Extracting hadron masses from fixed topology simulations

Lattice QCD simulations tend to become stuck in a single topological sector at fine lattice spacing or when using chirally symmetric overlap quarks. In such cases physical observables differ from their full QCD counterparts by finite volume corrections. These systematic errors need to be understood on a quantitative level and possibly removed. In this paper we extend an existing relation from the literature between two-point correlation functions at fixed and the corresponding hadron masses at unfixed topology by calculating all terms proportional to $1/V^2$ and $1/V^3$, where $V$ is the spacetime volume. Since parity is not a symmetry at fixed topology, parity mixing is comprehensively discussed. In the second part of this work we apply our equations to a simple model, quantum mechanics on a circle both for a free particle and for a square-well potential, where we demonstrate in detail, how to extract physically meaningful masses from computations or simulations at fixed topology.

Figure 1

Paths with topological charge $Q=0$ and $Q=1$.

Figure 2

The low lying spectrum for $U=0$; (left) $E_{n}I$ as a function of $\theta$ [Eq. (5.6)]; (right) ${\overline{E}}_{n}I$ as a function of $\theta$ [Eq. (5.7)].

Figure 3

The low lying energy eigenvalues ${\widehat{E}}_n$ for the square well potential (5.13) with ${\widehat{U}}_0=5.0$ and $\rho =0.9\times 2\pi$ as functions of $\theta$.

Figure 4

Effective masses ${\widehat{M}}_{Q,\widehat{T}}^{\text{eff}}$ as functions of the temporal separation $\widehat{t}$ for different topological sectors $Q$ and $\widehat{T}=6.0/{\widehat{\mathcal{E}}}_2\approx 930.2$.

Figure 5

Effective masses ${\widehat{M}}_{Q,\widehat{T}}^{\text{eff}}$ derived from the $1/V$ expansions of two-point correlation functions as functions of the temporal separation $\widehat{t}$ for different topological sectors $Q$ and $\widehat{T}=6.0/{\widehat{\mathcal{E}}}_2\approx 930.2$.

Figure 6

Effective masses ${\widehat{M}}_{Q,\widehat{T}}^{\text{eff}}$ derived from (3.18) and (3.20) restricted to the parameters ${\mathcal{E}}_2$, $M_H(0)$ and $M_H^{(2)}(0)$ as functions of the temporal separation $\widehat{t}$ for different topological sectors $Q$ and $\widehat{T}=6.0/{\widehat{\mathcal{E}}}_2\approx 930.2$.

Figure 7

Determining the physical mass ${\widehat{M}}_H(0)$ (i.e. the mass at unfixed topology) from a fixed topology computation; only those masses ${\widehat{M}}_{Q,\widehat{T}}$ are included in the fit, which fulfill $1/{\widehat{\mathcal{E}}}_{2}T$, $|Q|/{\widehat{\mathcal{E}}}_{2}T\le 0.5$ (red points).