ar X iv : h ep - p h / 94 06 36 3 v 2 1 5 Fe b 19 95 O ( αα 2 s ) correction to the electroweak ρ parameter

The three-loop QCD contributions to the vacuum polarization functions of the Z and W 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 ρ 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 two-loop and three-loop QCD corrections to the ρ parameter is performed. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna (Moscow Region), Russian Federation. E-mail: [email protected] Supported by Volkswagen-Stiftung. Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld 1, Germany. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]; On leave of absence from JINR, 141980 Dubna (Moscow Region), Russian Federation.; Supported by BMFT and RFFR grant No. 93-02-14428. Owing to the apparent discovery of the top quark [1] with a mass of 174± 10 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 top-mass 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 top-mass effects in higher-loop 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 mH was kept as an independent parameter, and both limits, mH ≪ mt and mH ≫ mt, were studied. In the Standard Model there are two different sources of corrections which become large (∼ Gμmt ) in the limit of a heavy top, owing to the large top-bottom mass ratio: the Z and W self-energies (affecting, in particular, the ρ parameter) [5] and the Zbb vertex [6]. Experimentally, these effects are best accessible in ee → f f̄ near the Z resonance measured at LEP1 and the on-resonance asymmetries measured at LEP1 and the SLAC ee linear collider SLC. In the present paper we are concerned with the heavy-top QCD-corrections to the electroweak ρ parameter in three-loop approximation. The ρ parameter is defined as the ratio of the neutral-current to charged-current amplitudes at zero momentum transfer: ρ = GNC(0) GCC(0) = 1 1−∆ρ (1) where the leading fermion contribution to ∆ρ is contained in the gauge-boson selfenergies ∆ρ = ΠZ(0) M Z − ΠW (0) M W . (2) In the approximation considered, we write ∆ρ = 3xt(1 + δ EW + δ + δ) ≃ 3xt(1 + ρxt)(1 + hδ (2) + hδ QCD (3) ), (3) with xt = √ 2 Gμ 16π2 mt , h = αs 4π , (4) αs being the QCD coupling constant. We have denoted by δ EW the pure electroweak, by δ the mixed electroweak-QCD, and by δ the pure QCD corrections. The two-loop electroweak correction ρ, due to virtual Higgs (ghost) effects, is small ρ|mH=0 = −0.74 for mH ≈ 0 [7] but reaches a maximum as large as −11.57 at mH ≈ 5.7mt [3, 4]. The one-loop correction to ∆ρ was first calculated in [5]. The two-loop QCD correction δ (2) has been calculated in [8]. It proved to be rather large. If one takes mt as the top-quark pole mass, then δ (2) = − 8 9 (π + 3). (5) Therefore, it is essential to evaluate the next, three-loop correction, in view of the high precision of modern experiments. To evaluate ∆ρ, the diagonal parts of the self-energies of the W and Z gauge bosons Π α (q) = gμνΠα(q ) + qμqνΠ̃α(q ) (6) (α = W,Z) at q = 0 are needed. Since at zero momentum and mt 6= 0 no infrared divergences appear in diagrams with only fermions and gluons, one may put q = 0 from the very beginning. Contracting Π α (0) with gμν , we obtain for Πα(0) an expression containing only bubble integrals. At the oneand two-loop level these are quite simple and for arbitrary space-time dimension d = 4− 2ε can be written in terms of Euler’s Γ function. Here we need only ∫ dk1 πd/2 (m) (k 1 +m 2)β = Γ(β − d/2) Γ(β) , ∫ ∫ dk1 d k2 πd (m) (k 1 +m )(k2 +m2)β ((k1 − k2)) = = Γ(α + β + γ − d) Γ( 2 − γ) Γ(β + γ − d 2 ) Γ(α + γ − d 2 ) Γ(α) Γ(β) Γ(α + β + 2γ − d) , ∫ ∫ dk1 d k2 πd (m) (k 1) ((k1 − k2))(k 2 +m2)γ = Γ(α+ β + γ − d) Γ(α + β − d/2) Γ(d/2− α) Γ(d/2− β) Γ(α) Γ(β) Γ(γ) Γ(d/2) . (7) At the three-loop level 22 diagrams of the Z-boson self-energy and 29 diagrams of the W -boson self-energy contribute to δ. The integrals that appear here are much more complicated than at the oneand two-loop level. The rather complicated task of computing massive three-loop 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 so-called 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 three-loop 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: ∫ ∫ ∫ dk1 d k2 d k3 [πd/2Γ(1 + ε)]3 (m) k 1[(k1 − k2) +m][(k2 − k3) +m2][k 3 +m2] = 1 ε3 + 15 4ε2 + 65 8ε + 135 16 + 81 4 S2 +O(ε), (8) ∫ ∫ ∫ dk1 d k2 d k3 [πd/2Γ(1 + ε)]3 (m) k 1(k1 − k2)k 3[(k1 − k3) +m2][k 2 +m][(k2 − k3) +m2] = 2ζ(3) 1 ε +D3 +O(ε) (9) with S2 = 4 9 √ 3 Cl2( π 3 ) = 0.260 434 137 632 162 098 955 729 . (10) We have not found a representation of D3 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],D3 can be calculated quite accurately. Here we give the first 22 digits, which is more than enough for a precise evaluation of ρ: D3 = −3.027 009 493 987 652 019 786. (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 MS scheme, we got for the W -boson propagator the following expression: Π (3) W (0) = 12xtM 2 W { ( − 1 2ε − 1 4 − 1 2 l̂ ) + CF ( 3 2ε2 − 5 4ε − 13 8 + ζ(2)− l̂ − 3 2 l̂ )