Elsevier

Nuclear Physics B

Volume 755, Issues 1–3, 30 October 2006, Pages 221-238
Nuclear Physics B

Single scale tadpoles and O(GFmt2αs3) corrections to the ρ parameter

https://doi.org/10.1016/j.nuclphysb.2006.08.007Get rights and content

Abstract

We present a new set of high precision numerical values of four-loop single-scale vacuum integrals, which we subsequently use to obtain the non-singlet corrections to the ρ parameter at O(GFmt2αs3). Our result for Δρ is in agreement with the recent calculation [K.G. Chetyrkin, M. Faisst, J.H. Kuhn, P. Maierhofer, C. Sturm, hep-ph/0605201].

Introduction

Single-scale four-loop vacuum integrals have attracted a lot of attention in recent years. Even though the first applications were connected to the calculation of anomalous dimensions [2], [3], the class of solved problems counts by now such topics as the pressure in hot QCD [4], coupling constant and mass decoupling relations [5], [6], moments of the hadronic production cross section [7], [8], and corrections to the ρ parameter [1], [9]. Studies of four-loop tadpoles have also led to new ideas in computational techniques, such as the introduction of special integral bases [10].

This impressive progress has been made possible to a large extent by the Laporta algorithm for the reduction of integrals to masters described in [11], and by the difference equation method for the numerical evaluation of the masters proposed in the same publication. The first sets of integrals have been evaluated precisely using these principles [12], [13]. At present, other methods are also available, see [14] and [1].

One of the applications of four-loop tadpoles mentioned at the beginning concerns a quantity of primary importance in the area of electroweak physics, namely the ρ parameter introduced by Veltman [15]. Defined as the ratio of the charged and neutral current strengths, it differs from its leading-order value of one, by a shift which can be expressed through the transverse parts of W and Z boson self-energiesΔρ=ΠZ(0)MZ2ΠW(0)MW2. This shift occurs as a universal correction in all electroweak observables and is thus related to the indirect prediction of the Higgs boson mass from the experimental data, and in particular from the W boson mass [16] and the effective weak mixing angle [17].

In view of the importance of Δρ, several corrections have been computed. In particular, the two- [18] and three-loop [19] QCD effects, and various electroweak effects in the limit of a large top quark mass [20] have been accounted for. At the three-loop level, the leading behavior in the limit of a large Higgs boson mass is also available [21]. From now on, we will denote the QCD corrections to Δρ in leading order in the electroweak interaction by δρ.

At the four-loop level, the singlet QCD corrections, i.e. corrections where the external gauge bosons couple to different fermion loops have been evaluated in [9]. Motivated by that publication, we started the calculation of the non-singlet contributions, which are obtained by attaching gluons (with possible fermion loop insertions) to the leading one-loop diagrams. The major obstacle to overcome is the calculation of the many new master integrals. It is the purpose of the present work to present our results for those integrals and apply them to the calculation of the four-loop non-singlet QCD corrections to Δρ.

Recently, Ref. [1] containing a result for the very same corrections appeared. Anticipating the content of Section 3, we can state that we agree with this calculation.

This paper is organized as follows. In the next section we present high precision numeric expansions of the master integrals. We then give our on-shell result and conclusions. An Appendix A contains the corrections expressed in the MS¯ scheme.

Section snippets

Master integrals

Upon reducing the complete set of scalar integrals occurring in the diagrams contributing to Δρ at the four-loop level, we are left with 65 masters. However, the latter number is only correct if we consider the integration-by-parts identities for a given prototype and not those of its parents.1 It has been noticed in the case of two [22] and three-loop [23]

On-shell results

Once the master integrals are calculated, it is just a matter of substituting their values into the reduced diagrams and performing MS¯ renormalization to obtain the corrections to Δρ in the MS¯ scheme. The latter can be found in Appendix A. As far as applications to electroweak physics are concerned, the on-shell scheme is more relevant. We used the relation between the MS¯ and on-shell masses of a heavy quark from [32], in order to perform the translation between the schemes.

In our result

Conclusions

In this work, we have computed a further subset of four-loop single scale vacuum integrals with enough terms in the ε expansion to even allow for five-loop calculations. Our results can be applied whenever the physical process shows large scale differences and the large mass procedure can be applied, leading naturally to expansion (Wilson) coefficients expressed through tadpoles.

The immediate motivation for this computation has been the calculation of the non-singlet corrections to the ρ

Acknowledgements

Parts of the presented calculations were performed on the DESY Zeuthen Grid Engine computer cluster. This work was supported by the Sofja Kovalevskaja Award of the Alexander von Humboldt Foundation sponsored by the German Federal Ministry of Education and Research.

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