Resonances and fluctuations at SPS and RHIC
a r X i v :n u c l -t h /0702020v 1 6 F e
b 2007EPJ manuscript No.
(will be inserted by the editor)
Resonances and ?uctuations at SPS and RHIC
Giorgio Torrieri
McGill University
Received:29July 2006/Revised version:date Abstract.We perform an analysis of preliminary data on hadron yields and ?uctuations within the Statis-tical hadronization ansatz.We describe the theoretical disagreements between di?erent statistical models currently on the market,and show how the simultaneous analysis of yields and ?uctuations can be used to determine if one of them can be connected to underlying physics.We perform such an analysis on preliminary RHIC and SPS A-A data that includes particle yields,ratios and event by event ?uctuations.We show that the equilibrium statistical model can not describe the K/π?uctuation measured at RHIC and SPS,unless an unrealistically small volume is assumed.Such small volume then makes it impossible to describe the total particle multiplicity.The non-equilibrium model,on the other hand,describes both the K/π?uctuation and yields acceptably due to the extra boost to the π?uctuation provided by the high pion chemical potential.We show,however,that both models signi?cantly over-estimate the p/π?uctuation measured at the SPS,and speculate for the reason behind this.PACS.2 5.75.-q,24.60.-k,24.10.Pa 1Introduction One of the main objectives of heavy ion physics is to study the collective properties of QCD matter.It’s equation of state,transport coe?cients and phase structure,and the dependence of these on energy and system size.Thus,the natural approach to study soft particle pro-duction in heavy ion collisions is through statistical me-chanics techniques.Such an approach has a long and il-
lustrious history [1,2,3,4].A consensus has developed that
the statistical hadronization model can indeed ?t most or
all particles for AGS,SPS and RHIC energies [5,6,7,8,9,
10,11].
The statistical model obtains particle yields by as-
suming entropy to be maximized given the constraints
imposed by energy and quantum number conservation.
These constraints can either be imposed rigorously,as
required for closed equilibrated systems,or on average,
as required for a sub-system equilibrated with an unob-
served “bath”.Full energy and quantum number conser-
vation is usually referred to as the micro-canonical ensem-
ble,while the Canonical (C)and Grand-canonical (GC)
ensembles assume that,respectively,energy and other con-
served quantities can vary via system-bath exchange.In
this work,we shall concentrate on the GC ensemble,as we
see it as most appropriate for describing the statistically
hadronizing ?reballs produced in heavy ion collisions.
Our approach is not universally agreed on by the heavy
ion community;In fact,noteworthy attempts were made
to explain the dependence of certain observables w.r.t.
en-
2Giorgio Torrieri:Resonances and?uctuations at SPS and RHIC tures.The present work is a step in this direction,focus-
ing on a Grand Canonical analysis of top energy RHIC
(
√s=17.6
GeV/A Pb-Pb collisions)heavy ion data.
Yields and?uctuations in the GC ensemble are calcu-
lated via the quantum statistic“ideal gas”formulae
N i = g i V
λ?1i e
√
?λi N i (2)
Where g i is the degeneracy,V the system volume,the up-per sign is for Fermions and the lower sign for bosons.It is sometimes confusing that states in a“strongly inter-acting”system can be accurately predicted via the ideal gas ansatz.The key insight[4]is that accounting for all strong excitations(resonances)is equivalent to counting the QCD“energy levels”.Since hadronic interactions are resonance dominated(a consequence of the con?ning na-ture of QCD)this is a good https://www.sodocs.net/doc/0d1330167.html,ttice studies have con?rmed the validity of this approach quantitatively [20].
The?nal state yield of particle i is computed by adding the direct yield and all resonance decay feed-downs.
N i tot= N i + all j→i B j→i N j (3) (?N i)2 tot= (?N i)2 + all j→i B2j→i (?N j)2 (4)
+ all j→i B j→i(1?B j→i) N j
The parameterλi corresponds to the particle fugacity,re-lated to the chemical potential byλi=eμi/T.Provided the law of mass action holds,it should be given by the product of charge fugacities(?avor,isospin etc.).It can be parametrized it in terms of equilibrium fugacitiesλeq i and phase space occupanciesγi.A hadron i with q(
s)strange quarks and isospin I3has fugacity
λi=λeq iγq+s
s ,λeq i=λq?s
s
λI3
I3
(5)
The temperature and chemical potentials can be obtained from data by doing aχ2?t[21](The chemical potentials for strangeness,λs,and isospin,λI3,are usually obtained by requiring strangeness and Q/B to be conserved).
