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Gaussian Effective Potential for the

arXiv:hep-ph/9207276v1 31 Jul 1992July, 1992DOE/ER/05096-51Gaussian Effective Potential for theU(1) Higgs ModelR. Iba˜nez-Meier, I. Stancu, and P. M. StevensonT.W.

Bonner Laboratory, Physics Department,Rice University, Houston, TX 77251, USAAbstract:In order to investigate the Higgs mechanism nonperturbatively, we compute the Gaussianeffective potential (GEP) of the U(1) Higgs model (“scalar electrodynamics”). We show that thesame simple result is obtained in three different formalisms.

A general covariant gauge is used,with Landau gauge proving to be optimal. The renormalization generalizes the “autonomous”renormalization for λφ4 theory and requires a particular relationship between the bare gaugecoupling eB and the bare scalar self-coupling λB.

When both couplings are small, then λ isproportional to e4 and the scalar/vector mass-squared ratio is of order e2, as in the classic 1-loop analysis of Coleman and Weinberg. However, as λ increases, e reaches a maximum value andthen decreases, and in this “nonperturbative” regime the Higgs scalar can be much heavier thanthe vector boson.

We compare our results to the autonomously renormalized 1-loop effectivepotential, finding many similarities. The main phenomenological implication is a Higgs mass ofabout 2 TeV.1

1IntroductionThe Higgs mechanism [1] is a vital, but problematic, aspect of the Standard Model. At theclassical level it is clear that spontaneous symmetry breaking (SSB) in the λφ4 scalar sector,through its coupling to the gauge sector, generates gauge-boson mass terms.

The issue of how –or whether – this works in the full quantum theory can be addressed using the effective potential[2], and traditionally the 1-loop approximation [3, 4] has been used. However – at least as it isconventionally renormalized – the 1-loop effective potential (1LEP) is closely tied to perturbationtheory.

The possibility that a perturbative approach is totally misleading must be raised by theclaims that the (λφ4)4 theory is actually “trivial” [5], and by the failure of lattice Monte-Carlocalculations to find a non-trivial, interacting theory [6]. Thus, it is very important to study(λφ4)4 theory and the Higgs mechanism with nonperturbative methods.A simple, nonperturbative method, founded upon intuitive ideas familiar in ordinary quan-tum mechanics, is the Gaussian effective potential (GEP) [7, 8].

In the appropriate limitingcases it contains the one-loop and leading-order 1/N effective potential results [7, 8, 9, 10]. TheGEP for O(N)-symmetric (λφ4)4 theory can be renormalized in two different ways [11]: the“precarious” renormalization, with a negative infinitesimal λB [12, 9, 10], yields essentially theleading-order 1/N result [13].

The resulting effective potential does not, however, display SSB.The other, “autonomous”, renormalization [14, 15, 10], which can have SSB, is characterized bya positive infinitesimal λB and an infinite re-scaling of the classical field. The resulting theory isasymptotically free [16], which can explain why the “triviality” proofs [5] do not apply.

Particlemasses turn out to be proportional to ⟨φ⟩, so in the unbroken-symmetry phase the particlesmust be massless. This can perhaps explain the negative findings of most lattice calculations[6].

(See Ref. [17] for an interesting comparison of recent lattice results [18] with the Gaussianapproximation.) The “autonomous” theory cannot be obtained in the 1/N expansion becauseλB must behave as 1/√N, not as 1/N, when N →∞[10, 19].We used to believe that the “autonomous” theory could only be seen with the Gaussian(or some still-better) approximation.However, it has been shown recently by Consoli andcollaborators [20] that the unrenormalized 1LEP can also be renormalized in an “autonomous”-like way.

This result generalizes to more complicated theories [21]. Applied to the SU(2)×U(1)electroweak theory it predicts a Higgs mass of about 2 TeV [20, 21].In this paper we calculate the GEP for the U(1)-Higgs model, which is O(2) λφ4 theorycoupled to a U(1) gauge field.

We show that it can be renormalized in an “autonomous”-likefashion, and that the vector boson aquires a mass proportional to ⟨φ⟩, just as in the traditionaldescription of the Higgs mechanism. The bare gauge coupling constant e2B and the bare scalarself-coupling λB, both infinitesimal, are related such that for a given e2B (below some maximumvalue) there are two allowed values of λB.

One of these lies in a “perturbative” regime in whichλ ∼e4, where the results agree with the classic 1-loop analysis of Coleman and Weinberg [3].The other lies in a “nonperturbative” regime, where it is possible to have a Higgs particle which2

is arbitrarily heavy compared to the vector boson. (See Figs.

1 and 2.) Our results have muchin common with the “autonomously renormalized” 1LEP [20, 21], and thus tend to support theexpectation of a 2 TeV Higgs mass.The layout of the paper is as follows: After some preliminaries in Sect.

2, we outline threeseparate calculations of the GEP for the U(1) Higgs model using three different formalisms: Sect.3 describes a canonical, Hamiltonian-based calculation, as in [8, 9]; Sect. 4 gives a covariant“δ expansion” calculation, as in [22] (see also [23, 24, 25]); and Sect.

5 outlines a covariantvariational calculation, as in [26]. We think it is very instructive, as well as reassuring, to seehow the same result emerges from these very different approaches.Some comments on theunrenormalized result are given in Sect.

6, where we show that the optimal gauge parameter isξ = 0 (Landau gauge). The renormalization of the GEP is carried out in Sect.

7. To conclude,we discuss the comparison to the 1LEP results and the implications for the Higgs mass in Sect.8.2PreliminariesWe first recall the integrals which play a central rˆole in the GEP.

The expectation value of φ2for a single scalar field yields the quadratically divergent integralI0(Ω) =Zd3p(2π)312ωp,ωp ≡q⃗p 2 + Ω2,(2.1)which is equivalent to the contracted Euclidean propagator G(x, x) (“tadpole diagram”) integralI0(Ω) =Zd4p(2π)41p2 + Ω2(2.2)that arises in the manifestly covariant formalism. The vacuum energy for a free scalar field ofmass Ωis given by the quartically divergent integralI1(Ω) =Zd3p(2π)312ωp,(2.3)which is the sum of the zero-point energies for each momentum mode.

This integral is familiarfrom the 1LEP and in the covariant formalism it arises in the form12Tr lnhG−1(x, y)i/V = I1(Ω) = 12Zd4p(2π)4 lnp2 + Ω2,(2.4)where V is the spacetime volume. (Actually, this form of I1 is only equivalent to the canonicalform (2.3) up to an infinite constant [9].) The GEP also naturally involves the combinationJ(Ω) ≡I1(Ω) −12Ω2I0(Ω),(2.5)3

which arises from the expectation value of a massless scalar-theory Hamiltonian (i.e., kineticterms only), evaluated in the vacuum of a free field theory with mass Ω.The GEP is essentially a variational calculation: one first obtains a function VG of theclassical field ϕc and of the mass parameters, and then one has to minimize with respect tothe mass parameters.This leads to coupled “optimization equations” for the optimal massparameters (denoted by overbars).In carrying out the minimization one needs the formalresult:dI1(Ω)dΩ2= 12I0(Ω). (2.6)Further discussion of these divergent integrals is postponed until Section 7.The quantization of gauge theories in a covariant gauge always involves Faddeev-Popovghosts.

However, in the U(1) case the ghosts are free. Since they do not couple to the otherfields, they have no effect, except for their contribution to the vacuum energy [27].

