Cumulants and factorial cumulants in the 3-dimensional Ising universality class
aa r X i v : . [ nu c l - t h ] J a n Cumulants and factorial cumulants in the 3-dimensional Ising universality class
Xue Pan, ∗ Mingmei Xu, and Yuanfang Wu † School of Electronic Engineering, Chengdu Technological University, Chengdu 611730, China Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China
The high-order cumulants and factorial cumulants of conserved charges are suggested to studythe critical dynamics in heavy ion collisions. In this paper, using parametric representation of the3-dimensional Ising model, the sign distribution on the phase diagram and temperature dependenceof the cumulants and factorial cumulants is studied and compared. In the vicinity of the criticalpoint, the cumulants and factorial cumulants can not be distinguished. Far away from the criticalpoint, sign changes occur in the factorial cumulants comparing with the same order cumulants. Thecause of these sign changes is analysed. They may be used to measure the distance to the criticalpoint.
PACS numbers: 25.75.Nq; 05.50.+q; 64.60.-i; 24.60.-k;
1. Introduction
Searching for the critical point and locating the phase boundary of quantum chromo-dynamics(QCD) phase transition are main goals of current relativistic heavy-ion collision experiments [1].The high-order cumulants of conserved charges, especially for net-baryon number, have been studiedfor many years. They reflect the fluctuations of the conserved charges. They are more sensitive to thecorrelation length and can be measured and calculated by experiments and theory, respectively [2–5].Lots of results of the high-order cumulants have been presented from experiments and theory [6–14]. The sign change and non-monotonic behavior of the high-order cumulants are considered to berelated to the critical signals.Recently, the other kind of moments, the factorial cumulants, which are also known as the inte-grated multi-particle correlations, get a lot of attention [15–23]. On one hand, they have advantagesin the analysis of the acceptance dependence in experiments [17]. They are useful in the reconstruc-tion of the original cumulants from the incomplete information obtained experimentally [21, 24].The uses of the factorial cumulants in the analysis of data in the heavy ion collisions have beenproposed [19, 22, 23]. On the other hand, multi-proton correlations have been found in the STARdata, at least at the lower energies [19, 22]. In Ref. [19], using the parametric representation of Isingmodel, the authors showed that the signs of the second to fourth order factorial cumulants are auseful tool to exclude regions in the QCD phase diagram close to the critical point. In the othermodel, very large values for the fifth and sixth order factorial cumulants have been predicted, whichcan be tested in experiments [25].The high-order cumulants are related to the generalized susceptibilities which can be got from thederivatives of the free energy with respect to the chemical potential. Their properties are explicablewith that clear definition. While the n -th order factorial cumulant of particle distribution reflectsthe integrated n -particle correlation, which can not be calculated directly. One way is to expressand calculate the factorial cumulants indirectly by the cumulants [19, 21]. In fact, cumulants andfactorial cumulants can not be distinguished and can be seen as identical in the vicinity of the critical ∗ Electronic address: [email protected] † Electronic address: [email protected] point in a model of critical fluctuations in Ref. [17].To make clear the meaning of the signs of factorial cumulants, and the relation to the cumulants,further study and comparison should be useful.The QCD critical point, if exists, is expected to belong to the 3-dimensional Ising universalityclass [26–29]. Critical behavior of the thermodynamics which is controlled by the critical exponents isthe same for different systems in the same universality class. By using the parametric representationof the 3-dimensional Ising model, we study the signs of the fifth and sixth order cumulants andfactorial cumulants on the phase diagram. The reason of sign change of factorial cumulants relative tocumulants is analysed. The temperature dependence of the second to sixth order factorial cumulantsand cumulants is discussed and compared at different distances to the phase boundary.The paper is organized as following. In section 2, the Ising model in the parametric representationis introduced. The parametric expressions of the second to sixth order cumulants are derived. Therelation between the factorial cumulants and cumulants is presented. In section 3, the signs ofthe fifth and sixth order factorial cumulants and cumulants on the phase diagram are studied andcompared. In section 4, the temperature dependence of the second to the sixth order cumulantsand factorial cumulants is shown and discussed. Finally, the conclusions and summary are given insection 5.
