Table Of ContentRelating the description of gluon
production in pA collisions and parton
7 energy loss in AA collisions
0
0
2 Yacine Mehtar-Tani(1)
n
a February 2, 2008
J
2
2
1. Laboratoire de Physique Th´eorique
2 Universit´e Paris Sud, Bˆat. 2101
v 91405, Orsay cedex
6
3 Abstract
2
6 Wecalculatetheclassicalgluonfieldofafastprojectilepassingthrough
0 a dense medium. We show that this allows us to calculate both the ini-
6 tialstategluonproductioninproton-nucleuscollisions andthefinalstate
0 gluon radiation off a hard parton produced in nucleus-nucleus collisions.
/ This unified description of these two phenomena makes the relation be-
h
p tween thesaturation scale Qs and thetransport coefficient qˆmore trans-
- parent. Also, we discuss the validity of the eikonal approximation for
p gluonpropagationinsidethenucleusinproton-nucleuscollisionsatRHIC
e energy.
h
:
v
i LPT-Orsay06-41
X
r
a 1 Introduction
The advent of a new generation of colliders, RHIC and LHC, has stimulated
the development of new tools for the understanding of high energy and high
density systems. The purpose of studying heavy ion collisions at high energy is
to create such dense systems, leading eventually to evidences for the existence
oftheQuark-Gluon-Plasma(QGP)predictedbytheQCDphasediagram. Very
early, it appeared that one has to distinguish between final state interactions,
occurringafterthehardpartonproduction,andinitialstateinteractionsrespon-
sibleforhardpartonproductioninnucleus-nucleuscollisions;thismotivatedthe
study of deuteron-gold collisions at RHIC where final state interaction are ab-
sent.
1Unit´emixtederechercheduCNRS(UMR8627)
1
The physics of proton-nucleus collisions at high energy turned out to be very
rich in new features compared to proton-proton collisions, for instance: high
p suppressionat forwardrapidities, Croninenhancement (also observedat low
t
energy), centrality dependence of spectra, etc. [3,4]. These results have been
confronted with the theory of the Color Glass Condensate (CGC) which de-
scribes the nuclear wave function at high energy [1,2]. Basically, it extends
small coupling QCD calculations to a region, the so-called saturation regime,
characterizedbythehardscaleQ (saturationscale),wherehighdensityeffects
s
do not allow one to apply the usual perturbative QCD.
Historically the idea of saturation of the gluon distribution at high energy has
been introduced in the early 80’s [5–7] as a necessary condition for unitarity.
Indeed at high energy gluons dominate the dynamics and the growth of the
number of gluons, driven by the BFKL evolution equation, violates unitarity.
This equation resums large log(s) effects but disregards gluon recombination
[8,9]. A first version of the CGC known as the McLerran-Venugopalan model
pointed out the interest of the semi-classicalpicture for the description of high
density systems. [10–12]. A couple of years later, a more sophisticated theory
has been built which incorporates the main BFKL features and extends them
to the non-linear regime: the saturation regime, leading to the BK-JIMWLK
equation [13–24].
Onthe other hand, severalworksinspiredby the Landau-Pomeranchuk-Migdal
(LPM) effect in QED [38–40]: the BDMPS formalism, based on Feynman di-
agrams calculations [25–31], and the Zakharov formalism (Z) based on path
integrals [32–37], have been dedicated to the study of final state interactions
in nucleus-nucleus collisions, especially to understand the large suppression of
observed large p hadron spectra compared to proton-proton collisions. The
t
basic idea is to explain this suppression by the energy loss of the initially pro-
ducedhardparton. Itloosespartofitsenergybyradiatinggluonswhenpassing
through the dense medium formed after the collision. In [41–44], Wiedemann
(W) has provided another treatment of the problem also in terms of Feynman
diagrams.
