UC Berkeley

Precision in semileptonic weak nuclear interactions has become an important topic, with applications to both high-energy and solar physics. We can search for physics beyond the standard model by checking CKM unitarity using superallowed Fermi beta decay. This is one of the most precise tests of the standard model. With better measurements of the axial form factor $g_A$, the neutron lifetime can also become a competitive test of CKM unitarity. In solar physics, the rate of pp-fusion can be used in the luminosity constraint to test for new physics in the sun.

The radiative correction to superallowed Fermi beta decay comes from a box diagram involving the axial-vector weak current. It is known that the spin-flip transitions generated by the magnetic moment and Gamow-Teller operators are modified by the nuclear environment. Nuclear shell model calculations of these transitions consistently give larger results than what is seen in experiments. This is corrected by phenomenological ``quenching factors" which suppress the rates to match experiment. In the analysis done by Towner and Hardy, they accounted for this effect by modifying the free-nucleon Born correction by a product of these quenching factors.

Their analysis was challenged in a 2019 paper by Seng Gorchtein and Ramsey-Musolf, who claim that this analysis is flawed due to the fact that the quasi-elastic contribution has been shown not to require this quenching correction. They provide a formula for the quasi-elastic part of the box diagram, which involves nuclear structure corrections due to Pauli blocking and the nucleon removal energy.

The goal of this thesis is to focus on a smaller nuclear system, in which the calculation does not suffer from these issues. Instead of focusing on Fermi decay, which has been the focus of much of the field, we analyze the case of $pp$-fusion which is mediated by a Gamow-Teller interaction. We also are able to confirm the approximation claimed in a 2003 analysis done by Kurylov Ramsey-Musolf and Vogel. In particular, we will be able to directly calculate the two-body contribution, Figure 1b in their paper.

Since much of the focus has been on Fermi decays, and much less has been written about Gamow-Teller transitions, we go through the analysis for both the one-body Born contribution and the two-body nuclear structure correction. We give a new analysis of the Born correction for Gamow-Teller transitions, which is slightly different from that of Hayen 2021. This contribution is important for comparing the measured value of $g_A$ in neutron decay to the one calculated in lattice QCD. We also derive new formulas for the two-body nuclear structure correction, in analogy with the analysis of Fermi decays by Towner 1992. Using the standard two-body density matrix technique, these formulas can be used to calculate the nuclear structure correction to Gamow-Teller transitions in larger nuclear systems.

We calculate the nuclear structure correction in $pp$-fusion in three different ways. First, we apply our new formulas for the two-body nuclear structure correction. This result depends on the approximation scheme outlined in Jaus and Rasche 1990 and Towner 1992, where we ignore the nuclear environment effects on the Green's function. Our result is slightly smaller than the estimate given by Kurylov and collaborators, due to a partial cancellation between the spin and the tensor terms and the narrow momentum space wavefunction of the initial $pp$-state.

We then give a new method which does not rely on approximating the nuclear Green's function. This involves expanding the box diagram in terms of all possible intermediate states. We are able to check that this works by verifying the completeness relation, which must be exactly satisfied if all intermediate states are included. This is the main reason we are limited to small nuclear systems, such as the two and three body system.

Once we expand the box diagram in terms of intermediate states, we are then able to directly calculate the effect of the nuclear environment through the nuclear Green's function. We find a significant enhancement over the previous calculation - much larger than one would expect from the non-relativistic $1/M_N$ power counting. This effect is due to a persistent energy gap at low momentum transfer, coming from the binding energy of the deuteron.

Using our detailed knowledge of the system, we are then able to derive an approximate form of the result - the ``modified normal ordering" scheme. We pick an average value of the nuclear energy, which is allowed to depend on the loop momentum. This approximation scheme allows us to remove the intermediate states and normal order the currents, thus allowing us to separate out the one-body and two-body parts.

In addition to narrowing the uncertainty in the $pp$-fusion cross section, the analysis done here is one concrete demonstration of the method outlined in Seng Gorchtein and Ramsey-Musolf's 2019 paper. While we were not able to resolve the issue of how to handle the phenomenological quenching factors in larger systems, the work done here can provide some insight into what an alternative method might look like.