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Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 1 Speculated 1 st order phase boundary M.

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Presentation on theme: "Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 1 Speculated 1 st order phase boundary M."— Presentation transcript:

1 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 1 Speculated 1 st 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 as quantified by the moments: μ, σ 2, S, K… Products Sσ and Kσ 2 less volume-dependent… Net-p Moments Products W.J. Llope, Rice University I analysis meeting, July 2013

2 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 2 STAR net-p Results at QM2012 √s NN * 7.7421 11.5316 19.6206 27156 39112 62.473 20024 * Cleymans et al. PRC 73, 034905 (2006) No strong non-monotonicity seen …. But what is the significance of the apparent dip?!? net-protons X. Luo for STAR, QM2012 UrQMD simulation Poisson Statistics

3 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 3 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 √s NN ~15 GeV

4 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 4 Baselines At QM2012, the ``baselines” we provided were poisson statistics – the simplest possible baseline. URQMD – a transport model. While those data were also consistent with a straight-line fit, the appearance of statistically-significant deviations resulted in some provocative inferences/discussions. ___________________________________________________________________ We definitely need more events, with “iTPC” (& new forward tracking?) BES Phase-II increased efficiency, increased  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  B 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 σ 2 ) Sampled Singles- Data-driven, breaks intra-event correlations via sampling… Will focus on efficiency-uncorrected results here. Efficiency-corrected results are also in hand.

5 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 5 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 (σ 2 ) Then, the values of C k, Sσ, & Kσ 2 are predicted. μ<σ 2 …NBD μ=σ 2 …Poisson μ>σ 2 …BD

6 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 6 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 Npbar 7.7 11.5 19.6 27 39 62.4 200 refmultXcorr

7 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 7 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 Nneg Then form Nnet = Npos-Nneg and Ntot = Npos+Nneg Fill similar 2D plots of Nnet and Ntot vs. centrality And 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 Npbar 7.7 11.5 19.6 27 39 62.4 200 refmult2corr cf. G. Torrieri et al., J. Phys. G, 37, 094016 (2010)

8 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 8 net-p Sσ vs. centrality by √s NN cyan bands are sampled singles… uncertainties from delta theorem…

9 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 9 net-p Kσ 2 vs. centrality by √s NN Note: TH2Ds from Xiaofeng at 200GeV include 40M events, net-p paper uses 240M.

10 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 10 net-p Sσ & Kσ 2 vs. √s NN 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

11 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 11 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 Npbar multiplicity distributions separately, then I should also be able to accurately describe the measured net-p moments products by exploiting the additivity properties of cumulants… Sσ(net-p) = C 3 (net-p)/C 2 (net-p)Kσ 2 (net-p) = C 4 (net-p)/C 2 (net-p)

12 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 12 Independent Random Variable Cumulant Arithmetic We are interested in measuring S  and K  2 for net-protons here. These quantities are related to the cumulants, C k, as follows. S  = C 3 /C 2 and K  2 = C 4 /C 2 (C 1 =mean, C 2 =variance) where C k 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 C k (u+v) = C k (u) + C k (v) But here, we are interested in S  and K  2 for net-p, i.e. “u-v” with u=Np and v=Npbar In this case, C k (u-v) = C k (u) + (-1) k ×C k (v) This relation will only hold if u (Np) and v (Npbar) are random and independent variables. So, here I’ll calculate S  and K  2 using the values of C k (u-v) via C k (u) and C k (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?

13 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 13 net-p Sσ vs centrality by √s NN using nrepeats=81 for the sampled singles here

14 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 14 net-p Kσ 2 vs centrality by √s NN

15 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 15 net-p Sσ and Kσ 2 vs √s NN for 0-5% centrality

16 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 16 S  Ratios

17 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 17 K   Ratios

18 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 18 Ratios of Sσ and Kσ 2 data to Sampled Singles, IRV C k Math, & (N)BD… C 3 /C 2 C 4 /C 2 C 4 /C 2

19 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 19 Latest version of net-p paper… Figures 3 and 4 now include the sampled singles baselines… Fig. 3… Fig. 4… Last sentence of the present version: That (N)BD also leads to a much better description than does the Poisson (Skellam) is also mentioned in the present version.

20 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 20 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 pbar multiplicity distributions separately… That is… Kσ 2 (net-p) = C 4 (net-p)/C 2 (net-p) = [C 4 (p)+C 4 (pbar)] / [C 2 (p)+C 2 (pbar)] Four terms there. Are the experimental values of Kσ 2 (net-p) driven by all four terms equally? Or does one term dominate?

21 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 21 Charge-separated K  2 vs. centrality by √s NN √s NN ≤ 27 GeV … K  2 (net-p) = K  2 (p) √s NN ≥ 39 GeV … K  2 (pbar) > K  2 (net-p) > K  2 (p)

22 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 22 C 2 (variance) vs. centrality by √s NN C 2 smoothly… increasing w/ Npart decreasing w/ √s NN

23 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 23 C 4 vs. centrality by √s NN proton C 4 … sags for 0-5% @ 19&27 pbar C 4 … increases “normally”

24 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 24 Summary Sampled singles approach quantitatively reproduces the experimental data points… One can also calculate the values of S  and K  2 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 the sampled 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 C k 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 C k arithmetic approaches… Seems to come entirely from the proton C 4 … proton C 2 increases ~normally (N)BD does not show this dip – but note that the input to the (N)BD is only C 1 and C 2 … Kσ 2 (net-p)= C 4 (net-p)/C 2 (net-p) = [C 4 (p)+C 4 (pbar)] / [C 2 (p)+C 2 (pbar)] ____________________________________________________________________________________________________________________________________________________________________________ I am now exploring these aspects with UrQMD, re: b-resn & efficiency effects…

25 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 25 BACKUP SLIDES

26 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 26 ν dyn (pbar/p) from Gary Westfall… ν dyn =0 …independent random variables ν dyn <0 …“correlations” ν dyn >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)

27 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 27 C 1 (mean) vs. centrality by √s NN

28 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 28 C 3 vs. centrality by √s NN

29 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 29 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 can be fixed easily with only expense being CPU time… Now sample each slice (pos or neg) NREPEATS * NEVT times and fill resulting Nnet and Ntot TH1Ds with a weight of 1/ NREPEATS Same statistical uncertainties Same moments uncertainties (Δ thm.) Stable sampled singles results will now show some plots for NREPEATS = 1,2,4,8,36,49,81

30 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 30 Sampled Singles method: Oversampling RMS of SampSing values vs NREPEATS by root-s RMS( ) decreases with increasing root-s, and with decreasing centrality at any NREPEATS Increasing NREPEATS decreases RMS as a power law.. Values stable for NREPEATS ≥ ~36 0.1 0.01 0.001

31 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 31 Sampled Singles method: Oversampling SampSing err( ) vs NREPEATS by root-s 0.1 0.01 0.001 0.0001

32 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 32 Sampled Singles method: Oversampling Delta theorem’s Kσ 2 uncertainty for many runs with NREPEATS =1 Average value of Δ-thm uncertainty exactly reproduces that from data even for NREPEATS =1 But there is some variance. largest at lowest root-s.

33 Net-p Moments Products Click to edit Master subtitle style Plenary Session, STAR Analysis Meeting, July, 2013 33 Sampled Singles method: Oversampling Delta theorem’s Kσ 2 uncertainty for many runs with NREPEATS =36 Average value of Δ-thm uncertainty still reproduces that from data Variance in SampSing uncertainty now tiny.

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