Constraints from $v_2$ fluctuations for the initial state geometry of heavy-ion collisions

0.26 MB·

by  Thorsten Renk

Constraints from v fluctuations for the initial state geometry of heavy-ion collisions
2
Thorsten Renk and Harri Niemi
Department of Physics, P.O. Box 35, FI-40014 University of Jyva¨skyla¨, Finland and
Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland
Theabilitytoaccuratelycomputetheseriesofcoefficientsvn characterizingthemomentumspace
anisotropiesofparticleproductioninultrarelativistic heavyioncollisions asafunctionofcentrality
is widely regarded as a triumph of fluid dynamics as description of the bulk matter evolution. A
key ingredient to fluid dynamical modeling is however the initial spatial distribution of matter as
createdbyayetnotcompletelyunderstoodequilibrationprocess. Ameasurementdirectlysensitive
to this initial state geometry is therefore of high value for constraining models of pre-equilibrium
4
dynamics. Recently, it has been shown that such a measurement is indeed possible in terms of the
1 eventbyeventprobabilitydistributionof thenormalized vn distributionasafunctionof centrality,
0
which is to high accuracy independent on the details of the subsequent fluid dynamical evolution
2
and hence directly reflects the primary distribution of spatial eccentricities. We present a study of

n thisobservableusingavarietyofGlauber-basedmodelsandarguethattheexperimentaldataplace
a very tight constraints on the initial distribution of matter and rule out all simple Glauber-based
J models.

