Matter-Antimatter Coexistence Method for Finite Density QCD

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 ${\bf R}^4$-sheets in five-dimensional Euclidean space-time. For the matter system $M$ with a chemical potential $μ\in {\bf C}$ on a ${\bf R}^4$-sheet, we also prepare the anti-matter system $\bar M$ with $-μ^*$ on the other ${\bf R}^4$-sheet shifted in the fifth direction. In the lattice QCD formalism, we introduce a correlation term between the gauge variables $U_ν\equiv e^{iagA_ν}$ in $M$ and $\tilde U_ν\equiv e^{iag \tilde A_ν}$ in $\bar M$, such as $S_λ\equiv \sum_{x,ν} 2λ\{N_c-{\rm Re~tr} [U_ν(x) \tilde U_ν^\dagger(x)]\} \simeq \sum_x \frac{1}{2}λa^2 \{A_ν^a(x)-\tilde A_ν^a(x)\}^2$ with a real parameter $λ$. In the limit of $λ\rightarrow \infty$, a strong constraint $\tilde U_ν(x)=U_ν(x)$ is realized, and the total fermionic determinant is real and non-negative. In the limit of $λ\rightarrow 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $μ$ and $-μ^*$. On a finite-volume lattice, if one takes an enough large value of $λ$, $\tilde U_ν(x) \simeq U_ν(x)$ is realized and there occurs a phase cancellation approximately between two fermionic determinants in $M$ and $\bar M$, 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 are evaluated through their measurement only for the matter part $M$. By the calculations with gradually decreasing $λ$ and their extrapolation to $λ=0$, physical quantities in finite density QCD are expected to be estimated.