Four-Loop QCD Corrections to the ElectroweakρParameter

Abstract

The four-loop QCD corrections to the electroweak ρ parameter arising from top and bottom quark loops are computed. Specifically we evaluate the missing "non-singlet" piece. Using algebraic methods the amplitude is reduced to a set of around 50 new master integrals which are calculated with various analytical and numerical methods. Inclusion of the newly completed term halves the final value of the four-loop correction for the minimally renormalized top-quark mass. The predictions for the shift of the weak mixing angle and the W-boson mass is thus stabilized.PACS numbers: 14.65. Ha, 14.70.Fm, 12.38.Bx Electroweak precision measurements and calculations provide stringent and decisive tests of the quantum fluctuations predicted from quantum field theory. As a most notable example, the indirect determination of the top quark mass, m t , mainly through its contribution to the ρ parameter [1], coincides remarkably well with the mass measurement performed by the CDF and D0 experiments at the TEVATRON [2]. Along the same line, the bounds on the mass of the Higgs-boson depend critically on the knowledge of m t and the control of the top-mass dependent effects on precision observables.A large group of dominant radiative corrections can be absorbed in the shift of the ρ parameter from its lowest order value ρ Born = 1. The result for the one-loop approximationhence quadratic in m t , was first evaluated in [3] and used to establish a limit on the mass splitting within one fermion doublet. In order to make full use of the present experimental precision, this one-loop calculation has been improved by two-loop [4,5,6] and even threeloop QCD corrections [7,8]. Also important are two-loop [9,10,11,12,13] An important ingredient for the interpretation of these results in terms of top mass measurements performed at hadron colliders or at a future linear collider is the relation between the pole mass and the MS-mass definitions, the former being useful for the determination of m t at colliders, the latter being employed in actual calculations and in short-distance considerations. To match the present three-loop precision of the ρ parameter, this relation must be know in two-loop approximation [16,17,18,19], and for the four-loop calculation under discussion the corresponding three-loop result [20,21,22] must be employed.For fixed pole mass of the top quark, the three-loop result leads to a shift of about 10 MeV in the mass of the W -boson as discussed in [15,23]. (This applies both to the pure QCD corrections and the mixed QCDelectroweak one.) Conversely, the corresponding shift of the top quark pole mass amounts to 1.5 GeV. (Similar considerations apply to the effective weak mixing angle and other precision observables.) These values are comparable to the experimental precision anticipated for topand W -mass measurements at the International Linear Collider [24]. In addition, there exists a disagreement (on the level of 3σ) between the values of the so-called onshell week mixing angle, sin 2 θ W , as measured by NuTeV collabo...

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

“…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 propagators that the identities of the parents can further reduce the set of master integrals.…”

“…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 [9,1]. Studies of four-loop tadpoles have also led to new ideas in computational techniques, such as the introduction of special integral bases [10].…”

“…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].…”

The m W − m Z interdependence in the Standard Model: a new scrutiny

“…(A.4) of the appendix. As before we give the higher order terms via a simple formula: [19,20]. It approximates the exact result to better than 0.075% forŝ 2 on the interval (0.23 − 0.232) when the other parameters in eq.…”

“…The latter turned out to be comparatively large and allowed for a drastic reduction of the scheme dependence. Leading three and four-loop effects related to δρ, in particular the O(α 3 M 6 t /m 6 W ), O(α 2 α s M 4 t /m 4 W ) [17,18] and the O(αα 3 s M 2 t /m 2 W ) [19,20] contributions, were also investigated. The complete calculation of ∆r at the two-loop level was accomplished in several steps.…”