^{3}

We propose a new method to construct canonical partition functions of quantum chromodynamics (QCD) from net number distributions, such as net baryon, net charge, and net strangeness, by using only the CP symmetry. To demonstrate the method, we apply it to the net-proton number distribution

^{3}

When temperature and density are varied, quantum chromodynamics (QCD) is expected to have a rich phase structure [

The Relativistic Heavy Ion Collider (RHIC) was built to explore the properties of QCD matter [

Usually, data obtained at a given colliding energy are assigned a set of temperature

In this paper, we propose a method by which we can obtain information on the QCD phase diagram not only at the experimental

In addition, even without direct lattice QCD calculations in physical chemical potential regions, we can calculate the canonical partition functions, which help us to understand QCD at finite density. The method provides us with an approach beyond the Taylor expansion method; namely, it is possible to calculate large-

In Sect. 2, we describe how to extract the canonical partition functions,

The grand partition function

Because of the charge-parity symmetry,

Experimental multiplicity data

Fugacity

The fugacity

Freeze-out (Ref. [ |
Freeze-out (Ref. [ |
||
---|---|---|---|

11.5 | 7.483 31 |
8.040 | 11.1 |

19.6 | 3.203 76 |
3.623 | 3.659 |

27 | 2.439 56 |
2.615 | 2.573 |

39 | 1.883 36 |
1.981 | 1.936 |

62.4 | 1.533 77 |
1.551 | 1.5573 |

200 | 1.174 99 |
1.152 | 1.1800 |

Figure

We assume that the net-proton multiplicity data are approximately proportional to those of the baryon.^{1} This approximation is justified if (i) after the chemical freeze-out, the net-proton number is effectively constant, or (ii) a created fireball is approximately isoneutral. See also Sect. 3 in Ref. [

From the grand partition function given by Eq. (

the values of the final three

the final two

As an example, we plot the number susceptibility

Number susceptibility,

Let us suppose that, as

Ratio of the moments

Potential regions of hidden phase transition. The regions are estimated as the complement of areas where no transition is evident.

We take the points where the lower curve of

Next we extend the fugacity

The first pioneering work to calculate the LYZs of the lattice QCD was carried out by Barbour and Bell [

Although there have been many phenomenological attempts to extract information on the QCD phase [

For this purpose it is important to reliably determine the LYZs; i.e., all zeros must be found without ambiguity, and their positions in the complex fugacity plane must be determined with high accuracy. We obtain the LYZs as follows. We first map the problem to the calculation of the residue of

Schematic of the cBK contours in the divide-and-conquer search for residues.

Figure

All calculations were performed using the multiple-precision package FMlib [

First, we study the LYZs obtained by lattice QCD simulation. Here we do not distinguish between

We update 11 000 trajectories, including 3000 thermalization trajectories. The measurement is performed every 10 (20) trajectories for an

LYZ diagram from the lattice QCD.

The LYZ diagram of the lattice QCD above the phase-transition temperature is shown in Fig.

In Ref. [

Because the

Roberge and Weiss discussed the regions of pure imaginary chemical potential and found that, at

Using the same lattice setup as that in the present study, the quantity

The fugacity,

In the left panel of Fig.

The grand partition function

Using the relation
^{2} . The symmetry of

Whether zeros appear in Eq. (

LYZ diagram from the RHIC data at

LYZ diagram from the RHIC data at

We next consider the LYZ diagrams obtained from the RHIC data. We construct the grand partition function, Eq. (

Although some zeros exist on the negative real axis, they do not form a line that clearly characterizes the RW transition, which suggests that the data correspond to temperatures below

In the LYZ diagrams obtained from the RHIC data, no zero appears on the positive real-

Several LYZ points appear on the unit circle for all the RHIC data. To understand the meaning of this, we calculate the LYZ diagram for the Skellam model. This is a simple probabilistic model based on the difference between two Poisson distribution variables

LYZ diagram of the Skellam distribution, in which

Finally, we estimate where the LYZs would intercept or approach the positive real axis as the volume increases, which indicates the QCD phase transition. These zones are indicated by double-headed arrows in the inset of Fig.

A simple but important relation discussed in this paper is

We have shown how to construct

In the lattice QCD simulation, we consider the path integral formula of the grand partition function,

The canonical partition functions,

This is very important for exploring the QCD phase boundary: So far, analyses such as the moments have been done on the freeze-out points. But the freeze-out points realized in the beam energy scan experiments are in the confinement regions, and not very near to the phase transition. Indeed, Fukushima estimated the baryon density on the freeze-out line using the hadron resonance gas model, and found that the maximum of the baryon density is realized at

We calculated the moments (

The formula (

There have been claims that the RW transition formulated in Euclidean space cannot be interpreted as a physics object in Minkowski space by considering its cosmological consequences [^{3}

The approach investigated here is based on simple statistical mechanics, and is easy to use for extracting information from experimental data. It works equally well in analyzing experimental data and lattice QCD simulations. In the latter case, we can study the real chemical potential regions without Taylor expansions.

