HTML
--> --> --> -->A.Cumulants and mixed-cumulants
In statistics, any distribution can be characterized by different order moments or cumulants. The rth-order moment of variable N is defined by the rth order derivative of moment generating function $ G(\theta) = \sum\limits_{N}{\rm e}^{N\theta}P(N) = {\langle{{\rm e}^{N\theta}}\rangle} , $ | (1) |
$ {\langle{N^{r}}\rangle} = \frac{{\rm d}^{r}}{{\rm d}\theta^{r}}G(\theta)\Bigl|_{\theta = 0}, $ | (2) |
$ T(\theta) = {\rm{ln}}G(\theta), $ | (3) |
$ {\langle{N^{r}}\rangle} _{\rm{c}} = \frac{{\rm d}^{r}}{{\rm d}\theta^{r}}T(\theta)\Bigl|_{\theta = 0}, $ | (4) |
From Eqs. (1)–(4), the 1st and 2nd-order cumulants are expressed in terms of moments
$ {\langle{N}\rangle} _{\rm{c}} = {\langle{N}\rangle} , $ | (5) |
$ {\langle{N^{2}}\rangle} _{\rm{c}} = {\langle{N^{2}}\rangle} - {\langle{N}\rangle} ^{2}, $ | (6) |
$ G(\theta_{1},\theta_{2}) = \sum\limits_{N_{1},N_{2}}{\rm e}^{\theta_{1}N_{1}}{\rm e}^{\theta_{2}N_{2}}P(N_{1},N_{2}) = {\langle{{\rm e}^{\theta_{1}N_{1}}{\rm e}^{\theta_{2}N_{2}}}\rangle} , $ | (7) |
$ {\langle{N_{1}^{r_{1}}N_{2}^{r_{2}}}\rangle} = \frac{\partial^{r_{1}}}{\partial\theta_{1}^{r_{1}}}\frac{\partial^{r_{2}}}{\partial\theta_{2}^{r_{2}}} G(\theta_{1},\theta_{2})\Big|_{\theta_{1} = \theta_{2} = 0}. $ | (8) |
$ T(\theta_{1},\theta_{2}) = {\rm{ln}}G(\theta_{1},\theta_{2}), $ | (9) |
$ {\langle{N_{1}^{r_{1}}N_{2}^{r_{2}}}\rangle} _{\rm{c}} = \frac{\partial^{r_{1}}}{\partial\theta_{1}^{r_{1}}}\frac{\partial^{r_{2}}}{\partial\theta_{2}^{r_{2}}} T(\theta_{1},\theta_{2})\Big|_{\theta_{1} = \theta_{2} = 0}. $ | (10) |
$ {\langle{N_{1}N_{2}}\rangle} _{\rm{c}} = {\langle{N_{1}N_{2}}\rangle} - {\langle{N_{1}}\rangle} {\langle{N_{2}}\rangle} . $ | (11) |
2
B.Binomial model
The particle detection efficiency of each detector is always limited. The event-by-event particle multiplicity distributions are convoluted owing to this finite detector efficiency. The efficiency correction needs to be performed to recover the true multiplicity distribution. For simplicity, we assume that the detection efficiency can be approximated by the binomial efficiency response function [46, 47]. The mean value (first-order moment/cumulant) can be easily reconstructed by division with the binomial efficiency response; however, its influence on higher-order cumulants is complicated and depends on the probability distribution of efficiency [48-50]. Throughout this paper, we focus on a simple assumption of the binomial distribution given by $ \tilde{P}(n) = \sum\limits_{N}P(N)B_{\varepsilon,N}(n), $ | (12) |
$ B_{\varepsilon,N}(n) = \frac{N!}{n!(N-n)!}\varepsilon^{n}(1-\varepsilon)^{N-n}, $ | (13) |
$ {\langle\!\langle{ K_{(x)}K_{(y)} }\rangle\!\rangle} _{\rm{c}} = {\langle{\kappa_{(1,0,1)}\kappa_{(0,1,1)}}\rangle} _{\rm{c}} + {\langle{\kappa_{(1,1,1)}}\rangle} _{\rm{c}} - {\langle{\kappa_{(1,1,2)}}\rangle} _{\rm{c}}, $ | (14) |
$ K_{(x)} = \sum\limits_{i}^{M}x_{i}n_{i},\quad K_{(y)} = \sum\limits_{i}^{M}y_{i}n_{i}, $ | (15) |
$ \kappa{(r,s,t)} = \sum\limits_{i = 1}^{M}\frac{x_{i}^{r}y_{j}^{s}}{\varepsilon_{i}^{t}}n_{i} , $ | (16) |
