Multiple Parton Scattering in Nuclei: Quark-quark Scattering
Multiple Parton Scattering in Nuclei:
Quark-quark Scattering
Andreas Schäfer, a Xin-Nian Wang b and Ben-Wei Zhang c,d,a
aInstitut für Theoretische Physik, Universität Regensburg
D-93040 Regensburg, Germany
bNuclear Science Division, MS 70R0319
Lawrence Berkeley National Laboratory, Berkeley, CA 94720
cCyclotron Institute and Physics Department, Texas A&M University
College Station, Texas 77843-3366
dInstitute of Particle Physics, Central China Normal University
Wuhan 430079, China
Abstract
Modifications to quark and antiquark fragmentation functions due to quark-quark
(antiquark) double scattering in nuclear medium are studied systematically up to
order O(α2s) in deeply inelastic scattering (DIS) off nuclear targets. At the order
O(α2s), twist-four contributions from quark-quark (antiquark) rescattering also ex-
hibit the Landau-Pomeranchuck-Midgal (LPM) interference feature similar to gluon
bremsstrahlung induced by multiple parton scattering. Compared to quark-gluon
scattering, the modification, which is dominated by t-channel quark-quark (anti-
quark) scattering, is only smaller by a factor of CF /CA = 4/9 times the ratio of
quark and gluon distributions in the medium. Such a modification is not negli-
gible for realistic kinematics and finite medium size. The modifications to quark
(antiquark) fragmentation 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 nuclei will lead
to different modifications of quark and antiquark fragmentation functions inside
a nucleus, which qualitatively explains the experimentally observed flavor depen-
dence of the leading hadron suppression in semi-inclusive DIS off nuclear targets.
The quark-antiquark annihilation processes also mix quark and gluon fragmentation
functions in the large fractional momentum region, leading to a flavor dependence
of jet quenching in heavy-ion collisions.
Key words: Jet quenching, modified fragmentation, parton energy loss.
PACS: 24.85.+p, 12.38.Bx, 13.87.Ce, 13.60.-r
Preprint submitted to Elsevier 30 October 2018
http://arxiv.org/abs/0704.0106v2
1 Introduction
Multiple parton scattering in a dense medium can be used as a useful tool to study
properties of both hot and cold nuclear matter. The success of such an approach has been
demonstrated by the discovery of strong jet quenching phenomena in central Au + Au
collisions at the Relativistic Heavy-ion Collider (RHIC) [1,2,3] and their implications
on the formation of a strongly coupled quark-gluon plasma at RHIC [4,5]. However,
for a convincing phenomenological study of the existing and future experimental data,
a unified description of all medium effects in hard processes involving nuclei, such as
electron-nucleus (e + A), hadron-nucleus (h + A) and nucleus-nucleus collisions (A +
A) has to be developed [6,7]. This must include the physics of transverse momentum
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
scattering [13,14,15,16,17,18,19].
There exist many different frameworks in the literature to describe multiple scattering
in a nuclear medium [20,21,22]. Among them the twist expansion approach is based on
the generalized factorization 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-parton correlations in analogy to the parton distribution op-
erators in leading twist processes. Though the corresponding higher twist corrections are
suppressed by powers of 1/Q2, they are enhanced at least by a factor of A1/3 due to
multiple scattering in a large nucleus. This framework has been applied recently to study
medium modification of the fragmentation functions as the leading parton propagates
through the medium [18,19]. Because of the non-Abelian Landau-Pomeranchuck-Midgal
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
by A2/3, quadratic in the nuclear size [18,19]. Phenomenological study of parton energy
loss and nuclear modification of the fragmentation functions in cold nuclear matter [24]
gives a good description of the nuclear modification of the leading hadron spectra in semi-
inclusive deeply inelastic lepton-nucleus scattering observed by the HERMES experiment
[25,26]. The same framework also gives a compelling explanation for the suppression of
large transverse momentum hadrons discovered at RHIC [27].
The emphasis of recent studies of medium modification of fragmentation functions has
been on radiative parton energy loss induced by multiple scattering with gluons. Such
processes indeed are dominant relative to multiple scattering with quarks because of the
abundance of soft gluons in either cold nuclei or hot dense matter produced in heavy-ion
collisions. Since gluon bremsstrahlung induced by scattering with medium gluons is the
same for quarks and anti-quarks, one also expects the energy loss and fragmentation
modification to be identical for quarks and anti-quarks. However, in a medium with fi-
nite baryon density such as cold nuclei and the forward region of heavy-ion collisions, the
difference between quark and anti-quark distributions in the medium should lead to differ-
ent energy loss and modified fragmentation functions for quarks and antiquarks through
quark-antiquark annihilation processes. To study such an asymmetry, one must 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
functions (FF) due to quark-quark (antiquark) double scattering in a nuclear medium,
working within the LQS framework for generalized factorization in perturbative QCD.
For a complete description of nuclear modification of the single inclusive hadron spec-
tra, one still have to consider medium modification of gluon fragmentation functions in
addition to modified quark fragmentation function due to quark-gluon scattering [18].
The theoretical results presented in this paper will be a second step toward a complete
description of medium modified fragmentation functions. However, one can already find
that quark-quark (antiquark) double scattering will give different corrections to quark
and antiquark FF, depending on antiquark and quark density of the medium, respec-
tively. This difference between modified quark and antiquark FF may shed light on the
interesting observation by the HERMES experiment [25,26] of a large difference between
nuclear suppression of the leading proton and antiproton spectra in semi-inclusive DIS
off large nuclei. Such a picture of quark-quark (antiquark) scattering can provide a com-
peting mechanism for the experimentally observed phenomenon in addition to possible
absorption of final state hadrons inside nuclear matter [28,29].
The paper is organized as follows. In the next section we will present the general for-
malism of our calculation including the generalized factorization of twist-4 processes.
In Section III we will illustrate the procedure of calculating the hard partonic parts of
quark-quark double scattering in nuclei. In Section IV we will discuss the modifications
to quark and antiquark fragmentation functions due to quark-quark (antiquark) dou-
ble scattering in nuclei. In Section V, we will focus on the flavor dependent part of the
medium modification to the quark FF’s due to quark-antiquark annihilation and we will
discuss the implications for the flavor dependence of the leading hadron spectra in both
DIS off a nucleus and heavy-ion collisions. We will summarize our work in Section VI. In
the Appendix A-1, we collect the complete results for the hard partonic parts for different
cut diagrams of quark-quark (antiquark) double rescattering in nuclei. We also provide
an alternative calculation of the hard parts of the central-cut diagrams in Appendix A-3
through elastic quark-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-
nucleus scattering, we consider the following processes,
e(L1) + A(p) −→ e(L2) + h(ℓh) +X ,
Ap Ap
Fig. 1. Lowest order and leading-twist contribution to semi-inclusive DIS.
where L1 and L2 are the four momenta of the incoming and outgoing leptons, and ℓh
is the observed hadron momentum. The differential cross section for the semi-inclusive
process can be expressed as
EL2Eℓh
dσhDIS
d3L2d3ℓh
LµνEℓh
dW µν
, (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
the lepton-nucleon center-of-mass energy and αEM is the electromagnetic (EM) coupling
constant. The leptonic tensor is given by Lµν = 1/2Tr(γ · L1γµγ · L2γν) while the semi-
inclusive hadronic tensor is defined as,
〈A|Jµ(0)|X, h〉〈X, h|Jν(0)|A〉2πδ4(q + p− pX − ℓh) (2)
where
X runs over all possible final states and Jµ =
q eqψ̄qγµψq is the hadronic EM
current.
Assuming collinear factorization in the parton model, the leading-twist contribution to
the semi-inclusive cross section can be factorized into a product of parton distributions,
parton fragmentation functions and the partonic cross section. Including all leading log
radiative corrections, the lowest order contribution [O(α0s)] from a single hard γ∗ + q
scattering, as illustrated in Fig. 1, can be written as
dW Sµν
dxfAq (x, µ
µν (x, p, q)Dq→h(zh, µ
2) ; (3)
H(0)µν (x, p, q) =
Tr(γ · pγµγ · (q + xp)γν)
2p · q
δ(x− xB) , (4)
where the momentum fraction carried by the hadron is defined as zh = ℓ
−, xB =
Q2/2p+q− is the Bjorken scaling variable, µ2I and µ
2 are the factorization scales for
the initial quark distributions fAq (x, µ
I) in a nucleus and the fragmentation functions
in vacuum Dq→h(zh, µ
2), respectively. The renormalized quark fragmentation function
Dq→h(zh, µ
2) satisfies the DGLAP QCD evolution equations [30]:
∂Dq→h(zh, µ
∂ lnµ2
γq→qg(z)Dq→h(zh/z, µ
+ γq→gq(z)Dg→h(zh/z, µ
; (5)
∂Dg→h(zh, µ
∂ lnµ2
γg→qq̄(z)Dq→h(zh/z, µ
+ γg→gg(z)Dg→h(zh/z, µ
, (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 effectively give rise to additional terms in the DGLAP evolution equation
leading to a modification of the fragmentation functions in a medium. These are power-
suppressed higher-twist corrections and they involve higher-twist parton matrix elements.
We will only consider those contributions that involve two-parton correlations from two
different nucleons inside the nucleus. They are proportional 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 will limit our study to such
double scattering processes in a nuclear medium. These are twist-four processes and give
leading contributions to the nuclear effects. The contributions of higher twist processes or
contributions not enhanced by the nuclear medium will be neglected for the time being.
When considering double scattering with nuclear enhancement, a very important process
is quark-gluon double scattering as illustrated in Fig. 2. Such processes give the dominant
contribution to the leading quark energy loss and have been studied in detailed in Refs.