If the system is in chemical equilibrium then detailed balance requires thatγq=γs=1.Assumingγq=1and ?tting ratios gives the T～160?170MeV freeze-out tem-perature typical of chemical equilibrium freeze-out models at SPS and RHIC[6,7,8].Some?ts[7,11]also allow for a kinetically out of equilibrium strangeness quantum num-ber,which is found to be at chemical equilibrium(γs=1) at RHIC and slightly below equilibrium(γs<1)at SPS.
In a system expanding and undergoing a phase tran-sition,however,the condition of chemical equilibrium no longer automatically holds,so one has to allow for the possibility that bothγs andγq=1.In particular,if the expanding system undergoes a fast phase transition from a QGP to a hadron gas,chemical non-equilibrium[5]and super-cooling[22]can arise due to entropy conservation: By dropping the hadronization temperature to～140MeV and oversaturating the hadronic phase space above equi-librium(γq～1.5,γs～2),it is possible to match the entropy of a hadron gas with that of a system of nearly massless partons[5].These are exactly the values found forγq and T at SPS and RHIC in?ts whereγq was a?t parameter[9,10].
The“γq=1”and“γq?tted”approaches are di?erent models of how hadrons are produced in heavy ion colli-sions.They are both based on statistical mechanics,yet di?er in the physics underlying it.They are,in principle, distinguishable experimentally but have not been to date.
Fig.1in[9]illustrates why an experimental test of either of these models is non-trivial:An increase in tem-perature acts in a very similar way on particles and anti-particles as an increase inγ:The abundances of both go up.The amount in which relative particle abundances go up is di?erent,making the two models distinguishable by yields alone.However,this di?erence is not enough to con-vincingly disentangle the two models,given the large ex-perimental errors in measurements of yields and ratios.
To falsify one of the models it is necessary to con-sider event-by-event?uctuations as well as average parti-cle yields[23].Ifγq becomes as large as claimed in[9],the pion chemical potentialλπapproaches e mπ/T.Near this limit(corresponding to the critical density for pion B-E condensation)it can be shown[23]that Nπ converges but (?Nπ)2 diverges as
(?N)2 ～??1/2.(6)
where?=1?λπe?mπ/T.thus,while T andγq are cor-related in yields,they are anti-correlated in?uctuations. The measurement of a yield and?uctuation constrains both to a high precision.At least one of the models will be ruled out by inclusion of a?uctuation in the?t.Inclu-sion of more than one?uctuation can be used to test the remaining model.
The question of chemical equilibration is is closely re-lated to the accounting for hadronization when modeling freeze-out.While a quantitative treatment of this topic is still lacking,it is obvious that if the freeze-out temperature is higher(i.e.,γq=1),the e?ect of hadronic interactions between chemical and thermal freeze-out is greater than if the system freezes out from a super-cooled state(γq>1).
