Matter-Antimatter Coexistence Method for Finite Density QCD Presented at the International Workshop on “Excited QCD 2017” (eQCD 2017), May 7-13, 2017, Sintra, Portugal

# Matter-Antimatter Coexistence Method for Finite Density QCD ††thanks: Presented at the International Workshop on “Excited QCD 2017” (eQCD 2017), May 7-13, 2017, Sintra, Portugal

Hideo Suganuma Department of Physics, Graduate School of Science, Kyoto University,
Kitashirakawaoiwake, Sakyo, Kyoto 606-8502, Japan
###### Abstract

We propose a “matter-antimatter coexistence method” for finite-density lattice QCD, aiming at a possible solution of the sign problem. In this method, we consider matter and anti-matter systems on two parallel -sheets in five-dimensional Euclidean space-time. For the matter system with a chemical potential on a -sheet, we also prepare the anti-matter system with on the other -sheet shifted in the fifth direction. In the lattice QCD formalism, we introduce a correlation term between the gauge variables in and in , such as with a real parameter . In the limit of , a strong constraint is realized, and the total fermionic determinant is real and non-negative. In the limit of , this system goes to two separated ordinary QCD systems with the chemical potential of and . On a finite-volume lattice, if one takes an enough large value of , is realized and there occurs a phase cancellation approximately between two fermionic determinants in and , which is expected to suppress the sign problem and to make the lattice calculation possible. For the obtained gauge configurations of the coexistence system, matter-side quantities are evaluated through their measurement only for the matter part . By the calculations with gradually decreasing and their extrapolation to , physical quantities in finite density QCD are expected to be estimated.

\PACS

11.15.Ha, 12.38.Gc, 12.38.Mh

## 1 Introduction

The lattice QCD Monte Carlo calculation has revealed many aspects of the QCD vacuum and hadron properties in both zero and finite temperatures. At finite density, however, lattice QCD is not yet well investigated, because of a serious problem called the “sign problem” [1, 2], which originates from the complex value including minus sign of the QCD action and the fermionic determinant at finite density, even in the Euclidean metric [3]. In fact, the Euclidean QCD action at finite density with the chemical potential is generally complex,

 S[A,ψ,¯ψ;μ]=SG[A]+∫d4x{¯ψ(⧸D+m+μγ4)ψ}∈C, (1)

with the gauge action and covariant derivative . Then, the action factor cannot be identified as a probability density in the QCD generating functional, unlike ordinary lattice QCD calculations.

In this paper, aiming at a possible solution of the sign problem, we propose a new approach of a “matter-antimatter coexistence method” for lattice QCD at finite density with a general chemical potential .

## 2 Matter-Antimatter Coexistence Method

Our strategy is to use a cancelation of the phase factors of the fermionic determinants between a matter system with and an anti-matter system with , and our method is based on the general property [3],

 S[A,ψ,¯ψ;μ]∗=S[A,ψ,¯ψ;−μ∗], (2)

for the Euclidean QCD action in the presence of the chemical potential . Actually, the fermionic kernel corresponding to generally satisfies in lattice QCD, so that one finds

 [¯ψ(DF+μγ4)ψ]∗=¯ψ(DF−μ∗γ4)ψ, (3)

which leads to the relation (2), and

 Det(DF+μγ4)∗=Det(DF−μ∗γ4). (4)

### 2.1 Definition and Setup

In the “matter-antimatter coexistence method”, we consider matter and anti-matter systems on two parallel -sheets in five-dimensional Euclidean space-time. For the matter system with a chemical potential on a -sheet, we also prepare the anti-matter system with on the other -sheet shifted in the fifth direction, as shown in Fig. 1.

We put an ordinary fermion field with the mass and the gauge variable at on the matter system , and we put the other fermion field with the same mass and the gauge variable on the anti-matter system .

In the lattice QCD formalism, we introduce a correlation term between the gauge variables in and in at , such as

 Sλ≡∑x,ν2λ{Nc−Re tr[Uν(x)~U†ν(x)]} (5)

with a real parameter (), which connects two different situations: in and two separated QCD systems in . Near the continuum limit, this additional term becomes

 Sλ≃∑x12λa2{Aaν(x)−~Aaν(x)}2≃∫d4x 12λphys{Aaν(x)−~Aaν(x)}2 (6)

with .

In fact, the total lattice action in this method is written as

 S = SG[U]+∑x¯ψ(DF[U]+μγ4)ψ+SG[~U]+∑x¯Ψ(DF[~U]−μ∗γ4)Ψ (7) + ∑x,ν2λ{Nc−Re tr[Uν(x)~U†ν(x)]} (8)

with the gauge action and the fermionic kernel in lattice QCD. After integrating out the fermion fields and , the generating functional of this theory reads

 Z = ∫DUe−SG[U]Det(DF[U]+μγ4)∫D~Ue−SG[~U]Det(DF[~U]−μ∗γ4) (10) e−∑x,ν2λ{Nc−Re tr[Uν(x)~U†ν(x)]} = ∫DU∫D~Ue−(SG[U]+SG[~U])Det{(DF[U]+μγ4)(DF[~U]−μ∗γ4)} (12) e−∑x,ν2λ{Nc−Re tr[Uν(x)~U†ν(x)]}.

