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RESOLUTION OF THE λΦ4 PUZZLE

arXiv:hep-ph/9303256v2 3 Aug 1993DE-FG05-92ER40717-5RESOLUTION OF THE λΦ4 PUZZLEAND A 2 TeV HIGGS BOSONM. ConsoliIstituto Nazionale di Fisica Nucleare, Sezione di Catania, Catania, ItalyCorso Italia 57, 95129 Catania, ItalyandP.

M. StevensonT. W. Bonner Laboratory, Physics DepartmentRice University, Houston, TX 77251, USAABSTRACTWe argue that massless (λΦ4)4 is “trivial” without being entirely trivial.It has anon-trivial effective potential which leads to spontaneous symmetry breaking, but theparticle excitations above the broken vacuum are non-interacting.

The key to this pictureis the realization that the constant background field (the mode with zero 4-momentum)renormalizes differently from the fluctuation field (the finite-momentum modes).Thispicture reconciles rigorous results and lattice calculations with one-loop and Gaussian-approximation analyses. Because of “triviality”, these latter two methods should be ef-fectively exact.

Indeed, they yield the same renormalized effective potential and the samerelation m2h = 8π2v2 between the particle mass and the physical vacuum expectationvalue.This relation predicts a Higgs mass mh ∼2 TeV in the standard model.Thenon-interacting nature of the scalar sector implies, by the equivalence theorem, that Higgsand gauge bosons interact only weakly, through their gauge and Yukawa couplings.

1INTRODUCTIONThe standard model of electroweak interactions [1] makes use of spontaneous symmetrybreaking (SSB) to explain the origin of vector-boson masses [2]. The traditional descrip-tion relies on an essentially classical treatment of a λΦ4 scalar sector, with perturbativequantum corrections.

In this picture the Higgs mass mh is proportional to the “renor-malized coupling” λR, so if the Higgs is heavy (mh ≥0.7 TeV or so), perturbation theoryclearly breaks down. However, in that case it is usually inferred that longitudinal gaugebosons would necessarily become strongly interacting at TeV energies [3].However, one should really ask if a full quantum treatment of the scalar sector can giveSSB, and if so what the consequences are.

Here the whole question is thrown open by thestrong evidence that the λΦ4 theory is “trivial” [4, 5, 6, 7]. Some authors interpret this tomean that the scalar sector of the standard model can only be an effective theory, validonly up to some cutoffscale.

Without a cutoff, the argument goes, there would be noscalar self-interactions, and without scalar self-interactions there would be no symmetrybreaking [8]. This view also leads to upper bounds on the Higgs mass [8, 9].However, symmetry breaking is not incompatible with “triviality”.

One could have anon-zero vacuum expectation value for the field, yet find only non-interacting, free-particleexcitations above the vacuum. We argue that this is what happens in pure λΦ4 theory.Our picture [10, 11, 12, 13] reconciles the evidence for “triviality” with the evidence for anon-trivial effective potential.A non-trivial effective potential with SSB [14] emerged naturally from an analysis ofthe Gaussian effective potential [15, 16] of λΦ4 theory.

More recently it was realized thatthe same results emerge from the one-loop effective potential [17]. The key features are“asymptotic freedom” (i.e., a flow of the bare coupling constant λB, as a function of thelattice spacing a, in which λ →0 as a →0, such that a macroscopic correlation length isobtained) and masslessness in the symmetric phase.Initially, it was thought [15] that the Gaussian-approximation results implied a fullynon-trivial, interacting theory.

However, since the effective potential contains informa-tion only about zero-momentum Green’s functions, it does not by itself provide informa-tion about the existence or otherwise of a momentum-dependent renormalized coupling“λR(Q2).”Consequently, these results do not necessarily imply the existence of non-trivial interactions in the “shifted” theory. (In fact, the finite-temperature generalization[18] implies that the shifted field is non-interacting, and apparent paradoxes in the exten-sion to the effective action, variously interpreted [19], may be seen as pointing to the same1

conclusion. )In this paper we elaborate the following picture for pure λΦ4 theory [10, 11, 20]:Writing the field Φ(x) as φ+h(x), we expect the fluctuation field h(x) to be non-interacting,but its mass will be proportional to the background constant field φ.