Several problems that should be clarified in future are as follows:

We can study finite real chemical potential regions in a lattice QCD simulation and the sign problem does not appear here. A possible obstacle is the overlap problem. In our lattice Monte Carlo simulations, the gauge configurations are produced at ^{4} On this circle,

Since ^{5} , or the

It is difficult to measure experimentally the net baryon multiplicity. One possible approach is to study the difference between net-proton multiplicity and net-baryon multiplicity.

Net strangeness and net charge multiplicity are analyzed in the same way.

The relation between the order of the RW transition and the Lee–Yang zeros should be studied more quantitatively near the end-point. The volume dependence of Lee–Yang zeros in the vicinity of a transition point depends on the order of the transition. If the transition is of second order, Lee–Yang zeros approach the RW phase points according to the volume dependence with a critical exponent.

The Lee–Yang zeros calculated in the lattice QCD simulation at high temperature suggest the RW transition. But there might be a danger of misidentifying the thermal singularity with the Lee–Yang zeros. The former appears as a branch point at

The Lee–Yang zeros are approaching to the real positive axis if the system indicates the phase transition. Therefore, the volume dependence of the Lee–Yang zeros is interesting. Such information may be obtained by varying the atomic number of the beam/target.

Recently, based on the canonical partition functions, detailed analyses of the effects of

Both the experimental and lattice QCD data include errors. It is important to check whether the obtained Lee–Yang zeros are stable or not against these errors, since it requires extreme care to calculate zeros of high-order polynomials. In lattice QCD, we have investigated the statistical errors of Lee–Yang zeros caused by the statistical error of

Open Access funding: ^{3}

This work grew out of a stimulus provided to one of the authors (A.N.) by L. McLerran and N. Xu at the “QCD Structure” workshop in Wuhan. We thank N. Xu, X. Luo, C. Sasaki, K. Shigaki, M. Kitazawa, and V. Skokov for valuable discussions. We are indebted to Ph. de Forcrand and K. Morita for critical reading of the manuscript and valuable comments. We wish to thank S. Aoki, T. Hatsuda, K. Redlich, and M. Yahiro for their continuous interest and encouragement. The work was completed after very stimulating discussion with B. Friman, J. Knoll, V. Koch, K. Morita, and J. Wambach at GSI. The calculations were performed on SX-9, SX-ACE, and SAHO, at RCNP Osaka, RICC at Riken, and SR16000 at KEK. This work was supported by Grants-in-Aid for Scientific Research 20105003-A02-0001, 23654092, and 24340054.

