Abstract
The threeloop QCD contributions to the vacuum polarization functions of the and bosons at zero momentum are calculated. The top quark is considered to be massive and the other quarks massless. Using these results, we calculate the correction to the electroweak parameter.
All computations are done in the framework of dimensional regularization as well as regularization by dimensional reduction. We use recurrence relations obtained by the method of integration by parts to reduce all integrals to a small set of master integrals.
A comparison of the twoloop and threeloop QCD corrections to the parameter is performed.
correction to the electroweak parameter
L. Avdeev^{1}^{1}1Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna (Moscow Region), Russian Federation.^{2}^{2}2Email: ^{3}^{3}3Supported by VolkswagenStiftung., J. Fleischer^{4}^{4}4Fakultät für Physik, Universität Bielefeld, D33615 Bielefeld 1, Germany.^{5}^{5}5Email: , S. Mikhailov^{6}^{6}6Email: , O. Tarasov^{7}^{7}7Email: ; On leave of absence from JINR, 141980 Dubna (Moscow Region), Russian Federation.; Supported by BMFT and RFFR grant No. 930214428.
Owing to the apparent discovery of the top quark [1] with a mass of GeV, the prospects grow to test the Standard Model on an even higher level of precision than it was possible by now. In particular, the precise knowledge of topmass effects will allow us to obtain better limits on virtual Higgs effects (and thus, indirectly, on the Higgs mass), and possibly, on new physics. For this reason, a great deal of work has been devoted to the study of the topmass effects in higherloop radiative corrections of various electroweak parameters. Mostly, in these studies the top mass is assumed to be large compared to all other masses so that the latter can be put equal to zero from the very beginning (see [2]). In [3, 4] also the Higgs mass was kept as an independent parameter, and both limits, and , were studied.
In the Standard Model there are two different sources of corrections which become large () in the limit of a heavy top, owing to the large topbottom mass ratio: the and selfenergies (affecting, in particular, the parameter) [5] and the vertex [6]. Experimentally, these effects are best accessible in near the resonance measured at LEP1 and the onresonance asymmetries measured at LEP1 and the SLAC linear collider SLC.
In the present paper we are concerned with the heavytop QCDcorrections to the electroweak parameter in threeloop approximation. The parameter is defined as the ratio of the neutralcurrent to chargedcurrent amplitudes at zero momentum transfer:
(1) 
where the leading fermion contribution to is contained in the gaugeboson selfenergies
(2) 
In the approximation considered, we write
(3) 
with
(4) 
being the QCD coupling constant. We have denoted by the pure electroweak, by the mixed electroweakQCD, and by the pure QCD corrections. The twoloop electroweak correction , due to virtual Higgs (ghost) effects, is small for [7] but reaches a maximum as large as 11.57 at [3, 4].
The oneloop correction to was first calculated in [5]. The twoloop QCD correction has been calculated in [8]. It proved to be rather large. If one takes as the topquark pole mass, then
(5) 
Therefore, it is essential to evaluate the next, threeloop correction, in view of the high precision of modern experiments.
To evaluate , the diagonal parts of the selfenergies of the and gauge bosons
(6) 
() at are needed. Since at zero momentum and no infrared divergences appear in diagrams with only fermions and gluons, one may put from the very beginning. Contracting with , we obtain for an expression containing only bubble integrals. At the one and twoloop level these are quite simple and for arbitrary spacetime dimension can be written in terms of Euler’s function. Here we need only
(7) 
At the threeloop level 22 diagrams of the boson selfenergy and 29 diagrams of the boson selfenergy contribute to . The integrals that appear here are much more complicated than at the one and twoloop level.
The rather complicated task of computing massive threeloop Feynman diagrams is accomplished by applying the method of recurrence relations [9, 10]. This method allows us to relate various scalar Feynman integrals of the same prototype which differ by powers of their scalar propagators. As a result, by means of plain algebra, any diagram is reduced to a limited number of socalled master integrals. They need to be evaluated once and for all, and can then be used in any renormalizable quantum field theory. Some of the integrals that we need for the present threeloop calculation were considered in [10]. Here, however, more types of integrals are required. In addition to the master integrals evaluated in [10], two more nontrivial master integrals are encountered:
(8)  
(9) 
with
(10) 
We have not found a representation of in terms of known transcendental numbers, though we do not exclude its existence. By means of the numerical method for the evaluation of Feynman diagrams proposed in [12], can be calculated quite accurately. Here we give the first 22 digits, which is more than enough for a precise evaluation of :
(11) 
Calculations were mostly done using FORM 1.1 [11]. All the diagrams were computed in the covariant gauge with an arbitrary gauge parameter. Performing charge and mass renormalization in the scheme, we got for the boson propagator the following expression:
(12) 
Here is the total number of quarks, , is the renormalized mass in the scheme, and the constant
(13) 
was defined in [10]. The result for the boson propagator is
(14) 
In the sum of the bare diagrams contributing to (), as well as in the corresponding counterterms separately, the gauge parameter cancels, which is a partial check of our result.
