Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting,...

Net-p Moments Products

Click to edit Master subtitle style

Plenary Session, STAR Analysis Meeting, July, 2013 1

Speculated 1st order phase boundary

M. Stephanov, Rice Workshop, May 23-25, 2012

We study the moments products of “Net-protons”as they might be a proxy for baryon number

If so…Experimentally-measured moments products may be directly related to the susceptibility ratios (QCD order parameters) from the lattice. Values may relate to HG vs QGP phases…

In the NLSM, experimentally-measured moments products may also be proportional to powers of the correlation length. (critical opalescence)Divergent values may indicate a Critical Point…

Measure the shapes of multiplicity distributions asquantified by the moments: μ, σ2, S, K…

Products Sσ and Kσ2 less volume-dependent…

Net-p Moments ProductsW.J. Llope, Rice UniversityIanalysis meeting, July 2013




STAR net-p Results at QM2012

√sNN <mB>*

7.7 421

11.5 316

19.6 206

27 156

39 112

62.4 73

200 24 * C

leym

ans

et a

l. P

RC

73,

034

905

(200

6)

No strong non-monotonicity seen …. But what is the significance of the apparent dip?!?

net-protonsX. Luo for STAR, QM2012

UrQMD simulation

Poisson Statistics




Significance of net-p dip near 19.6 GeV?

J. Nagle, last talk at QM2012

what the NLSM would actually expect for a CP at √sNN~15 GeV




Baselines

At QM2012, the ``baselines” we provided werepoisson statistics – the simplest possible baseline. URQMD – a transport model.

While those data were also consistent with a straight-line fit, the appearance ofstatistically-significant deviations resulted in some provocative inferences/discussions.___________________________________________________________________

We definitely need more events, with “iTPC” (& new forward tracking?) BES Phase-IIincreased efficiency, increased h reach, better centrality resn, better low-pt PID…

We all also very much look forward to the 15 GeV Au+Au to come in Run-14! squarely in the middle of a wide (110 MeV) gap in mB near ~260 MeV…

___________________________________________________________________

What can we learn about our existing data using different sorts of baselines?

(N)BD - Negative Binomial/Poisson/Binomial (depending on μ<=>σ2)

Sampled Singles - Data-driven, breaks intra-event correlations via sampling…

Will focus on efficiency-uncorrected results here. Efficiency-corrected results are also in hand.




Negative Binomial / Poisson / Binomial

T..J. Tarnosky & G. Westfall, Oct. 2012 http://arxiv.org/pdf/1210.8102v1.pdf

Functional form describes the (particle identified) multiplicity distributions ranging from NA22 & UA5 to PHENIX

Inputs:mean (μ) &variance (σ2)

Then, the values of Ck , Sσ , & Kσ2 are predicted.

μ<σ2 …NBDμ=σ2 …Poissonμ>σ2 …BD




Sampled Singles

The only input is the 2D distributions of Npos vs. centrality and Nneg vs. centrality.where pos = p, K+, q+ & neg = pbar, K-, q-

With Nnet and Ntot vs. centrality I can also independently produce the experimental results,with delta theorem error bars, efficiency corrections, etc…

Xiaofeng (net-p) and Daniel McDonald (net-p,-K,-q) have shared these 4×7 TH2Ds with me.

similar plots for K±, q±…

Filled at exactly the same spot in the analysis codes where the deviates are saved

i.e. TH2Ds include the same track cuts, PID, and run&evt QA as the local analysis…

Np

N

pbar 7.7 11.5 19.6 27 39 62.4 200

refmultXcorr




Sampled Singles method

In every slice of rmXcorr, sample one value of Npos and one value of Nneg, Nevt times.

In each sample at a given rmXcorr, one then has a value for Npos and NnegThen form Nnet = Npos-Nneg and Ntot = Npos+NnegFill similar 2D plots of Nnet and Ntot vs. centralityAnd then extract the moments (products) and do the CBW corrections as usual…

Destroys all intra-event correlations between Npos and Nneg, reproduces singles distributions, & has the same statistical certainty as the data by construction…

Np

N

pbar 7.7 11.5 19.6 27 39 62.4 200

refmult2corrcf. G. Torrieri et al., J. Phys. G, 37, 094016 (2010)




net-p Sσ vs. centrality by √sNN

cyan bands are sampled singles…uncertainties from delta theorem…




net-p Kσ2 vs. centrality by √sNN

Note: TH2Ds from Xiaofengat 200GeV include 40Mevents, net-p paper uses 240M.




net-p Sσ & Kσ2 vs. √sNN for 0-5% centrality

Sampled Singles reproduces the data values…

(N)BD also does a decent job for net-p…slight overprediction near 19.6 GeV




Importance of intra-event correlations to the measured net-p moments

Sampled singles approach accurately reproduces the experimental results.

Our experimental results on the net-p moments products can be understood quantitatively when considering only the Np and Npbar distributions separately.

Intra-event correlations between Np and Npbar do not have any measurable impact on the net-p moments products values._________________________________________________________________________

(N)BD approach also basically assumes that there are no intra-event correlations, as the input quantities are only μ+, μ-, σ+

2, and σ-2

(N)BD also reproduces the data much more accurately than the Poisson baseline shown at QM2012… …although there are some interesting deviations with the data…

_________________________________________________________________________

If our measured moments products are truly understandable from the Np and Npbarmultiplicity distributions separately, then I should also be able to accurately describethe measured net-p moments products by exploiting the additivity properties of cumulants…

Sσ(net-p) = C3(net-p)/C2(net-p) Kσ2(net-p) = C4(net-p)/C2(net-p)




Independent Random Variable Cumulant Arithmetic

We are interested in measuring Ss and Ks2 for net-protons here. These quantities are related to the cumulants, Ck, as follows.

