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Results for Precision Observables in the Electroweak Standard Model at Two-Loop Order and Beyond

Higher-order contributions to the precision observables \Delta r, \sin^2\theta_eff and \Gamma_l are discussed. The Higgs-mass dependence of the observables is investigated at the two-loop level, and exact results are derived for the Higgs-dependent two-loop corrections associated with fermions. The top-quark corrections are compared with the results obtained by an expansion in the top-quark mass up to next-to-leading order. For the pure fermion-loop contributions to \Delta r results up to four-loop order are derived. They allow to investigate the validity of the commonly used resummation of the leading fermionic contributions to \Delta r.


Introduction
Confronting the predictions of the electroweak Standard Model (SM) with the precision data [1] allows to derive indirect constraints on the mass of the Higgs boson, which is the last missing ingredient of the minimal SM, and also sets the basis for the investigation of possible effects of new physics via radiative corrections. The bounds on the Higgs-boson mass, M H , obtained from fitting the SM predictions to the data, are strongly affected by the theoretical uncertainty due to unknown higher-order corrections. In particular the two-loop electroweak corrections have to be under control in order to obtain reliable bounds. At this order, the resummations of the leading one-loop contributions are known [2], the leading and next-to-leading term in an expansion for asymptotically large values of the top-quark mass, m t , have been evaluated [3,4,5], and also the leading term of an asymptotic expansion in the Higgs-boson mass has been obtained [6]. The terms in the m t expansion were found to be sizable, and the next-to-leading term turned out to be about equally large as the (formally) leading term [4]. Exact results have been derived for the Higgs-mass dependence of the fermionic two-loop corrections to muon decay [7]. In the present paper the Higgs-mass dependence of ∆r (which is derived from muon decay [8]), of the effective weak mixing angle at the Z-boson resonance, sin 2 θ eff , and of the leptonic width of the Z boson, Γ l , is briefly discussed. The results are compared with those obtained by the expansion in the top-quark mass. Furthermore the pure fermion-loop contributions, which constitute the leading corrections in one-loop order, are analyzed in higher orders. Exact results for these contributions to ∆r are derived up to four-loop order. They are compared with the resummations of the leading one-loop terms according to Ref. [2].
2. Higgs-mass dependence of ∆r, sin 2 θ eff and Γ l at two-loop order In order to study the Higgs-mass dependence of the precision observables, we consider subtracted quantities of the form Potentially large M H -dependent contributions are the ones associated with the top quark due to the large Yukawa coupling ttH, and the corrections proportional to ∆α. Further contributions are the ones of the light fermions (except the corrections already contained in ∆α), and purely bosonic contributions. The latter are expected to be rather small [7]. We therefore have focussed on the Higgs-dependent fermionic contributions to the precision observables, for which we have obtained exact two-loop results [7,9].  Higgs-mass dependent fermionic contributions to ∆r at two-loop order. The different curves show the contribution from the diagrams involving the top/bottom doublet (∆r tbh ), the contribution proportional to ∆α (∆r ∆α ), the contribution of the light fermions (∆r lfh ), and the approximation of the top/bottom correction by the leading term proportional to m 4 t (∆r(m 4 t )).
The methods used for these calculations have been outlined in Ref. [10]. The generation of the diagrams and counterterm contributions is performed with the help of computer-algebra tools [11]. For the renormalization the complete on-shell scheme (with the conventions as in Ref. [12]) has been used, i.e. physical parameters are used throughout. The two-loop scalar integrals are evaluated numerically with one-dimensional integral representations [13]. given in Ref. [5] (with the input parameters as in Ref. [5]). Besides the difference in the two-loop results also higher-order effects due to differences in the renormalization procedure and in the treatment of non-leading higher-order corrections enter the comparison (in the comparison performed in Ref. [7] the QCD corrections have been omitted). A discussion about different options for treating these higherorder corrections will be given in Ref. [9]. Over the range of Higgs-mass values from 65 GeV to 1 TeV the difference between the results amounts to about 8 MeV, which corresponds to 50% of the two-loop top/bottom contribution. The difference in the Higgs-mass dependence of 8 MeV is slightly larger than the theoretical uncertainty quoted in Ref. [14] of 2 MeV from electroweak and 5 MeV from QCD corrections. The contribution of the light fermions has not been included in the comparison in Tab. 1, since it is not contained in the result of Ref. [5]. As mentioned above, its Higgs-mass dependence gives rise to a further shift of up to 4 MeV in the considered range of M H -values.
We have performed an analogous analysis also for sin 2 θ eff and Γ l . The detailed results will be presented in a forthcoming publication [9]. For the shift in sin 2 θ eff induced by varying M H from 65 GeV to 1 TeV we find (leaving out again the light-fermion contribution) sin 2 θ top,∆α eff,subtr (M H = 1 TeV) = 13.5 · 10 −4 , which differs from the value given in Ref. [5] by 1.0 · 10 −4 . This difference amounts to about 80% of the two-loop top/bottom contribution and is larger than the uncertainty quoted in Ref. [14] of 4 · 10 −5 (electroweak) and 3 · 10 −5 (QCD). It turns out that the difference in the predictions for sin 2 θ eff (M H ) is mainly induced by the difference in the prediction for M W (M H ). We have calculated sin 2 θ top,∆α eff,subtr (M H = 1 TeV) using the value for M W (M H = 1 TeV) from Ref. [5] instead of our result given above, and found a value for sin 2 θ top,∆α eff,subtr (M H = 1 TeV) that differs from the one given in Ref. [5] only by 2 · 10 −5 . The Higgs-mass dependence of the light-fermion contributions, which is not included in Ref. [5], amounts to a shift in sin 2 θ eff of up to 2 · 10 −5 . For the Higgs-mass dependence of the leptonic width of the Z boson, Γ l , we find good agreement with the result of the expansion in m t [15]. The difference amounts to about 15% of the two-loop top/bottom contribution.