In this appendix, we give

0 | 0.100 00E+01 | 0.621 12E |

1 | 0.623 61E+00 | 0.697 41E |

2 | 0.332 22E+00 | 0.668 95E |

3 | 0.125 97E+00 | 0.456 72E |

4 | 0.402 92E |
0.263 01E |

5 | 0.104 19E |
0.122 45E |

6 | 0.239 49E |
0.506 81E |

7 | 0.468 82E |
0.178 63E |

8 | 0.835 46E |
0.573 15E |

9 | 0.131 10E |
0.161 94E |

10 | 0.187 49E |
0.416 98E |

11 | 0.245 29E |
0.982 22E |

12 | 0.295 22E |
0.212 85E |

13 | 0.327 86E |
0.425 61E |

14 | 0.337 56E |
0.789 01E |

15 | 0.332 51E |
0.139 94E |

16 | 0.300 00E |
0.227 32E |

17 | 0.261 63E |
0.356 94E |

18 | 0.210 94E |
0.518 18E |

19 | 0.158 16E |
0.699 54E |

20 | 0.112 58E |
0.896 53E |

21 | 0.837 12E |
0.120 03E |

22 | 0.566 68E |
0.146 30E |

23 | 0.385 19E |
0.179 05E |

24 | 0.181 80E |
0.152 15E |

25 | 0.702 47E |
0.105 85E |

26 | 0.821 36E |
0.222 85E |

27 | 0.313 60E |
0.153 20E |

28 | 0.349 22E |
0.307 17E |

0 | 0.100 00E+01 | 0.113 60E |

1 | 0.884 57E+00 | 0.161 50E |

2 | 0.655 21E+00 | 0.199 45E |

3 | 0.402 03E+00 | 0.198 13E |

4 | 0.205 50E+00 | 0.144 53E |

5 | 0.903 48E |
0.114 42E |

6 | 0.345 69E |
0.709 07E |

7 | 0.116 65E |
0.387 55E |

8 | 0.352 14E |
0.189 49E |

9 | 0.968 01E |
0.843 71E |

10 | 0.238 49E |
0.336 68E |

11 | 0.544 73E |
0.124 56E |

12 | 0.113 84E |
0.421 60E |

13 | 0.220 56E |
0.132 31E |

14 | 0.401 06E |
0.389 68E |

15 | 0.674 42E |
0.106 14E |

16 | 0.109 94E |
0.280 23E |

17 | 0.160 66E |
0.663 33E |

18 | 0.214 51E |
0.143 45E |

19 | 0.276 32E |
0.299 29E |

20 | 0.395 70E |
0.694 20E |

21 | 0.465 94E |
0.132 40E |

22 | 0.461 70E |
0.212 50E |

23 | 0.576 45E |
0.429 73E |

0 | 0.100 00E+01 | 0.554 68E |

1 | 0.903 62E+00 | 0.835 94E |

2 | 0.681 68E+00 | 0.105 15E |

3 | 0.431 57E+00 | 0.108 29E |

4 | 0.232 45E+00 | 0.954 00E |

5 | 0.108 13E+00 | 0.738 58E |

6 | 0.442 16E |
0.588 95E |

7 | 0.159 62E |
0.308 83E |

8 | 0.520 22E |
0.167 18E |

9 | 0.153 12E |
0.817 33E |

10 | 0.412 35E |
0.365 60E |

11 | 0.100 14E |
0.147 48E |

12 | 0.227 25E |
0.555 89E |

13 | 0.476 93E |
0.193 78E |

14 | 0.936 67E |
0.632 14E |

15 | 0.167 94E |
0.188 26E |

16 | 0.307 99E |
0.573 46E |

17 | 0.471 06E |
0.145 69E |

18 | 0.699 13E |
0.359 14E |

19 | 0.115 24E |
0.983 28E |

20 | 0.932 32E |
0.132 13E |

21 | 0.178 34E |
0.419 84E |

0 | 0.100 00E+01 | 0.206 03E |

1 | 0.913 88E+00 | 0.292 89E |

2 | 0.697 79E+00 | 0.349 96E |

3 | 0.451 49E+00 | 0.350 06E |

4 | 0.250 20E+00 | 0.305 16E |

5 | 0.120 24E+00 | 0.224 48E |

6 | 0.508 05E |
0.149 14E |

7 | 0.191 22E |
0.932 27E |

8 | 0.646 36E |
0.512 54E |

9 | 0.197 62E |
0.220 24E |

10 | 0.548 91E |
0.953 11E |

11 | 0.140 64E |
0.380 48E |

12 | 0.327 79E |
0.138 16E |

13 | 0.733 96E |
0.481 97E |

14 | 0.156 25E |
0.159 86E |

15 | 0.282 34E |
0.450 05E |

16 | 0.508 70E |
0.126 34E |

17 | 0.107 89E |
0.417 46E |

18 | 0.184 28E |
0.111 09E |

19 | 0.127 62E |
0.119 87E |

20 | 0.338 82E |
0.495 81E |

21 | 0.119 93E |
0.273 44E |

0 | 0.100 00E+01 | 0.473 05E |

1 | 0.921 68E+00 | 0.796 53E |

2 | 0.722 16E+00 | 0.113 85E |

3 | 0.481 91E+00 | 0.139 29E |

4 | 0.277 92E+00 | 0.146 14E |

5 | 0.140 35E+00 | 0.133 33E |

6 | 0.624 29E |
0.111 01E |

7 | 0.248 31E |
0.785 10E |

8 | 0.892 21E |
0.517 45E |

9 | 0.288 87E |
0.338 28E |

10 | 0.869 15E |
0.158 45E |

11 | 0.233 70E |
0.834 09E |

12 | 0.606 94E |
0.395 64E |

13 | 0.140 52E |
0.167 30E |

14 | 0.301 32E |
0.655 23E |

15 | 0.569 45E |
0.226 16E |

16 | 0.103 96E |
0.754 08E |

17 | 0.266 27E |
0.352 77E |

18 | 0.473 47E |
0.114 57E |

0 | 0.100 00E+01 | 0.165 46E |

1 | 0.920 30E+00 | 0.260 94E |

2 | 0.720 70E+00 | 0.350 16E |

3 | 0.483 88E+00 | 0.402 86E |

4 | 0.280 66E+00 | 0.400 41E |

5 | 0.142 65E+00 | 0.348 75E |

6 | 0.639 67E |
0.267 97E |

7 | 0.256 82E |
0.184 36E |

8 | 0.937 39E |
0.115 31E |

9 | 0.305 91E |
0.644 84E |

10 | 0.957 99E |
0.346 04E |

11 | 0.243 94E |
0.150 99E |

12 | 0.572 85E |
0.607 60E |

13 | 0.181 42E |
0.329 74E |

14 | 0.319 23E |
0.994 25E |

15 | 0.929 47E |
0.496 06E |

The ratio of the moments

The ratio of the moments

The proportionality factor has no effect on any of the results reported in this paper.

This holds for both

The authors thank the referee for pointing out this argument.

This is, of course, on the condition that the number operator

We thank K. Splittorf for drawing our attention to this point.