Abstract
Partial quenching allows one to consider correlation functions and amplitudes that do not arise in the corresponding unquenched theory. For example, physical $s$-wave pion scattering can be decomposed into $I=0$ and 2 amplitudes, while, in a partially-quenched extension, the larger symmetry group implies that there are more than two independent scattering amplitudes. It has been proposed that the finite-volume quantization condition of L\"uscher holds for the correlation functions associated with each of the two-particle amplitudes that arise in partially-quenched theories. Using partially-quenched chiral perturbation theory, we show that this proposal fails for those correlation functions for which the corresponding one-loop amplitudes do not satisfy $s$-wave unitarity. For partially-quenched amplitudes that, while being unphysical, do satisfy one-loop $s$-wave unitarity, we argue that the proposal is plausible. Implications for previous work are discussed.
Highlights
Simulations of lattice QCD (LQCD) and related theories naturally separate into the generation of gauge fields and the calculation of quark propagators on these gauge fields
This paper has demonstrated that the violation of unitarity in PQ theories at best restricts the application of the two-particle quantization condition to a subset of the unphysical two-particle channels
This has been demonstrated using PQχPT, and our initial criteria C1 is stated within this effective field theory (EFT) framework
Summary
Simulations of lattice QCD (LQCD) and related theories naturally separate into the generation of gauge fields and the calculation of quark propagators on these gauge fields. The second part of the claim is that we can insert the resulting energies En into Lüscher’s quantization condition and correctly obtain the scattering amplitude in the corresponding irrep We know that this procedure is valid for the 84 irrep, as it corresponds to a physical correlation function in QCD. This proposal is quite powerful, as, combined with PQχPT, it allows the determination of results from types of contractions that are difficult to calculate numerically from those that are simpler to evaluate [7,13,14,18] It is, prima facie surprising, since both the existence of a normal spectral decomposition and the derivation of the two-particle quantization condition rely on unitarity, which the PQ theory violates [1,19].
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