleon, y−

1 + y−2 , however, can be anywhere within the nucleus. Therefore, any con-

tributions from double parton scattering that have unrestricted integration over y−1 and y−

2 shouldbe proportional to the nuclear size of the target A1/3 and therefore are nuclear enhanced. In thispaper, we will only keep the nuclear enhanced contributions and neglect the contact contribu-tions. This will greatly simplify the final results for double parton scattering.

4.1. qq → g annihilation

For the lowest order of quark–antiquark annihilation in Eqs. (14)–(16), the hard parts fromthe three cut diagrams are almost the same except for the parton fragmentation functions. Thecentral-cut diagram is proportional to the gluon fragmentation function while the left and right-cut diagrams are proportional to quark fragmentation functions. Rearranging the contributionsfrom the three cut diagrams and neglecting the contact term that is proportional to the path-ordered integral as in Eq. (33), the total contribution can be written as

dWD(0)μν

dzh

=∑q

∫dx T

A(H)qq (x,0)

2παs

Nc

2CF

xB

Q2H(0)

μν (x,p, q)

× [Dg→h(zh) − Dq→h(zh)

]. (34)

According to our definition in Eq. (13) of the twist-four correction to the quark fragmentationfunctions, the modification to the quark fragmentation function from the lowest order quark–antiquark annihilation is then,

�D(qq→g)q→h (zh) = 2παs

Nc

2CF

xB

Q2

[Dg→h(zh) − Dq→h(zh)

]TA(H)qq (x,0)

f Aq (x)

. (35)

Here the effective quark–antiquark correlation function TA(H)

(x,0) is defined as,
qq


TA(H)qqi

(x, xL) ≡∫

p+ dy−

2πdy−

1 dy−2 eixp+y−−ixLp+(y−

2 −y−1 )θ(−y−

2 )θ(y− − y−1 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉, (36)

with the antiquark qi carrying momentum fraction xL. This two-parton correlation functionis always associated with double-hard rescattering processes. Similarly, we define three otherquark–antiquark correlation matrix elements

TA(S)qqi

(x, xL) ≡∫

p+ dy−

2πdy−

1 dy−2 ei(x+xL)p+y−

θ(y− − y−1 )θ(−y−

2 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉, (37)

TA(I−L)qqi

(x, xL) ≡∫

p+ dy−

2πdy−

1 dy−2 ei(x+xL)p+y−−ixLp+(y−−y−

1 )θ(y− − y−1 )θ(−y−

2 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉, (38)

TA(I−R)qqi

(x, xL) ≡∫

p+ dy−

2πdy−

1 dy−2 ei(x+xL)p+y−−ixLp+y−

2 θ(y− − y−1 )θ(−y−

2 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉, (39)

that are associated with hard–soft rescattering and interference between double hard and hard–soft rescattering. In the first parton correlation T

A(H)qqi

(x, xL), the antiquark qi carries momentumfraction xL while the initial quark has the momentum fraction x. The two-parton correlationT

A(S)qqi

(x, xL) corresponds to the case when the leading quark has x +xL but the antiquark carrieszero momentum. The two interference matrix elements are approximately the same for smallvalue of xL and will be denoted as T

A(I)qqi

(x, xL).

4.2. qq → qi qi annihilation

Contributions from the next-to-leading order quark–antiquark annihilation or quark–quark(antiquark) scattering are more complicated since they involve many real and virtual correc-tions. The simplest real correction comes from qq → qi qi annihilation (qi �= q) [Fig. 10 andEqs. (A.25) and (A.26)] which has only a central-cut diagram,

�D(qq→qi qi )q→h (zh) = CF

NC

α2s xB

Q2

∫d�2

T

�2T

1∫zh

dz

z

[z2 + (1 − z)2]

×∑qi �=q

[Dqi→h(zh/z) + Dqi→h(zh/z)

]TA(H)qq (x, xL)

f Aq (x)

. (40)

This kind of qq annihilation is truly a hard processes and thus requires the second antiquark tocarry finite initial momentum fraction xL. Furthermore, there are no other interfering processes.


4.3. qqi(qi) → qqi(qi) scattering

Contributions from non-identical quark–quark scattering qqi → qqi (qi �= q) are a little com-plicated because they involve all three cut diagrams (central, left and right) [Eqs. (A.28)–(A.32)].One can factor out the θ -functions in the hard parts according to Eq. (32) and re-organize thephase factors in each cut diagram,

I11,C = ei(x+xL)p+y−(1 − e−ixLp+y−

2)(

1 − e−ixLp+(y−−y−1 )

)= ei(x+xL)p+y−[

1 − e−ixLp+y−2 − e−ixLp+(y−−y−

1 ) + e−ixLp+(y−+y−2 −y−

1 )];

I11,L = ei(x+xL)p+y−(1 − e−ixLp+(y−−y−

1 ))

= ei(x+xL)p+y−[1 − e−ixLp+y−

2 − e−ixLp+(y−−y−1 ) + e−ixLp+y−

2];

I11,R = ei(x+xL)p+y−(1 − e−ixLp+y−

2)

= ei(x+xL)p+y−[1 − e−ixLp+y−

2 − e−ixLp+(y−−y−1 ) + e−ixLp+(y−−y−

1 )], (41)

such that the first three terms in each amplitude are the same. These three common phase factorswill give rise to a contact contribution for all similar hard parts from the three cut diagrams,which we will neglect since they are not nuclear enhanced. The remaining part will have thefollowing phase factors,

I11 = ei(x+xL)p+y−[θ(−y−

2 )θ(y− − y−1 )e−ixLp+(y−+y−

2 −y−1 )

− θ(y−1 − y−

2 )θ(y− − y−1 )e−ixLp+y−

2 − θ(−y−2 )θ(y−

2 − y−1 )e−ixLp+(y−−y−

1 )]. (42)

Note that the phase factors of the last two terms in the above equation give identical contributionsto the matrix elements when intergated over y−

1 and y−2 as they differ only by the substitution

y−2 ↔ y−

1 − y−. One therefore can combine them with θ(−y−2 )θ(y− − y−

1 )e−ixLp+(y−−y−1 ) to

form another contact contribution (path-ordered) which can be neglected. The final effectivephase factor is then

I11 = eixp+y−−ixLp+(y−2 −y−

1 )(1 − eixLp+y−

2). (43)

Using the above effective phase factor, one can obtain the effective modification to the quarkfragmentation function due to quark–antiquark scattering, qqi → qqi ,

�D(qqi→qqi )q→h (zh)

= CF

Nc

α2s xB

Q2

∫d�2

T

�2T

1∫zh

dz

z

∑qi �=q

{[Dq→h

(zh

z

)1 + z2

(1 − z)2

+ Dg→h

(zh

z

)1 + (1 − z)2

z2

]T

A(HI)qqi

(x, xL)

f Aq (x)

+[Dqi→h

(zh

z

)− Dg→h

(zh

z

)]1 + (1 − z)2

z2

TA(HS)qqi

(x, xL)

f Aq (x)

}

= CF

Nc

α2s xB

Q2

∫d�2

T

�2T

1∫dz

z

∑qi �=q

{[Dq→h

(zh

z

)1 + z2

(1 − z)2
zh


+ Dqi→h

(zh

z

)1 + (1 − z)2

z2

]T

A(HI)qqi

(x, xL)

f Aq (x)

+[Dqi→h

(zh

z

)− Dg→h

(zh

z

)]1 + (1 − z)2

z2

TA(SI)qqi

(x, xL)

f Aq (x)

}, (44)

where three types of two-parton correlations are defined:

TA(HI)qqi

(x, xL) ≡ TA(H)qqi

(x, xL) − TA(I)qqi

(x, xL)

=∫

p+ dy−

2πdy−


2 −y−1 )

(1 − eixLp+y−

2)

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉θ(−y−

2 )θ(y− − y−1 ),

(45)

TA(SI)qqi

(x, xL) ≡ TA(S)qqi

(x, xL) − TA(I)qqi

(x, xL)

=∫

p+ dy−

2πdy−


1 − e−ixLp+y−2)

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉θ(−y−

2 )θ(y− − y−1 ),

(46)

TA(HS)qqi

(x, xL) ≡ TA(HI)qqi

(x, xL) + TA(SI)qqi

(x, xL)

=∫

p+ dy−

2πdy−



× (1 − e−ixLp+(y−−y−

1 ))θ(−y−

2 )θ(y− − y−1 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉. (47)

One can similarly obtain the modification of quark fragmentation from non-identical quark–quark scattering by replacing qi → qi in Eq. (44),

�D(qqi→qqi )q→h (zh)

= CF

Nc

α2s xB

Q2

∫d�2

T

�2T

1∫zh

dz

z

∑qi �=q

{[Dq→h

(zh

z

)1 + z2

(1 − z)2

+ Dqi→h

(zh

z

)1 + (1 − z)2

z2

]T

A(HI)qqi

(x, xL)

f Aq (x)