2. Cumulants and factorial cumulants in the parametric representation
There are two parameters in the 3-dimensional Ising model, the temperature T and the externalmagnetic field H . Using T c representing the critical temperature, in the parametric representation,the magnetization M (order parameter of the Ising model) and reduced temperature t = ( T − T c ) /T c can be parameterized by two variables R and θ [30, 31], M = R β θ, t = R (1 − θ ) . (1)The equation of state of the 3-dimensional Ising model can be given by the parametric represen-tation in terms of R and θ as H = R βδ h ( θ ) . (2)Where β and δ are critical exponents of the three-dimensional Ising universality class with values0.3267(10) and 4.786(14), respectively [32]. Because M is an odd function of the magnetic field M ( − H ) = − M ( H ), the function h ( θ ) should be analytic and an odd function of θ . One simplechoice of h ( θ ) in the form of the linear parametric model is as follows [30], h ( θ ) = θ (3 − θ ) . (3)The n -th order cumulant can be got from the derivatives of M with respect to H , κ n ( t, H ) = ( ∂ n − M∂H n − ) (cid:12)(cid:12)(cid:12)(cid:12) t . (4)When taking the approximate values of the critical exponents β = 1 / δ = 5, the first sixth-ordercumulants in the parametric presentations are as follows, TABLE I: Values for the expansion coefficients a n . a a a a a a a -18225 86670 -33966 8244 -741 14 4TABLE II: Values for the expansion coefficients b n . b b b b b b b b b b -98415 3306744 -11234619 7120872 -2736261 501120 -53001 1560 -8 8 κ ( t, H ) = R / θ,κ ( t, H ) = 1 R / (2 θ + 3) ,κ ( t, H ) = 4 θ ( θ + 9) R ( θ − θ + 3) ,κ ( t, H ) = 12 (2 θ − θ + 105 θ − θ + 81) R / ( θ − (2 θ + 3) ,κ ( t, H ) = 48 θ P n =6 n =0 a n θ n R / ( θ − (2 θ + 3) ,κ ( t, H ) = 240 P n =9 n =0 b n θ n R ( θ − (2 θ + 3) , (5)where a n and b n are the expansion coefficients of the fifth and sixth-order cumulants, respectively.Their values are listed in TABLE I and TABLE II.The first sixth-order factorial cumulants ( C n ) can be expressed by the cumulants as follows [21], C = κ ,C = κ − κ ,C = κ − κ + 2 κ ,C = κ − κ + 11 κ − κ ,C = κ − κ + 35 κ − κ + 24 κ ,C = κ − κ + 85 κ − κ + 274 κ − κ . (6)From Eq. (5), it is clear that the odd-order cumulants are odd functions of θ , while the even-ordercumulants are even functions of θ . Because H is also an odd function of θ , so the odd-order andeven-order cumulants are odd and even functions of H , respectively. The relation can be expressedas follows, κ n − ( − H ) = − κ n − ( H ) , κ n ( − H ) = κ n ( H ) , n = 1 , , .... (7)In Ref. [9] and [33], it has shown that the sign distributions of the fourth and sixth order cumulantson the phase diagram are symmetric around the axis H = 0.Turn to the high-order factorial cumulants, because they mix each order of cumulants, they are notodd or even function of the magnetic field any more. Sign distributions of the second to the fourthorder factorial cumulants do not have symmetries about the magnetic field any more as showedin Ref. [19]. What is more, because the higher the order of the cumulants, the more sensitive tothe correlation length, one can predict from Eq. (6) that the highest cumulant will dominate the behavior of the corresponding factorial cumulant in the vicinity of the critical point. When far awayfrom the critical point, this dominance will disappear.
3. Sign distributions of the fifth and sixth-order cumulants andfactorial cumulants
To compare the cumulants and factorial cumulants clearly, using the parametric representationof the 3-dimensional Ising universality class, the signs of the fifth and sixth-order cumulants andfactorial cumulants, κ , κ , C and C on the phase diagram are studied and shown in Fig. 1(a), (b),(c) and (d), respectively. The red point at H = 0 , t = 0 in each sub-figure represents the criticalpoint. The black solid line is the first order phase transition line while the black dash line representsthe crossover. In the green area, the values of the corresponding cumulants are negative. While theyare positive in the white area. κ >0 κ <0 κ <0 κ >0 (a) κ >0 κ <0 κ <0 κ >0 (a) H -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t -1-0.8-0.6-0.4-0.200.20.40.60.81 κ >0 κ >0 κ <0 κ <0 (b) H -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t -1-0.8-0.6-0.4-0.200.20.40.60.81 C >0C >0C <0 C >0 (c) H -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t -1-0.8-0.6-0.4-0.200.20.40.60.81 C >0 C <0 C <0C <0 (d) H -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t -1-0.8-0.6-0.4-0.200.20.40.60.81 FIG. 1: (Color online). Sign distributions of κ (a), κ (b), C (c) and C (d) in H − t plane. Their values are positive in thewhite areas and negative in the green areas, respectively. Sign distribution of κ and κ conform to the parity in Eq. (7). The sign of κ is opposite on thetwo sides of the axis H = 0, while it is the same for κ as showed in Fig. 1(a) and (b), respectively. InFig. 