Dipartimento di Fisica dell’Universit`a di Pisa

arXiv:hep-ph/9205238v1 27 May 1992May 1992Dipartimento di Fisica dell’Universit`a di PisaIFUP-TH 20/92Radiative correction effects of a very heavy topRiccardo Barbieri 1, 2, †Matteo Beccaria 1, 2Paolo Ciafaloni 1, 2Giuseppe Curci 2, 1Andrea Vicer´e 3, 2(1) Dipartimento di Fisica, Universit´a di PisaPiazza Torricelli 2, I-56100 Pisa, Italy(2) I.N.F.N., sez. di PisaVia Livornese 582/a, I-56010 S. Piero a Grado (Pisa) Italy(3) Scuola Normale SuperiorePiazza dei Cavalieri 7, I-56100 Pisa, ItalyAbstractIf the top is very heavy, mt ≫MZ, the dominant radiative cor-rection effects in all electroweak precision tests can be exactly char-acterized in terms of two quantities, the ρ-parameter and the GIMviolating Z →b¯b coupling.

These quantities can be computed usingthe Standard Model Lagrangian with vanishing gauge couplings. Thisis done here up to two loops for arbitrary values of the Higgs mass.† Address from May 1, 1992: CERN, Geneva, Switzerland

1. The effects of virtual heavy top quark exchanges in electroweak pre-cision tests are recognized as being very important.

In the Standard Modelof the electroweak interactions these effects are used to get a significant con-straint on the top quark mass, mt. Perhaps even more importantly, theseeffects are so large that, until the value of mt will not be measured directlyand independently, they will obscure the comparison of different models, in-cluding the SM itself, in their predictions of electroweak precision tests.

Wehave especially in mind the effects that grow like m2t at one loop level, whichaffect the electroweak ρ−parameter [1] ( and all related quantities ) and theGIM-violating coupling of the Z boson to a b¯b pair [2]. The main contribu-tion of this work is the explicit calculation of the two loop m4t−correctionsto these quantities in the SM for arbitrary values of the Higgs mass, mH.

Inthe literature [3] one finds already the m4t contributions to ρ for mH ≪mt,which we confirm.The actual calculation of these corrections is greatly simplified by theobservation that to obtain them it is enough to consider the lagrangian ofthe SM in the limit of vanishing coupling constants: the gauge bosons play therˆole of external sources and the relevant propagating fields are the top quark,the massless bottom quark, the Higgs field and the charged and neutralGoldstone bosons, φ±, χ. We call this the Gaugeless Limit of the StandardModel.2.A way to relate proper quantities computed in the reduced modelto physical observables is to consider the Ward Identities satisfied by thecharged weak current J±µ and by the usual combination of neutral currents,Jµ = J3µ −sin2 θWJemµ , that couples to the Z boson.

When the bare neutralcomponent of the Higgs doublet acquires a vev v, both these currents get apiece proportional to the derivative of the Goldstone fieldsJµ=ˆJµ +v√2 ∂µχ(1)J±µ=ˆJ±µ ∓i v ∂µϕ±1

From the current conservation eq.s ∂µJµ = ∂µJ±µ = 0, it is easy to derive thefollowing Ward Identities (s2 ≡sin2 θW, PL,R = 12(1 ± γ5))(p −p′)µ Γµ(p, p′)=iv√2Γ(p, p′) ++121 −23s2 S−1(p′) PL −PR S−1(p)+−23s2 S−1(p′) PR −PL S−1(p)qµqν Πµν(q)=v22 Π(q)(2)qµqν Π±µν(q)=v2 Π±(q)which involve the b-quark propagator S(p); the self energies, Π(q), Π±(q), ofthe neutral and charged Goldstones; the correlation functions Πµν(q), Π±µν(q)of the currents ˆJµ, ˆJ±µ respectively; the vertex Γµ(p, p′) between b-quarks ofmomentum p, p′ and the current ˆJµ; the vertex Γ(p, p′) between b-quarks ofmomentum p, p′ and the neutral Goldstone χ. Both the Π’s and the Γ’s aremeant to be irreducible.To get the physical observables we now define the following constantsΓµ(p, p′)≃−i21 −23s2Z1 γµPL −23s2γµPRat p ≃p′Γ(p, p′)≃Zχ1/p′ −/p√2 v PL at p ≃p′S−1b (p)≃i Zb2 /pPL + i/pPR at p2 ≃0Πµν(q)≃v22 (Z −1) δµν at q ≃0(3)Π±µν(q)≃v2 (Z± −1) δµν at q ≃0Π(q)≃(Zχ2 −1) q2 at q2 ≃02

