Wilson chiral perturbation theory for dynamical twisted mass fermions vs lattice data - a case study

We compute the low lying eigenvalues of the Hermitian Dirac operator in lattice QCD with $N_{\rm f} = 2+1+1$ twisted mass fermions. We discuss whether these eigenvalues are in the $\epsilon$-regime or the $p$-regime of Wilson chiral perturbation theory ($\chi$PT) for twisted mass fermions. Reaching the deep $\epsilon$-regime is practically unfeasible with presently typical simulation parameters, but still the few lowest eigenvalues of the employed ensemble evince some characteristic $\epsilon$-regime features. With this conclusion in mind, we develop a fitting strategy to extract two low energy constants from analytical $\epsilon$-regime predictions at a fixed index. Thus, we obtain results for the chiral condensate and the low energy constant $W_8$. We also discuss how to improve both the theoretical calculation and the lattice computation.