Effect of running coupling on photons from jet - plasma interaction in relativistic heavy ion collisions

0.21 MB·English

by  Lusaka Bhattacharya

Effect of running coupling on photons from jet - plasma interaction in relativistic
heavy ion collisions
Lusaka Bhattacharya∗ and Pradip Roy†
Saha Institute of Nuclear Physics, Kolkata - 700064, India
We discuss the role of collisional energy loss on high pT photon data measured by PHENIX collaboration
by calculating photon yield in jet-plasma interaction. The phase space distribution of the participating jet is
dynamically evolved by solving Fokker-Planck equation. We treat thestrong coupling constant (αs) as function
of momentum and temperature while calculating the drag and diffusion coefficients. It is observed that the
quenchingfactorissubstantiallymodifiedascomparedtothecasewhenαs istakenasconstant. Itisshownthat
the data is reasonably well reproduced when contributions from all the relevant sources are taken into account.
Predictions at higher beam energies relevant for LHC experiment havebeen made.
1
1 PACSnumbers:
0 Keywords: energy-loss,quark-gluon-plasma
2

n
I. INTRODUCTION interaction) [2] which occurs when a high energy jet in-
a
teracts with the medium constituents via annihilation
J
and Compton processes. It might be noted that this
0 Heavy ion collisions have received significant atten-
phenomenon (for Compton process) has been illustrated
2 tion in recent years. Various possible probes have been
quite sometime ago[3]inthe contextofestimating pho-
studied in order to detect the signatures of quark gluon
] tons from equilibrating plasma. There, it is assumed
h plasma(QGP).Studyofdirectphotonanddileptonspec-
that because of the larger cross-section, gluons equili-
p traemanatingfromhotanddensematterformedinultra-
bratefasterprovidingaheatbathtotheincomingquark-
- relativistic heavy ion collisions is a field of considerable
p jet. Acomparisonofthenon-equilibriumphotons(equiv-
current interest. Electromagnetic probes have been pro-
e alent to photons from jet-plasma interaction) with the
h posed to be one of the most promising tools to charac-
directphotons(thermal)showsthatthiscontributionre-
[ terizetheinitialstateofthecollisions[1]. Becauseofthe
mains dominant for photons with pT upto 6 GeV. How-
very nature of their interactions with the constituents of
1 ever, while evaluating jet-photon the authors in Ref. [2]
the system they tend to leave the system almost unscat-
v assumes that the largest contribution to photons corre-
9 tered. In fact, photons (dilepton as well) can be used spondstop ∼p (p ). Thisimpliesthattheannihilating
6 to determine the initial temperature, or equivalently the γ q q¯
quark (anti-quark) directly converts into a photon. In
8 equilibration time. These are related to the final multi-
the present work, we calculate photons from jet-plasma
3 plicity of the produced hadrons in relativistic heavy ion
. interaction relaxing the above assumption and including
1 collisions (HIC). By comparing the initial temperature
the energy loss of the participating jet where the strong
0 with the transition temperature from lattice QCD, one
coupling constant (α ) is taken as ’running’ as given in
1 can infer whether QGP is formed or not. s
Ref. [4].
