Pion and Kaon masses in Staggered Chiral Perturbation Theory

TL;DR

This work extends Lee–Sharpe chiral perturbation theory to multiple staggered flavors, addressing previous errors in the multi-flavor generalization and incorporating taste-symmetry breaking and the fourth-root trick. By formulating a corrected $4n\times4n$ chiral Lagrangian with a two-part taste-violating potential, the authors derive the one-loop chiral logarithms for pion and kaon masses in full, partially quenched, and quenched scenarios, including hairpin mixing in flavor-neutral channels. The analysis yields explicit expressions for the self-energies in terms of residues $R^{[n,k]}_j$ and chiral logs, and shows how to project the $4+4+4$ results to the physical $1+1+1$ theory with automatic accounting of disconnected diagrams and sea-quark effects. The final results, expressed with Gasser–Leutwyler $L_i$ constants and new parameters $\delta'_V$ and $\delta'_A$, enable controlled chiral and continuum extrapolations of staggered lattice QCD data and provide a framework for future inclusion of heavy quarks. This work thus strengthens the foundational basis forQuantitative lattice analyses using staggered fermions and clarifies the interplay between taste violations, hairpin contributions, and the rooting trick in chiral dynamics.

Abstract

We show how to compute chiral logarithms that take into account both the $\cO(a^2)$ taste-symmetry breaking of staggered fermions and the fourth-root trick that produces one taste per flavor. The calculation starts from the Lee-Sharpe Lagrangian generalized to multiple flavors. An error in a previous treatment by one of us is explained and corrected. The one loop chiral logarithm corrections to the pion and kaon masses in the full (unquenched), partially quenched, and quenched cases are computed as examples.

Figures (4)

• Figure 1: The two-point mixing vertex (among taste vectors) coming from the new $\mathcal{U}\,'$ term. (a) corresponds to the chiral theory. (b) shows the corresponding quark level diagram. We also have $U$-$S$ and $D$-$S$ mixings and diagonal terms ($U$-$U$etc.). There are similar vertices among the axial tastes (with $a^2\delta'_V \to a^2\delta'_A$), as well as the singlet tastes (with $a^2\delta'_V \to 4m_0^2/3)$.
• Figure 2: The only diagrams contributing to the flavor-nonsinglet meson self energy. (a) includes all contributions where the propagator in the loop contains no two-point vertex insertions. (b) subsumes the graphs which have disconnected insertions on the propagator. The cross represents one or more insertions of either the $m_0^2$, $\delta'_V$, or $\delta'_A$ vertices.
• Figure 3: The quark level diagrams that could contribute to the 1-loop meson self energy. Diagrams (b) and (c), which require vertices of the form of Fig. \ref{['fig:vertices']}(c), do not occur in the case of interest. Similarly, (d) only contributes to flavor-neutral propagators. Note that (f), (h) and (j) correspond to (e), (g), and (i), respectively, with iteration of either $m_0^2$, $\delta'_V$, or $\delta'_A$ vertices. These diagrams are to be taken as including any number of iterations, thus multiple internal quark loops.
• Figure 4: The possible quark level diagrams for $2\rightarrow 2$ meson scattering, where one incoming and one outgoing particle (shown horizontally) are fixed to be valence mesons, $P^+ =x\bar{y}$. The indices $i$ and $j$ represent arbitrary quark flavors. There are two additional diagrams (not shown), which are like (a) and (d) but have the roles of $x$ and $y$ interchanged.