$ K_{(x)} = \sum\limits_{j}^{n^{\rm{tot}}}x_{j},\quad K_{(y)} = \sum\limits_{i}^{n^{\rm{tot}}}y_{j}, $ | (17) |
$ \kappa{(r,s,t)} = \sum\limits_{j = 1}^{n^{\rm{tot}}}\frac{x_{j}^{r}y_{j}^{s}}{\varepsilon_{j}^{t}}, $ | (18) |
In the rest of this section, we consider two efficiency bins for simplicity. Particles for each bin have the same efficiency values,
2
C.Mutually exclusive variable
Let's consider the correlation between two mutually exclusive variables. From Eq. (10), the mixed-cumulant of generated particles are expanded in terms of moments as $ {\langle{N_{1}N_{2}}\rangle} _{\rm{c}} = {\langle{N_{1}N_{2}}\rangle} - {\langle{N_{1}}\rangle} {\langle{N_{2}}\rangle} . $ | (19) |
$ x = (x_{1},x_{2}) = (1,0), $ | (20) |
$ y = (y_{1},y_{2}) = (0,1), $ | (21) |
$\begin{aligned}[b] {\langle\!\langle{ K_{(1,0)}K_{(0,1)} }\rangle\!\rangle} _{\rm{c}} =& {\langle{\kappa{(1,0,1)}\kappa{(0,1,1)}}\rangle} _{\rm{c}} + {\langle{\kappa{(1,1,1)}}\rangle} _{\rm{c}}\\& - {\langle{\kappa{(1,1,2)}}\rangle} _{\rm{c}}= {\Bigl\langle{\frac{n_{1}}{\varepsilon_{1}}\frac{n_{2}}{\varepsilon_{2}}}\Bigr\rangle} _{\rm{c}}\\ =& \frac{1}{\varepsilon_{1}\varepsilon_{2}}{\langle{n_{1}n_{2}}\rangle} - \frac{1}{\varepsilon_{1}\varepsilon_{2}}{\langle{n_{1}}\rangle} {\langle{n_{2}}\rangle} ,\end{aligned} $ | (22) |
2
D.Mutually inclusive variables
31.Problem
Next, we consider the correlation between $ {\langle{N_{1}(N_{1}+N_{2})}\rangle} _{\rm{c}} = {\langle{N_{1}N_{2}}\rangle} - {\langle{N_{1}}\rangle} {\langle{N_{2}}\rangle} + {\langle{N_{1}^{2}}\rangle} - {\langle{N_{1}}\rangle} ^{2}, $ | (23) |
$\begin{aligned}[b] {\langle\!\langle{K_{(1,0)}K_{(0,1)}}\rangle\!\rangle} _{\rm{c}} =& \frac{1}{\varepsilon_{1}\varepsilon_{2}'}{\langle{n_{1}(n_{1}+n_{2})}\rangle} - \frac{1}{\varepsilon_{1}\varepsilon_{2}'}{\langle{n_{1}}\rangle} {\langle{n_{1}+n_{2}}\rangle} , \\ =& \frac{1}{\varepsilon_{1}\varepsilon_{2}'}{\langle{n_{1}n_{2}}\rangle} - \frac{1}{\varepsilon_{1}\varepsilon_{2}'}{\langle{n_{1}}\rangle} {\langle{n_{2}}\rangle} \\&+ \frac{1}{\varepsilon_{1}\varepsilon_{2}'}{\langle{n_{1}^{2}}\rangle} - \frac{1}{\varepsilon_{1}\varepsilon_{2}'}{\langle{n_{1}}\rangle} ^{2},\end{aligned}$ | (24) |
$ \varepsilon_{2}' = \frac{{\langle{N_{1}}\rangle} \varepsilon_{1}+{\langle{N_{2}}\rangle} \varepsilon_{2}}{{\langle{N_{1}}\rangle} +{\langle{N_{2}}\rangle} }, $ | (25) |
$ \varepsilon_{2}' = 0.44,\; {\langle{n_{1}}\rangle} = 2,\; {\langle{n_{2}}\rangle} = 2.4, $ | (26) |
$ {\langle{n_{1}^{2}}\rangle} _{\rm{c}} = {\langle{n_{1}^{2}}\rangle} - {\langle{n_{1}}\rangle} ^{2} = 2, $ | (27) |
$ {\langle{n_{1}n_{2}}\rangle} = {\langle{n_{1}}\rangle} {\langle{n_{2}}\rangle} , $ | (28) |
$ {\langle\!\langle{K_{(1,0)}K_{(0,1)}}\rangle\!\rangle} _{\rm{c}} = 4.55. $ | (29) |
3
2.Solution
The solution is to adopt the appropriate indices for x and y in Eqs. (16) and (15). To consider $ x = (x_{1},x_{2}) = (1,0), $ | (30) |
$ y = (y_{1},y_{2}) = (1,1), $ | (31) |
