2016/03/31 by Yiming Cai, Thomas D. Cohen, Thomas Cohen +2
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Combinatorics #Euclidean space #Mathematical analysis #Mathematical physics #Mathematics #Particle physics theoretical and experimental studies #Partition function (quantum field theory) #Physics #Quantum Chromodynamics and Particle Interactions #Quantum mechanics #Sign (mathematics) #hep-lat #hep-ph #hep-th #nucl-th
paper · pdf · doi:10.1103/physrevd.93.114510
published as Phys. Rev. D 93, 114510 (2016)
arxiv created 2016/05/29 · openalex publication_date 2016/06/17 · arxiv updated 2016/06/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
QCD and related gauge theories have a sign problem when a \ensuremathθ 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, ZQ, are summed weighted by exp(iQ\ensuremathθ) to obtain the partition function for fixed \ensuremathθ, Z(\ensuremathθ). 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 \ensuremathθ, ϵ(\ensuremathθ)=\ensuremath-log(Z(\ensuremathθ))/V. This is expected to be well defined in the large four-volume limit. Similarly, one expects the energy density for a fixed topological density \stackrel\texttildelowϵ(Q/V)=\ensuremath-log(ZQ)/V to be well defined in the limit of large four volumes. Intuitively, one might expect that if one had the infinite volume expression for \stackrel\texttildelowϵ(Q/V) to arbitrary accuracy, then one could reconstruct ϵ(\ensuremathθ) 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 ϵ(\ensuremathθ) is negative.