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

The three-particle $K$-matrix, ${\mathcal{K}}_{\mathrm{d}\mathrm{f},3}$, is a scheme-dependent quantity that parametrizes short-range three-particle interactions in the relativistic-field-theory three-particle 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 ${\mathcal{K}}_{\mathrm{d}\mathrm{f},3}$ is significantly improved with respect to leading order once NLO effects are incorporated.