Thus, its vacuumenergy is φ dependent and this, together with the λφ4 term, produces a non-trivial one-loop effective potential, which should be exact, because there are no h-particle interactions.The renormalization requires the bare coupling constant to go to zero in the continuumlimit, and involves an infinite re-scaling of the φ field. The h(x) field, however, is notre-scaled, and this provides a simple way to understand why it is non-interacting.

Mostof the paper is concerned with pure λΦ4 theory, but we consider the implications for theelectroweak theory in the final section.2“TRIVIALITY” AND SSBAnalytical and numerical studies of (λΦ4)4 theory [4, 5, 6, 7] have overwhelmingly beendriven to the conclusion that it is a “generalized free field theory.” This means that allrenormalized Green’s functions of the continuum theory are expressible in terms of thefirst two moments of a Gaussian distribution [21]:τ(x) = v,(2.1)τ(x, y) = v2 + G(x −y),(2.2)so thatτ(x, y, z) = v3 + v(G(x −y) + G(x −z) + G(y −z)),(2.3)τ(x, y, z, w) = v4 + v2(G(x −y) + perm.) + G(x −y)G(z −w) + perm.,(2.4)and so on.

Here, v is a constant (since we assume that translational invariance is notbroken), and G(x −y) is just a free propagator with some mass mh and residue Zh = 1,since it must satisfy a K¨allen-Lehmann representation with spectral function δ(s −m2h).The index “h” in Zh and mh refers to the shifted field h(x) introduced by means of asuitably de-singularized, renormalized field operator ΦR(x), such that ⟨ΦR(x)⟩= v andh(x) ≡ΦR(x) −v. The above equations imply that all connected three- and higher-pointGreen’s functions of the h(x) field vanish; i.e., “triviality”.However, although the generalized free-field structure implies a trivially free shiftedtheory it does not forbid a non-zero value of v.Indeed, there are explicit studies of“triviality” in the broken phase (v ̸= 0) [7].This suggests that it is possible for the2

theory to have a non-trivial effective potential Veffwith SSB minima. Indeed, the latticecalculations of Huang et al [22] do find a non-trivial effective potential, even while findingno sign of particle interactions: they conclude that the theory cannot be entirely trivial[22, 23].

The effective potential, when expanded about the origin, generates the zero-momentum Green’s functions of the symmetric phase [24]. Thus, for the effective potentialto be non-trivial and give SSB, there has to be non-triviality in the zero-momentum-modesector of the symmetric phase, which somehow acts to de-stabilize the symmetric vacuum.This immediately suggests that we concentrate on massless λΦ4 theories, for whichzero-momentum (pµ = 0) represents a physical, on-shell point.

For a massless theory, anon-trivial effective potential would imply a non-trivial scattering matrix. There is, infact, good evidence that massless λΦ4 theory is non-trivial in this sense [25].

[In Ref. [25] the conformal symmetry of the massless λΦ4 theory is exploited, allowing a mappingof the Minkowski space into the Einstein universe.

Asymptotic states are restricted tothose which carry “Einstein quantum numbers”. In Minkowski space these are collec-tive states containing an indefinite number of massless particles.

As in the well-knownBloch-Nordsieck procedure in QED, such states are needed because of the serious infraredproblems. The effective potential also involves a sum of diagrams with an arbitrarily largenumber of external legs, with infrared divergences cancelling only in the sum [24].

]It is quite natural to expect that any non-trivial dynamics of the zero-momentummode would induce SSB: The classical potential of the massless theory is very flat at theorigin, and ripe for instability. Moreover, this would be the simplest way for the theoryto escape its infrared difficulties.

Since the massless theory contains no intrinsic scale,the physical scale, v (with mh proportional to v), must be spontaneously generated by“dimensional transmutation” from a theory with no intrinsic scale in its symmetric phase.This is exactly the philosophy of the classic paper of Coleman and Weinberg [24].“Dimensional transmutation” requires ultraviolet divergences to produce a non-zeroCallan-Symanzik β function. A familiar example is the origin of the Λ scale in QCD.However, in that case one can obtain the β function perturbatively, from the momentumdependence of the renormalized coupling constant.In (λΦ4)4 theory that approach isdoomed to failure since “triviality” means that any “renormalized coupling constant”,defined from the 4-point function at non-zero external momenta, must vanish.

