Table Of ContentFrom EMC- and Cronin-effects to signals of
quark-gluon plasma
Wei Zhu1,2, Jianhong Ruan1 and Fengyao Hou2
2 1Department of Physics, East China Normal University, Shanghai 200062, P.R. China
1
0 2Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, P.R. China
2
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e
h Abstract
[
The EMC- and Cronin-effects are explained by a unitarized evolution equation,
2
v where the shadowing and antishadowing corrections are dynamically produced by
4
gluon fusions. For this sake, an alternative form of the GLR-MQ-ZRS equation
2
2 is derived. The resulting integrated and unintegrated gluon distributions in proton
4 andnucleiareusedtoanalyze thecontributions oftheinitialpartondistributionsto
.
2 the nuclear suppression factor in heavy ion collisions. A simulation of the fractional
1
energy loss is extracted from the RHIC and LHC data, where the contributions of
0
the nuclear shadowing and antishadowing effects are considered. We find a rapid
1
: crossover from week energy loss to strong energy loss at a universal critical energy
v
i of gluon jet Ec 10GeV.
X ∼
r
a
PACS number(s): 24.85.+p; 12.38.-t; 13.60.Hb
keywords: Quark gluon plasma; Nuclear gluon distribution; Energy loss
1 Introduction
One of the important findings at RHIC and LHC is that high transverse momentum k
t
hadronproduction incentral heavy ioncollisions is suppressed compared to p+p collisions
[1, 2]. This suppression can be attributed to energy loss of high-k partons that traverse
t
the hot and dense medium (i.e., quark-gluon plasma QGP) formed in these collisions. An
important goal of the study of heavy ion collisions is therefore to determine the properties
QGPfromthemeasured fractionalenergy lossafterdeducting nuclear effects ontheinitial
parton distributions.
The parton densities in a bound nucleon differ from that in a free nucleon. One of
such examples is that the ratio of nuclear and deuterium’s structure functions is smaller
or larger than unity at Bjorken variable x < 0.1 or 0.1 < x < 0.3. These two facts are
called as the nuclear shadowing and antishadowing in the EMC effect [3]. The nuclear
shadowingandantishadowingeffectsoriginfromthegluonfusion(recombination)between
two different nucleons in a nucleus, which changes the distributions of gluon and quarks,
but does not change their total momentum [4]. In consequence, the lost gluon momentum
in the shadowing range should be compensated in terms of new gluons with larger x,
which forms the antishadowing effect.
An other example is the Cronin effect: the ratio of particle yields in d+A (scaled by
the number of collisions)/p + p, exceeds unity in an intermediate transverse momentum
range (Cronin enhancement) and is bellow unity at smaller transverse momentum (anti-
Cronin suppression). This effect was first seen at lower fixed target energies [5] and has
been confirmed in √s = 200GeV d + Au collisions at the BNL Relativistic Heavy Ion
Collider (RHIC) [6].
The Cronin effect is more complicated than the EMC effect, the former mixes the
shadowing-antishadowing corrections at initial state and the medium modifications at
final state. The later is an important information for understanding the properties of
dense and hot matter formed in high-energy heavy-ion collisions. Therefore, the nuclear
shadowing and antishadowing effects, which have appeared in the EMC effect, should be
extractedfromtheCronineffect, thentheremainingresultexposesthemediumproperties.
The saturation models are broadly used to study the Cronin effect. The satura-
tion is a limiting behavior of the shadowed gluon distribution in the Jalilian-Marian-
Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) equation [7], where the uninte-
grated gluon distribution is absolutely flat in k -space at k < Q , Q is the saturation
t t s s
scale. An elemental QCD process, which arises nonlinear corrections in the JIMWLK
equation is also the gluon fusion gg g. As we have pointed out that the antishadowing
→
effect always coexists with the shadowing effect in any gluon fusion processes due to a
general restriction of momentum conservation [8]. However, such antishadowing effect is
completely neglected in the original saturation models. The Cronin enhancement in these
models, (i) is additionally explained as multiple scattering [9] using the Glauber-Mueller
model [10], or the McLerran-Venugopalan model [11]; (ii) is produced by special initial
gluon distributions of proton and nucleus [12]. A following question is: what are the
nuclear antishadowing contributions to the Cronin effect?