9
PACSnumbers: 25.75.-q,25.75.Gz

]
h
t I. INTRODUCTION pactparameter[20, 21]. The event-by-eventfluctuations
-
l ofε ’sthentranslateintotheevent-by-eventfluctuations
c n
u Itisnowcommonlyagreedthatultrarelativisticheavy- of vn’s. Fluid dynamical calculations have established
n that the relation hv i=C ε , where the angular brack-
ion (A-A) collisions create a transient state of collective n n n
[ ets hi denote the average over many collisions with the
QCD matter. In modeling the dynamics of this droplet,
sameeccentricity,holdsverywell[22–25]. Forthesecond
1 the essential input to the models is an initial distribu-
v tion of matter density as createdin a yet not completely harmonics v2 and ε2 it has been found that the correla-
9 understood equilibration process [1]. One of the clearest tionisevenstrongerandarelationv2 =C2ε2 holdsaccu-
6 rately also in individual nuclear collisions [25], not only
signal of collective (or fluid dynamical) behavior of such
0 a system is the appearance of non-trivial patterns in the onaverage. Thismeansthatinagivencentralityclassε2
2
istheonlycharacteristicsoftheinitialconditionthatde-
. azimuthal distribution of final state hadron spectra [2].
1 Suchpatternsarecreatedbythefluiddynamicalresponse termines v2, while the proportionality coefficient C2 de-
0 pendsonthedetailsofthefluiddynamicalevolutionina
to pressuregradientswhich inturn aregivenby the geo-
4 complicatedway [26]. The simple relationmeans that in
1 metricshapeoftheinitialstate[3,4]. Theprecisedetails
: of this response then depend strongly on the transport relative fluctuations δv2 = (v2 −hv2i)/hv2i the propor-
v tionality coefficient cancels. Therefore, the probability
properties of the matter, e.g. shear viscosity [5–15].
i distribution P(δε ) is the same as the probability distri-
X The azimuthal asymmetries of the measured hadron n
bution P(δv ). In other words P(δv ) is determined by
n n
r momentum spectra are usually characterized by the set
a the properties of initial state alone and is independent
ofFouriercoefficientsv ,andsimilarlytheazimuthaldis-
n of the fluid dynamical evolution. Thus, by measuring
tributionofmatterinpositionspacecanbecharacterized
P(δv ) one gets an immediate access to the fluctuations
n
by its eccentricity coefficients ε . The determination of
n in the initial geometry [25].
the viscosity of the stronglyinteracting matter is largely
Recently, the event-by-event distributions of v have
based on measured v ’s. However, v ’s do not only de- n
n n been measured by the ATLAS [27] and ALICE [28] Col-
pend on the fluid dynamical response to ε ’s , but also
n laborations. Making use of the result P(δε ) = P(δv )
on the initial values of ε ’s. Therefore, it is essential n n
n we use different variants of the Monte-Carlo Glauber
thatthe rightinitial conditionis used: determining both
(MCG) model to calculate P(δε ), and compare with
the transport properties and the initial geometry simul- n
ATLAS data. Furthermore, we study the sensitivity of
taneously from the available data is a very complicated
the distributions to several assumptions underlying the
task[16,17]. Thus,findinganobservablethatissensitive
MCG model and its extensions.
to the initial geometry, but independent of the fluid dy-
namical response would simplify the task considerably.
Another phenomena where the detailed knowledge of
the initial geometrybecomes importantis jet quenching, II. THE MODEL
where the observed azimuthal jet suppression patterns
depend strongly on the assumed initial state [18, 19]. Wecomputethenormalizedfluctuationsofv byeval-
n
Inrealisticmodeling,the initialstate geometryfluctu- uating the spatial eccentricity ε of a set of randomly
n
atesfromonecollisiontotheanotherevenforafixedim- generated initial states for a given centrality class.
2
Westartbydistributingthepotentiallyinteractingob- then consider three different possibilities to initialize the
jects in the initial states of the colliding nuclei. In the initial entropy density,
default scenario, these are the nucleons, but in an alter-
s(x)=Nρbc(x)α, (3)
native constituent quark scattering (CQS) scenario, we
assumethatthesubstructureofnucleonsintermsofcon- s(x)=Nρwn(x)β, (4)
stituent quarks is the relevant level of description. s(x)=N[(fρwn(x)+(1−f)ρbc(x)], (5)
In the default case, we use a Woods-Saxon
where the parameters α, β and f are fixed to reproduce
parametrization of the measured nuclear charge density
the centrality dependence of the multiplicity, by assum-
[29] to distribute nucleons randomly in a 3-dim volume.
ingthatthefinalmultiplicityisproportionaltotheinitial
