Applications of Effective Field Theories in Atomic and Nuclear Problems

Abstract

The effective field theory (EFT) approach is employed to study problems in nuclear and atomic physics. First, we revisit Pionless EFT in the two-nucleon system. Special focus is given to the analysis of residual cutoff dependence and its implications for power counting. In the two $S$ waves, we show that the four-derivative contact interactions only come in at next-to-next-to-next-to-leading order (N$^3$LO), in contrast with the conventional power counting where they appear at next-to-next-to-leading order. We present results up to N$^3$LO in distorted-wave perturbation theory for phase shifts and some low-energy properties of deuteron. Comparison with empirical results suggests convergence up to center-of-mass momenta on the order of the pion mass in the two $S$ waves. The convergence patterns in the two $S$ waves show that an expansion around the unitarity limit works well for momenta greater than the magnitude of the inverse of the scattering length and smaller than the pion mass. Second, in an EFT for short-range forces, we study correlations between the four-body excited- and ground-state properties, such as binding energies and radii, up to next-to-leading order in an expansion around the two-body unitarity limit. We obtain the parameters in these correlations from similar correlations arising from existing precise calculations based on short-range potentials. We also derive correlations among excited-state properties that emerge from the proximity of the state to the break-up threshold into a boson and a three-boson bound state, using an EFT for ``halo'' states. Finally, within the framework of Chiral EFT, we relate the unknown short-range contribution to neutrinoless double-beta decay ($0\nu \beta \beta$) to charge-independence breaking (CIB) in nuclear systems and pion-nucleus reactions. We construct different CIB quantities from pion-nucleus reactions, which have the potential to determine the short-range contribution. By considering elastic scattering between pions and fictitious two-nucleon systems in the $^1S_0$ channel, we calculate one of these CIB quantities and resolve the corresponding renormalization problem. It is shown that the renormalization in pion-nucleus scattering is consistent with that in $0\nu \beta \beta$. As a proof of principle, we demonstrate that the short-range contribution to $0\nu \beta \beta$ could be determined from pion-nucleus reactions. To eventually fix the short-range contribution, experimental input from pion-nucleus reactions and solid many-body calculations are needed.