The 0.2% experimental accuracy of the 1968 Beers and Hughesmeasurement of the annihilation lifetime of ortho-positroniummotivates the attempt to compute the first order quantum electrodynamiccorrections to this lifetime. The theoretical problemsarising in this computation are here studied in detail up to thepoint of preparing the necessary computer programs and using themto carry out some of the less demanding steps -- but the computationhas not yet been completed. Analytic evaluation of the contributingFeynman diagrams is superior to numerical evaluation, and for thisprocess can be carried out with the aid of the Reduce algebramanipulation computer program.
The relation of the positronium decay rate to the electronpositronannihilation-in-flight amplitude is derived in detail, andit is shown that at threshold annihilation-in-flight, Coulomb divergencesappear while infrared divergences vanish. The thresholdCoulomb divergences in the amplitude cancel against like divergencesin the modulating continuum wave function.
Using the lowest order diagrams of electron-positronannihilation into three photons as a test case, various pitfalls ofcomputer algebraic manipulation are discussed along with ways ofavoiding them. The computer manipulation of artificial polynomialexpressions is preferable to the direct treatment of rationalexpressions, even though redundant variables may have to be introduced.
Special properties of the contributing Feynman diagramsare discussed, including the need to restore gauge invariance tothe sum of the virtual photon-photon scattering box diagrams bymeans of a finite subtraction.
A systematic approach to the Feynman-Brown method ofDecomposition of single loop diagram integrals with spin-relatedtensor numerators is developed in detail. This approach allowsthe Feynman-Brown method to be straightforwardly programmed in theReduce algebra manipulation language.
The fundamental integrals needed in the wake of theapplication of the Feynman-Brown decomposition are exhibited andthe methods which were used to evaluate them -- primarily dispersion techniques are briefly discussed.
Finally, it is pointed out that while the techniquesdiscussed have permitted the computation of a fair number of thesimpler integrals and diagrams contributing to the first ordercorrection of the ortho-positronium annihilation rate, furtherprogress with the more complicated diagrams and with the evaluationof traces is heavily contingent on obtaining access to adequatecomputer time and core capacity.
