Observations on discretization errors in twisted-mass lattice QCD

TL;DR

The paper develops and applies twisted-mass chiral perturbation theory (tmχPT) to quantify discretization errors in two-flavor twisted-mass lattice QCD, covering both the generic small mass (GSM) and Aoki regimes. It analyzes multiple definitions of maximal twist, predicting an $O(a)$ interception δω between definitions and showing how a nonperturbative vacuum shift sums infrared-like divergences into finite, gauge-invariant predictions, thereby preserving the validity of the Symanzik expansion. By extending tmχPT to next-to-leading order in the Aoki regime, it identifies new low-energy constants and derives modified phase diagrams and lines of maximal twist, enabling consistent fits across mass planes. The work concludes that there is no fundamental barrier to simulating at maximal twist even in the Aoki regime, provided one uses an optimal critical mass and accounts for nonperturbative vacuum effects; it also clarifies the interpretation of IR divergences and reinforces practical guidance for lattice simulations. Overall, tmχPT provides a robust framework to diagnose, sum, and control discretization errors in tmLQCD, with direct implications for defining and achieving maximal twist in simulations.

Abstract

I make a number of observations concerning discretization errors in twisted-mass lattice QCD that can be deduced by applying chiral perturbation theory including lattice artifacts. (1) The line along which the PCAC quark mass vanishes in the twisted mass-twisted mass plane makes an angle to the untwisted mass axis which is a direct measure of O(a) terms in the chiral Lagrangian, and is found numerically to be large; (2) Numerical results for pionic quantities in the mass plane show the qualitative properties predicted by chiral perturbation theory, in particular an asymmetry in slopes between positive and negative untwisted quark masses; (3) By extending the description of the ``Aoki regime'' (where m_q is of size a^2 Lambda_QCD^3) to next-to-leading order in chiral perturbation theory I show how the phase transition lines and lines of maximal twist (using different definitions) extend into this region, and give predictions for the functional form of pionic quantities; (4) I argue that the recent claim that lattice artifacts at maximal twist have apparent infrared singularities in the chiral limit results from expanding about the incorrect vacuum state. Shifting to the correct vacuum (as can be done using chiral perturbation theory) the apparent singularities are summed into non-singular, and furthermore predicted, forms. I further argue that there is no breakdown in the Symanzik expansion in powers of lattice spacing, and no barrier to simulating at maximal twist in the Aoki regime.

Figures (6)

• Figure 1: Illustration of different methods for working at maximal twist. See text for description [method (iv) is discussed in sec. \ref{['sec:IR']}]. The arrows represent the direction one moves to approach the chiral limit (which occurs at $\mu_0=0$). The plot represents the GSM regime in which $m_q\sim a$. The Aoki regime, in which $m_q\sim a^2$ is represented by the shaded region. It is discussed in sec. \ref{['sec:aoki']}. The angle $\delta\omega$ is defined to be positive for the situation shown in the Figure.
• Figure 2: $m_{\pi^\pm}^2$ as a function of $m"$ for fixed $\mu=0,0.01,0.015,0.025$, for the parameter set described in the text. Here $m"$ is the NLO quark mass in the Aoki regime, defined in eq. (\ref{['eq:mprprdef']}). For the GSM regime [i.e. most of fig. (a)] $m"$ is equivalent to $m'$ at NLO accuracy. All quantities are in units of appropriate powers of GeV. The curves in (a) can be identified from the enlargement (b): the lines decrease in thickness as $\mu$ increases, with the $\mu=0.025$ curve dashed.
• Figure 3: $m_{\rm PCAC}$ as a function of $m"$, with parameters and legend as in Fig. \ref{['fig:mpi']}.
• Figure 4: Illustration of phase structure and lines of maximal twist at NLO in the Aoki regime for both Aoki-phase and first-order scenarios. The solid lines indicates first-order transition lines, with the solid circles being second-order end-points. Both masses $m"$ and $\mu$ range up to values of $O(a^2)$ in this figure, with other features (the asymmetry in the phase line in the Aoki-phase, the offset of the $\omega_P=\pm\pi/2$ lines, and the horizontal extent of the phase-line in the first-order scenario) being of $O(a^3)$. See the text for more details. The angle $\delta\omega$ is the same as that in Fig. \ref{['fig:maxtwist']}.
• Figure 5: Examples of quark-mass dependence when using a critical mass with an error of $O(a)$ [method (iv) of Fig. \ref{['fig:maxtwist']}]: (a) $f_A/f$ and (b) $m_{\pi^\pm}^2$ (in GeV${}^2$) versus $\mu$ (in GeV) The parameter-set is described in the text, and corresponds to the first-order transition scenario. $m"$ is fixed to $\delta m$, with values $0.005$, $0.015$ and $0.025$ GeV. These are distinguished by the line width, which increases with the magnitude of $\delta m$. The dashed line shows the continuum result.