For all of its successes, the Standard Model (SM) of particle physics cannot explain the observed asymmetry of the matter and antimatter contents of the universe. Toward a so- lution for this problem, Andrei Sakharov proposed in 1967 three necessary and sufficient conditions for any extension of the accepted model to be able to produce such an imbal- ance. In particular, the combined parity (P) and charge conjugation (C) symmetry must be significantly violated by fundamental interactions. While there is some CP violation in the electroweak sector of the Standard Model, it is grossly insufficient to account for the observed difference. A historically attractive probe into sources of CP violation beyond the Standard Model (BSM) has been the neutron electric dipole moment (nEDM). The exper- imental upper bound on its value lies several order of magnitude above the lower bound imposed by the Standard Model, providing a large window to search for CP-violating BSM phenomena. There are many potential sources. At hadronic scales, these interactions may be encoded by effective local operators of SM fields. In order to disentangle their contri- butions, their hadronic matrix elements must be precisely precisely determined, which is currently only possible within the framework of lattice quantum chromodynamics (LQCD). The primary difficulty in the computation of these matrix elements is their renormalization, which mixes the effective operators. Since the only available scale to parametrize the mixing is the lattice spacing, these computations are prone to potential power divergences related to lower-dimensional operators in the continuum limit. In this thesis, we propose to use the gradient flow to temper these divergences. The gradient flow is essentially a gauge-covariantsmearing of the quantum fields. It introduces a fifth dimension, the flow time, that controls the extent of the smearing. Critically, the flow time also provides an alternative scale to the lattice spacing. This allows us to define the effective operators through a short-flow- time expansion, which enjoys a smooth continuum limit for fixed, nonzero flow times. the expansion coefficients can be determined on the lattice, so long as their ultraviolet behav- ior is constrained in some manner. The natural way to do this is through perturbation theory, though the calculations are made much more difficult by the introduction of Gaus- sian damping factors. In this thesis, we comprehensively construct the perturbation theory and renormalization of the gradient flow from the ground up, introducing along the way a new method for calculating dimensionally-regularized loop integrals with difficult angular dependence. This method relies heavily on the Schwinger proper time representation of propagators and handles the angular pieces through a combinatorial tensor decomposition. Using this novel technique, we calculate the renormalization constants and short-flow-time coefficients of a handful of physically interesting operators, including the topological charge density and chromoelectric dipole moments. We further use the gradient flow to define a pure-lattice renormalization scheme along with an induced renormalization group flow, which we connect to more phenomenologically amenable renormalization schemes using our new perturbative techniques.
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