[18,19]. The modification to the vacuum quark fragmentation function from quark-gluon
scattering is,
qg→qg
q→h (zh)=
α2sCA
Dq→h(zh/z)
1 + z2
(1− z)+
TAqg(x, xL)
fAq (x)
+ δ(z − 1)
∆TAqg(x, ℓ
fAq (x)
+Dg→h(zh/z)
1 + (1− z)2
TAqg(x, xL)
fAq (x)
, (7)
Fig. 2. A typical diagram for quark-gluon double scattering with three possible cuts [central(C),
left(L) and right(R)].
where the +function is defined as
F (z)
(1 − z)+
F (z)− F (1)
for any F (z) that is sufficiently smooth at z = 1 and the twist-four quark-gluon correla-
tion function,
TAqg(x, xL) =
dy−1 dy
i(x+xL)p
+y−(1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
×〈A|ψ̄q(0)
F +σ (y
+σ(y−1 )ψq(y
−)|A〉θ(−y−2 )θ(y− − y−1 ), (9)
has explicit interference included. The matrix element in the virtual correction [the term
with δ(z − 1)] is defined as
∆TAqg(x, ℓ
T ) ≡
2TAqg(x, xL)|z=1 − (1 + z2)TAqg(x, xL)
. (10)
Since TAqg(x, xL)/f
q (x) is proportional to gluon distribution and independent of the flavor
of the 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 for quark and antiquark fragmentation. It was shown in Ref. [24] that
such modification of parton fragmentation functions by quark-gluon double scattering
and gluon bremsstrahlung in a nuclear medium describes very well the recent HERMES
data [25] on semi-inclusive DIS off nuclear targets.
Fig. 3. Diagram for leading order quark-antiquark annihilation with three possible cuts [cen-
tral(C), left(L) and right(R)].
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)].
In this paper, we will consider quark-quark (antiquark) double scattering such as the
process shown in Fig. 3 and its radiative corrections at order O(α2s) in Fig. 4. The
contributions of quark-quark double scattering is proportional to the quark density in
a nucleon, while the contribution of quark-gluon double scattering is proportional to
the gluon density in a nucleon; and the gluon density 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 and gluon fragmentation functions and
therefore gives rise to new nuclear effects. The annihilation processes as shown in Figs. 3
and 4 will lead to different modifications of quark and antiquark fragmentation functions
in a medium with finite baryon density (or valence quarks). Such differences will in
turn lead to flavor dependence of the nuclear modification of leading hadron spectra as
observed in HERMES experiment [25,26].
Quark-quark double scattering as well as quark-gluon double scattering are twist-4 pro-
cesses. We will apply the same generalized factorization procedure for twist-4 processes
as developed by LQS [23] for semi-inclusive processes in DIS. In general, the twist-four
contributions can be expressed 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 following form,
dWDµν
p+dy−
dy−1 dy
−, y−1 , y
2 , p, q, zh)
×〈A|ψ̄q(0)
−)ψ̄q(y
2 )|A〉. (11)
Here we have neglected transverse momenta of all quarks in the hard partonic part.
Transverse momentum dependent contributions are higher twist and are suppressed by
〈k2⊥〉/Q2, Therefore, all quarks’ momenta are assumed collinear, k2 = x2p and k3 = x3p.
−, y−1 , y
2 , p, q, zh) is the Fourier transform of the partonic hard part H̃µν(x, x1, x2, p, q, zh)
in momentum space,
−, y−1 , y
2 , p, q, zh) =
eix1p
+y−+ix2p
+i(x−x1−x2)p
×H̃Dµν(x, x1, x2, p, q, zh)
dxH(0)µν (x, p, q)H
(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
leading twist without multiple parton scattering can be factorized out of the high-twist
hard part H̃Dµν(x, x1, x2, p, q, zh). The momentum fractions x, x1, and x2 are fixed by
δ-functions of the on-shell conditions of the final state partons and poles of parton prop-
agators in the partonic hard part. The phase factors in H
−, 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 or two-parton correlations. The quark-quark
double scattering corrections in Eq. (11) can then be factorized as the convolution of
fragmentation functions, twist-four partonic matrix elements and the partonic hard scat-
tering cross sections. For scatterings (versus the annihilation) with quarks (antiquarks),
a summation over the flavor of the secondary quarks (antiquarks) should be included
in two-quark correlation matrix elements and both t, u channels and their interferences
should 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
fragmentation function in the same form [Eq. (3)],
dWDµν
dxfAq (x)H
µν (x, p, q)∆Dq→h(zh) . (13)
3 Quark-quark double scattering processes
In this section we will discuss the calculation of the hard part of quark-quark double
scattering in detail. The lowest order process of quark-quark (antiquark) double scattering
in nuclei is quark-antiquark annihilation (or quark-gluon conversion) as shown in Fig. 3.
The hard partonic parts from the three cut diagrams in this figure are [18]:
0,C(y
−, y−1 , y
2 , x, p, q, zh)=Dg→h(zh)
×θ(−y−2 )θ(y− − y−1 ) , (14)
0,L(y
−, y−1 , y
2 , x, p, q, zh)=Dq→h(zh)
×θ(y−1 − y−2 )θ(y− − y−1 ) , (15)
0,R(y
−, y−1 , y
2 , x, p, q, zh)=Dq→h(zh)
×θ(−y−2 )θ(y−2 − y−1 ) . (16)
The main focus of this paper is about contributions from the next-leading order correc-
tions to 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 the Appendix A-1, each having up
to three different cuts. In this section, we demonstrate the calculation of the hard parts
from the quark-antiquark annihilation in Fig. 4 in detail as an example. We will list the
complete results of all diagrams in Appendix A.
One can write down the hard partonic part of the central-cut diagram of Fig. 4 (Fig. 5
in Appendix A-1) according to the conventional Feynman rule,
C µν(y
−, y−1 , y
2 , p, q, zh)=
Dg→h(
eix1p
+y−+ix2p
×ei(x−x1−x2)p+y
(2π)4
γµĤγν
2πδ+(ℓ
2)2πδ+(ℓ
g) δ(1− z −
γ · (q + x1p)
(q + x1p)2 − iǫ
γ · (q + x1p− ℓ)
(q + x1p− ℓ)2 − iǫ
γ · (q + xp− ℓ)
(q + xp− ℓ)2 + iǫ
γ · (q + xp)
(q + xp)2 + iǫ
εαρ(ℓ) εβσ(ℓg) , (17)
where δ+ is a Dirac delta-function with only the positive solution in its functional variable,
εαρ(ℓ) = −gαρ + (nαℓρ + nρℓα)/n · ℓ is the polarization tensor of a gluon propagator in
an axial gauge (n · A = 0) with n = [1, 0−,~0⊥], ℓ and ℓg = q + (x1 + x2)p − ℓ are the
4-momenta carried by the two final gluons respectively. The fragmenting gluon carries
a fraction, z = ℓ−g /q
−, of the initial quark’s longitudinal momentum (the large minus
component).
To simplify the calculation in the case of small transverse momentum ℓT ≪ q−, p+, we
can apply the collinear approximation to complete the trace of the product of γ-matrices,
Ĥ ≈ γ · ℓq
γ · ℓqĤ
. (18)
According to the convention in Eqs. (11) and (12), contributions from quark-quark double
scattering in the nuclear medium to the semi-inclusive hadronic tensor in DIS off a nucleus
can be expressed in the general factorized form:
dWDqq̄,µν
dxH(0)µν (x, p, q)
p+dy−
dy−1 dy
(y−, y−1 , y
2 , x, p, q, zh)
× 〈A|ψ̄q(0)
ψq̄(y
−)ψ̄q(y
ψq̄(y
2 )|A〉 . (19)
After carrying out the momentum integration in x, x1, x2 and ℓ
± in Eq. (17) with the
help of contour integration and δ-functions, one obtains the hard partonic part, H
the rescattering for the central-cut diagram in Fig. 4 (Fig. 5) as
5,C(y
−, y−1 , y
2 , x, p, q, zh) =
α2sxB
Dg→h(zh/z)
× 2(1 + z
z(1− z)
I5,C(y
−, y−1 , y
2 , x, xL, p) , (20)
I5,C(y
−, y−1 , y
2 , x, xL, p) = e
i(x+xL)p
+y−θ(−y−2 )θ(y− − y−1 )
× (1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (21)
where the momentum fractions xL is defined as
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
combine these phase factors with the matrix elements of the quark fields to define a
special two-quark correlation function
A(5,C)
qq̄ (x, xL) =
p+dy−
dy−1 dy
2 〈A|ψ̄q(0)
ψq̄(y
−)ψ̄q(y
ψq̄(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 the hadronic tensor can then be expressed as
dWDqq̄,µν
dxH(0)µν (x, p, q)
α2sxB
Dg→h(
2(1 + z2)
z(1 − z)
A(5,C)
qq̄ (x, xL). (24)
Contributions from all quark-quark (antiquark) double scattering processes can be cast
in the above factorized form.
