We argue that high energy proton-nucleus (p+A) collisions provide an
excellent laboratory for studying nuclear size enhanced parton multiple
scattering where power corrections to the leading twist perturbative QCD
factorization approach can be systematically computed. We identify and resum
these corrections and calculate the centrality and rapidity dependent nuclear
suppression of single and double inclusive hadron production at moderate
transverse momenta. We demonstrate that both spectra and dihadron correlations
in p+A reactions are sensitive measures of such dynamical nuclear attenuation
effectsComment: 5 pages, 3 figures. Updated figures and reference

Adams

Albacete

Angelis

Arsene

Basu

Binnewies

Brandt

Collins

Di'Lieto

Dokshitzer

Doria

Filimonov

Frankfurt

Gluck

Gyulassy

Ivan Vitev

Jian-Wei Qiu

McLerran

Owens

Ritter

Vitev

Wang

Zhang

Physics Letters B

English

arXiv

AbstractWe argue that high energy proton–nucleus (p + A) collisions provide an excellent laboratory for studying nuclear size enhanced parton multiple scattering where power corrections to the leading twist perturbative QCD factorization approach can be systematically computed. We identify and resum these corrections and calculate the centrality- and rapidity-dependent nuclear suppression of single and double inclusive hadron production at moderate transverse momenta. We demonstrate that both spectra and dihadron correlations in p + A reactions are sensitive measures of such dynamical nuclear attenuation effects

