Exploring the QCD phase diagram via the collision energy dependence of multi-particle femtoscopy with PHENIX
EExploring the QCD phase diagram via the collision energydependence of multi-particle femtoscopy with PHENIX
M´at´e Csan´ad for the PHENIX CollaborationE¨otv¨os Lor´and University, H-1117 Budapest, P´azm´any P. s. 1/A, HungaryAugust 25, 2020
Abstract
Exploration of the rich structure of the QCD phase diagram is an important topic in the RHICheavy ion program. One of the ultimate goals of this program is to search for the critical endpoint.Investigation of the space-time structure of hadron emissions at various phase transition points usingBose-Einstein correlations of identical bosons may provide insight on the location of the criticalendpoint. PHENIX has performed comprehensive measurements of the Bose-Einstein correlation inAu+Au collisions at √ s NN = 15, 19, 27, 39, 62.4, and 200 GeV, where we incorporated L´evy-typesource functions to describe the measured correlation functions. We put particular focus on one ofthe parameters of the L´evy-type source functions, the index of stability α , which is related to one ofthe critical exponents (the so-called correlation exponent η ). We have measured its collision energyand centrality dependence. We have also extended our analysis from two-particle to three-particlecorrelations to characterize the nature of the hadron emission source. The three particle correlationsconfirmed the findings of the two-particle correlations, and also provide insight on the pion productionmechanism beyond the core-halo model. Femtoscopy (coined by Ledniczky [1]) is an important subfield of high energy nuclear and particle physics,as it allows us to investigate the space-time structure of femtometer scale processes. This subfield orig-inates from the work of Hanbury Brown and Twiss, who investigated the angular diameter of starsusing radio and optical telescopes, based on intensity correlations; this work was later on understood byGlauber, and the technique independently discovered by Goldhaber and collaborators (see more detailsand references in Ref. [2]). Nowadays we understand femtoscopic correlations to be caused by Bose-Einstein statistics, and hence use the name Bose-Einstein or quantum statistical correlations as well.The most important equation utilized in two-particle femtoscopic correlations is one that relates thepair source D ( r, K ) (describing the probability density of creating a pair with spatial separation of r and average momentum K ) and the correlation function C ( q, K ) (indicating the amount of correlationof pairs with momentum difference q and average momentum K ): C ( q, K ) = 1 + (cid:102) D ( q, K ) (cid:102) D (0 , K ) , (1)where (cid:102) D denotes the Fourier-transform of D in its first variable. The C correlation function can alsobe written up with the single particle source S ( r, p ) ( D ( r, K ) is the autoconvolution of S ( r, p ) in thefirst variable with p = K ): C ( q, K ) = 1 + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:101) S ( q, K ) (cid:101) S (0 , K ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2)There are several approximations behind these equations, see more details and references e.g. in sectionII.D of Ref. [3]: some of these are related to the kinematics of the pair (e.g. requiring the particle momentato be close to each other), others to thermal emission. Given the validity of these assumptions, theabove equations indicate the importance of femtoscopy: by measuring C , one can obtain information onfemtometer-scale processes via the above mentioned S or D source functions. If one assumes a Gaussiansource for example, then the correlation function will also be Gaussian, and hence the Gaussian source1 a r X i v : . [ nu c l - e x ] A ug /R0 2 4 6 8 S (r) R - - - - -
10 1 =0.6 a =1.0 a =1.2 a =1.5 a =2.0 a Figure 1: Spherically symmetric L´evy sources L ( α, R, r ) for various α = values. The variable on thehorizontal axis is r/R , since the R dependence can be scaled out that way.radius can be directly measured. Generally, for expanding sources the obtained radius will be related tothe homogeneity length of the source for particles of given momenta [3].It was observed [3–7] that when investigating two-particle correlation functions, one has to go beyondthe Gaussian approximation. L´evy-distributed sources with power-law tails may appear (among otherphenomena) due to anomalous diffusion [8–10]. These are characterized by the L´evy index α , and includeGaussian ( α = 2) as well as Cauchy ( α = 1) distributions as well. The three-dimensional L´evy source isgiven as L ( α, R, r ) = 1(2 π ) (cid:90) d q e i qr e − | qR q | α/ , (3)where R is the matrix of homogeneity lengths (also known as femtoscopic radii), which (in case ofspherical symmetry) can be assumed to be the product of a single L´evy scale squared ( R ) and the3 × | q | R ) α . In this case L ( α, R, r ) becomesspherically symmetric. Such spherically symmetric L´evy sources with various α values are illustrated inFig. 