TTP95–03
hepph/9502291
February 1995
Corrections of order to the
parameter
K.G. Chetyrkin, J.H. Kühn, M. Steinhauser

Institute for Nuclear Research
Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow, 117312, Russia

Institut für Theoretische Teilchenphysik
Universität Karlsruhe, Kaiserstr. 12, Postfach 6980, D76128 Karlsruhe, Germany
The threeloop QCD corrections to the parameter from top and bottom quark loops are calculated. The result differs from the one recently calculated by Avdeev et al. As function of the pole mass the numerical value is given by
1 Introduction
The precision of electroweak observables measured at LEP, SLC and the TEVATRON has stimulated a variety of theoretical calculations. These are required to match the experimental accuracy and to pin down the parameters of the Standard Model, in particular the top and the Higgs mass and to search for new physics.
A cornerstone in this analysis is the evaluation of top quark contributions to the parameter. With the high value of as suggested by the CDFcollaboration [1] and the strong sensitivity of to small variations of through the quadratic dependence [2] precise theoretical predictions become mandatory.
Top mass corrections to the vertex, the only other place with a strong dependence on [3, 4], are specific to this reaction, while those to the parameter enter numerous relations between observables. This is the second justification for a precision calculation.
In addition to the two loop contribution [5] (for a related calculation based on dispersion relations see [6], for a recent review see [7]), the twoloop electroweak corrections have also been evaluated, for vanishing [8] and even for arbitrary [9].
In this paper the threeloop result of order is presented. A similar calculation has been performed by Avdeev, Fleischer, Mikhailov and Tarasov [10]. However, our calculation disagrees with their formula. In the following section 2 details of the calculation will be described, in section 3 the result will be presented and a brief numerical discussion given.
2 The calculation
Quantum corrections to the parameter can be connected to the gauge boson selfenergies through
(1) 
Here denotes the transverse part of the polarisation tensor for vanishing momentum . The evaluation of these selfenergy diagrams is performed for and within the framework of dimensional regularisation. Large intermediate expressions are treated with the help of FORM 2.0 [11]. For anticommuting was used, except for the double triangle diagram. In order to evaluate this diagram, which is related to the axial anomaly, the definition of ’t Hooft and Veltman [12], formalized in [13], was applied. Its contribution is finite and the result coincides with a previous calculation [14] where from the very beginning. A covariant gauge with arbitrary gauge parameter for the gluon propagator was chosen.
The tadpole integrals required to calculate the one or twoloop corrections are easy to evaluate even for arbitrary powers of the propagators. This does not hold true for the threeloop case. After performing the traces the reduction to scalar integrals is performed by decomposing the scalar products of the numerator in appropriate combinations of the denominator. Subsequently, recurrence relations provided by the integrationbyparts (IP) method [15] are used in order to reduce every scalar Feynman integral to a small number of socalled master diagrams which have to be calculated explicitly. The IP method was first applied in [16] to threeloop tadpole integrals. There the subclass of those diagrams which contain a continuous massive quark line and which are relevant for the boson selfenergy was considered. For the calculation of the parameter the method has to be extended to a second class of integrals originating from the selfenergies. One thus arrives at three master integrals
(The same variables and conventions as those of [10] are adopted. Following standard practice we discarded terms proportional to and in the r.h.s.) The first master integral has been calculated analytically in [16]; the results for the last two integrals are given [10] ( is presently only known numerically.). We checked the result for numerically. For we reproduced the analytical result. The evaluation of is described in [17]. Employing a different method, we obtain the result given below, which is consistent with [17]. The values for the constants , and are as follows:
3 Results and Discussion
After performing mass and charge renormalization in the scheme the following result for the boson propagator is obtained:
The coefficient of differs from the recent result of [10]. Its value has to be compared with . This leads to a significant modification of the numerical predictions to be discussed below. In this expression denotes the total number of quark species and . The result is expressed in terms of the renormalized top mass . The variable is defined as
(3) 
From now on the explicit dependence both in and will be suppressed. From the context it should be evident which scale is adopted.
For the boson propagator the following result is obtained
in agreement with [10]. Both and are independent of the gauge parameter. As expected this holds true even before mass and charge renormalization are performed.
Eqs. (3) and (3) immediately lead to in terms of the renormalized top mass. The poles which are still present in the and selfenergies individually cancel.
With the help of the relation between the OS and running top mass [18] for
the result is easily expressed in terms of the OS mass:
Here is the pole mass and . The residual terms are cancelled by the dependence of .
At this point two consistency checks should be mentioned which were performed in order to test the correctness of our result. The first one is a different method of calculating respectively the polarisation functions for the and boson. It relies on the axial Ward identity which connects the axial part of the polarisation tensor with the pseudoscalar polarisation function. (The double triangle diagram was not considered in this context.) If the fermions in the loop have the masses and the following identity holds:
(8) 
The second term of the r.h.s. is independent of and therefore not relevant in this context. The l.h.s. in lowest order is already . was evaluated up to . Because of the different tensor structure, different recurrence relations have to be applied to compute the parameter. has to be expanded in up to order . Additional propagators are generated and different parts of the FORM programs become relevant. In addition it was checked that the independent part of cancels against the second term such that the r.h.s. indeed is of . as given in eq. (3) was reproduced with this method.
The second check is also connected with an expansion in the external momentum. The polarisation tensor of the vector bosons was expanded up to order . The external momentum was routed through the graphs in two different ways. Again the same result was obtained for every diagram although the intermediate steps are very different.
Substituting and or in eqs. (3) and (3) respectively a fairly compact form for is obtained:
To evaluate these results numerically the values for , , and the functions are inserted. For convenience of the reader we display separately four different contributions: (i ) the contribution from the double triangle diagrams related to the axial anomaly [14], (ii ) the contributions with exactly one fermion loop which together with the previous one would give the result in the “quenched” approximation, (iii ) the contribution from light quark loops proportional to , and finally (iv ), the contribution with the light quarks replaced by top quarks.
(11)  
(12)  
The coefficient in front of in eq. (12) is in reasonable agreement with the numerical result in [19]. Eqs. (11) and (12) in particular the separation of the various contributions allow the use or test of a variety of optimization schemes which is left to the reader.
For the final result and after setting we obtain
(13)  
(14) 
The coefficient in front of in the result differs significantly from [10]: versus . The OS results differ by the same amount versus .
In [10] it was mentioned that the contribution from the double triangle diagram, associated with the axial anomaly [14], alone amounts to about of the total threeloop correction. Here a fraction of approximately is still traceable back to this single diagram in the OS scheme. In the scheme the corrections are completely dominated by this diagram.
Finally the numerical effect on the prediction for and from and will be discussed. If subleading terms are neglected the following relations hold [20]:
Here GeV, and . The relative size of the one, two and threeloop corrections is given in Table 1. The first row indicates the size of itself. Rows two and three give the relative contribution of the one, two and threeloop corrections with respect to the Born result in the OS and scheme. The numbers are obtained with the following input data: , GeV, GeV, GeV and GeV. The value of was obtained from with the help of the threeloop function [21] and the matching condition between and at the scale [22]. (The numbers in brackets indicate thereby the numbers of active flavours.) The top mass was derived from eq. (3). This value serves as starting point in order to calculate using the corresponding threeloop evolution equation [23]. (Unlike the coupling constant the running top quark mass is only defined in full theory.)