Qiu, Jian-Wei

Vitev, Ivan

Elsevier - Publisher Connector 

Coherent QCD multiple scattering in proton–nucleus collisions 

Physics Letters B 632 (2006) 507–511
www.elsevier.com/locate/physletb
Coherent QCD multiple scattering in proton–nucleus collisions
Jian-Wei Qiu a, Ivan Vitev b,∗
a Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
b Los Alamos National Laboratory, Theory Division and Physics Division, Los Alamos, NM 87545, USA
Received 16 October 2005; accepted 26 October 2005
Available online 8 November 2005
Editor: H. Georgi
Abstract
We argue that high energy proton–nucleus (p + A) collisions provide an excellent laboratory for studying nuclear size enhanced parton multiple
scattering where power corrections to the leading twist perturbative QCD factorization approach can be systematically computed. We identify and
resum these corrections and calculate the centrality- and rapidity-dependent nuclear suppression of single and double inclusive hadron production
at moderate transverse momenta. We demonstrate that both spectra and dihadron correlations in p + A reactions are sensitive measures of such
dynamical nuclear attenuation effects.
© 2005 Elsevier B.V.
PACS: 12.38.Cy; 12.39.St; 24.85.+p
Open access under CC BY license.Copious experimental data [1] from central Au + Au reac-
tions at the Relativistic Heavy Ion Collider (RHIC) has gener-
ated tremendous excitement by pointing to the possible creation
of a deconfined state of QCD with energy density as high as 100
times normal nuclear matter density [2]. In order to diagnose
its properties [3], we first need to quantitatively understand the
multiple scattering between a partonic probe and the partons of
the medium in simpler strongly interacting systems, for exam-
ple p + A. The importance of such theoretical investigations has
been recently stressed by the measured intriguing deviation [4]
of the moderate-pT single inclusive hadron spectra from the in-
dependent multiple nucleon–nucleon scattering limit [5] in d +
Au reactions at RHIC.
In this Letter we present a systematic calculation of the co-
herent multiple parton scattering with several nucleons in p +
A collisions, which has so far been considered a challenge for
the perturbative QCD factorization approach [6]. Coherent mul-
tiple scattering and the well-studied elastic multiple scattering
coexist in nuclear collisions and their relative role depends on
the probes (or observables) and the underlying dynamics. Elas-
* Corresponding author.
E-mail addresses: jwq@iastate.edu (J.-W. Qiu), ivitev@lanl.gov (I. Vitev).0370-2693 © 2005 Elsevier B.V.
doi:10.1016/j.physletb.2005.10.073
Open access under CC BY license.tic scattering has played an important role in understanding the
kT -broadening and the Cronin effect [5]. It dominates when
the probe is so localized that the quantum correlation between
the different scattering centers can be neglected. In terms of
the factorization approach, contributions from coherent multi-
ple scattering to a physical cross section are power suppressed
in comparison to the leading hard partonic processes [7]. How-
ever, as we demonstrate below, when enhanced by the large
nuclear size these may become important at small and moderate
transverse momenta at RHIC.
Hard scattering in nuclear collisions requires one large mo-
mentum transfer Q ∼ xP  ΛQCD with parton momentum
fraction x and beam momentum P . A simple example, shown
in Fig. 1(a), is the lepton–nucleus deeply inelastic scattering
(DIS). The effective longitudinal interaction length probed by
the virtual photon of momentum qμ is characterized by 1/xP .
If the momentum fraction of an active initial-state parton x 
xc = 1/2m rN 0 ∼ 0.1 with nucleon mass mN and radius r0, it
could cover several Lorentz contracted nucleons of longitudinal
size ∼ 2r (m /P )0 N in a large nucleus [8]. In the photon–nucleus
frame, Fig. 1(a), the scattered parton of momentum  will inter-
act coherently with the partons from different nucleons at the