+[Dqi→h

(zh

z

)− Dg→h

(zh

z

)]1 + (1 − z)2

z2

TA(SI)qqi

(x, xL)

f Aq (x)

}. (48)

The two-quark correlations, TA(HI)qqi

(x, xL) and TA(SI)qqi

(x, xL) can be obtained from

TA(HI)qqi

(x, xL) and TA(SI)qqi

(x, xL), respectively, by making the replacements ψqi(y2) → ψqi

(y2)

and ψqi(y1) → ψqi

(y1) in Eqs. (45) and (46),


T A(HI)qqi

(x, xL)

≡∫

p+ dy−

2πdy−


2 −y−1 )

(1 − eixLp+y−

2)

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−2 )

γ +

2ψqi

(y−1 )|A〉θ(−y−

2 )θ(y− − y−1 ), (49)

T A(SI)qqi

(x, xL)

=∫

p+ dy−

2πdy−



× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−2 )

γ +

2ψqi

(y−1 )|A〉θ(−y−

2 )θ(y− − y−1 ), (50)

and TA(HS)qqi

(x, xL) = TA(HI)qqi

(x, xL) + TA(SI)qqi

(x, xL).Note that the contribution from fragmentation of quark qi or antiquark qi only comes from

the central-cut diagram. This contribution is positive and is proportional to TA(HI)qqi

(x, xL) +T

A(SI)qqi

(x, xL), containing all four terms: hard–soft, double-hard and both interference terms.The gluon fragmentation comes only from the single–triple interferences (left and right-cut dia-grams). Its contribution is therefore negative and partially cancels the production of qi(qi) fromthe hard–soft rescattering. The cancellation is not complete since the gluon and quark fragmen-tation functions are different. The structure of this hard–soft rescattering (quark plus gluon) isvery similar to the lowest order result of qq → g in Eq. (35). It contributes to the modificationof the effective fragmentation function but does not contribute to the energy loss. The energyloss of the leading quark comes only from double-hard rescattering, since the leading quark frag-mentation comes both from the central-cut and single–triple interferences, and the single–tripleinterference terms cancel the effect of hard–soft scattering for the leading fragmentation. Its netcontribution is therefore proportional to T

A(HI)qqi (qi )

. Since the double-hard rescattering amounts toelastic qqi(qi) scattering, the effective energy loss is essentially elastic energy loss as shown inRef. [34]. There is, however, LPM suppression due to partial cancellation by single–triple in-terference contributions. For long formation time, 1/xLp+ � RA, the cancellation is complete.Therefore, LPM interference effectively imposes the lower limit xL � 1/p+RA on the fractionalmomentum carried by the second quark (antiquark).

4.4. qq → qq scattering

For identical quark–quark scattering, qq → qq , one has to include both t- and u-channels,their interferences, and the related single–triple interference contributions. Using the same tech-nique to identify and neglect the contact contributions, one can find the corresponding modifica-tion to the quark fragmentation function from Eqs. (48) and (A.45)–(A.49),

�D(qq→qq)q→h (zh)

= CF

Nc

α2s xB

Q2

∫d�2

T

�2T

1∫zh

dz

z

{T

A(HS)qq (x, xL)

f Aq (x)

×[Dq→h

(zh

)− Dg→h

(zh

)](1 + (1 − z)2

2− 1 1

)
z z z Nc z(1 − z)


+[Dq→h

(zh

z

)(1 + z2

(1 − z)2− 1

Nc

1

z(1 − z)

)

+ Dg→h

(zh

z

)(1 + (1 − z)2

z2− 1

Nc

1

z(1 − z)

)]T

A(HI)qq (x, xL)

f Aq (x)

}

= CF

Nc

α2s xB

Q2

∫d�2

T

�2T

1∫zh

dz

z

{T

A(SI)qq (x, xL)

f Aq (x)

P (s)qq→qq(z)

[Dq→h

(zh

z

)− Dg→h

(zh

z

)]

+ Dq→h

(zh

z

)Pqq→qq(z)

TA(HI)qq (x, xL)

f Aq (x)

}, (51)

where the effective splitting functions are defined as

P (s)qq→qq(z) = 1 + (1 − z)2

z2− 1

Nc

1

z(1 − z), (52)

Pqq→qq(z) = 1 + (1 − z)2

z2+ 1 + z2

(1 − z)2− 2

Nc

1

z(1 − z). (53)

4.5. qq → qq , gg annihilation

The most complicated twist-four processes involving four quark field operators are quark–antiquark annihilation into two gluons or an identical quark–antiquark pair. We have to considerthem together since they have similar single–triple interference processes and they involve thesame kind of quark–antiquark correlation matrix elements, T

(i)qq (x, xL), (i = HI,SI,HS).

For notation purpose, we first factor out the common factor (CF /Nc)α2s xB/Q2/f A

q (x) andthe integration over �T and z and define

�D(qq→gg,qq)q→h (zh) ≡ CF

Nc

α2s xB

Q2f Aq (x)

∫d�2

T

�2T

1∫zh

dz

z�D

(qq→gg,qq)q→h (zh, z, x, xL). (54)

After rearranging the phase factors and identifying (by combining central, left and right cut dia-grams) and neglecting contact contributions we can list in the following the twist-four correctionsto the quark fragmentation from the hard partonic parts of each cut diagram (see Appendix A):

Fig. 5 (t-channel qq → gg),

�D(qq→gg,qq)

q→h(5) = Dg→h

(zh

z

)2CF

[1 + (1 − z)2

z(1 − z)+ 1 + z2

z(1 − z)

]T

A(HI)qq (x, xL)

+[Dg→h

(zh

z

)− Dq→h

(zh

z

)]2CF

1 + z2

z(1 − z)T

A(SI)qq (x, xL); (55)

Fig. 6 (interference between u and t-channel of qq → gg),

�D(qq→gg,qq)

q→h(6) = Dg→h

(zh

z

)−4(CF − CA/2)

z(1 − z)T

A(HI)qq (x, xL)

+[Dg→h

(zh

z

)− Dq→h

(zh

z

)]−2(CF − CA/2)

z(1 − z)T

A(SI)qq (x, xL); (56)


Fig. 7 (s-channel of qq → gg),

�D(qq→gg,qq)

q→h(7) = Dg→h

(zh

z

)4CA

(1 − z + z2)2

z(1 − z)T

A(H)qq (x, xL); (57)

Figs. 8 and 9 (interference of s and t-channel qq → gg),

�D(qq→gg,qq)

q→h(8+9) = Dg→h

(zh

z

)(−2CA)

[1 + z3

z(1 − z)+ 1 + (1 − z)3

z(1 − z)

]T

A(HI)qq (x, xL)

+ CA

[Dq→h

(zh

z

)1 + z3

z(1 − z)+ Dg→h

(zh

z

)1 + (1 − z)3

z(1 − z)

]

× [T

A(I2)qq (x, xL) − T

A(I)qq (x, xL)

]; (58)

Fig. 10 (s-channel of qq → qq),

�D(qq→gg,qq)

q→h(10) =[Dq→h

(zh

z

)+ Dq→h

(zh

z

)][z2 + (1 − z)2]T A(H)

qq (x, xL); (59)

Fig. 11 (t-channel of qq → qq), similar to Eq. (44),

�D(qq→gg,qq)

q→h(11) = Dq→h

(zh

z

)1 + z2

(1 − z)2T

A(HI)qq (x, xL)

+ Dq→h

(zh

z

)1 + (1 − z)2

z2T

A(HS)qq (x, xL)

− Dg→h

(zh

z

)1 + (1 − z)2

z2T

A(SI)qq (x, xL); (60)

Figs. 12 and 13 (interference between s and t-channel qq → qq),

�D(qq→gg,qq)

q→h(12+13)

= −4

(CF − CA

2

)[Dq→h

(zh

z

)z2

1 − z+ Dq→h

(zh

z

)(1 − z)2

z

]T

A(HI)qq (x, xL)

+ 2

(CF − CA

2

)[Dq→h

(zh

z

)z2

1 − z+ Dg→h

(zh

z

)(1 − z)2

z

]

× [T

A(I2)qq (x, xL) − T

A(I)qq (x, xL)

]; (61)

Figs. 15 and 16 (two additional single–triple interference diagrams),

�D(qq→gg,qq)

q→h(15+16)

= −2CF

[Dq→h

(zh

z

)1 + z2

1 − z+ Dg→h

(zh

z

)1 + (1 − z)2

z

]T

A(I2)qq (x, xL). (62)

Most processes involve both T A(HI)(x, xL) for double-hard rescattering with interferenceand T A(SI)(x, xL) for hard–soft rescattering with interference. All the s-channel (Figs. 7and 10) processes involve double-hard scattering only. Therefore, they depend only on theT