A global Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) analysis of nuclear par-
ton distribution functions (for example, the ESP09-set [13]) was proposed, where the data
from deep inelastic scattering (DIS), Drell-Yan dilepton production, and inclusive high-k
t
hadron production data measured at RHIC are used. They found that a strong gluon
antishadowing effect is necessary to support the RHIC data. However, the shadowing
and antishadowing effects in the DGLAP analysis are phenomenologically assumed in the
initial conditions, since the DGLAP equation [14] does not contain the nonlinear cor-
rections of gluon fusion. Due to the lack of the experimental data about nuclear gluon
distribution, the above mentioned status are undetermined. For example, a similar global
DGLAP analysis shows that the available data are not enough to fix these complicated
input distributions and even appearing or not the antishadowing effect are uncertain
[15]. Besides, the DGLAP equation in the collinear factorization scheme evolves the inte-
grated parton distributions, the behavior of the unintegrated gluon distributions, which
contain information of the transverse momentum distribution is completely undefined in
this method. Therefore, the ESP09-set of nuclear parton distributions can not predict
the RHIC data at lower k , where the contributions from intrinsic transverse momentum
t
become important [13].
The modification of the gluon recombination to the standard DGLAP evolution equa-
tionwas first proposed by Gribov-Levin-Ryskin and Mueller-Qiu (the GLR-MQequation)
in [16,17], and it is naturally regarded as the QCD dynamics of the nuclear shadowing
since the same gluon fusion exists both in proton and in nucleus, and they differ only
by the strength of the nonlinear terms [18]. However, the GLR-MQ equation does not
predict the nuclear antishadowing effect since the momentum conservation is violated in
this equation. This defect of the GLR-MQ equation is corrected by a modified equation
(the GLR-MQ-ZRS equation) by Zhu, Ruan and Shen in [19,20], where the corrections
of the gluon fusion to the DGLAP equation lead to the shadowing and antishadowing
effects. Unfortunately, the integral solutions in the present GLR-like equations need the
initial distributions on a boundary line (x,Q2) at a fixed Q2, and they still contain the
0 0
unknown input with nuclear shadowing and antishadowing effects at small x.
Motivation of this work is trying to improve the above questionable methods. We shall
study the nuclear shadowing and antishadowing effects in the EMC- and Cronin-effects,
which are dynamically arisen from the gluon recombination. Then we use the resulting
nuclear gluon distributions to predict the contributions of the initial parton distributions
to the nuclear suppression factor in heavy ion collisions and to extract fractional energy
lossfromtheRHICandLHCdata. Forthissake, analternativeformoftheGLR-MQ-ZRS
equation at the double-leading-logarithmic-approximation (DLLA) is derived in Sec. 2.
The evolution of this equation is along small x-direction. The nonlinear corrections to the
input distributions may neglected if the value of thestarting point x islarge enough. The
0
shadowing and antishadowing effects are naturally grown up in the evolution at x < x .
0
Thus, we avoid the unperturbative nuclear modifications in the input distributions and
simplify the initial conditions. We use the existing data about the EMC- and Cronin-
effects to fix a few of free parameters in the solutions. Then the resulting integrated
and unintegrated gluon distributions in proton and nuclei are used to analyze the nuclear
suppression factor in heavy ion collisions.
Ourmainconclusionsare: (i)wesupportthestrongershadowing-antishadowing effects
in the heavy nucleus both in the unintegrated and integrated gluon distributions due to
a strong A-dependence of the nonlinear corrections; (ii) the anti-Cronin suppression and
Cronin enhancement mainly origin from the same gluon recombination mechanism as in
the nuclear shadowing and antishadowing effects of the EMC effect; (iii) fractional energy
loss is rapid crossover from week energy loss to strong energy loss at a universal critical
energy of gluon jet E 10GeV.
c
∼
The organizations of this work are as follows. We shall derive the GLR-MQ-ZRSequa-
tion in a new form in Sec.2. Using this equation we study the shadowing and antishadow-
ing effects in the EMC effect in Sec.3. These resulting unintegrated gluon distributions
in proton and nuclei are used to expose the nuclear shadowing and antishadowing contri-
butions to the Cronin effect in Sec.4. The nuclear shadowing and antishadowing effects
to the nuclear suppression factor are predicted and a simulations of fractional energy loss
is extracted from the RHIC and LHC data in Sec. 5.
2 A new form of the GLR-MQ-ZRS equation
Inhistory, theDGLAPevolutionisderivedbyusingtherenormalizationgroupmethod
for the integrated distributions. The resulting equation evolves with factorization scale
µ. In this section, we try to write the DGLAP equation with the nonlinear modifications
beginning from the unintegrated distributions. Then, we get an alternative form of these
equations, which evolves with the Bjorken variable x.