It is also unclear what,if any,is the e?ect on ob-servables of the evolution between hadronization(the for-mation of hadrons as e?ective degrees of freedom)and freeze-out(the moment when all hadrons decouple)[24]. As shown in[25],the discrepancy with experiment would lessen if chemical and kinetic freeze-out were to coincide. While,as claimed in[25],such a high freeze-out tem-perature would spoil agreement with particle spectra,?ts based on a single freeze-out model prescription have shown that,provided a correct treatment of resonances is main-
Giorgio Torrieri:Resonances and ?uctuations at SPS and RHIC 3
tained,particle spectra are compatible with simultaneous freeze-out [26,27,28].A way to distinguish between these scenarios is to di-rectly
measure
the abundance of resonances.Here,the sit-uation becomes even more ambiguous:As pointed out in [29],resonance abundance generally depends on two quan-tities:m/T ,where m is resonance mass and the chemical freeze-out temperature,as well as τΓ,where Γis the res-onance width and τis the reinteraction time.Observing
two ratios where the two particles have the same chemical
composition,but di?erent m and Γ,such as Λ(1520)/Λvs
K ?/K ,or Σ(1385)/Λvs K ?/K ,could therefore be used
to extract the magnitude of the freeze-out temperature
and the re-interaction time.Studies of this type are still
in progress;As we will show,Λ(1520)/Λand K ?/K seem
to be compatible with sudden freeze-out,provided freeze-
out happens in a super-cooled over-saturated state [5,9,
10].Other preliminary results,such as ρ/π,?/p [8],and
now Σ?/Λ[31]seem to be produced in excess of the statis-
tical model [8,30,31],both equilibrium and not.It is di?-
cult to see how a long re-interacting phase would produce
such a result:Resonances whose interaction cross-section
is small w.r.t.the timescale of collective expansion would
generally be depleted by the dominance of rescattering
over regeneration processes at the detectable (on-shell)
mass range.More strongly interacting resonances would
be re-thermalized at a smaller,close to thermal freeze-
out temperature.Both of these scenarios would generally
result in a suppression,rather than an enhancement,of
directly detectable resonances.Transport model studies
done on resonances generally con?rm this [32,33].Yet the
only resonance,so far,found to be strongly suppressed
w.r.t.expectations is the SPS Λ(1520)[34],and even that
measurement is to date preliminary.
The observation of an enhanced μ+μ?continuum around
the ρpeak [35]has been pointed to as evidence of ρbroad-ening,which in turn would signify a long hadronic re-
interaction phase [35].The absence of a broadening in
the nominal peak itself,prevents us from considering this as the unique interpretation of experimental data.More-over,even a conclusive link of broadening with hadronic re-interactions would still give no indication to the length of the hadronic rescattering period.Nor it would resolve the discrepancies pertaining hadronic resonances encoun-
tered in the previous paragraph;The lack of modi?cation in either mass or width,between p-p and Au-Au seen so far [30,31]is only compatible with the NA60result pro-vided the ρis very quick to thermalize,so [30]sees only the ρs formed close to thermal freeze-out.But,as argued in the previous paragraph,that under-estimates the abun-dance of the ρand other resonances,since the ρ/π,K ?/K ,and even Λ(1520)/Λratios point to a freeze-out tempera-ture signi?cantly above the 100MeV,commonly assumed to be the “thermal freeze-out”temperature in a staged freeze-out scenario.To help resolve this ambiguity,we aim to directly in-fer the magnitude of hadronic reinteraction by combining,
within the same analysis,an observable sensitive to the chemical freeze-out resonance abundance with the direct observation of resonances.As shown in [36],the measure-ment of ?uctuations of a ratio is such an observable.
The ?uctuation of a ratio N 1/N 2can be computed from the ?uctuation of the denominator and the numera-tor [36](σ2X = (?X )2 / X ):
σ2N 1/N 2= (?N 1)2 N 2 2?2 ?N 1?N 2 σ2?σ2stat (9)σstat ,usually obtained through a Mixed event approach [37],includes a baseline Poisson component,which for a ratio N 1/N 2can be modeled as
σ2stat =1 N 2 (10)as well as a contribution from detector e?ciency and kine-matic cuts.Provided certain assumptions for the detec-tor response function hold (see appendix A of [37]),sub-tracting σstat from σshould yield a “robust”detector-independent observable.More complicated to deal with are detector acceptance e?ects a?ecting particle correlations (the probability for both resonance decay products to be within the detector acceptance region)[23].These were not corrected for in the present study;RHIC data-points do not include this term,while at SPS,due to it’s large acceptance,this term
is likely to be less important.
Related to the time-scale of the thermal freeze-out is the question of the total normalization of the system.This quantity can be obtain in ?ts by ?tting yields of particles
4Giorgio Torrieri:Resonances and ?uctuations at SPS and RHIC
rather than ratios.It is physically important because it
is connected to the system’s volume at chemical freeze-
out,which is in turn important to gauge the relative im-portance of the hadronic phase in the system’s
dynamics
[24].It is also necessary to obtain the thermal energy and entropy content of the system [9],and it’s scaling with
centrality.
On the other hand,normalization introduces an addi-
tional source of inter-parameter correlation,since it scales
in the same way as T and γq .Measuring ?uctuations of
ratios in addition to yields and rations,once again,can be
used to disentangle these three quantities.Note,from Eqs.