A(H)qq (x, xL) = T

A(HI)qq (x, xL) + T

A(I)qq (x, xL). For interference between single and triple scat-

tering (left and right-cut diagrams in Figs. 8, 9, 12 13, 15 and 16), where a hard rescattering with


the second quark (antiquark) follows a soft rescattering with the third antiquark (quark), onlyinterference matrix elements, T

A(I)qq (x, xL) and T

A(I2)qq (x, xL), are involved. Here,

TA(I2)qq (x, xL) ≡

∫p+ dy−

2πdy−

1 dy−2 eixp+y−+ixLp+y−

2

× 〈A|ψq(0)γ +

2ψq(y−)ψq(y−

1 )γ +

2ψq(y−

2 )|A〉θ(−y−2 )θ(y− − y−

1 )

=∫

p+ dy−

2πdy−

1 dy−2 eixp+y−+ixLp+(y−−y−

1 )

× 〈A|ψq(0)γ +

2ψq(y−)ψq(y−

1 )γ +

2ψq(y−

2 )|A〉θ(−y−2 )θ(y− − y−

1 ),

(63)

is a new type of interference matrix elements that is only associated with this type of single–tripleinterference processes. One can categorize the above contributions according to the associatedtwo-quark correlation matrix elements and rewrite the above contributions as,

�Dqq→qq,gg

q→h(HI)= T

A(HI)qq (x, xL)

[Dg→h

(zh

z

)Pqq→gg(z) + Dq→h

(zh

z

)Pqq→qq (z)

+ Dq→h

(zh

z

)Pqq→qq (1 − z)

], (64)

�Dqq→qq,gg

q→h(SI) = TA(SI)qq (x, xL)

{[CA

z(1 − z)+ 2CF

z

1 − z− 1 + (1 − z)2

z2

]Dg→h

(zh

z

)

− Dq→h

(zh

z

)[CA

z(1 − z)+ 2CF

z

1 − z

]+ Dq→h

(zh

z

)1 + (1 − z)2

z2

}

= TA(SI)qq (x, xL)

{[Dq→h

(zh

z

)− Dg→h

(zh

z

)][P (s)

qq→qq(z)

− 2CF

1 + z2

z(1 − z)

]+

[Dq→h

(zh

z

)− Dq→h

(zh

z

)]1 + (1 − z)2

z2

}, (65)

�Dqq→qq,gg

q→h(I) = TA(I)qq (x, xL)

{[CA

4(1 − z + z2)2 − 1

z(1 − z)− 2CF

(1 − z)2

z

]Dg→h

(zh

z

)

+ [z2 + (1 − z)2]Dq→h

(zh

z

)

+ Dq→h

(zh

z

)[z2 + (1 − z)2 − CA

z(1 − z)− 2CF

z2

1 − z

]}

+ TA(I2)qq (x, xL)

{Dq→h

(zh

z

)[CA

z(1 − z)− 2CF

1 − z

]

+ Dg→h

(zh

z

)[CA

z(1 − z)− 2CF

z

]}, (66)

where P(s)qq→qq(z) is given in Eq. (52) and the effective splitting functions for qq → gg and

qq → qq are defined as


Pqq→gg(z) = 2CF

z2 + (1 − z)2

z(1 − z)− 2CA

[z2 + (1 − z)2], (67)

Pqq→qq (z) = z2 + (1 − z)2 + 1 + z2

(1 − z)2+ 2

Nc

z2

1 − z, (68)

which come from the complete matrix elements of qq → gg and qq → qq scattering (see Ap-pendix A.3). Again, double-hard rescattering corresponds to the elastic scattering of the leadingquark with another antiquark in the medium and the interference contributions. The structure ofthe hard–soft rescattering contribution we identify above shows the same kind of gluon–quark(or quark–antiquark) mixing in the fragmentation functions and does not contribute to the en-ergy loss of the leading quark. The unique contributions in the qq → qq, gg processes are theinterference-only contributions. They mainly come from single–triple interference processes inthe multiple parton scattering.

5. Modification due to quark–gluon mixing

We have so far cast the modification of the quark fragmentation function due to quark–quark(antiquark) scattering (or annihilation) in a form similar to the DGLAP evolution equation invacuum. In fact, one can also view the evolution of fragmentation functions in vacuum as mod-ification due to final-state gluon radiation. In both cases, the modification at large zh is mainlydetermined by the singular behavior of the splitting functions for z → 1, whereas the modifica-tions at mall zh is dominated by the singular behavior of the splitting function for z → 0.

Let us first focus on the modification at large zh. A careful examination of the contributionsfrom all possible processes shows that the dominant modification to the effective quark fragmen-tation function comes from the t-channel of double hard quark–quark scattering processes,

�Dq→h(zh)

∼ CF

Nc

α2s xB

Q2

∑qi

∫d�2

T

�2T

1∫zh

dz

zDq→h

(zh

z

)[T

A(HI)qqi

(x, xL)

f Aq (x)

1 + z2

(1 − z)2+

+ δ(1 − z)Δqi

(�2T

)]

= CF

Nc

α2s

∑qi

∫d�2

T

�4T

1∫zh

dz

zDq→h

(zh

z

)

×[xLT

A(HI)qqi

(x, xL)

f Aq (x)

z(1 + z2)

(1 − z)++ δ(1 − z)Δqi

(�2T

)], (69)

where the summation is over all possible quark and antiquark flavors including qi = q, q andΔqi

(�2T ) represents the contribution from virtual corrections. We have expressed the modification

in a form that it is proportional to the matrix elements xLTA(HI)qqi

(x, xL)/f Aq (x) ∼ A1/3xLf N

qi(xL)

as compared to the modification from quark–gluon scattering where the corresponding matrix el-ement [Eq. (9)] is T

A(HI)qg (x, xL)/f A

q (x) ∼ A1/3xLGN(xL). Here, f Nqi

(x) and GN(x) are quarkand gluon distributions, respectively, in a nucleon. This leading contribution to the modificationfrom quark–quark scattering is very similar in form to that from quark–gluon scattering [seeEq. (7)]. However, it is smaller due to the different color factors CF /CA = 4/9 and the different


quark and gluon distributions, f Nqi

(xL) and GN(xL) in a nucleon. Because of LPM interference,small angle scattering with long formation time τf = 1/xLp+ is suppressed, leading to a mini-mum value of xL � xA = 1/mNRA = 0.043 for a Kr target. For this value of xL, the ratio∑

qif N

qi(xL,Q2)

GN(xL,Q2)� 1.40

1.85∼ 0.75, (70)

at Q2 = 2 GeV2 according the CTEG4HJ parameterization [35]. Therefore, one has to includethe effect of quark–quark scattering for a complete calculation of the total quark energy loss andmedium modification of quark fragmentation functions.

In a weakly coupled and fully equilibrated quark–gluon plasma, quark to gluon number den-sity ratio is ρq/ρg = nf (3/2)Nc/(N

2c − 1) = 9nf /16. An asymptotically energetic jet in an

infinitely large medium actually probes the small x = 〈q2T 〉/2ET regime, where quark–antiquark

pairs and gluons are predominantly generated by thermal gluons through pQCD evolution. Inthis ideal scenario one expects Nq/Ng ∼ 1/4CA = 1/12 and therefore can neglect quark–quarkscattering. The modification of quark fragmentation function will be dominated by quark–gluonrescattering. However, for moderate jet energy E ≈ 20 GeV and a finite medium L ∼ 5 fm, par-ton distributions in a quark–gluon plasma are close to the thermal distribution. In particular, ifquark and gluon production is dominated by non-perturbative pair production from strong colorfields in the initial stage of heavy-ion collisions [36], the quark to gluon ratio is comparable tothe equilibrium value. In this case, we should take into account the medium modification of thequark fragmentation functions by quark–quark scattering.

An important double hard process in quark–quark (antiquark) scattering is qq → gg

[Eq. (64)]. In this process, the annihilation converts the initial quark into two final gluons thatsubsequently fragment into hadrons. This will lead to suppression of the leading hadrons notonly because of energy loss (energy carried away by the other gluon) but also due to the softerbehavior of gluon fragmentation functions at large zh. Even though the leading behavior of theeffective splitting function [Eq. (67)]

Pqq→gg(z) ≈ 2CF

z2 + (1 − z)2

z(1 − z)(71)

is not as dominating as that of t-channel quark–quark scattering, it is enhanced by a color factor2CF = 8/3. One expects this to make a significant contribution to the medium modification atintermediate zh.