1
Photons are produced at various stages of the evolu-
:
v tion process. The initial hard scatterings (Compton and The phenomena of jet quenching vis-a-vis energy loss
i
X annihilation)ofpartonsleadtophotonproductionwhich have been studied by several authors [5] taking into ac-
we call hard photons. If quark gluon plasma (QGP) is count both collisional and radiative losses. However, in
r
a produced initially, there are QGP-photons from thermal the present work, our main concern is to examine the
Compton plus annihilation processes. Photons are also role of collisional loss (where α is running) in the con-
s
producedfromdifferenthadronicreactionsfromhadronic text of photon production from jet-plasma interaction
matter either formed initially (no QGP scenario) or re- for the following reason. The suppression of single elec-
alized as a result of a phase transition (assumed to be tron data [6] is more than expected which led to the re-
first order in the present work) from QGP. In addition thinkingoftheimportanceofcollisionalenergylossinthe
to that there is a large background of photons coming contextofRHICdata. Asubstantialamountofworkhas
fromπ0 andη0 decays. If this decaycontributionis sub- been done to look into this issue in recent times [4, 7–
tracted from the total photon yield what is left is the 11]. Few comments about the recent developments of
direct (excess) photons. collisional energy loss are in order here. It is argued in
These apart, there exits another class of photon emis- Ref.[12] thatthe collisionalenergylossis approximately
sionprocessviathejetconversionmechanism(jet-plasma of the same order as the radiative loss. It is also shown
by Braun et. al [4] that the collisional energy loss in-
creases substantially if the strong coupling is treated as
function of temperature and momentum and if, in addi-
∗Electronicaddress: [email protected] tiontot-channelprocess,theinverseComptonreactionis
†Electronicaddress: [email protected] considered. Inamostrecentcalculation,usingareduced
2
screening mass and running coupling the collisional en- participating partons. Apart from the thermal interac-
ergy loss is six times larger than that with the constant tions of the plasma partons, interaction of a leading jet
coupling [13]. It explains single electron R quite well. partonwiththeplasmawasfoundtobeaveryimportant
AA
However,itfailsto accountforthe elliptic flow, v ofthe source of photons.
2
electron. Effective resonance with LO-pQCD model [14] Thedifferentialphotonproductionrateforthisprocess
also improves the collisional energy loss and the single is given by:
electron data is well reproduced. It is also important to
note that only the radiative energy loss fails to account dR N d3p d3p d3p
E = 1 2 3 f (p )
for the single electron data at RHIC [15]. On the other d3p 2(2π)3 Z 2E (2π)32E (2π)32E (2π)3 jet 1
1 2 3
hand, the authors of Ref. [16] claims that the collisional
×f (E )(2π)4δ(p +p −p −p)|M|2(1±f (E )) (1)
energy loss is sub-leading. However, in order to see the 2 2 1 2 3 3 3
effects of energy loss on jet-photon one should also in-
where,|M|2 representsthespinaveragedmatrixelement
corporate the radiative energy loss for completeness and
squared for one of those processes which contributes in
this has to be done in the same formalism in a realistic
the photon rate and N is the degeneracy factor of the
scenario.
correspondingprocess. f ,f andf aretheinitialstate
Thus, it is apparent that the issue of the relative im- jet 2 3
and final state partons. f and f are the Bose-Einstein
2 3
portance of the mechanism of energy loss in the context
or Fermi-Dirac distribution functions.
of RHIC data is not settled yet. We shall re-visit the
importance of collisional energy loss in the context of 1
f (E )= (2)
photons from jet-plasma interactions. Moreover,in high 2 2,3 exp(E /kT)±1
2,3
temperature (T) effective field theory the coupling con-
stant, α , is taken to be a function of temperature only,
s
which may be justified when T >> Λ . However, in
QCD B. Fokker - Planck Equation: Parton transverse
the case of relativistic heavy ion collisions, temperature
momentum spectra
is not the only scale, there is the momentum scale (k)
also. One has to take into account the case when k ∼T
In the photon production rate (from jet-plasma inter-
and treat α to be function of both k and T [4]. By
s action) one of the collision partners is assumed to be
incorporating this fact it is shown that the energy loss
in equilibrium and the other (the jet) is executing ran-
is by a factor of 2−4 more than the case when α is
s dom motion in the heat bath provided by quarks (anti-
constant. It is for this purpose we concentrate on the
quarks) and gluons. Furthermore, the interaction of the
collisionalenergy loss [7, 17] with the formalismgivenin
jet is dominated by small angle scattering. In such sce-
Ref.[4]to calculatephotonsfromjet-plasmainteraction.
nario the evolution of the jet phase space distribution is
Forcompleteness,wealsoincludetheradiativelossinan
governedby Fokker-Planck(FP) equationwherethe col-
effective way.