same impact parameter [9]. The on-shell condition for  fixes
the initial-state parton’s momentum fraction to the Bjorken
508 J.-W. Qiu, I. Vitev / Physics Letters B 632 (2006) 507–511Fig. 1. Coherent multiple scattering of the struck parton in deeply inelastic scat-
tering (a) and in proton–nucleus collisions (b). A set of resummed two gluon
exchange diagrams (c) in p + A reactions.
variable x = xB = Q2/(2P · q) but additional final state scat-
tering forces the rescaling x = xB(1 + ξ2(A1/3 − 1)/Q2) [9].
Here, A is the nuclear atomic weight and the scale of power
corrections
(1)ξ2 ≈ 3παs(Q)
8r20
〈p|Fˆ 2|p〉,
with the matrix element 〈p|Fˆ 2|p〉 = 12 limx→0 xG(x,Q2). In
Ref. [9] we demonstrated that the hard parton interactions for
any number of coherent multiple scattering in DIS are infrared
safe, which explicitly verifies the factorization theorem. For
ξ2 = 0.09–0.12 GeV2 the calculated reduction in the DIS struc-
ture functions, known as nuclear shadowing, is consistent with
the xB -, Q2- and A-dependence of the data and the small Q2
modification of the QCD sum rules [9]. Coherence should play
a similar role in virtual quark and gluon exchanges in hadronic
collisions, as shown in Fig. 1(b).
Before we resum the power corrections in p + A reactions
and derive the modification to the single and double inclusive
hadron production cross sections we need to address the va-
lidity of factorization in a hadronic environment. The colored
soft gluon interactions between the incoming and/or outgoing
hadrons can potentially ruin the factorization [10,11]. Neverthe-
less, factorization in hadronic collisions was proved to be valid
at the leading power [6] as well as the leading power correc-
tions O(1/Q2) [12]. It has been argued [13] that factorization
can be extended to the calculation of (A1/3/Q2)N -type nuclear
size enhanced power corrections in p + A collisions at all pow-
ers of N .
In DIS, as shown in Fig. 1(a), only diagrams with multiple
final-state interactions to the scattered parton lead to medium
size enhanced power corrections [9]. On the other hand, in p +
A collisions, shown in Fig. 1(b), all diagrams with either final-
state and/or initial-state multiple interactions could in principle
contribute. However, the coherent multiple scattering is a result
of an extended probe size, which is determined by the momen-
tum exchange of the hard collisions. As shown in Fig. 1(b),
once we fix the momentum fractions xa and z1, the effectiveinteraction region is determined by the momentum exchange
qμ = (xaPa − Pc/z1)μ. In the head-on frame of q − Pb , the
scattered parton of momentum  interacts coherently with par-
tons from different nucleons at the same impact parameter. In-
teractions that have taken place between the partons from the
nucleus and the incoming parton of momentum xaPa and/or the
outgoing parton of momentum Pc/z1 at a different impact pa-
rameter are much less coherent and actually dominated by the
independent elastic scattering [14].
We pursue the analogy with the DIS results of Ref. [9] and
express the lowest order single and double inclusive hadron pro-
duction cross sections as follows:
dσ
h1
NN
dy1 d2pT1
=
∑
abcd
∫
dz1
z21
Dh1/c(z1)
×
∫
dxa
φa/N(xa)
xa
[
1
xaS + U/z1
]
α2s
S
(2)×
∫
dxb δ(xb − x¯b)Fab→cd(xb),
dσ
h1h2
NN
dy1 dy2 d2pT1 d
2pT2
= δ(
ϕ − π)
pT1pT2
∑
abcd
∫
dz1
z1
Dh1/c(z1)Dh2/d(z2)
φa/N (x¯a)
x¯a
α2s
S2
(3)×
∫
dxb δ(xb − x¯b)Fab→cd(xb),
where
∑
abcd runs over all parton flavors, φa/N(xa), φb/N(xb)
are the parton distribution functions (PDFs) [15] and Dh1/c(z1),
Dh2/d(z2) are the fragmentation functions (FFs) [16]. The de-
pendence on the small momentum fraction xb is isolated in the
function
(4)Fab→cd(xb) ≡ φb/N(xb)
xb
|M¯ab→cd |2,
where the 2 → 2 on shell squared partonic matrix elements
|M¯ab→cd |2 are given in [17]. In Eq. (2) the invariants T =
−pT1
√
S e−y1 , U = −pT1
√
S ey1 and the momentum frac-
tion from the nucleus x¯b = −T/(z1S + U/xa). In Eq. (3)

