UC Berkeley

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.