lision integral is approximated by appropriately defined
The organization of the paper is as follows. We give
drag and diffusion coefficients.
a brief description of jet-photon production in QGP in
As mentioned already in the introduction that the
section IIA. The evolution of jet quark and photon p
T
quark jet here is not in equilibrium. Therefore the cor-
distributions are discussed in sections IIB and IIC re-
responding distribution function (f ) that appears in
spectively. Section III is devoted to the discussions of jet
Eq.(1)iscalculatedbysolvingtheFPequation. Theen-
results and finally, we summarize in section IV.
ergylossisrepresentedbythedragcoefficient(seelater).
The FP equation, can be derived from Boltzmann equa-
tion if one of the partners of the binary collisions is in
II. FORMALISM
thermal equilibrium and the collisions are dominated by
the small angle scattering involving soft momentum ex-
A. Jet-Photon Rate
change [8, 14, 20–24]. For a longitudinally expanding
plasma, FP equation reads [8, 25]:
The lowest order processes for photon emission from
QGP are the Compton scattering (q(q¯)g → q(q¯)γ) and ∂ p ∂ ∂
− z f(p ,p ,t)= A (p)f(p)+
annihilation(qq¯ → gγ) process. The totalcross-section (cid:18)∂t t ∂p (cid:19) T z ∂p i
z i
diverges in the limit t or u → 0. These singularities
1 ∂
have to be shielded by thermal effects in order to ob- [Bij(p)f(p)], (3)
2∂p ∂p
tain infrared safe calculations. It has been argued in i j
Ref. [18] that the intermediate quark acquires a ther-
where [8]
malmass in the medium, whereasthe hardthermalloop
(HTL) approach of Ref. [19] shows that very soft modes
ν d3k′d3kd3q
are suppressed in a medium providing a natural cut- A = dωq |M|2 f(k)(1+f(k′))
i 16p(2π)5 Z k′ k p′ i t→0
off k ∼ gT. We assume that the singularities can be
c
shielded by the introduction of thermal masses for the δ3(q−k′+k)δ(ω−vk′ ·q)δ(ω−vk·q) (4)
3
ν d3k′d3kd3q
B = dωq q |M|2 f(k)
ij 16p(2π)5 Z k′ k p′ i j t→0 100 Without E-loss
×He(1re+νfi(skt′)h)eδ3a(pqp−rokp′ri+atke)dδ(eωge−nevrakc′y·qfa)δc(tωor−. Nvkot·eqt)h,(a5t) -2V) 10-1 RHIC τττiii === τ01c..55 f mffmm/c//cc
thecoefficientA isrelatedthedragcoefficientη byA = Ge 10-2
i i
ηpNi,owwh,erBe η =can(1/bEe)ddEec/odmxp.osed into longitudinal and pdy(T10-3
ij 2
d
transverse components: N/ -4
d 10
p p p p
i j i j
B =B (δ − )+B (6)
ij t ij p2 l p2 10-5
(a)
Explicit calculation shows that the off diagonal compo- -6
10
nents of B vanish and we have, 2 4 6 8 10 12 14
ij
p (GeV)
T
ν d3kd3qdω
B = δ(ω−v ·q)δ(ω−v ·q)
t,l (2π)4 Z 2k2k′2p2p′ p k
∂f
hMi2 f(k)[1+f(k)+ω ]q2 , (7)
t→0 ∂k t,l 101
Without E-loss
where, Bl = dh(∆pz)2i/dt, Bt = dh(∆pT)2i/dt, repre- 100 LHC ττi == 01..55 ffmm//cc
sent diffusion constants along parallel and perpendicu- i
lar directions of the propagating partons. Evidently, Ai -2V) 10-1 τi = τc fm/c
(B ) is infrared singular. Such divergences do not arise e
t,l G
if close and distant collisions are treated separately. For y(
very low momentum transfer the concept of individual pdT10-2
2
collisionbreaksdownandonehasto takecollectiveexci- d
tationsoftheplasmaintoaccount. Hencethereshouldbe dN/ 10-3
a lowermomentumcut offabovewhichbare interactions
-4
might be considered. While for soft collisions medium 10
(b)
modified hard thermal loop correctedpropagator should
-5
be used [7, 26]. It is evident that Eq. (4) actually gives 10 2 4 6 8 10 12 14 16 18 20
dE/dt or the energy loss rate [7] that can be related to p (GeV)
T
the drag coefficient.