The structure of the phase factors in I5,C(y
−, y−1 , y
2 , x, xL, p) is exactly the same as for
gluon bremsstrahlung induced by quark-gluon scattering as studied in Ref. [18,19]. It
resembles the cross section of dipole scattering and represents contributions from two
different processes and 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
×[1 + e−ixLp+(y−+y
) − e−ixLp+y
2 − e−ixLp+(y−−y
)] . (25)
The first term corresponds to the so-called hard-soft processes where the gluon emission
is induced 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 from the nucleus that carries zero momentum and converts into a real
gluon in the final state. The second term corresponds to a process in which the initial
quark with momentum xp is on-shell after the first hard γ∗-quark scattering. It then
annihilates with another antiquark and produces two final gluons in the final state. In
this process, the antiquark carries finite (hard) momentum xLp. Therefore one often refers
to this process as double-hard scattering as compared to the first process in which the
antiquark carries zero momentum. Set aside the change of flavors in the initial and final
states, the double-hard scattering corresponds essentially to two-parton elastic scattering
with finite momentum and energy transfer. This is in contrast to the hard-soft scattering
which is essentially the final state radiation of the γ∗-quark scattering and the total
energy and momentum of the two final state gluons all come from the initial quark. The
corresponding 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 interfer-
ences between 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 gives rise 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
. (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 the other gluon (close to the γ∗-quark interaction) fragments, the contribution
to the semi-inclusive hadronic tensor is similar except that the corresponding effective
“splitting function” should be replaced by
1 + z2
z(1 − z)
→ 1 + (1− z)
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 an-
nihilation processes and their interferences are summed. Therefore, one can simply mul-
tiply the final results by a factor 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-
cut diagrams 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 only differences are in the phase factors and the fragmentation functions
since the fragmenting parton can be the final-state quark or gluon. These hard parts can
be calculated following a similar procedure and one gets,
5,L(R)(y
−, y−1 , y
2 , x, p, q, zh) =
α2sxB
Dq→h(
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
×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
×θ(−y−2 )θ(y−2 − y−1 ) . (31)
In the asymmetrical cut diagrams, the above contributions come from the fragmentation
of the final-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 contribution can be obtained by simply replacing the quark fragmentation
function by the gluon fragmentation function Dg→h(zh/z) and replacing z by 1 − z.
Summing the contributions from three 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 by Wang and Guo [18] and is a unique
feature of quark-quark (antiquark) double scattering among all 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(α2s), which are listed in Ap-
pendix A-1. There are three types of processes: two annihilation processes, qq̄ → gg
(central-cut diagrams in Figs. 5, 6, 7, 8 and 9), qq̄ → qiq̄i (central-cut diagram in Fig. 10)
and quark-quark (antiquark) scattering, qqi(q̄i) → qqi(q̄i) (central-cut diagram in Fig. 11).
One also has to consider the interference of s and t-channel amplitude for annihilation
into an identical quark pair, qq̄ → qq̄ (central-cut diagrams in Figs. 12 and 13) and the
interference between t and u channels of identical quark scattering qq → qq (central-cut
diagram in Fig. 14).
Contributions from left and right-cut diagrams correspond to interference between the
amplitude of gluon radiation from single γ∗-quark scattering and triple quark scattering.
The amplitudes of gluon radiation via triple quark scattering essentially come from ra-
diative corrections to the left and right-cut diagrams of the lowest-order quark-antiquark
annihilation in Fig. 3 (as shown in left and right-cut diagrams in Figs. 5, 6, 8, 9, 12,
13, 15 and 16). Two other triple quark scatterings with gluon radiation, shown as the
left and right-cut diagrams in Figs. 11 and 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 (annihi-
lation), 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 of θ-functions,
= HDC θ(−y−2 )θ(y− − y−1 )−HDL 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
the integration range along the light-cone. For contributions from central, left and right-
cut diagrams that have identical hard partonic parts, H
C = H
L = H
R , they will
have a common combination of θ-functions that produces a path-ordered integral,
dy−2 =−
dy−1 dy
θ(−y−2 )θ(y− − y−1 )− θ(−y−2 )θ(y−2 − y−1 )
−θ(y− − y−1 )θ(y−1 − y−2 )
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 very small. Those contributions that are proportional to the above path-ordered
integral are referred to as contact contributions (or contact interactions).
Similarly, y−1 − y−2 is the spatial spread of the second parton and can only be limited
by the spatial size of its host nucleon even for small value of momentum fraction. The
spatial position of its host nucleon, y−1 + y
2 , however, can be anywhere within the nu-
cleus. Therefore, any contributions from double parton scattering that have unrestricted
integration over y−1 and y
2 should be proportional to the nuclear size of the target A
and therefore are nuclear enhanced. In this paper, we will only keep the nuclear enhanced
contributions and neglect the contact contributions. This will greatly simplify the final
results for double parton scattering.
4.1 qq̄ → g annihilation
For the lowest order of quark-antiquark annihilation in Eqs. (14)-(16), the hard parts
from the three cut diagrams are almost the same except for the parton fragmentation
functions. The central-cut diagram is proportional to the gluon fragmentation function
while the left and right-cut diagrams are proportional to quark fragmentation functions.
Rearranging the contributions from 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
qq̄ (x, 0)
H(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 fragmen-
tation functions, the modification to the quark fragmentation function from the lowest
order quark-antiquark annihilation is then,
(qq̄→g)
q→h (zh) =
[Dg→h(zh)−Dq→h(zh)]
qq̄ (x, 0)
fAq (x)
. (35)
Here the effective quark-antiquark correlation function T
qq̄ (x, 0) is defined as,
qq̄i (x, xL)≡
p+dy−
dy−1 dy
ixp+y−−ixLp
)θ(−y−2 )θ(y− − y−1 )
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (36)
with the antiquark q̄i carrying momentum fraction xL. This two-parton correlation func-
tion is always associated with double-hard rescattering processes. Similarly, we define
three other quark-antiquark correlation matrix elements
qq̄i (x, xL)≡
p+dy−
dy−1 dy
i(x+xL)p
+y−θ(y− − y−1 )
× θ(−y−2 )〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (37)
A(I−L)
qq̄i (x, xL)≡
p+dy−
dy−1 dy
i(x+xL)p
+y−−ixLp
+(y−−y
)θ(y− − y−1 )
× θ(−y−2 )〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (38)
A(I−R)
qq̄i (x, xL)≡
p+dy−
dy−1 dy
i(x+xL)p
+y−−ixLp
2 θ(y− − y−1 )
× θ(−y−2 )〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(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
qq̄i (x, xL), the antiquark q̄i
carries momentum fraction xL while the initial quark has the momentum fraction x. The
two-parton correlation T
qq̄i (x, xL) corresponds to the case when the leading quark has
x+ xL but the antiquark carries zero momentum. The two interference matrix elements
are approximately the same for small value of xL and will be denoted as T
qq̄i (x, xL).
4.2 qq̄ → qiq̄i 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 vir-
tual corrections. The simplest real correction comes from qq̄ → qiq̄i annihilation (qi 6= q)
[Fig. 10 and Eqs. (A-25) and (A-26)] which has only a central-cut diagram,
(qq̄→qiq̄i)
q→h (zh) =
α2sxB
[z2 + (1− z)2]
qi 6=q
[Dqi→h(zh/z) +Dq̄i→h(zh/z)]
qq̄ (x, xL)
fAq (x)
. (40)
This kind of qq̄ annihilation is truly a hard processes and thus requires the second an-
tiquark to carry finite initial momentum fraction xL. Furthermore, there are no other
interfering processes.
4.3 qqi(q̄i) → qqi(q̄i) scattering
Contributions from non-identical quark-quark scattering qq̄i → qq̄i (qi 6= q) are a little
complicated 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 the phase factors in each cut diagram,
I11,C = e
i(x+xL)p
+y−(1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
= ei(x+xL)p
+y−[1− e−ixLp+y
2 − e−ixLp+(y−−y
) + e−ixLp
+(y−+y
I11,L = e
i(x+xL)p
+y−(1− e−ixLp+(y−−y
= ei(x+xL)p
+y−[1− e−ixLp+y
2 − e−ixLp+(y−−y
) + e−ixLp
2 ] ;
I11,R = e
i(x+xL)p
+y−(1− e−ixLp+y
= ei(x+xL)p
+y−[1− e−ixLp+y
2 − e−ixLp+(y−−y
) + e−ixLp
+(y−−y
)] , (41)
such that the first three terms in each amplitude are the same. These three common phase
factors will 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 the following phase factors,
I11= e
i(x+xL)p
+y−[θ(−y−2 )θ(y− − y−1 )e−ixLp
+(y−+y
−θ(y−1 − y−2 )θ(y− − y−1 )e−ixLp
2 − θ(−y−2 )θ(y−2 − y−1 )e−ixLp
+(y−−y
)] . (42)
Note that the phase factors of the last two terms in the above equation give identi-
cal contributions to the matrix elements when intergated over y−1 and y
2 as they dif-
fer 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
) to form another contact contribution (path-ordered)
which can be neglected. The final effective phase factor is then
I11 = e
ixp+y−−ixLp
)(1− eixLp+y
2 ) . (43)
Using the above effective phase factor, one can obtain the effective modification to the
quark fragmentation function due to quark-antiquark scattering, qq̄i → qq̄i,
(qq̄i→qq̄i)
q→h (zh) =
α2sxB
q̄i 6=q̄
Dq→h(zh/z)
1 + z2
(1− z)2
+ Dg→h(zh/z)
1 + (1− z)2
A(HI)
qq̄i (x, xL)
fAq (x)
+ [Dq̄i→h(zh/z))−Dg→h(zh/z)]
1 + (1− z)2
A(HS)
qq̄i (x, xL)
fAq (x)
α2sxB
q̄i 6=q̄
Dq→h(zh/z)
1 + z2
(1− z)2
+ Dq̄i→h(zh/z)
1 + (1− z)2
A(HI)
qq̄i (x, xL)
fAq (x)
+ [Dq̄i→h(zh/z)−Dg→h(zh/z)]
1 + (1− z)2
A(SI)
qq̄i (x, xL)
fAq (x)
 , (44)
where three types of two-parton correlations are defined:
A(HI)
qq̄i (x, xL)≡T
qq̄i (x, xL)− T
qq̄i (x, xL)
p+dy−
dy−1 dy
ixp+y−−ixLp
)(1− eixLp+y
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉θ(−y−2 )θ(y− − y−1 ) , (45)
A(SI)
qq̄i (x, xL)≡T
qq̄i (x, xL)− T
qq̄i (x, xL)
p+dy−
dy−1 dy
i(x+xL)p
+y−(1− e−ixLp+y
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉θ(−y−2 )θ(y− − y−1 ) , (46)
A(HS)
qq̄i (x, xL)≡T
A(HI)
qq̄i (x, xL) + T
A(SI)
qq̄i (x, xL)
p+dy−
dy−1 dy
i(x+xL)p
+y−(1− e−ixLp+y
× (1− e−ixLp+(y−−y
))θ(−y−2 )θ(y− − y−1 )
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 . (47)
One can similarly obtain the modification of quark fragmentation from non-identical
quark-quark scattering by replacing q̄i → qi in Eq. (44),
(qqi→qqi)
q→h (zh) =
α2sxB
qi 6=q
Dq→h(zh/z)
1 + z2
(1− z)2
+ Dqi→h(zh/z)
1 + (1− z)2
TA(HI)qqi (x, xL)
fAq (x)
+ [Dqi→h(zh/z)−Dg→h(zh/z)]
1 + (1− z)2
TA(SI)qqi (x, xL)
fAq (x)
. (48)
The two-quark correlations, TA(HI)qqi (x, xL) and T
A(SI)
(x, xL) can be obtained from T
A(HI)
qq̄i (x, xL)
and T
A(SI)
qq̄i (x, xL), respectively, by making the replacements ψqi(y2) → ψ̄qi(y2) and ψ̄qi(y1) →
ψqi(y1) in Eqs. (45) and (46),
TA(HI)qqi (x, xL)≡
p+dy−
dy−1 dy
ixp+y−−ixLp
)(1− eixLp+y
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
1 )|A〉θ(−y−2 )θ(y− − y−1 ) , (49)
TA(SI)qqi (x, xL) =
p+dy−
dy−1 dy
i(x+xL)p
+y−(1− e−ixLp+y
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
1 )|A〉θ(−y−2 )θ(y− − y−1 ) , (50)
and TA(HS)qqi (x, xL) = T
A(HI)
(x, xL) + T
A(SI)
(x, xL).