1(c) and (d), there are no symmetries of the sign distributions of C and C about the magnetic field.Comparing the sign distribution of κ and C in Fig. 1(a) and (c), it is clear that their signdistribution is similar in the vicinity of the critical point, such as in the area surrounded by redcircles. The same cases occur for κ and C . This is because that in the vicinity of the critical point,the behavior of C n is dominated by κ n .When far away from the critical point, the similarity of C n and κ n disappears. For example, atthe lower temperature side and positive magnetic field, signs of C and κ are opposite as shownin the lower right corner of Fig. 1(a) and Fig. 1(c), so as C and κ as shown in the lower rightcorner in Fig. 1(b) and Fig. 1(d). For the second, third and fourth-order cumulants and factorialcumulants, their signs are also opposite at the lower temperature side and positive magnetic field [9,19]. One need to keep in mind that κ is positive and κ is negative in the whole phase diagram,sign distribution of κ is shown in Ref. [9] and sign distributions of C to C can be inferred fromRef. [19]. In fact, in the lower temperature side, the sign of C n is dominated by the term related to κ in Eq. (6).Take C as an example and analyze the reason in detail. C is made up of six terms in Eq. (6). Atthe lower temperature side and a positive magnetic field, signs of C and the six terms are shown inTABLE III. TABLE III: Sign of C and each term at the lower temperature side and positive magnetic field. C κ -15 κ κ -225 κ κ -120 κ - + + + + + - It is clear that the sign of C is up to the last term which is related to κ . As we know, thelower the temperature, the more ordered the system, the bigger the value of magnetization. Whenapproaching to the lower temperature side, the absolute value of κ (the magnetization) gets biggerwhile the higher order cumulants are approaching to zero. The last term in Eq. (6) determines thatthe even order factorial cumulants have different sign with κ and the odd order have the same signwith κ . What is more, κ is positive at H >
H <
0. That explains why C isnegative at H >
H <
0, while C is reversed.Overall, at the lower temperature side far away from the critical temperature, the sign of C n depends on the last term in Eq. (6) which is related to κ . Comparing C n and κ n , if their signs areopposite for each order, it may predict that the system is in the ordered phase and far away fromthe critical point.At the higher temperature side, κ or the magnetization is approaching to zero, the last term in C n will not dominate its sign any more. The influences of the other terms related to the lower ordersof cumulants show up. For example, in the up right corner of Fig. 1(a) and (c), the sign of κ and C is opposite. Signs of C and each term made up of C in Eq. (6) are shown in TABLE IV. It isclear that the sign of C is consistent with the terms which are related to κ and κ . It is oppositewith the terms related to higher order cumulants κ and κ , and also κ . TABLE IV: Sign of C and each term at the higher temperature side and positive magnetic field. C κ -10 κ κ -50 κ κ - + + - - + In summary, in the vicinity of the critical point, sign distributions of C and κ , C and κ arestill consistent. The higher the order of the cumulants, the more sensitive to the correlation length,the stronger of the singularity. The first term of C n at the right of Eq. (6), i.e. κ n , dominates itssign. Far away from the critical point, the terms on the right side of Eq. (6) which related to thelower orders of cumulants will influence the sign of C n , resulting the sign difference between C n and κ n . The comparison of the signs of C n and κ n can give some information about the distance to thecritical point. To learn more about their behavior, at different distances to the phase boundary,their temperature dependence is studied in the next section.
4. Temperature dependence of the cumulants and factorial cumulants
The temperature dependence of the second to sixth-order cumulants and factorial cumulants atdifferent positive and negative magnetic fields is studied, respectively. The bigger the absolute valueof H , the further away from the phase boundary.Temperature dependence of the even-order cumulants κ n and factorial cumulants C n for n =1 , , H = 0 . , . , . , − .
05 are shown in Fig. 2(a), (b), (c) and (d), respec-tively. For the sake of comparison, they have been normalized by their maximum. For κ n , thequalitative temperature dependence does not change with magnetic field. κ keeps a peak structureand positive. At all of the magnetic fields, temperature dependence of κ and κ oscillates. Theratios of maximum to minimum for κ and κ approximate −
28 and −
6, respectively [9, 33]. t -0.3 -0.2 -0.1 0 0.1 0.2 0.3 κ o r C -0.200.20.40.60.81 (a) H=0.01 κ C κ C κ C t -0.4 -0.2 0 0.2 0.4 κ o r C -0.200.20.40.60.81 (b) H=0.05 t -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 κ o r C -0.200.20.40.60.81 (c) H=0.2 t -0.4 -0.2 0 0.2 0.4 κ o r C -0.200.20.40.60.81 (d) H=-0.05 FIG. 2: (Color online). Temperature dependence of κ n and C n for n = 1 , , H = 0 .