In high-energy heavy-ion collisions, the ratios of initial production rates for valence quarks,gluons and antiquarks vary with the transverse momentum pT . Gluon production rate dominatesat low pT while the fraction of valence quark jets increases at large pT . Quarks are more likelyto fragment into protons than antiprotons, while gluons fragment into protons and antiprotonswith equal probabilities. Therefore, the ratio of large pT antiproton and proton yields in p + p

collisions is smaller than 1 and decreases with pT as the fraction of valence quark jets increases.Since gluons are expected to lose more energy than quark jets, one would naively expect to seethe antiproton to proton ratio p/p becomes smaller due to jet quenching. However, if the quark–gluon conversion due to qq → gg becomes important, one would expect that the fractions ofquark and gluon jets are modified toward their equilibrium values. The final p/p ratio could belarger than or comparable to that in p + p collisions. Such a scenario of quark–gluon conversionwas recently considered in Ref. [37] via a master rate equation.

The mixing between quark and gluon jets also happens at the lowest order of quark–antiquarkannihilation as shown in Fig. 3. At NLO, all hard–soft quark–quark (antiquark) scattering


processes have this kind of mixing between quark and gluon fragmentation functions. Theircontributions generally have the form,

CF

Nc

α2s xB

Q2

∑qi

∫d�2

T

�2T

1∫zh

dz

z

[Dqi→h

(zh

z

)− Dg→h

(zh

z

)]Pqqi→qqi

(z)T

A(SI)qqi

(x, xL)

f Aq (x)

,

(72)

where again the summation over the quark flavor includes qi = q, q . This mixing does not occuron the probability but rather on the amplitude level since it involves interferences between singleand triple scattering. Therefore, this contribution depends on the difference between gluon andquark fragmentation functions [Eq. (35)] and can be positive or negative in different region of zh.Nevertheless, they contribute to the modification of the effective quark fragmentation functionand the flavor dependence of the final hadron spectra.

6. Flavor dependence of the medium modified fragmentation

Summing all contributions to quark–quark (antiquark) double scattering as listed in Section 4,we can express the total twist-four correction up to O(α2

s ) to the quark fragmentation function as

�Dq→h(zh) = CF

Nc

2παsxB

Q2

{2[Dg→h(zh) − Dq→h(zh)

]TA(H)qq (x,0)

f Aq (x)

+ αs

2π

∫d�2

T

�2T

1∫zh

dz

z

∑a,b,i

Db→h

(zh

z

)P

(i)qa→b(z)

TA(i)qa (x, xL)

f Aq (x)

}, (73)

where the summation is over all possible q + a → b + X processes and all different matrixelements T

A(i)qa (x, xL) (i = HI,SI, I, I2), which will be four basic matrix elements we will

use. The effective splitting functions P(i)qa→b(z) are listed in Appendix A.2. One should also

include virtual corrections which can be constructed from the real corrections through unitarityconstraints [18].

Similarly, we can also write down the twist-four corrections to antiquark fragmentation in anuclear medium,

�Dq→h(zh) = CF

Nc

2παsxB

Q2

{2[Dg→h(zh) − Dq→h(zh)

]TA(H)qq (x,0)

f Aq (x)

+ αs

2π

∫d�2

T

�2T

1∫zh

dz

z

∑a,b,i

Db→h

(zh

z

)P

(i)qa→b(z)

TA(i)qa (x, xL)

f Aq (x)

}, (74)

where the matrix elements TA(i)qa (x, xL) and the effective splitting functions P

(i)qa→b(z) can be

obtained from the corresponding ones for quarks. Given a model for the two-quark correlationfunctions, one will be able to use the above expressions to numerically evaluate twist-four cor-rections to the quark (antiquark) fragmentation functions. In this paper, we will instead give aqualitative estimate of the flavor dependence of the correction in DIS off a large nucleus.

For the purpose of a qualitative estimate, one can assume that all the twist-four two-quarkcorrelation functions can be factorized, as has been done in Refs. [18,19,23,33],


∫p+ dy−

2πdy−

1 dy−2 eix1p

+y−+ix2p+(y−

1 −y−2 )θ(−y−

2 )θ(y− − y−1 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉

≈ C

xA

f Aq (x1)f

Nqi

(x2), (75)∫p+ dy−

2πdy−

1 dy−2 eix1p

+y−+ix2p+(y−

1 −y−2 )±ixLp+y−

2 θ(−y−2 )θ(y− − y−

1 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉

≈ C

xA

f Aq (x1)f

Nqi

(x2)e−x2

L/x2A, (76)

where xA = 1/mNRA, mN is the nucleon mass, RA the nucleus size, f Nqi

(x2) is the antiquarkdistribution in a nucleon and C is assumed to be a constant, parameterizing the strength oftwo-parton correlations inside a nucleus. The integration over the position of the antiquark(y−

1 + y−2 )/2 in the twist-four two-quark correlation matrix elements gives rise to the nuclear

enhancement factor 1/xA = mNRA = 0.21A1/3.We should note that we set kT = 0 for the collinear expansion. As a consequence, the sec-

ondary quark field in the twist-four parton matrix elements will carry zero momentum in thesoft–hard process. Finite intrinsic transverse momentum leads to higher-twist corrections. If asubset of the higher-twist terms in the collinear expansion can be resummed to restore the phasefactors such as eixT p+y−

, where xT ≡ 〈k2T 〉/2p+q−z(1 − z), the soft quark fields in the parton

matrix elements will carry a finite fractional momentum xT .Under such an assumption of factorization, one can obtain all the two-quark correlation matrix

elements:

TA(HI)qqi

(x, xL) ≈ C

xA

f Aq (x)f N

qi(xL + xT )

[1 − e−x2

L/x2A], (77)

TA(SI)qqi

(x, xL) ≈ C

xA

f Aq (x + xL)f N

qi(xT )

[1 − e−x2

L/x2A], (78)

TA(I)qqi

(x, xL) ≈ TA(I2)qqi

(x, xL) ≈ C

xA

f Aq (x + xL)f N

qi(xT )e−x2

L/x2A

≈ C

xA

f Aq (x)f N

qi(xL + xT )e−x2

L/x2A. (79)

In the last approximation, we have assumed xL ∼ xT x. Similarly, one can obtain TA(i)qqi

(x, xL),

TA(i)qqi

(x, xL) and TA(i)qqi

(x, xL). With these forms of two-quark correlation matrix elements, wecan estimate the flavor dependence of the nuclear modification to the quark (antiquark) fragmen-tation functions.

The lowest order corrections [O(αs)] are very simple

�D(LO)q→h(zh) ∝ CA1/3[Dg→h(zh) − Dq→h(zh)

]f N

q (xT ), (80)

�D(LO)

q→h(zh) ∝ CA1/3[Dg→h(zh) − Dq→h(zh)

]f N

q (xT ). (81)

We consider the dominant contribution from the fragmentation of a quark (antiquark) which isone of the valence quarks (antiquarks) of the final particle h (antiparticle h). The gluon frag-


mentation functions into h and h are the same. For large zh, the gluon fragmentation function isalways softer than the valence quark (antiquark) fragmentation [38]. Therefore, the lowest ordertwist-four corrections are always negative for large zh, leading to a suppression of the valencequark (antiquark) fragmentation function, Dqv→h(zh) [Dqv→h(zh)]. Consider those quarks thatare also valence quarks of a nucleon:

n = udd,

p = uud, p = uud, (82)

K+ = us, K− = us, (83)

π+,π0,π− = ud, (uu − dd)/√

2, du. (84)

One can find the following flavor dependence of the lowest order twist-four corrections to thequark (antiquark) fragmentation functions,

�D(LO)

qv→h(zh)

�D(LO)qv→h(zh)

=−|�D

(LO)

qv→h(zh)|

−|�D(LO)qv→h(zh)|

= f Nqv

(xT )

fqv (xT )> 1, (85)

or

R(LO)

qv→h(zh)

R(LO)qv→h(zh)

=1 + �D

(LO)

qv→h(zh)/Dq→h(zh)

1 + �D(LO)qv→h(zh)/Dq→h(zh)

< 1, (86)

where R(LO)qv→h is the corresponding leading order suppression of the fragmentation function at

large zh for proton (antiproton) and K+ (K−). Since pions contain both valence quark and an-tiquark, the suppression factors should be similar for all pions. For xT � 0.043, u(x)/u(x) � 3and d(x)/d(x) � 2 [35]. Therefore, the modification of antiquark fragmentation functions due toquark–antiquark annihilation is significantly larger than that of a quark.