In the present work, we give a new and simple derivation of the radiated gluon
spectrum, recovering the BDMPS-Z-W result. We provide a universal formu-
lation for both gluon production in proton-nucleus collisions [45], and gluon
radiation off a produced hard parton in nucleus-nucleus collisions. The basic
ideaistocalculategluonradiationoffahighenergyprojectilepassingthrougha
densemediuminthesemi-classicalpicture. Todothat,wesolvetheYang-Mills
equationsforthe radiatedgluonfieldδAµ,whichis treatedasaperturbationof
the background medium field Aµ. This medium could be a nucleus (cold mat-
0
ter) in proton-nucleus collisions or a hot medium produced after the collision
of two heavy nuclei. This approach turns out to be a very useful framework,
which avoids many technical problems, making the picture clear. Obviously,
the obtained classical field is a function of the medium source density ρ (or
0
equivalently, the medium background field Aµ). The source distribution ρ is
0 0
a random quantity. Thus, to calculate observables one has to average over all
2
possible source configurations with a given statistical weight [ρ ]:
0
W
[ρ ] [ρ ] [ρ ]. (1)
0 0 0
hOi≡ D W O
Z
The differential averagenumber of produced gluons is given by the formula:
d N 1
ω h i = d4x(cid:3) δA (x)ǫµ (q)eix.q 2 (2)
dωd2q 16π3h | x µ (λ) | i
⊥ λ Z
X
where ǫµ is the polarization vector of the radiated gluon, and q (q+ =
(λ) ≡
ω,q− =q2/2ω,q ) its momentum.
⊥ ⊥
In section 2, we derive the gauge field induced after the interaction of a fast
projectile with an unspecified medium characterized by the statistical weight
, by solving the Yang-Mills equations. Choosing the gauge as the light cone
W
gauge of the projectile allows us to give a simple derivation, and we show, in
section3,thatitisstraightforwardtodeducefromtheclassicalfieldtheinduced
radiativegluonspectruminnucleus-nucleuscollisions,andintheapproximation
of independent scattering centers we reproduce a well-known formula derived
in the more formal approach mentioned above (see BDMPS-Z-W papers listed
above).
In section4, we consider gluonproduction in proton-nucleuscollisions. Assum-
ingtheinteractiontimetobemuchsmallerthananyothertimescaleappearing
intheproblem(thatamountstoconsidergluonpropagationinsidethenucleusto
be eikonal) we recover the k -factorization formula which resums high density
t
effects in the nucleus [45,46,48–51]. Then we discuss the validity of this ap-
proximation, and show that it fails in some kinematic regime probed at RHIC
energies. Finally, we summarize in section 5.
2 The gauge field of a fast moving parton (or
hadron) passing through a dense medium
Weconsideramasslessparton(orahadrondescribedasacollectionofpartons)
moving with the velocity of light, in the x+ direction. At some time it passes
through a dense static medium of size L 2. The medium is described by the
following current, in the medium rest frame,
Jµ =ρ (x3,x )δµ0. (3)
0(r.f) 0 ⊥
We are interested in the induced radiative spectrum off the hard parton inside
the medium. Knowingthe simplicity ofsolving the Yang-Mills equations in the
light-cone gauge of the parton A+ = 0 [45], we perform a boost of velocity
β 1 (the cross-section is Lorentz invariant) which affects only the medium,
∼−
2The assumption of a static medium (static scattering centers) in parton energy loss has
beenusedinBDMPS-W-ZformalismandintheGyulassy-Wangmodel[52].
3
namely the parton remains in the x+ direction and the medium is pushed very
close to the light-cone in the x− direction.