For Pb-nuclei as appropriate for the LHC, our distribu-
entropy. We have tested that scaling s → s4/3, corre-
tion is given by
sponding to the approximate difference between s and e
ρ (r)= ρ0 (1) scaling, does not change any of our results.
N
1+exp((r−c)/z) In the default scenario, we set N = 1, thus assuming
withc=6.61fmandthe skinthicknessz =0.51fm. We that the multiplicity created in each N-N collision is a
checkedthat a slightchangesin these parametersdo not constant. In real N-N collisions, the multiplicity fluctu-
affect our result significantly. We do not correct for the ates and the relative distribution of multiplicity around
nucleonhardcore,i.e.we permitconfigurationsinwhich the mean value exhibits a near universal behaviour, the
individual nucleons overlap in 3d space. After generat- so-called KNO scaling [30]. In order to account for this,
ing a 3d ensemble of nucleons, we project their position we also take into account a scenario where N is dis-
intotransverse(x,y)space. IntheCQSscenario,wedis- tributed according to the KNO distribution.
tributethreeconstituentquarksinsideaGaussianradius Thevalue ofσ ischaracteristicforthe interactionpro-
of 0.6 fm around the nominal position of each nucleon, cess,andreflectstheprecisephysicsofmatterproduction
thenprojectconstituentquarkpositionsinto(x,y)space. insecondaryinteractions. Generalconsiderationssuggest
In order to test alternative scalings, we also explore that it should be of the order of the nucleon radius. We
a Hard Sphere scenario (HS) in which we set the skin test in the following scenarios involving both constant
thickness parameter z = 0 and a Sheet (S) scenario in values σ = 0.6 fm, σ = 1.0 fm and a Gaussian distri-
which we mimick a strongly saturated picture in which bution of width ∆σ = 0.3 fm centered around σ = 0.6
we distribute nucleons a priori into a 2d circular surface fm.
bounded by the nuclear radius parameter c (in such a Theeventsaredividedintocentralityclassesaccording
picture, the center of the nucleus is as dense as the pe- to the total entropy, which is the closest to the central-
riphery). BothHSandScanbe combinedwiththe CQS ity selectionin the realexperiments. We further checked
scenario. thesensitivityoftheresultstothecentralityselection,by
Once the transverse position of the colliding objects consideringalsotheselectionaccordingtoimpactparam-
have been specified for two nuclei, we displace the two eter, number of collision participants or the number of
distributions by a randomly sampled impact parameter. binarycollisions. Itturnsoutthatnoneoftheseschemes
Collisions are evaluated according to a transverse dis- to determine centralitychangesourresults substantially,
tance criterion d2 < σ /π. In the case of nucleon- i.e. the details of centrality determination do not matter
NN
nucleon collisions we take σNN = 64 mb, in the case of for the question of vn fluctuations as long as we do not
interacting constituent quarks we use 1/9 of this value consider ultra-central events.
to get back to the same cross section in the case of p-p Given ρ(x,y), we compute the center of gravity of the
collisions. distribution and shift coordinates such that (0,0) coin-
There are four common ways in which EbyE hy- cides with the center of gravity. Next we determine the
drodynamics is commonly initialized. Matter can be angular orientation of the εn plane from
distributed either according to collision participants 2 2
1 dxdy(x +y )sin(nφ)ρ(x,y)
(wounded nucleon, WN) or according to binary colli- Ψ = arctan +π/n
n n R dxdy(x2+y2)cos(nφ)ρ(x,y)
sions(BC)andthematterdistributioncanbespecifiedin
terms of entropy s or energy density e, leading to sWN, R (6)
and the eccentricity of the event as
eWN, sBC and eBC scenarios. We note that none of
thesealoneisabletogiveacorrectcentralitydependence dxdy(x2+y2)cos[n(φ−Ψ )]ρ(x,y)
n
of the multiplicity. εn = R dxdy(x2+y2)ρ(x,y) (7)
For each event, we associate a binary collision and a
R
wounded nucleon density according to AveragingoveralargenumberO(20.000)ofevents,we
determine the mean eccentricity hε i for each centrality
Nbin/wn 1 x2 n
ρ(x)bin/wn = 2πσ2 exp(cid:18)−2σi2(cid:19) (2) celcacsesntarnicditeyxapsress the fluctuations in terms of the scaled
Xi=1
where x is the binary collision point or the position of ε −hε i
i δε = n n (8)
the woundednucleon,andσ isafree parameter. We will n hε i
n
3
where εn is the eccentricity determined for a particular • For central collisions, the scaled fluctuations in v2
event. become universal, i.e. show the same pattern inde-
pendent of the underlying geometry. For less cen-