Π±(q)≃(Zϕ2 −1) q2 at q2 ≃0They satisfy, from the Ward Identities (2), the relations1 −23s2Z1 = Zχ1 +1 −23s2Zb2 ;Z = Zχ2 ;Z± = Zϕ2(4)By recalling now that, in the full SM Lagrangian, the W- and Z-bosonscouple to the current J±µ , Jµ via∆L = g√2J+µ W −µ + J−µ W +µ+ gcJµZµ(5)these relations enable us to compute, to leading order in the SU(2) gaugecoupling g, the physical Zµ →b¯b vertexVµ = −i g2 c" 1 −23s2 + Zχ1Zb2!γµPL −23s2γµPR#(6)and the vector boson massesM2Z=g2v22 c2 Z = g2v22 c2 Zχ2(7)M2W=g2v22Z± = g2v22Zϕ2(8)These are indeed the physical masses, as g →0, because the displacementbetween q2 = 0 and the pole at q2 = M2 is irrelevant and because there isno wave function renormalization of the vector bosons in this limit. The ρparameter is therefore [4]ρ = M2WM2Z c2 = Zϕ2Zχ2(9)Notice that the Ward Identities (2) are identically true, as they stand, alsoin the full theory, with the gauge couplings switched on, if one works in the3

Background Gauge. Let us also remark that, if we had used an effectiveLagrangian formalism, the constants Zχ1 , Zχ2 , Zϕ2 would have appeared infront of terms involving derivatives of the Goldstone bosons.

Eq.s (6-8) couldhave then be simply obtained by proper covariantization of these derivatives.Since the W and the Z are treated as external sources, there is never theneed to fix the gauge and break gauge invariance.What remains to be done at this point is the re-expression of the variousparameters appearing in (6) and (9) in favor of physically measurable quan-tities. As usual, g, v and c are traded for MZ, eq.

(7-8), the fine structureconstant α and the Fermi constant Gµ as measured in µ-decay, which, in ourapproximation, are given byα =g24πs2,Gµ =√2g28M2W=√24v2Zϕ2(10)From eq.s (6-10), by retaining only the non vanishing corrections as g →0,the radiative effects in all electroweak precision tests can be “non-perturbatively”characterized in terms of the two quantities ρ and τ ≡Zχ1 /Zb2 appearing inthe GIM-violating Z →b¯b vertex. In particular, wherever s2 appears, it mustbe replaced, from eq.s (7-10), withs2 = 121 −vuut1 −4πα√2GµM2Zρ(11)As an example, in terms of ρ and τ, the width of the Z into a b¯b pair is givenbyΓZ →b¯b=ρGµM3Z8π√2vuut1 −4m2bM2Z"g2bV + g2bA 1 + 2 m2bM2Z!−6g2bAm2bM2Z#gbV=1 −43s2 + τ,gbA = 1 + τ(12)where the kinematical dependence on the b quark mass, mb, is also intro-duced.

Of course, hidden in ρ and τ there are the top Yukawa coupling gt4

and the quartic Higgs coupling λ, which must also be expressed in favor ofthe top and of the Higgs mass, defined as the positions of the poles of thecorresponding propagators. From the top propagatorS−1t (p) ≃i Zt2L /p PL + i Zt2R /p PR + gt v Bat /p ≃mt(13)one hasmt = gtvB/(Zt2LZt2R)1/2(14)whereas, up to the order we are working, m2H = 4λv23.Fig 1 contains the relevant one loop diagrams contributing to theconstants Zχ2 , Zϕ2 and Zχ1 which allow to determine the coefficients of thefirst term in an expansion in powers of Gµm2t both of ρ and of the GIMviolating Z →b¯b vertex.

The diagrams contributing to Zχ2 and Zϕ2 and onlycontaining Higgs or Goldstone boson lines, but no quark lines, have not beendrawn since they do not affect ρ or more precisely, the difference Zχ2 −Zϕ2 ,in terms of which ρ can be identically written, using eq.s (9, 10), as1ρ −1 = 4Gµv2√2(Zχ2 −Zϕ2 )(15)In the literature [5], this is often called the irreducible part of the correctionsto the ρ parameter. At order (Gµm2t)2, other than the contributions of thegenuine two loops irreducible diagrams, one has to compute the Gµm2t cor-rections to the wave function renormalizations of the top and bottom quarksas specified in eq.s (6, 14).In a decently compact way, the results of the calculations can be givenin an analytic form in two different asymptotic regimes ( x = Gµm2t8π2√2, Nc = 3,r = m2tm2H )(a) mt ≫mH1ρ −1=−Ncxh1 + x19 −2π2iZχ1Zb2=−2x1 + x327 −π2(16)5

(b) mH ≫mt1ρ −1 =−Ncx1 + x494 + π2 + 272 log r + 32 log2 r++r32 −12π2 + 12 log r −27 log2 r++ r2481613 −240π2 −1500 log r −720 log2 r#)Zχ1Zb2=−2x1 + x144h311 + 4π2 + 282 log r + 90 log2 r+−4r40 + 6π2 + 15 log r + 18 log2 r++ 3r210024209 −6000π2 −45420 log r −18000 log2 r#)(17)In the expression for mH ≫mt we have made explicit some sub-asymptoticterms vanishing in the limit r →0 in such a way that the expansion can betrusted even for mH close to mt. For practical purposes, the combined useof eq.s (16, 17) allows to control numerically the m4t terms for all values ofmH.

The effects of these corrections, in view of the attainable experimentalprecision, start being significant for mt ∼> 200 GeV. These corrections arebeing implemented in the ZFITTER code [6].A detailed description of the calculation will be given elsewhere.Acknoledgements: Useful conversations with D. Bardin and L. Maiani aregratefully acknowledged.6

Figure 1: relevant one loop diagrams contributing to the constants Zχ2 , Zϕ2and Zχ17

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