Analytic evaluation of the amplitudes for orthopositronium decay to three photons to one-loop order

The matrix element for the decay of orthopositronium to three photons can be expressed in terms of three independent amplitudes. We describe the analytic evaluation of these amplitudes, both to lowest order and with the inclusion of all one-loop corrections. We use these amplitudes to find precise values for the one-loop correction to the orthopositronium decay rate Gamma_1=-10.286606(10) (alpha/pi) Gamma_{LO}, and for the order-alpha^2 "square" correction to the decay rate Gamma_2(square) = 28.860(2) (alpha/pi)^2 Gamma_{LO}, where Gamma_{LO} is the lowest order rate. We give in explicit form the function describing the one-loop correction to the distribution in phase space of the final state photons.