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...

We present results for the three-loop top quark contributions to the ρ parameter in the limit of large top quark mass. The simultaneous dependence on the mass of the Higgs boson M H and the mass of the top quark m t is obtained from expansions in the range of M H around m t and in the limit M H ≫ m t . In combination with the previous result for M H = 0 the dependence of the ρ parameter on the leading Yukawa contributions, i.e. on m t and M H , is well under control for all mass values of practical importance. The effects lead to a shift in the W mass in the order of 5 MeV and are relevant for precision measurements at TESLA.

We present analytical results for the leading contributions of the top quark to the electroweak ρ parameter at order G 3 F M 6 t and G 2 F M 4 t α s . The Higgs boson and the gauge bosons are taken to be massless in this limit. The correction of order G 3 F M 6 t is found to be sizeable in comparison to the the leading two-loop O(G 2 F M 4 t ) correction, however it is much smaller than the subleading O(G 2 F M 2 t M 2 Z ) correction.

It is shown that for every problem within dimensional regularization, using the Integration-By-Parts method, one is able to construct a set of master integrals such that each corresponding coefficient function is finite in the limit of dimension equal to four. We argue that the use of such a basis simplifies and stabilizes the numerical evaluation of the master integrals. As an example we explicitly construct the ǫ-finite basis for the set of all QED-like four-loop massive tadpoles. Using a semi-numerical approach based on Padé approximations we evaluate analytically the divergent and numerically the finite part of this set of master integrals.The calculations confirm the recent results of Schröder and Vuorinen. All the contributions found there by fitting the high precision numerical results have been confirmed by direct analytical calculation without using any numerical input.

Two different two-loop relations between the pole- and the MS-mass of the top quark have been derived in the literature which were based on different treatments of the tadpole diagrams. In addition, the limit M_W^2/m_t^2 \to 0 was employed in one of the calculations. It is shown that, after appropriate transformations, the results of the two calculations are in perfect agreement. Furthermore we demonstrate that the inclusion of the non-vanishing mass of the W-boson leads to small modifications only.Comment: 6 pages, 6 figures; references added, figure 1 change