Note that the contribution from fragmentation of quark qi or antiquark q̄i only comes from
the central-cut diagram. This contribution is positive and is proportional to T
A(HI)
qq̄i (x, xL)+
A(SI)
qq̄i (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 diagrams). Its contribution is therefore negative and partially cancels the pro-
duction of qi(q̄i) from the hard-soft rescattering. The cancellation is not complete since
the gluon and quark fragmentation functions are different. The structure of this hard-soft
rescattering (quark plus gluon) is very similar to the lowest order result of qq̄ → g in
Eq. (35). It contributes to the modification of the effective fragmentation function but
does not contribute to the energy loss. The energy loss of the leading quark comes only
from double-hard rescattering, since the leading quark fragmentation comes both from
the central-cut and single-triple interferences, and the single-triple interference terms can-
cel the effect of hard-soft scattering for the leading fragmentation. Its net contribution is
therefore proportional to T
A(HI)
qq̄i(q̄i)
. Since the double-hard rescattering amounts to elastic
qqi(q̄i) scattering, the effective energy loss is essentially elastic energy loss as shown in
Ref. [34]. There is, however, LPM suppression due to partial cancellation by single-triple
interference 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 fractional momentum 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
technique to identify and neglect the contact contributions, one can find the correspond-
ing modification to the quark fragmentation function from Eqs. (48) and (A-45)-(A-49),
(qq→qq)
q→h (zh) =
α2sxB
TA(HS)qq (x, xL)
fAq (x)
× [Dq→h(zh/z))−Dg→h(zh/z)]
1 + (1− z)2
z(1 − z)
Dq→h(zh/z)
1 + z2
(1− z)2
z(1 − z)
+Dg→h(zh/z)
1 + (1− z)2
z(1 − z)
TA(HI)qq (x, xL)
fAq (x)
α2sxB
TA(SI)qq (x, xL)
fAq (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)
fAq (x)
, (51)
where the effective splitting functions are defined as
P (s)qq→qq(z) =
1 + (1− z)2
z(1 − z)
, (52)
Pqq→qq(z) =
1 + (1− z)2
1 + z2
(1− z)2
z(1− z)
. (53)
4.5 qq̄ → qq̄, 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
consider them together since they have similar single-triple interference processes and
they involve the same kind of quark-antiquark correlation matrix elements, T
qq̄ (x, xL),
(i = HI, SI,HS).
For notation purpose, we first factor out the common factor (CF/Nc)α
sxB/Q
2/fAq (x) and
the integration over ℓT and z and define
(qq̄→gg,qq̄)
q→h (zh) ≡
α2sxB
Q2fAq (x)
(qq̄→gg,qq̄)
q→h (zh, z, x, xL). (54)
After rearranging the phase factors and identifying (by combining central, left and right
cut diagrams) and neglecting contact contributions we can list in the following the twist-
four corrections to the quark fragmentation from the hard partonic parts of each cut
diagram (see Appendix A):
Fig. 5 (t-channel qq̄ → gg),
(qq̄→gg,qq̄)
q→h(5) =Dg→h(zh/z)2CF
1 + (1− z)2
z(1 − z)
1 + z2
z(1− z)
A(HI)
qq̄ (x, xL)
+ [Dg→h(zh/z)−Dq→h(zh/z)] 2CF
1 + z2
z(1− z)
A(SI)
qq̄ (x, xL) ; (55)
Fig. 6 (interference between u and t-channel of qq̄ → gg),
(qq̄→gg,qq̄)
q→h(6) =Dg→h(zh/z)
−4(CF − CA/2)
z(1 − z)
A(HI)
qq̄ (x, xL)
+ [Dg→h(zh/z)−Dq→h(zh/z)]
−2(CF − CA/2)
z(1 − z)
A(SI)
qq̄ (x, xL) ; (56)
Fig. 7 (s-channel of qq̄ → gg),
(qq̄→gg,qq̄)
q→h(7) = Dg→h(zh/z)4CA
(1− z + z2)2
z(1 − z)
qq̄ (x, xL) ; (57)
Figs. 8 and 9 (interference of s and t-channel qq̄ → gg),
(qq̄→gg,qq̄)
q→h(8+9) =Dg→h(zh/z)(−2CA)
1 + z3
z(1− z)
1 + (1− z)3
z(1 − z)
×TA(HI)qq̄ (x, xL) + CA
Dq→h(zh/z)
1 + z3
z(1 − z)
+Dg→h(zh/z)
1 + (1− z)3
z(1 − z)
× [TA(I2)qq̄ (x, xL)− T
qq̄ (x, xL)] ; (58)
Fig. 10 (s-channel of qq̄ → qq̄),
(qq̄→gg,qq̄)
q→h(10) = [Dq→h(zh/z) +Dq̄→h(zh/z)] [z
2 + (1− z)2]
×TA(H)qq̄ (x, xL) , (59)
Fig. 11 (t-channel of qq̄ → qq̄), similar to Eq. (44),
(qq̄→gg,qq̄)
q→h(11) =Dq→h(zh/z)
1 + z2
(1 − z)2
A(HI)
qq̄ (x, xL)
+Dq̄→h(zh/z)
1 + (1− z)2
A(HS)
qq̄ (x, xL)
−Dg→h(zh/z)
1 + (1− z)2
A(SI)
qq̄ (x, xL) ; (60)
Figs. 12 and 13 (interference between s and t-channel qq̄ → qq̄),
(qq̄→gg,qq̄)
q→h(12+13) =−4(CF − CA/2)
Dq→h(zh/z)
1 − z
+ Dq̄→h(zh/z)
(1− z)2
A(HI)
qq̄ (x, xL)
+2(CF − CA/2)
Dq→h(zh/z)
+Dg→h(zh/z)
(1− z)2
× [TA(I2)qq̄ (x, xL)− T
qq̄ (x, xL)] ; (61)
Figs. 15 and 16 (two additional single-triple interference diagrams),
(qq̄→gg,qq̄)
q→h(15+16) =−2CF
Dq→h(zh/z)
1 + z2
+ Dg→h(zh/z)
1 + (1− z)2
A(I2)
qq̄ (x, xL) . (62)
Most processes involve both TA(HI)(x, xL) for double-hard rescattering with interference
and TA(SI)(x, xL) for hard-soft rescattering with interference. All the s-channel (Figs. 7
and 10) processes involve double-hard scattering only. Therefore, they depend only on the
qq̄ (x, xL) = T
A(HI)
qq̄ (x, xL) + T
qq̄ (x, xL). For interference between single and triple
scattering (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), only interference matrix elements, T
qq̄ (x, xL) and T
A(I2)
qq̄ (x, xL), are
involved. Here,
A(I2)
qq̄ (x, xL)≡
p+dy−
dy−1 dy
ixp+y−+ixLp
×〈A|ψ̄q(0)
−)ψ̄q(y
2 )|A〉θ(−y−2 )θ(y− − y−1 )
p+dy−
dy−1 dy
ixp+y−+ixLp
+(y−−y
×〈A|ψ̄q(0)
−)ψ̄q(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-triple interference processes. One can categorize the above contributions according
to the associated two-quark correlation matrix elements and rewrite the above contribu-
tions as,
qq̄→qq̄,gg
q→h(HI) = T
A(HI)
qq̄ (x, xL)[Dg→h(zh/z)Pqq̄→gg(z) +Dq→h(zh/z)Pqq̄→qq̄(z)
+Dq̄→h(zh/z)Pqq̄→qq̄(1− z)] (64)
qq̄→qq̄,gg
q→h(SI) = T
A(SI)
qq̄ (x, xL)
z(1 − z)
+ 2CF
1 + (1− z)2
×Dg→h(zh/z)−Dq→h(zh/z)
z(1− z)
+ 2CF
+ Dq̄→h(zh/z)
1 + (1− z)2
A(SI)
qq̄ (x, xL)
[Dq→h(zh/z)−Dg→h(zh/z)]
P (s)qq→qq(z)
− 2CF
1 + z2
z(1− z)
+ [Dq̄→h(zh/z)−Dq→h(zh/z)]
1 + (1− z)2
qq̄→qq̄,gg
q→h(I) = T
qq̄ (x, xL)
4(1− z + z2)2 − 1
z(1 − z)
− 2CF
(1− z)2
×Dg→h(zh/z) + [z2 + (1− z)2]Dq̄→h(zh/z)]
+ Dq→h(zh/z)
z2 + (1− z)2 −
z(1 − z)
− 2CF
A(I2)
qq̄ (x, xL)
Dq→h(zh/z)
z(1 − z)
− 2CF
+ Dg→h(zh/z)
z(1 − z)
− 2CF
, (66)
where P (s)qq→qq(z) is given in Eq. (52) and the effective splitting functions for qq̄ → gg and
qq̄ → qq̄ are defined as
Pqq̄→gg(z) = 2CF
z2 + (1− z)2
z(1− z)
− 2CA[z2 + (1− z)2] ; (67)
Pqq̄→qq̄(z) = z
2 + (1− z)2 +
1 + z2
(1− z)2
, (68)
which come from the complete matrix elements of qq̄ → gg and qq̄ → qq̄ scattering (see
Appendix A-3). Again, double-hard rescattering corresponds to the elastic scattering of
the leading quark with another antiquark in the medium and the interference contribu-
tions. The structure of the 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 energy loss of the leading quark. The unique contributions
in the qq̄ → qq̄, gg processes are the interference-only contributions. They mainly come
from single-triple interference processes in the 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 in vacuum. In fact, one can also view the evolution of fragmentation functions in
vacuum as modification due to final-state gluon radiation. In both cases, the modification
at large zh is mainly determined by the singular behavior of the splitting functions for
z → 1, whereas the modifications 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 contribu-
tions from all possible processes shows that the dominant modification to the effective
quark fragmentation function comes from the t-channel of double hard quark-quark scat-
tering processes,
∆Dq→h(zh)∼
α2sxB
Dq→h(
TA(HI)qqi (x, xL)
fAq (x)
× 1 + z
(1 − z)2+
+δ(1− z)∆qi(ℓ2T )
Dq→h(
A(HI)
(x, xL)
fAq (x)
×z(1 + z
(1 − z)+
+ δ(1− z)∆qi(ℓ2T )
, (69)
where the summation is over all possible quark and antiquark flavors including qi = q, q̄
and ∆qi(ℓ