In the continuum limit, this generating functional is expressed as

 Zcont = ∫DA∫D~Ae−(SG[A]+SG[~A])Det{(⧸D+m+μγ4)(⧸~D+m−μ∗γ4)} (14) e−∫d4x12λphys{Aaν(x)−~Aaν(x)}2

with the continuum gauge action and .

In the practical lattice calculation with the Monte Carlo method, the fermionic determinant in is factorized into its amplitude and phase factor as

 Z = ∫DU∫D~Ue−(SG[U]+SG[~U])e−∑x,ν2λ{Nc−Re tr[Uν(x)~U†ν(x)]} (16) |Det{(DF[U]+μγ4)(DF[~U]−μ∗γ4)}|⋅Ophase[U,~U],

and the phase factor of the total fermionic determinant

 Ophase[U,~U]≡eiarg[Det{(DF[U]+μγ4)(DF[~U]−μ∗γ4)}]∈C (17)

is treated as an “operator” instead of a probability factor, while all other real non-negative factors in are treated as the probability density.

### 2.2 Property and Procedure

The additional term connects the following two different situations as the two limits of the parameter .

1. In the limit of , a strong constraint is realized, and the phase factors of two fermionic determinants and are completely cancelled, owing to Eq. (4). Therefore, the total fermionic determinant is real and non-negative,

 Det{(DF[U]+μγ4)(DF[~U=U]−μ∗γ4)}≥0, (18)

and the sign problem is absent [4]. Note however that this system resembles QCD with an isospin chemical potential [5].

2. In the limit of , this system goes to two separated ordinary QCD systems with the chemical potential of and , although the cancellation of the phase factors cannot be expected between the two fermionic determinants and for significantly different and , which are independently generated in the Monte Carlo simulation.

On a four-dimensional finite-volume lattice, if an enough large value of is taken, is realized and there occurs the phase cancellation approximately between the two fermionic determinants and in and , so that one expects a modest behavior of the phase factor in Eq.(17), which leads to feasibility of the numerical lattice calculation with suppression of the sign problem.

Once the lattice gauge configurations of the coexistence system are obtained with the most importance sampling in the Monte Carlo simulation, matter-side quantities can be evaluated through their measurement only for the matter part with .

By performing the lattice calculations with gradually decreasing and their extrapolation to , we expect to estimate the physical quantities in finite density QCD with the chemical potential .

## 3 Summary, Discussion and Outlook

We have proposed a “matter-antimatter coexistence method” for the lattice calculation of finite density QCD. In this method, we have prepared matter with and anti-matter with on two parallel -sheets in five-dimensional Euclidean space-time, and have introduced a correlation term between the gauge variables in and in . In the limit of , owing to , the total fermionic determinant is real and non-negative, and the sign problem is absent. In the limit of , this system goes to two separated ordinary QCD systems with the chemical potential of and .

For an enough large value of , is realized and a phase cancellation approximately occurs between two fermionic determinants in and , which is expected to suppress the sign problem and to make the lattice calculation possible. For the obtained gauge configurations of the coexistence system, matter-side quantities can be evaluated by their measurement only for the matter part . By gradually reducing and the extrapolation to , it is expected to obtain estimation of the physical quantities in finite density QCD with .

In this paper, we have demonstrated this method with taking in Eq.(5) as the simplest correlation between in and in . In this method, however, there is some variety on the choice of the correlation between and . For instance, it may be interesting to consider the other correlation term like

 ¯Sξ ≡ ∑x8ξ(∑ν{Nc−Re tr[Uν(x)~U†ν(x)]})3 (19) ≃ ∫d4x 18a2ξ[{Aaν(x)−~Aaν(x)}2]3 (20)

with a dimensionless non-negative real parameter . At the classical level, this correlation is an irrelevant interaction and it gives vanishing contributions in the continuum limit , like the Wilson term .

The next step is to perform the actual lattice QCD calculation at finite density using this method. It would be useful to combine this method with the other known ways such as the hopping parameter expansion [6], the complex Langevin method [1] and the reweighting technique [2]. For example, if one utilizes the hopping parameter expansion, the quenched-level analysis becomes possible in this method, since the additional term only includes gauge variables.

Efficiency of this method would strongly depend on the system parameters, such as the space-time volume , the quark mass , the temperature and the chemical potential . In any case, this method is expected to enlarge calculable area of the QCD phase diagram on .

## Acknowledgements

The author would like to thank Prof. P. Bicudo for his useful comment and his hospitality during the Workshop. The author is supported in part by the Grant for Scientific Research [(C)15K05076] from the Ministry of Education, Science and Technology of Japan.

## References

• [1] G. Parisi, Phys. Lett. 131B (1983) 393.
• [2] Z. Fodor and S.D. Katz, Phys. Lett. B534 (2002) 87.
• [3] P. Hasenfratz and F. Karsch, Phys. Rept. 103 (1984) 219.
• [4] H. Suganuma, arXiv:1705.07516.
• [5] D.T. Son and M.A. Stephanov, Phys. Rev. Lett. 86 (2001) 592.
• [6] G. Aarts, E. Seiler, D. Sexty and I.-O. Stamatescu, Phys. Rev. D90 (2014) 114505.
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