How to Measure Specific Heat Using Event-by-Event Average p T Fluctuations

DNP 2005 Maui M. J. Tannenbaum 1/18

How to Measure Specific Heat How to Measure Specific Heat Using Event-by-Event Using Event-by-Event

Average pAverage pTT Fluctuations Fluctuations

M. J. TannenbaumBrookhaven National Laboratory

Upton, NY 11973 USA

Division of Nuclear Physics Meeting 2005 Maui, Hawaii September 20, 2005

PHENIX Collaboration


Something New: cSomething New: cVV/T/T33

• R. Gavai, S.Gupta and S. Mukherjee, hep-lat/0412036, PRD 71, 074013 (2005) predict in “quenched QCD” at 2Tc and 3Tc that cV/T3 differs significantly from the ideal gas. Can this be measured?


It is generally agreed that It is generally agreed that ccVV is related to temperature fluctuations is related to temperature fluctuations

• Ntot is the total number of particles on an event

• the ``temperature’’ T varies event-by-event with T and T.

• R.Korus, St.Mrowczynski, M.Rybczynski, Z.Wlodarczyk, PRC 64, 054908 (2004).

• Also see: L. Stodolsky, PRL 75, 1044 (1995) , S.A.Voloshin, V.Koch, H.G.Ritter, PRC 60, 024901 (1999).

• For a more nuanced (i.e. complicated) view see M. Stephanov, K. Rajagopal, E. Shuryak, PRD 60, 114028 (1999)


p < 2

p=2p > 2

dN/x

dx

x =

Inclusive pInclusive pTT spectra are Gamma Distributions spectra are Gamma Distributions note: dN/xdx is shown. p shown is for dN/dx

• This is inclusive, averaged over all events. T=1/b is the Temperature parameter.

x =pT


Event-by-Event Average pEvent-by-Event Average pTT

• If all the pTi on all events are random samples of the same distribution:

• For events with n charged particles of transverse momentum pTi, the event-by-event average transverse momentum is defined:

•By its definition <MpT>=<pT> but you must work hard to make sure that your data have this property to <<< 1%.


What e-by-e tells you that you don’t What e-by-e tells you that you don’t learn from the inclusive averagelearn from the inclusive average

• A nice example I like is by R.Korus, St.Mrowczynski, M.Rybczynski, Z.Wlodarczyk, PRC 64, 054908 (2004).

• Suppose the temperature T~1/b varies event-by-event with T and T:

• Also see: L. Stodolsky, PRL 75, 1044 (1995) , S.A.Voloshin, V.Koch, H.G.Ritter, PRC 60, 024901 (1999).


PHENIX MpPHENIX MpTT vs centrality vs centrality

200 GeV Au+Au 200 GeV Au+Au PRL PRL 9393, 092301 (04), 092301 (04)

• compare Data to Mixed events for random.

• Must use exactly the same n distribution for data and mixed events and match inclusive <pT> to <MpT>

• best fit of real to mixed is statistically unacceptable

• deviation expressed as:

FpT= MpTdata / MpTmixed -1 ~ few %

MpT (GeV/c)

MpT (GeV/c)


Measures of non-random fluctuationsMeasures of non-random fluctuations

random means pT of all particles on all events are independent samples of the inclusive pT distribution (averaged over all events). non-random is the difference between measured and random.

PHENIX

STAR

CERES

NA49 Mroczynski


For small non-random/randomFor small non-random/randomAll measures are equivalentAll measures are equivalent


Fluctuation is a few percent of Fluctuation is a few percent of MpMpT T ::

Interesting variation with N Interesting variation with Npartpart and p and pTmax Tmax

n >3 0.2 < pT < 2.0 GeV/c 0.2 GeV/c < pT < pTmax

PHENIX nucl-ex/0310005 PRL PRL 9393, 092301 (2004), 092301 (2004)

Errors are totally systematic from run-run r.m.s variations


Npart and pNpart and pTTmaxmax dependences explained by jet dependences explained by jet

correlations with measured jet suppressioncorrelations with measured jet suppression

20-25% centrality

Other explanations proposed include percolation of color strings E.G.Ferreiro, et al, PRC69, 034901 (2004)


Assuming all fluctuations are from Assuming all fluctuations are from TT//TT Very small and relatively constant with Very small and relatively constant with ssNNNN

T/T

CERES tabulation H.Sako, et al, JPG 30, S1371 (04) QM2004

Where is the critical point?