T ) represents the contribution from virtual corrections. We have expressed the
modification in a form that it is proportional to the matrix elements xLT
A(HI)
(x, xL)/f
q (x) ∼
A1/3xLf
(xL) as compared to the modification from quark-gluon scattering where the
corresponding matrix element [Eq. (9)] is TA(HI)qg (x, xL)/f
q (x) ∼ A1/3xLGN(xL). Here,
fNqi (x) and G
N(x) are quark and gluon distributions, respectively, in a nucleon. This
leading contribution to the modification from quark-quark scattering is very similar in
form to that from quark-gluon scattering [see Eq. (7)]. However, it is smaller due to the
different color factors CF/CA = 4/9 and the different quark and gluon distributions,
fNqi (xL) and G
N(xL) in a nucleon. Because of LPM intereference, small angle scattering
with long formation time τf = 1/xLp
+ is suppressed, leading to a minimum value of
xL ≥ xA = 1/mNRA = 0.043 for a Kr target. For this value of xL, the ratio
fNqi (xL, Q
GN(xL, Q2)
≥ 1.40/1.85 ∼ 0.75, (70)
at Q2 = 2 GeV2 according the CTEG4HJ parameterization [35]. Therefore, one has to
include the effect of quark-quark scattering for a complete calculation of the total quark
energy loss and medium modification of quark fragmentation functions.
In a weakly coupled and fully equilibrated quark-gluon plasma, quark to gluon number
density ratio is ρq/ρg = nf (3/2)Nc/(N
c − 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. In this ideal scenario one expects Nq/Ng ∼ 1/4CA = 1/12 and therefore
can neglect quark-quark scattering. The modification of quark fragmentation function will
be dominated by quark-gluon rescattering. However, for moderate jet energy E ≈ 20 GeV
and a finite medium L ∼ 5 fm, parton distributions in a quark-gluon plamsa are close to
the thermal distribution. In particular, if quark and gluon production is dominated by
non-perturbative pair production from strong color fields in the initial stage of heavy-ion
collisions [36], the quark to gluon ratio is comparable to the equilibrium value. In this
case, we should take into account the medium modification of the quark fragmentation
functions by quark-quark scattering.
An important double hard process in quark-quark (antiquark) scattering is qq̄ → gg
[Eq. (64)]. In this process, the annihilation converts the initial quark into two final gluons
that subsequently fragment into hadrons. This will lead to suppression of the leading
hadrons not only because of energy loss (energy carried away by the other gluon) but
also due to the softer behavior of gluon fragmentation functions at large zh. Even though
the leading behavior of the effective splitting function [Eq. (67)]
Pqq̄→gg(z) ≈ 2CF
z2 + (1− z)2
z(1− z)
is not as dominating as that of t-channel quark-quark scattering, it is enhanced by a color
factor 2CF = 8/3. One expects this to make a significant contribution to the medium
modification at intermediate 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 dominates at low pT while the fraction of valence quark jets increases at large pT .
Quarks are more likely to fragment into protons than antiprotons, while gluons fragment
into protons and antiprotons with 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 see the antiproton to proton ratio
p̄/p becomes smaller due to jet quenching. However, if the quark-gluon conversion due
to qq̄ → gg becomes important, one would expect that the fractions of quark and gluon
jets are modified toward their equilibrium values. The final p̄/p ratio could be larger than
or comparable to that in p+ p collisions. Such a scenario of quark-gluon conversion was
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-
antiquark annihilation 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. Their contributions generally have the form,
α2sxB
Dqi→h(
)−Dg→h(
×Pqqi→qqi(z)
TA(SI)qqi (x, xL)
fAq (x)
, (72)
where again the summation over the quark flavor includes qi = q, q̄. This mixing does
not occur on the probability but rather on on the amplitude level since it involves in-
terferences between single and triple scattering. Therefore, this contribution depends on
the difference between gluon and quark fragmentation functions [Eq. (35)] and can be
positive or negative in different region of zh. Nevertheless, they contribute to the mod-
ification of the effective quark fragmentation function and 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 Sec-
tion 4, we can express the total twist-four correction up to O(α2s) to the quark fragmen-
tation function as
∆Dq→h(zh)=
2 [Dg→h(zh)−Dq→h(zh)]
qq̄ (x, 0)
fAq (x)
a,b,i
Db→h(zh/z)P
qa→b(z)
TA(i)qa (x, xL)
fAq (x)
 , (73)
where the summation is over all possible q+a→ b+X processes and all different matrix
elements TA(i)qa (x, xL) (i = HI, SI, I, I2), which will be four basic matrix elements we will
use. The effective splitting functions P
qa→b(z) are listed in Appendix A-2. One should also
include virtual corrections which can be constructed from the real corrections through
unitarity constraints [18].
Similarly, we can also write down the twist-four corrections to antiquark fragmentation
in a nuclear medium,
∆Dq̄→h(zh)=
2 [Dg→h(zh)−Dq̄→h(zh)]
q̄q (x, 0)
fAq̄ (x)
a,b,i
Db→h(zh/z)P
q̄a→b(z)
q̄a (x, xL)
fAq̄ (x)
 , (74)
where the matrix elements T
q̄a (x, xL) and the effective splitting functions P
q̄a→b(z) can
be obtained from the corresponding ones for quarks. Given a model for the two-quark
correlation functions, one will be able to use the above expressions to numerically evaluate
twist-four corrections to the quark (antiquark) fragmentation functions. In this paper,
we will instead give a qualitative 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-quark
correlation functions can be factorized, as has been done in Refs. [18,19,23,33],
p+dy−
dy−1 dy
+y−+ix2p
)θ(−y−2 )θ(y− − y−1 )
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉
fAq (x1) f
(x2) , (75)
p+dy−
dy−1 dy
+y−+ix2p
)±ixLp
2 θ(−y−2 )θ(y− − y−1 )
×〈A|ψ̄q(0)
ψq̄(y
−)ψ̄qi(y
ψqi(y
2 )|A〉
fAq (x1) f
(x2)e
A , (76)
where xA = 1/mNRA, mN is the nucleon mass, RA the nucleus size, f
(x2) is the
antiquark distribution in a nucleon and C is assumed to be a constant, parameterizing
the strength of two-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 = 0.21A
We should note that we set kT = 0 for the collinear expansion. As a consequence, the
secondary quark field in the twist-four parton matrix elements will carry zero momentum
in the soft-hard process. Finite intrinsic transverse momentum leads to higher-twist cor-
rections. If a subset of the higher-twist terms in the collinear expansion can be resummed
to restore the phase factors 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:
A(HI)
qq̄i (x, xL)≈
fAq (x) f
(xL + xT )[1− e−x
A], (77)
A(SI)
qq̄i (x, xL)≈
fAq (x+ xL) f
(xT )[1− e−x
A], (78)
qq̄i (x, xL)≈T
A(I2)
qq̄i (x, xL) ≈
fAq (x+ xL) f
(xT )e
fAq (x) f
(xL + xT )e
A. (79)
In the last approximation, we have assumed xL ∼ xT ≪ x. Similarly, one can obtain
TA(i)qqi (x, xL), T
q̄qi (x, xL) and T
q̄q̄i (x, xL). With these forms of two-quark correlation
matrix elements, we can estimate the flavor dependence of the nuclear modification to
the quark (antiquark) fragmentation functions.
The lowest order corrections [O(αs)] are very simple
q→h (zh) ∝ CA1/3[Dg→h(zh)−Dq→h(zh)]fNq̄ (xT ) , (80)
q̄→h̄
(zh) ∝ CA1/3[Dg→h̄(zh)−Dq̄→h̄(zh)]fNq (xT ) . (81)
We consider the dominant contribution from the fragmentation of a quark (antiquark)
which is one of the valence quarks (antiquarks) of the final particle h (antiparticle h̄).