It is interesting to mention that the scheme dependence decreases enormously when taking higher loop corrections successively into account.
One observes that the threeloop corrections are (at least for ) not at all small compared with the twoloop QCD contribution. Furthermore they are approximately of the same order of magnitude as the twoloop electroweak result because for the contribution from the twoloop electroweak correction amounts [10, 9] to be compared with the threeloop QCD corrections of . It is also possible to translate the corrections directly in a change of the top mass contained in . In the onshell scheme this corresponds to a change of approximately GeV.
The dependence of the result on the renormalization scale is shown in Figs. 1a and 1b for and respectively. The same input parameters have been used as before. The dotted line in Fig. 1b gives the oneloop prediction (which is constant for and completely offscale in Fig. 1a), dashed and solid lines represent the two and threeloop results. The prediction is clearly stabilized through inclusion of higher orders.
Another possibility would be to absorb the contribution in the choice of an effective scale of the correction:
(15) 
To summarize: The evaluation of the threeloop QCD correction to the parameter has been repeated with a result different from the one of [10]. The numerical difference is sizeable.
Acknowledgments
We would like to thank G. Passarino and A. Sirlin for discussions and A. Czarnecki for advice in the numerical evaluation of Feynman integrals. One of the authors (M.S.) would like to thank B.A. Kniehl for the occasion to present the results of this paper at the Ringberg Workshop on “Perspectives for electroweak interaction in collisions”, February 58, 1995.
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