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The nonrelativistic Schrodinger equation is used with confinement potentials that are either fractional power laws or logarithmic functions of the radial coordinate to investigate the spectrum of states, leptonic decay widths, and radiative decays of the 4 family of resonances. The spectrum of states and the leptonic decay widths are in good agreement with the data for the entire class of potentials considered here. The radiative decays are still somewhat large for the standard model but could be brought into agreement with the data when threshold and relativistic effects are taken into account.

Radiative corrections to the two-gamma decay of parapositronium are examined. Special care is taken in the handling of the so-called binding diagram; in particular, the limiting procedure related to the infrared divergence is considered carefully. The general covariant gauges and the Fried-Yennie gauge are used in the computation to see that gauge invariance is accounted for. The order 01 correction of Harris and Brown is confirmed. In addition, from a sharp peak of the matrix element at low momentum and the low-momentum correction to the wave function, an ae In M-I correction is derived.

One of the most remarkable consequences of grand unified gauge theories is that the proton decays. Several authors have made estimates of the proton decay rate in the SU(5) and SO(10) models 1 * 2 and have found the lifetime to be in the range of 10 31~1 0 33 yr, 3 ' 4 which is close to the present experimental lower bound 5 (~ 10 30 yr). Renewed attempts to observe such decays are under way and are expected to give some results soon if the proton lifetime is in the above-mentioned range.Among various decay modes, the mode p-*e + + TT° seems to be the most appropriate one for detection in most experiments in progress if its branching ratio is significantly large. However, there is some uncertainty in the theoretical estimates for this process since the estimates are made based on either the SU(6) wave functions or the Massachusetts Institute of Technology bag model wave functions, which are either nonrelativistic or noncovariant. The proton, consisting of three light quarks, is likely to be a relativistic system and the relativistic corrections may not be negligible. In this article, I present a rela-