Exact Differential O ( α 2 ) Results for Hard Bremsstrahlung in e + e − Annihilation to Two Fermions At and Beyond LEP 2 Energies ∗

We present the exact O(α) correction to the process e+e− → f f̄ + γ, f 6= e, for ISR⊕FSR at and beyond LEP2 energies. We give explicit formulas for the completely differential cross section. As an important application, we compute the size of the respective sub-leading corrections of O(αL) to the f f̄ cross section, where L is the respective big logarithm in the renormalization group sense so that it is identifiable as L = ln |s|/me when s is the squared e+e− cms energy. Comparisons are made with the available literature. We show explicitly that our results have the correct infrared limit, as a cross-check. Some comments are made about the implementation of our results in the framework of the Monte Carlo event generator KK MC. Work supported in part by the US DoE, contract DE-FG05-91ER40627, by the Polish Government, grants KBN 2P03B08414 and KBN 2P03B14715, and by the US-Poland Maria Sklodowska-Curie Fund II PAA/DOE-97-316. Permanent address: Institute of Nuclear Physics, ul. Kawiory 26a, PL 30-059 Cracow, Poland Permanent Address: Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200, USA Currently, the final LEP2 data analysis is in its beginning stages, and the desired total precision tags on the important LEP2 physics processes ee → f f̄ , f 6= e, are already called out in the LEP2 MC Workshop in Ref. [1]. It has been demonstrated in Ref. [1] that the Monte Carlo (MC) event generator program KK [2], hereafter referred to as KK MC, and the semi-analytical program ZFITTER [3] realize these precisions (.2 − 1%) in most channels for inclusive cross sections and that for the fully differential distributions, the KK MC again meets most of the requirements for the LEP2 final data analysis. In this paper, we present exact results on the O(α) correction to the single hard bremsstrahlung processes ee → f f̄ + γ, f 6= e. This correction is an important contribution to the differential distributions as they are realized in the KK MC which allows the very demanding precisions just cited to be achieved. Specifically, the exact results for the O(α) corrections to s-channel annihilation hard bremsstrahlung processes under study here were also considered in Refs. [4, 5]. We differ from these results as follows. Concerning Ref. [4], the entire result was given only for the case in which the photon angle variables are all integrated out; here, we give the fully differential results. With regard to Ref. [5], the completely differential results were given as well but the mass corrections were omitted. In our work, the masses of the electrons and positrons and the masses of the final state fermion and anti-fermion are exactly taken into account in contrast to the literature. Thus, by comparing with the two calculations in Refs. [4, 5] as we do here, we get a measure of the size of the mass corrections as well as cross checks on both our differential and our integrated results. Our work is organized as follows. In Section 1, we set our notational conventions. In Section 2, we present our exact amplitudes for the O(α) virtual corrections to initialstate and final-state real radiation. In Section 3, we derive the differential cross-sections corresponding to these amplitudes in a form useful for comparisons. In Section 4, we compare these results with those in Refs. [4, 5] while illustrating our results as they are used in the KK MC in Ref. [2]. Section 5 contains our summary remarks. The Appendix contains technical details about the scalar integrals. 1 Preliminaries In this section we set our notational conventions. We will use the conventions of Refs. [2, 6, 7] for our spinors. These conventions are based on the Kleiss-Stirling [8] Weyl spinors augmented as described in Refs. [2,6,7] with the rules for controlling their complex phases, or equivalently, the three axes of the fermion rest frame in which the spin of that fermion is quantized. We sometimes refer to this fermion rest frame as the global positioning of spin