F. Jegerlehner

The muon g−2

Abstract The muon anomalous magnetic moment is one of the most precisely measured quantities in particle physics. In a recent experiment at Brookhaven it has been measured with a remarkable 14-fold improvement of the previous CERN experiment reaching a precision of 0.54 ppm. Since the first results were published, a persistent “discrepancy” between theory and experiment of about 3 standard deviations is observed. It is the largest “established” deviation from the Standard Model seen in a “clean” electroweak observable and thus could be a hint for New Physics to be around the corner. This deviation triggered numerous speculations about the possible origin of the “missing piece” and the increased experimental precision animated a multitude of new theoretical efforts which lead to a substantial improvement of the prediction of the muon anomaly a μ = ( g μ − 2 ) / 2 . The dominating uncertainty of the prediction, caused by strong interaction effects, could be reduced substantially, due to new hadronic cross section measurements in electron-positron annihilation at low energies. Also the recent electron g − 2 measurement at Harvard contributes substantially to progress in this field, as it allows for a much more precise determination of the fine structure constant α as well as a cross check of the status of our theoretical understanding. In this report we review the theory of the anomalous magnetic moments of the electron and the muon. After an introduction and a brief description of the principle of the muon g − 2 experiment, we present a review of the status of the theoretical prediction and in particular discuss the role of the hadronic vacuum polarization effects and the hadronic light-by-light scattering correction, including a new evaluation of the dominant pion-exchange contribution. In the end, we find a 3.2 standard deviation discrepancy between experiment and Standard Model prediction. We also present a number of examples of how extensions of the electroweak Standard Model would change the theoretical prediction of the muon anomaly a μ . Perspectives for future developments in experiment and theory are briefly discussed and critically assessed. The muon g − 2 will remain one of the hot topics for further investigations.

Hadronic contributions to (g−2) of the leptons and to the effective fine structure constant α(MZ2)

The new experiment planned at Brookhaven to measure the anomalous magnetic moment of the muonaμ≡(gμ−2)/2 will improve the present accuracy of 7 ppm by about a factor of 20. This requires a careful reconsideration of the theoretical uncertainties of theg−2 predictions, which are dominated by the error of the contribution from the light quarks to the photon vacuum polarization. This issue is cruicial also for the precise determination of the running fine structure constant at theZ-peak as LEP/SLC experiments continue to increase their precision. In this paper we present an updated analysis of the hadronic vacuum polarization using all presently availablee+e− data. This seems to be justified because previous work on the subject was based to some extent on preliminary or incomplete experimental data. Contributions from different energy ranges are presented separately forg−2 of the muon and the τ-lepton and for α(MZ2). We obtain the resultsaμhad*=(725±16)×10−10 andaτhad*=(351±10)×10−8, where the asterisk indicates the dressed (renormalization group improved) value. For the effective fine structure constant atMZ=91.1888 GeV we obtainΔαhad(5)=0.0280±0.0007 and α(MZ2)−1=128.896±0.090. Further improvement in the accuracy of theoretical predictions which depend on the hadronic vacuum polarization requires more precise measurements ofe+e− cross-sections at energies below about 12 GeV in future experiments.

Quest for precision in hadronic cross sections at low energy: Monte Carlo tools vs. experimental data

We present the achievements of the last years of the experimental and theoretical groups working on hadronic cross section measurements at the low-energy e+e− colliders in Beijing, Frascati, Ithaca, Novosibirsk, Stanford and Tsukuba and on τ decays. We sketch the prospects in these fields for the years to come. We emphasise the status and the precision of the Monte Carlo generators used to analyse the hadronic cross section measurements obtained as well with energy scans as with radiative return, to determine luminosities and τ decays. The radiative corrections fully or approximately implemented in the various codes and the contribution of the vacuum polarisation are discussed.

Two-loop heavy top corrections to the ϱ parameter. A simple formula valid for arbitrary Higgs mass

We present a compact analytic result for the two-loop leading heavy top contribution to the p-parameter which is valid for arbitrary Higgs mass. Our results confirm a recent calculation by Barbieri et al. [Phys. Lett. B 288 (1992) 95; preprint CERN-TH.6713/92 (1992)]. We directly calculate the physical W and Z amplitudes and explicitly check the validity of the Ward identities on which the calculation of Barbieri et al. was based. This also checks that the use of an anticommuting γ5 preserves the Ward identities. A corresponding formula for the Zbb-vertex is given as well.