Because theghosts correspond to two free, massless, anticommuting degrees of freedom, their contribution iseasily seen to be −2I1(0). (In the covariant formalism this term would come from performing thefunctional integral over the ghost fields.) Since this contribution is ϕc independent, it will dropout when the infinite vacuum-energy constant is subtracted off.

This happens automatically indimensional regularization, which effectively sets I1(0) = 0. We shall therefore ignore ghosts inthe following calculations.3Canonical GEP calculationThe Lagrangian for the U(1) Higgs model (ignoring ghosts) is:L = LGauge + LScalar,(3.1)where LGauge is discussed below, and LScalar is the Lagrangian for a complex scalar field, φ, withthe derivative replaced by the covariant derivative: [28]LScalar = (Dµφ)∗(Dµφ) −m2Bφ∗φ −4λB (φ∗φ)2 ,(3.2)withDµ = ∂µ + ieBAµ,(3.3)where eB is the bare gauge coupling constant.

Replacing the complex field by two real fields:φ =1√2 (φ1 + iφ2) ,(3.4)we find the O(2)-symmetric λφ4-theory Lagrangian plus coupling terms to the gauge field:LScalar = 12(∂µφ1 −eBAµφ2)2 + 12(∂µφ2 + eBAµφ1)2 −12m2B(φ21 + φ22) −λB(φ21 + φ22)2. (3.5)4

Forming the Hamiltonian densityHScalar ≡˙φ1Π1 + ˙φ2Π2 −LScalar,(3.6)with Πi ≡δL/δ ˙φi, we obtain:HScalar = HO(2) −eB ⃗A. (φ1⃗∇φ2 −φ2⃗∇φ1) −12e2BAµAµ(φ21 + φ22),(3.7)where HO(2) is the Hamiltonian density for O(2)-symmetric λφ4 theory.Without loss of generality, we can choose the classical field ϕc to lie in the φ1 direction.Our trial vacuum | 0>G is a direct product of the free-field vacua for the ˆφ1 “radial” field (withˆφ1 ≡φ1 −ϕc), with mass Ω; for the φ2 “transverse” field, with mass ω; and for the gauge fields(to be discussed below).

The middle term in (3.7) therefore gives no contribution when we takethe expectation value of HScalar in the trial state | 0>G. Hence, we find:< HScalar > = V O(2)G−12e2B (ϕ2c + I0(Ω) + I0(ω)),(3.8)where the first term is the O(2) λφ4-theory result [29, 10]:V O(2)G=J(Ω) + J(ω) + 12m2B(ϕ2c + I0(Ω) + I0(ω))+ λBh3(I0(Ω) + ϕ2c)2 + 2I0(ω)(I0(Ω) + ϕ2c) + 3I0(ω)2 −2ϕ4ci.

(3.9)The gauge-field Lagrangian, including gauge-fixing terms, can be written as:LGauge = −14FµνF µν + (∂µB)Aµ + 12ξB2,(3.10)where Fµν ≡∂µAν −∂νAµ. The last two terms, involving the Nakanishi-Lautrup [30] auxiliaryfield B, are equivalent to the usual covariant gauge-fixing term −12ξ(∂·A)2, where ξ is the gaugeparameter.

[To see this one integrates by parts to get −B(∂· A) + 12ξB2, and then eliminates Bby its equation of motion B = (∂· A)/ξ. ]By itself LGauge would just describe a set of free massless fields.

We want to consider ageneralization of this Lagrangian that includes a mass term:LTrial = LGauge + 12∆2AµAµ. (3.11)The ground state of this “trial theory” will provide us with our trial vacuum state, with themass ∆playing the rˆole of a variational parameter.

To construct the GEP we shall then needto take the expectation value of HGauge (which we can obtain from HTrial by setting ∆= 0) inthe vacuum state of HTrial.The content of the “trial theory” is made plain by definingAµ ≡Aµ + 1∆2 ∂µB,(3.12)5

which de-couples LTrial into separate Aµ and B sectors:LTrial =−14FµνFµν + 12∆2AµAµ−1∆212∂µB∂µB −12ξ∆2B2. (3.13)The Aµ field is thus a free, massive vector field, and its equation of motion ∂µFµν + ∇2Aν = 0yields both (∂2+∆2)Aν = 0 and ∂·A = 0.

The B field is a normal scalar field, mass √ξ∆, exceptthat its Lagrangian has the “wrong sign” and has an overall factor 1/∆2. It is now a relativelystraightforward exercise to obtain the Hamiltonian and canonically quantize the theory, and wejust list some of the key steps below.The plane-wave expansion for the Aµ field is:Aµ =XλZd3k(2π)3 2ωk(∆)ha(⃗k, λ)ǫµ(⃗k, λ)e−ik.x + h.c.i,(3.14)in which k0 = ωk(∆) ≡q⃗k2 + ∆2, and the three polarization vectors ǫµ(⃗k, λ), with λ = −1, 0, 1being the helicity label, satisfy the usual completeness relation:Xλǫ∗µ(⃗k, λ)ǫν(⃗k, λ) = −gµν −kµkν∆2.

(3.15)The creation-annihilation operators obey[a(⃗k, λ), a†(⃗k′, λ′)] = 2ωk(∆)(2π)3δ(3)(⃗k −⃗k′)δλλ′. (3.16)The plane-wave expansion for B isB = ∆Zd3k(2π)3 2ωk(√ξ∆)ha(⃗k, B)e−ik.x + h.c.i,(3.17)in which k0 = ωk(√ξ∆) ≡q⃗k2 + ξ∆2, and the operators obey a “wrong-sign” commutationrelation:[a(⃗k, B), a†(⃗k′, B)] = −2ωk(pξ∆)(2π)3δ(3)(⃗k −⃗k′).

(3.18)Our trial vacuum state | 0 >G is, by definition, annihilated by the operators a(⃗k, λ) (λ =−1, 0, 1) and a(⃗k, B). To construct the GEP we need to substitute the above plane-wave expan-sions into HGauge (conveniently obtained from HTrial by dropping all terms involving ∆) andthen sandwich the result between G<0 | and | 0>G.

From the Aµ fields one obtains = 3J(∆),(3.19)which is three times (because of the three polarization states) the usual GEP result for a massless-scalar Hamiltonian evaluated in a free-field vacuum state of mass ∆. The B fields give = J(pξ∆),(3.20)6

which is positive because the “wrong sign” of the Hamiltonian is compensated by the “wrongsign” of the commutator. Therefore, in total we have = 3J(∆) + J(pξ∆).

(3.21)[As a check, note that if we were considering the gauge sector by itself, minimization of(3.21) would yield ¯∆= 0 and the result would reduce to 4J(0) = 4I1(0). Recalling that theghosts contribute −2I1(0), the total is 2I1(0), which is the vacuum energy associated with twomassless, bosonic degrees of freedom.

These correspond to the two transverse polarizations ofthe massless vector field. Thus, we see explicitly, for any value of the gauge parameter ξ, howthe ghosts act to cancel out the vacuum-energy contributions from the unphysical componentsof the gauge field [27].

]To obtain the total GEP we combine from (3.21) with from (3.8). Ashort calculation gives:= + 1∆4 <∂µB∂µB >=−3I0(∆) −ξI0(pξ∆),(3.22)so we obtain finally:VG(ϕc; Ω, ω, ∆)=V O(2)G+ 3J(∆) + J(pξ∆)+ 12e2B(3I0(∆) + ξI0(pξ∆))(ϕ2c + I0(Ω) + I0(ω)).