7and 10that ?uctuations of ratios,while independent of
the volume ?uctuation scale as the inverse of the absolute
normalization,σ
N 1/N 2～( V T 3)?1.Hence an increase in volume makes the yields go up,
does not a?ect the average ratios,but decreases the event-
by-event ?uctuation of ratios .An increase in temperature
generally a?ects both yields and ratios,increases the yields
and decreases the ?uctuations.An increase in γ
q increases
yields,baryon to meson ratios and ?uctuations.It is there-
fore not di?cult to ?x all three parameters with a data-
sample of relatively few data-points,provided both yields,
ratios and ?uctuations are present.
2Fit to RHIC and SPS data
We have performed ?ts for both SPS and RHIC energies,
using publicly available statistical hadronization software
[38,39].For RHIC,we have used the same data sample as
in [40],including σdyn K +/π+and σdyn
K ?/π?[41].For SPS yields,
we used the same data sample as in [9],augmented by pre-
liminary ?uctuation measurements of σdyn
(K ++K ?)/(π++π?)
and σdyn
(p +
s =
0±0.01),charge and baryon number ( Q / B =Z/A ±
0.01)conservation.
The results of the analysis are shown in table 1and Fig.
1.An equilibrium analysis (γq =1,empty blue squares in
Fig.1)?ts the particle yields and ratios but under-predicts
σdyn K/πby many standard deviations.If σdyn
K/πis ?tted to-
gether with ratios (red triangles down in Fig.1)but no
yields,it forces the system volume (all other parameters
being the same within error)to be unrealistically small
(～500fm 3at RHIC,～1000fm 3at SPS),thereby under-
predicting particle yields by several standard deviations.
It is only the addition of γq (circles in Fig 1)that allows
?uctuations to be driven to a high enough value while maintaining su?ciently high volume to describe the par-ticle multiplicities,and su?ciently high temperature to describe ratios.It is important to underline that both yields and ?uc-tuations contribute to such a precise determination:Equi-librium statistical models can describe most yields and ra-tios acceptably with γq =γs =1,but fail to describe the event-by-event ?uctuation.Conversely,transport models provide an acceptable description of event-by-event ?uc-tuations [43],but fail to describe the yield of multi-strange particles [44].Unlike what is sometimes asserted,the Λ(1520)and K ?are acceptably described by the statistical model at RHIC.The strongest disagreement arises from Σ?under-prediction,at the level of 1.5standard deviations.We await for more measurements of resonances such as ?and ρin central collisions before trying to interpret this under-prediction.The acceptable description of the Λ(1520)and K ?yield using the same freeze-out temperature as the stable par-ticles,and the under-prediction of the Σ?,makes a case for the proposition that the re-interaction period between hadronization and freeze-out might be not as signi?cant as generally thought.However,the current data is not ca-pable to rule out such a long-reinteraction period,since the crucial ?uctuations,of ratios correlated by resonances (e.g.σK +/π?dyn vs σK +/π?dyn )are still not available at RHIC and are not precise enough at for such a falsi?cation at SPS.It will also be interesting to see if the SPS resonance results (preliminary since 2001[34],shown in the plot but not included in the ?t)are con?rmed:Here,while the K ?abundance is acceptably described by both the equilib-rium and the non-equilibrium model,Λ(1520)is consid-erably over-predicted by both through the disagreement with the equilibrium model is larger,due to the higher freeze-out temperature.The discrepancies between this ?t,and earlier ?ts with γq [9]are due to data-set choice.The lower temperature,and higher γq at SPS is due to the Σ?,K ?and Λ(1520)resonance yields at RHIC that push for a higher tempera-ture,both in equilibrium and non-equilibrium.Until now,unfortunately,no published resonance result exists in SPS,although Λ(1520)and K ?results are available in confer-ence proceedings [34].These are displayed in Fig.1,but were not used in the ?t.Qualitatively,the only major di?erence between SPS and RHIC systems is the higher baryo-chemical potential (μB ),expected due to the higher initial transparency at the higher energy RHIC collisions.The introduction of γq as a ?t parameter makes the freeze-out temperature drop to a value compatible with the QGP super-cooling hypothesis,and decreases the volume by ～30%(a re-sult that goes in the right direction to explain the “HBT puzzle”[25]).Due to the very di?erent acceptances in the experiments at RHIC and SPS,a direct comparison of the volumes would not be meaningful.The main disagreement between the data and all mod-els is the over-prediction of σdyn
