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 ${\mathbf{R}}^4$-sheets in five-dimensional Euclidean space-time. We put a matter system $M$ with a chemical potential $\mu \in {\bf C}$ona${\bf R}^4$-sheet,andalsoputananti-mattersystem$\bar M$with$-\mu^*$ontheother${\bf R}^4$-sheet shifted in the fifth direction. Between the gauge variables $U_{\nu }\equiv e^{iag{A}_{\nu }}$ 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 $\lambda \to \mathrm{\infty }$, 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 $\lambda \to 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $\mu$ and $-{\mu }^{\ast }$. 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.