FIG.1: (Coloronline) pT distribution ofpartonat(a) RHIC
However, in the above treatment, the infra-red cut-off energy(Ti =0.446GeVandτi =0.147fm/c)andat(b)LHC
isfixedbyplasmaeffects,whereonlythe mediumpartis energy (Ti =0.897 GeV, τi =0.073 fm/c).
considered, completely neglecting the vacuum contribu-
tion leading to ambiguity in the energy loss calculation. v
1
+ , (9)
Ifthelatterpartistakenintoaccountthestrongcoupling (1+exp(v k −v ))(ln(e+( k )a+( k )b)
should be running. Thus for any consistent calculation 2T 3 λs λs
one has to take into consideration this fact. In that case
with k = |ω2−q2| in this case. The parameters a, b
α = α (k,T) (k = |ω2−q2| in this case), and the
s s
aboveintegralsmustbpe evaluatednumericallywherethe and λs arepgiven by a = 9.07, b = 5.90 and λs = 0.263
GeV.Forthelimiting behavior(k <<T)ofthe coupling
infra-redcut-off is fixed by Debye mass to be solvedself-
we choose,
consistently:
m (T)=4π 1+ NF α (m (T),T)T2 (8) u1 =α∗3d(1+exp(−u3)) (10)
D s D
(cid:18) 6 (cid:19)
Here α∗ and α∗ denote the values of the IR fixed
3d s
WereiteratethatthematrixelementsinEqs. (4)and(7) point of SU(3) Yang-Mills theory in d = 3 and d = 4
contains the strong coupling which we take as running, dimensions, respectively. The remaining four parame-
i. e. α = α ( |ω2−q2|,T). We chose the follow- ters (u = 5.47,u = 6.01,v = 10.13 and v = 9.27) fit
s s 2 3 2 3
ing parameterizatpion of αs which respects the perturba- thenumericalresultsforpureYang-Millstheoryobtained
tiveultra-violet(UV)behaviorandthe3Dinfra-red(IR) from the RG equations in Ref. [27].
point [4]: So far we have discussed about the collisional energy
loss which, although, dominates at lower energies, is not
α (k,T)= u1Tk the only mechanism of energy loss. As the energy in-
s 1+exp(u k −u ) creasesradiativeenergylossstartstodominateandhence
2T 3
4
cannot be neglected. In case of jet-photon production, two effects on the parton energy loss: (i) The expanding
since the photon energy is almost equal to the jet en- geometry will increase the duration of propagation, (ii)
ergy, one has to include the radiative loss to account for the same expansion will cause the parton density to fall
the high p photons. However, in order to see the ef- along its path. These two effects partially compensate
T
fects of both the collisional and radiative energy losses, each other and the energy loss is almost the same as in
one must develop a formalism in which both the mech- the case without transverse expansion [30]. Since we are
anisms can be taken into account in a consistent man- considering the central rapidity region (|η| < 0.35) the
ner. The two mechanisms are not entirely independent, arguments given above is not applicable in the case of
i.e., the collisional loss may influence the radiative loss. longitudinal expansion scenario.