(15)  
If we perform mass renormalization in such a way that the renormalized mass is the pole mass , then we obtain
(16)  
In the above formula . Ultraviolet finiteness of (15) and (16) is an additional check of our result. Expression (16) can also be obtained from our result in the scheme by using the relation between in the scheme and the pole mass [13].
In the scheme for QCD [i.e. for the SU(3) gauge group with , ] we get the following expression:
(17) 
Substituting numerical values for all the constants and taking with the minimally subtracted mass we obtain
(18) 
which at turns into
(19) 
The smallness of this correction in the scheme confirms expectations about higher order effects in electroweak parameters (see e.g. [14]).
With the definition of the renormalized mass as the pole mass of the top quark we obtain:
(20)  
Substituting numerical values for all the constants and putting we get
(21) 
and at we have
(22) 
Both two and threeloop QCD contributions are negative, and their effect is a screening of the bare mass splitting, so that only a reduced “effective” quantity enters the parameter.
Following the method of fastest apparent convergence [15], we can absorb our threeloop correction into a rescaling of . With , will be zero, if we take . When we apply the BLM procedure [16], the dependent term in (20) can be absorbed into the rescaling of , if we choose . The same value was also obtained in [17]. We conclude that the expression for the term proportional to , given in [17], agrees with the analytical result for this term given in (20). As an important result of our calculation, we stress the stability of : the usual perturbation theory, the FAC and BLM procedures give rather close results for . Taking , we have 0.1125, 0.1159 and 0.1154 for the perturbation theory, FAC and BLM procedures, respectively.
For our calculations we used the anticommuting . To check our result obtained with this prescription at least partially, we calculated again, using the regularization by dimensional reduction [18] which keeps the algebra of matrices fourdimensional. For propagatortype diagrams in the threeloop approximation the inconsistency of this recipe is not yet revealed. As was expected, the result that we obtained in the regularization by dimensional reduction agrees with in the conventional dimensional regularization after the following recalculation of the coupling constant:
(23) 
which just corresponds to a change in the renormalization scheme. The relation can be derived, for example, by equating invariant charges in these two schemes.
Special attention was paid to the evaluation of the diagram with the axial anomaly. Only one such diagram contributes to (namely, to ) with two triangles and only the top quark running around. We evaluated it in the framework of the regularization by dimensional reduction and using the prescription given in [19]. The contribution from this diagram is finite, and the calculations yielded the same value in both approaches (although the evanescent parts were different). Our result also agrees with the one given in [20]. At this contribution amounts to about 30% of the total threeloop correction (22).
Some observables that are affected by our result are briefly mentioned here. One of them is the mass of the boson as predicted from , and [21]
(24) 
where and is the shift of the fine structure constant due to photon vacuum polarization effects. The ellipsis stands for the nonleading remainder terms. Another physical quantity is the effective weak mixing parameter relevant to resonance physics. It is given by
(25) 
Numerical values illustrating how these observables are affected by various corrections are given in Table 1. QCD corrections were calculated using (22).
Observable  Twoloop  Threeloop  Twoloop electroweak  

QCD  QCD  1.5  5.7  10  
0.065  0.011  0.002  0.018  0.025  0.023  
0.130  0.022  0.003  0.036  0.051  0.046 
Table 1 demonstrates that the threeloop QCD correction is comparable with the twoloop electroweak correction for sizable Higgs masses (for =1.5 the former amounts to more than 60% of the latter). Thus, we conclude that it makes sense to evaluate subleading electroweak twoloop corrections to (or another observable) only if the threeloop QCD corrections are taken into account as well.
Acknowledgments. L.Avdeev, S.Mikhailov and O.Tarasov are grateful to the Physics Department of the Bielefeld University for warm hospitality. L.Avdeev and S.Mikhailov are thankful to the VolkswagenStiftung and O.Tarasov to the BMFT and RFFR grant for financial support. The authors are grateful to F.Jegerlehner for carefully reading the manuscript.
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