Staggered Chiral Perturbation Theory at Next-to-Leading Order

TL;DR

This paper extends staggered chiral perturbation theory to next-to-leading order by enumerating all analytic operators up to ${ m O}(a^2p^2)$, ${ m O}(a^4)$, and ${ m O}(a^2m)$, enabling a full NLO treatment of pseudo-Goldstone boson masses and decay constants with discretization effects. It shows that SO(4) taste symmetry breaking appears at NLO and derives explicit relations among mass splittings, dispersion relations, and decay-constant splittings, including hairpin contributions for flavor-singlet states. Several testable predictions are made for non-singlet and singlet PGBs, including simple ratios between masses and decay constants and correlations between dispersion and SO(4) breaking, which can be checked against lattice QCD data with the fourth-root trick. These results provide a framework to validate the long-distance equivalence of staggered lattice theories to QCD in the continuum limit and to assess the validity of the ${ m Det}^{1/4}$ procedure in practical simulations.

Abstract

We study taste and Euclidean rotational symmetry violation for staggered fermions at nonzero lattice spacing using staggered chiral perturbation theory. We extend the staggered chiral Lagrangian to O(a^2 p^2), O(a^4) and O(a^2 m), the orders necessary for a full next-to-leading order calculation of pseudo-Goldstone boson masses and decay constants including analytic terms. We then calculate a number of SO(4) taste-breaking quantities, which involve only a small subset of these NLO operators. We predict relationships between SO(4) taste-breaking splittings in masses, pseudoscalar decay constants, and dispersion relations. We also find predictions for a few quantities that are not SO(4) breaking. All these results hold also for theories in which the fourth-root of the fermionic determinant is taken to reduce the number of quark tastes; testing them will therefore provide evidence for or against the validity of this trick.

Figures (6)

• Figure 1: NLO contribution to the $\langle PP\rangle$ flavor-disconnected correlator. The two black squares represent insertions of the pseudoscalar density, while the cross represents the hairpin vertex. For tensor taste, this vertex comes from both ${\cal O}(a^2 p^2)$ and ${\cal O}(a^4)$ two supertrace operators, but only two of the ${\cal O}(a^2 p^2)$ operators lead to $SO(4)$ breaking. The analogous diagram, with pseudoscalar sources changed to axial currents, contributes to the $\langle AA \rangle$ flavor-disconnected correlator.
• Figure 2: NLO contribution to the $\langle AA\rangle$ flavor-disconnected correlator from axial current renormalization. The single black square represents an insertion of the LO axial current, while the double box represents an insertion of the NLO axial current. It is shown as two boxes because the hairpin vertex is in the current itself.
• Figure 3: The product of two bifundamentals in $SU(4N | 4M)$. The result also applies for $SU(N)$ for $N > 3$ (with the dashed lines removed).
• Figure 4: The symmetric product of two bifundamentals in $SU(4N | 4M)$. The result also applies to $SU(N)$ for $N > 3$ (with the dashed lines removed).
• Figure 5: The symmetric product of two adjoints in $SU(4N | 4M)$. The undotted boxes are fundamental representations and the dotted boxes are anti-fundamental representations. The result also applies to $SU(N)$ for $N>3$ (with the dashed lines removed).