FFactorial cumulants from global baryon number conservation
Micha(cid:32)l Barej ∗ and Adam Bzdak † AGH University of Science and Technology,Faculty of Physics and Applied Computer Science, 30-059 Krak´ow, Poland
The proton, antiproton and mixed proton-antiproton factorial cumulants originating fromthe global conservation of baryon number are calculated analytically up to the sixth order.Our results can be directly tested in experiments.
I. INTRODUCTION
Many effective models of quantum chromodynamics (QCD) predict the first-order phase tran-sition and the associated critical end point between the hadronic matter and quark-gluon plasma[1–4]. One of the main approaches to search for such structures in the QCD phase diagram is basedon the investigation of fluctuations of, e.g., net-baryon number, net-charge or net-strangeness num-ber [1, 5–24] measured in relativistic heavy ion collisions, see also a recent review in Ref. [4].Higher-order cumulants, κ n , of the multiplicity distribution can be used to quantify the proper-ties of such fluctuations since they are proportional to the higher powers of the correlation length[10]. However, the cumulants mix the correlation functions of different orders, and thus in ex-perimental situations might be challenging to interpret. Also, in practice, the cumulants mightbe dominated by the trivial term representing the average number of particles. To avoid thesedifficulties, the factorial cumulants, ˆ C n , can be used as they represent the integrated genuinemulti-particle correlation functions [4, 25–27].The factorial cumulants have already been successfully applied to the STAR data on net-protonfluctuations [28–30], which unveiled rather unexpected source of strong three- and four-protoncorrelations in central Au+Au collisions at √ s NN = 7 . P ( n p , ¯ n p ) contain more information than the cumulants of the net-proton distribution P ( n p − ¯ n p ) [27]. Our results extend the so far published results and will allow for more sophisticatedtests of the global baryon conservation effects in experiments.In the next Section, we discuss our derivation of the proton, antiproton, and mixed proton-antiproton factorial cumulants. In Section III we present the exact results up to the sixth order ∗ michal.barej@fis.agh.edu.pl † adam.bzdak@fis.agh.edu.pl In this paper we adopt the notation of Ref. [4]. a r X i v : . [ nu c l - t h ] J un and discuss some relations among them. We also provide very simple approximate expressionsapplicable at high energies. This is followed by the numerical results in Section IV. We finishthe paper with comments and a summary. In Appendixes A-D some additional formulae andderivations are given. II. CALCULATION
In this Section we derive analytically the factorial cumulants of proton and antiproton multiplic-ity distribution, originating from the global conservation of baryon number. We assume that theonly source of correlations is given by the global conservation law. By B we denote the conservedbaryon number, N b and ¯ N b are the event-by-event total numbers of baryons and anti-baryons, re-spectively, and n p and ¯ n p are the numbers of observed protons and antiprotons in a given rapidityand/or transverse momentum interval. The probability distribution of n p and ¯ n p is given by P ( n p , ¯ n p ) = A ∞ (cid:88) N b = n p ∞ (cid:88) ¯ N b =¯ n p δ N b − ¯ N b ,B (cid:20) (cid:104) N b (cid:105) N b N b ! e −(cid:104) N b (cid:105) (cid:21) (cid:34) (cid:104) ¯ N b (cid:105) ¯ N b ¯ N b ! e −(cid:104) ¯ N b (cid:105) (cid:35) × (cid:20) N b ! n p !( N b − n p )! p n p (1 − p ) N b − n p (cid:21) (cid:20) ¯ N b !¯ n p !( ¯ N b − ¯ n p )! ¯ p ¯ n p (1 − ¯ p ) ¯ N b − ¯ n p (cid:21) , (1)where p = (cid:104) n p (cid:105) / (cid:104) N b (cid:105) is the probability that the initial baryon is observed as a proton and ¯ p = (cid:104) ¯ n p (cid:105) / (cid:104) ¯ N b (cid:105) is the probability that the initial antibaryon is observed as an antiproton in a givenacceptance region. (cid:104) x (cid:105) denotes an event average value of x . The normalization constant is: A = (cid:16) (cid:104) ¯ N b (cid:105)(cid:104) N b (cid:105) (cid:17) B e (cid:104) Nb (cid:105) + (cid:104) ¯ Nb (cid:105) I B (cid:16) (cid:113) (cid:104) N b (cid:105) (cid:10) ¯ N b (cid:11)(cid:17) , (2)where I ν ( x ) is a modified Bessel function of the order ν . As already emphasized, our goal is tocalculate the factorial cumulants assuming that the only source of correlation is given by the con-servation of baryon number. Consequently, we start with N b and ¯ N b following Poisson distributionsand the multiplicities of observed protons and antiprotons are governed by binomial distributions[4], which do not introduce any new correlations (see also footnote 3). The global baryon con-servation is obviously enforced by δ N b − ¯ N b ,B . Without this term, P ( n p , ¯ n p ), would be given by aproduct of two Poisson distributions, and all the factorial cumulants would vanish. Note that Eq.(1) can be derived from a more general expression including protons, antiprotons, neutrons, andantineutrons. This is demonstrated in Appendix A.Using Eqs. (1) and (2), it is straightforward to calculate the factorial moment generatingfunction (a.k.a. probability generating function) H ( x, ¯ x ) = ∞ (cid:88) n p =0 ∞ (cid:88) ¯ n p =0 x n p ¯ x ¯ n p P ( n p , ¯ n p ) , (3)and the factorial cumulant generating function G ( x, ¯ x ) = ln[ H ( x, ¯ x )] . (4) Experimentally, one is usually restricted to the measurement of protons, however, the connection with baryonscan be made [43, 44]. This derivation is slightly different than the one from Ref. [38], where the total volume was divided into observedand unobserved systems and the joint multiplicity distribution was written as a product of distributions fromthe two subvolumes (Eq. (5) in [38]), see also [42]. Both procedures lead to identical results if the underlyingdistributions are Poissons.