In the boosted frame the medium field can be checked to be 3 [46]:
1
Aµ = δ−µ ρ (x ,x+), (4)
0 − ∂2 0 ⊥
⊥
In the light-cone gauge of the fast parton, A+ = 0, the Yang-Mills equations
read
∂+(∂ Aµ) ig[Ai,∂+Ai]=J+,
µ
− −
[D−,∂+A−] [Di,Fi−]=J−,
−
∂+F−i+[D−,∂+Ai] [Dj,Fji]=0. (5)
−
We assume that when the parton traverses the medium, it induces a perturba-
tion δAµ of the strong medium field A (linear response):
0
Aµ =Aµ+δAµ . (6)
0
Similarly for the conserved current
Jµ =Jµ+δJµ . (7)
0
Keeping only terms in the Yang-Mills equations which are linear in the fluctu-
ation δAµ:
∂+(∂ δAµ)=δJ+,
µ
−
(cid:3)δAi 2ig[A−,∂+δAi]=∂i(∂ δAµ),
− 0 µ
(cid:3)δA− 2ig[A−,∂+δA−]=δJ−+2ig[∂iA−,δAi]+∂−(∂ δAµ)
− 0 0 µ
ig[A−,∂ δAµ]. (8)
− 0 µ
The parton current obeys the conservation relation
∂+δJ−+D−δJ+ =∂+δJ−+∂−δJ+ ig[A−,δJ+]=0. (9)
− 0
With the initial condition δJ+(x+ = t ) = δ(x−)ρ(x ), where t is the source
0 ⊥ 0
production time, namely the production time of the hard parton in nucleus-
nucleus collisions. In writing this current, we assume that the hard projectile
propagates in the x+ direction of the light cone; therefore, its interaction with
the medium iseikonal: itonly getsa colorprecessionwhenpassingthroughthe
medium. The solution of (9) reads
δJ+ = U(x+,t ,x )δ(x−)ρ(x )θ(x+ t ),
0 ⊥ ⊥ 0
−
δJ− = θ(x+)ρ(x )δ(x+ t ), (10)
⊥ 0
− −
3ForanygaugechoiceforthemediumfieldinitsrestframetheA+componentissuppressed
bytheboost.
4
where U is a Wilson line in the adjoint representation of the gauge group:
x+
U(x+,t ,x ) exp ig dz+ A−a(z+,x )T , (11)
0 ⊥ ≡T+ " Zt0 0 ⊥ a#
where denotes the time orderingofthe integrals alongz+. The currentδJ−
+
T
correspondsto the propagationof an antiparticle moving in the opposite direc-
tion of the hard parton we are interested in. It is necessary to get a conserved
current. Makinguseofthethefirstequationof(8)(seenasaconstraintbecause
it contains no time derivative) in the last equation for δA− we get :
1
(cid:3)δA− 2ig[A−,∂+δA−]=2ig[∂iA−,δAi]+δJ− ∂−δJ+ ig[A−,δJ+] ,
− 0 0 −∂+ − 0
(12)
(cid:0) (cid:1)
Thanks to the current conservation (9), δJ+ cancels out in right hand side of
equation (12). Finally, we simplify equations (8) which reduce to
∂i
(cid:3)δAi 2ig[A−,∂+δAi]= δJ+,
− 0 −∂+
(cid:3)δA− 2ig[A−,∂+δA−]=2ig[∂iA−,δAi]+2δJ−. (13)
− 0 0
The first equation of (8) is solved leading to 4:
δAi(x)= d4zθ(z−)G(x,z)∂i(U(z+,t ,z )ρ(z ))θ(z+ t ). (14)
0 ⊥ ⊥ 0
− −
Z
The gluon propagator G is the retarded Green’s function obeying the equation
of motion:
((cid:3) 2ig(A−.T)∂+)G(x,y)=δ(x y), (15)
x− 0 x −
with the initial condition, the free retarded propagator
1 d4p e−ip.(x−y)
G0(x,y)= θ(x+ y+)θ(x− y−)δ((x y)2)= . (16)
2π − − − (2π)4 p2+iǫp+
Z
The propagation of the emitted gluon, contrary to the hard projectile, is not
necessarily eikonal. Eq. (14) has a simple diagrammatic representation, shown
in fig. (1). The color precession of the source before the gluon emission is
accounted for by the U that multiplies ρ; while the rescatterings of the gluon
afteritisemittedarehiddenintheGreen’sfunctionG. ThecomponentδA−can
beextractedfromtheconstraint(firstequationin(8)),butitisnotrelevantfor
gluon production since only transverse polarizations are physical. The integral
overz− isrestrictedtothepositivevalues: thiscomesfromtheterm(1/∂+)δJ+
(which contains a δ(z−)) appearing in the second equation in (8) after the
substitution of δA− from the constraint.
The radiated gluon transverse field (14) is the main result of this section, it
contains,aswewillshow,thephysicsofproton-nucleuscollisionsathighenergy
and of the induced radiative gluon spectrum in nucleus-nucleus collisions.
4In light cone gauge, we can forget about the δA− since it plays no role in the gluon
spectrum.
