Bayesian method for fitting the low-energy constants in chiral perturbation theory

The values of the low-energy constants (LECs) are very important in the chiral perturbation theory. This paper adopts a Bayesian method with the truncation errors to globally fit eight next-to-leading order (NLO) LECs L ^r_i and next-to-next-leading order (NNLO) LECs C ^r_i . With the estimation of the truncation errors, the fitting results of L ^r_i in the NLO and NNLO are very close. The posterior distributions of C ^r_i indicate the boundary-dependent relations of these C ^r_i . Ten C ^r_i are weakly dependent on the boundaries and their values are reliable. The other C ^r_i are required more experimental data to constrain their boundaries. Some linear combinations of C ^r_i are also fitted with more reliable posterior distributions. If one knows some more precise values of C ^r_i , some other C ^r_i can be obtained by these values. With these fitting LECs, most observables provide a good convergence, except for the πK scattering lengths a ^3/2₀ and a ^1/2₀ . An example is also introduced to test the improvement of the method. All the computations indicate that considering the truncation errors can improve the global fit greatly, and more prior information can obtain better fitting results. This fitting method can be extended to the other effective field theories and the perturbation theory.