The result is: G ( x, ¯ x ) = ln (cid:18) px + 1 − p ¯ p ¯ x + 1 − ¯ p (cid:19) B I B (cid:16) (cid:113) (cid:104) N b (cid:105) (cid:10) ¯ N b (cid:11) ( px + 1 − p )(¯ p ¯ x + 1 − ¯ p ) (cid:17) I B (cid:16) (cid:113) (cid:104) N b (cid:105) (cid:10) ¯ N b (cid:11)(cid:17) . (5)The factorial cumulants ˆ C ( n,m ) which are the integrated (over a given acceptance region) cor-relation functions for (in our context) n protons and m antiprotons are given byˆ C ( n,m ) = ∂ n ∂x n ∂ m ∂ ¯ x m G ( x, ¯ x ) (cid:12)(cid:12)(cid:12)(cid:12) x =¯ x =1 . (6)By definition, the factorial cumulants ˆ C ( n,m ) = 0 for all n ≥ m ≥
1, if there are no correlationsin the system [4], i.e., if P ( n p , ¯ n p ) factorizes and both n p and ¯ n p are distributed according to Poissondistributions. The global baryon number conservation, being a long-range correlation, results innon-zero ˆ C ( n,m ) . We note that the cumulants, which are usually measured in experiments, see,e.g., [28–30, 37, 41, 45, 46], can be expressed by ˆ C ( n,m ) . We will discuss this issue later on. Here weonly emphasize that the cumulants mix the factorial cumulants of different orders and in general,the factorial cumulants contain more information than the cumulants.Before we present our results let us introduce additional notation: z = (cid:113) (cid:104) N b (cid:105)(cid:104) ¯ N b (cid:105) , (7) (cid:104) N b (cid:105) c = z I B − (2 z ) I B (2 z ) , (cid:104) ¯ N b (cid:105) c = z I B +1 (2 z ) I B (2 z ) , (8) z c = (cid:113) (cid:104) N b (cid:105) c (cid:104) ¯ N b (cid:105) c , (9)where (cid:104) N b (cid:105) is the mean number of baryons (present in Eq. (1)) before the baryon number con-servation is enforced, and (cid:104) N b (cid:105) c is the mean number of baryons with the conservation of baryonnumber (and analogously for antibaryons). The baryon number conserved averages obviously sat-isfy (cid:104) N b (cid:105) c − (cid:104) ¯ N b (cid:105) c = B (see Eq. (8) and footnote 4). III. RESULTSA. Exact formulae
In this Section we present analytic expressions for ˆ C ( n,m ) up to the sixth order. It is natural todefine: (cid:104) N (cid:105) c = (cid:104) N b (cid:105) c + (cid:104) ¯ N b (cid:105) c , (10)which is the total average number of baryons. To present the formulae in a more compact way weidentified commonly appearing terms and denoted them as:∆ = z c − z , (11) γ = z c + ∆ (cid:104) N (cid:105) c , (12) β = γ ( (cid:104) N (cid:105) c + 2) + 2∆ , (13)where (cid:104) N (cid:105) c , ∆, γ and β depend on B and z only, see Eqs. (8) and (9). The factorial cumulantsread :ˆ C (1 , = p (cid:104) N b (cid:105) c (14)ˆ C (2 , = − p ( (cid:104) N b (cid:105) c + ∆) (15)ˆ C (1 , = − p ¯ p ∆ (16)ˆ C (3 , = p (cid:2) (cid:0) (cid:104) N b (cid:105) c + ∆ + γ (cid:1)(cid:3) (17)ˆ C (2 , = p ¯ p γ (18)ˆ C (4 , = − p (cid:2) (cid:0) (cid:104) N b (cid:105) c + ∆ + γ (cid:1) + β (cid:3) (19)ˆ C (3 , = − p ¯ pβ (20)ˆ C (2 , = − p ¯ p ( β − γ ) (21)ˆ C (5 , = p (cid:2) (cid:0) (cid:104) N b (cid:105) c + ∆ + γ (cid:1) + ( (cid:104) N (cid:105) c + 7) β + 6 γ ∆ (cid:3) (22)ˆ C (4 , = p ¯ p [( (cid:104) N (cid:105) c + 3) β + 6 γ ∆] (23)ˆ C (3 , = p ¯ p [( (cid:104) N (cid:105) c + 1) β + 6 γ ∆] (24)ˆ C (6 , = − p (cid:2) (cid:0) (cid:104) N b (cid:105) c + ∆ + γ (cid:1) + { ( (cid:104) N (cid:105) c + 5)( (cid:104) N (cid:105) c + 7) + 12 } β + 6 γ + 16∆ +2 γ ∆(7 (cid:104) N (cid:105) c + 35)] (25)ˆ C (5 , = − p ¯ p (cid:2) ( (cid:104) N (cid:105) c + 3)( (cid:104) N (cid:105) c + 4) β + 6 γ + 16∆ + 2 γ ∆(7 (cid:104) N (cid:105) c + 20) (cid:3) (26)ˆ C (4 , = − p ¯ p (cid:2) ( (cid:104) N (cid:105) c + 1)( (cid:104) N (cid:105) c + 3) β + 6 γ + 16∆ + 2 γ ∆(7 (cid:104) N (cid:105) c + 11) (cid:3) (27)ˆ C (3 , = − p ¯ p (cid:2) ( (cid:104) N (cid:105) c + 1)( (cid:104) N (cid:105) c + 2) β + 6 γ + 16∆ + 2 γ ∆(7 (cid:104) N (cid:105) c + 8) (cid:3) (28)Having ˆ C ( n,m ) , one can easily obtain ˆ C ( m,n ) :ˆ C ( m,n ) = ˆ C ( n,m ) ( p → ¯ p, ¯ p → p ) for n m (cid:54) = 0 , (29)ˆ C (0 ,n ) = ˆ C ( n, (cid:0) p → ¯ p, (cid:104) N b (cid:105) c → (cid:104) ¯ N b (cid:105) c (cid:1) , (30)that is, to obtain ˆ C ( m,n ) from ˆ C ( n,m ) with both n and m larger than zero, it is enough to exchange p with ¯ p . To obtain ˆ C (0 ,n ) from ˆ C ( n, it is also necessary to replace (cid:104) N b (cid:105) c by (cid:104) ¯ N b (cid:105) c . For example,ˆ C (0 , = ¯ p (cid:104) ¯ N b (cid:105) c and ˆ C (1 , = p ¯ p γ . B. Relations
As seen from Eqs. (14)–(28), ˆ C ( n,m ) is proportional to p n ¯ p m . Therefore it is natural to studythe following ratios ˆ R ( n,m ) = ˆ C ( n,m ) p n ¯ p m , (31) In this calculation we extensively use I ν − ( x ) − I ν +1 ( x ) = νx I ν ( x ). This is not unexpected. As argued in, e.g., Refs. [27, 36] the long-range correlation, such as global baryonconservation, naturally results in ˆ C ( n,m ) being proportional to (cid:104) n p (cid:105) n (cid:104) ¯ n p (cid:105) m , where (cid:104) n p (cid:105) = p (cid:104) N b (cid:105) and (cid:104) ¯ n p (cid:105) = ¯ p (cid:104) ¯ N b (cid:105) . which are independent of the size of the chosen acceptance bin.Using Eqs. (14)–(28) we find several simple relations between various ˆ R ( n,m ) :ˆ R (2 , = ˆ R (1 , − ˆ R (1 , , (32)ˆ R (3 , = ˆ R (2 , − R (2 , , (33)ˆ R (4 , = ˆ R (3 , − R (3 , , (34)ˆ R (5 , = ˆ R (4 , − R (4 , , (35)ˆ R (6 , = ˆ R (5 , − R (5 , , (36)ˆ R (3 , = ˆ R (2 , − ˆ R (2 , , (37)ˆ R (4 , = ˆ R (3 , − R (3 , , (38)ˆ R (5 , = ˆ R (4 , − R (4 , , (39)ˆ R (4 , = ˆ R (3 , − ˆ R (3 , , (40)or in general ( n > m > R ( n +1 ,m ) = ˆ R ( n,m +1) − ( n − m ) ˆ R ( n,m ) , (41)which we verified by direct calculations up to n + m < C. Approximate formulae for B = 0 Here, we analyze in detail the special case of B = 0, meaning the same total number of baryonsand antibaryons, which characterizes large energy conditions, such as at the LHC CERN. In thiscase (cid:104) N b (cid:105) c = (cid:104) ¯ N b (cid:105) c , z c = (cid:104) N b (cid:105) c and (cid:104) N (cid:105) c = 2 (cid:104) N b (cid:105) c . All components appearing in Eqs. (14)–(28),that is, (cid:104) N (cid:105) c , (cid:104) N b (cid:105) c , ∆, γ and β depend on z only. Next, we apply to Eq. (8) the asymptotic (largeargument) expansion of the modified Bessel function [47]: I ν ( x ) ∼ e x √ πx (cid:32) ∞ (cid:88) n =1 ( − n (cid:81) ni =1 (4 ν − (2 i − ) n !(8 x ) n (cid:33) . (42)After eliminating the Bessel functions (the higher the order of the factorial cumulant, the moreterms are needed in Eq. (42)) we expand ˆ R ( n,m ) ( z ) into a power series for large z and obtain thedependency of the form ˆ R ( n,m ) ( z ) ∼ a z + a + a − z − + a − z − + ... , (43)where the coefficients a i depend on n and m . It is worth noting that ˆ R ( n,m ) ( z ) grows linearly with z for large z . The details and explicit expressions for ˆ R ( n,m ) ( z ) are presented in Appendix B.It can be proved (see Appendix B) that ˆ R ( n,m ) ( z c ) is also of the same form, that is, the highest-order term is proportional to z c and the coefficients of the series can be easily calculated. Theobtained asymptotic expressions for ˆ R ( n,m ) ( z c ) at large z c are given below ( z c = (cid:104) N b (cid:105) c = (cid:104) ¯ N b (cid:105) c ):ˆ R (2 , ( z c ) ∼ − z c + + z − c + ... (44)ˆ R (1 , ( z c ) ∼ z c + + z − c + ... (45) Here we introduce z = 1 /y and expand about y = 0 and then substitute back y = 1 /z . ˆ R (3 , ( z c ) ∼ z c − − z − c + ... (46)ˆ R (2 , ( z c ) ∼ − z c − − z − c + ... (47)ˆ R (4 , ( z c ) ∼ − z c + + z − c + ... (48)ˆ R (3 , ( z c ) ∼ z c + + z − c + ... (49)ˆ R (2 , ( z c ) ∼ z c + + z − c + ... (50)ˆ R (5 , ( z c ) ∼ z c − − z − c + ... (51)ˆ R (4 , ( z c ) ∼ − z c − − z − c + ... (52)ˆ R (3 , ( z c ) ∼ − z c − − z − c + ... (53)ˆ R (6 , ( z c ) ∼ − z c + + z − c + ... (54)ˆ R (5 , ( z c ) ∼ z c + + z − c + ... (55)ˆ R (4 , ( z c ) ∼ z c + + z − c + ... (56)ˆ R (3 , ( z c ) ∼ z c + + z − c + ... (57)We checked, see Section IV, that the obtained approximate formulae work with very goodaccuracy already from z c = (cid:104) N b (cid:105) c > IV. NUMERICAL RESULTS
