Regularisation in Nonperturbative Extensions of Effective Field Theory

Abstract

The process of renormalisation in nonperturbative Hamiltonian Effective Field Theory (HEFT) is examined in the $Δ$-resonance scattering channel. As an extension of effective field theory incorporating the Lüscher formalism, HEFT provides a bridge between the infinite-volume scattering data of experiment and the finite-volume spectrum of energy eigenstates in lattice QCD. HEFT also provides phenomenological insight into the basis-state composition of the finite-volume eigenstates via the state eigenvectors. The Hamiltonian matrix is made finite through the introduction of finite-range regularisation. The extent to which the established features of this regularisation scheme survive in HEFT is examined. In a single-channel $πN$ analysis, fits to experimental phase shifts withstand large variations in the regularisation parameter, $Λ$, providing an opportunity to explore the sensitivity of the finite-volume spectrum and state composition on the regulator. While the Lüscher formalism ensures the eigenvalues are insensitive to $Λ$ variation in the single-channel case, the eigenstate composition varies with $Λ$; the admission of short distance interactions diminishes single-particle contributions to the states. In the two-channel $πN$, $πΔ$ analysis, $Λ$ is restricted to a small range by the experimental data. Here the inelasticity is particularly sensitive to variations in $Λ$ and its associated parameter set. This sensitivity is also manifest in the finite-volume spectrum for states near the opening of the $πΔ$ scattering channel. Finally, HEFT has the unique ability to describe the quark-mass dependence of the finite-volume eigenstates. The robust nature of this capability is presented and used to confront current state-of-the-art lattice QCD calculations.