Comparisons of Fully Differential Exact Results for O ( α ) Virtual Corrections to Single Hard Bremsstrahlung in e + e − Annihilation at High Energies †

We present comparisons of the fully differential exact virtual correction to the important single hard bremsstrahlung process in e + e − annihilation at high energies , which is essential for precision studies of the Standard Model from 1 GeV to 1 TeV, as calculated by two completely independent methods and groups. We show that the two sets of results are in excellent agreement. Phenomenological implications are discussed. Now that the Standard Model electroweak theory has been established at the one-loop level [1, 2], the stage is set for studying its consequences as both signal and background to the physics objectives of current and planned high energy colliding beam devices, from 1 GeV to 1 TeV for the cms energy in e + e − annihilations for example. The attendant per mille level studies necessitate control of the EW higher order radiative corrections at least to order O(α 3 L 3) for the leading log effects and to the exact O(α 2). One set of the important contributions to the latter exact results are the virtual corrections to the single hard bremsstrahlung in e + e − annihilations, which therefore have been studied by several groups [3–6]. Comparisons of these results are essential in order to gain some confidence in their correctness and some facility in their use to confront the SM with precision data. For example, in the effort to exploit the radiative return from cms energies in the 1-2 GeV regime to the resonance regime of the ππ system in the Daphne environment, precision predictions of the type compared in this paper are essential. Similarly, in order to use the radiative return from 200 GeV to the Z in the final LEPII data analysis for precision EW tests, again precision predictions of the type compared in the following for the respective return processes are essential. have presented comparisons of the results in Refs. [3–5] and in general a very good agreement was found. However, if one looks at the comparisons in Ref. [5], one can see that at the level of the NNLL (next-to-next leading log), there was a difference in the results that was consistent with the different levels of " exactness " in the calculations. Specifically, the mass corrections are included in Ref. [5] in a fully differential way, whereas in Ref. [3] the mass corrections are included but the photon angular variable is integrated over …