The interplay of the sign problem and the infinite volume limit: gauge theories with a theta term

Abstract: QCD and related gauge theories have a sign problem when a $\theta$-term is included; this complicates the extraction of physical information from Euclidean space calculations as one would do in lattice studies. The sign problem arises in this system because the partition function for configurations with fixed topological charge $Q$, $\mathcal{Z}_Q$, are summed weighted by $\exp(i Q \theta)$ to obtain the partition function for fixed $\theta$, $ \mathcal{Z}(\theta)$. The sign problem gets exponentially worse numerically as the space-time volume is increased. Here it is shown that apart from the practical numerical issues associated with large volumes, there are some interesting issues of principle. A key quantity is the energy density as a function of $\theta$, $\varepsilon(\theta) = -\log \left( \mathcal{Z}(\theta) \right )/V$. This is expected to be well defined in the large 4-volume limit. Similarly, one expects the energy density for a fixed topological density $\tilde{\varepsilon}(Q/V) = -\log \left(\mathcal{Z}_Q \right )/V$ to be well defined in the limit of large 4-volumes. Intuitively, one might expect that if one had the infinite volume expression for $\tilde{\varepsilon}(Q/V)$ to arbitrary accuracy, that one could reconstruct $\varepsilon(\theta)$ by directly summing over the topological sectors of the partition function. We show here that there are circumstances where this is not the case. In particular, this occurs in regions where the curvature of $\varepsilon(\theta)$ is negative.