1. If one aims to describe more general sources, a convenient assumption is to introduce the radiiin the Bertsch-Pratt coordinate system R out , R side and R long , which represent the square roots of thediagonal elements of the R matrix. A 3D femtoscopy analysis usually uses this definition. One may goeven beyond this approximation and introduce off-diagonal elements, but we refrain from doing that inthe analyses presented in this paper.Besides anomalous diffusion, other effects can also cause the appearance of L´evy sources. One im-portant possibility is the QCD critical point, where large scale spatial fluctuations can be present, andthe spatial correlation function becomes a power-law [11, 12]. In this case the η critical exponent maybe identical to the L´evy exponent α . Finite size effects or dynamical criticality may distort this simplepicture; nevertheless, this connection between η and α yields a strong motivation to measure the L´evyindex precisely. For more details, see Section IV.B of Ref. [3].As seen in Eqs. (1) and (2), the momentum correlation function takes the value 2 for q = 0. Experi-mentally however, one cannot measure correlations down to q = 0 due to momentum resolution effects.The extrapolated value of C ( q →
0) differs generally from 2, and takes the value 1 + λ . This can beinterpreted in the core-halo model easily: it appears that the quantity λ = C ( q → − λ = f c , where f c = core / (core+halo). This relation makes it possible to study thein-medium mass modification of the η (cid:48) meson, as detail in Section IV.A of Ref. [3].Furthermore, it is important to note that the above noted λ parameter may differ from unity dueto other reasons as well. One such possibility is coherent pion production [13–15]. In this case theabove mentioned approximations are not valid, and hence the strength of multi-particle Bose-Einstein2orrelations ( C n ) will be modified; these correlation strengths are given by λ n = lim q ij → C n ( { q ij } ) − , (4)where λ ≡ λ for simplicity. In a higher order correlation function, lower order correlations will also givecontributions, for example pairs within triplets or quadruplets. Hence a given λ n has contributions fromlower order correlations as well. If p c is defined as the fraction of coherently produced pions (within thecore), then for 2- and 3-particle correlations one obtains [15] λ = f c (cid:2) (1 − p c ) + 2 p c (1 − p c ) (cid:3) = f c (1 − p c ) , (5) λ = 2 f c (cid:2) (1 − p c ) + 3 p c (1 − p c ) (cid:3) + 3 f c (cid:2) (1 − p c ) + 2 p c (1 − p c ) (cid:3) . (6)This means that simultaneous measurement of 2- and 3-particle correlations may shed light on the valueof p c and f c [2]. Furthermore, one may define a core-halo independent parameter as κ = λ − λ √ λ (7)which is independent of the value of f c , and it takes the value κ = 1 if p c = 0. Hence by measuring κ ,one can investigate if the simple core-halo model interpretation of λ is consistent. Two-pion measurements were performed with the Au+Au data taken by the PHENIX detector systemduring the 2010-2011 running period. Correlation functions were measured using the event mixing tech-nique, and fits based on L´evy-distributed sources were done to obtain the statistically most probable L´evyparameters α , R and λ . Details about the analyses presented in this paper are given in Refs. [2,3,16–20].The L´evy exponent α was measured in collisions of various center of mass energies ( √ s NN rangingfrom 14.5 GeV to 200 GeV) and with various centrality selections, as shown in Fig. 2. These plots showthat: • Results on α from the 1D and the 3D analyses are compatible with each other, confirming thevalidity of the applied approach. • While α ( m T ) is approximately constant for each centrality, there is a clear centrality dependence,with α decreasing for more central collisions. • There does not seem to be a strong √ s NN dependence in this range - but this conclusion can beaffected by the different m T windows used for each of the analysis (this had to be done as therewas not enough statistics at the lower collision energies).The L´evy scale R was also measured in both 1D and 3D analyses, and the results are shown in Fig. 3.The 1D (spherically symmetric) analysis is mostly compatible with the 3D results, but there are cleardifferences in R out in the m T ∈ [0 . − .
7] GeV /c range, as well as for R long for the m T < .