5
x+
0 z+ L
Figure1: Aschematicrepresentationofeq. (14). Thefastprojectile(thickline)
passes through the medium of thickness L and emits a gluon at the time z+.
The gluon emissioncould also occur outside the medium, at z+ <0 or z+ >L.
3 Induced radiative spectrum in A-A collisions
We assume that the fast projectile is a hard parton produced in a nucleus-
nucleus collision at t , and take the origin of times when the parton enters the
0
medium. In coordinate space, (for x+ > L) the gluon field (14), amputated of
its final free propagator,reads
(cid:3) δAi(x) = θ(x−)∂i(U(x+,t ;x )ρ(x ))
x − x 0 ⊥ ⊥
d4zθ(z−)θ(L z+)θ(z+ t )δ(x+ L)
0
− − − −
Z
2∂+G(x,z)∂i(U(z+,t ;z )ρ(z )), (17)
× x z 0 ⊥ ⊥
where we used in the second term the following remarkable property of the
Green’s function (valid for y+ <L<x+)[46]:
G(x,y)= dz−d2z G(x,z)2∂+G(z,y). (18)
⊥ z
Z
z+=L
In equation (17), the first term corresponds to gluon emission occurring after
the hard parton left the medium and the second term corresponds to gluon
emission before the hard partonleft the medium (if the partonis produced out
side the medium, this emission could also occur before it enters the medium).
The invariance by translation, with respect to the - variables, of this Green’s
function is due to the fact that the medium field A− is independent of x−,
0
this can be seen in eq. (15). This invariance implies that G depends on the
coordinates only via x− y−.
−
It is useful to introduce a new Green’s function, defined as
(x+,x ;y+,y )=2 dl−∂+G(x+,x ;y+,y ;l− =x− y−)eil−ω. (19)
Gω ⊥ ⊥ x ⊥ ⊥ −
Z
6
Itiseasytoverifythat isaGreen’sfunctionofthetwo-dimensionalSchr¨odinger
ω
G
operator:
∂2
(i∂−+ ⊥ +ig(A−.T)) (x+,x ;y+,y )=iδ(x+ y+)δ(x y ), (20)
2ω 0 Gω ⊥ ⊥ − ⊥− ⊥
therefore, it can be written in terms of a path integral for a quantum particle
of mass ω moving in a potential:
iω x+
(x+,x ;y+,y ) = r (ξ)exp dξr˙2(ξ)
Gω ⊥ ⊥ Z D ⊥ " 2 Zy+ ⊥ #
U(x+,y+;r ), (21)
⊥
×
wherer (x+)=x andr (y+)=y . ThispathintegraldescribestheBrown-
⊥ ⊥ ⊥ ⊥
ianmotionofthe emitted gluoninthe transverseplane. Therefore,the emitted
gluon follows a non-eikonal trajectory inside the medium 5. We recover the
eikonal case by taking ω or equivalently by assuming x+ y+ 0:
→∞ − →
(x+,x ;y+,y )=δ(x y )θ(x+ y+)U(x+,y+;x ), (22)
Gω ⊥ ⊥ ⊥− ⊥ − ⊥
The Fourier transform of (17) gives
ieiq−L qi
q2δAi(q)= d2x e−iq⊥.x⊥ U(L,t ;x )ρ(x )
ω+iε ⊥ q−+iε 0 ⊥ ⊥
Z
L (cid:2)
+ d2z dz+ (L,x ;z+,z )∂i(U(z+,t ;z )ρ(z )) . (23)
⊥ Gω ⊥ ⊥ z 0 ⊥ ⊥
Z Zt0
(cid:3)
The amplitude for the gluon radiation reads
=q2δAi(q)ǫi (q), (24)
M(λ) (λ)
where the gluon is taken on-shell i.e. q2 = 0 (or q− = q2/2ω) . To get the
⊥
averagenumber ofthe produced gluons,we square the amplitude andsum over
the polarization vectors with the help of the completeness identity
ǫi (q)ǫj∗ (q)= gij. (25)
(λ) (λ) −
λ
X
In the same way as in the CGC treatment we have to perform the averageover
the sources. First, for the fast parton, we write
ρa(x )ρb(x′ ) =µ2(x )δabδ(x x′ ), (26)
⊥ ⊥ p p ⊥ ⊥− ⊥
5Sincethemedium(cid:10)isstatictheseare(cid:11)theonlynon-eikonalcorrectionstogluonpropagation.
