Pseudoscalar Decay Constants in Staggered Chiral Perturbation Theory

TL;DR

The paper develops a staggered chiral perturbation theory framework to compute one-loop chiral logarithms and analytic terms for pseudoscalar decay constants $f_{\pi^+_5}$ and $f_{K^+_5}$, incorporating taste violations across partially quenched, full three-flavor dynamical, and quenched QCD. It uses a $4+4+4$ KS-theory and a quark-flow mapping to a physical $1+1+1$ theory, with detailed treatment of mixed flavor-neutral sectors via the Lee–Sharpe Lagrangian and its taste-breaking potential. The authors derive comprehensive NLO results, including lattice-spacing dependent analytic terms and standard $L_i$-driven chiral logs, across multiple mass schemes (1+1+1, 2+1, and quenched), recovering continuum chiral perturbation theory in the appropriate limits. The work provides a practical path to fitting MILC lattice data to extract $f_\pi$, $f_K$, $m_s$, $(m_u+m_d)/2$, and low-energy constants, and outlines extensions to heavy-light mesons and baryons.

Abstract

In a continuation of an ongoing program, we use staggered chiral perturbation theory to calculate the one-loop chiral logarithms and analytic terms in the pseudoscalar meson leptonic decay constants, $f_{π^+_5}$ and $f_{K^+_5}$. We consider the partially quenched, ``full QCD'' (with three dynamical flavors), and quenched cases.

Figures (5)

• Figure 1: The two-point mixing vertex coming from the $\mathcal{U}\,'$ term. (a) corresponds to the chiral theory (we also have similar $U-S$ and $D-S$ mixing terms). (b) shows the corresponding quark level diagram. Here we only show the mixing among the taste-vectors, but there are similar vertices among the axial tastes, as well as the singlet tastes (with $a^2\delta'_V \to 4m_0^2/3)$.
• Figure 2: The S$\chi$PT diagrams contributing to the pion decay constant, coming from wave-function renormalization. The box represents the the axial current. (a) is the connected piece, where the propagator in the loop contains no two-point vertex insertions. (b) subsumes the graphs which have disconnected insertions within the loop. The cross represents one or more insertions of the $\delta'$ vertex, with $\delta'$ given in Eq. (\ref{['eq:dp_def']}).
• Figure 3: Same as Fig. \ref{['fig:tadZ']}, but these contributions to the decay constant are from axial current corrections.
• Figure 4: The quark level diagrams that contribute to the one-loop pion decay constant. The box represents an insertion of the axial current. The diagrams on the left correspond to the wavefunction renormalization while those on the right correspond to the current corrections.
• Figure 5: The quark level diagrams for $2\rightarrow 2$ meson scattering which contribute to $f_{P^+_5}$. The indices $i$ and $j$ represent arbitrary quark flavors. There are two additional diagrams (not shown), which are like those in (a) but have the roles of $x$ and $y$ interchanged. The box stands for the axial current.