(3.23)Minimization with respect to the mass parameters ∆, Ω, and ω leads to:¯∆2 = e2B[ϕ2c + I0(¯Ω) + I0(¯ω)],(3.24)¯Ω2 = m2B + 4λB[3I0(¯Ω) + I0(¯ω) + 3ϕ2c] + e2B[3I0( ¯∆) + ξI0(pξ ¯∆)],(3.25)¯ω2 = m2B + 4λB[I0(¯Ω) + 3I0(¯ω) + ϕ2c] + e2B[3I0( ¯∆) + ξI0(pξ ¯∆)]. (3.26)4Covariant δ-expansion CalculationIn this section we perform the calculation in the Euclidean functional-integral formalism in themanner of Ref.

[22]. Note that in passing to the Euclidean formalism the Minkowski scalarproduct aµbµ goes to −aµbµ; thus terms with just one pair of contracted indices change signrelative to other terms.

The Euclidean action reads:S=Zd4x14 FµνFµν + 12ξ (∂µAµ)2+ (Dµφ)∗(Dµφ) + m2Bφ∗φ + 4λB (φ∗φ)2. (4.1)7

Rewritten in terms of the real scalar fields φ1 and φ2, the action becomesS=Zd4x 14 FµνFµν + 12ξ (∂µAµ)2 + 12 φ1−∂2 + m2Bφ1+ 12 φ2−∂2 + m2Bφ2 + λBφ21 + φ222+12 e2BAµAµφ21 + φ22+ eBAµ (φ1∂µφ2 −φ2∂µφ1),(4.2)where ∂2 ≡∂µ∂µ. Introducing a source for the φ1 field, the generating functional is given by:Z[j] =ZD [φ1, φ2, Aµ] exp−S +Zd4x j(x)φ1(x),(4.3)and the effective action is obtained by the Legendre transformationΓ[ϕc] = ln Z[j] −Zd4x j(x)ϕc(x),(4.4)where the classical field, ϕc(x), is the vacuum expectation value of the field φ1(x) in the presenceof the source j(x).

The effective potential itself, Veff(ϕc), is obtained from Γ[ϕc] by setting ϕc(x)to be a constant, ϕc, and dividing out a minus sign and a spacetime volume factor.Generalizing the procedure of Ref. [22] (see also [23, 24, 25]), we can calculate the GEPfrom a first-order expansion in a nonstandard kind of perturbation theory.

First we introducethe shifted fields:ˆφ1(x) = φ1(x) −ϕc,ˆφ2(x) = φ2(x). (4.5)(Notice that we have taken the shift parameter to be exactly ϕc, the vacuum expectation valueof φ(x): Although not obligatory [22], this simplifies the calculation.

)We then split the (Euclidean) Lagrangian into two parts:L =L0 + Lintδ=1,(4.6)where L0 is a sum of three free-field Lagrangians: one for the vector field Aµ, of mass ∆; one forthe radial scalar field, ˆφ1, with mass Ω; and one for the transverse scalar field, ˆφ2, with mass ω:L0=12 Aµ(x)−∂2 + ∆2δµν +1 −1ξ∂µ∂νAν(x)+ 12ˆφ1(x)−∂2 + Ω2 ˆφ1(x) + 12ˆφ2(x)−∂2 + ω2 ˆφ2(x). (4.7)The interaction Lagrangian is then:Lint=δv0 + v1 ˆφ1 + v2 ˆφ21 + v3 ˆφ31 + λB ˆφ41 + v′2 ˆφ22 + λB ˆφ42+ 4λBϕc ˆφ1 ˆφ22 + 2λB ˆφ21 ˆφ22 + 12e2Bϕ2c −∆2AµAµ+ eBϕc Aµ ∂µ ˆφ2 + eBAµˆφ1∂µ ˆφ2 −ˆφ2∂µ ˆφ1+ e2Bϕc ˆφ1AµAµ + 12e2BAµAµˆφ21 + ˆφ22 .

(4.8)8

The “coupling constants” v0, v1, v2, v′2 and v3, which are ϕc dependent, are the same as in theλφ4 case [22]:v0=12m2Bϕ2c + λBϕ4c,v1=m2B + 4λBϕ2cϕc,v2=12m2B −Ω2+ 6λBϕ2c,(4.9)v′2=12m2B −ω2+ 2λBϕ2c,v3=4λBϕc.The artificial expansion parameter δ has been introduced in Lint in order to keep track ofthe order of approximation, which consists in obtaining a (truncated) Taylor series in δ, aboutδ = 0, which is then used to extrapolate to δ = 1.The expansion in Lint (or equivalently in δ) is now quite straightforward, following standardperturbation theory procedures. To first order in δ it yields:Γ[ϕc]=ΓO(2)[ϕc] −12 Tr lnhG−1µν (x, y)i−δZd4x12e2Bϕ2c −∆2Gµµ(x, x)+ 12e2B [I0(Ω) + I0(ω)] Gµµ(x, x)+ Oδ2,(4.10)where ΓO(2)[ϕc] is the first-order action for the scalar sector, andGµν(x, y) =Zd4p(2π)41p2 + ∆2δµν + (ξ −1)pµpνp2 + ξ∆2e−ip·(x−y)(4.11)is the gauge field propagator, .

The Trace-log term givesTr lnhG−1µν (x, y)i= 2Vh3I1(∆) + I1(pξ ∆)i,(4.12)where V is the infinite spacetime volume, and the contracted propagator givesGµµ(x, x) = 3I0(∆) + ξI0(pξ ∆). (4.13)To obtain the GEP we discard the O(δ2) terms and set δ = 1 and divide through −V toobtainVG = V O(2)G+ 3I1(∆) + I1(pξ∆) + 12e2Bϕ2c −∆2 3I0(∆) + ξI0(pξ ∆)+ 12e2B [I0(Ω) + I0(ω)]3I0(∆) + ξI0(pξ ∆).

(4.14)Recalling that J(∆) ≡I1(∆) −12∆2I0(∆), one sees that this result coincides with the resultobtained from the canonical calculation, Eq. (3.23).9

5Covariant Variational CalculationIn this section we use the method developed in [26] based on Feynman’s variational principle [31]applied to the Euclidean action. This in turn follows from Jensen’s inequality for expectationvalues of convex functions; in particular, exponential functions:Zdµ(φ) exp g(φ) ≥expZdµ(φ)g(φ),(5.1)for a normalized integration measure dµ(φ).

The inequality applies only to commuting fields,but happily in the U(1) case the anticommuting ghost fields can be integrated out exactly. Theremaining action can be written in the following form (using the shifted fields ˆφ1 = φ1 −ϕc,ˆφ2 = φ2, as in the last section):S[Aµ, ˆφ1, ˆφ2] = SA[Aµ] + SA,φ[Aµ, ˆφ1, ˆφ2] + Sφ[ˆφ1, ˆφ2] ,(5.2)whereSA = 12Zd4xAµ(x)−∂2δµν + (1 −1ξ )∂µ∂νAν(x),(5.3)SA,φ=Zd4x12e2B(ϕ2c + ˆφ21 + ˆφ22)Aµ(x)Aµ(x)+ eBAµh(ˆφ1 + ϕc)∂µ ˆφ2 + ˆφ2∂µ(ˆφ1 + ϕc)i+ e2Bϕc ˆφ1AµAµo,(5.4)and Sφ is given by the usual O(2) λφ4 action.Following Ref.