7
µ2(x ) is the parton charge density in the transverse plane.
p ⊥
From (24) (or equivalently (2)), the gluon spectrum reads
d N 1 1
ω h i = 2 = e d2x d2y e−i(x⊥−y⊥).q⊥
dωd2q 16π3 |Mλ| (2π)3ωℜ ⊥ ⊥
⊥ λ Z Z
X
L 1 z+
dz+ d2z µ2(z ) dz′+
× ω ⊥ p ⊥
Zt0 h Z Zt0
× trU(z+,z′+,z⊥)∂zi′Gω†(L,y⊥;z′+,z′⊥)∂ziGω(L,x⊥;z+,z⊥) |z⊥=z′⊥
2(cid:10)µ2(y ) qi trU†(L,z+,y )∂i (L,x ;z+,y ) (cid:11)
− p ⊥ q2 ⊥ yGω ⊥ ⊥
⊥
2(N2 1) (cid:10) (cid:11)i
+ c − d2x µ2(x ).
(2π)3q2 ⊥ p ⊥
⊥ Z
(27)
For a single parton produced at x =0 it reduces to
⊥ ⊥
g2C
µ2(x )= R δ(x ), (28)
p ⊥ N2 1 ⊥
c −
where R=A for a gluon and R=F for a quark.
In eq. (27), x end y are the coordinates of the emitted gluon in the trans-
⊥ ⊥
verseplan. Thefirsttermin(27)correspondstotheprobabilityofproducingthe
gluon inside the medium, whereas the second term corresponds to the interfer-
ence between the amplitudes for producing the gluon outside (after the parton
left the medium),-first term in (23)-, and inside the medium,-second term in
(23)-; this is illustrated in fig. 2. In the second term, note that the transverse
coordinates of the emitted gluon and of the hard parton are the same (see the
first term in (23) where the final gluon free propagator has been amputated).
The last term is the probability of radiating a gluon in the vacuum, it has to
be removed to get the medium induced gluon spectrum. This formula is quite
general, in the sense that we do not specify the nature of the medium, so one
still has to find a model for averaging over the medium sources.
NowwewillshowthatthisformulaleadstothewellknownBDMPS-Z-Wspec-
(a) (b)
Figure 2: The diagrammatic representation of the two first terms in (27).
8
trum,inthe caseofuncorrelatedsources. This approximationassumesthatthe
scattering centers at different times in the medium are independent6, therefore
the medium sources can be treated as Gaussian:
ρa(x+,x )ρb(y+,y ) =n(x+)δ(x+ y+)δabδ(x y ), (29)
0 ⊥ 0 ⊥ − ⊥− ⊥
(cid:10) (cid:11)
Namely, that amounts to choose the statistical weight as follows
W
1 ρa(x+,x )ρa(x+,x )
[ρ ]= [ρ ]exp dx+d2x 0 ⊥ 0 ⊥ , (30)
W 0 D 0 −2 ⊥ n(x+)
Z (cid:26) Z (cid:27)
where n(x+) is the medium scattering center density at the time x+. The
corresponding correlator for A−, given by (4), reads:
0
A−a(x+,x )A−b(y+,y ) =n(x+)δ(x+ y+)δabγ(x ,y ), (31)
0 ⊥ 0 ⊥ − ⊥ ⊥
where γ((cid:10)x ,y ) can be expressed a(cid:11)s follows:
⊥ ⊥
1 d2k
γ(x ,y )= ⊥ei(x⊥−y⊥).k⊥, (32)
⊥ ⊥ (2π)2 k4
Z ⊥
and is related to the dipole cross-sectionby the relation7
C
σ(x y )= A [γ(x ,x )+γ(y ,y )+2γ(x ,y )]. (33)
⊥− ⊥ 2 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
In this approximation the average of two Wilson lines is a color singlet:
δ
U (z+,z′+,x )U†(z+,z′+,y ) = ab trU(z+,z′+,x )U†(z+,z′+,y ) ,
ac ⊥ cb ⊥ N2 1 ⊥ ⊥
D E c − (cid:10) (34)(cid:11)
and the two-point function reads
1 z+
trU(z+,z′+,x )U†(z+,z′+,y ) =exp dξn(ξ)σ(x y ) (35)
⊥ ⊥ "−2Zz′+ ⊥− ⊥ #
(cid:10) (cid:11)
The second term in (27) is easily evaluated leading to (see Appendix A for
details):
1
tr (z′+,y ;z+,x )U†(z′+,z+,z )
N2 1 Gω ⊥ ⊥ ⊥
c −
(cid:10)= (z′+,y z ;z+,x z ) (cid:11)
Kω ⊥− ⊥ ⊥− ⊥
z+ iω 1
= r (ξ)exp dξ( r˙2(ξ) n(ξ)σ(r )) , (36)
Z D ⊥ "Zz′+ 2 ⊥ − 2 ⊥ #
6This approximation holds when the range of one scattering center is much smaller than
themeanfreepathoftheradiatedgluonandofthehardpartoninsidethemedium.