The flavor dependence of the NLO results are more complicated since they involve scatteringwith both quarks and antiquarks in the medium. One can observe first that effective splittingfunctions (or quark–quark scattering cross section) are the same for the t-channel qq ′ → qq ′ andqq ′ → qq ′ (q ′ �= q) scatterings,

P(i)

qq ′→b(z) = P

(i)

qq ′→b(z) = P

(i)

qq ′→b(z) = P

(i)

qq ′→b(z). (87)

For identical quark–quark scattering or quark–antiquark annihilation, one can separate the qq

annihilation splitting functions (or cross sections) into singlet and non-singlet contributions bysingling out the t-channel contributions,

P(i)qq→b(z) ≡ P

(i)qq→b(z) + �P

N(i)qq→b(z), (88)

P(i)qq→b(z) ≡ P

(i)qq→b(z) + �P

N(i)qq→b(z). (89)

These singlet contributions to the modified fragmentation functions are,

�DS(NLO)q→h (zh) ∝ αs

2πA1/3

∫d�2

T

�2T

∑b,q ′,i

Db→h ⊗ P(i)

qq ′→b(zh)

[f N

q ′ (xT ) + f Nq ′ (xT )

]C(i),

(90)

�DS(NLO)

q→h(zh) ∝ αs

2πA1/3

∫d�2

T

�2T

∑b,q ′,i

Db→h ⊗ P(i)

qq ′→b(zh)

[f N

q ′ (xT ) + f Nq ′ (xT )

]C(i),

(91)


where the summation over q ′ now includes q ′ = q and C(i)(xL) are flavor-independent functionsdetermined from Eqs. (77)–(79),

C(HI) = C(SI) = C(xL)(1 − e−x2

L/x2A),

C(I) = C(I2) = C(xL)e−x2L/x2

A, (92)

and C(xL) is a common coefficient that is a function of xL. Using P(i)

qq→b(z) = P

(i)qq→b(z), one

can conclude that the singlet contributions to the modified quark and antiquark fragmentationfunctions are the same, �D

S(NLO)q→h (zh) = �D

S(NLO)

q→h(zh).

The non-singlet contributions, mainly from s-channel and s-t interferences, are,

�DN(NLO)q→h (zh) ∝ αs

2πA1/3

∫d�2

T

�2T

∑b,i

Db→h ⊗ �PN(i)qq→b(zh)f

Nq (xT )C(i), (93)

�DN(NLO)

q→h(zh) ∝ αs

2πA1/3

∫d�2

T

�2T

∑b,i

Db→h ⊗ �PN(i)

qq→b(zh)f

Nq (xT )C(i), (94)

where again �PN(i)qq→b(z) = �P

N(i)

qq→b(z) due to crossing symmetry. We have listed all non-

vanishing nonsinglet splitting functions �PN(i)qq→b(z) in Appendix A.2.

We again consider the limit zh → 1. In this region the convolution in the modified fragmenta-tion function is dominated by the large z → 1 behavior of the effective splitting functions. Fromthe listed �P

N(i)qq→b(z) in Appendix A.2, we can obtain the leading contributions,

∑i

C(i)�PN(i)qq→q(z) ≈ −4CF

C(xL)

1 − z,

∑i

C(i)�PN(i)qq→g(z) ≈ 2

[2CF + CF

(1 − e−x2

L/x2A) + CAe−x2

L/x2A]C(xL)

1 − z, (95)

where we have also neglected terms proportional to 1/Nc. All �PN(i)qq→q (z) are non-leading in

the limit z → 1 and therefore can be neglected. With these leading contributions, the non-singletmodification to the quark and antiquark fragmentation functions can be estimated as

�DN(NLO)q→h (zh) ∝ αs

πA1/3

∫d�2

T

�2T

1∫zh

dz

z

{Dg→h

(zh

z

)[C(xL)

(1 − z)+

× (CF

(1 − e−x2

L/x2A) + CAe−x2

L/x2A) + δ(1 − z)Δ1(�T )

]

+[Dg→h

(zh

z

)− Dq→h

(zh

z

)]

×[

2CF

C(xL)

(1 − z)++ δ(1 − z)�2(�T )

]}f N

q (xT ), (96)


�DN(NLO)

q→h(zh) ∝ αs

πA1/3

∫d�2

T

�2T

1∫zh

dz

z

{Dg→h

(zh

z

)[C(xL)

(1 − z)+

× (CF

(1 − e−x2

L/x2A) + CAe−x2

L/x2A) + δ(1 − z)Δ1(�T )

]

+[Dg→h

(zh

z

)− Dq→h

(zh

z

)]

×[

2CF

C(xL)

(1 − z)++ δ(1 − z)Δ2(�T )

]}f N

q (xT ), (97)

where Δ1(�T ) and Δ2(�T ) are from virtual corrections,

Δ1(�T ) =1∫

0

dz

1 − z

{CF C(xL)|z=1 − [

CF

(1 − e−x2

L/x2A) + CAe−x2

L/x2A]C(xL)

}, (98)

Δ2(�T ) =1∫

0

dz

1 − z2CF

[C(xL)|z=1 − C(xL)

]. (99)

Because of momentum conservation, C(xL) = 0 when xL → ∞ for z = 1. Therefore, the abovevirtual corrections are always negative. At large zh, these virtual corrections dominate over thereal ones.

There are two kinds of non-singlet contributions in the expressions given above. One that isproportional to gluon fragmentation functions is due to quark–antiquark annihilation into gluonswhich then fragment. The fragmenting gluon not only carries less energy than the initial quarkbut also has a softer fragmentation function, leading to suppression of the final leading hadrons.The second type of contributions is proportional to Dg→h(zh) − Dq→h(zh) and therefore mixesquark and gluon fragmentation functions, similarly as the lowest order quark–antiquark annihi-lation processes [see Eqs. (80) and (81)]. Since a gluon fragmentation function is softer than aquark one, the real corrections from this type of processes are positive for small zh and nega-tive for large zh. The virtual corrections have just the opposite behavior. Therefore, the secondtype of contributions will reduce the total net modification. For intermediate values of zh where2Dg→h(zh) > Dq→h(zh), the net effect is still the suppression of the effective fragmentationfunctions for leading hadrons.

Since f Nq (xT ) > f N

q (xT ), we can conclude that the LO and NLO combined non-singletsuppression for antiquark fragmentation into valence hadrons is larger than that for quark frag-mentation into valence hadrons. This qualitatively explains the flavor dependence of nuclearsuppression of leading hadrons in DIS off heavy nuclear targets as measured by the HERMES ex-periment [25,26]. The ratio of differential semi-inclusive cross sections for nucleus and deuterontargets were used to study the nuclear suppression of the fragmentation functions. It was observedthat suppression of leading antiproton is stronger than for leading proton and K− suppression isstronger than K+. In the valence quark fragmentation picture, the leading proton (K+) is pro-duced mainly from u, d (u) quark fragmentation while antiprotons come primarily from u, d (u)fragmentation. Therefore, HERMES data are consistent with stronger suppression of antiquarkfragmentation.

Since gluon bremsstrahlung and the singlet qqi(qi) scattering also suppress quark and anti-quark fragmentation, but independently of quark flavor, one has to include all the processes in


order to have a complete and quantitative numerical evaluation of the flavor dependence of thenuclear modification of the quark fragmentation functions. Furthermore, the NLO contributionsare proportional to αs ln(Q2)/2π that are as important as the lowest order correction for largevalues of Q2. In principle, one should resum these higher order corrections via solving a setof coupled DGLAP evolution equations, including medium modification for gluon fragmenta-tion functions. The contributions from quark–quark (antiquark) scattering derived in this paperwill be an important part of the complete description. Detailed numerical study of the effect ofquark–quark (antiquark) scattering will be possible only after the completion of this completedescription in the future.

7. Summary

Utilizing the generalized factorization framework for twist-four processes we have studiedthe nuclear modification of quark and antiquark fragmentation functions (FF) due to quark–quark (antiquark) double scattering in dense nuclear matter up to order O(α2

s ). We calculatedand analyzed the complete set of all possible cut diagrams. The results can be categorized intocontributions from double-hard, hard–soft processes and their interferences. The double-hardrescatterings correspond to elastic scattering of the leading quark with another medium quark. Itrequires the second quark to carry a finite fractional momentum xL. Therefore, the energy loss ofthe leading quark through such processes can be identified as elastic energy loss at order O(α2

s ).The quark energy loss and modification of quark fragmentation functions are dominated by thet-channel of quark–quark (antiquark) scattering and are shown to be similar to that caused byquark–gluon scattering. The contribution from quark–quark scattering is smaller than that fromquark–gluon scattering by a factor of CF /CA times the ratio of quark and gluon distributionfunctions in the medium. We have shown that such contributions are not negligible for realistickinematics and finite medium size. The soft-hard rescatterings mix gluon and quark scattering,in the same way as the lowest order qq → g processes. Such processes modifies the final hadronspectra or effective fragmentation functions but do not contribute to energy loss of the leadingquark. For qq → qq, gg processes, there also exist pure interference contributions mainly com-ing from single–triple-scattering interference.