p/πat SPS.Other works [42]
Giorgio Torrieri:Resonances and ?uctuations at SPS and RHIC
5Table 1.Best ?t parameters at SPS Pb-Pb √
s =200GeV collisions RHIC Au-Au √
s =17.3GeV γq =1
γq =1T [MeV]159.6±6.9
151.3±3.8μB =3T ln(λq )[MeV]23.2±3.4
248.0±25.1μs =T ln(λq λ?1s [MeV] 5.1±0.8
60.5±9.3μI 3=T ln λI 3[MeV]-0.7±0.2
-8.8±2.1γq 1
1.γs 1.083±0.112
0.815±0.067Normalization (fm 3)1689±488
3746±44410?410?310
?210
?110
010
110
210
3?10?505σd y n (%
)10
?1100101102103?6?303
σd y n (%
)
Fig.1.(color online)Best ?t yields,ratios and ?uctuations for RHIC (left panel)and SPS (right panel).Green circles represent a ?t where γq was ?tted.Red triangles down,γq =1and yields not ?tted.Blue squares,γq =1and ?uctuation not ?tted.Red triangles,γq =1,normalization determined from the ?uctuation and yields not ?tted.The magenta diamond refers to a calculation with halved yields,as per Eq.11.The blue triangle up (right panel only)refers to enhanced ?production (m ?=m p )
have suggested that the low value of σdyn p/πis indicative
of a rich ?resonance abundance at chemical freeze-out.
The preliminary observation of an enhanced ?production
at RHIC [8]ties in well with this picture.However,this
study puts this conjecture in doubt:As can be seen from
the ?gure,even if one assumes that m ?=m p ,or ～80%
of the photons come from ?s (blue triangle up in the right
panel of Fig.1),it is still not enough to account for the
discrepancy.
It is intriguing that halving the ?uctuation term in Eq.
7σN 1/N 2=1 N 1 2+ (?N 2)2 N 1 N 2 .(11)produces a result compatible with observations (magenta diamond in Fig.1)for σdyn p/π.This is the scaling expected,in the thermodynamic limit,if the ensemble physically rel-evant for baryons were canonical rather than GC [17,18].The roughness of this estimation,and the preliminary na-ture of the data-points in question prevents us from draw-ing conclusions from the result.We also point out that if this scaling is applied to Kaons (suggesting C ensemble for strangeness)rather than protons (suggesting C ensem-ble for Baryon number),the model fails miserably at both
SPS and RHIC,as Fig.1shows.We eagerly await more complete and rigorous studies using the canonical ensem-ble [19]to further clarify these issues.In conclusion,we have used preliminary experimental data to show that,at both 200GeV Au-Au collisions and 17.6GeV Pb-Pb collisions,the equilibrium model is un-able to describe both yields and ?uctuations within the same statistical parameters.The non-equilibrium model,in contrast,succeeds in describing almost all of the yields and ?uctuations measured so far at SPS and RHIC,with the parameters expected from a scenario where non-equilibrium arises through a phase transition from a high entropy state,with super-cooling and oversaturation of phase space.
Some preliminary SPS data-points,such as the Λ(1520)yield and the σdyn p/π,are however signi?cantly over-predicted
in both equilibrium and non-equilibrium models.We await
more published data to determine weather the non-equilibrium
model is really capable of accounting for both yields and
?uctuations in all light and strange hadrons produced in
heavy ion collisions.
Work supported in part by grants from the Natural
Sciences and Engineering research council of Canada,the
Fonds Nature et Technologies of Quebec,and the Tom-
linson foundation.We would like to thank J.Rafelski,C.
6Giorgio Torrieri:Resonances and?uctuations at SPS and RHIC
Gale and S.Jeon for helpful discussions and continued sup-port.
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0.20.3
0.4
R e s o n a n c e s ?4
?2
2
4
σd y n (%
)
1 1.
2 1.4 1.6γq 00.2
0.4
0.6
0.8
1
P t r u
e