[26], we now apply the Feynman-Jensen inequality to Z[j], Eq (4.3), withdµ(φ) = N −1DAµD ˆφ1D ˆφ2e−SG and g(φ) = SG −S +R jφ, whereN =ZDAµD ˆφ1D ˆφ2 e−SG,(5.5)and where SG is a quadratic “trial action”:SG = 12Zd4xnAµG−1µν Aν + ˆφ1G−11 ˆφ1 + ˆφ2G−12 ˆφ2o,(5.6)involving adjustable kernels G−1. Taking the Legendre transform, (4.4), we obtain the “Gaussianeffective action” [26]:¯ΓGEA[ϕc] = maxGlog(N) + N −1ZDAµD ˆφ1D ˆφ2 e−SG (SG −S),(5.7)as a lower bound on the exact effective action (which will hence yield an upper bound on theeffective potential).

Since the kernels involve differential operators, it is convenient to go tomomentum space, using Fourier transforms (indicated by tildes) and the convenient notationRp =R d4p/(2π)4, ¯δ(p) = (2π)4δ(p). We can then write the trial action asSG = 12ZpZqn ˜Aµ(p) ˜G−1µν (p, q) ˜Aν(q) + ˜φ1(p) ˜G−11 (p, q)˜φ1(q) + ˜φ2(p) ˜G−12 (p, q)˜φ2(q)o,(5.8)10

in which the ˜G−1’s are the inverses of the momentum-space propagators.Evaluation of the Gaussian functional integrals involved in (5.7) is straightforward, and yields¯ΓGEA = max˜GΓO(2)[ ˜ϕc, ˜G1, ˜G2] −12Tr ln [ ˜G−1µν (p, q)]−12Zpp2δµν + e2BZr( ˜G1(r, −r) + ˜G2(r, −r))δµν −(1 −1ξ )pµpν˜Gµν(p, −p)−12e2BZpqrs˜ϕc(r) ˜ϕc(s) ˜Gµµ(p, q)¯δ(p + q + r + s). (5.9)Maximization yields optimization equations determining the optimal ˜G propagators, denoted by¯G(p, q):¯G−1µν (p, q)=p2δµν + e2BZr( ¯G1(r, −r) + ¯G2(r, −r))δµν −(1 −1ξ )pµpν¯δ(p + q)+ e2BZrs˜ϕc(r) ˜ϕc(s)¯δ(p + q + r + s)δµν,(5.10)¯G−11 (p, q)=p2 + m2B + e2BZr¯Gµµ(r, −r)¯δ(p + q)+ 4λBZrs[3 ¯G1(r, s) + ¯G2(r, s) + 3 ˜ϕc(r) ˜ϕc(s)]¯δ(p + q + r + s),(5.11)¯G−12 (p, q)=p2 + m2B + e2BZr¯Gµµ(r, −r)¯δ(p + q)+ 4λBZrs[ ¯G1(r, s) + 3 ¯G2(r, s) + ˜ϕc(r) ˜ϕc(s)]¯δ(p + q + r + s).

(5.12)For a spatially constant classical field we have ˜ϕc(p) = ϕc¯δ(p), and the above equations thendictate that the propagators all become proportional to ¯δ(p + q), so we may write them in theform¯G−1µν (p, q) =(p2 + ¯∆2)δµν −(1 −1ξ )pµpν¯δ(p + q),(5.13)¯G−11 (p, q) = (p2 + ¯Ω2)¯δ(p + q),(5.14)¯G−12 (p, q) = (p2 + ¯ω2)¯δ(p + q),(5.15)where the optimal mass parameters ¯∆, ¯Ω, and ¯ω are given by¯∆2 = e2B[ϕ2c + I0(¯Ω) + I0(¯ω)],(5.16)¯Ω2 = m2B + 4λB[3I0(¯Ω) + I0(¯ω) + 3ϕ2c] + e2BV−1Zp¯Gµµ(p, −p),(5.17)¯ω2 = m2B + 4λB[I0(¯Ω) + 3I0(¯ω) + ϕ2c] + e2BV−1Zp¯Gµµ(p, −p). (5.18)As usual, factors of “¯δ(0)” have been interpreted as spacetime volume factors V. The integralRp ¯Gµµ(p, −p) , where¯Gµν(p, −p) =Vp2 + ¯∆2δµν + (ξ −1)pµpνp2 + ξ ¯∆2,(5.19)11

can be evaluated in terms of I0 integrals:Zp¯Gµµ(p, −p) = V[3I0( ¯∆) + ξI0(pξ ¯∆)]. (5.20)The Trace-log term can be taken from Eq.

(4.12), so that we obtain finally the same result asin Eq. (3.23).6Comments on the Unrenormalized ResultThe Gaussian-approximation result shows a dependence upon the gauge parameter ξ.Thismeans that our Gaussian approximation does not fully respect gauge invariance.

However, weargue that this is inevitable and not fatal. It is inevitable because, for the O(2) scalars, we haveto use “Cartesian-coordinate” fields ˆφ1, ˆφ2 rather than “polar-coordinate” fields, so that, whenϕc ̸= 0, the O(2) symmetry is not being fully respected.

In pure λφ4 theory this produces anapparent conflict with Goldstone’s theorem, in that the transverse mass parameter, ω, is non-zero [10, 29]. However, the point is that the transverse field ˆφ2 is not the true “polar-angle”,Goldstone field.

In the U(1) Higgs model, in a covariant gauge, the Goldstone field becomesan unphysical degree of freedom [1], but the problem remains in the form of gauge-parameterdependence. This just means, though, that we have a “non-invariant” approximation – whichis where the exact result is known to be independent of some parameter, but the approximateresult has a dependence on that parameter.

This is actually quite a common occurrence, andcan be dealt with by “optimizing” the unphysical parameter; requiring the approximate resultto be stationary, or more generally “minimally sensitive,” to the unphysical parameter [32]. Oneis in a still better position when the approximation has a variational character, because thenthe optimal choice for the unphysical parameter is unquestionably determined by minimization.Our calculation here does indeed have a variational character.

[One might well have beenunsure, with the canonical calculation alone, whether or not the variational inequality is validin the presence of a “wrong-sign” field (and hence negative-norm states). However, this doubtis allayed by the covariant variational calculation: the Jensen inequality just depends on the theconvexity of the exponential function and is equally valid for exp(−g(φ)) and exp(+g(φ)).] Bydifferentiating ¯VG one finds that the optimal gauge is the Landau gauge, ξ = 0.

This can easilybe seen by noting that, by virtue of the ¯∆equation, the ξ-dependence in ¯VG comes only froman I1(√ξ ¯∆) term. Since I1 is (formally) an increasing function of its argument, the energy isminimized when ξ = 0.With ξ = 0, and discarding a vacuum-energy contribution I1(0), the GEP and its optimiza-tion equations simplify to:VG(ϕc; Ω, ω, ∆) = V O(2)G+ 3J(∆) + 32e2BI0(∆)(ϕ2c + I0(Ω) + I0(ω)),(6.1)12

with V O(2)Ggiven by (3.9), and¯∆2 = e2B[ϕ2c + I0(¯Ω) + I0(¯ω)],(6.2)¯Ω2 = m2B + 4λB[3I0(¯Ω) + I0(¯ω) + 3ϕ2c] + 3e2BI0( ¯∆),(6.3)¯ω2 = m2B + 4λB[I0(¯Ω) + 3I0(¯ω) + ϕ2c] + 3e2BI0( ¯∆). (6.4)Note that if the ¯∆equation, (6.2), is substituted back into VG, (6.1), then we can write the GEPas¯VG(ϕc) = V O(2)G+ 3I1( ¯∆),(6.5)with separate contributions from the scalar and gauge sectors.