With a simple model of a factorized two-quark correlation functions, we further investigatedthe flavor dependence of the medium modified quark fragmentation functions in a large nucleus.We identified the flavor dependent part of the modification and find that the nuclear modificationfor an antiquark fragmentation into a valence hadron is larger than that of a quark. This offersan qualitative explanation for the flavor dependence of the leading hadron suppression in semi-inclusive DIS off nuclear targets as observed by the HERMES experiment [25,26].

Acknowledgements

The authors thank Jian-Wei Qiu and Enke Wang for helpful discussion. This work was sup-ported by NSFC under project No. 10405011, by MOE of China under project IRT0624, byAlexander von Humboldt Foundation, by BMBF, by the Director, Office of Energy Research,Office of High Energy and Nuclear Physics, Divisions of Nuclear Physics, of the US Depart-ment of Energy under Contract No. DE-AC02-05CH11231, and by the US NSF under GrantNo. PHY-0457265, the Welch Foundation under Grant No. A-1358.


Appendix A

A.1. Hard partonic parts for quark–quark double scattering

In Section 3 we have discussed the calculation of the hard part of one example cut-diagram(Fig. 5) in detail. In this appendix we list the results for all possible real corrections to quark–quark (antiquark) double scattering in the next-to-leading order O(α2

s ). There are a total of 12diagrams as illustrated in Figs. 5–16. For the purpose of abbreviation, we will suppress the vari-ables in the notations of partonic hard parts

HD ≡ HD(y−, y−1 , y−

2 , x,p, q, zh), (A.1)

and phase factor functions

I ≡ I (y−, y−1 , y−

2 , x, , xL,p). (A.2)

We first consider all qq → gg annihilation diagrams with different possible cuts. The contri-butions of Fig. 5 are:

HD5,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI5,CDg→h

(zh

z

)

[2

1 + z2

z(1 − z)+ 2

1 + (1 − z)2

z(1 − z)

]C2

F

Nc

, (A.3)

I5,C = θ(−y−2 )θ(y− − y−

1 )ei(x+xL)p+y−(1 − e−ixLp+y−

2)(

1 − e−ixLp+(y−−y−1 )

), (A.4)

HD5,L(R) = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI5,L(R)

[Dq→h

(zh

z

)2

1 + z2

z(1 − z)

+ Dg→h

(zh

z

)2

1 + (1 − z)2

z(1 − z)

]C2

F

Nc

, (A.5)

Fig. 5. The t -channel of qq → gg annihilation diagram with three possible cuts, central (C), left (L) and right (R).


Fig. 6. The interference between t - and u-channel of qq → gg annihilation.

Fig. 7. The s-channel of qq → gg annihilation diagram with only a central-cut.

Fig. 8. The interference between t - and s-channel of qq → gg annihilation.


Fig. 9. The complex conjugate of Fig. 8.

Fig. 10. s-channel qq → qi qi annihilation.

Fig. 11. t -channel qqi (qi ) → qqi (qi ) scattering.


Fig. 12. Interference between s- and t -channel of qq → qq scattering.


Fig. 14. The interference between t - and u-channel of identical quark–quark scattering qq → qq .


Fig. 15. Interference between final-state gluon radiation from single and triple-quark scattering.


I5,L = −θ(y−1 − y−

2 )θ(y− − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+(y−−y−1 )

), (A.6)

I5,R = −θ(−y−2 )θ(y−

2 − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+y−2). (A.7)

Here we have included the fragmentation of both final-state partons.The contributions from Fig. 6 are:

HD6,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI6,C2Dg→h

(zh

z

) −2

(1 − z)z

CF (CF − CA/2)

Nc

, (A.8)

HD6,L(R) = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI6,L(R)

[Dg→h

(zh

z

)+ Dq→h

(zh

z

)]

× −2 CF (CF − CA/2), (A.9)

(1 − z)z Nc


I6,C = θ(−y−2 )θ(y− − y−

1 )ei(x+xL)p+y−

× (1 − e−ixLp+y−

2)(

1 − e−ixLp+(y−−y−1 )

), (A.10)

I6,L = −θ(y−1 − y−

2 )θ(y− − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+(y−−y−1 )

), (A.11)

I6,R = −θ(−y−2 )θ(y−

2 − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+y−2). (A.12)

Note that the central-cut diagram in Fig. 6 corresponds to the interference between t andu-channel of the qq → gg annihilation processes in Fig. 5. Since the splitting function is sym-metric in z and 1−z, a factor of 2 comes from the fragmentation of both gluons in the central-cutdiagram.

The s-channel of qq → gg is shown in Fig. 7 which has only one central-cut. Its contributionto the partonic hard part is,

HD7,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI7,C2Dg→h

(zh

z

)2(z2 − z + 1)2

z(1 − z)

CF CA

Nc

, (A.13)

I7,C = θ(−y−2 )θ(y− − y−

1 )ei(x+xL)p+y−e−ixLp+(y−−y−

1 )e−ixLp+y−2 . (A.14)

Note that the splitting function 2(z2 − z+ 1)2/z(1 − z) = 2[1 − z(1 − z)]2/z(1 − z) is symmetricin z and 1 − z. Therefore, fragmentation of the two final gluons gives rise to the factor of 2 infront of the gluon fragmentation function.

The interferences between t- and s-channel of qq → gg processes are shown in Figs. 8 and 9.There are only two possible cuts in these diagrams. The contributions from Fig. 8 are:

HD8,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI8,CDg→h

(zh

z

)[2

1 + z3

z(1 − z)+ 2

1 + (1 − z)3

z(1 − z)

]CF CA

2Nc

,

(A.15)

HD8,L = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI8,L

[Dq→h

(zh

z

)2

1 + z3

z(1 − z)

+ Dg→h

(zh

z

)2

1 + (1 − z)3

z(1 − z)

]CF CA

2Nc

, (A.16)

I8,C = θ(−y−2 )θ(y− − y−


2)e−ixLp+(y−−y−

1 ), (A.17)

I8,L = θ(y−1 − y−

2 )θ(y− − y−1 )ei(x+xL)p+y−

× (e−ixLp+(y−−y−

2 ) − e−ixLp+(y−−y−1 )

). (A.18)

Contributions from Fig. 9, which are just the complex conjugate of Fig. 8, are:

HD9,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI9,CDg→h

(zh

z

)[2

1 + z3

z(1 − z)+ 2

1 + (1 − z)3

z(1 − z)

]CF CA

2Nc

,

(A.19)


HD9,R = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI9,R

[Dq→h

(zh

z

)2

1 + z3

z(1 − z)

+ Dg→h

(zh

z

)2

1 + (1 − z)3

z(1 − z)

]CF CA

2Nc

, (A.20)

I9,C = θ(−y−2 )θ(y− − y−

1 )ei(x+xL)p+y−(1 − e−ixLp+(y−−y−

1 ))e−ixLp+y−

2 , (A.21)

I9,R = θ(−y−2 )θ(y−

2 − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+(y−2 −y−

1 ))e−ixLp+y−

1 . (A.22)

One can collect all contributions of the double hard qq → gg processes from the central-cutdiagrams, which should have the common phase factor

IC = θ(−y−2 )θ(y− − y−

1 )eixp+y−e−ixLp+(y−

2 −y−1 ), (A.23)

and obtain the total effective splitting function in the hard partonic part,

CF

Nc

Pqq→gg(z) = 2

z(1 − z)

1

Nc

{C2

F

[1 + z2 + 1 + (1 − z)2] − 2CF

(CF − CA

2

)

+ 2CF CA

(1 − z + z2)2 − CF CA

[1 + z3 + 1 + (1 − z)3]}

= CF

Nc

[2CF

z2 + (1 − z)2

z(1 − z)− 2CA

[z2 + (1 − z)2]]. (A.24)

We will find later in Appendix A.3 that the above result can also be obtained from the total matrixelements squared for qq → gg annihilation.