In this Section we present numerical results for ˆ R ( n,m ) ( z c ) = ˆ C ( n,m ) / ( p n ¯ p m ) for two specialcases: B = 0 corresponding to large energies, and B = 300 corresponding to central collisions atlow energies in heavy-ion collisions. A. B = 0 For B = 0, z c = (cid:104) N b (cid:105) c = (cid:104) ¯ N b (cid:105) c = (cid:104) N (cid:105) c / R ( n,m ) ( z c ) equals ˆ R ( n,m ) ( (cid:104) N b (cid:105) c ). FromEqs. (44)–(57) it is clear that the dominant contribution is linear with z c = (cid:104) N b (cid:105) c and thereare certain deviations for small (cid:104) N b (cid:105) c . Therefore, for B = 0, it is natural to divide ˆ R ( n,m ) by (cid:104) N b (cid:105) c so that the leading term is simply constant. In Fig. 1 we present ˆ R ( n,m ) ( (cid:104) N b (cid:105) c ) divided by (cid:104) N b (cid:105) c for all the discussed factorial cumulants. Markers represent exact formulae for the factorialcumulants ˆ C ( n,m ) given by Eqs. (14)–(28), whereas lines represent our asymptotic expressions(large (cid:104) N b (cid:105) c ) given by Eqs. (44)–(57). These functions are essentially constant, in agreement withour asymptotic results, except for small values of (cid:104) N b (cid:105) c . The approximated formulae work verywell starting from (cid:104) N b (cid:105) c ≈
2. The precision better than 1% is obtained starting from (cid:104) N b (cid:105) c ≈ For the exact results we first take (cid:104) N b (cid:105) c and solve Eq. (8) for z , which we substitute to Eqs. (14)–(28). h N b i c − . − . − . . . . . . R (2 , / h N b i c ˆ R (1 , / h N b i c ˆ R (3 , / h N b i c ˆ R (2 , / h N b i c h N b i c − . . . . . . R (4 , / h N b i c ˆ R (3 , / h N b i c ˆ R (2 , / h N b i c h N b i c − . − . − . . . . R (5 , / h N b i c ˆ R (4 , / h N b i c ˆ R (3 , / h N b i c h N b i c − . . . . . . R (6 , / h N b i c ˆ R (5 , / h N b i c ˆ R (4 , / h N b i c ˆ R (3 , / h N b i c FIG. 1: ˆ R ( n,m ) / (cid:104) N b (cid:105) c as a function of (cid:104) N b (cid:105) c for B = 0, where ˆ R ( n,m ) = ˆ C ( n,m ) / ( p n ¯ p m ). Markersrepresent exact formulae for the factorial cumulants ˆ C ( n,m ) given by Eqs. (14)–(28), whereas linesrepresent our asymptotic formulae (large (cid:104) N b (cid:105) c ) given by Eqs. (44)–(57). Markers are plotted for (cid:104) N b (cid:105) c = 1, 2, 5, 10, 15, .... For (cid:104) N b (cid:105) c > (cid:104) N b (cid:105) c between 2 and 7 depending on the order of thefactorial cumulant. Some of the functions were scaled by a factor of 0.1 to improve readability. B. B = 300 Here we investigate the case of B (cid:54) = 0 and, as an example, we choose B = 300. In this case,obviously (cid:104) N b (cid:105) c = (cid:104) ¯ N b (cid:105) c + B and now z c = [ (cid:104) N b (cid:105) c ( (cid:104) N b (cid:105) c − B )] / . In general ˆ R ( n,m ) is morecomplicated than for B = 0 and only for very large z c or (cid:104) N b (cid:105) c it asymptotically approaches alinear function. This is demonstrated in Fig. 2, where we plot ˆ R ( n,m ) divided by z c as a functionof (cid:104) ¯ N b (cid:105) c . We were unable to obtain a simple approximated formula and thus in Fig. 2 we presentonly exact ˆ R ( n,m ) /z c based on Eqs. (14)–(28). In the