We begin from a deeply inelastic scattering process, where the unintegrated gluon
distribution is measured. In the k -factorization scheme, the cross section is decomposed
t
into
∗
dσ(probe P kX)
→
k2 x
= f(x ,k2 ) t , ,α dσ(probe∗k k)
1 1t ⊗K k12t x1 s!⊗ 1 →
∆f(x,k2) dσ(probe∗k k), (1)
≡ t ⊗ 1 →
which contains the evolution kernel , the unintegrated gluon distribution function f
K
and the probe∗-parton cross section dσ(probe∗k k). For simplicity, we take the fixed
1
→
QCD coupling at the leading order (LO) approximation in this work. According to the
scale-invariant parton picture of the renormalization group [21], we regard ∆f(x,k2) as
t
the increment of the distribution f(x ,k2 ) when it evolves from (x ,k2 ) to (x,k2). Thus,
1 1t 1 1t t
the connection between two functions f(x ,k2 ) and f(x,k2) via Eq. (1) is
1 1t t
f(x,k2) = f(x ,k2 )+∆f(x,k2)
t 1 1t t
k2 dk2 1 dx k2 x
= f(x ,k2 )+ t 1t 1 t , ,α f(x ,k2 ). (2)
1 1t Zk12t,min k12t Zx x1 K k12t x1 s! 1 1t
In the case of the LO DGLAP evolution, we adopt a physical (axial) gauge, which sums
over only the transverse gluon polarizations, so that the ladder-type diagrams dominate
the evolution, the unintegrated distributions satisfy the normalization relation
µ2 dk2 µ2 dk2
G(x,µ2) xg(x,µ2) = t xf(x,k2) t F(x,k2), (3)
≡ Zkt2,min kt2 t ≡ Zkt2,min kt2 t
where the possible non-logarithmictail for k > µ arebeyond NLO accuracy. These distri-
t
butions correspond to the density of partons in the proton with longitudinal momentum
fraction x, integrated over the parton transverse momentum up to k = µ.
t
From Eqs. (2) and (3), we have
µ2 dk2 k2 dk2 1 dx 1 k2 x
∆g(x,µ2) = t t 1t 1 t , ,α F(x ,k2 ), (4)
Zkt2,min kt2 Zk12t,min k12t Zx x1 x1K k12t x1 s! 1 1t
or
µ2 dk2 k2 dk2 1 dx x k2 x
∆G(x,µ2) = ∆xg(x,µ2) = t t 1t 1 t , ,α F(x ,k2 )
ZkT2,min kt2 Zk12t,min k12t Zx x1 x1K k12t x1 s! 1 1t
µ2 dk2 1 dx x x
= t 1 ,α G(x ,k2), (5)
Zkt2,min kt2 Zx x1 x1K(cid:18)x1 s(cid:19) 1 t
where the last step is held when the k -strong ordered. Usually DGLAP evolution is
t
written in terms of the virtuality k2 rather than k2 , but at LO level this is the same since
t
the difference is a NLO effect. Therefore, we take
G(x,k2) G(x,µ2). (6)
t →
Thus, in
G(x,µ2) = G(x ,µ2)+∆G(x,µ2), (7)
1 1
we write
∆G(x,µ2)
= µ2 dµ21 1 dx1 x LL(µ2) ( x ,α )G(x ,µ2), (8)
Zµ21,min µ21 Zx x1 x1KDGLAP x1 s 1 1
at the leading logarithmic µ2 approximation, in which the unregularized splitting kernels
dx1 LL(µ2) αsNcdx1 1 z z
= [z(1 z)+ − + ]
x KDGLAP π x − z 1 z
1 1
−
α N dx x(x x) x x x
s c 1 1 1
= − + − + (9a)
π x1 " x21 x x1 −x#
dx α N dx
x≪x1 1 DLL = s c 1. (9b)
−→ x KDGLAP π x
1
We add the contributions of the nonlinear recombination terms in Eq. (8) according
to Refs. [19, 20] (See appendix A),
∆G(x,µ2)
= µ2 dµ21 1 dx1 x LL(µ2) ( x ,α )G(x ,µ2)
Zµ21,min µ21 Zx x1 x1KDGLAP x1 s 1 1
2 Q2 dµ21 1/2 dx1 x GG→GG, LL(µ2) x ,α G(2)(x ,µ2)
− Zµ21min µ41 Zx x1 x1KGLR−MQ−ZRS (cid:18)x1 s(cid:19) 1 1
+ µ2 dµ21 1/2 dx1 x GG→GG, LL(µ2) x ,α G(2)(x ,µ2), (10)
Zµ21min µ41 Zx/2 x1 x1KGLR−MQ−ZRS (cid:18)x1 s(cid:19) 1 1
where
dx1 GG→GG, LL(µ2)
x KGLR−MQ−ZRS
1
α2 N2 (2x x)(72x4 48x3x+140x2x2 116x x3 +29x4)
= s c 1 − 1 − 1 1 − 1 dx (11a)
8 N2 1 x5x 1
c − 1
dx N2 dx