Thusbothshouldbeincludedtocalculatetransportcoef- The solution with an arbitrary initial momentum dis-
ficients. Sincethereisnorigorouswaytoimplementthis, tribution can now be written as [14],
theapproximatewayistodefineeffectivedrag(diffusion)
in the following manner: EdN = d3p P(p~,t|p~ ,t )E dN (16)
d3p Z 0 0 i 0d3p
0
η = η +η
coll rad
We use the initial partonp distributions (at the forma-
1 dE dE T
= + (11) tion time t ) taken from [2, 31]:
E (cid:20)(cid:18)dx(cid:19) (cid:18)dx(cid:19) (cid:21) i
coll rad
dN KN
wmhinedret(cid:0)hdadExt(cid:1)ηcpoill=caAnib.eTchaelccuolalltiesidonfraolm(dEiffqe.re(4n)tikael)epeinnegrginy d2p0Tdy0|y0=0 = (1+ p0β0T)α, (17)
loss(dE/dx) canbe calculatedfromEq.(4)using the
coll where K is a phenomenologicalfactor (∼1.5−2) which
method described in [8]. For running α it is given by,
s takesintoaccountthe higherordereffects. Thevaluesof
the parameters are listed in Table. I. We note that the
dE 4 N s α 2( |t|,T)
= π(1+ f)T2 d|t| s |t|
(cid:18)dx(cid:19)coll 3 6 Z mD2 p|t|2 RHIC LHC
(12) q q¯ q q¯
N0 [1/GeV2] 5.0×102 1.3×102 1.4×104 1.4×105
Where, s = 2ET, E is the energy of the incident quark.
β [GeV] 1.6 1.9 0.61 0.32
The radiative energy loss is given by,
α 7.9 8.9 5.3 5.2
dE C α (E,T)Lµ2 E
F s
= ln( ), (13)
(cid:18)dx(cid:19)rad N(E) λg µ TABLEI: Parameters for initial parton pT distribution.
where N(E) is an energy dependent factor and N(E →
parametric form of Eq. (17) may not represent the true
∞) = 4 if kinematic bounds are neglected [28]. It is
picture of the jet p distribution. In recent years, more
important to point out here that N(E) = 7.3,10.1,24.4 T
sophisticatedcalculationshave been done using different
for E =500,50,5 GeV respectively and N(E →∞)=4
parameterizations of the parton distribution functions.
(see the Ref. [28]). L is the distance traversed by the
The p distribution used here might differ substantially
jets in the plasma. Similarly we can define the effective T
from these calculations and the results presented here is
diffusion coefficients.
correct upto a factor that might come from using the
Having known the drag and diffusion, we solve the
more state of the art calculation.
FP equation using Green’s function techniques: If
P(p~,t|p~ ,t ) is a solution to Eq. (3) with the initial con-
0 i
dition
C. Space time evolution
P(p~,t=t |p~ ,t )=δ(3)(p~−p~ ) (14)
i 0 i 0
In order to obtain the space-time integrated rate we
the full solution with an arbitrary initial condition can first note that the phase space distribution function for
be obtained as [22] the incoming jet in the mid rapidity region is given by
(see Ref. [32] for details)
f(t,p~)= d3p~ P(p~,t|p~ ,t )f (p~ ) (15)
Z 0 0 i 0 0 f (~r,~p,t′)| = (2π)3P(|w~r|) ti 1
jet y=0
where for the initial condition f(t = 0,~p) = f (p ) and νq ti2−z02 pT
0 0
P(p~,t|p~0,ti) is the Green’s function of the partial differ- × dNp (p ,t′)δ(z ) (18)
ential Eq. (2). d2p dy T 0
T
We assume here that the plasma expands only lon-
gitudinally (Bjorken expansion scenario [29]). The rea- where d2dpNTdy(pT,t′)canbe obtainedfromEq. (16). ti is
son is the following. The transverse expansion will have thejetformationtimeandν isthespin-colordegeneracy
q
5
factor. z0 is thejetformationpositioninthe directionof 10-4
QGP expansion and P(|w~r|) is the initial jet production JTeht-etrhm (awlithout e-loss + const. αs)
probability distribution at the initial radial position w~r 10-5 Jet-th (with coll e-loss + running αs)
in the plane z =0, where Jet-th (with coll e-loss + rad e-loss
0 -2V) 10-6 + running αs)
|w~ | = (~r−(t′−t ) p~ )·rˆ Ge
r i |~p| dy( 10-7
T
p
= (rcosφ−t′)2+r2sin2φ for t ∼0 (19) 2d
q i N/ 10-8
d
and φ is the angle in the plane z = 0 between the di-
0 -9
10
rectionofthe photonandthepositionwherethis photon
hasbeenproduced. We assumetheplasmaexpandsonly
-10
longitudinally. Thususingd4x=rdrdt′dφdz andthe ex- 10 4 6 8 10 12 14
pression for f from Eq. 17 we obtain the transverse p (GeV)
jet T
momentum distribution of photon as follows [32, 33]:
FIG. 2: (Color online) pT distribution of photons at RHIC
dNγ = d4x dNγ energy with Ti=0.446 GeV and τi =0.147 fm/c. The violet
d2pTdy Z d4xd2pTdy (magenta)curvedenotesthephotonyieldfromjet-plasmain-
teractionwithcollisional (collisional+radiative)energyloss.