The gluon fragmentation functions into h and h̄ are the same. For large zh, the gluon
fragmentation function is always softer than the valence quark (antiquark) fragmenta-
tion [38]. Therefore, the lowest order twist-four corrections are always negative for large
zh, leading to a suppression of the valence quark (antiquark) fragmentation function,
Dqv→h(zh) [Dq̄v→h̄(zh)]. Consider those quarks that are also valence quarks of a nucleon:
n = udd
p = uud , p̄ = ūūd̄ , (82)
K+ = us̄ ,K− = ūs . (83)
π+, π0, π− = ud̄ , (uū − dd̄ )/
2 , dū . (84)
One can find the following flavor dependence of the lowest order twist-four corrections
to the quark (antiquark) fragmentation functions,
q̄v→h̄
−|∆D(LO)
q̄v→h̄
(zh)|
−|∆D(LO)qv→h(zh)|
fNqv (xT )
fq̄v(xT )
> 1 , (85)
q̄v→h̄
1 + ∆D
q̄v→h̄
(zh)/Dq̄→h̄(zh)
1 + ∆D
(zh)/Dq→h(zh)
< 1 , (86)
where R
is the corresponding leading order suppression of the fragmentation function
at large zh for proton (anti-proton) and K
+ (K−). Since pions contain both valence
quark and antiquark, the suppression factors should be similar for all pions. For xT ≥
0.043, u(x)/ū(x) ≥ 3 and d(x)/d̄(x) ≥ 2 [35]. Therefore, the modification of antiquark
fragmentation functions due to quark-antiquark annihilation is significantly larger than
that of a quark.
The flavor dependence of the NLO results are more complicated since they involve scatter-
ing with both quarks and antiquarks in the medium. One can observe first that effective
splitting functions (or quark-quark scattering cross section) are the same for the t-channel
qq′ → qq′ and qq̄′ → qq̄′ (q′ 6= q) scatterings,
qq′→b(z) = P
q̄q′→b(z) = P
qq̄′→b(z) = P
q̄q̄′→b(z) . (87)
For identical quark-quark scattering or quark-antiquark annihilation, one can separate
the qq̄ annihilation splitting functions (or cross sections) into singlet and non-singlet
contributions by singling out the t-channel contributions,
qq̄→b(z)≡P
qq→b(z) + ∆P
qq̄→b(z), (88)
q̄q→b(z)≡P
q̄q̄→b(z) + ∆P
q̄q→b(z). (89)
These singlet contributions to the modified fragmentation functions are,
S(NLO)
q→h (zh)∝
b,q′,i
Db→h ⊗ P (i)qq′→b(zh)
×[fNq′ (xT ) + fNq̄′ (xT )]C(i) , (90)
S(NLO)
q̄→h̄
(zh)∝
b,q′,i
Db̄→h̄ ⊗ P
q̄q̄′→b̄
×[fNq′ (xT ) + fNq̄′ (xT )]C(i) , (91)
where the summation over q′ now includes q′=q and C(i)(xL) are flavor-independent
functions determined from Eqs. (77)-(79),
C(HI) =C(SI) = C(xL)(1− e−x
C(I) =C(I2) = C(xL)e
A , (92)
and C(xL) is a common coefficient that is a function of xL. Using P
q̄q̄→b̄
(z) = P
qq→b(z)
, one can conclude that the singlet contributions to the modified quark and antiquark
fragmentation functions are the same, ∆D
S(NLO)
q→h (zh) = ∆D
S(NLO)
q̄→h̄
(zh).
The non-singlet contributions, mainly from s-channel and s-t interferences, are,
N(NLO)
q→h (zh)∝
Db→h ⊗∆PN(i)qq̄→b(zh)fNq̄ (xT )C(i) , (93)
N(NLO)
q̄→h̄
(zh)∝
Db̄→h̄ ⊗∆P
q̄q→b̄
(zh)f
q (xT )C
(i) , (94)
where again ∆P
qq̄→b(z) = ∆P
q̄q→b̄
(z) due to crossing symmetry. We have listed all non-
vanishing nonsinglet splitting functions ∆P
qq̄→b(z) in Appendix A-2.
We again consider the limit zh → 1. In this region the convolution in the modified
fragmentation function is dominated by the large z → 1 behavior of the effective split-
ting functions. From the listed ∆P
qq̄→b(z) in Appendix A-2, we can obtain the leading
contributions,
C(i)∆P
qq̄→q(z)≈−4CF
C(xL)
C(i)∆P
qq̄→g(z)≈ 2
2CF + CF (1− e−x
A) + CAe
] C(xL)
, (95)
where we have also neglected terms proportional to 1/Nc. All ∆P
qq̄→q̄(z) are non-leading
in the limit z → 1 and therefore can be neglected. With these leading contributions,
the non-singlet modification to the quark and antiquark fragmentation functions can be
estimated as
N(NLO)
q→h (zh)∝
C(xL)
(1− z)+
CF (1− e−x
A) + CAe
+ δ(1− z)∆1(ℓT )
−Dq→h
C(xL)
(1− z)+
+ δ(1− z)∆2(ℓT )
fNq̄ (xT ) , (96)
N(NLO)
q̄→h̄
(zh)∝
Dg→h̄
C(xL)
(1− z)+
CF (1− e−x
A) + CAe
+ δ(1− z)∆1(ℓT )
Dg→h̄
−Dq̄→h̄
C(xL)
(1− z)+
+ δ(1− z)∆2(ℓT )
fNq (xT ) , (97)
where ∆1(ℓT ) and ∆2(ℓT ) are from virtual corrections,
∆1(ℓT )=
CFC(xL)|z=1 − [CF (1− e−x
+ CAe
A ]C(xL)
, (98)
∆2(ℓT )=
2CF [C(xL)|z=1 − C(xL)] . (99)
Because of momentum conservation, C(xL) = 0 when xL → ∞ for z = 1. Therefore,
the above virtual corrections are always negative. At large zh, these virtual corrections
dominate over the real ones.
There are two kinds of non-singlet contributions in the expressions given above. One that
is proportional to gluon fragmentation functions is due to quark-antiquark annihilation
into gluons which then fragment. The fragmenting gluon not only carries less energy than
the initial quark but 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 mixes quark and gluon fragmentation functions, similarly as the
lowest order quark-antiquark annihilation processes [see Eqs. (80) and (81)]. Since a gluon
fragmentation function is softer than a quark one, the real corrections from this type of
processes are positive for small zh and negative for large zh. The virtual corrections
have just the opposite behavior. Therefore, the second type of contributions will reduce
the total net modification. For intermediate values of zh where 2Dg→h(zh) > Dq→h(zh),
the net effect is still the suppression of the effective fragmentation functions for leading
hadrons.
Since fNq (xT ) > f
q̄ (xT ), we can conclude that the LO and NLO combined non-singlet
suppression for antiquark fragmentation into valence hadrons is larger than that for quark
fragmentation into valence hadrons. This qualitatively explains the flavor dependence of
nuclear suppression of leading hadrons in DIS off heavy nuclear targets as measured by
the HERMES experiment [25,26]. The ratio of differential semi-inclusive cross sections
for nucleus and deuteron targets were used to study the nuclear suppression of the frag-
mentation functions. It was observed that suppression of leading anti-proton is stronger
than for leading proton and K− suppression is stronger than K+. In the valence quark
fragmentation picture, the leading proton (K+) is produced mainly from u, d (u) quark
fragmentation while anti-protons come primarily from ū, d̄ (ū) fragmentation. Therefore,
HERMES data are consistent with stronger suppression of antiquark fragmentation.
Since gluon bremsstrahlung and the singlet qqi(q̄i) scattering also suppress quark and
antiquark 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 the nuclear modification of the quark fragmentation functions. Further-
more, the NLO contributions are proportional to αs ln(Q
2)/2π. They are as important
as the lowest order correction for large values of Q2. In principle, one should resum these
higher order corrections via solving a set of coupled DGLAP evolution equations, in-
cluding medium modification for gluon fragmentation functions. The contributions from
quark-quark (antiquark) scattering derived in this paper will be an important part of the
complete dscription. Detailed numerical study of the effect of quark-quark (antiquark)
scattering will be possible only after the completion of this complete description in the
future.
7 Summary
Utilizing the generalized factorization framework for twist-four processes we have stud-
ied the nuclear modification of quark and antiquark fragmentation functions (FF) due
to quark-quark (antiquark) double scattering in dense nuclear matter up to order O(α2s).
We calculated and analyzed the complete set of all possible cut diagrams. The results
can be categorized into contributions from double-hard, hard-soft processes and their
interferences. The double-hard rescatterings correspond to elastic scattering of the lead-
ing quark with another medium quark. It requires the second quark to carry a finite
fractional momentum xL. Therefore, the energy loss of the leading quark through such
processes can be identified as elastic energy loss at order O(α2s). The quark energy loss
and modification of quark fragmentation functions are dominated by the t-channel of
quark-quark (antiquark) scattering and are shown to be similar to that caused by quark-
gluon scattering. The contribution from quark-quark scattering is smaller than that from
quark-gluon scattering by a factor of CF/CA times the ratio of quark and gluon distribu-
tion functions in the medium. We have shown that such contributions are not negligible
for realistic kinematics and finite medium size. The soft-hard rescatterings mix gluon and
quark scattering, in the same way as the lowest order qq̄ → g processes. Such processes
modifies the final hadron spectra or effective fragmentation functions but do not con-
tribute to energy loss of the leading quark. For qq̄ → qq̄, gg processes, there also exist
pure interference contributions mainly coming from single-triple-scattering interference.