Ss = C3/C2 and Ks2 = C4/C2 (C1=mean, C2=variance)

where Ck is a “cumulant.”

A feature of cumulants is their additivity for pairs of independent random variables. i.e. given independent random variables u and v, then

Ck (u+v) = Ck (u) + Ck (v)

But here, we are interested in Ss and Ks2 for net-p, i.e. “u-v” with u=Np and v=Npbar

In this case, Ck (u-v) = Ck (u) + (-1)k×Ck(v)This relation will only hold if u (Np) and v (Npbar) are random and independent variables.

So, here I’ll calculate Ss and Ks2 using the values of Ck (u-v) via Ck (u) and Ck(v)

Tests the importance of intra-event correlations of Np and Npbar that requires no stochastic sampling. The information used here comes only from the singles distributions.

How does this approach compare to the sampled singles approach? and to the data?




net-p Sσ vs centrality by √sNN

using nrepeats=81 forthe sampled singles here




net-p Kσ2 vs centrality by √sNN




net-p Sσ and Kσ2 vs √sNN for 0-5% centrality




Ss Ratios




Ks2 Ratios




Ratios of Sσ and Kσ2 data to Sampled Singles, IRV Ck Math, & (N)BD…

C3/C2 C4/C2 C4/C2




Latest version of net-p paper…

Figures 3 and 4 now includethe sampled singles baselines…

Fig. 3…

Fig. 4…

Last sentence of the present version:

That (N)BD also leads to a much betterdescription than does the Poisson (Skellam)is also mentioned in the present version.




But wait. There’s more…

Now we have proven (two ways) that there is no aspect of the net-proton moments products that cannot be understood in terms of the p and pbarmultiplicity distributions separately…

That is…Kσ2(net-p)

= C4 (net-p)/C2(net-p)

= [C4(p)+C4 (pbar)] / [C2 (p)+C2 (pbar)]

Four terms there.

Are the experimental values of Kσ2(net-p) drivenby all four terms equally? Or does one term dominate?




Charge-separated Ks2 vs. centrality by √sNN

√sNN ≤ 27 GeV … Ks2(net-p) = Ks2 (p)√sNN ≥ 39 GeV … Ks2 (pbar) > Ks2 (net-p) > Ks2 (p)




C2 (variance) vs. centrality by √sNN

C2 smoothly…increasing w/ Npartdecreasing w/ √sNN




C4 vs. centrality by √sNN

proton C4… sags for 0-5% @ 19&27

pbar C4… increases “normally”




Summary

Sampled singles approach quantitatively reproduces the experimental data points…

One can also calculate the values of Ss and Ks2 assuming Np and Npbar are random and independent variables via the additivity properties of cumulants. This approach requires no stochastic sampling.

The “IRV” (independent random variable) cumulant arithmetic reproduces thesampled singles results and the experimental values.

This should lend confidence to the sampled singles approach and underscore the unimportance of (Np,Npbar) intra-event correlations to the net-p moments products values.

Sampled singles (=IRV Ck math) bands now included in present draft of net-p paper.____________________________________________________________________________________________________________________________________________________________________________

Re: the “apparent dip” for 0-5% and 19.6 & 27 GeV….Perfectly reproduced by the Sampled Singles and IRV Ck arithmetic approaches…Seems to come entirely from the proton C4… proton C2 increases ~normally(N)BD does not show this dip – but note that the input to the (N)BD is only C1 and

C2…

Kσ2(net-p) = C4 (net-p)/C2(net-p)= [C4(p)+C4 (pbar)] / [C2 (p)+C2 (pbar)]

____________________________________________________________________________________________________________________________________________________________________________

I am now exploring these aspects with UrQMD, re: b-resn & efficiency effects…




BACKUP SLIDES




νdyn(pbar/p) from Gary Westfall…

νdyn=0 …independent random variablesνdyn<0 …“correlations”νdyn>0 …“fluctuations” (w.r.t. Poisson)

pbar and p are ~independent for ~central…Correlations for ~peripheral…

Relevance to Sσ data/SampSing deviations?(see slide 16)




C1 (mean) vs. centrality by √sNN




C3 vs. centrality by √sNN




Sampled Singles method: Oversampling

Approach used up to a ~1 week ago sampled NEVT times for a given slice of refmultNcorr.

Was observed however that the resulting CBW-corrected moments products had a noticeable variance (lowest root-s generally) with the choice of TRandom3 seed…

This is not physical but canbe fixed easily with only expensebeing CPU time…

Now sample each slice (pos or neg)NREPEATS*NEVT times and fillresulting Nnet and Ntot TH1Ds witha weight of 1/NREPEATS

Same statistical uncertaintiesSame moments uncertainties (Δ thm.)Stable sampled singles results

will now show some plots forNREPEATS = 1,2,4,8,36,49,81





RMS of SampSing <Kσ2> values vs NREPEATS by root-s

RMS(<Kσ2>) decreases with increasing root-s, and with decreasing centrality at any NREPEATSIncreasing NREPEATS decreases RMS as a power law.. Values stable for NREPEATS ≥ ~36

0.1

0.01

0.001





SampSing err(<Kσ2>) vs NREPEATS by root-s

0.1

0.01

0.001

0.0001





Delta theorem’s Kσ2 uncertainty for many runs with NREPEATS=1

Average value of Δ-thmuncertainty exactly reproduces that from dataeven for NREPEATS=1

But there is some variance.largest at lowest root-s.





Delta theorem’s Kσ2 uncertainty for many runs with NREPEATS=36

Average value of Δ-thmuncertainty still reproduces that from data

Variance in SampSinguncertainty now tiny.

Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting,...

Documents

Transcript of Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting,...