ϕ = ϕ2 − ϕ1, z2 = z1pT2/pT1 , and the solution for the mo-
mentum fractions carried by the partons from the nucleon and
the nucleus are x¯a = pT1(ey1 +ey2)/z1
√
S and x¯b = pT1(e−y1 +
e−y2)/z1
√
S, respectively. In both Eqs. (2) and (3) the possible
K-factor for the lowest order formula and all renormalization
and factorization scale dependences are suppressed. Ratios of
cross sections in p + A and p + p reactions are insensitive to
their choice.
As in [9], the leading A1/3-enhanced contribution to the
cross sections comes from the coherent scatterings with N nu-
cleons shown in Fig. 1(b) and (c). The two gluon exchange with
an individual nucleon gives rise to the characteristic scale of
high twist corrections ξ2 = 0.09–0.12 GeV2, defined in Eq. (1).
For fixed N , the final state interactions of parton d , illustrated
in Fig. 1(c), replace the integral over the small xb-dependent
J.-W. Qiu, I. Vitev / Physics Letters B 632 (2006) 507–511 509function Fab→cd(xb) in Eqs. (2) and (3) by
∫
dxb
(−1)N
N !
[
Cd
x¯bξ
2
−t
(
A1/3 − 1)
]N
(5)× d
Nδ(xb − x¯b)
dxNb
Fab→cd(xb).
Here, the hard scale t = q2 = (xaPa −Pc/z1)2 and Cd is a color
factor depending on the flavor of parton d . Cq(q¯) = 1 and Cg =
CA/CF = 9/4 for quark (antiquark) and gluon, respectively.
By resumming all contributions in Fig. 1(c), N = 0, . . . ,∞ in
Eq. (5), we derive the nuclear modified cross sections for single
and double inclusive hadron production in p + A collisions.
These have the same form as Eqs. (2) and (3) with the following
replacement:
Fab→cd(xb) ⇒ exp
[
Cd
ξ2
−t
(
A1/3 − 1)xb d
dxb
]
Fab→cd(xb)
(6)= Fab→cd
(
xb
[
1 + Cd ξ
2
−t
(
A1/3 − 1)
])
.
Similarly to the DIS case, resumming the coherent scattering
with multiple nucleons is equivalent to a shift of the momentum
fraction of the active parton from the nucleus and leads to a net
suppression of the cross sections. The t -dependence of this shift
in Eq. (6) indicates that the attenuation increases in the forward
rapidity region.
We consider d + Au reactions at RHIC at √S = 200 GeV
with the deuteron and the nucleus moving with rapidities
ymax = ±5.4, respectively. Small pT1 = 1 GeV and forward
y1 = 4 hadron production corresponds to nuclear xb  1.3 ×
10−4 where coherence and our calculated power corrections
will certainly become important. Dynamical nuclear effects in
multi-particle production can be studied through the ratio
(7)R(n)AB =
dσ
h1···hn
AB /dy1 · · ·dynd2pT1 · · ·d2pTn
〈NcollAB 〉dσh1···hnNN /dy1 · · ·dynd2pT1 · · ·d2pTn
.
Centrality dependence is implicit in Eq. (7) and the modified
cross section per average collision dσh1···hndAu /〈NcolldAu〉 can be
calculated from Eqs. (2) and (3) via the substitution Eq. (6).
The necessary scaling with impact parameter, b, of the en-
hanced power corrections can be obtained through the nuclear
thickness function, A1/3 → A1/3TA(b)/TA(bmin.bias), where
TA(b) =
∫∞
−∞ ρA(z, b) dz.
The top panels of Fig. 2 show the rapidity and transverse
momentum dependence of Rh1dAu(b) and R
h1h2
dAu (b) for minimum
bias collisions. The amplification of the suppression effect at
forward y1 comes from the build-up of coherence and the de-
crease of the Mandelstam variable (−t). At high pT1 , pT2 the at-
tenuation is found to disappears in accord with the QCD factor-
ization theorems [6,12]. BRAHMS data [4] on h− production
at forward rapidity is shown for comparison. Our result should
be contrasted with a recent numerical study of non-linear par-
ton evolution that found a factor of 2 suppression of the hadron
production in p + A collisions at pT = 102–104 GeV but negli-
gible rapidity dependence of Rh1 (b) at small pT  2 GeV [18].pAFig. 2. Top: suppression of the single and double inclusive hadron pro-
duction rates in d + Au reactions versus pT for rapidities y1 = 0,1,2,3
and 4. ξ2 = 0.12 GeV2. Data on forward rapidity h− attenuation is from
BRAHMS [4]. Bottom: impact parameter dependence of the calculated nuclear
modification for central b = 3 fm, minimum bias bmin.bias = 5.6 fm and pe-
ripheral b = 6.9 fm collisions. The trigger hadron pT1 = 1.5 GeV, y1 = 3 and
the associated hadron y2 = 0.
The bottom panels of Fig. 2 show the growth of the nuclear at-
tenuation effect with centrality.
In the double collinear approximation, Eq. (3), the leading
hadrons are produced strictly back-to-back. The acoplanarity,

