Approximations are inevitable in solving realistic physics problems, and reliability of calculations depends on evaluation of how much error is associated with the approximations. One method to quantify errors is building effective theories, which organize successive approximations as a power series in some small parameter. We apply effective theories to two problems in nuclear physics.

One is the calculation of atomic electric dipole moments (EDMs). EDMs are of interest as a probe of CP-violating physics. For atoms, EDM signals can be thought of as departures from Schiff theorem, which states that a neutral system of point-like, nonrelativistic charges that interact only electrostatically has no net EDM. We show how each of the conditions for Schiff theorem are violated in actual atoms by expanding the Breit interaction between the electrons and the nucleus in spherical multipoles. We see that EDM signals arising from violations of the Schiff limit can be organized as a power series in R_N/R_A, the ratio between the spatial sizes of the nucleus and the atom. This ratio is of order 10^-5, and the power series in this parameter would have quantifiable errors. We identify the contributions to atomic EDM that correspond to the so-called Schiff moment, and give the general considerations for other contributions that may be of the same order as the Schiff moment in powers of R_N/R_A.

The other problem is nucleon-nucleon (NN) interaction. The difficulty in describing this basic interaction is the fine tuning that exists between the long-range attraction and short-range repulsion in the NN potential. An effective theory of NN interaction must separate these two length scales. In order to achieve this separation, we introduce a harmonic oscillator (HO) basis, and restrict the calculation to a finite Hilbert space (P-space) of states with energies below some cutoff Λℏω. HO eigenstates contains a length scale, the oscillator length b, which we choose to be 1.7fm, as an intermediate scale. We show that, despite the short range nature of HO states, restricted wavefunctions contain enough information to reconstruct phase shifts. Projecting wavefunctions into this space throws away both the long-range physics due to the kinetic energy, and the short-range physics due to the strong interactions. We derive an equation in the P-space whose solution is the P-space restriction of the full-space scattering wavefunction, and identify the components of the equation that treat the long-range and short-range physics, respectively. The long-range information is encoded in what we call the tilde states, which are modification to the HO wavefunction with slower decay as r → ∞. The short-range information is modeled by a contact-gradient expansion, which is essentially a power series in a/b, where a is the length scale associated with the repulsive core in the NN potential. The behavior of the theory is investigated using a toy model of a spherical square well with a repulsive core.

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.

The coming decade promises exceptional experimental progress in searches for charged lepton flavor-violating (CLFV) $\mu\rightarrow e$ conversion due to efforts at Fermilab (Mu2e) and J-PARC (COMET). Branching ratio sensitivities for this process are expected to advance by four orders of magnitude, potentially extending the reach of these probes up to energies of $10^4$ TeV. A pressing challenge for theorists is to extract the maximum amount of information about possible sources of CLFV from these measurements, whether or not a signal of new physics is detected.

Efforts to observe $\mu\rightarrow e$ conversion share many similarities with other experimental programs where the nucleus is treated as a laboratory in which to search for beyond-standard-model (BSM) physics. These approaches are utilized because they present certain practical advantages: In searches for CLFV, the act of trapping a muon into the Coulomb field of a nucleus allows one to control the energy of the final state electron, ensuring that it is maximal compared to the energy of background electrons originating in standard-model free muon decays. The downside of employing the nucleus as part of the apparatus is that a host of complex nuclear physics consequently intervenes between the experimentalist and the desired observable. To extract meaningful constraints, one must have a method for disentangling the nuclear physics from the underlying BSM physics.

Another experimental setting in which the nucleus is treated as a laboratory is in direct detection searches for weakly-interacting massive particle (WIMP) dark matter, where one aims to discern the mass, spin, and fundamental interactions of WIMP dark matter through scattering off of atomic nuclei. Again, to access the sought-after information about BSM physics, one must be able to separate it cleanly from the nuclear physics. In the case of dark matter direct detection, this separation has been achieved through the development of an effective theory formulated at the nuclear scale, which factorizes the nuclear physics from the BSM dark matter physics, sequestering the latter quantity into unknown low-energy constants (LECs) that are probed directly by experiment. As the effective theory describes the most general coupling between the WIMPs and the nucleus, the LECs that specify the effective theory represent the maximum information about the nature of dark matter that can be obtained from scattering off of nuclei.

In this thesis, we introduce an analogous effective theory for the problem of $\mu\rightarrow e$ conversion. In order to adapt the existing framework to the problem at hand, several significant modifications are required, primarily stemming from the differing nature of the particles that couple to the nucleus in each scenario: non-relativistic plane-wave dark matter must be replaced by a bound muon in the initial state and an ultra-relativistic electron in the final state. We focus primarily on the case of elastic $\mu\rightarrow e$ conversion, wherein the nucleus remains in its ground state (as this ensures that the energy of the outgoing electron is maximal).

