We investigate non perturbatively scattering properties of Goldstone Bosons in an SU(2) gauge theory with two Wilson fermions in the fundamental representation. Such a theory can be used to build extensions of the Standard Model that unifies Technico
lor and pseudo Goldstone composite Higgs models. The leading order contribution to the scattering amplitude of Goldstone bosons at low energy is given by the scattering lengths. In the context of technicolor extensions of the Standard Model the scattering lengths are constrained by WW scattering measurements. We first describe our setup and in particular the expected chiral symmetry breaking pattern. We then discuss how to compute them on the lattice and give preliminary results using finite size methods.
We have calculated continuum limit step scaling functions of bilinear and four-fermion operators renormalized in a Rome-Southampton scheme using various smearing prescriptions for the gauge field. Also, for the first time, we have calculated non-pert
urbative anomalous dimensions of operators renormalized in a Rome-Southampton scheme. The effect of such smearing first enters connected fermionic correlation functions via radiative corrections. We use off-shell renormalisation as a probe, and observe that the upper edge of the Rome-Southampton window is reduced by link smearing. This can be interpreted as arising due to the fermions decoupling from the high momentum gluons and we observe that the running of operators with the scale at large lattice momenta shows enhanced lattice artefacts. We find that the effect is greater for HEX smearing than for Stout smearing, but that in both cases additional care must be taken when using off-shell renormalisation with smeared gauge fields compared to thin link simulations.
We show that the running of operators which mix under renormalization can be computed fully non-perturbatively as a product of continuum step scaling matrices. These step scaling matrices are obtained by taking the ratio of Z matrices computed at dif
ferent energies in an RI-MOM type scheme for which twisted boundary conditions are an essential ingredient. Our method allows us to relax the bounds of the Rome-Southampton window. We also explain why such a method is important in view of the light quark physics program of the RBC-UKQCD collaborations. To illustrate our method, using n_f=2+1 domain-wall fermions, we compute the non-perturbative running matrix of four-quark operators needed in K->pipi decay and neutral kaon mixing. Our results are then compared to perturbation theory.