We now consider the annihilation processes qq → qi qi with qi �= q . There is only thes-channel process with one central-cut diagram as shown in Fig. 10. Its contribution to the hardpart is

HD10,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI10,C

∑qi �=q

[Dqi→h

(zh

z

)+ Dqi→h

(zh

z

)]

× [z2 + (1 − z)2]CF

Nc

, (A.25)

I10,C = θ(−y−2 )θ(y− − y−

1 )ei(x+xL)p+y−e−ixLp+(y−−y−

1 )e−ixLp+y−2 . (A.26)

Here we define the effective splitting function for qq → qi qi annihilation as,

CF

Nc

Pqq→qi qi(z) = CF

Nc

[z2 + (1 − z)2]. (A.27)

Similarly, for qqi → qqi scattering with qi �= q , there is only the t-channel as shown inFig. 11. There are, however, three cut diagrams. Their contributions to the partonic hard partare:

HD11,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI11,C

×[Dq→h

(zh

)1 + z2

2+ Dqi→h

(zh

)1 + (1 − z)2

2

]CF

, (A.28)
z (1 − z) z z Nc


HD11,L(R) = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI11,L(R)

×[Dq→h

(zh

z

)1 + z2

(1 − z)2+ Dg→h

(zh

z

)1 + (1 − z)2

z2

]CF

Nc

, (A.29)

I11,C = θ(−y−2 )θ(y− − y−

1 )ei(x+xL)p+y−

× (1 − e−ixLp+y−

2)(

1 − e−ixLp+(y−−y−1 )

), (A.30)

I11,L = −θ(y−1 − y−

2 )θ(y− − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+(y−−y−1 )

), (A.31)

I11,R = −θ(−y−2 )θ(y−

2 − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+y−2). (A.32)

The twist-four two-parton correlation matrix element associated with the above quark–antiquarkscattering is the quark–antiquark correlator,

T Aqqi

(x, xL) ∝ eixp+y−−ixLp+(y−2 −y−

1 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−1 )

γ +

2ψqi

(y−2 )|A〉, (A.33)

and one should sum over all possible qi �= q flavors. Note that in the above matrix element, themomentum flow for the antiquark (qi ) is opposite to that of the quark (q) fields.

For quark–quark scattering, qqi → qqi , the hard part is essentially the same. The only differ-ence is the associated matrix element for the quark–quark correlator which is obtained from thatof the quark–antiquark correlator via the exchange ψqi

(y2) → ψqi(y2) and ψqi

(y1) → ψqi(y1),

T Aqqi

(x, xL) ∝ eixp+y−+ixLp+(y−1 −y−

2 )

× 〈A|ψq(0)γ +

2ψq(y−)ψqi

(y−2 )

γ +

2ψqi

(y−1 )|A〉. (A.34)

Note that the momentum flows of the two quarks (q and qi ) point in the same direction.The effective splitting function of this scattering process is defined through the fragmentation

of the quark in the central-cut diagram,

CF

Nc

Pqqi (qi )→qqi (qi )(z) = CF

Nc

1 + z2

(1 − z)2. (A.35)

For annihilation qq → qq into identical quark and antiquark pairs, in addition to the s-channel(Fig. 10 for qi = q) and t-channel (Fig. 11 for qi = q), one has also to consider the interferencebetween s- and t-channel amplitudes as shown in Figs. 12 and 13, each having two cuts. Theircontributions to the hard partonic parts are, respectively:

HD12,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI12,C

[Dq→h

(zh

z

)2z2

(1 − z)+ Dq→h

(zh

z

)2(1 − z)2

z

]

× CF (CF − CA/2), (A.36)

Nc


HD12,L = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI12,L

[Dq→h

(zh

z

)2z2

(1 − z)+ Dg→h

(zh

z

)2(1 − z)2

z

]

× CF (CF − CA/2)

Nc

, (A.37)

I12,C = θ(−y−2 )θ(y− − y−


2)e−ixLp+(y−−y−

1 ), (A.38)

I12,L = θ(y−1 − y−

2 )θ(y− − y−1 )ei(x+xL)p+y−

× (e−ixLp+(y−−y−

2 ) − e−ixLp+(y−−y−1 )

); (A.39)

HD13,C = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI13,C

[Dq→h

(zh

z

)2z2

(1 − z)+ Dq→h

(zh

z

)2(1 − z)2

z

]

× CF (CF − CA/2)

Nc

, (A.40)

HD13,R = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI13,R

[Dq→h

(zh

z

)2z2

(1 − z)+ Dg→h

(zh

z

)2(1 − z)2

z

]

× CF (CF − CA/2)

Nc

, (A.41)

I13,C = θ(−y−2 )θ(y− − y−

1 )ei(x+xL)p+y−(1 − e−ixLp+(y−−y−

1 ))e−ixLp+y−

2 , (A.42)

I13,R = θ(−y−2 )θ(y−

2 − y−1 )ei(x+xL)p+y−(

e−ixLp+y−1 − e−ixLp+y−

2). (A.43)

One can again collect contributions from the central-cut diagrams of the double scatteringprocesses in Figs. 10, 11, 12 and 13 and obtain the total effective splitting function for qq → qq ,

CF

Nc

Pqq→qq (z) = CF

Nc

[z2 + (1 − z)2] + CF

Nc

1 + z2

(1 − z)2− CF (CF − CA/2)

Nc

4z2

1 − z

= CF

Nc

[z2 + (1 − z)2 + 1 + z2

(1 − z)2+ 1

Nc

2z2

1 − z

]. (A.44)

Here we have used CF − CA/2 = −1/2Nc. For antiquark fragmentation, Pqq→qq (z) =Pqq→qq (1 − z). One can also obtain the above result from qq → qq scattering matrix squaredas shown in Appendix A.3.

Similarly, for scattering of identical quarks qq → qq , one should set qi = q in Fig. 11 [inEq. (A.28)]. In addition, one should also include interference between t- and u-channel of thescattering as shown in Fig. 14. The contributions from such interference diagram are,

HD14,C = α2

s xB

Q2

∫d�2

T

�2T

1∫dz

zI14,C2Dq→h

(zh

z

)2

z(1 − z)

CF (CF − CA/2)

Nc

, (A.45)

zh


HD14,L(R) = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI14,L(R)

×[Dq→h

(zh

z

)+ Dg→h

(zh

z

)]2

z(1 − z)

CF (CF − CA/2)

Nc

, (A.46)

I14,C = θ(−y−2 )θ(y− − y−


2)(

1 − e−ixLp+(y−−y−1 )

), (A.47)

I14,L = −θ(y−1 − y−

2 )θ(y− − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+(y−−y−1 )

), (A.48)

I14,R = −θ(−y−2 )θ(y−

2 − y−1 )ei(x+xL)p+y−(

1 − e−ixLp+y−2). (A.49)

Note again that the fragmentation of both quarks contributes to the factor 2 in Eq. (A.45) sincethe splitting function is symmetric in z and 1 − z. The twist-four two-quark correlation matrixelement associated with qq → qq scattering is T A

qq(x, xL) as compared to T Aqq(x, xL) for quark–

antiquark annihilation processes.We can sum contributions from the double hard scattering in all the central-cut diagrams in

Figs. 11 and 14 and obtain the total effective splitting function for qq → qq processes,

CF

Nc

Pqq→qq(z) = CF

Nc

[1 + z2

(1 − z)2+ 1 + (1 − z)2

z2

]+ CF (CF − CA/2)

Nc

4

z(1 − z)

= CF

Nc

[1 + z2

(1 − z)2+ 1 + (1 − z)2

z2− 1

Nc

2

z(1 − z)

]. (A.50)

There are two remaining cut diagrams that contribute to the quark–antiquark annihilation atthe order of O(α2

s ) as shown in Figs. 15 and 16. Their contributions are:

HD15,L = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI15,L

×[Dq→h

(zh

z

)2

1 + z2

1 − z+ Dg→h

(zh

z

)2

1 + (1 − z)2

z

]C2

F

Nc

, (A.51)

I15,L = −θ(y−1 − y−

2 )θ(y− − y−1 )ei(x+xL)p+y−

e−ixLp+(y−−y−2 ), (A.52)

HD16,R = α2

s xB

Q2

∫d�2

T

�2T

1∫zh

dz

zI16,R

×[Dq→h

(zh

z

)2

1 + z2

1 − z+ Dg→h

(zh

z

)2

1 + (1 − z)2

z

]C2

F

Nc

, (A.53)

I16,R = −θ(−y−2 )θ(y−

2 − y−1 )ei(x+xL)p+y−

e−ixLp+y−1 . (A.54)


A.2. Effective splitting functions

In this appendix, we list the effective splitting functions associated with each process qa → b

and the double-hard (HI ), hard–soft (SI ) or their interferences (I, I2) according to Eq. (73).

