Low-energy chiral field theory for pions and non-relativistic nucleons on a lattice

Abstract

We study the simplest version of the SU(2) ⊗ SU(2) chiral field theory for pions and nucleons on an 8 4 lattice in which we make a non-relativistic expansion in the nucleon momenta for the pion-nucleon interactions. The simulation is performed using a second-order Langevin simulation scheme with a slave equation to estimate the pion two-point function. The renormalised pion mass, M R, and decay constant, F R, are computed and their ratio is adjusted to be the empirical value by an appropriate choice of the bare parameters for a range of values of M R. Various static properties of the nucleon are computed as a function of M R using a transfer-matrix method. These properties include the nucleon charge-radius, the iso-vector form-factor and the renormalisation of the axial charge. The results of the simulation are compared with one-loop perturbation theory where we find agreement with the computation for quantities for which the perturbative corrections are power-law divergent but disagreement for the pion contribution to the charge-radius for which the associated loop calculation is log divergent. We conclude that unitarity corrections to tree-level quantities are not necessarily accurately obtained by one-loop perturbation theory and that this must be taken into account when constructing a full effective theory of pions and nucleons.