The isospin-3 three-particle $K$-matrix at NLO in ChPT

Abstract

The three-particle K-matrix, K df,3 , is a scheme-dependent quantity that parametrizes short-range three-particle interactions in the relativistic-field-theory threeparticle finite-volume formalism. In this work, we compute its value for systems of three pions at maximal isospin through next-to-leading order (NLO) in Chiral Perturbation Theory (ChPT). We compare the values to existing lattice QCD results and find that the agreement between lattice QCD data and ChPT in the first two coefficients of the threshold expansion of K df,3 is significantly improved with respect to leading order once NLO effects are incorporated. 4 Details of the calculation of K NLO df,3 4.1 General form of the threshold expansion 4.2 Threshold expansion of the non-OPE part of M 3 4.2.1 Threshold expansion of A J 4.2.2 Threshold expansion of A C 4.2.3 The full non-OPE threshold expansion 4.3 Bull's head subtraction contribution 4.3.1 Threshold expansion 4.3.2 Hadamard finite-part integration 4.3.3 Analytic approximation 4.3.4 Direct numerical evaluation 4.3.5 The full bull's head subtraction 4.4 OPE diagrams 4.4.1 Expression for Re M NLO 2,off 4.4.2 Decomposition of t 2 u 2 4.4.3 s-wave contributions 4.4.4 d-wave contributions 4.4.5 The full OPE contribution 5 Conclusions and outlook A Dependence on the cutoff B Loop integrals C Cancellation of imaginary parts 41 C.1 Bull's head diagram 41 C.2 OPE diagrams 44 C.3 Remaining diagrams 44 D Threshold expansion using single-parameter kinematic configurations 45 E An integration method for less well-behaved M 3 46 References 50