This observation applies to theGaussian effective action, too, since Eq. (5.10) substituted back into (5.9) yields¯ΓGEA = ΓO(2) −12Tr ln[ ¯G−1µν (p, q)] .

(6.6)Note, however, that the optimization equations for ¯∆, ¯Ω, and ¯ω, (6.2–6.4), remain coupled.We may also remark that the generalization of the result to ν + 1 dimensions is trivial: theintegrals need to be re-defined in an obvious way, and the factors of 3 associated with the ∆integrals need to be replaced by ν, since these factors correspond to the number of polarizationstates of a massive vector field.Finally, we briefly comment upon some previous work relating to the GEP and the U(1) Higgsmodel. (i) All`es and Tarrach [33] used a somewhat naive canonical approach which we believe isvalid in Feynman gauge (ξ =1) only.

Their treatment of the scalar sector effectively sets ω ≡Ω,which is sub-optimal. In renormalizing their result, All`es and Tarrach used a generalizationof the “precarious” λφ4 theory, which does not have SSB.

(ii) Cea [34] describes a temporal-gauge GEP calculation, but contents himself with demonstrating that the 1-loop terms arerecovered correctly. (iii) The papers of Ref.

[35] make a comprehensive study of the Schr¨odingerwavefunctional formalism, and try hard to maintain gauge invariance and compliance with theGoldstone theorem. Our view, as discussed above, is less puritanical.

(iv) Kovner and Rosenstein[36] use yet another formulation of the Gaussian approximation, based on truncating the Dyson-Schwinger equations. Their renormalization is quite different from ours: it appears to be relatedto the “precarious” λφ4 renormalization, but it somehow transfers the negative sign from λB towavefunction renormalization factors.The “precarious” renormalization of the U(1)-Higgs-model GEP is a topic which we do notpursue here, but it could be of theoretical interest: We would expect the results to be similar tothe 1/N-expansion analysis of Kang [37].7“Autonomous” Renormalization of the GEP13

7.1The divergent integralsThe GEP involves the quartically and quadratically divergent integrals I1 and I0.Anotherrelated integral:I−1(Ω) ≡−2 dI0dΩ2 =Zd3p(2π)312(ωp)3 = 2Zd4p(2π)41(p2 + Ω2)2 ,(7.1)which is logarithmically divergent, will play a crucial role.Ref. [9] derives useful formulasfor these integrals by Taylor-expanding the integrands about Ω2 = m2, and then re-summingthe terms that give convergent integrals.

From these we can obtain the still more convenientformulas:I1(Ω) = I1(0) + Ω22 I0(0) −Ω48 I−1(µ) + f(Ω2),(7.2)I0(Ω) = I0(0) −Ω22 I−1(µ) + 2f ′(Ω2),(7.3)I−1(Ω) = I−1(µ) −18π2 ln Ω2µ2 ,(7.4)wheref(Ω2) =Ω464π2"ln Ω2µ2 −32#,(7.5)and f ′(Ω2) is its derivative with respect to Ω2. These formulas are valid in any regularizationscheme that preserves the property dIn/dΩ2 = (n −12)In−1.

This allows one, at least in theλφ4 case, to discuss the renormalization procedure in a completely regularization-independentmanner. However, in gauge theories, most cutoff-based renormalizations – because they interferewith gauge invariance – have problems with quadratic divergences in the vector self-energy.

Herethese problems would manifest themselves as quadratic divergences in the vector-mass parameter¯∆2, Eq. (6.2) [35].

(Unlike the scalar case, these cannot be simply absorbed into a bare-mass. )It is well known in other contexts that, with sufficient technical virtuosity, these problems can beshown to be spurious [38].

However, it is much simpler to appeal to dimensional regularization,or some such scheme, in which one can justify setting the scale-less integrals, I0(0) and I1(0),equal to zero. This automatically elimates any problem with quadratic divergences.

All theremaining divergences can be written in terms of I−1, which has a 1/ǫ pole in dimensionalregularization: Explicitly:I−1(Ω) = Aǫ Ω−ǫ,A ≡14π2 Γ(1 + ǫ/2)(4π)ǫ/2. (7.6)7.2Renormalization: Part ITo renormalize ¯VG we use an “autonomous” renormalization (Cf.

[15, 10]), characterized by aninfinite re-scaling of the classical field and infinitesimal bare coupling constants:ϕ2c = ZφΦ2c = z0I−1(µ)Φ2c,(7.7)14

λB =ηI−1(µ) ,e2B =γI−1(µ) ,(7.8)where z0, η and γ are finite, and µ is a finite mass scale. For the present, we assume that allthe I−1 factors have the same argument, µ.

We shall also take m2B = 0. These simplifyingassumptions will be removed later in subsection 7.4.

We shall also postpone the determinationof the finite wavefunction-renormalization factor z0 to that subsection.First, we substitute the renormalization equations into the optimization equations (6.2 – 6.4)and use the key formula for I0, (7.3), setting I0(0) = 0. Keeping only the finite terms, for thepresent, we obtain:¯∆2=γ(z0Φ2c −12¯Ω2 −12 ¯ω2) + ǫ∆,¯Ω2=4η(−32¯Ω2 −12 ¯ω2 + 3z0Φ2c) −32γ ¯∆2 + ǫΩ,(7.9)¯ω2=4η(−12¯Ω2 −32 ¯ω2 + z0Φ2c) −32γ ¯∆2 + ǫω,where the ǫ∆, ǫΩ, ǫω terms are infinitesimal, O(1/I−1), terms.

Ignoring the ǫ terms the equationsare linear and homogeneous, so that each mass parameter is proportional to Φc. The equationscan be straightforwardly solved to yield:¯∆2=2γ(2 + 16η −3γ2)z0Φ2c + O(1/I−1),¯Ω2=[8η(3 + 16η) −3γ2(1 + 8η)](1 + 4η)(2 + 16η −3γ2)z0Φ2c + O(1/I−1),(7.10)¯ω2=(8η −3γ2)(1 + 4η)(2 + 16η −3γ2)z0Φ2c + O(1/I−1).Since ∂VG/∂Ω= 0, etc., by virtue of the gap equations, the total derivative of ¯VG withrespect to ϕc is equal to its partial derivative, and so can be calculated very easily:d ¯VGdϕc= ∂VG∂ϕc=ϕc[m2B + 4λB(3I0(¯Ω) + I0(¯ω) + ϕ2c) + 3e2BI0( ¯∆)]=ϕc(¯Ω2 −8λBϕ2c).

(7.11)The last equality follows from the optimization equation for ¯Ω, (6.3), and yields the sameexpression as in pure λφ4 theory [10]. In order for ¯VG to be finite in terms of the re-scaledfield Φc, we must have a cancellation between the finite part of ¯Ω2 and 8λBϕ2c = 8ηz0Φ2c.

Thiscondition implies a constraint on the coefficients η and γ of the λB and e2B coupling constants.This can be expressed as:γ2 = 8η3(1 −8η −64η2)(1 −32η2),(7.12)and will be discussed further in the next subsection.15

Using the constraint one can simplify the expressions for the optimal mass parameters to:¯∆2=γ 1 −32η21 + 4η z0Φ2c + O(1/I−1),¯Ω2=8ηz0Φ2c + O(1/I−1),(7.13)¯ω2=32η21 + 4η z0Φ2c + O(1/I−1).The renormalized GEP is most easily obtained from the expression for its first derivative,(7.11). The leading terms cancel, so one needs to obtain the infinitesimal, O(1/I−1), part of ¯Ω2.The calculation is straightforward, if tedious.