22TT//TT22Specific HeatSpecific Heat

• Korus, et al, PRC 64, 054908 (2001) discuss specific heat:

• n represents the measured particles while Ntot is all the particles, so n/Ntot is a simple geometrical factor for each experiment



• Gavai, et al, hep-lat/0412036 call this same quantity cV/T3

• These formulas with cvcV/T3 agree with Eq 9. in Korus, et al.



• R. Gavai, S.Gupta and S. Mukherjee, hep-lat/0412036, PRD 71, 074013 (2005) predict in “quenched QCD” at 2Tc and 3Tc that cV/T3 differs significantly from the ideal gas. Can this be measured?

• In PHENIX publication, PRL PRL 9393, 092301 (2004), 092301 (2004), n/Ntot~1/33, so FpT

~ 0.2% for cV/T3~15. This may be possible if we go to low pTmax out of the region where jets contribute.


Worth TryingWorth Trying

0.2 GeV/c < pT < pTmax

ccVV/T/T33~15~15

PRL PRL 9393, 092301 (2004), 092301 (2004)

• Concentrate on pTmax < 0.6 GeV/c where jets have least effect

• Error is totally systematic---run by run variation---can be improved.


IssuesIssues

• at fixed centrality, test for is FpT n, i.e. independent of n. Increase n by increasing solid angle, e.g. PHENIX vs STAR.

€

T2 / T

2


What We Have LearnedWhat We Have Learned• In central heavy ion collisions, the huge correlations in p-p collisions are washed out. The remaining correlations are:

Jets Bose-Einstein correlations Hydrodynamic flow

• These correlations saturate the fluctuation measurements. No other sources of non-random fluctuations are observed. This puts a severe constraint on the critical fluctuations that were expected for a sharp phase transition but is consistent with the present expectation from lattice QCD that the transition is a smooth crossover.

• In order to see temperature fluctuations predicted by cV/T3~15 in lattice gauge calculations, present sensitivity needs to be improved by an order of magnitude by removing other known sources of correlation and improving the measurement errors.


Some DetailsSome Details


Statistics--What you have to rememberStatistics--What you have to remember

• The mean and standard deviation of an average of n independent trials from the same population obey the rules:

where is the mean and x (or ) is the standard deviation of the population x .


Variance of MpVariance of MpTT

• If all the pTi are random samples of the same distribution:

• If the pTi are correlated, we find an identity with the pT-pT correlator variable of Voloshin, et al PRC 60, 024901 (1999)


A)---Detail of MpA)---Detail of MpTT Calculation Calculation


B)--Variance of MpB)--Variance of MpTT for T fluctuation for T fluctuation


<N<Ntottot>/<n> for PHENIX>/<n> for PHENIX


BACKUPBACKUP


NA49-First Measurement of MpNA49-First Measurement of MpTT distribution distribution

NA49 Pb+Pb central measurement PLB 459, 679 (1999)

• Points=data; hist=mixed; minimal, if any, difference

• Very nice paper, gives all the relevant information


Statistics at Work--Analytical Formula for MpStatistics at Work--Analytical Formula for MpT T for statistically independent Emissionfor statistically independent Emission

It depends on the 4 semi-inclusive parameters: b, p of the pT distribution (Gamma) <n>, 1/k (NBD), which are derived from the quoted means and standard deviations of the semi-inclusive pT and multiplicity distributions. The result is in excellent agreement with the NA49 Pb+Pb central measurement PLB 459, 679 (1999)

See M.J.Tannenbaum PLB 498, 29 (2001)


0-5 % Centrality

Black Points = Data

Blue curve = Gamma distribution derived from inclusive pT spectra

It’s not a Gaussian…it’s a Gamma distribution!

“ “Average pAverage pTT Fluctuations” Fluctuations”

PHENIX

From one of Jeff Mitchell’s talks:


Mortadella-NYTimes 2/10/2000Mortadella-NYTimes 2/10/2000

How to Measure Specific Heat Using Event-by-Event Average p T Fluctuations

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