Light from Cascading Partons in Relativistic Heavy-Ion Collisions

4 Pages·0.12 MB·English

by  Steffen A. Bass

Light from Cascading Partons in Relativistic Heavy-Ion Collisions
Steffen A. Bass,1,2 Berndt Mu¨ller,1 and Dinesh K. Srivastava3
1Department of Physics, Duke University, Durham, NC 27708-0305
2RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA
3Physics Department, McGill University, 3600 University Street, Montreal, H3A 2T8, Canada∗
(Dated: February 8, 2008)
Wecalculate theproductionofhighenergyphotonsfrom Comptonandannihilation processes as
well as fragmentation off quarks in the Parton Cascade Model. The multiple scattering of partons
isseen tolead toa substantial production of high energy photons, which rises furtherwhen parton
multiplication due to final state radiation is included. The photon yield is found to be directly
proportional to the number of hard collisions and thus provides valuable information on the pre-
equilibrium reaction dynamics.
3
0
High-energycollisionsofheavynucleiarestudiedwith portance. These are analogous to the the processes gov-
0
thegoaltoexplorethepropertiesofmatteratenergyden- erningthe energyloss ofenergetic partons,where gluons
2
sities many times that of normal nuclear matter. There are emitted instead of photons. Photons, however, once
n
are strong theoretical reasons to believe that matter un- emitted, almost always escape the system, and their en-
a
J der these conditions, when thermally equilibrated, forms ergy and momentum will remain unaffected [2] as they
aQuark-GluonPlasma(QGP),astateofmatterinwhich will only rarely interact with the surrounding medium.
4
1 quarks and gluons are not confined to hadrons. The dis- For this reason they are excellent and reliable probes of
coveryofthisstateofmatterandthestudyofitsproper- theevolutionofthepartoniccascadeinnuclearcollisions.

2 tieswouldconstituteamajoradvanceinourunderstand- Infact,onecanarguethatthe photonsconfirmthepres-
v
ing of quantum chromodynamics (QCD). ence of parton cascading processes after the initial pri-
0
The partoncascade model (PCM) [1] was proposedto mary parton-parton collisions.
3
0 provide a detailed description of the temporal evolution In this letter we address several questions relating to
9 ofnuclearcollisionsathighenergy,fromtheonsetofhard
the value of photons as probes of the dense matter cre-
0 interactionsamongthe partonsofthe colliding nucleiup
ated in a nuclear collision: Are photons sensitive to the
2
to the moment of hadronization. The PCM is based on
0 pre-equilibrium partonic reaction dynamics, i.e. parton
a relativistic Boltzmann equation for the time evolution
/ re-scattering? What effect do parton fragmentationpro-
h of the parton density in phase space due to perturbative
cesses have on photon production? Is there a direct
t
- QCD interactions including scattering and radiation in
connection between the photon yield and the number
l
c the leading logarithmic approximation. of hard perturbative parton-parton collisions during the
u Duringtheepochwherethepartonicpictureofsequen-
reaction? In view of our specific focus, we do not con-
n
tial perturbative interactions among QCD quanta may
: sider the production of photons from a late-stage, ther-
v be valid, important many-body effects, such as parton
malmixedphaseorhadronicphase. Thesecontributions
i shadowing, color screening, and Landau-Pomeranchuk-
X arequite wellunderstood[3], andcaneasily be addedto
Migdal(LPM)suppressionofgluonradiation,complicate
r our results.
a the single-particle dynamics of parton transport. Obvi-
The fundamental assumption underlying the PCM is
ously, the PCM involves the application of perturbative
that the state of the dense partonic system can be
QCDtoahithertountesteddomain. Thisposestheques-
characterized by a set of one-body distribution func-
tion - how canone test the correctnessof this approach?
tions F (xµ,pα), where i denotes the flavor index (i =
This is clearly not possible by studying the hadronic fi- i
g,u,u¯,d,d¯,...) and xµ,pα are coordinates in the eight-
nalstate,since–ifthedensematterequilibrates–allin-
dimensional phase space. The partons are assumed to
formation about its dynamics before equilibrium will be
be on their mass shell, except before the first scatter-
lost. One may look for remnants of the pre-equilibrium
ing. In our numerical implementation, the GRV-HO
dynamics in the form of very energetic partons, which
parametrization [4] is used, and the parton distribu-
did not lose a large fraction of their initial momentum
tion functions are sampled at an initialization scale Q2
duetointeractioninthedensemediumandproducejets. 0
( (pmin)2; see later)to createa discrete set ofparticles.
Evenbettersuitedtothispurposeareweaklyinteracting ≈ T
Partons generally propagate on-shell and along straight-
probes of the pre-equilibrium dynamics, such as quanta
line trajectories between interactions. Before their first
that interact only electromagnetically: photons and lep-
collision,partonsmayhaveaspace-likefour-momentum,
tons.
especially if they are assigned an “intrinsic” transverse
Photons, which are produced from Compton (qg
qγ), annihilation (qq gγ), and bremsstrahlung (q⋆ → momentum.
