Regularization in nonperturbative extensions of effective field theory

Abstract

The process of renormalization in nonperturbative Hamiltonian effective field theory (HEFT) is examined in the $\mathrm{\ensuremath{\Delta}}$-resonance scattering channel. As an extension of effective field theory incorporating the L\"uscher 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 regularization. The extent to which the established features of this regularization scheme survive in HEFT is examined. In a single-channel $\ensuremath{\pi}N$ analysis, fits to experimental phase shifts withstand large variations in the regularization parameter $\mathrm{\ensuremath{\Lambda}}$, providing an opportunity to explore the sensitivity of the finite-volume spectrum and state composition on the regulator. While the L\"uscher formalism ensures the eigenvalues are insensitive to $\mathrm{\ensuremath{\Lambda}}$ variation in the single-channel case, the eigenstate composition varies with $\mathrm{\ensuremath{\Lambda}}$; the admission of short-distance interactions diminishes single-particle contributions to the states. In the two-channel $\ensuremath{\pi}N$, $\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$ analysis, $\mathrm{\ensuremath{\Lambda}}$ is restricted to a small range by the experimental data. Here the inelasticity is particularly sensitive to variations in $\mathrm{\ensuremath{\Lambda}}$ and its associated parameter set. This sensitivity is also manifest in the finite-volume spectrum for states near the opening of the $\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$ scattering channel. Future high-quality lattice QCD results will be able to discriminate $\mathrm{\ensuremath{\Lambda}}$, describe the inelasticity, and constrain a description of the basis-state composition of the energy eigenstates. 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.