(It isnot surprising, then, that the perturbatively defined β function is positive and has nofixed point. This implies an unphysical Landau pole at high energies – which can only beavoided by requiring the renormalized coupling at finite energies to vanish as the ultraviolet3

regulator is removed. In this sense, perturbation theory itself signals “triviality” [8].) Toextract a more meaningful β function one should start from a quantity that will be finite,and non-vanishing in the infinite-cutofflimit.

This is not true for the 4-point function.However, in our picture, Veffis non-trivial and one may extract from it a nonperturba-tively defined β function that characterizes the dependence of the bare coupling constanton the ultraviolet regulator. For the picture to be consistent we would expect this β func-tion to be negative, giving asymptotic freedom (λB →0 in the continuum limit), since itseems [4] that asymptotic freedom is the only possibility to avoid “entire triviality”.3THE EFFECTIVE POTENTIALTo find β from Veffwe follow the usual Renormalization-Group (RG) procedure.

First,we introduce a generic ultraviolet regulator “r” (which may be a cutoffΛ, or an inverselattice spacing 1/a, or one may identify ln r2 with24−d in dimensional regularization, etc. ),so that the effective potential depends on r, and on the bare classical field φB and on thebare mass and coupling parameters:Veff= Veff(φB, λB, m2B, r).

(3.1)The bare mass is treated as a counterterm for the quantum theory such that, in anyapproximation and in any regularization scheme, the condition [24]∂2Veff∂φ2BφB=0= 0(3.2)is satisfied. This ensures that the theory has no intrinsic scale in its symmetric phase.

(Indimensional regularization this simply requires mB = 0 and the classical scale invarianceof the bare Lagrangian is manifest.) After implementing the masslessness condition, (3.2),the effective potential depends only on r itself and on the bare parameters φB = φB(r)and λB = λB(r) whose flow, in the limit r →∞, is dictated by the requirement of RGinvariance; that is:limr→∞r ∂∂r + β ∂∂λB−γφB∂∂φBVeff(φB, λB, r) = 0,(3.3)where β ≡r dλBdrand γ ≡−rφBdφBdr .

This anomalous dimension γ should not be confusedwith the more conventional quantity associated with the p2-dependence of the two-pointfunction. (That quantity would be associated with the cutoffdependence of Zh, and must4

vanish as r →∞since Zh →1.) Here, φB is a constant background field, and has nodependence on p2.One can easily disentangle β from γ as follows [16].

If the effective potential has itsnon-trivial minimum at a particular φB = vB; i.e.,∂Veff∂φBφB=vB= 0,(3.4)where vB will depend on r and λB, then the vacuum energy density:W(λB, r) = Veff(vB(r,λB), λB, r)(3.5)will satisfy (from Eq. (3.3))limr→∞r ∂∂r + β ∂∂λBW(λB, r) = 0.

(3.6)This equation will determine β and one can then go back to (3.3) to extract γ.Acceptance of “triviality” in the sense of Eqs. (2.1–2.4) means that this procedurerepresents the only hope for obtaining a not-entirely-trivial continuum limit.

(It alsomeans that we may expect a contradiction if we try to use an approximation to theeffective potential that is inherently inconsistent with generalized-free-field behaviour. Weshall return to this important point in Sect.

6. )4THE ONE-LOOP EFFECTIVE POTENTIALNow let us see what happens in the simplest approximation scheme.

We first write thebare field asΦB(x) = φB + hB(x),(4.1)where we have separated out the zero-momentum component φB = constant (demanding,in order for the separation to be unambiguous, that the field hB(x) has no Fourier pro-jection on to pµ = 0). Consider the approximation in which the field hB(x) is allowed tointeract to all orders in λB with φB but has no non-linear interaction with itself.

To thislevel of approximation hB(x) is just a free field whose mass-squared isω2(φB) ≡12λBφ2B. (4.2)However, through its zero-point energy, it will produce a non-trivial contribution to theeffective potential.

To compute this we take the shifted, linearized Hamiltonian:Ho(φB) =Zd3⃗xλB4! φ4B + 12Π2h + 12(⃗∇h)2 + 12ω2(φB) h2,(4.3)5

and find its lowest eigenvalue, as a function of φB. This gives the effective potential timesa volume factor.

