Pi-K Scattering in Full QCD with Domain-Wall Valence Quarks

TL;DR

This paper addresses the low-energy πK scattering problem in three-flavor QCD by performing a fully-dynamical lattice calculation with domain-wall valence quarks on MILC configurations and extracting scattering lengths through Lüscher's finite-volume method. The authors analyze pion-mass dependent lattice data at NLO SU(3) chiral perturbation theory to determine the I=3/2 and I=1/2 threshold amplitudes, yielding precise values for $m_π a_{3/2}$ and $m_π a_{1/2}$ at the physical point. They quantify statistical and systematic uncertainties, including truncation errors, and discuss implications for chiral low-energy constants $L_5$ and $L_{πK}$, as well as potential future improvements with finer lattices and mixed-action effects. The results provide stringent tests for hadronic interactions in the strange sector and offer benchmarks for πK-atom studies and resonance analyses in QCD.

Abstract

We calculate the pi+ K+ scattering length in fully-dynamical lattice QCD with domain-wall valence quarks on MILC lattices with rooted staggered sea-quarks at a lattice spacing of b=0.125 fm, lattice spatial size of L =2.5 fm and at pion masses of m_pi=290, 350, 490 and 600 MeV. The lattice data, analyzed at next-to-leading order in chiral perturbation theory, allows an extraction of the full pi K scattering amplitude at threshold. Extrapolating to the physical point gives m_pi a_3/2 = -0.0574 (+- 0.0016)(+0.0024 -0.0058) and m_pi a_1/2 = 0.1725 (+- 0.0017)(+0.0023 -0.0156) for the I=3/2 and I=1/2 scattering lengths, respectively, where the first error is statistical and the second error is an estimate of the systematic due to truncation of the chiral expansion.

Figures (6)

• Figure 1: The effective $\pi^+K^+$ scattering length times the reduced mass, $\mu_{\pi K}\ a_{\pi^+K^+}(t)$ as a function of time slice arising from smeared sinks. The solid black lines and shaded regions are fits with 1-$\sigma$ errors tabulated in Table \ref{['tab:LatResults']}. The dashed lines on the m010 ensemble plot are an estimate of a systematic error due to fitting.
• Figure 2: $\mu_{\pi K} \ a_{\pi^+K^+}$ vs. $\mu_{\pi K}/f_\pi$. The data points are the results of this lattice calculation, while the curve is the theoretical prediction at tree level in chiral perturbation theory Weinberg:1966kf. The dark error bar is statistical, while the lighter error bar corresponds to the systematic error. The vertical dashed line denotes the physical pion and kaon masses.
• Figure 3: ${\Gamma}$ vs. $m_K/m_\pi$. The dark error bar on the data points is statistical, while the lighter error bar corresponds to the systematic error. The lines correspond to the four linear fits (A,B,C,D). The bars on the y axis represent the 1-$\sigma$ errors in the determinations of $L_5=\Gamma(m_K/m_\pi =0)$ as given in Table \ref{['tab:FitResultsNNLO']}. (At 95% confidence level, these determinations are in agreement.)
• Figure 4: Error ellipses in the $L_5$-$L_{\pi K}$ plane for the four fits (A,B,C,D) at 68% (dotted lines) and 95% (solid lines) confidence level.
• Figure 5: Error ellipses for the four fits (A,B,C,D) at 95% confidence level. (Note that these results are derived from lattice data on a single lattice spacing of $b=0.125~{\rm fm}$.). The star corresponds to the current-algebra predictions ($\chi$PT $p^2$) from Ref. Weinberg:1966kf. We also display 1-$\sigma$ error ellipses from a $\chi$PT analysis at NLO Bernard:1990kw (denoted $\chi$PT $p^4$) and from a fit using the Roy-Steiner equations Buettiker:2003pp.