case of B (cid:54) = 0, ˆ R ( n, (cid:54) = ˆ R (0 ,n ) and we decidedto plot ( ˆ R ( n, − ( − n − ( n − (cid:104) N b (cid:105) c ) /z c because this is symmetric when baryons and antibaryonsare exchanged, see Eqs. (14)–(28). We note that for some ˆ R ( n,m ) /z c with n , m close to eachother (e.g., ˆ R (2 , , ˆ R (3 , ) we observe a maximum or minimum at (cid:104) ¯ N b (cid:105) c about 100. Experimentallyavailable cases at heavy-ion colliders cover the values of (cid:104) ¯ N b (cid:105) c of the order of 100 and in Fig. 3 weshow the results (except ˆ R ( n, ) in the range of 0 < (cid:104) ¯ N b (cid:105) c < h ¯ N b i c − . − . − . . . . R (2 , + h N b i c ) /z c ≡ ˆ R (1 , /z c ( ˆ R (3 , − h N b i c ) /z c ˆ R (2 , /z c h ¯ N b i c − . . . . . . . R (4 , + 3! h N b i c ) /z c ˆ R (3 , /z c ˆ R (2 , /z c h ¯ N b i c − . − . − . − . − . . . . R (5 , − h N b i c ) /z c ˆ R (4 , /z c
10 ˆ R (3 , /z c h ¯ N b i c − . . . . . .
01 ( ˆ R (6 , + 5! h N b i c ) /z c ˆ R (5 , /z c ˆ R (4 , /z c
10 ˆ R (3 , /z c FIG. 2: ˆ R ( n,m ) /z c as a function of (cid:104) ¯ N b (cid:105) c for B = 300 based on Eqs. (14)–(28). ˆ R ( n,m ) =ˆ C ( n,m ) / ( p n ¯ p m ). For m = 0 we present ( ˆ R ( n, − ( − n − ( n − (cid:104) N b (cid:105) c ) /z c because it gives thesame values for both ˆ R ( n, and ˆ R (0 ,n ) . Some of the functions were scaled by a factor of 10, 0.1 or0.01 to improve readability. Note the logarithmic scale on the horizontal axis. V. COMMENTS AND SUMMARY
In this paper we calculated the proton, antiproton and mixed proton-antiproton factorial cu-mulants, ˆ C ( n,m ) , up to the sixth order, n + m = 6, assuming that the only source of correlationsis the global conservation of baryon number. The exact formulae are given in Eqs. (14)–(28) andfor the case of B = 0 the asymptotic expressions are provided in Eqs. (44)–(57). The latter onesrepresent very good approximation already from (cid:104) N b (cid:105) c ≈
10 20 30 40 50 h ¯ N b i c − . − . . . .
10 ˆ R (2 , /z c
10 20 30 40 50 h ¯ N b i c − . − . . . .
10 ˆ R (3 , /z c ˆ R (2 , /z c
10 20 30 40 50 h ¯ N b i c − . − . . . .
10 ˆ R (4 , /z c ˆ R (3 , /z c
10 20 30 40 50 h ¯ N b i c − . − . . . .
10 ˆ R (5 , /z c ˆ R (4 , /z c
10 ˆ R (3 , /z c FIG. 3: Same as Fig. 2 but for nm (cid:54) = 0 and for small (cid:104) ¯ N b (cid:105) c . Note the linear scale on the horizontalaxis. ˆ R (3 , /z c was scaled by 10 to make the maximum at (cid:104) ¯ N b (cid:105) c ≈
15 visible.Several comments are in order.Recently the ALICE Collaboration measured [41] the second-order cumulant, κ , of the net-proton number and the result is consistent with the global baryon conservation. We note that, e.g., κ contains less information than the second-order factorial cumulants ˆ C (2 , , ˆ C (1 , and ˆ C (0 , . Itwould be instructive to see whether the second-order factorial cumulants are consistent with theALICE data. Also, the measurement of the higher-order factorial cumulants would be warranted.Having all the factorial cumulants we can immediately calculate the net-proton cumulants κ n .For example [27] κ = ˆ C (1 , + ˆ C (0 , + ˆ C (2 , + ˆ C (0 , − C (1 , , (58)and the expressions for the higher order κ n are shown in Appendix C. Here ˆ C (1 , and ˆ C (0 , arethe mean numbers of observed, e.g., protons and antiprotons, respectively.Finally, one possible way to measure factorial cumulants ˆ C ( n,m ) is to first measure factorialmoments F i,k ≡ (cid:104) n !( n − i )! ¯ n !(¯ n − k )! (cid:105) , which allow to directly obtain ˆ C ( n,m ) . Explicit relations betweenˆ C ( n,m ) and F i,k are given in Appendix D.0 ACKNOWLEDGMENTS
This work was partially supported by the Ministry of Science and Higher Education, and bythe National Science Centre, Grant No. 2018/30/Q/ST2/00101.