01 (a), H = 0 .
05 (b), H = 0 . H = − .
05, respectively.
Temperature dependence of C n changes with the variation of H . For example, the negative values of C and C at the lower temperature side become more and more obvious as the increasing H asshown in Fig. 2(a), (b) and (c), the negative valley of C at the higher temperature side changes topositive, the negative valley of C becomes flatter. What is more, the temperature dependence of C n also differs when the magnetic field turn to negative as shown in Fig. 2(b) and (d).Comparing κ n and C n , with the increasing H and far away from the phase boundary, the positivepeak structure in the vicinity of critical temperature does not change, is their common feature, whiletheir differences increase in the higher or lower temperature side, even sign changes occur.Temperature dependence of the odd order cumulants κ n − and factorial cumulants C n − for n = 2 , H = 0 . , . , . , − .
05 are shown in Fig. 3(a), (b), (c) and (d),respectively. For the sake of comparison, they have been normalized by their absolute values of theminimum for a positive H and maximum for a negative H , respectively. For κ n − , the qualitativetemperature dependence does not change with magnetic field. κ keeps a valley structure andnegative for a positive magnetic field as shown in Fig. 3(a), (b), (c), and a peak structure for anegative magnetic as shown in Fig. 3(d). Temperature dependence of κ oscillates. The ratios of theminimum to maximum for κ approximates −
10 at positive magnetic fields and − . t -0.2 -0.1 0 0.1 0.2 κ - o r C - -1-0.8-0.6-0.4-0.200.2 (a) H=0.01 κ C κ C t -0.4 -0.2 0 0.2 0.4 κ - o r C - -1-0.8-0.6-0.4-0.200.2 (b) H=0.05 t -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 κ - o r C - -1-0.8-0.6-0.4-0.200.2 (c) H=0.2 t -0.4 -0.2 0 0.2 0.4 κ - o r C - -0.200.20.40.60.81 (d) H=-0.05 FIG. 3: (Color online). Temperature dependence of κ n − and C n − for n = 2 , H = 0 .
01 (a), H = 0 .
05 (b), H = 0 . H = − .
05, respectively.
Temperature dependence of C n − changes with the variation of H . It is clear that at H = 0 .
01 as shown in Fig. 3(a), κ and κ almost overlap with C and C , respectively. When H increases and faraway from the phase boundary, the negative valley structure in the vicinity of critical temperaturedoes not change, is the common feature of κ n − and C n − at a positive magnetic field. For a negativemagnetic field, their common feature is a positive peak in the vicinity of the critical temperature.The difference between κ n − and C n − increases in the higher or lower temperature side when faraway from the phase boundary, even sign changes occur.In the vicinity of the critical point, the highest order cumulant dominates the temperature de-pendence of factorial cumulant, a peak or valley structure is their common feature. Cumulants andfactorial cumulants can not be distinguished in the 3-dimensional Ising universality class, consistentwith the model of critical fluctuations in Ref. [17].When it is far away from the phase boundary, at the lower or higher temperature side, the tem-perature dependence of factorial cumulants has great changes comparing with the correspondingcumulants, even sign changes. Comparison of the temperature dependence and sign of cumulantsand factorial cumulants may help to measure the distance to the critical point.
5. Summary and conclusions
By using the parametric representation of the 3-dimensional Ising model, sign distributions ofthe fifth and sixth order cumulants and factorial cumulants have been studied and compared. Wefound that in the vicinity of the critical point, sign distributions of the same order cumulant andfactorial cumulant are consistent, while far away from the critical point, sign of factorial cumulantswill change under the influence of the terms related with the lower order cumulants. Especially, atthe lower temperature side, sign of the factorial cumulants just depends on the term related to thefirst-order cumulant, i.e. the magnetization.Through the comparison of temperature dependence of cumulants and factorial cumulants atdifferent magnetic fields, we found that the behavior of cumulants are more stable. Its qualitativetemperature dependence does not change with the variation of the distance to the phase boundary.The ratio of the peak hight and valley depth in the temperature dependence of the fourth, fifth andsixth order cumuants keep the same at different magnetic fields. While the latter changes with thedistance to the phase boundary. Temperature dependence of the two kinds of cumulants is almostthe same in the vicinity of the critical point, showing a positive peak or negative valley structure.While obvious sign changes of factorial cumulants occur in the higher or lower temperature side.These features may help to evaluate the distance to the critical point. [1] Adams J et al (STAR Collaboration) 2005
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