Leptonic contribution to the effective electromagnetic coupling constant up to three loops

In this note the leptonic contribution to the running of the electromagnetic coupling constant is discussed up to the three-loop level. Special emphasize is put on the evaluation of the double-bubble diagrams. The dominant correction to electroweak observables is provided by the running of the electromagnetic coupling constant from its value at vanishing momentum transfer to high energies. The main part is delivered from the leptonic contribution, ∆αlep, where QED corrections to the photon polarization function, Π(q), have to be considered. The large logarithms of the ratio between the lepton masses, Ml, and the mass of the Z boson, MZ , is responsible for the size of the corrections. On the other hand the small ratio ensures that it is enough to consider an expansion of Π(q) for q ≫ M l . Below in addition to the leading term we will include the first order mass corrections for the oneand two-loop case and demonstrate that the numerical impact is small. In [1] it is shown that the two-loop term for ∆αlep(M 2 Z) amounts to approximately 0.78× 10 which though is less than 0.3 % of the one-loop value is of the same order of magnitude as the error of the hadronic contribution, ∆α (5) had [1]. Hence, it is interesting to have a look at the three-loop contributions also because there for the first time quadratic logarithms of M l /M 2 Z appear. In this note the three-loop diagrams contributing to Π(q), respectively, ∆αlep, are discussed. Thereby only the electron, muon and tau lepton are considered. Virtual quark loops which are present at this order are not taken into account. However, the formulae given below could in principle also be applied to this case. At oneand two-loop level and for the diagrams relevant for the so-called quenched QED only one mass scale — in the following called M1 — is involved. Concerning the double-bubble diagrams, however, in general two mass scales, M1 and M2, appear, where