Abstract
We study pion-pion scattering in partially-quenched twisted-mass lattice QCD using chiral perturbation theory. The specific partially-quenched setup corresponds to that used in numerical lattice QCD calculations of the $I=0$ scattering length. We study the discretization errors proportional to $a^2$, with $a$ the lattice spacing, and the errors that arise due to the use of L\"uscher's two-particle quantization condition in a theory that is not unitary. We argue that the former can be as large as $\sim 100\%$, but explain how they can be systematically subtracted using a calculation of the $I=2$ scattering amplitude in the same partially-quenched framework. We estimate the error from the violation of unitarity to be $\sim 25\%$, and argue that this error will be difficult to reduce in practice.
Highlights
Considerable progress has been made over the last decade in studying two-particle scattering at physical, or nearphysical, quark masses [1,2,3,4,5,6,7,8,9]
Using a partially quenched (PQ) theory implies that unitarity is violated [17,18], which introduces an additional source of systematic error
Ref. [4] used pions composed of OS quarks to study I 1⁄4 0 ππ scattering, since there is an exact valence isospin symmetry, and the abovedescribed mixing with the I 1⁄4 2 channel does not occur. This setup has not been studied previously using χPT, and we carry out the calculation here in order to determine the form of the leading discretization errors in this approach
Summary
Considerable progress has been made over the last decade in studying two-particle scattering at physical, or nearphysical, quark masses [1,2,3,4,5,6,7,8,9]. The ETM collaboration has used TM fermions to determine scattering lengths in the following channels: KþKþ [11], Kþπþ [12], and ππ scattering for I 1⁄4 2 [3,13] and I 1⁄4 0 [4] All these works are done at maximal twist, and make use of the two-particle quantization condition derived by Lüscher [14,15]. [4] used pions composed of OS quarks to study I 1⁄4 0 ππ scattering, since there is an exact valence isospin symmetry, and the abovedescribed mixing with the I 1⁄4 2 channel does not occur This setup has not been studied previously using χPT, and we carry out the calculation here in order to determine the form of the leading discretization errors in this approach. Some technical results concerning the NLO contributions, which we find do not give corrections to the scattering amplitudes or masses, are relegated to Appendix
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