This work focuses on construction of a bridge from QCD (quantum chromodynamics), the theory of quarks, gluons, and their interactions, to nuclear structure, an obvious but unattained objective ever since the introduction of QCD in 1973. The bridge footing on one side of the chasm is QCD in the non-perturbative regime, only now beginning to yield to massively parallel computation in a Monte-Carlo space-time lattice formulation of QCD called LQCD (lattice quantum chromodynamics) that is our only tool for such problems. The resulting trickle of information about the nucleon interaction comes in the form of a fuzzy spectrum for two nucleons in a periodic box. It can be expected that the spectrum will sharpen and even eventually include a spectrum for three nucleons in a box with the introduction of larger and faster supercomputers as well as more clever algorithms. Fundamentally though, limits on what can be accomplished in LQCD are set by the famous fermion sign problem. Results in LQCD are produced as a small residual of the sum of large positive and negative contributions from the Monte-Carlo trials and accuracy only improves slowly with the number of expensive trials.
The bridge footing on the other side of the chasm is the configuration interaction shell model, which is commonly used for nuclear structure calculations from a microscopic Hamiltonian expressed in the colorless degrees of freedom of QCD we call nucleons. As currently executed, this method is a model, the two- and possibly three-body interaction in use lacking a rigorous connection to QCD or direct accounting for contributions from scattering outside the model space. Nucleons, like quarks, are fermions and a fermion sign like problem exists in these calculations as well. The configuration interaction shell model is formulated in an antisymmetrized harmonic oscillator basis that grows with the number of permutations of identical nucleons in the model space. However, fantastically e cient parallel sparse matrix techniques for finding low lying eigenstates exist, allowing quite large problems to be solved.
One footing of the bridge is solid and the other is nearing completion. Construction of the bridge itself then faces three major problems addressed in this dissertation, construction of the effective nuclear interaction from observables, finite volume effects associated with the periodic volume in which LQCD results are calculated, and the construction of the A-body effective Hamiltonian from the two body effective interaction.
An effective theory is a organized and complete parameterized approximation limited to and preserving the known symmetries of an underlying theory (QCD in this case), constrained to some regime (energies below the mass of the pion in this case), and expressed in degrees of freedom suitable for solving the problem at hand (nucleons in a harmonic oscillator basis below an energy cutoff for the nuclear structure problem). An effective theory has a formal relationship to the underlying theory that a model does not. Unlike a model, a small number of observables may be used to fix the lowest order expansion parameters of the effective theory approximation with the expectation that the approximation remains valid in other situations for which observables are not available.
The first portion of this work focuses on the construction of a harmonic oscillator based effective theory (HOBET) from observables in a spherical harmonic oscillator basis. It builds on the prior work of Haxton, Song, and Luu in demonstrating the construction of an convergent effective theory from a known potential, establishing the form of the required effective theory expansion. The new work required the extension of HOBET to a theory no longer limited to bound states and with continuity in energy, enabling uniform treatment of bound and continuum states. Here the expansion parameters are instead derived from phase shift observables at continuum energies. A key insight developed during this work was the way in which the effective theory constructed at an energy is connected to the boundary constraints of the wave function. Using known techniques, Lu ̈scher’s method and the HAL QCD potential method, to transform the LQCD spectrum in periodic box to infinite volume phase shifts produces a successful mechanism for fitting the effective interaction without knowledge of the details of the potential.
The techniques for converting LQCD results to phase shifts have issues such as uncontrolled systematics related to the volume size and range of the interaction as well as suspect perturbative expansions. These issues motivated an investigation into the possibility of directly constructing the effective theory in a periodic volume. This new construction relies heavily on the previous insight about the connection of the effective theory to the wave function boundary constraints. A key result is that the kernel of the effective theory, which captures scattering through the excluded degrees of freedom, is in fact independent of the boundary conditions. It can be fit in the periodic volume context and then transplanted into an infinite volume spherical formulation of the effective theory by a straightforward basis transformation. Finite volume effects are automatically handled in the process. Of immediate interest to the LQCD community is that accurate phase shifts can be easily extracted from the effective theory, avoiding systematic and finite volume errors in existing methods.
With a two body effective interaction in hand the last step to a usable bridge is the construction of an A-body interaction in terms of the two body one. The exact form this construction is not settled yet, but one promising structure with leading contributions that can be calculated is explored.
The assembly of these three pieces completes the bridge, producing a way to perform nuclear structure calculations that is formally connected to the underlying theory of QCD.