ϕ = π [19], arises from soft gluon resummation [20], next-
to-leading order corrections and in the presence of nuclear
matter—transverse momentum diffusion [14]. We consider di-
hadron correlations associated with hard scattering partonic
2 → 2 processes that have the bulk many-body collision back-
ground subtracted and can be approximated by near-side and
away-side Gaussians:
C2(
ϕ) = 1Norm
dN
h1h2
dijet
d
ϕ
≈ ANear√
2πσNear
exp
{
− 
ϕ
2
2σ 2Near
}
(8)+ AFar√
2πσFar
exp
{
− (
ϕ − π)
2
2σ 2Far
}
.
In Eq. (8) the near-side width σNear of C2(
ϕ) is determined by
jet fragmentation [21]. If the strength of the away-side correla-
tion function in elementary N + N collisions is normalized to
unity, dynamical quark and gluon shadowing in p + A reactions
will be manifest in the attenuation of the area AFar = Rh1h2pA (b).
The away-side width σFar is related [14,21] to the accumulated
dijet transverse momentum squared in a plane perpendicular to
the collision axis:
(9)〈k2T 〉dijet = 2〈k2T 〉vac +
∑ μ2Leff
λ
(
1 + tanh2 yi
)
.i=a,c,d
510 J.-W. Qiu, I. Vitev / Physics Letters B 632 (2006) 507–511Fig. 3. Left: centrality dependence of C2(
ϕ) at moderate pT1 = 4 GeV,
pT2 = 2 GeV and rapidities y1 = 0,2 and y2 = 0. Central d + Au and
p + p data from STAR [22]. Right: C2(
ϕ) at small transverse momenta
pT1 = 1.5 GeV, pT2 = 1 GeV and rapidities y1 = 0,4 and y2 = 0.
In Eq. (9) yi are the interacting parton rapidities and μ2Leff/λ =
0.72 GeV2 characterizes the elastic scattering power of cold
nuclear matter in minimum bias d + Au collisions [5,14]
and varies with centrality ∼ TA(b)/TA(bmin.bias). The vacuum
broadening 〈k2T 〉vac is taken from the data [21,22].
In triggering on a large transverse momentum hadron, the
nuclear modification from enhanced power corrections and
transverse momentum diffusion is reflected in the away-side
correlation function. For large vacuum broadening, 〈k2T 〉vac =
2.5–3.5 GeV2, elastic scattering may lead to only a moderate
additional growth of σFar. The left panels of Fig. 3 show that
for pT1 = 4 GeV, pT2 = 2 GeV the dominant effect in C2(
ϕ)
is a small increase of the broadening with centrality, compati-
ble with the PHENIX [21] and STAR [22] measurements. Even
at forward rapidity, such as y1 = 2, the effect of power correc-
tions in this transverse momentum range is not very significant.
At small pT1 = 1.5 GeV, pT2 = 1 GeV, shown in the right-hand
side of Fig. 3, the apparent width of the away-side C2(
ϕ) is
larger. In going from midrapidity, y1 = y2 = 0, to forward ra-
pidity, y1 = 4, y2 = 0, we find a significant reduction by a factor
of 3–4 in the strength of dihadron correlations from the nuclear
enhanced power corrections. The suppression in the rate of in-
clusive two particle production is similar to the suppression in
the inclusives. We emphasize again that this result depends sen-
sitively on centrality and the choice of transverse momenta and
disappears at high pT .
The dynamical cross section attenuation calculated here does
not contradict, instead, complements the effects from a pos-
sible modification of the nuclear parton distribution functions
(nPDFs), known as leading twist shadowing [23]. In our for-
malism, one can include both effects by replacing the PDFs in
Eqs. (2) and (3) with the corresponding nPDFs and applyingEq. (6). The weaker scale dependence of nPDFs will thus slow
down the disappearance of the nuclear suppression at high pT
in Fig. 2. A combined global QCD analysis would, however, be
required and is beyond the scope of this Letter.
In conclusion, we presented a systematic approach to the cal-
culation of coherent QCD multiple scattering and resummed the
nuclear enhanced power correction to the single and double in-
clusive hadron production cross sections in p + A reactions.
At low pT we find a sizable suppression, which grows with
rapidity and centrality. At high pT the nuclear modification dis-
appears in accord with the QCD factorization theorems. We
demonstrated that both particle spectra and dihadron correla-
tions are sensitive measures of such dynamical attenuation of
the parton interaction rates.
Our approach, with its intuitive and transparent results, can
be easily applied to study the nuclear modification of other
physical observables in p + A reactions and its predictions can
be readily tested against RHIC data. The systematic incorpo-
ration of coherent power corrections provides a tool to address
the most interesting transition region between “hard” and “soft”
physics in hadron–nucleus collisions. It allows for the extension
of perturbative QCD calculations to relatively small transverse
momenta and for bridging the gap between the parton model
picture and the possible onset of gluon saturation [2,24] at very
large collider energies and very small values of nuclear x.
Acknowledgements
This work is supported in part by the US Department of
Energy under Grant No. DE-FG02-87ER40371 and by the
J. Robert Oppenheimer fellowship of the Los Alamos National
Laboratory.
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Coherent QCD multiple scattering in proton-nucleus collisions

Coherent QCD multiple scattering in proton-nucleus collisions

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