One needs to obtain the explicit form of ǫ∆, ǫΩ, ǫω,in Eqs. (7.9) by going back to Eqs.

(6.2 – 6.4). One can then solve for the O(1/I−1) correctionto ¯Ωin Eq.

(7.13). After some algebra, one finds that the coefficients of the three f ′ termsmatch those in (7.13) above, so that one can write:d ¯VGdΦc= 2Φc"3 d ¯∆2dΦ2c!f ′( ¯∆2) + d¯Ω2dΦ2c!f ′(¯Ω2) + d¯ω2dΦ2c!f ′(¯ω2)#.

(7.14)Thus, by integrating with respect to Φc, one obtains the renormalized GEP as just:¯VG = 3f( ¯∆2) + f(¯Ω2) + f(¯ω2),(7.15)where f is the function defined in Eq. (7.5).

The GEP is thus a sum of Φ4c ln Φ2c and Φ4c terms.If we swap the parameter µ for the vacuum value Φv (defined as the position of the minimumof ¯VG), we can write the GEP simply as¯VG = Kz20Φ4c ln Φ2cΦ2v!−12!,(7.16)whereK =η8π2"(1 + 8η)(1 −8η + 32η2 + 256η3)(1 + 4η)2#. (7.17)7.3DiscussionThe constraint (7.12) arises from the requirement that the divergent I−1 terms in ¯VG cancel.The equivalent constraint in pure O(2) λφ4 analysis [10, 29] would fix the coefficient η to be thepositive root of the numerator factor, (1 −8η −64η2), which isη0 ≡14(1 +√5) = 0.0773.

(7.18)Here, however, one has instead a relationship between the two coupling coefficients, which isshown in Fig. 1.

It is easily established that only the region between η = 0 and η = η0 isphysically relevant. This is because (i) γ, being proportional to e2B, must be positive; and (ii)the vector mass-squared ¯∆2 must be positive, which precludes η2 from being larger than 1/3216

(see Eq. (7.13)).

From the figure we see that there is a “perturbative region” in which both ηand γ are small, with γ2 ≈(8/3)η. This corresponds to e4 ∼O(λ), as in Coleman and Weinberg(CW) [3].

However, as η increases, γ2 reaches a maximum and then starts to decrease, goingto zero at η = η0. This extreme case corresponds to a free vector theory completely decoupledfrom a self-interacting λφ4 theory.The vector-boson and Higgs masses come directly from Eq.

(7.13), evaluated at Φc = Φv.Their ratio is given by:M2HM2V=¯Ω2v¯∆2v= 8η(1 + 4η)γ(1 −32η2),(7.19)which is just a function of η, since γ is determined by the constraint (7.12). The mass-squaredratio is plotted in Fig.

2. In the “perturbative regime” the Higgs is much lighter than the vectorboson, by a factor of 3γ, which is O(e2) as in CW.

However, for most of the range of η theHiggs has a mass comparable to the vector. When η becomes close to η0 the Higgs can be muchheavier than the vector.The other mass parameter, ¯ω2, does not have a direct physical meaning.

It correspondsto the mass of the transverse scalar field, which is, approximately, the Goldstone field. In thecovariant-gauge Higgs mechanism [1] the Goldstone field is an unphysical degree of freedom.As discussed in Sect.

6, the fact that ¯ω2 is non-zero is due to our approximation being unableto fully respect the O(2) symmetry. We can therefore be pleased by the fact that ω2 is small(dashed line in Fig.

2).7.4Renormalization: Part IIWe initially assumed that the mass-scale in the I−1 denominator of e2B was the same as themass-scale µ in λB (see Eq. (7.8).

If this is not so then, using (7.4), we can re-write e2B as:e2B =γI−1(µ) +γ2(I−1(µ))2 + . .

. ,(7.20)where γ2 is a coefficient proportional to the logarithm of the ratio of the two mass-scales.The subleading γ2/(I−1(µ))2 term leads to an extra contribution, proportional to Φ2c, in theinfinitesimal part of ¯Ω2.

Thus, when ¯VG is obtained by integrating (7.11), we obtain an extrafinite contribution proportional to Φ4c in Eq. (7.15).

However, if we then re-parametrize theGEP in terms of the vacuum value Φv, we obtain Eq. (7.16) unchanged: all the differences areabsorbed into the relationship of Φv to µ and γ2.

Exactly the same argument applies if the scalein the I−1 factor of Zφ is different from that in λB [15]. [The argument also applies if one wantsto insist upon replacing the factors of 3 in the GEP, representing the number of polarizationstates of a massive vector field, by 3 −ǫ in dimensional regularization.

]Note that, for m2B = 0, the bare Lagrangian is characterized by just two bare parameters;λB and eB. Thus, we expect the renormalized GEP to be characterized by two parameters.This is indeed the case, and in the final form, (7.16), these are η and Φv.

(We shall shortly see17

that z0 is fixed in terms of η by Eq. (7.24) below.) The Φv parameter has dimensions of mass,and its appearance constitutes the “dimensional transmutation” phenomenon [3].

Originally, the“autonomous” renormalization conditions (7.7) and (7.8) introduced a superfluity of parameters;η, γ, z0, and the scale arguments of the I−1 factors. As just discussed, it does not matter if allthese mass-scales are different, since they are eventually subsumed in a single scale, Φv.

We sawearlier that γ was fixed in terms of η by the constraint (7.12), required for the I−1 divergencesto cancel. It remains to show how z0 is determined, and we turn to this topic next.The “autonomous” renormalization involves a wavefunction renormalization constant Zφ =z0I−1(µ).

The λφ4 analysis in Refs. [15, 10] set z0 = 1 arbitrarily (although the possibility offurther finite re-scalings of the field was considered).

However, as Ref. [20] has pointed out,z0 is actually fixed uniquely by the following argument.

The bare and renormalized two-pointfunctions are related byΓ(2)B = Z−1φ Γ(2)R . (7.21)Let us consider this relation at zero momentum in the vacuum ϕc = ϕv.

Γ(2)Bis then given bythe second derivative of the effective potential, with respect to the bare field, at ϕc = ϕv. Thisis easily calculated from (7.16):d2 ¯VGdϕ2cϕc=ϕv=1Zφd2 ¯VGdΦ2cΦc=Φv= 1Zφ8Kz20Φ2v.

(7.22)The renormalized (Euclidean) two-point function (i.e., inverse propagator), Γ(2)R , is just p2 + ¯Ω2in the Gaussian approximation. At zero momentum and at ϕc = ϕv it therefore becomes thephysical Higgs mass squared M2H = ¯Ω2v = 8ηz0Φ2v.

Hence, Eq. (7.21) gives1Zφ8Kz20Φ2v = 1Zφ8ηz0Φ2v,(7.23)which impliesz0(mB =0) = ηK = 8π2"(1 + 4η)2(1 + 8η)(1 −8η + 32η2 + 256η3)#.

(7.24)The factor in square brackets varies between 1 and 1.536 for η between 0 and η0. (See Fig.

3. )Finally, we remove our initial simplifying assumption that the bare mass vanishes identically.A finite bare mass would spoil the cancellation of I−1 divergences, but an infinitesimal bare mass,m2B = m20/I−1(µ),(7.25)is allowed (Cf.

