We assume the stability of vacuum under radiative corrections in the context
of the standard electroweak theory. We find that this theory behaves as a good
effective model already at cut off energy scales as low as 0.7 TeV. This
stability criterion allows to predict m_H= 318 +- 13 GeV for the Higgs boson
mass.Comment: Latex, 5 pages, 1 Postscript figure include

Castro, G. Lopez

Fang, Zhen Yun

Lucio, J. L.

Pestieau, J.

English

arXiv

We assume the stability of vacuum under radiative corrections in the context
of the standard electroweak theory. We find that this theory behaves as a good
effective model already at cut off energy scales as low as 0.7 TeV. This
stability criterion allows to predict m_H= 318 +- 13 GeV for the Higgs boson
mass.
 Comment: Latex, 5 pages, 1 Postscript figure include

Lopez Castro, G.

Lucio, J.L.

Pestieau, Jean

DIAL UCLouvain

Effective SU(2)-L x U(1) theory and the Higgs boson mass

Zhen Yun Fang

G. López Castro

J. Pestieau

J. L. M. Lucio

Crossref

Effective
                    SU

arXiv:hep-ph/9612430v2  29 Dec 1996UCL-IPT-96-22Effective SU(2)L ⊗ U(1) theory and the Higgs bosonmassZhen Yun Fanga, G. Lo´pez Castrob, 1, J. L. Lucio M.c and J. Pestieauba Department of Applied Physics, Chongqing University, Chongqing,Sichuan 630044, People’s Republic of Chinab Institut de Physique The´orique, Universite´ catholique de Louvain,B-1348 Louvain-la-Neuve, Belgiumc Instituto de Fisica, Universidad de Guanajuato, ApartadoPostal E-143, 37150 Leo´n, Guanajuato, Me´xicoAbstractWe assume the stability of vacuum under radiative corrections in the contextof the standard electroweak theory. We find that this theory behaves as a goodeffective model already at cut off energy scales as low as 0.7 TeV. This stabilitycriterion allows to predict mH = 318 ± 13 GeV for the Higgs boson mass.PACS: 14.80.Bn, 11.10.Jj, 11.15.BtKeywords: Higgs, tadpole, effective, divergencies.1Permanent address: Departamento de F´isica, CINVESTAV del IPN, Apartado Postal 14-740,07000 Me´xico D.F., Me´xicoSince the standard electroweak theory is renormalizable, the ultraviolet (quadraticand logarithmic) divergencies appearing in loop calculations can be removed by properredefinitions of a small number of parameters (e.g., masses). The renormalizationprogram allows to get finite predictions for physical quantities except for those fewparameters which require redefinitions. In this way, we renounce to give some physicalcontent to the divergent behaviour of the theory at very high energies [1].Instead of following the renormalization program, we can consider that SU(2)L ⊗U(1) is an effective theory with an ultraviolet cut off at the energy scale Λ and try totrigger some information on the model itself from the behaviour of divergent terms. Inother words, we assume that a more complete theory than SU(2)L ⊗ U(1) introducesnew physical effects at high energy that cut off the ultraviolet divergencies. Below Λ,SU(2)L ⊗ U(1) is supposed to describe all the electroweak processes in a satisfactoryway.In this letter we consider SU(2)L⊗U(1) as an effective theory below Λ and assumethe vanishing of the common divergent contributions (tadpoles) to the masses of allthe particles. We find that the electroweak theory becomes effective even at cut offenergies as small as Λ ∼ 0.7 TeV. This criterion for vacuum stability under radiativecorrections allows to derive a mass for the Higgs boson that differs from its asymptoticvalue (Λ → ∞) mH =√4(m2t +m2b)− 2m2W −m2Z [2]- [3] by less than 0.6 % alreadyat Λ ∼ 0.7 TeV.Let us start by considering the divergent one-loop tadpole contributions. Thesetadpoles give a universal contribution to self-mass corrections because the bare massesof all the particles are proportional to the vacuum expectation value v of the Higgsfield. The tadpoles are gauge-dependent and are given by [4]δvtv=(α16π)1m2Hm2W sin2 θW{−2m4Z − 4m4W −32m2HA0(mH)− 3m2ZA0(mZ)−6m2WA0(mW )−12m2HA0(√ξmZ)−m2HA0(√ξmW )+12[m2tA0(mt) +m2bA0(mb)]}(1)1where mi (i = W,Z,H, t, b) denote the masses of gauge bosons, the Higgs boson andtop and bottom quarks (we have neglected fermion masses other than mt,b), ξ is thegauge parameter (ξ = 1 in the t’Hooft-Feynman gauge and ξ = 0 in the Landau gauge)and [4]A0(m) =−m2(2ǫ − γ + log(4πµ2) + 1− logm2),Λ2 −m2 log Λ2m2−m2(2)The two expressions in Eq. (2) are obtained by using the n-dimensional and corre-sponding cut off regularization methods, respectively.In a similar way, we can compute the corrections to the neutrino mass (we assumea non-vanishing value for the neutrino mass). We getδmνmν=(α16π)1m2W sin2 θW{32m2W +B0(0, 0,√ξmW )+34m2Z +12B0(0, 0,√ξmZ)}+δvtv(3)where [4]B0(0, 0, m) =−m2(2ǫ − γ + log(4πµ2) + 1− logm2),−m2 log Λ2m2−m2,(4)respectively, using the n-dimensional and cut off methods. Eq. (3) is in agreementwith corresponding results in Ref. [5] where finite terms of the form m2 and m4 areneglected.As it can be easily checked from the previous expressions, δmν/mν is explicitlygauge-independent. This means that if we write the tadpoles in the Landau gauge weobtainδvtv∣∣∣∣∣ξ=0=δmνmν−34(α16π)1m2W sin2 θW{2m2W +m2Z}, (5)i.e., this quantity is SU(2)L ⊗ U(1) gauge-invariant. As is well known, the tadpolescan be computed from an effective potential only in the Landau gauge [6].Now let us assume that tadpole contributions in the Landau gauge vanish, namelyδvt = 0. According to Eq. (5), δmν/mν becomes negligible small (∼ 1.6×10−3), i.e. the2radiative corrections to the small neutrino mass are very small. Since tadpole radiativecorrections affect the masses of all the particles in an universal way, the stability ofthe vacuum under radiative corrections (δvt = 0) implies that the masses of all theparticles in the effective model scale as vclass, the classical vacuum expectation valueof the Higgs boson.In figure 1, we plot the Higgs boson mass as a function of Λ when we impossethe condition δvt = 0 in the Landau gauge. The three different curves correspond tomt = 169, 175 and 181 GeV and we have used mW = 80.35 GeV, mZ = 91.187 GeVand mb = 4.5 GeV [7]. When Λ > 0.7 TeV, we observe that the mass of the Higgsboson is approximately given by the relationm2H = 4(m2t +m2b)− 2m2W −m2Z (6)≈ 318± 13 GeV. (7)Eq. (6) corresponds to the condition for the cancellation of quadratic divergencies[2]-[3] in the self-masses of all the particles in the standard electroweak theory.If we have chosen δmν/mν = 0 instead of δvt/v = 0, the plots obtained would havebeen almost identical for Λ ≥ 0.7 TeV because finite terms in Eq. (5) are negligibleabove these cut off energies. It is interesting to note that our predicted value for mH issomewhat higher than the one predicted in another version of the effective electroweaktheory [8] where the Higgs potential is reduced to its quartic term.In conclusion, by assuming the vanishing of tadpole radiative corrections we findthat the standard electroweak theory behaves as a good effective model with a relativelysmall cut off energy scale. The mass obtained for the Higgs boson departs by less than0.6 % from the value obtained by requiring cancellation of quadratic divergencies [2]-[3]already at Λ ∼ 0.7 TeV. Thus, no fine-tunning of physical masses [1] are required tosuppress the quadratic divergent corrections to self-masses in the effective version ofthe standard electroweak theory.3AcknowledgementsWe thank Abdelhouahed Bernicha, Jean-Marc Ge´rard, Thomas Hambye, Mat´ıasMoreno and Jacques Weyers for valuable discusions.4References[1] See for example: M. Chanowitz, Ann. Rev. Nucl. Part. Sci. 38 (1988) 323.[2] R. Decker and J. Pestieau, preprint UCL-IPT-79-19; DESY Workshop, October22-24, 1979; Mod. Phys. Lett. A7 (1992) 3773; G. Lo´pez Castro and J. Pestieau,Mod. Phys. Lett A10 (1995) 1155.[3] M. Veltman, Acta Phys. Polonica B12 (1981) 437.[4] J. Fleischer and F. Jegerlehner, Phys. Rev. D23 (1981) 2001.[5] I.-Hsiu Lee and S. D. Drell, in: M.A.B. Be´g Memorial Volume, Eds. A. Ali and P.Hoodbhoy, (World Scientific,Singapore, 1991) p.13.[6] S. Weinberg, Phys. Rev. D7 (1973) 2887.[7] Review of Particle Properties, Phys. Rev. D54 (1996) Part I.[8] T. Hambye, Phys. Lett. B371 (1996) 87.5FIGURE CAPTIONFigure 1: Mass of the Higgs boson as a function of Λ by imposing δvt = 0. Theplotted curves (from bottom to top) correspond to mt = 169, 175 and 181 GeV,respectively. Units in both axis are TeV.61 2 3 4 5Lambda0.260.280.30.320.34m_Higgs

Effective SU(2)_L x U(1) theory and the Higgs boson mass

Effective SU(2)_L x U(1) theory and the Higgs boson mass

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