(2π)3 tc R
= dt′ rdr dφP(w~ ) The blue curve corresponds to the case without any energy
r
νq Zti Z0 Z loss andthegreen curverepresentsthethermalcontribution.
N
× i dsˆdtˆ|M |2 dE dE
16(2π)7Eγ Z i Z 1 2 the hadronic phase) where the quenching factor is the
× 1 dN (p ,t′)f2(E2)(1±f3(E3)) largest. AtLHCenergies(seeFig.1b)thisfactorismore
p1T dp21Tdy 1T aE22+2bE2+c asthetemperatureinthiscaseislargecomparedtoRHIC
energies. It is seen from Eq. (20) the photon p distri-
p (20) T
bution is directly proportional to the quark p spectra.
T
Thus the photon yield will be affected as we shall see in
f is the distribution function of the jet quark (see
jet
the following.
Eq. (18)) and rest of the distribution functions i.e f ,f
2 3
areFermi-DiracorBose-Einsteindistributions. φdepen-
dence occurs only in P(w~ ). So the φ integrationcan be
r -3
doneanalyticallyasinRef.[32]. Thetemperatureprofile 10 Jet-th (without e-loss + const. α)
s
is taken from Ref. [32]. Thermal
10-4 Hard
Besides the thermal photons from QGP and hadronic Phenix data
Jet-th (with coll e-loss + running α)
mteraitntgerfrwoematlshoecraealccutliaotneopfhtohteontyspferohmAhinBiti→al hγaXrdussciantg- -2V) 10-5 J +e tr-uthn n(winigth α cso)ll e-loss + rad e-loss s
perturbativeQCD.Weincludethetransversemomentum Ge 10-6 Total
broadeningintheinitialstatepartons[34,35]. Thecross- dy(
section for this process can then be written in terms of pT 10-7
2
d
elementaryparton-partoncross-sectionmultipliedbythe N/
partonic flux which depends on the parton distribution d 10-8
functions (PDF) for whichwe takeCTEQ parameteriza-
-9
10
tion [36]. A phenomenological factor K is used to take
into account the higher order effects. We also include -10
10
4 6 8 10 12 14
photons from fragmentation process.
p (GeV)
T
III. RESULTS FIG.3: (Coloronline)pT distributionofphotonsatRHICen-
ergywithTi =0.446GeVandτi =0.147fm/c. Themagenta
(blue) curve denotes the photon yield from jet-plasma inter-
To obtain the quark momentum distribution we use action running(constant) αs. Theblack (green) curvecorre-
Eqs. (3), (4), (7), (8) and (16). The transverse mo- spondstohard(thermal)photons. Theorangerepresentsthe
mentum distributions of quarks are shown in Fig. 1 for total photon yield compared with the Phenix measurements
different times (proper) at RHIC and LHC energies re- of photon data [37].
