Z. Phys. C - Particles and Fields 32, 195-207 (1986) 9 Partkzi for Phys~k C andF ds 9 Springer-Verlag 1986 Hadronic Contributions to Electroweak Parameter Shifts A Detailed Analysis F. Jegerlehner tnstitut fiJr theoretischePhysik, Universit~it,CH-3012 Bern, Switzerland, and Fakult~it fiir Physik, Universitfit 1, D-4800Bielefeld 1, Federal Republic of Germany Received 30 January 1986 Abstract. Using e § e--data, an updated analysis of hadronic contributions to electroweak parameter re- normalizations is presented. We emphasize the esti- mate of uncertainties which is important for preci- sion tests at LEP and SLC. For Mz = 93 GeV and sin2 @o = 0.22 hadronic contributions from 5 flavors are found to be d r~ d = 0.0326 • 0.0007(A r~)D,had = 0.0286 • 0.0007) and za- vhaut't 5 ~ = 0.0602 • 0.0016(A g~3~ ), had= 0.0619 _+ 0.0016) for the renormalization of~ and % = ~/sin 2 6)0, respec- tively. Parameter shifts are calculated and uncer- tainties due to higher order effects are estimated. 1. Introduction Properties of the intermediate vector bosons [ 1] will be determined with high accuracy at LEP and SLC in the near future. Of particular interest are precision tests of the mass-coupling relations, characteristic to mass generation by the Higgs mechanism, and their modifi- cation by radiative corrections. Within the standard model of Glashow-Weinberg-Salam (GWS) [2] one- loop-calculations of vector boson masses and widths in terms of low energy parameters have been presented by many authors I-3-7]. Since higher order predictions necessarily include effects from as yet unknown parti- cles like the Higgs-boson or the top-quark, other "new physics" may be included as well, precision measure- ments yield important information on hitherto unob- served effects. Unfortunately, the precision of theoreti- cal predictions is limited due to uncertainties of the hadronic (quark) vacuum polarization effects involved. The answer to the question of how much can be learned from precision experiments crucially depends on a precise evaluation of hadronic effects. To the level of accuracy we are interested in here, results published so far seem not to be in sufficient agreement, in particular concerning the estimated uncertainties [8-13]. Fur- thermore, I think that attempts which are frequently made to predict low energy hadronic contributions by means of perturbation theory [3, 12] are rather ques- tionable* even if used in conjunction with analytic continuation methods [13]. The aim of this article is to present a detailed account of an up-date dispersion theoretic computation of hadronic effects using experimental data up to 40 GeV. I adapt the method used by Kinoshita et al. [14] in a recent analysis of the anomalous magnetic moment of the muon. A number of hadronic vacuum polarization functions needed for the calculation of the vector boson masses in terms of low energy parameters are deter- mined. In case of the electromagnetic vacuum polariz- ation (which is directly determined by e§ previous analogous calculations by Paschos [8], Ellis etal. [9], Wetzel [6] and Penso [10] are improved and/or essentially confirmed. In Sect. 2, we specify the parameter shift formulae to be considered. The hadronic contributions are given in terms of QCD vacuum polarization amplitudes in Sect. 3. A detailed account of the contributions from the electromagnetic hadron current follows in Sect. 4 whereas the remaining terms are estimated in Sect. 5. Conclusions and applications to parameter shift cal- culations are contained in Section 6. The presentation is extensive in order to allow for detailed comparison with similar work. 2. Parameters and Parameter Shifts We consider the standard electroweak theory [2] parametrized in terms of the fine structure constant =e2/4n (from low momentum transfer Coulomb scattering) the Fermi constant G~ (from the muon 1 Permanent address * Related ambiguities have been illustrated by Ellis et al. 1-9]