The Decay Rate of Bound Pion to Bound Muon

Abstract

The decay rate of exotic atoms is of great importance due to their short lifetimes, which presents a challenge when attempting to probe them. The study of pionic atoms can open up a new area of physics, and experiments such as the PiHe collaboration have found a new method of laser spectroscopy for pionic helium. [1]. The first part of this thesis provides a rigorous calculation of the atomic form factor, which involves evaluating an integration of a slowly convergent and oscillatory integrand. The main goal is to transform the integral into a summation which is easy to compute from the computational perspective. Using the plane wave expansion, the integral is separated into two separate integral, which are radial and angular integrals. The integral involving the product of Bessel functions and associated Laguerre polynomials is used to calculate the radial integral. The angular integral is evaluated using the Wigner’s 3−j symbol. An explicit analytical expression for the discrete-discrete transition form factor is presented, exactly in the way they are implemented in the program. Atomic form factor is an important factor in the calculation of decay rate and scattering amplitude. The second part of the thesis investigates the decay rate of the pion bound in a Coulomb potential, which is an exotic atom. The bound pion in a hydrogen atom is formed when an electron of an ordinary hydrogen atom is replaced by a negative pion. We consider the bound pion decaying into a bound muon, producing muonic hydrogen and a muon antineutrino. Using the form factor calculation from the first part, we present the decay rate of the bound pion and the most probable final state of the muon. In the calculation of decay rate, we considered the non-relativistic wave function of the initial and final particles, since the velocity of pion in hydrogen is of the order of αc, where α is the fine structure constant.