→ →
qγ) processes (see Fig. 1), thus assume considerable im- The time-evolution of the parton distribution is gov-
2
a) b) 10-1
compton: 5 VNI/BMS BMS:prim-prim&prim-2nd
BMS:fullrescattering
2
collinear radiation: -2)V10-2 Au+Au;E =200AGeV
e 5 CM
G
( 2
annihilation: dpt10-3
N/ 5
d
p)t 2
10-4
2
1/( 5
2
photons
FIG.1: Photonproductionprocessesincludedinourcalcula- 10-5
tion. 5 |yCM|<0.5
0 1 2 3 4 5 6
p (GeV)
t
erned by a relativistic Boltzmann equation: FIG. 2: Transverse momentum distribution of photons from
centralcollision ofgoldnucleiatRHIC(seetextforanexpla-
∂
pµ F (x,p~)= [F] (1) nation of thesymbols).
∂xµ i Ci
where the collision term is a nonlinear functional of
i
C
the phase-space distribution function. The calculations
take a smaller value for the cut-off µ for a quark frag-
discussed below include all lowest-order QCD scattering 0
menting into a photon [9], but we have not done so as
processesbetweenmasslessquarksandgluons[5],aswell
we are only interested in high energy photons here. The
as all (2 2) processes involving the emission of pho-
→ soft-gluon interference is included as in [7], namely by
tons (qg qγ, q¯g q¯γ, qq¯ gγ, qq¯ γγ). A low
momentum→-transfer→cut-off pm→in is neede→d to regularize selecting the angular ordering of the emitted gluons. An
T
essential difference between emission of a photon and a
theinfrareddivergenceoftheperturbativeparton-parton
parton in these processes is that the parton encounters
cross sections. A more detailed descriptionof our imple-
further interactions and contributes to the build-up of
mentation is in preparation [6].
the cascade, while the photon almost always (in our ap-
We account for the final state radiation [7] following
proximation, always) leaves the system along with the
a hard interaction in the parton shower approach. In
information about the interaction.
the leading logarithmic approximation, this picture cor-
Some of these aspects were discussed by authors of
responds to a sequence of nearly collinear 1 2 branch-
→ Ref.[11]andexploredwithinframeworkofthe PCMim-
ings: a bc. Here a is called the mother parton, and
→ plementation VNI [12]. Unfortunately, several compo-
b and c are called daughters. Each daughter can branch
nents of their numerical implementation were incorrect.
again, generating a structure with a tree-like topology.
Our present work is based on a thoroughly revised and
Weincludeallsuchbranchingsallowedbythestrongand
extensively tested implementation of the parton cascade
electromagnetic interactions. The probability for a par-
ton to branch is given in terms of the variables Q2 and model by the present authors, called VNI/BMS [6].
z. Q2 is the momentum scale of the branching and z As a first step, we discuss the results obtained when
describes the distribution of the energy of the mother the parton cascade is restricted to two-body scattering.
partona among the daughters b and c, such that b takes We are considering central collisions (b = 0) of Au nu-
the fraction z and c the remaining fraction 1 z. The clei at full RHIC energy (√s = 200 GeV per nucleon
differential probability to branch is: − pair). We have confirmed the accuracy of our code by
comparing the analytical pQCD prediction for prompt
αabc dQ2 photon production, due to Compton and annihilation
Pa =X 2π Pa→bc Q2 dz (2) processes, with our results, when we use an eikonal ap-
b,c
proximation in the cascade. Our results are shown in
where the sum runs over all allowed branchings. The Fig. 2, where we compare the contributions from inter-
α is α for branchings involving emission of a pho- actions involving at least one primary parton (triangles)
abc em
ton and α for the QCD branchings. The splitting ker- with the result obtained with full binary cascading (di-
s
nels Pa→bc are given in [8]. The collinear singularities amonds). Note that contribution from primary partons
in the showers are regulated by terminating the branch- does not reflect the momentum broadening (Cronin ef-
ings when the virtuality of the (time-like) partons drops fect) due to multiple soft scattering. The figure shows
to µ , which we take as 1 GeV. In principle, one could that most of the photons between 2 and 4 GeV have
0
3
100 increaseinthe photonemissionbyaboutafactor4. The
5 VNI/BMS withpartonradiation reason for this surprisingly dramatic effect is that the
partonand radiation
fragmentation leads to a large multiplication of partons
2
-2V)10-1 and thus to an enhanced photon yield to the increase in
e 5 the number of multiple scatterings. In comparison, the
G
( 2 increase in the photon yield due to the fragmentation
pt
d10-2 of photons off the scattered quarks (triangles) is small,
N/ 5 except below 1 GeV, filling up the artificial gap caused
d
p)t 2 by the cut-off pmTin in the binary collisions. The strong
10-3 enhancementinthelowpT regionisnotsurprising,since
2
1/( 5 radiativeprocessesgenerallyfavorlow-energyfinalstates.