Fermion-loop corrections
The dominant contributions to the electroweak precision observables at one-loop order are the fermion-loop corrections ∆α and ∆ρ = N C Gµm 2 t 8π 2 √ 2 , which arise from the renormalization of the electric charge and the weak mixing angle, respectively. One of the main issues in improving the oneloop predictions has therefore been to find a proper resummation of these leading one-loop contributions. In the case of ∆r, for instance, it could be shown [2] that the resummation correctly takes into account the terms of the form (∆α) 2 , (∆ρ) 2 , (∆α∆ρ), ∆α∆r rem at two-loop order. However, even with the knowledge of the recently evaluated terms of O(G 2 µ m 2 t M 2 Z ) [4,5] different treatments of contributions whose correct resummation is not known (i.e. non-leading two-loop terms and the higher-order terms except the leading term in ∆α) may result in predictions for M W that differ from each other by several MeV.
We have investigated the pure fermion-loop contributions to the precision observables, i.e. contributions containing n fermion loops at n-loop order (which in the case of quark loops are therefore proportional to the nth power of the color factor N C ). 1 The relevant diagrams in a given loop order are reducible diagrams of vacuum-polarization type, which can easily be taken into account by Dyson summation, but also graphs with counterterm insertions into fermion-loop diagrams (see Fig. 2). We have derived recursive relations which allow to express the n-loop results in terms of one-loop one-and two-point functions [16]. Using the on-shell renormalization scheme we have worked out explicit results for the fermion-loop contributions to the precision observables up to four-loop order [16]. We have compared our result for the two-loop contribution to ∆r with the MS result derived in Ref. [17], and found very good agreement within less than 1 MeV for the M W prediction. Our exact results allow to investigate the validity of the resummation eq. (2) for the non-leading two-loop and the higher-order contributions. We found that already the first non-leading two-loop term of O(∆ρ log(m 2 t /M 2 Z )) is not correctly produced by the resummation eq. (2). The terms of the form ∆ρ∆r top,rem and (∆r top,rem ) 2 generated by eq. (2) give rise to a deviation in M W of about 7 MeV compared to the exact result for the top/bottom doublet. It turns out, however, that large numerical cancellations occur between the non-leading two-loop terms that are not correctly resummed in eq. (2). While separately these terms differ from the corresponding terms in the exact result by several MeV, their sum approximates the full result remarkably well, up to less than 1 MeV.
From our exact result for the three-and four-loop contributions we find that eq. (2) in fact correctly produces the leading terms in ∆α, ∆ρ of the form (∆α) a (∆ρ) b , where a + b = n, and n is the number of loops. The three-loop term of the form (∆α) 2 ∆r α ferm,rem is also correctly generated, while the three-loop term ∆α∆r α 2 ferm,rem is not correctly produced. In the full fermion-loop contribution at three-loop order again accidental numerical cancellations occur, which give rise to the fact that the total contribution is much smaller than the individual terms and affects the prediction for M W by less than 1 MeV.
In total we find that due to accidental numerical cancellations the resummation of the fermion-loop contributions according to eq. (2) yields a very good numerical approximation of the complete fermion-loop result. The exact result is approximated within about 2 MeV. The results for the pure fermion-loop contributions to sin 2 θ eff and Γ l will be discussed in Ref. [16].