Use of Effective Theories in Nuclear Physics
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 RN/RA, 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 RN/RA.
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.