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 $\mu \in {\bf C}$ on a ${\bf R}^4$-sheet, we also prepare the anti-matter system $\bar M$ with $-\mu^*$ 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 $U_\nu \equiv e^{iagA_\nu}$ in $M$ and $\tilde U_\nu \equiv e^{iag \tilde A_\nu}$ in $\bar M$, such as $S_\lambda \equiv \sum_{x,\nu} 2\lambda \{N_c-{\rm Re~tr} [U_\nu(x) \tilde U_\nu^\dagger(x)]\} \simeq \sum_x \frac{1}{2}\lambda a^2 \{A_\nu^a(x)-\tilde A_\nu^a(x)\}^2$ with a real parameter $\lambda$. In the limit of $\lambda \rightarrow \infty$, a strong constraint $\tilde U_\nu(x)=U_\nu(x)$ is realized, and the total fermionic determinant is real and non-negative. In the limit of $\lambda \rightarrow 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $\mu$ and $-\mu^*$. On a finite-volume lattice, if one takes an enough large value of $\lambda$, $\tilde U_\nu(x) \simeq U_\nu(x)$ 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 $\lambda$ and their extrapolation to $\lambda=0$, physical quantities in finite density QCD are expected to be estimated.