Abstract
Recently, the formalism needed to relate the finite-volume spectrum of systems of nondegenerate spinless particles has been derived. In this work we discuss a range of issues that arise when implementing this formalism in practice, provide further theoretical results that can be used to check the implementation, and make available codes for implementing the three-particle quantization condition. Specifically, we discuss the need to modify the upper limit of the cutoff function due to the fact that the left-hand cut in the scattering amplitudes for two nondegenerate particles moves closer to threshold; we describe the decomposition of the three-particle amplitude mathcal{K} df,3 into the matrix basis used in the quantization condition, including both s and p waves, with the latter arising in the amplitude for two nondegenerate particles; we derive the threshold expansion for the lightest three-particle state in the rest frame up to mathcal{O} (1/L5); and we calculate the leading-order predictions in chiral perturbation theory for mathcal{K} df,3 in the π+π+K+ and π+K+K+ systems. We focus mainly on systems with two identical particles plus a third that is different (“2+1” systems). We describe the formalism in full detail, and present numerical explorations in toy models, in particular checking that the results agree with the threshold expansion, and making a prediction for the spectrum of π+π+K+ levels using the two- and three-particle interactions predicted by chiral perturbation theory.
Highlights
We discuss the need to modify the upper limit of the cutoff function due to the fact that the left-hand cut in the scattering amplitudes for two nondegenerate particles moves closer to threshold; we describe the decomposition of the three-particle amplitude Kdf,3 into the matrix basis used in the quantization condition, including both s and p waves, with the latter arising in the amplitude for two nondegenerate particles; we derive the threshold expansion for the lightest three-particle state in the rest frame up to O(1/L5); and we calculate the leading-order predictions in chiral perturbation theory for Kdf,3 in the π+π+K+ and π+K+K+ systems
We describe the formalism in full detail, and present numerical explorations in toy models, in particular checking that the results agree with the threshold expansion, and making a prediction for the spectrum of π+π+K+ levels using the two- and three-particle interactions predicted by chiral perturbation theory
Appendix A collects some technical details related to the implementation of the 2 + 1 quantization condition; appendix B outlines the derivation of the threshold expansion for three nondegenerate particles; and appendix C derives the relationship between Kdf,3 and M3 for a 2 + 1 system, which is needed for the χPT calculation of section 3
Summary
In the last few years, considerable progress has been made in the development of the formalism needed to connect the three-particle finite-volume spectrum in quantum field theories to infinite-volume scattering amplitudes [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25], and in the implementation of this formalism and its application to the results from lattice QCD (LQCD) simulations [20, 26,27,28,29,30,31,32,33,34,35,36]. We have determined the first three nontrivial terms in the 1/L expansion of energy of the threshold three-particle state (the “threshold expansion”) for nondegenerate particles This extends previous results for identical and 2 + 1 systems, and provides useful checks on our implementation of the formalism. The first involves a quantization condition, an equation whose solutions give the spectrum in terms of two- and three-particle contact interactions or K matrices These latter quantities are defined in infinite volume, but are not, in general, physical, since they depend on the details of cutoff functions and other technical choices. Appendix A collects some technical details related to the implementation of the 2 + 1 quantization condition; appendix B outlines the derivation of the threshold expansion for three nondegenerate particles; and appendix C derives the relationship between Kdf, and M3 for a 2 + 1 system, which is needed for the χPT calculation of section 3. In appendix D, we provide examples of the use of our codes
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