With a simple model of a factorized two-quark correlation functions, we further investi-
gated the 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 modification for an antiquark fragmentation into a valence hadron is larger
than that of a quark. This offers an 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 supported by NSFC under project No. 10405011, by MOE of China under project
IRT0624, by Alexander 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 U.S. Department of Energy under Contract No. DE-AC02-05CH11231,
and by the US NSF under Grant No. PHY-0457265, the Welch Foundation under Grant
No. A-1358.
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(α2s). There are
a total of 12 diagrams as illustrated in Figs. 5-16. For the purpose of abbreviation, we
will suppress the variables in the notations of partonic hard parts
D ≡ 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 qq̄ → gg annihilation diagrams with different possible cuts. The
contributions of Fig. 5 are:
5,C =
α2sxB
I5,CDg→h(zh/z)
1 + z2
z(1 − z)
1 + (1− z)2
z(1 − z)
, (A-3)
Fig. 5. The t-channel of qq̄ → gg annihilation diagram with three possible cuts, central(C),
left(L) and right(R).
Fig. 6. The interference between t and u-channel of qq̄ → gg annihilation.
I5,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (A-4)
5,L(R) =
α2sxB
I5,L(R)
Dq→h(zh/z)2
1 + z2
z(1 − z)
+Dg→h(zh/z)2
1 + (1− z)2
z(1 − z)
, (A-5)
I5,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−(1− e−ixLp+(y−−y
)) , (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:
Fig. 7. The s-channel of qq̄ → gg annihilation diagram with only a central-cut.
6,C =
α2sxB
I6,C 2Dg→h(zh/z)
(1− z)z
CF (CF − CA/2)
, (A-8)
6,L(R) =
α2sxB
I6,L(R) [Dg→h(zh/z) +Dq→h(zh/z)]
(1− z)z
CF (CF − CA/2)
, (A-9)
I6,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
× (1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (A-10)
I6,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−(1− e−ixLp+(y−−y
)) . (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 and
u-channel of the qq̄ → gg annihilation processes in Fig. 5. Since the splitting function is
symmetric in z and 1 − z, a factor of 2 comes from the fragmentation of both gluons in
the central-cut diagram.
The s-channel of qq̄ → gg is shown in Fig. 7 which has only one central-cut. Its contri-
bution to the partonic hard part is,
7,C =
α2sxB
I7,C 2Dg→h(zh/z)
2(z2 − z + 1)2
z(1 − z)
, (A-13)
I7,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
+y−e−ixLp
+(y−−y−
)e−ixLp
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
symmetric in z and 1− z. Therefore, fragmentation of the two final gluons gives rise to
the factor of 2 in front of the gluon fragmentation function.
Fig. 8. The interference between t and s-channel of qq̄ → gg annihilation.
Fig. 9. The complex conjugate of Fig. 8.
The interferences between t and s-channel of qq̄ → gg processes are shown in Figs. 8 and
9. There are only two possible cuts in these diagrams. The contributions from Fig. 8 are:
8,C =
α2sxB
I8,C Dg→h(zh/z)
1 + z3
z(1 − z)
1 + (1− z)3
z(1 − z)
, (A-15)
α2sxB
Dq→h(zh/z)2
1 + z3
z(1− z)
+ Dg→h(zh/z)2
1 + (1− z)3
z(1 − z)
, (A-16)
I8,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+y
2 )e−ixLp
+(y−−y−
) , (A-17)
I8,L = θ(y
1 − y−2 )θ(y− − y−1 )ei(x+xL)p
×(e−ixLp+(y−−y
) − e−ixLp+(y−−y
)) . (A-18)
Contributions from Fig. 9, which are just the complex conjugate of Fig. 8, are:
9,C =
α2sxB
I9,C Dg→h(zh/z)
1 + z3
z(1 − z)
1 + (1− z)3
z(1 − z)
, (A-19)
9,R =
α2sxB
Dq→h(zh/z)2
1 + z3
z(1 − z)
+ Dg→h(zh/z)2
1 + (1− z)3
z(1 − z)
, (A-20)
I9,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+(y−−y
))e−ixLp
2 , (A-21)
I9,R = θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
×(1− e−ixLp+(y
))e−ixLp
1 . (A-22)
One can collect all contributions of the double hard qq̄ → gg processes from the central-
cut diagrams, which should have the common phase factor
ĪC = θ(−y−2 )θ(y− − y−1 )eixp
+y−e−ixLp
) , (A-23)
and obtain the total effective splitting function in the hard partonic part,
Pqq̄→gg(z) =
z(1 − z)
{C2F [1 + z2 + 1 + (1− z)2]− 2CF (CF − CA/2)
+2CFCA(1− z + z2)2 − CFCA[1 + z3 + 1 + (1− z)3]}
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 matrix elements squared for qq̄ → gg annihilation.
We now consider the annihilation processes qq̄ → qiq̄i with qi 6= q. There is only the
s-channel process with one central-cut diagram as shown in Fig. 10. Its contribution to
the hard part is
10,C =
α2sxB
I10,C
qi 6=q
[Dqi→h(zh/z) +Dq̄i→h(zh/z)]
Fig. 10. s-channel qq̄ → qiq̄i annihilation.
Fig. 11. t-channel qqi(q̄i) → qqi(q̄i) scattering.
×[z2 + (1− z)2]
, (A-25)
I10,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
+y−e−ixLp
+(y−−y
)e−ixLp
2 . (A-26)
Here we define the effective splitting function for qq̄ → qiq̄i annihilation as,
Pqq̄→qiq̄i(z) =
[z2 + (1− z)2] . (A-27)
Similarly, for qq̄i → qq̄i scattering with qi 6= q, there is only the t-channel as shown in
Fig. 11. There are, however, three cut diagrams. Their contributions to the partonic hard
part are:
11,C =
α2sxB
I11,C
Dq→h(zh/z)
1 + z2
(1− z)2
+Dq̄i→h(zh/z)
1 + (1− z)2
, (A-28)
11,L(R) =
α2sxB
I11,L(R)
Dq→h(zh/z)
1 + z2
(1− z)2
+Dg→h(zh/z)
1 + (1− z)2
, (A-29)
I11,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
× (1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (A-30)
I11,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−(1− e−ixLp+(y−−y
)) , (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-
antiquark scattering is the quark-antiquark correlator,
TAqq̄i(x, xL)∝ e
ixp+y−−ixLp
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (A-33)
and one should sum over all possible qi 6= q flavors. Note that in the above matrix element,
the momentum flow for the antiquark (q̄i) is opposite to that of the quark (q) fields.
For quark-quark scattering, qqi → qqi, the hard part is essentially the same. The only
difference is the associated matrix element for the quark-quark correlator which is ob-
tained from that of the quark-antiquark correlator via the exchange ψqi(y2) → ψ̄qi(y2)
and ψ̄qi(y1) → ψqi(y1),
TAqqi(x, xL)∝ e
ixp+y−+ixLp
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(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 fragmen-
tation of the quark in the central-cut diagram,
Pqqi(q̄i)→qqi(q̄i)(z) =
1 + z2
(1− z)2
. (A-35)
For annihilation qq̄ → qq̄ into identical quark and antiquark pairs, in addition to the
s-channel (Fig. 10 for qi = q) and t-channel (Fig. 11 for qi = q̄), one has also to consider
the interference between s and t-channel amplitudes as shown in Figs. 12 and 13, each
having two cuts. Their contributions to the hard partonic parts are, respectively:
Fig. 12. Interference between s and t-channel of qq̄ → qq̄ scattering
Fig. 13. The complex conjugate of Fig. 12.
12,C =
α2sxB
I12,C
Dq→h(zh/z)
(1− z)
+Dq̄→h(zh/z)
2(1− z)2
CF (CF − CA/2)
, (A-36)
12,L =
α2sxB
I12,L
Dq→h(zh/z)
(1− z)
+Dg→h(zh/z)
2(1− z)2
CF (CF − CA/2)
, (A-37)
I12,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+y
2 )e−ixLp
+(y−−y−
) , (A-38)
I12,L = θ(y
1 − y−2 )θ(y− − y−1 )ei(x+xL)p
×(e−ixLp+(y−−y
) − e−ixLp+(y−−y
)) ; (A-39)
Fig. 14. The interference between t and u-channel of identical quark-quark scattering qq → qq.
13,C =
α2sxB
I13,C
Dq→h(zh/z)
(1− z)
+Dq̄→h(zh/z)
2(1− z)2
CF (CF − CA/2)
, (A-40)
13,R =
α2sxB
I13,R
Dq→h(zh/z)
(1− z)
+Dg→h(zh/z)
2(1− z)2
CF (CF − CA/2)
, (A-41)
I13,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+(y−−y
))e−ixLp
2 , (A-42)
I13,R = θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
×(e−ixLp+y
1 − e−ixLp+y
2 ) . (A-43)
One can again collect contributions from the central-cut diagrams of the double scattering
processes in Figs. 10, 11 12 and 13 and obtain the total effective splitting function for
qq̄ → qq̄,
Pqq̄→qq̄(z) =
[z2 + (1− z)2] +
1 + z2
(1− z)2
CF (CF − CA/2)
z2 + (1− z)2 +
1 + z2
(1− z)2
. (A-44)
Here we have used CF − CA/2 = −1/2Nc. For antiquark fragmentation, Pqq̄→q̄q(z) =
Pqq̄→qq̄(1 − z). One can also obtain the above result from qq̄ → qq̄ scattering matrix
squared as shown in Appendix A-3.