Appendix A: A comment on Eq. (1)
Let in each heavy-ion collision event B = N p + N n − ¯ N p − ¯ N n be the net-baryon number. Here N p and ¯ N p are the total numbers of protons and antiprotons, respectively, N n and ¯ N n are thetotal numbers of neutrons and antineutrons. Moreover, by n p and ¯ n p we denote the numbers ofobserved protons and antiprotons in a given acceptance bin. p = (cid:104) n p (cid:105) / (cid:104) N p (cid:105) is the probability toobserve a proton in a given acceptance region and p = (cid:104) ¯ n p (cid:105) / (cid:104) ¯ N p (cid:105) is the probability to observe anantiproton. The probability distribution of n p and ¯ n p is given by P ( n p , ¯ n p ) = A ∞ (cid:88) N p = n p ∞ (cid:88) ¯ N p =¯ n p ∞ (cid:88) N n =0 ∞ (cid:88) ¯ N n =0 δ N p + N n − ¯ N p − ¯ N n ,B (cid:20) (cid:104) N p (cid:105) N p N p ! e −(cid:104) N p (cid:105) (cid:21) (cid:34) (cid:104) ¯ N p (cid:105) ¯ N p ¯ N p ! e −(cid:104) ¯ N p (cid:105) (cid:35) × (cid:20) (cid:104) N n (cid:105) N n N n ! e −(cid:104) N n (cid:105) (cid:21) (cid:34) (cid:104) ¯ N n (cid:105) ¯ N n ¯ N n ! e −(cid:104) ¯ N n (cid:105) (cid:35) × (cid:20) N p ! n p !( N p − n p )! p n p (1 − p ) N p − n p (cid:21) (cid:20) ¯ N p !¯ n p !( ¯ N p − ¯ n p )! p ¯ n p (1 − p ) ¯ N p − ¯ n p (cid:21) , (A1)where A is a normalization factor. In this expression we assume that the only source of correlationis given by the global conservation of baryon number implemented by δ N p + N n − ¯ N p − ¯ N n ,B .Next, N b = N p + N n is the total number of baryons, and ¯ N b = ¯ N p + ¯ N n is the total number ofanti-baryons. Using relations N p = N b − N n , ¯ N p = ¯ N b − ¯ N n , (A2)and summing over N n and ¯ N n leads to our starting Eq. (1). Appendix B: Asymptotic expansion for B = 0 Here we present more details leading to the asymptotic Eqs. (44)–(57). As already mentionedin Section III C, in all Eqs. (14)–(28) we eliminate the Bessel functions (the higher the order of thefactorial cumulant, the more terms are needed in Eq. (42) and it is enough to take the first 7 termsfor the sixth order ˆ C ( n,m ) ) and expand ˆ R ( n,m ) ( z ) into a power series for large z . We obtain:ˆ R (2 , ( z ) ∼ − z + + z − + ... (B1)ˆ R (1 , ( z ) ∼ z + z − + ... (B2)ˆ R (3 , ( z ) ∼ z − − z − + ... (B3)ˆ R (2 , ( z ) ∼ − z − z − + ... (B4)ˆ R (4 , ( z ) ∼ − z + + z − + ... (B5)ˆ R (3 , ( z ) ∼ z + z − + ... (B6)ˆ R (2 , ( z ) ∼ z + z − + ... (B7)1ˆ R (5 , ( z ) ∼ z − − z − + ... (B8)ˆ R (4 , ( z ) ∼ − z − z − + ... (B9)ˆ R (3 , ( z ) ∼ − z − z − + ... (B10)ˆ R (6 , ( z ) ∼ − z + 30 + z − + ... (B11)ˆ R (5 , ( z ) ∼ z + z − + ... (B12)ˆ R (4 , ( z ) ∼ z + z − + ... (B13)ˆ R (3 , ( z ) ∼ z + z − + ... (B14)Note that all the ˆ R ( n,m ) ( z ) can be written asˆ R ( n,m ) ( z ) ∼ a z + a + a − z − + a − z − + ... , (B15)where the coefficients a i depend on n and m and a (cid:54) = 0 for m = 0 only.It is easy to see that ˆ R ( n,m ) ( z c ) is also of the same form, that is, the highest term is proportionalto z c and the coefficients of the series can be easily calculated. First, let us expand z c in a seriesof z : z c ( z ) ∼ z − − z − − z − ... (B16)It is clear that ˆ R ( n,m ) ( z c ) cannot have a z c term (or higher order) because it would generate a z term in ˆ R ( n,m ) ( z ) and we know that this term is not present, see Eq. (B15). Thus ˆ R ( n,m ) ( z c )can be written as ˆ R ( n,m ) ( z c ) ∼ b z c + b + b − z − c + b − z − c + ... , (B17)where the coefficients b i are to be determined. Substituting Eq. (B16) into Eq. (B17) andcomapring with Eq. (B15) we obtain: b = a , (B18) b = a + a , (B19) b − = a − + a , (B20) b − = a − − a − + a . (B21)Clearly, this procedure may be easily extended to obtain more terms if needed. These relationscombined with Eqs. (B1)–(B14) lead to our Eqs. (44)–(57). Appendix C: Net-proton cumulants
The cumulant generating function for two species of particles reads K ( t, ¯ t ) = G (cid:16) e t , e ¯ t (cid:17) , (C1)where G ( x, ¯ x ) is given by Eq. (4). In particular, the net-particle (e.g. net-proton) cumulants aregiven by (¯ t = − t ) κ i = d i dt i K ( t, − t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 . (C2)2Combining Eqs. (C1) and (C2), we have: κ i = d i dt i G ( x ( t ) , ¯ x ( t )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 , (C3)where x ( t ) = e t and ¯ x ( t ) = e − t and hence derivatives x ( n ) ( t =0) = 1, ¯ x ( n ) ( t =0) = ( − n . Usingthis and Eq. (6), we obtain the formulae for the net-proton cumulants in terms of the factorialcumulants: κ = ˆ C (1 , − ˆ C (0 , , (C4) κ = ˆ C (1 , + ˆ C (0 , + ˆ C (2 , + ˆ C (0 , − C (1 , , (C5) κ = ˆ C (1 , − ˆ C (0 , + 3 (cid:16) ˆ C (2 , − ˆ C (0 , (cid:17) + ˆ C (3 , − ˆ C (0 , − (cid:16) ˆ C (2 , − ˆ C (1 , (cid:17) , (C6) κ = ˆ C (1 , + ˆ C (0 , + 7 (cid:16) ˆ C (2 , + ˆ C (0 , (cid:17) − C (1 , + 6 (cid:16) ˆ C (3 , + ˆ C (0 , (cid:17) − (cid:16) ˆ C (2 , + ˆ C (1 , (cid:17) + ˆ C (4 , + ˆ C (0 , − (cid:16) ˆ C (3 , + ˆ C (1 , (cid:17) + 6 ˆ C (2 , , (C7) κ = ˆ C (1 , − ˆ C (0 , + 15 (cid:16) ˆ C (2 , − ˆ C (0 , (cid:17) + 25 (cid:16) ˆ C (3 , − ˆ C (0 , (cid:17) − (cid:16) ˆ C (2 , − ˆ C (1 , (cid:17) + 10 (cid:16) ˆ C (4 , − ˆ C (0 , (cid:17) − (cid:16) ˆ C (3 , − ˆ C (1 , (cid:17) + ˆ C (5 , − ˆ C (0 , − (cid:16) ˆ C (4 , − ˆ C (1 , (cid:17) + 10 (cid:16) ˆ C (3 , − ˆ C (2 , (cid:17) , (C8) κ = ˆ C (1 , + ˆ C (0 , + 31 (cid:16) ˆ C (2 , + ˆ C (0 , (cid:17) − C (1 , + 90 (cid:16) ˆ C (3 , + ˆ C (0 , (cid:17) − (cid:16) ˆ C (2 , + ˆ C (1 , (cid:17) + 65 (cid:16) ˆ C (4 , + ˆ C (0 , (cid:17) − (cid:16) ˆ C (3 , + ˆ C (1 , (cid:17) + 30 ˆ C (2 , + 15 (cid:16) ˆ C (5 , + ˆ C (0 , (cid:17) − (cid:16) ˆ C (4 , + ˆ C (1 , (cid:17) + 30 (cid:16) ˆ C (3 , + ˆ C (2 , (cid:17) + ˆ C (6 , + ˆ C (0 , − (cid:16) ˆ C (5 , + ˆ C (1 , (cid:17) + 15 (cid:16) ˆ C (4 , + ˆ C (2 , (cid:17) −
20 ˆ C (3 , , (C9)where ˆ C (1 , and ˆ C (0 , are the mean numbers of, e.g., protons and antiprotons, respectively. Theseresults extend the formulae provided in Appendix A of Ref. [27]. Appendix D: ˆ C ( n,m ) vs F i,k The factorial moments for two variables (two species of particles) are defined via the factorialmoment generating function H ( x, ¯ x ) (see Eq. (3)): F i,k ≡ (cid:28) n !( n − i )! n !( n − k )! (cid:29) = d i dx i d k d ¯ x k H ( x, ¯ x ) (cid:12)(cid:12)(cid:12)(cid:12) x =¯ x =1 . (D1)Using Eqs. (4) and (6), and the normalization condition H (1 ,