Ref. [15] with I0(0) = 0).

This produces an extra, Φc-independent, contributionto the 1/I−1(µ) part of ¯Ω2. Thus, when we integrate (7.11), we obtain an extra finite contribu-tion, proportional to Φ2c, in the GEP, Eq.

(7.15). The result, conveniently re-parametrized byΦv and a new parameter m2 (trivially related to m20), takes the form:¯VG = Kz20Φ4c ln Φ2cΦ2v!−12!+ 12m2z0Φ2c 1 −12Φ2cΦ2v!.

(7.26)18

As before, K is given by (7.17) and Φv corresponds to the position of the minimum of ¯VG [39].Nothing else is affected except the determination of z0. The second derivative of the GEPat the vacuum is now given by:d2 ¯VGdϕ2cϕc=ϕv=1Zφd2 ¯VGdΦ2cΦc=Φv= 1Zφ8Kz20Φ2v −2m2z0,(7.27)which replaces the left-hand side of Eq.

(7.23), so that we obtainz0 = 1K η + 14m2Φ2v!. (7.28)Note that m is not a particle mass.

In the symmetric vacuum all the particles would be massless,for any m2. In the SSB vacuum the particle masses are affected by m2 only through its effecton z0.8Summary, Comparison to 1LEP, and Implications for the HiggsMassWe have calculated, with three different formalisms, the GEP of the U(1) Higgs model.

Theunrenormalized result, in a general covariant gauge, is given at the end of Sect. 3.

In the optimalgauge, ξ = 0, the result is given in Sect. 6.To renormalize the GEP we postulated the infinitesimal forms λB = η/I−1, e2B = γ/I−1,m2B = m20/I−1 for the bare parameters, and an infinite re-scaling of the classical field, ϕ2c =z0I−1Φ2c, where I−1 is a log-divergent integral.

The cancellation of I−1 divergences in the GEPgave the constraintγ2 = 8η3(1 −8η −64η2)(1 −32η2). (8.1)(See Fig.

1.) The vector and Higgs masses were found to be given byM2V=γ 1 −32η21 + 4η z0Φ2v,(8.2)M2H=8ηz0Φ2v.

(8.3)(See Fig. 2.) The z0 factor in the ϕ2c re-scaling was obtained in Eq.

(7.28): in the mB = 0 caseit varies between 8π2 and (8π2) × (1.536) (see (7.24) and Fig. 3).

The renormalized GEP, Eq. (7.26) is a sum of Φ4c ln Φ2c, Φ4c, and Φ2c terms.At the unrenormalized level, we can recover the 1LEP simply by discarding all the I20 termsin Eq.

(6.1), since each I0 and I1 is really accompanied by an ¯h factor.Consequently, theoptimization equations, (6.2 – 6.4), would be reduced to the classical expressions, ∆2c = e2Bϕ2c,19

Ω2c = m2B + 12λBϕ2c,ω2c = m2B + 4λBϕ2c, and Eq. (6.5) would reduce to the familiar (unrenor-malized) 1-loop result [3, 4]:V1l = 12m2Bϕ2c + λBϕ4c + 3I1(∆c) + I1(Ωc) + I1(ωc).

(8.4)Conventionally, the 1LEP is renormalized in a perturbative fashion, with λR = λB(1 +O(λB¯hI−1) + . .

. ), etc..

However, it has been realized recently [20, 21] that the 1LEP can alsobe renormalized in an “autonomous” fashion. The analysis exactly parallels the GEP case, andcan be made even simpler by directly using (7.2) for I1 [21].

In the 1-loop case the constraintneeded to cancel the I−1 divergences is:˜γ2 = 83 ˜η(1 −20˜η),(8.5)with tilde’s distinguishing the 1-loop quantities from their GEP counterparts. The vector andHiggs masses are given byM2V=˜γ˜z0Φ2c,(8.6)M2H=12˜η˜z0Φ2c.

(8.7)The ˜z0 factor in the massless case is 12π2 (so one can regard 12π2˜γ as the renormalized e2).The renormalized 1LEP emerges (modulo the qualifications mentioned below) as:V1l = 3f(∆2c) + f(Ω2c) + f(ω2c),(8.8)in terms of the function f defined in Eq. (7.5), and so is a mixture of Φ4c ln Φ2c and Φ4c terms.

[Actually, this result assumes mB = 0, and that all the I−1 factors have the same scale µ. Theseassumptions are easily removed, as discussed in “Part II” of the GEP analysis (Sect. 7.4): onesimply gets additional Φ2c and Φ4c terms with free-parameter coefficients.

The final result canagain be parametrized in the form (7.26). ]Clearly, the autonomously renormalized 1LEP and GEP results have much in common.

The1LEP constraint equation and M2H/M2V ratio are plotted in Figs. 4 and 5.

Qualitatively, theseclosely resemble the GEP results in Figs.1 and 2.In the 1-loop case the maximum ˜η is1/20 = 0.05, rather than η0 = 0.077, but a rescaling of the η and γ axes almost entirely absorbsthe differences between the 1LEP and GEP results. The form of the renormalized potentials isalso remarkably similar, both when we compare (8.8) with (7.15), and when we note that theyshare the same final form (7.26).

In the 1-loop case the coefficient ˜K would be given by ˜η/(8π2)instead of by Eq. (7.17).

These differ by the same factor that occurs in the GEP’s z0; a factorthat lies between 1 and 1.536.To see the implications for phenomenology, we can consider Φv = (√2GF )−1/2 = 246 GeV,and MV ∼90 GeV. This implies a very small γ, and hence we must either be in the perturbativeregime where both η and γ are small, or near to the maximum allowed η.

The former case gives20

a light Higgs, as in CW [3]: The latter case gives a Higgs that is much heavier than the vectorboson. In fact, the Higgs mass would be almost exactly that of a pure O(2) λφ4 theory whoseΦv was 246 GeV.

For definiteness let us assume the attractive possibility that the bare mass iszero [3]. From the 1-loop result (8.7), with ˜η = 1/20, ˜z0 = 12π2, we would obtain MH = 2.07TeV.

From the GEP result (8.3), with η = η0 = 0.0773, z0 = (8π2) × (1.533), we would obtainMH = 2.13 TeV. These results agree remarkably well.Of course, these results are for the U(1) Higgs model, not the actual SU(2)×U(1) theory.However, the 1-loop analysis is easily extended to that theory [21], and yields MH = 1.89 TeV.The GEP calculation for SU(2)×U(1) is a more difficult matter.

However, it is clear that thephenomenological result would be essentially governed by the scalar sector, which is an O(4),rather than an O(2), λφ4 theory. The GEP results for the O(N) case can be obtained from Ref.

[10], supplemented by a quick calculation of the proper z0 factor, as explained in Sect. 7.4.

Forzero bare mass, this gives:z0[O(N) λφ4] = π22η0(1 + 4η0)(1 −4η0),(8.9)where η0 in the O(N) case isη0 =14(1 +√N + 3). (8.10)The Higgs mass is again given by the form (8.3).

[If the bare mass is non-zero, then the resultis affected only by an O(m2/Φ2v) correction to z0.] The O(4) case gives us a GEP prediction forthe Standard-Model Higgs mass:MH = 2.05 TeV.Consoli et al.

