Form of the effective interaction in harmonic-oscillator-based effective theory

Abstract

I explore the form of the effective interaction in harmonic-oscillator-based effective theory (HOBET) in leading order (LO) through next-to-next-to-next-to-leading order (N$^{3}\mathrm{LO}$). Because the included space in a HOBET (as in the shell model) is defined by the oscillator energy, both long-distance (low-momentum) and short-distance (high-momentum) degrees of freedom reside in the high-energy excluded space. A HOBET effective interaction is developed in which a short-range contact-gradient expansion, free of operator mixing and corresponding to a systematic expansion in nodal quantum numbers, is combined with an exact summation of the relative kinetic energy. By this means the very strong coupling of the included ($P$) and excluded ($Q$) spaces by the kinetic energy is removed. One finds a simple and rather surprising result, that the interplay of $\mathit{QT}$ and $\mathit{QV}$ is governed by a single parameter \ensuremath{\kappa}, the ratio of an observable, the binding energy $|E|$, to a parameter in the effective theory, the oscillator energy $\ensuremath{\hbar}\ensuremath{\omega}$. Once the functional dependence on \ensuremath{\kappa} is identified, the remaining order-by-order subtraction of the short-range physics residing in $Q$ becomes systematic and rapidly converging. Numerical calculations are used to demonstrate how well the resulting expansion reproduces the running of ${H}^{\mathrm{eff}}$ from high scales to a typical shell-model scale of $8\ensuremath{\hbar}\ensuremath{\omega}$. At N$^{3}\mathrm{LO}$ various global properties of ${H}^{\mathrm{eff}}$ are reproduced to a typical accuracy of 0.01%, or about 1 keV, at $8\ensuremath{\hbar}\ensuremath{\omega}$. Channel-by-channel variations in convergence rates are similar to those found in effective field theory approaches. The state dependence of the effective interaction has been a troubling problem in nuclear physics and is embodied in the energy dependence of ${H}^{\mathrm{eff}}(|E|)$ in the Bloch-Horowitz formalism. It is shown that almost all of this state dependence is also extracted in the procedures followed here, isolated in the analytic dependence of ${H}^{\mathrm{eff}}$ on \ensuremath{\kappa}. Thus there exists a simple, Hermitian ${H}^{\mathrm{eff}}$ that can be use in spectral calculations. The existence of a systematic operator expansion for ${H}^{\mathrm{eff}}$, depending on a series of short-range constants augmented by \ensuremath{\kappa}, will be important to future efforts to determine the HOBET interaction directly from experiment, rather than from an underlying $\mathit{NN}$ potential.