$ \begin{aligned}[b]{\langle\!\langle{ K_{(1,0)}K_{(1,1)} }\rangle\!\rangle} _{\rm{c}} =& {\langle{\kappa{(1,0,1)}\kappa{(0,1,1)}}\rangle} _{\rm{c}} \\&+ {\langle{\kappa{(1,1,1)}}\rangle} _{\rm{c}} - {\langle{\kappa{(1,1,2)}}\rangle} _{\rm{c}}\\ =& \frac{1}{\varepsilon_{1}\varepsilon_{2}}{\langle{n_{1}n_{2}}\rangle} - \frac{1}{\varepsilon_{1}\varepsilon_{2}}{\langle{n_{1}}\rangle} {\langle{n_{2}}\rangle} + \frac{1}{\varepsilon_{1}^{2}}{\langle{n_{1}^{2}}\rangle} \\&- \frac{1}{\varepsilon_{1}^{2}}{\langle{n_{1}}\rangle} ^{2} + \frac{1}{\varepsilon_{1}}{\langle{n_{1}}\rangle} - \frac{1}{\varepsilon_{1}^{2}}{\langle{n_{1}}\rangle} ,\end{aligned}$ | (32) |
We summarize this section as follows. The efficiency correction formula for the 2nd-order mixed-cumulant was fully expanded for two cases: one is for two mutually exclusive variables, and the other case assumes that one variable is a subset of the other, to consider the self-correlation, as expressed in Eqs. (22) and (32). Both cases were determined to be incompatible with each other. The proper correction formulas needs to be obtained by substituting appropriate indices into Eqs. (14)–(16). This implies that the efficiency values have to be handled properly for each variable, without averaging them, especially when considering the self-correlation. It should be noted that the risk of using the averaged efficiency has already been pointed out in Ref. [55] for higher-order cumulants of single-variables. The efficiency bins always need to be carefully handled. The track-by-track efficiency via the identified particle approach expressed in Eqs. (17) and (18) would be a better way to handle all possible variations of efficiencies [56]. However, the particle identification needs to be applied to determine the efficiencies for different particle species, which does not discard a small amount of particles, depending on the overlapping area of the variables for the particle identification. This effect will be studied by numerical simulations in the next section [25, 57].
-->
A.Closure test using UrQMD model
To validate the discussion from the previous section, we have analyzed the second-order mixed cumulants from the UrQMD event generator atFigure1. (color online) Centrality dependence of second-order mixed-cumulants of net-charge (
In the next step, we correct the efficiency using input efficiency values. In this case, we adopted unidentified charged particles for the net-charge (Q), and applied Eq. (22), similar to the STAR measurement. The p-k mixed cumulant "true" value can be reproduced via this method, as they are mutually exclusive variables. However, the efficiency correction for Q-p and Q-k fails to reproduce the "true" values, as we discussed in Sec. II. We end up with a higher value for unidentified charge correlators. This leads to a higher value in cumulant ratios
Figure2. (color online) Centrality dependence of second-order off-diagonal to diagonal cumulant ratios for Au+Au collisions at 200 GeV, using the UrQMD model. The efficiency corrections are performed assuming the variables are mutually exclusive (Eq. (22)).