P(HI)qqi (qi )→qi (qi )

(z) = 1 + (1 − z)2

z2, P

(HI)qqi (qi )→q(z) = 1 + z2

(1 − z)2,

P(SI)qqi (qi )→qi (qi )

(z) = 1 + (1 − z)2

z2, P

(SI)qqi (qi )→g(z) = −1 + (1 − z)2

z2, (A.55)

P(HI)qq→qi

(z) = P(HI)qq→qi

(z) = z2 + (1 − z)2,

P(I)qq→qi

(z) = P(I)qq→qi

(z) = z2 + (1 − z)2, (A.56)

P (HI)qq→q(z) = 1 + (1 − z)2

z2+ 1 + z2

(1 − z)2− 2

Nc

1

z(1 − z),

P (SI)qq→g(z) = −P (SI)

qq→q(z),

P (SI)qq→q(z) = 1 + (1 − z)2

z2− 1

Nc

1

z(1 − z), (A.57)

P(HI)qq→q(z) = z2 + (1 − z)2 + 1 + z2

(1 − z)2+ 2

Nc

z2

1 − z,

P(HI)qq→q (z) = P

(HI)qq→q(1 − z),

P(HI)qq→g(z) = 2CF

z2 + (1 − z)2

z(1 − z)− 2CA[z2 + (1 − z)2], (A.58)

P(SI)qq→q(z) = −

[CA

z(1 − z)+ 2CF

z

1 − z

],

P(SI)qq→q (z) = 1 + (1 − z)2

z2,

P(SI)qq→g(z) = CA

z(1 − z)+ 2CF

z

1 − z− 1 + (1 − z)2

z2, (A.59)

P(I)qq→q(z) = z2 + (1 − z)2 − CA

z(1 − z)− 2CF

z2

1 − z,

P(I)qq→q (z) = z2 + (1 − z)2,

P(I)qq→g(z) = CA

4(1 − z + z2)2 − 1

z(1 − z)− 2CF

(1 − z)2

z, (A.60)

P(I2)qq→q(z) = CA

z(1 − z)− 2CF

1 − z, P

(I2)qq→g(z) = CA

z(1 − z)− 2CF

z. (A.61)

The non-singlet splitting functions for qq → b, defined as

�PN(i)qq→b(z) ≡ P

(i)qq→b(z) − P

(i)qq→b(z), (A.62)

are listed as below:


�PN(HI)qq→qi (qi )

(z) = P(HI)qq→qi (qi )

(z), �PN(I)qq→qi (qi )

(z) = P(I)qq→qi (qi )

(z), (A.63)

�PN(HI)qq→q (z) = − (1 − z2)(1 + z2 + (1 − z)2)

z2+ 2

Nc

1 + z3

z(1 − z),

�PN(HI)qq→q (z) = P

(HI)qq→q (z), �P

N(HI)qq→g (z) = P

(HI)qq→g(z), (A.64)

�PN(SI)qq→q (z) = −

[2CF

1 + z2

1 − z+ 1 + (1 − z)2

z2

],

�PN(SI)qq→q (z) = P

(SI)qq→q (z),

�PN(SI)qq→g (z) = 2CF

1 + z2

1 − z+ 2

Nc

1

z(1 − z)− 2

1 + (1 − z)2

z2, (A.65)

�PN(I)qq→b(z) = P

(I)qq→b(z), �P

N(I2)qq→b (z) = P

(I2)qq→b(z) (b = q, q, g). (A.66)

A.3. Alternative calculations of central-cut diagrams

As a cross-check of the hard partonic parts calculated from different cut diagrams in Appen-dix A.1, we provide an alternative calculation of all the central-cut diagrams, which correspondto quark–quark (antiquark) scattering.

Considering a parton (a) with momentum q scattering with another parton (b) that carries afractional momentum xp, a(q) + b(xp) → c(�) + d(p′), the cross section can be written as

dσab = g4

2s|M|2ab→cd

(t

s,u

s

)d3�

(2π)32�02πδ

[(p + q − �)2]

= g4

(4π)2|M|2ab→cd

(t

s,u

s

)π

s2

dz

z(1 − z)d�2

T δ

(1 − xL

x

), (A.67)

where q = [0, q−,0] and p = [xp+,0,0] are momenta of the initial partons and

� =[

�2T

2zq− , zq−, ��T

](A.68)

is the momentum of one of the final partons. With the given kinematics, the on-shell condition inthe cross section can be recast as

(xp + q − �)2 = 2(1 − z)xp+q−(

1 − xL

x

), xL = �2

T

2z(1 − z)p+q− . (A.69)

The Mandelstam variables of the collision are,

s = (q + xp)2 = 2xp+q− = �2T

z(1 − z), u = (� − xp)2 = −zs,

t = (� − q)2 = −(1 − z)xL

xs = −(1 − z)s, (A.70)

where we have used the on-shell condition x = xL.With Eq. (A.67) and parton distribution functions f N

b (x), one can obtain the parton–nucleoncross section,


dσaN =∑

b

dσab f Nb (x) dx

=∑

b

f Nb (xL)xL|M|2ab→cd

(t

s,u

s

)πα2

s

s2

dz

z(1 − z)d�2

T

=∑

b

f Nb (xL)

πα2s

sC0Pab→cd (z)dz

d�2T

�2T

, (A.71)

where s = 2p+q− is the center-of-mass energy for aN collision, C0 is some common colorfactor in the scattering matrix elements and

Pab→cd(z) =(

1

C0

)|M|2ab→cd

(t

s,u

s

)(A.72)

is what we have defined as the effective splitting function for the corresponding processes.One can therefore easily obtain these effective splitting functions from the corresponding ma-trix elements for elementary parton–parton scattering [39]. We will list them in the following.A common color factor for all quark–quark (antiquark) scattering is C0 = CF /Nc.

qq → qi qi annihilation:

|M|2qq→qi qi= CF

Nc

t 2 + u2

s2,

Pqq→qi qi(z) = z2 + (1 − z)2. (A.73)

qq → qq annihilation:

|M|2qq→qq = CF

Nc

[u2 + s2

t 2+ u2 + t 2

s2− 1

Nc

2u2

s t

],

Pqq→qq (z) = 1 + z2

(1 − z)2+ z2 + (1 − z)2 + 2

Nc

z2

1 − z. (A.74)

qq → gg annihilation:

|M|2qq→gg = CF

Nc

[2CF

(u

t+ t

u

)− 2CA

u2 + t 2

s2

],

Pqq→gg(z) = 2CF

z2 + (1 − z)2

z(1 − z)− 2CA

(z2 + (1 − z)2). (A.75)

qqi(qi) → qqi(qi) scattering:

|M|2qqi (qi )→qqi (qi )= CF

Nc

u2 + s2

t 2,

Pqqi (qi )→qqi (qi )(z) = 1 + z2

(1 − z)2. (A.76)

qq → qq scattering:

|M|2qq→qq = CF

Nc

[u2 + s2

t 2+ s2 + t 2

u2− 1

Nc

2s2

t u

],

Pqq→qq(z) = 1 + z2

2+ 1 + (1 − z)2

2− 2 1

. (A.77)
(1 − z) z Nc z(1 − z)


For quark–gluon Compton scattering, the relevant gluon distribution function is xLGN(xL).One can therefore rewrite contribution from qg → qg to Eq. (A.71) as,

dσqN = xLGN(xL)πα2s z(1 − z)|M|2qg→qg

(t

s,u

s

)dz

d�2T

�4T

≡ xLGN(xL)πα2s

CF

Nc

Pqg→qg(z)dzd�2

T

�4T

. (A.78)

We have then for qg → qg scattering,

|M|2qg→qg = CA

Nc

s2 + u2

t 2− CF

Nc

u2 + s2

us,

Pqg→qg(z) = z(1 − z)

[CA

CF

1 + z2

(1 − z)2+ 1 + z2

z

]. (A.79)

Comparing this result with that in Ref. [18] for the quark–gluon rescattering, we can see that theyagree in the limit 1 − z → 0. This is a consequence of the collinear approximation employed inRef. [18] in the calculation of the hard partonic part in quark–gluon rescattering.

We can also extend this calculation to the case of gluon–nucleon scattering. One can useEq. (A.71) to define the splitting function for gq → gq scattering,

|M|2gq→gq = CA

Nc

s2 + t 2

u2− CF

Nc

t 2 + s2

t s,

Pgq→gq(z) = z(1 − z)

[CA

Nc

1 + (1 − z)2

z2+ CF

Nc

1 + (1 − z)2

(1 − z)

]. (A.80)

Here for gluon–parton scattering, there is no common color factor.gg → qq annihilation:

|M|2gg→qq = 1

2Nc

t 2 + u2

t u− 1

2CF

t 2 + u2

s2,

Pgg→qq (z) = z(1 − z)

{1

2Nc

z2 + (1 − z)2

z(1 − z)− 1

2CF

[z2 + (1 − z)2]}. (A.81)

gg → gg scattering:

|M|2gg→gg = 2CA

CF

[3 − t u

s2− us

t 2− t s

u2

],

Pgg→gg(z) = 2CA

CF

(1 − z + z2)3

z(1 − z). (A.82)

One can use this technique to extend the study of modified fragmentation functions to propa-gating gluons. Since the modification is dominated by quark–gluon and gluon–gluon scattering,comparing the effective splitting functions,

CF

Nc

Pqg→qg(z) ≈ CA

Nc

2

1 − z, (A.83)

Pgg→gg(z) ≈ 2CA 1, (A.84)

CF 1 − z


in the limit z → 1, one can conclude that a gluon’s radiative energy loss is larger than a quark bya factor of Nc/CF = CA/CF = 9/4. We will leave the complete derivation of medium modifi-cation of gluon fragmentations to a future publication.

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Multiple parton scattering in nuclei: Quark–quark scattering

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