Towards a formalism for $ππ$ scattering from staggered lattice QCD

Abstract

Scattering processes featuring the strong interactions can be studied using lattice QCD by means of the Lüscher formalism. This approach relies on analyticity and unitarity of the $S$-matrix to relate infinite-volume scattering amplitudes to finite-volume energy levels. However, lattice QCD simulations employing rooted staggered fermions manifest unitarity violation as an $\mathcal{O}(a^2)$ lattice artifact. Moreover, the meson sector of this theory contains multiple non-mass-degenerate pions (due to the so-called taste splitting), which only reduce to the physical pion in the continuum limit. These features restrict the applicability of the Lüscher formalism to staggered lattice data at non-zero lattice spacing. Hence, in this work, we discuss two complementary approaches to deal with the challenges of extracting $ππ$ scattering amplitudes from lattice QCD with staggered quarks: (1) using the corresponding effective theory, Rooted Staggered Chiral Perturbation Theory, to calculate one-loop amplitudes for the first time. These amplitudes can be used to explicitly check the validity of the quantization condition. And (2) generalizing the formalism to incorporate taste-splitting as well as fourth-rooting effects. We focus on the simpler case of $ππ$ scattering in the isospin-2 channel, and discuss prospects for other channels.

Figures (3)

• Figure 1: (a) Example of $s$-channel diagram contributing to the $I=2$$\pi\pi$ scattering. (b,c) Two examples of $t$-channel diagrams that are relevant for this scattering. The diagram (b) has an internal taste loop (inner green line) and requires a rooting factor (thus violating unitarity), whereas the diagram (c) involves disconnected propagators due to the hairpin vertex.
• Figure 2: From left to right, Bethe-Salpeter kernels for the processes $\pi^+\pi^+\to\pi^+\pi^+$ ($I=2$), $\pi^+\pi^-\to\pi^+\pi^-$, and $\pi^0\pi^0\to\pi^0\pi^0$. As usual, the green blobs represent a sum of all diagrams that are not in the $s$-channel.
• Figure 3: Modified skeleton expansion to include $N$$s$-channel topologies $T_n$ rescaled by an arbitrary factor $\alpha_n$. This factor is set to $1/4$ if the diagram contains an internal taste loop, or to $1$ otherwise.