$ \begin{aligned}[b] {\langle\!\langle{N_{Q}N_{k}}\rangle\!\rangle} _{\rm{c}} =& {\langle\!\langle{(N_{\pi}+N_{p}+N_{k})N_{k}}\rangle\!\rangle} _{\rm{c}} \\ =& \frac{1}{\varepsilon_{1}\varepsilon_{3}}{\langle{N_{\pi} N_{k}}\rangle} - \frac{1}{\varepsilon_{1}\varepsilon_{3}}{\langle{N_{\pi}}\rangle} {\langle{N_{k}}\rangle} \\&+ \frac{1}{\varepsilon_{2}\varepsilon_{3}}{\langle{N_{p} N_{k}}\rangle} - \frac{1}{\varepsilon_{2}\varepsilon_{3}}{\langle{N_{p}}\rangle} {\langle{N_{k}}\rangle} \\&+ \frac{1}{\varepsilon_{3}^{2}}{\langle{N_{k}^{2}}\rangle} - \frac{1}{\varepsilon_{3}^{2}}{\langle{N_{k}}\rangle} ^{2} + \frac{1}{\varepsilon_{3}}{\langle{N_{k}}\rangle} - \frac{1}{\varepsilon_{3}^{2}}{\langle{N_{k}}\rangle} , \end{aligned} $ | (33) |
Figure3. (color online) Centrality dependence of second-order mixed-cumulants of identified net-charge (
Figure4. (color online) Centrality dependence of second-order off-diagonal to diagonal cumulant ratios of identified charged particles for Au+Au collisions at 200 GeV, using the UrQMD model. The efficiency corrections are performed assuming the variables are mutually inclusive (Eq. (32)).
However, owing to the particle identification with optimal purity, we may lose a few pions, protons, and kaons. We also studied the effects of these tracks for mixed cumulants via UrQMD simulations. Charged particle identification is performed using the ionization energy loss inside the time projection chamber (TPC) detector subsystem. We mimic the ionization energy loss curve in UrQMD simulation using the STAR TPC resolution. Figure 5(a) presents the measured
Figure5. (color online) (a)
$ n\sigma_{X} = \frac{1}{R} \ln \frac{[{\rm d}E/{\rm d}x]_{\rm obs}}{[{\rm d}E/{\rm d}x]_{th,X}}, $ | (34) |
Figure 6 presents the event-by-event average of net-charge, net-kaon, and net-proton multiplicities as a function of
Figure6. (color online)
Figure7. (color online)
For | Legend | Particle Code | Baseline | ||
Signal | S1 | Used | Applied | N/A | B1 |
S2 | Used | Applied | Applied | B2 | |
S3 | N/A | Applied | Applied | B2 | |
Baseline | B1 | Used | N/A | N/A | N/A |
B2 | Used | N/A | Applied | N/A |
Table1.This table describes the information that is used to select the particles for mixed-cumulant calculations in Fig. 7, in terms of the particle species code given by UrQMD,
S1 : Information on the particle species is provided by UrQMD. The
S2 : Information on the particle species is provided by UrQMD. Both
S3 : Particles are identified by both
We note that "Q" is defined as the summation of the identified π, K, and p. The "
B1 : Information on the particle species is provided by UrQMD.
B2 : Information on the particle species is provided by UrQMD. The
The difference between two baselines is owing to the particle multiplicity. There are more particles for B1 than B2 becuase the
It is determined that the values of mixed-cumulants are close to the corresponding baselines with a loose
