Matter-antimatter coexistence method for finite density QCD toward a solution of the sign problem

Toward the lattice QCD calculation at finite density, we propose"matter-antimatter coexistence method", where matter and anti-matter systems are prepared on two parallel ${\bf R}^4$-sheets in five-dimensional Euclidean space-time. We put a matter system $M$ with a chemical potential $\mu \in {\bf C}$ on a ${\bf R}^4$-sheet, and also put an anti-matter system $\bar M$ with $-\mu^*$ on the other ${\bf R}^4$-sheet shifted in the fifth direction. Between the gauge variables $U_\nu \equiv e^{iagA_\nu}$ in $M$ and $\tilde U_\nu \equiv e^{iag \tilde A_\nu}$ in $\bar M$, we introduce a correlation term with a real parameter $\lambda$. In one limit of $\lambda \rightarrow \infty$, a strong constraint $\tilde U_\nu(x)=U_\nu(x)$ is realized, and therefore the total fermionic determinant becomes real and non-negative, due to the cancellation of the phase factors in $M$ and $\bar M$, although this system resembles QCD with an isospin chemical potential. In another limit of $\lambda \rightarrow 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $\mu$ and $-\mu^*$. For a given finite-volume lattice, if one takes an enough large value of $\lambda$, $\tilde U_\nu(x) \simeq U_\nu(x)$ is realized and phase cancellation approximately occurs between two fermionic determinants in $M$ and $\bar M$, which suppresses the sign problem and is expected 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 $M$. The physical quantities in finite density QCD are expected to be estimated by the calculations with gradually decreasing $\lambda$ and the extrapolation to $\lambda=0$. We also consider more sophisticated improvement of this method using an irrelevant-type correlation.


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 S[A, ψ, ψ; µ] at finite density with the chemical potential µ is generally complex, with the gauge action S G [A] ∈ R 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 µ ∈ C.

Matter-Antimatter Coexistence Method
Our strategy is to use a cancelation of the phase factors of the fermionic determinants between a matter system with µ and an anti-matter system with −µ * , and our method is based on the general property [3], for the Euclidean QCD action S[A, ψ, ψ; µ] in the presence of the chemical potential µ ∈ C. Actually, the fermionic kernel D F corresponding to D + m generally satisfies D † F = γ 5 D F γ 5 in lattice QCD, so that one finds which leads to the relation (2), and

Definition and Setup
In the "matter-antimatter coexistence method", we consider matter and anti-matter systems on two parallel R 4 -sheets in five-dimensional Euclidean space-time.For the matter system M with a chemical potential µ ∈ C on a R 4 -sheet, we also prepare the anti-matter system M with −µ * on the other R 4 -sheet shifted in the fifth direction, as shown in Fig. 1.
We put an ordinary fermion field ψ(x) with the mass m and the gauge variable U µ (x) ≡ e iagAν (x) at x ∈ R 4 on the matter system M , and we put the other fermion field Ψ(x) ≡ ψ(x + 5) with the same mass m and the gauge variable Ũµ (x) ≡ e iag Ãν (x) ≡ U µ (x + 5) on the anti-matter system M .Fig. 1.The matter-antimatter coexistence system in five-dimensional Euclidean space-time.We put the matter system M with µ, U ν (x) and ψ(x) on a R 4 -sheet, and the anti-matter system M with −µ * , Ũν (x) = U ν (x + 5) and Ψ(x) = ψ(x + 5) on the other R 4 -sheet shifted in the fifth direction.
In the lattice QCD formalism, we introduce a correlation term between the gauge variables U ν (x) in M and Ũν (x) in M at x ∈ R 4 , such as with a real parameter λ (≥ 0), which connects two different situations: Ũν (x) = U ν (x) in λ → ∞ and two separated QCD systems in λ → 0. Near the continuum limit, this additional term becomes with λ phys ≡ λa −2 .In fact, the total lattice action in this method is written as with the gauge action S G [U ] ∈ R and the fermionic kernel D F [U ] in lattice QCD.After integrating out the fermion fields ψ and Ψ, the generating functional of this theory reads In the continuum limit, this generating functional is expressed as with the continuum gauge action S G [A] ∈ R and Dν ≡ ∂ ν + ig Ãν .
In the practical lattice calculation with the Monte Carlo method, the fermionic determinant in Z is factorized into its amplitude and phase factor as and the phase factor of the total fermionic determinant is treated as an "operator" instead of a probability factor, while all other real non-negative factors in Z are treated as the probability density.

Property and Procedure
The additional term S λ connects the following two different situations as the two limits of the parameter λ.  4).Therefore, the total fermionic determinant is real and non-negative, 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 λ → 0, this system goes to two separated ordinary QCD systems with the chemical potential of µ and −µ * , although the cancellation of the phase factors cannot be expected between the two fermionic determinants Det(D F [U ] + µγ 4 ) and Det(D for significantly different U ν (x) and Ũν (x), which are independently generated in the Monte Carlo simulation.
On a four-dimensional finite-volume lattice, if an enough large value of λ is taken, Ũν (x) ≃ U ν (x) is realized and there occurs the phase cancellation approximately between the two fermionic determinants Det(D F [U ] + µγ 4 ) and Det(D F [ Ũ ]− µ * γ 4 ) in M and M , so that one expects a modest behavior of the phase factor O phase [U, Ũ ] in Eq.( 11), 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 M with µ.
By performing the lattice calculations with gradually decreasing λ and their extrapolation to λ = 0, we expect to estimate the physical quantities in finite density QCD with the chemical potential µ.

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 M with µ and anti-matter M with −µ * on two parallel R 4 -sheets in five-dimensional Euclidean space-time, and have introduced a correlation term )} 2 between the gauge variables U ν = e iagAν in M and Ũν = e iag Ãν in M .In the limit of λ → ∞, owing to Ũν (x) = U ν (x), the total fermionic determinant is real and non-negative, and the sign problem is absent.In the limit of λ → 0, this system goes to two separated ordinary QCD systems with the chemical potential of µ and −µ * .
For an enough large value of λ, Ũν (x) ≃ U ν (x) is realized and a phase cancellation approximately occurs between two fermionic determinants in M and M , 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 M .By gradually reducing λ and the extrapolation to λ = 0, it is expected to obtain estimation of the physical quantities in finite density QCD with µ.