The three-momentum transferred from the leptons to the nucleus $q\approx m_{\mu}$ is comparable to the inverse nuclear size, allowing significant angular momentum to be transferred between the leptons and the nucleus. As a result, the nuclear multipole expansion cannot be truncated at any order. This decomposition is complicated by the fact that the outgoing electron interacts with the nuclear charge through the Coulomb potential. Nonetheless, the nuclear multipole expansion can be performed in a straightforward manner by replacing the Coulomb-distorted electron wave function with a plane-wave form parameterized by a suitable local momentum.

The effective theory is then specified by a controlled expansion in terms of the relevant velocity operators for the nucleons $\vec{v}_N$ and the bound muon $\vec{v}_{\mu}$. (The electron velocity is, in essence, ``integrated out'' of the theory by the assumption that it is ultra-relativistic.) The construction of the nucleon-scale effective theory proceeds in two steps: First, we specify a complete set (through a given order in power-counting) of CLFV operators that couple the leptons to single-nucleon charges and currents. Next, after performing the nuclear multipole decomposition, the resulting nucleon-level theory is embedded into the target nucleus, where the approximate parity and time-reversal symmetries of the nuclear ground state restrict the operators that can contribute to elastic $\mu\rightarrow e$ conversion.

A valid effective theory can be constructed at three distinct degrees of complexity: The most basic theory is generated by including neither $\vec{v}_N$ nor $\vec{v}_{\mu}$. Relativistic corrections to the muon wave function and effects stemming from nuclear compositeness are completed ignored, and the CLFV amplitude depends on just three nuclear response functions, those of a point-like nucleus. Next, we extend the theory by considering $\vec{v}_N$ to first order, and consequently the set of nuclear responses is enlarged by the addition of three velocity-dependent response functions. Finally, we formulate the most complete effective theory, including both velocity operators, $\vec{v}_N$ and $\vec{v}_{\mu}$, to first order. This corresponds to the inclusion of relativistic muon effects, in the form of the muon's lower Dirac component, and introduces six additional nuclear responses. The muon's lower component always appears as a correction to the upper-component contribution, and therefore we consider the second of these constructions---containing $\vec{v}_N$ but not $\vec{v}_\mu$---to be the prototypical effective theory, complete through leading order in the nuclear response.

The various nuclear responses can be understood as the ``nuclear dials'' that an experimentalist can tune through nuclear target selection in order to access different regions of CLFV parameter space. The nucleus $^{27}$Al, the target of the Mu2e and COMET experiments, has ground-state angular momentum $J=5/2$ and provides good sensitivity across a range of responses that are spin- and velocity-dependent/independent. On the other hand, a target such as Ca, whose natural abundance consists (almost) entirely of isotopes with ground-state angular momentum $J=0$, will not couple to non-scalar operators. A detailed understanding of the interplay between the various nuclear responses is prerequisite to carrying out an experimental program---across a multitude of targets---in order to fully constrain the unknown CLFV parameters of the nuclear-scale effective theory.

Much of the previous literature has focused on a narrow special case in which the leading operator that mediates $\mu\rightarrow e$ conversion couples equally to protons and neutrons and is spin- and velocity-independent. Such an operator sums coherently in the conversion amplitude and receives an enhancement by the atomic mass number $A$ relative to incoherent operators, thereby dominating the CLFV response in cases where it is present. The primary advantage of working in this limited case is that the nuclear physics, which is a source of significant complication in general, becomes exceedingly simple. In fact, the coherent nuclear response is governed entirely by the scalar nucleon density, a quantity that is accurately determined by experiments. When considering specific extensions of the standard model that yield a leading coherent response, the $\mu\rightarrow e$ branching ratio can be predicted with a well-understood uncertainty. However, in the initial discovery phase of CLFV searches, one should not assume anything about the underlying nature of flavor-violating operators. The proper approach, which we pursue in this thesis, is to constrain the most general interaction as specified by the effective theory.

Effective theories are a controlled approach to making approximations in physics. We describe here work on two applications of effective theory techniques in few-body physics, specifically to nuclear and atomic systems.

First, we analytically reduce the chiral three-nucleon interaction at NNLO to a density-dependent effective two-body potential by summing the third particle over the states of a spin-symmetric Fermi gas. Results are given for the potential in momentum space and in coordinate space, where the potential is seen to be nonlocal. An expansion of the potential in the difference of the nonlocal coordinates is made in order to arrive at a fully local effective two-body potential.

We then explore the two-body spectra of spin-1/2 fermions in isotropic harmonic traps with external spin-orbit potentials and short range two-body interactions. Using a truncated basis of total angular momentum eigenstates, which is known to be equivalent to an effective theory when the atomic gas is harmonically confined, nonperturbative results are presented for experimentally realistic forms of the spin-orbit coupling: a pure Rashba coupling, Rashba and Dresselhaus couplings in equal parts, and a Weyl-type coupling. The technique is easily adapted to bosonic systems and other forms of spin-orbit coupling.