2 The lower frame of Fig. 3 shows the rapidity distribu-
photons
10-4 tion of photons with p >2 GeV for the same two cases
5 |yCM|<0.5 as are discussed in thTe upper frame, but includes also
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
p (GeV) the distribution obtained when only binary interactions
t
are allowed (diamonds, corresponding to the uppermost
0.2
curveinFig.2). The enhancementofphotonproduction
Au+Au;E =200AGeV binarycollisions
CM
0.18 withpartonradiation due to the inclusion of fragmentation processes occurs
partonand radiation
0.16 mainly aroundmid-rapidity, where a dense partonic sys-
tem is created and most of the fragmentation processes
0.14
contribute. We chose the cut-off p > 2p0, because the
T T
0.12 photon yield below this value is expected to receive a
y
N/d 0.1 largecontributionfromhadronicreactions,whichdonot
d interest us here.
0.08
The strength of the source of single photons is often
0.06 discussed in terms of the ratio (γ/π0), as most of the
0.04 photons backgroundtothemcomesfromthedecayofπ0. Wecan
0.02 pt>2GeV not geta precise value for this unless we supplement our
descriptionwithamodelforthehadronization. Areason-
0.0
-6 -4 -2 0 2 4 6 able alternative is provided by the ratio of (γ/partons)
y
as the multiplicity of pions will be proportional to the
multiplicity of partons. We find that this quantity is of
FIG. 3: Upper frame: photon transverse momentum spectra the order 0.04% when only multiple scatterings among
for contributions from collisions and fragmentations (trian-
partons is considered and rises to 0.15% when fragmen-
gles) and from collisions alone (squares), when parton frag-
tations are allowed and which considerably enhance the
mentations are included in the calculations. Results for the
multiple scatterings.
calculationswithonlythebinarycollisions arealsogiven(di-
amonds). Lower frame : The multiplicity density of photons The results shown so far do not account for the pos-
havingp > 2 GeV. sible LPM suppression of radiation from the scattered
T
partons. An accurate implementation of this effect in a
semi-classic treatment is difficult. To explore the pos-
sible magnitude of this effect, we consider here a treat-
their origin in the multiple semi-hard scattering of par- ment in which the scattered parton, having energy E
a
tons. Thislendssupporttoarecentpredictionthatthere and time-like virtuality m2, is forbidden to fragment if
a
would be a substantial production of photons from the it re-scatters before the formation time of the radiated
passage of high energy quarks through the QGP which parton- E /m2 has expired. This ansatz should provide
a a
may be produced in such secondary collisions [13]. We us with an upper limit of the suppression as it does not
findtheaveragesquaredtransversemomentumofthein- account for the relative phases of the two scatterings.
coming rescattered partons at the photon vertex to be We find that the emission of photons at about 2 GeV
p2 2.1 (GeV/c)2, confirming previous phenomeno- is reduced by a factor of 3.5 if the LPM suppression is
h ti ≈
logical analyses of nuclear broadening effects in prompt treated in this manner. Most of the enhancement due
photon production [14]. to fragmentation processes visible in Fig. 3 disappears
Wenextallowforfragmentationprocessesafterbinary in this limit. Also, increasing the pQCD cut-off (pmin)2
T
interactions as discussed above. The lower set of points to 1 GeV2 reduces the number of binary collisions and
intheupperframeofFig.3(squares)includesonlyQCD thebuild-upofcascadingactivity. Thenetresultisade-
induced fragmentations of partons. A comparison with creaseinthephotonyieldbyafactorofabout1.5beyond
Fig. 2 (diamonds) shows that this already leads to an p =1 GeV near central rapidities.
T
4
functions. We intend to study these effects in a forth-
Au+Au;E =200AGeV
CM
coming publication.
5.0
binarycollisions
Insummary,wehavecalculatedtheproductionofhigh
4.5 withfinalstaterad.
Npart4/3 energy photons in collision of gold nuclei at 200 A GeV
4.0
inthePartonCascadeModel. Multiplescatteringofpar-
3.5 tons is seen to lead to a substantial production of high
d
el3.0 energy photons, which rises further when parton mul-
yi tiplication due to final state radiation is included. The
n2.5
hoto2.0 pphorottioonnaylietlodtihsefonuunmdbteorsocaflheaarsdNpp4aa/rr3ttoann-pdairstdoinreccotlllyispiornos-
p
1.5 inthesystem,providingvaluableinformationonthepre-
1.0 equilibrium reactiondynamics of the system. Our calcu-
lations suggest that that at the much higher energies to
0.5
VNI/BMS bereachedattheLargeHadronCollider,thepartoncas-
0.0 cades developing in the nuclear collision will produce an
50 100 150 200 250 300 350 400
Npart even larger “firework”of photons.
This work was supported in part by DOE grants
FIG.4: IntegratedphotonmultiplicityasafunctionofNpart. DE-FG02-96ER40945andDE-AC02-98CH10886andthe
Squares denote a calculation with only binary parton-parton
Natural Sciences and Engineering Research Council of
collisionswhereasdiamondsshowacalculationincludingfrag-
mentation processes. The scaling with N4/3 is indicative of Canada.
part
thephotonyieldbeingdirectlyproportionaltothenumberof
hard collisions.
∗
on leave from: Variable Energy Cyclotron Centre, 1/AF
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