tral events, differences between the four different
III. RESULTS scenarios become readily apparent.
A. Centrality dependence from the Glauber model • As evidenced by the differences between Glauber
and HS, the surface diffuseness of the nucleus is a
key parameter determining the width of the distri-
First, we test the centrality dependence of P(δε2) of
bution.
the different initial states given by Eqs. (3)–(5). The
distributions are shown in Fig. 1 for several centrality
• As indicated by the differences between HS and
classes and compared to the ATLAS data [27]. The val-
sheet,differencesincentraldensityarealsoprobed.
ues of the free parameters α, β and f are shown in the
This suggests a scenario in which the wide fluc-
figure, and we use σ =0.4 fm and N =1. We can make
tuations are driven by nucleons at the edge of
the following observations:
one nucleus, passing (for large impact parameters)
• In the mostcentralcollisions allthe different mod- throughthe centralregionofthe othernucleus,i.e.
els give the same distribution, and is in practice in what matters is both the probability to have a nu-
perfect agreement with the ATLAS data. cleonfarfromthecenterofnucleusAandtheeffect
of its passage through nucleus B. This would sug-
• WhileitwasobservedinRef.[25]thatthedistribu- gestthatanykindofsaturationgenericallynarrows
tions are same for sWN and sBC initializations at the width of the distribution as compared with an
RHIC, the same does not hold for the LHC energy unsaturated scenario.
due to the larger nucleon-nucleon cross-section. In
general sWN initialization at the LHC gives wider • The more realistic scenarios Glauber and CQS are
distributions than sBC initializations. This differ- closer to the data, but no scenario can account
enceisevenenhancedbythepowersinEqs.(3)and for the full centrality dependence. In particular,
(4)requiredtoreproducethecentralitydependence Glauber becomes too narrow above 20% centrality
of the multiplicity. and CQS is too wide between 10 and 25% central-
ity.
• Although the binary collision based initialization,
Eq.(3),givesagoodagreementwiththedatainthe
Wehavesimilarlystudiedthecentralitydependenceof
central collisions and the binary/participant mix-
v3 and v4 fluctuations, however these appear to follow
ture, Eq. (5), in the peripheral collisions, none of
the same generic scaling as observed for v2 in central
thesesimplemodelscanfullyaccountthecentrality
collisionsanddonotallowtodistinguishdifferentmodels.
dependence of the distributions.
These results raise the question if a modified version
of the Glauber scenario, for instance size scale fluctua-
tions or KNO multiplicity fluctuations could not bring
B. Initial nuclear geometry
the model in agreementwith the data. We explore these
possibilitiesfor5-10%centrality(wherethe Glaubersce-
Next, we aim at testing the assumptions for the ini-
nario gives a wider distribution than the data) and for
tialnucleargeometry. Inparticularwetestfourdifferent
35-40%centrality(where the width ofthe data is under-
scenarios across the whole centrality range: 1) a stan-
estimated).
dardMCGlauberscenariobasedonnucleonsdistributed
with a realistic Woods-Saxon nuclear density (Glauber),
2) a standard Glauber scenario based on scattering con-
C. Size scale and multiplicity fluctuations
stituent quarks instead (CQS), 3) a Glauber scenario
based on nucleons sampled from a hard sphere distri-
bution (HS) 4) a scenario mimicking strong saturation In Fig. 3, we againconsider the sBC Glauber scenario
effects in the initial density based on a 2d nucleon sheet and try variations of the parameter σ which represents
distribution(S).Inallthesecases,σ =0.6fmandN =1 thesizeofthematterspotgeneratedinanindividualN-N
isassumed. Here,weusesimplesBCmodel,withentropy collision in combination with possible KNO-type multi-
densitydirectlyproportionaltothe densityofthe binary plicity fluctuations. As the figure demonstrates, there
collisions. is no significantdependence oneither ofthese factors,in
The centrality dependence of the v2 or ε2 fluctuations particularnocombinationofparametersisabletoshrink
from the 0-5% most central to 35-40% peripheral colli- the distribution at central collisions while at the same
sionsis shownforthese fourdifferentscenariosandcom- time widen it at large centralities.
pared with ATLAS data in Fig. 2. Similar results (not shown here) can be obtained for
Several observations are readily apparent: the CQS scenario.
4
ATLAS ρα
0 BC
10
-1
10
0 5% 5 10%
− −
-2
10
0 ρWβN fρWN+(1−f)ρBC
10
)
-1
210
ǫ 10 15% 15 20%
δ − −
(
P 10-2
α=0.55
,
)2 100 β=1.65
v f=0.84
δ
( -1
P 10
20 25% 25 30%