To cancel the trivial volume dependence, the normalized mixed-cumulants are also calculated in Fig. 8 as a function of the
Figure8. (color online)
$ \tag{A2}\begin{aligned}[b] {\langle\!\langle{ K_{(x)}^2K_{(y)}^2 }\rangle\!\rangle} _{\rm{c}} = &{\langle{\kappa{(1,0,1)}^{2}\kappa{(0,1,1)}^{2}}\rangle} _{\rm{c}} + {\langle{\kappa{(1,0,1)}^{2}\kappa{(0,2,1)}}\rangle} _{\rm{c}} - {\langle{\kappa{(1,0,1)}^{2}\kappa{(0,2,2)}}\rangle} _{\rm{c}} + {\langle{\kappa{(0,1,1)}^{2}\kappa{(2,0,1)}}\rangle} _{\rm{c}} \\ & - {\langle{\kappa{(0,1,1)}^{2}\kappa{(2,0,2)}}\rangle} _{\rm{c}} +4{\langle{\kappa{(1,0,1)}\kappa{(0,1,1)}\kappa{(1,1,1)}}\rangle} _{\rm{c}} -4{\langle{\kappa{(1,0,1)}\kappa{(0,1,1)}\kappa{(1,1,2)}}\rangle} _{\rm{c}} \\ &+2{\langle{\kappa{(1,0,1)}\kappa{(1,2,1)}}\rangle} _{\rm{c}} -6{\langle{\kappa{(1,0,1)}\kappa{(1,2,2)}}\rangle} _{\rm{c}} +4{\langle{\kappa{(1,0,1)}\kappa{(1,2,3)}}\rangle} _{\rm{c}} \\ & +2{\langle{\kappa{(0,1,1)}\kappa{(2,1,1)}}\rangle} _{\rm{c}} -6{\langle{\kappa{(0,1,1)}\kappa{(2,1,2)}}\rangle} _{\rm{c}} +4{\langle{\kappa{(0,1,1)}\kappa{(2,1,3)}}\rangle} _{\rm{c}} \\ & -4{\langle{\kappa{(1,1,1)}\kappa{(1,1,2)}}\rangle} _{\rm{c}} +2{\langle{\kappa{(1,1,1)}^{2}}\rangle} _{\rm{c}} +2{\langle{\kappa{(1,1,2)}^{2}}\rangle} _{\rm{c}} \\ &+ {\langle{\kappa{(2,0,1)}\kappa{(0,2,1)}}\rangle} _{\rm{c}} - {\langle{\kappa{(2,0,1)}\kappa{(0,2,2)}}\rangle} _{\rm{c}} - {\langle{\kappa{(2,0,2)}\kappa{(0,2,1)}}\rangle} _{\rm{c}} + {\langle{\kappa{(2,0,2)}\kappa{(0,2,2)}}\rangle} _{\rm{c}} \\ & + {\langle{\kappa{(2,2,1)}}\rangle} _{\rm{c}} -7{\langle{\kappa{(2,2,2)}}\rangle} _{\rm{c}} +12{\langle{\kappa{(2,2,3)}}\rangle} _{\rm{c}} -6{\langle{\kappa{(2,2,4)}}\rangle} _{\rm{c}}, \end{aligned} $ |
$ \tag{A3}\begin{aligned}[b] {\langle\!\langle{ K_{(x)}^3 K_{(y)} }\rangle\!\rangle} _{\rm{c}} =& {\langle{\kappa{(1,0,1)}^{3}\kappa{(0,1,1)}}\rangle} _{\rm{c}} +3{\langle{\kappa{(1,0,1)}^{2}\kappa{(1,1,1)}}\rangle} _{\rm{c}} -3{\langle{\kappa{(1,0,1)}^{2}\kappa{(1,1,2)}}\rangle} _{\rm{c}} +3{\langle{\kappa{(2,0,1)}\kappa{(1,0,1)}\kappa{(0,1,1)}}\rangle} _{\rm{c}} \\ &-3{\langle{\kappa{(2,0,2)}\kappa{(1,0,1)}\kappa{(0,1,1)}}\rangle} _{\rm{c}} +3{\langle{\kappa{(1,0,1)}\kappa{(2,1,1)}}\rangle} _{\rm{c}} -9{\langle{\kappa{(1,0,1)}\kappa{(2,1,2)}}\rangle} _{\rm{c}} \\ &+6{\langle{\kappa{(1,0,1)}\kappa{(2,1,3)}}\rangle} _{\rm{c}} +3{\langle{\kappa{(2,0,1)}\kappa{(1,1,1)}}\rangle} _{\rm{c}} -3{\langle{\kappa{(2,0,1)}\kappa{(1,1,2)}}\rangle} _{\rm{c}} -3{\langle{\kappa{(2,0,2)}\kappa{(1,1,1)}}\rangle} _{\rm{c}} \\ &+3{\langle{\kappa{(2,0,2)}\kappa{(1,1,2)}}\rangle} _{\rm{c}} + {\langle{\kappa{(3,0,1)}\kappa{(0,1,1)}}\rangle} _{\rm{c}} -3{\langle{\kappa{(3,0,2)}\kappa{(0,1,1)}}\rangle} _{\rm{c}} +2{\langle{\kappa{(3,0,3)}\kappa{(0,1,1)}}\rangle} _{\rm{c}} \\& + {\langle{\kappa{(3,1,1)}}\rangle} _{\rm{c}} -7{\langle{\kappa{(3,1,2)}}\rangle} _{\rm{c}} +12{\langle{\kappa{(3,1,3)}}\rangle} _{\rm{c}} -6{\langle{\kappa{(3,1,4)}}\rangle} _{\rm{c}}. \end{aligned} $ |