Approach to the thermodynamic limit in lattice QCD atμ≠0

The expectation value of the complex phase factor of the fermion determinant is computed to leading order in the $p$ expansion of the chiral Lagrangian. The computation is valid for $\mu <m_{\pi }/2$ and determines the dependence of the sign problem on the volume and on the geometric shape of the volume. In the thermodynamic limit with $L_i\to \infty$ at fixed temperature $1/L_0$, the average phase factor vanishes. In the low temperature limit where $L_i/L_0$ is fixed as $L_i$ becomes large, the average phase factor approaches 1 for $\mu <m_{\pi }/2$. The results for a finite volume compare well with lattice results obtained by Allton et al. After taking appropriate limits, we reproduce previously derived results for the $ϵ$ regime and for one-dimensional QCD. The distribution of the phase itself is also computed.