− −
10-2

0
10

-1
10
30 35% 35 40%
− −
-2
10
−1.0 −0.5 0.0 0.5 1.0 1.5−1.0 −0.5 0.0 0.5 1.0 1.5
δǫ , δv δǫ , δv
2 2 2 2
FIG.1. Centralitydependenceofδv2orδε2fluctuationsinvariousscenariostogeneratetheinitialstatefromtheinitialnucleon
distributions.
D. Surface thickness • While P(δv2) is constrained to fulfill
dδv2P(δv2) = 1 and dδv2δv2P(δv2) = 0
In contrast, we demonstrate in Fig. 4 that there is a bRy construction, the shapRe is, given these con-
straints, free. Looking closely at Fig. 4, one may
characteristic dependence of the width of the P(δv2) on
notethatadifferentvalueofthesurfacediffuseness
the surface diffuseness assumed for the nuclear density
reproduces the left and side and the right hand
distribution Eq. (1) — the distribution widens with in-
side of the distribution, i.e. data and model do not
creased surface diffuseness and shrinks with decreased
match in shape.
surfacediffuseness. OfallinfluencestestedforaGlauber
model based on colliding nucleons, this is the only one
clearly leading to an effect above the statistical uncer-
tainty. IV. CONCLUSIONS
However, even assuming yet unknown physics allows
to make the surface diffuseness a free parameter, there
We calculated the centrality dependence of the ec-
are two further obstacles:
centricity fluctuation spectra from several MC Glauber
model based initial states. First, we found that the v2
• The surfacediffuseness alwayscorrelatespositively fluctuations are universal in the most central collisions,
with the width of P(v2). However, the central- i.e. independent of the model details, and well described
ity dependence of the mismatch between data and by all the models. The same holds for the higher har-
model is non-trivial, i.e. in order to account for monicsinallthe centralityclasses. However,noneofthe
the data one would have to assume that the sur- models tested here were able to reproduce the centrality
facediffusenessofanucleus(whichisapropertyof dependenceofP(δv2)observedbytheATLASCollabora-
the particular nucleus) depends on atwhat impact tion. In particular, a simple mixture of binary collisions
parameter that nucleus will later collide, which is and wounded nucleons fails to reproduce the fluctuation
conceptually very problematic. spectra, except in the most peripheral centrality classes
5
1
ATLAS Glauber
0.1
0-5% 5-10%
0.01
1
) CQS HS
2
ε
δ 0.1
(
P 10-15% 15-20%
0.01
1
) sheet
2
v
δ 0.1
(
P 20-25% 25-30%
0.01
1
0.1
30-35% 35-40%
0.01
-1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 1.5 2
δ δε δ δε
v , v ,
2 2 2 2
FIG. 2. Centrality dependence of δv2 or δε2 fluctuations in various scenarios to generate the initial distributions inside the
nuclei (see text).
5-10% centrality 35-40% centrality
100 100
ATLAS data ATLAS data
σ = 0.6 fm σ = 0.6 fm
10 σ = 1.0 fm 10 σ = 1.0 fm
σ = 0.6 fm, KNO σ = 0.6 fm, KNO
σ = 0.6 fm, ∆σ = 0.3 fm σ = 0.6 fm, ∆σ = 0.3 fm
)2 σ = 0.6 fm, ∆σ = 0.3 fm, KNO )2 σ = 0.6 fm, ∆σ = 0.3 fm, KNO
δv 1 δv 1
( (
P P
), ),
2 2
ε0.1 ε0.1
δ δ
( (
P P
0.01 0.01
0.001 0.001
-1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2
δε , δv δε , δv
2 2 2 2
FIG. 3. Dependenceof δv2 fluctuations on multiplicity or N-Ncollision geometry size scale fluctuations.
considered here. These findings suggest that the geometrical fluctua-
tions in the positions of the nucleons are not enough to
We also identified several parameters that do not af-
explain the data, but some non-linear dynamics in the
fect the distribution, like KNO fluctuations and the size
creation of the matter and/or additional sources of fluc-
of the matter spots generated in the individual NN col-
tuations arenecessary. Bothof these propertiesarereal-
lisions. We further demonstrated that the distributions
izedintheQCDbasedinitialstatemodels. Forexample,
are sensitive to simple non-linear parametrizationsgiven
pQCD+saturationmodelinRef.[31,32]leadstoasim-
byEqs.(3)and(4),aswellasbythechangesintheinitial
ilarnon-linearbehavioroftheentropydensityasEq. (3)
distributions of the interacting objects.
6
35-40% centrality and the sub-nucleon color fluctuations in Refs. [33] pre-
sumably lead to a similar effect on the distributions as
100
the CQS model above.
ATLAS data
z = 0.51 fm Overall, reproducing the observed centrality depen-
10 z = 0.612 fm
z = 1.02 fm dence of the v2 fluctuation distributions is a non-trivial
task and gives very tight constraints for the modeling of
)
2
δv 1 theinitialstate. Allthesimplemodelsconsideredinthis
(
P workcanalreadybe ruledoutasvalidrepresentationsof
), the initial state geometry.
2
ε0.1
δ
(
P
0.01
0.001 ACKNOWLEDGMENTS
-1 -0.5 0 0.5 1 1.5 2
δε , δv
2 2
We thank G. Denicol for useful discussions. This
FIG. 4. Dependence of δv2 fluctuations on the assumed sur- work is supported by the Academy researcher program
face diffuseness parameter in the Woods-Saxon model of the of the Academy of Finland, Project No. 130472 and the
nuclear density distribution. Academy of Finland, Project No. 133005.
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