[20] argue that the nonperturbative renormalization used here implies a van-ishing renormalized scalar self-coupling (see also [18]).That would drastically suppress theHiggs-to-longitudinal-W, Z couplings, leaving the Higgs with a relatively narrow width. Thephenomenology of such a Higgs deserves urgent attention.AcknowledgementsWe would like to thank Maurizio Consoli, Jos´e Latorre, Anna Okopi´nska, and Rolf Tarrachfor very valuable discussions.This work was supported in part by the U.S. Department of Energy under Contract No.

DE-AS05-76ER05096.21

References[1] P. W. Higgs, Phys. Lett.

12, 132 (1964); Phys. Rev.

Lett. 13, 508 (1964); F. Englert andR.

Brout, ibid, 321; G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, ibid, 585; T. W. B.Kibble, Phys. Rev.

155, 1554 (1967). [2] J. Goldstone, A. Salam, and S. Weinberg, Phys.

Rev. 127, 965 (1962); G. Jona-Lasinio,Nuovo Cimento 34, 1790 (1964).

[3] S. Coleman and E. Weinberg, Phys. Rev.

D 7, 1888 (1973). [4] S. Weinberg, Phys.

Rev. D 7, 2887 (1973); R. Jackiw, Phys.

Rev. D 9, 1686 (1974).

[5] M. Aizenman, Phys. Rev.

Lett. 47,1 (1981); J. Fr¨ohlich, Nucl.

Phys. B200, 281 (1982).

[6] D. J. E. Callaway and R. Petronzio, Nucl. Phys.

B240, 577 (1984); I. A.

Fox and I. G.Halliday, Phys. Lett.

B 159, 148 (1985); C. B. Lang, Nucl. Phys.

B 265, 630 (1986); M.L¨uscher and P. Weisz, Nucl. Phys.

B 290, 25 (1987); ibid. 295, 65 (1988); D. J. E. Callaway,Phys.

Rep. 167, 241 (1988). [7] T. Barnes and G. I. Ghandour, Phys Rev.

D 22, 924 (1980); S. J. Chang, ibid. 12, 1071(1975); G. Rosen, Phys.

Rev. 173, 1632 (1968); L. I. Schiff, ibid.

130, 458 (1963); J.M.Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev.

D10, 2428 (1974). A more completelist can be found in Refs.

[8, 9]. [8] P. M. Stevenson, Phys.Rev.

D 30, 1712 (1984). [9] P. M. Stevenson, Phys.

Rev. D 32, 1389 (1985).

[10] P. M. Stevenson, B. All`es, and R. Tarrach, Phys. Rev.

D35, 2407 (1987). [11] The existence of two distinct renormalizations can be understood in terms of distinct limitsas the spacetime dimension d tends to 4 from above or from below: see P. M. Stevenson,Z.

Phys. C 35, 467 (1987).

[12] P. M. Stevenson, Z. Phys. C 24, 87 (1984).

[13] S. Coleman, R. Jackiw, and H. D. Politzer, Phys. Rev.

10, 2491 (1974); L.F. Abbott, J. S.Kang, and H. J. Schnitzer, ibid 13, 2212 (1976); W. A. Bardeen and M. Moshe, Phys. Rev.D 28, 1372 (1983).

[14] M. Consoli and A. Ciancitto, Nucl. Phys.

B 254, 653 (1985). [15] P. M. Stevenson and R. Tarrach, Phys.

Lett. 176B, 436 (1986).22

[16] V. Branchina, P. Castorina, M. Consoli, and D. Zappal`a, Phys. Rev.

D 42, 3587 (1990); P.Castorina and M. Consoli, Phys. Lett.

235B, 302 (1990). [17] A. K. Kerman, C. Martin, and D. Vautherin, MIT preprint MIT-CTP-2082, April 1992.

[18] K. Huang, E. Manousakis, and J. Polonyi, Phys. Rev.

D 35, 3187 (1987). [19] U. Ritschel, I. Stancu, and P. M. Stevenson, Rice preprint DOE/ER/05096-47.

(to bepublished in Z. Phys. C).

[20] V. Branchina, P. Castorina, M. Consoli and D. Zappal`a, Phys. Lett.

B 274, 404 (1992);Proceedings of the Moriond meeting, March 1991; V. Branchina, N. M. Stivala and D.Zappal`a, Diagrammatic expansion for the effective potential and renormalization of masslessλφ4 theory, INFN, Catania preprint; M. Consoli, Large Higgs mass, triviality and asymptoticfreedom. Contributed paper to the Conference on Gauge Theories – Past and Future, inhonor of the 60th birthday of M. G. J. Veltman.

Ann Arbor, May 16-18, 1991. WorldScientific, to appear; M. Consoli, Spontaneous symmetry breaking and the Higgs mass,INFN, Catania preprint, June 1992; M. Consoli, Is there any upper limit on the Higgsmass?, INFN Catania preprint, July 1992.

[21] R. Iba˜nez-Meier and P. M. Stevenson, Autonomous renormalization of the one-loop effectivepotential and a 2 TeV Higgs in SU(2)×U(1), Rice preprint DOE/ER/05096-51, July, 1992(hep-ph 9207241). [22] I. Stancu and P. M. Stevenson, Phys.

Rev. D 42, 2710 (1990); I. Stancu, ibid 43, 1283(1991).

[23] A. Okopi´nska, Phys. Rev D 35, 1835 (1987).

[24] A. Duncan and M. Moshe, Phys. Lett.

215B, 352 (1988); H. F. Jones, Nucl. Phys.

B (Proc.Suppl.) 16, 592 (1990).

[25] S. K. Gandhi, H. F. Jones, and M. B. Pinto, Nucl. Phys.

B359, 429 (1991). [26] R. Iba˜nez-Meier, L. Polley, and U. Ritschel, Phys.

Lett. B279, 106 (1992); R. Iba˜nez-Meier,Rice preprint DOE/ER/05096-49.

[27] J. Ambjørn and R. J. Hughes, Nucl. Phys.

B 217, 336 (1983). [28] The factor of 4 in the λB term is included so that the O(2) λφ4 potential in Eq.

(3.5) hasthe exactly the same form as in Ref. [10].

[29] Y. Brihaye and M. Consoli, Phys. Lett.

157B, 48 (1985); Nuovo Cimento A 94,1 (1986).23

[30] N. Nakanishi, Prog. Theor.

Phys. 35, 1111 (1966); B. Lautrup, Kg.

Dansk. Vid.

Selsk.Mat.-fys. Medd.

35, 1 (1967). [31] R. P. Feynman, Phys.

Rev. 97, 660 (1955); Statistical Mechanics, Benjamin, New York,1972.

[32] P. M. Stevenson, Phys. Rev.

D23, 2916 (1981). [33] B. All`es and R. Tarrach, J. Phys.

A 19, 2087 (1986). [34] P. Cea, Phys.

Lett. B 165, 197 (1985).

[35] B. All`es, R. Mu˜noz-T`apia, and R. Tarrach, Ann. Phys.

204, 432 (1990); R. Mu˜noz-T`apiaand R. Tarrach, ibid, 468. [36] A. Kovner and B. Rosenstein, Phys.

Rev. D 40, 515 (1989).

[37] J.S. Kang, Phys.

Rev. D 14, 1587 (1976).

[38] J. S. Schwinger, Phys. Rev.

74, 1439 (1948), especially Sec. 2; J. M. Rauch and F. Rohrlich,The theory of photons and electrons, Springer-Verlag (New York, 1955).

[39] This parametrization is only appropriate if an SSB minimum exists, which need not be thecase if m20 is sufficiently large and positive.24

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