1) = 1, we can express the factorialcumulants through the factorial moments:ˆ C (1 , = F , (D2)ˆ C (0 , = F , (D3)ˆ C (2 , = − F , + F , (D4)ˆ C (1 , = − F , F , + F , (D5)3ˆ C (3 , = 2 F , − F , F , + F , (D6)ˆ C (2 , = 2 F , F , − F , F , − F , F , + F , (D7)ˆ C (4 , = − F , + 12 F , F , − F , − F , F , + F , (D8)ˆ C (3 , = − F , F , + 6 F , F , + 6 F , F , F , − F , F , − F , F , − F , F , + F , (D9)ˆ C (2 , = (cid:0) − F , + 2 F , (cid:1) F , + 8 F , F , F , − F , − F , F , + (cid:0) F , − F , (cid:1) F , − F , F , + F , (D10)ˆ C (5 , = 24 F , − F , F , + 30 F , F , + 20 F , F , − F , F , − F , F , + F , (D11)ˆ C (4 , = 24 F , F , − F , F , − F , F , F , + 24 F , F , F , + 6 F , F , + 12 F , F , − F , F , + 8 F , F , F , − F , F , − F , F , − F , F , + F , (D12)ˆ C (3 , = 2 (cid:0) F , − F , (cid:1) F , − F , F , F , + 12 F , F , + 6 F , F , − (cid:0) F , − F , (cid:1) F , F , + 12 ( F , F , + F , F , ) F , − F , F , − F , F , − F , F , + (cid:0) F , − F , (cid:1) F , − F , F , + F , (D13)ˆ C (6 , = − F , + 360 F , F , − F , F , − F , F , + 30 F , F , + 120 F , F , F , − F , F , + 30 F , − F , − F , F , + F , (D14)ˆ C (5 , = − F , F , + 120 F , F , + 240 F , F , F , − F , F , − F , F , F , − F , F , F , + 20 F , F , − F , F , F , + 60 F , F , F , + 40 F , F , F , + 10 F , F , F , − F , F , + 30 F , F , + 20 F , F , F , − F , F , − F , F , − F , F , − F , F , + F , (D15)ˆ C (4 , = − (cid:0) F , − F , (cid:1) F , + 192 F , F , F , + 12 (cid:0) F , − F , (cid:1) F , F , − (cid:0) F , − F , (cid:1) F , F , − (cid:0) F , − F , (cid:1) F , − (cid:0) F , F , + 12 F , F , (cid:1) + 24 F , F , F , − F , (cid:0) F , F , + 2 F , F , F , (cid:1) + 12 (cid:0) F , F , + 4 F , F , F , + (cid:0) F , + 2 F , F , (cid:1) F , (cid:1) − (cid:0) F , + 2 F , F , (cid:1) + 16 F , ( F , F , + F , F , ) − F , F , + 2 F , F , + F , F , )+ (cid:0) F , − F , (cid:1) F , − F , F , + F , (D16)ˆ C (3 , = 2 (cid:0) − F , + 36 F , F , − F , (cid:1) F , + 18 (cid:0) F , − F , (cid:1) F , F , − (cid:0) − F , + 18 F , F , − F , (cid:1) F , F , − F , (cid:0) F , F , + 6 F , F , (cid:1) + 2 (cid:0) F , + 18 F , F , F , + 3 F , F , (cid:1) − (cid:0) F , − F , (cid:1) ( F , F , + F , F , )+ 18 F , ( F , F , + 2 F , F , + F , F , ) − F , F , + 3 F , F , + 3 F , F , + F , F , ) + (cid:0) − F , + 6 F , F , − F , (cid:1) F , + 3 (cid:0) F , − F , (cid:1) F , − F , F , + F , . (D17)These results extend the formulae provided in Appendix A of Ref. [27]. [1] M. A. Stephanov, Non-perturbative quantum chromodynamics. Proceedings, 8th Workshop,Paris, France, June 7-11, 2004 , Prog. Theor. Phys. Suppl. , 139 (2004), [Int. J. Mod. Phys.A20,4387(2005)], arXiv:hep-ph/0402115 [hep-ph].[2] P. Braun-Munzinger and J. Wambach, Rev. Mod. Phys. , 1031 (2009), arXiv:0801.4256 [hep-ph].[3] P. Braun-Munzinger, V. Koch, T. Sch¨afer, and J. Stachel, Phys. Rept. , 76 (2016), arXiv:1510.00442[nucl-th].[4] A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov, and N. Xu, Phys. Rept. , 1 (2020),arXiv:1906.00936 [nucl-th].[5] S. Jeon and V. Koch, Phys. Rev. Lett. , 2076 (2000), arXiv:hep-ph/0003168 [hep-ph].[6] M. Asakawa, U. W. Heinz, and B. Muller, Phys. Rev. Lett. , 2072 (2000), arXiv:hep-ph/0003169[hep-ph].[7] M. Gazdzicki, M. I. Gorenstein, and S. Mrowczynski, Phys. Lett. B585 , 115 (2004), arXiv:hep-ph/0304052 [hep-ph].[8] M. I. Gorenstein, M. Gazdzicki, and O. S. Zozulya, Phys. Lett.
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