Algebraic reduction of one-loop Feynman graph amplitudes

Abstract An algorithm for the reduction of one-loop n -point tensor integrals to basic integrals is proposed. We transform tensor integrals to scalar integrals with shifted dimension [A.I. Davydychev, Phys. Lett. B 263 (1991) 107] and reduce these by recurrence relations to integrals in generic dimension [O.V. Tarasov, Phys. Rev. D 54 (1996) 6479]. Also the integration-by-parts method [F.V. Tkachov, Phys. Lett. B 100 (1981) 65; K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. B 192 (1981) 159] is used to reduce indices (powers of scalar propagators) of the scalar diagrams. The obtained recurrence relations for one-loop integrals are explicitly evaluated for 5- and 6-point functions. In the latter case the corresponding Gram determinant vanishes identically for d =4, which greatly simplifies the application of the recurrence relations.

vs. pole masses of gauge bosons II: two-loop electroweak fermion corrections

We have calculated the fermion contributions to the shift of the position of the poles of the massive gauge boson propagators at two-loop order in the Standard Model. Together with the bosonic contributions calculated previously the full two-loop corrections are available. This allows us to investigate the full correction in the relationship between MS and pole masses of the vector bosons Z and W. Two-loop renormalization and the corresponding renormalization group equations are discussed. Analytical results for the master-integrals appearing in the massless fermion contributions are given. A new approach of summing multiple binomial sums has been developed.Abstract We have calculated the fermion contributions to the shift of the position of the poles of the massive gauge boson propagators at two-loop order in the Standard Model. Together with the bosonic contributions calculated previously the full two-loop corrections are available. This allows us to investigate the full correction in the relationship between MS and pole masses of the vector bosons Z and W . Two-loop renormalization and the corresponding renormalization group equations are discussed. Analytical results for the master-integrals appearing in the massless fermion contributions are given. A new approach of summing multiple binomial sums has been developed.

Two-loop QCD corrections of the massive fermion propagator

Abstract The off-shell two-loop correction to the massive quark propagator in an arbitrary covariant gauge is calculated and results for the bare and renormalized propagator are presented. The calculations were performed by means of a set of new generalized recurrence relations proposed recently by one of the authors [Nucl. Phys. B 502 (1997) 455]. From the position of the pole of the renormalized propagator we obtain the relationship between the pole mass and the MS mass. This relation confirms the known result by Gray et al. [Z. Phys. C 48 (1990) 673; C 52 (1991) 111]. The bare amplitudes are given for an arbitrary gauge group and for arbitrary space-time dimensions.

B-meson properties from lattice QCD☆

Abstract A high statistics computation of the decay constant ƒ B is presented to study scaling and finite size effects. Smooth wave functions for the B-meson allow to suppress excited state contributions effectively. We observe the scaling violation in the static approximation to be 15% between β = 5.7 and 6.0. The conventional light quark methods appear to remain applicable, at β = 6.0 even beyond the D-meson mass. The value obtained in the static approximation, ƒ B = 366 ± 22( stat. ) ± 55 ( syst. ) MeV , cannot be reached by smooth extrapolation from the D-meson range, where we find ƒ D = 198 ± 17 MeV .

The effect of the top quark on the MW−MZ interdependence and possible decoupling of heavy fermions from low energy physics

Abstract The resummation of the effects due to fermion doublets with a large mass splitting and their interplay with the photon vacuum polarization are discussed. The possibility of reexpressing the M W − M Z interdependence in a form which is valid to all orders in perturbation theory is proposed. Some considerations on the possible decoupling of heavy fermions from low energy physics are presented.

Uncertainties in the Hadronic Contribution to the QED Vacuum Polarization

We describe an updated evaluation of the hadronic contribution to the QED vacuum polarization. It is obtained from a dispersion integral over the measured cross section ofe+e−→hadrons.