spectively where the initial distributions are taken from
Eq. (17). It is observed that the spectra are more re- In order to obtain the photon p distribution we nu-
T
ducedasthe time increases. It isgenerallyassumedthat merically integrate Eq. (20) using Eq. (16). The results
quarksfragmentinto hadronsaroundτ (the begining of for jet-photons for RHIC energies are plotted in Fig. 2
c
6
wherewe havetakenT =446MeVandt =0.147fm/c. former case. The yield further reduces when both the
i i
Asindicatedearlier,theradiativeenergylossstartsdom- mechanisms of energy loss are included. The total pho-
inating at higher energies of the jet, we include this in ton yield consisting of jet-photon, photons from initial
the calculation of photon p distribution. We find that hard collisions, jet-fragmentation and thermal photons
T
the yield is decreased with the inclusion of both the en- is compared with the PHENIX photon data [37]. It is
ergylossmechanismsascomparedtothe casewhenonly seen that the data is well reproduced in our model (see
collisionalenergylossisconsidered. Itistobenotedthat Fig. 3). To cover the uncertainties in the initial condi-
whenoneconsiderscollisionalenergylossalonethe yield
withconstantα ismorecomparedtothesituationwhen
s
running α is taken into account. This is due to the fact
s
that the energy loss in the later case is more [4]. On the
-3
10
other hand, with the inclusion of the radiative loss the
Phenix data
yield decreases further. 10-4 JJeett--tthh ((WWiitthho cuotl lE e--lolossss + + c rounnsnt.i nαgs )α)
s
Jet-th (with coll e-loss + rad e-loss
-5 + running α)
-2V) 10 Total s
Ge 10-6
0.001 y(
d
Jet-th (with coll e-loss + const. α) 2pT 10-7
0.0001 s d
Jet-th (with coll e-loss + running α) N/
s d 10-8
2) 1e-05
-V 10-9
e
G 1e-06
y( -10
dT 10 4 6 8 10 12 14
2p 1e-07 p (GeV)
d T
N/
d 1e-08
FIG. 5: (Color online) pT distribution of photons at RHIC
1e-09 energieswithTi =0.350GeVandτi =0.25fm/c. Theorange
line corresponds to total photon yield from all the sources
as in Fig. 3 with both energy losses (collisional + radiative)
1e-10
4 6 8 10 12 14 included in thejet-photon contribution.
p (GeV)
T
tions foragivenbeam energy,weconsideranothersetof
FIG. 4: (Color online) pT distribution of jet-th photons at
initialconditionsatalowertemperatureT =0.350GeV
RHIC energy with Ti = 0.446 GeV and τi = 0.147 fm/c. i
and somewhat later initial time of τ = 0.25 fm/c. The
The red (blue) curve denotes the photon yield coming from i
jet-plasma interaction with collisional energy loss + running yieldforthissetisshowninFig.5. We seethatthe data
(constant) αs. is reproduced reasonably well.
In Fig. 6 we plot the p distribution of photons for
T
In order to compare our results with high pT photon the RHIC energy (Ti = 0.236 GeV and τi = 0.5 fm/c)
data measured by the PHENIX collaboration [37], we for a lower value of dN/dy = 600. It is clearly visible
have to evaluate the contributions to the photons from from Fig. 6 that we can not explain Phenix photon data
other sources,thatmight contribute in this pT range. In satisfactorily in the pT range 4−8 GeV. For the higher
Fig. 3 the results for jet-photons corresponding to the pT range Phenix photon data is well reproduced.