Similarly, for scattering of identical quarks qq → qq, one should set qi = q in Fig. 11[in
Eq. (A-28)]. In addition, one should also also include interference between t and u-channel
of the scattering as shown in Fig. 14. The contributions from such interference diagram
14,C =
α2sxB
I14,C
× 2Dq→h(zh/z)
z(1 − z)
CF (CF − CA/2)
, (A-45)
14,L(R) =
α2sxB
I14,L(R)
× [Dq→h(zh/z) +Dg→h(zh/z)]
z(1− z)
CF (CF − CA/2)
, (A-46)
I14,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
× (1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (A-47)
I14,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−(1− e−ixLp+(y−−y
)) , (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) since the splitting function is symmetric in z and 1 − z. The twist-four two-quark
correlation matrix element associated with qq → qq scattering is TAqq(x, xL) as compared
to TAqq̄(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,
Pqq→qq(z) =
1 + z2
(1− z)2
1 + (1− z)2
CF (CF − CA/2)
z(1 − z)
1 + z2
(1− z)2
1 + (1− z)2
z(1 − z)
. (A-50)
There are two remaining cut diagrams that contribute to the quark-antiquark annihilation
at the order of O(α2s) as shown in Figs. 15 and 16. Their contributions are:
15,L =
α2sxB
I15,L
Dq→h(zh/z)2
1 + z2
+Dg→h(zh/z)2
1 + (1− z)2
, (A-51)
I15,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−e−ixLp
+(y−−y
) , (A-52)
Fig. 15. Interference between final-state gluon radiation from single and triple-quark scattering.
Fig. 16. The complex conjugate of Fig. 15.
16,R =
α2sxB
I16,R
Dq→h(zh/z)2
1 + z2
+Dg→h(zh/z)2
1 + (1− z)2
, (A-53)
I16,R =−θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
+y−e−ixLp
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).
qqi(q̄i)→qi(q̄i)
(z) =
1 + (1− z)2
qqi(q̄i)→q
(z) =
1 + z2
(1− z)2
qqi(q̄i)→qi(q̄i)
(z) =
1 + (1− z)2
qqi(q̄i)→g
(z) = −1 + (1− z)
(A-55)
qq̄→qi(z) =P
qq̄→q̄i(z) = z
2 + (1− z)2,
qq̄→qi(z) =P
qq̄→q̄i(z) = z
2 + (1− z)2, (A-56)
P (HI)qq→q(z) =
1 + (1− z)2
1 + z2
(1− z)2
z(1 − z)
P (SI)qq→g(z) =−P (SI)qq→q(z) , (A-57)
P (SI)qq→q(z) =
1 + (1− z)2
z(1 − z)
qq̄→q(z) = z
2 + (1− z)2 + 1 + z
(1− z)2
qq̄→q̄(z) =P
qq̄→q(1− z) ,
qq̄→g(z) = 2CF
z2 + (1− z)2
z(1 − z)
− 2CA[z2 + (1− z)2], (A-58)
qq̄→q(z) =−
z(1 − z)
+ 2CF
qq̄→q̄(z) =
1 + (1− z)2
qq̄→g(z) =
z(1 − z)
+ 2CF
− 1 + (1− z)
(A-59)
qq̄→q(z) = z
2 + (1− z)2 −
z(1 − z)
− 2CF
qq̄→q̄(z) = z
2 + (1− z)2 ,
qq̄→g(z) =CA
4(1− z + z2)2 − 1
z(1 − z)
− 2CF
(1− z)2
, (A-60)
qq̄→q(z) =
z(1 − z)
− 2CF
qq̄→g(z) =
z(1 − z)
− 2CF
. (A-61)
The non-singlet splitting functions for qq̄ → b, defined as
qq̄→b(z) ≡ P
qq̄→b(z)− P
qq→b(z), (A-62)
are listed as below:
N(HI)
qq̄→qi(q̄i)
(z) =P
qq̄→qi(q̄i)
(z), ∆P
qq̄→qi(q̄i)
(z) = P
qq̄→qi(q̄i)
(z), (A-63)
N(HI)
qq̄→q (z) =−
(1− z2)(1 + z2 + (1− z)2)
1 + z3
z(1− z)
N(HI)
qq̄→q̄ (z) =P
qq̄→q̄(z), ∆P
N(HI)
qq̄→g (z) = P
qq̄→g(z), (A-64)
N(SI)
qq̄→q (z) =−
1 + z2
1 + (1− z)2
N(SI)
qq̄→q̄ (z) =P
qq̄→q̄(z)
N(SI)
qq̄→g (z) = 2CF
1 + z2
z(1 − z)
1 + (1− z)2
(A-65)
qq̄→b(z) =P
qq̄→b(z), ∆P
N(I2)
qq̄→b (z) = P
qq̄→b(z) (b = q, q̄, 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
Appendix A-1, we provide an alternative calculation of all the central-cut diagrams,
which correspond to quark-quark (antiquark) scattering.
Considering a parton (a) with momentum q scattering with another parton (b) that
carries a fractional momentum xp, a(q) + b(xp) → c(ℓ) + d(p′), the cross section can be
written as
dσab =
|M |2ab→cd(t̂/ŝ, û/ŝ)
(2π)32ℓ0
2πδ[(p+ q − ℓ)2]
(4π)2
|M |2ab→cd(t̂/ŝ, û/ŝ)
z(1− z)
dℓ2T δ
, (A-67)
where q = [0, q−, 0] and p = [xp+, 0, 0] are momenta of the initial partons and
, zq−, ~ℓT
(A-68)
is the momentum of one of the final partons. With the given kinematics, the on-shell
condition in the cross section can be recast as
(xp + q − ℓ)2 = 2(1− z)xp+q−
1− xL
, xL =
2z(1− z)p+q−
. (A-69)
The Mandelstam variables of the collision are,
ŝ=(q + xp)2 = 2xp+q− =
z(1 − z)
, û = (ℓ− xp)2 = −zŝ,
t̂=(ℓ− q)2 = −(1 − z)
ŝ = −(1− z)ŝ, (A-70)
where we have used the on-shell condition x = xL.
With Eq. (A-67) and parton distribution functions fNb (x), one can obtain the parton-
nucleon cross section,
dσaN =
dσabf
b (x)dx
fNb (xL)xL|M |2ab→cd(t̂/ŝ, û/ŝ)
z(1− z)
fNb (xL)
C0Pab→cd(z)dz
, (A-71)
where s = 2p+q− is the center-of-mass energy for aN collision, C0 is some common color
factor in the scattering matrix elements and
Pab→cd(z) = (1/C0)|M |2ab→cd(t̂/ŝ, û/ŝ) (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
matrix elements for elementary parton-parton scattering [39]. We will list them in the fol-
lowing. A common color factor for all quark-quark(antiquark) scattering is C0 = CF/Nc.
qq̄ → qiq̄i annihilation:
|M |2qq̄→qiq̄i =
t̂2 + û2
Pqq̄→qiq̄i(z) = z
2 + (1− z)2 . (A-73)
qq̄ → qq̄ annihilation:
|M |2qq̄→qq̄ =
û2 + ŝ2
û2 + t̂2
Pqq̄→qq̄(z) =
1 + z2
(1− z)2
+ z2 + (1− z)2 +
. (A-74)
qq̄ → gg annihilation:
|M |2qq̄→gg =
− 2CA
û2 + t̂2
Pqq̄→gg(z) = 2CF
z2 + (1− z)2
z(1− z)
− 2CA(z2 + (1− z)2) . (A-75)
qqi(q̄i) → qqi(q̄i) scattering:
|M |2qqi(q̄i)→qqi(q̄i)=
û2 + ŝ2
Pqqi(q̄i)→qqi(q̄i)(z) =
1 + z2
(1− z)2
. (A-76)
qq → qq scattering:
|M |2qq→qq =
û2 + ŝ2
ŝ2 + t̂2
Pqq→qq(z) =
1 + z2
(1− z)2
1 + (1− z)2
z(1− z)
. (A-77)
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)πα
sz(1 − z)|M |2qg→qg(t̂/ŝ, û/ŝ)dz
≡xLGN(xL)πα2s
Pqg→qg(z)dz
. (A-78)
We have then for qg → qg scattering,
|M |2qg→qg =
ŝ2 + û2
û2 + ŝ2
Pqg→qg(z) = z(1 − z)
1 + z2
(1− z)2
1 + z2
. (A-79)
Comparing this result with that in Ref. [18] for the quark-gluon rescattering, we can see
that they agree in the limit 1 − z → 0. This is a consequence of the collinear approxi-
mation employed in Ref. [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 use
Eq. (A-71) to define the splitting function for gq → gq scattering,
|M |2gq→gq =
ŝ2 + t̂2
t̂2 + ŝ2
Pgq→gq(z) = z(1 − z)
1 + (1− z)2
1 + (1− z)2
(1− z)
. (A-80)
Here for gluon-parton scattering, there is no common color factor.
gg → qq̄ annihilation,
|M |2gg→qq̄ =
t̂2 + û2
t̂2 + û2
Pgg→qq̄(z) = z(1 − z)
z2 + (1− z)2
z(1− z)
[z2 + (1− z)2]
. (A-81)
gg → gg scattering
|M |2gg→gg =2
3− t̂û
− ûŝ
− t̂ŝ
Pgg→gg(z) = 2
(1− z + z2)3
z(1− z)
. (A-82)
One can use this technique to extend the study of modified fragmentation functions to
propagating gluons. Since the modification is dominated by quark-gluon and gluon-gluon
scattering, comparing the effective splitting functions,
Pqg→qg(z)≈
, (A-83)
Pgg→gg(z)≈
, (A-84)
in the limit z → 1, one can conclude that a gluon’s radiative energy loss is larger than
a quark by a factor of Nc/CF = CA/CF = 9/4. We will leave the complete derivation of
medium modification of gluon fragmentations to a future publication.
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	Introduction
	General formalism
	Quark-quark double scattering processes
	Modified Fragmentation Functions
	 q"7016q g annihilation
	q"7016q qi"7016qi annihilation
	qqi("7016qi) qqi("7016qi) scattering
	qqqq scattering
	q"7016q q"7016q, gg annihilation
	Modification due to quark-gluon mixing
	Flavor dependence of the medium modified fragmentation
	Summary
	Hard partonic parts for quark-quark double scattering
	Effective splitting functions
	Alternative calculations of central-cut diagrams
	References