35 GeV /c range. This may be due to the non-spherical nature of the source in the longitudinally comoving system(LCMS). This may also be the reason for the small discrepancy between the 1D and the 3D results on α ( m T ) seen in Fig. 2.The correlation strength parameter λ was measured versus m T from √ s NN = 39 GeV to 200 GeV, andthe results are shown in Fig. 4. The results show a reduced λ and small m T values, as compared to larger m T values; this seems to be an universal feature of λ ( m T ) across collision energies and centralities inthe investigated range. As discussed in Ref. [3], this may be attributed to an in-medium mass reductionof the η (cid:48) meson. This statement relies heavily on the core-halo model assumption. This was furthertested in a simultaneous analysis of 2- and 3-particle correlations, in particular the measurement of the κ parameter defined above in Eq. (7). The lowest panel of Fig. 4 shows that this parameter is compatiblewith κ = 1, i.e. the value predicted by the core-halo model with fully chaotic emission. The speaker was supported by the ´UNKP-19-4 New National Excellence Program of the HungarianMinistry for Innovation and Technology, the J´anos Bolyai Research Scholarship of the Hungarian Academyof Sciences, as well as grant FK-123842 of the National Research, Development and Innovation Office ofHungary. 3 [GeV/c T m a - p - p + p + p
1D Phys. Rev. C 97, 064911 - p - p
1D Phys. Rev. C 97, 064911 + p + p PHENIX 0-30% Centrality = 200 GeV NN sAu+Au PH ENIX preliminary ] [GeV/c T m a = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 200 GeV NN sPHENIX Au+Au + p + p + - p - p > = 1.096 a a a a a a PH ENIX preliminary part N > a < = 200 GeV NN sPHENIX Au+Au + p + p + - p - p PH ENIX preliminary [GeV] NN s a [GeV]: NN s ]: [MeV/c T m D + p + p + - p - p , = 0.420 GeV/c æ T m Æ PHENIX 0-30% Au+Au,
PH ENIX preliminary
Figure 2: Measurements of the L´evy exponent α in 0-30% centrality √ s NN = 200 GeV Au+Au collisions(top left panel), various 10% centrality bins also at 200 GeV (versus m T in the top right panel, versus N part in the bottom left panel), and in 0-30% centrality collisions in for various √ s NN values (bottomright panel). ] [GeV/c T m0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R [f m ] PHENIX out ) 3D - p - p ( out R ) 3D + p + p ( out R ) 1D Phys. Rev. C 97, 064911 - p - p R ( ) 1D Phys. Rev. C 97, 064911 + p + p R ( ] [GeV/c T m0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 [f m ] R side0-30 % Centrality ) 3D - p - p ( side R ) 3D + p + p ( side R ) 1D Phys. Rev. C 97, 064911 - p - p R ( ) 1D Phys. Rev. C 97, 064911 + p + p R ( ] [GeV/c T m0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 [f m ] R long = 200 GeV NN sAu+Au ) 3D - p - p ( long R ) 3D + p + p ( long R ) 1D Phys. Rev. C 97, 064911 - p - p R ( ) 1D Phys. Rev. C 97, 064911 + p + p R (
PH ENIX preliminary
Figure 3: Transverse momentum dependence of the Bose-Einstein correlation radii (L´evy scale parame-ters) R out , side , long in √ s NN = 200 GeV Au+Au collisions. Results of the 1D analysis (with a sphericallysymmetric source, employing only one L´evy scale R ) are also shown in each of the panels for comparison.4 [GeV/c T m m a x l / l = 200 GeV NN sPHENIX 0-30% Au+Au (syst)), -0.14+0.23 – H=(0.59 /NDF=83/60, CL=2.7% c , (syst)) GeV/c -0.09+0.08 – =(0.30 s =55 MeV -1' h *=958 MeV, B ' h m =168 MeV -1' h *=530 MeV, B ' h m =55 MeV -1' h *=530 MeV, B ' h m =55 MeV -1' h *=250 MeV, B ' h m )] s )/(2 p -m PRL105,182301(2010),PRC83,054903(2011),resonance model:Kaneta and Xu (0.55-0.9) GeV/c ælÆ = max l - p - p + p + p ] [GeV/c T m m a x l / l = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 200 GeV NN sPHENIX Au+Au + p + p + - p - p = 1.096 fixed a a a a a a PH ENIX preliminary l + p + p + - p - p = 39 GeV, NN sPHENIX Au+Au ] [GeV/c T m l / l d -0.4-0.200.20.4 rel. syst. uncertaintiesrel. syst. uncertainties PH ENIX preliminary l + p + p + - p - p = 62 GeV, NN sPHENIX Au+Au + p + p + - p - p = 62 GeV, NN sPHENIX Au+Au + p + p + - p - p = 62 GeV, NN sPHENIX Au+Au ] [GeV/c T m l / l d -0.4-0.200.20.4 rel. syst. uncertaintiesrel. syst. uncertaintiesrel. syst. uncertaintiesrel. syst. uncertainties PH ENIX preliminary [GeV/c] T m / l ) / l - l . ( - - - NN sPHENIX 0-30% Au+Au @ - p - p - p + p + p + p Core-Halo + chaotic emission value = 200 GeV NN sPHENIX 0-30% Au+Au @ T vs. m l )/ l -3 l =0.5( k = 200 GeV NN sPHENIX 0-30% Au+Au @ PH ENIX preliminary
Figure 4: Results on λ normalized by its large m T average ( λ max ) in √ s NN = 200 GeV Au+Au collisionsare shown in the top two plots. The top left plot also shows calculations based on various η (cid:48) masses.The second row shows λ ( m T ) for various centralities in √ s NN = 39 and 62 GeV Au+Au collisions. Thebottom plot shows the core-halo independent κ parameter as a function of m T (here for clarity, λ ≡ λ is the strength of two-particle correlations, while λ is the strength of three-particle correlations).5 eferences [1] Lednicky R “ Femtoscopy with unlike particles ” (2001) [arXiv:nucl-th/0112011][2] Csan´ad M (PHENIX Collaboration) 2018
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