RHIC energiesareshown,where we havetakenT =446 We also consider the high p photon production at
i T
MeV and t = 0.147 fm/c. The individual contribu- LHC energies. The contributions from various sources
i
tions from hard and bremsstrahlung processes [38] are are shown in Figs. 7 where the jet-plasma contribution
also shown for comparison. These are estimated using is calculated with running coupling constant (α ) (con-
s
the formalism given in Ref. [38]. The total yield com- sideringboththe collisionalandradiativeenergylosses).
prises of photons from jet-plasma interaction (with en- Since the initial temperature in this case is higher, the
ergy loss), hard and bremsstrahlung processes, thermal plasma lives for longer time. Thus the energy loss suf-
Compton and annihilation processes. We also show in a fered by the parton is more. As a result, the difference
separate plot (in Fig. 4) photons from jet-plasma inter- betweenthecaseswithandwithoutenergylossisslightly
action corresponding to the cases with constant α and morethanwhatis obtainedatRHIC. Itis observedthat
s
running α with collisional energy loss alone. It is ob- due to the inclusion of radiative energy loss along with
s
served that the spectra in the case of collisional energy collisional energy loss the jet photon yield is suppressed
losswith running couplingis depleted bya factor1.7−2 significantly. Also due to the inclusion of running cou-
compared to the case where the strong coupling is con- pling constant the jet photon yield is suppressed by a
stant. This is expected as the energy loss is more in the factor of 3−4.
7
IV. SUMMARY
-3
10
Thermal
Hard
10-4 PJehte-tnhi x( wdiatthaout e-loss+ const. α) Wehavecalculatedthetransversemomentumdistribu-
s
jet-th (with coll e-loss+running α) tionofphotonsfromjetplasmainteractionwithrunning
-2V) 10-5 J +e tr-uthn n(winigth α cso)ll e-loss+ rad e-losss coupling, i. e. with αs = αs(k,T) where we have in-
Ge 10-6 Total cluded both collisional and radiative energy losses. It is
y( found that the assumption made in Ref. [2] while calcu-
d
pT 10-7 lating photons from jet-plasma interactions may not be
2
d good at LHC energies as we observe a difference is by a
N/
d 10-8 factor of 2−3. Using running coupling we find that the
depletioninthephotonp spectraisbyafactorof2−2.5
T
10-9 moreas comparedto the case with constantcoupling for
RHIC energies. This is due to the fact that the energy
-10
10 4 6 8 10 12 14 loss (and hence drag and diffusion coefficients) is more
p (GeV) by similarfactor inthe caseofrunning coupling. Phenix
T
photon data have been contrasted with the present cal-
FIG. 6: (Color online) pT distribution of photons at RHIC culationandthedataseemtohavebeenreproducedwell
energywithTi =0.236GeVandτi =0.5fm/c. Themagenta in the low pT domain. The energy of the jet quark to
(blue) curve denotes the photon yield from jet-plasma inter- produce photons in this range (4 < p < 14) is such
T
action running (constant) αs. The black (green) curve cor- that collisional energy loss plays important role here. It
responds to hard (thermal) photons. The orange represents
is shown that inclusion of radiative energy loss also de-
the total yield compared with the Phenix measurements of
scribes the data reasonable well.
photon data [37].
To check the sensitivity with the initial conditions, we
consider two sets of initial conditions. In both the cases
the data can be described quite well. This is due to the
10-2 fact that both the initial conditions corresponds to the
Jet-th (without E-loss + const. α) same dN/dy =1150.
s
10-3 Jet-th (with coll e-loss + running αs) As we validate our model through the description of
Jet-th (with coll e-loss + rad e-loss
2) + running αs) Phenix photon data we also predict the high pT photon
-V yield that might be expected in the future experiment
Ge 10-4 at LHC. We notice that the inclusion of the radiative
y(
d energylossfurther reducesthe yieldathighp . Itis ob-
T T
2dp 10-5 served that the contribution from jet-plasma interaction
N/ is slightly more reduced as compared to the RHIC case
d
as the initial temperature is higher at LHC.
-6
10
We do not consider transverse expansion as the en-
-7 ergy loss of the partons remains just about the same as
10
4 6 8 10 12 14 16 18 the case without transverse expansion. Finally, we con-
p (GeV)
T cludebynotingthattheroleofrunningcouplingconstant
should be explored in the context of other observables
FIG. 7: (Color online) Same as Fig.(3) at LHC energy (Ti = such as thermal photons, dileptons and so on.
0.897 GeV and τi =0.073 fm/c).
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