Twisted mass chiral perturbation theory at next-to-leading order

We study the properties of pions in twisted mass lattice QCD (with two degenerate flavors) using chiral perturbation theory ($\chi \mathrm{PT}$). We work to next-to-leading order (NLO) in a power-counting scheme in which $m_q\sim a{\Lambda }_{\mathrm{QCD}}^2$, with $m_q$ the physical quark mass and $a$ the lattice spacing. We argue that automatic $O(a)$ improvement of physical quantities at maximal twist, which has been demonstrated in general if $m_q\gg a{\Lambda }_{\mathrm{QCD}}^2$, holds even if $m_q\sim a{\Lambda }_{\mathrm{QCD}}^2$, as long as one uses an appropriate nonperturbative definition of the twist angle, with the caveat that we have shown this only through NLO in our chiral expansion. We demonstrate this with explicit calculations, for arbitrary twist angle, of all pionic quantities that involve no more than a single pion in the initial and final states: masses, decay constants, form factors, and condensates, as well as the differences between alternate definitions of twist angle. We also calculate the axial and pseudoscalar form factors of the pion, quantities which violate flavor and parity, and which vanish in the continuum limit. These are of interest because they are not automatically $O(a)$ improved at maximal twist. They allow a determination of the unknown low-energy constants introduced by discretization errors, and provide tests of the accuracy of $\chi \mathrm{PT}$ at NLO. We extend our results into the regime where $m_q\sim a^2{\Lambda }_{\mathrm{QCD}}^3$, and argue in favor of a recent proposal that automatic $O(a)$ improvement at maximal twist remains valid in this regime.