7formoredetailsonWilsonlineaverages inthegaussianapproximationseeref.[47,48]
9
where r (z′+)=y z and r (z+)=x z .
⊥ ⊥− ⊥ ⊥ ⊥− ⊥
Using the following property of the Green’s functions, equivalent to (18),
(x+,x ;y+,y )= d2u (x+,x u+,u ) (z+,u ;y+,y ),
Gω ⊥ ⊥ ⊥Gω ⊥ ⊥ Gω ⊥ ⊥
Z
y+<u+<x+
(37)
and the locality in time of the source average, the first term in (27) can be
factorized as follows
1
trU(z+,z′+,z ) †(L,y ;z′+,z′ ) (L,x ;z+,z )
N2 1 ⊥ Gω ⊥ ⊥ Gω ⊥ ⊥
c −
(cid:10) 1 (cid:11)
= d2u Uab(z+,z′+,z ) †bc(z+,u ;z′+,z′ )
N2 1 ⊥ ⊥ Gω ⊥ ⊥
c − Z
†cd(L,y ;(cid:10)z+,u ) da(L,x ;z+,z ) , (cid:11)
× Gω ⊥ ⊥ Gω ⊥ ⊥
1
= (cid:10) d2u trU(z+,z′+,z ) †(z+,(cid:11)u ;z′+,z′ )
(N2 1)2 ⊥ ⊥ Gω ⊥ ⊥
c − Z
tr †(L,y ;z+(cid:10),u ) (L,x ;z+,z ) . ((cid:11)38)
× Gω ⊥ ⊥ Gω ⊥ ⊥
(cid:10) (cid:11)
Putting everything together, and following the calculations in Appendix A, we
recoverthe well-known induced radiative gluon spectrum (see for instance [30],
where it is shown to be equivalent to Wiedemann’s formulation [44])
d N α C L
ω h i = s R 2 e dz+ d2u e−iq⊥.u⊥
dωd2q (2π)2ω ℜ ⊥
⊥ Zt0 Z
× ω1 z+dz′+e−21RzL′+dξn(ξ)σ(u⊥)∂⊥y.∂⊥uKω(z+,u⊥;z′+,y⊥ =0)
hq Zt0
2 ⊥.∂ (L,u ;z+,y =0) . (39)
− q2 ⊥yKω ⊥ ⊥
⊥
i
4 gluon production in proton-nucleus collisions
Inthissection,wewillre-derivethegluonproductioninproton-nucleuscollisions
inthehighenergylimit[45,46,49–51]. Weendupbyadiscussiononthevalidity
of the eikonal approximationat RHIC.
4.1 Gluon production in the high energy limit
At high energy the nucleus is Lorentz contracted, so that we take the limit
L 0, and put t = . As a consequence the produced gluon is eikonal
0
→ −∞
during the interaction time L, and the retarded gluon propagator (15) simply
reads
1
G(x,y)= θ(x+ y+)θ(x− y−)δ(x y )U(x+,y+,x ). (40)
2 − − ⊥− ⊥ ⊥
10
