  The process of renormalisation in nonperturbative Hamiltonian Effective Field
Theory (HEFT) is examined in the $\Delta$-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 regularisation. The extent
to which the established features of this regularisation scheme survive in HEFT
is examined. In a single-channel $\pi N$ analysis, fits to experimental phase
shifts withstand large variations in the regularisation parameter, $\Lambda$,
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 $\Lambda$ variation in the
single-channel case, the eigenstate composition varies with $\Lambda$; the
admission of short distance interactions diminishes single-particle
contributions to the states. In the two-channel $\pi N$, $\pi \Delta$ analysis,
$\Lambda$ is restricted to a small range by the experimental data. Here the
inelasticity is particularly sensitive to variations in $\Lambda$ and its
associated parameter set. This sensitivity is also manifest in the
finite-volume spectrum for states near the opening of the $\pi \Delta$
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.