x≪x1 1 GG→GG, DLL = 18α2 c 1. (11b)
−→ x KGLR−MQ−ZRS sN2 1 x
1 c −
G(2)(x,µ2) = R G2(x,µ2), (12)
G
whereR = 1/(πR2)isacorrelationcoefficient withthedimension[L−2], Ristheeffective
G
correlation length of two recombination gluons. One can easily get the GLR-MQ-ZRS
equation at DLL approximation
∂G(x,µ2)
∂lnµ2
α N 1 dx 36α2 N2 1/2 dx
= s c 1G(x ,µ2) s c 1G2(x ,µ2)
π x 1 − πµ2R2N2 1 x 1
Zx 1 c − Zx 1
18α2 N2 1/2 dx
+ s c 1G2(x ,µ2). (13)
πµ2R2N2 1 x 1
c − Zx/2 1
It is interesting to compare this small-x version of the GLR-MQ-ZRS equation with
the GLR-MQ equation, which is [17]
∂G(x,µ2)
∂lnµ2
α N 1 dx 36α2 N2 1/2 dx
= s c 1G(x ,µ2) s c 1G2(x ,µ2), (14)
π x 1 − 8µ2R2N2 1 x 1
Zx 1 c − Zx 1
where
9
G(2)(x,µ2) = G2(x,µ2), (15)
8πR2
is assumed.
Comparing with the GLR-MQ equation (14), there are several features in the GLR-
MQ-ZRS equation (13): (i) the momentum conservation of partons is restored in Eq.
(13); (ii) because of the shadowing and antishadowing effects in Eq. (13) have different
kinematic regions, the net effect depends not only on the local value of the gluon distri-
bution at the observed point, but also on the shape of the gluon distribution when the
Bjorken variable goes from x to x/2. In consequence, the shadowing effect in the evolu-
tion process will be obviously weakened by the antishadowing effect if the distribution is
steeper. Therefore, the antishadowing effect can not be neglected in the pre-saturation
range.
According to the definition Eq. (3), one can roughly estimate the unintegrated gluon
distribution using
∂G(x,µ2)
F(x,k2) µ2 . (16)
t ≃ ∂µ2 (cid:12)(cid:12)µ2=kt2
(cid:12)
(cid:12)
However, Eq. (16) cannot remain true as x increases,(cid:12) since the negative virtual DGLAP
term may exceed the real emission DGLAP contribution and it would give negative values
of F. In fact, due to strong k ordering in DGLAP evolution, the transverse momentum
t
of the final parton is obtained, to leading-order accuracy, just at the final step of the evo-
lution. Thus, the k -dependent distribution can be calculated directly from the DGLAP
t
equation keeping only the contribution which corresponds to a single real emission, while
all the virtual contributions from a scale equal to k up to the final scale µ of the hard
t
subprocess are resummed into a Sudakov factor T. The factor T describes the probability
that during the evolution there are no parton emissions. However, at the small x range,
the virtual contributions to the gluon distribution in the DGLAP kernel can be neglected.
Besides, we have indicated that the contributions of the virtual processes in the nonlin-
ear kernels of the GLR-MQ-ZRS equation are canceled [19], therefore, the Sudakov form
factors are same in nucleon and nucleus [22] and they are really canceled in the following
ratio form. Thus, we use
∆G(x,µ2)
µ2 dk2 k2 dk2 1 dx x x
= t t 1t 1 DLL ( ,α )F(x ,k2 )
Zkm2in kt2 Zk12t,min k12t Zx x1 x1KDGLAP x1 s 1 1t
µ2 dk2 1/2 dx x x
2 t 1 DLL ,α G(2)(x ,k2)
− Zµ21min kt4 Zx x1 x1KMD−DGLAP (cid:18)x1 s(cid:19) 1 t
µ2 dk2 1/2 dx x x
+ t 1 DLL ,α G(2)(x ,k2), (17)
Zµ21min kt4 Zx/2 x1 x1KMD−DGLAP (cid:18)x1 s(cid:19) 1 t
and obtain
∂∆G(x,µ2)
∆F(x,k2) = µ2
t ∂µ2 (cid:12)(cid:12)µ2=kt2
(cid:12)
(cid:12)
(cid:12)
