Matter-Antimatter Coexistence Method for Finite Density QCD

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 ${\mathbf{R}}^4$-sheets in five-dimensional Euclidean space-time. For the matter system $M$ with a chemical potential $\mu \in \mathbf{C}$ on a ${\mathbf{R}}^4$-sheet, we also prepare the anti-matter system $\bar{M}$ with $-{\mu }^{\ast }$ on the other ${\mathbf{R}}^4$-sheet shifted in the fifth direction. In the lattice QCD formalism, we introduce a correlation term between the gauge variables $U_{\nu }\equiv e^{iag{A}_{\nu }}$ in $M$ and ${\tilde{U}}_{\nu }\equiv e^{iag{\tilde{A}}_{\nu }}$ in $\bar{M}$, such as $S_\lambda \equiv \sum_{x,\nu} 2\lambda {N_c-{\rm Re~tr} [U_\nu(x) \tilde U_\nu^\dagger(x)]} \simeq \sum_x \frac{1}{2}\lambda a^2 {A_\nu^a(x)-\tilde A_\nu^a(x)}^2$witharealparameter$\lambda$\mathrm{.}Inthelimitof$\lambda \rightarrow \infty$,astrongconstraint$\tilde U_\nu(x)=U_\nu(x)$ is realized, and the total fermionic determinant is real and non-negative. In the limit of $\lambda \to 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $\mu$ and $-{\mu }^{\ast }$. 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.