Dropping a φB-independent constant term, this gives:V 1−loop(φB)=λB4! φ4B + ω4(φB)64π2 ln ω2(φB)Λ2−12!,(4.4)=λB24 φ4B + λ2Bφ4B256π2 ln λBφ2B2Λ2−12!,(4.5)where we use an ultraviolet cutoffΛ, as in Ref.

[24].This is, of course, the familiar “one-loop effective potential” [24, 26, 27]. In Ref.

[24]it was obtained, using Feynman diagrams of the unshifted theory, from the sum of all one-loop 1PI graphs with external lines φB and massless propagators running round the loop. (This diagrammatic interpretation links the effective potential with the dynamics of theunderlying massless theory, as discussed earlier.) Although we use the traditional name,“one-loop”, for this approximation, we stress that it is the above linearization procedure,and not the loop expansion, that is our rationale for it.Taking this effective potential and following the well-defined procedure described inEqs.

(3.3–3.6) above, one finds straightforwardly [17, 10]:β = −3λ2B16π2 ,(4.6)γ = −3λB32π2 =β2λB. (4.7)This result is crucial, and the reader is urged to check it for him- or her-self.Thus, this β is indeed negative.

These RG functions allow the limit Λ →∞to betaken such that the vacuum energy density W = −m4h/(128π2) is finite and RG invariant.Hence, the particle mass:m2h ≡ω2(vB) = 12λBv2B = Λ2 exp(−32π23λB),(4.8)is finite and RG invariant. When re-arranged, this equation shows the asymptotic-freedomproperty explicitly:λB = 32π231ln(Λ2/m2h).

(4.9)Eliminating Λ in favour of vB, V 1−loop can be expressed as:V 1−loop(φB) = λ2Bφ4B256π2 ln φ2Bv2B−12!. (4.10)6

To make the finiteness and RG invariance of V 1−loop manifest we may re-express itin terms of a renormalized field φR = Z−12φφB, where an infinite Zφ absorbs the cutoffdependence. This Zφ must satisfy γ = −12ΛddΛ ln Zφ where γ is the anomalous dimensionof Eqs.

(3.3, 4.7). The absolute normalization of Zφ is fixed by requiring the physicalmass m2h of the fluctuation field hR(x) = hB(x) to be equal to the second derivative, atthe minimum, of V 1−loop with respect to φR.

(We shall return to justify this condition ina moment. )As shown in Refs.

[10, 12, 11] one obtainsZ1−loopφ= 16π2λB= 32 ln(Λ2/m2h),(4.11)so that (with v ≡vR = Z−12φvB)V 1−loop = π2φ4R ln φ2Rv2 −12!,(4.12)andm2h = 8π2v2,(4.13)as advertised earlier.5THE FIELD RE-SCALINGWe now want to discuss two apparently unconventional aspects of this analysis. Firstly,it is crucial to this picture that the Z1/2φre-scaling of the constant background field φBis quite distinct from the Z1/2h= 1 re-scaling of the fluctuation field h(x).

We can, infact, express this as a single, overall re-scaling of the whole field, provided that we use amomentum-dependent Z1/2(p):Z12 (p) = Z12φ P + Z12h P,(5.1)whereP ≡¯δ4(p)¯δ4(0)andP = 1 −P(5.2)are orthogonal projections (P2 = P, P2 = P, PP = 0) which select and remove the pµ = 0mode, respectively. [Here ¯δ4(p) ≡(2π)4δ4(p), and ¯δ4(0) has the usual interpretation as thespacetime volume.] Note that “pµ = 0” is a Lorentz-invariant statement so our momentum-dependent Z(p) does not violate any sacred principles.

In terms of the Fourier transform7

of the field operators it works as follows:Z12(p)˜ΦR(p) =Z12φ P + Z12h P φR¯δ4(p) + ˜hR(p)(5.3)= Z12φ φR¯δ4(p) + Z12h ˜hR(p) = φB¯δ4(p) + ˜hB(p) = ˜ΦB(p). (5.4)(Note that P˜h(p) = 0 by definition.) It is easy to check that Z−1/2(p), Z(p) = (Z1/2(p))2,etc., work properly.

For consistency, the renormalized momentum-space propagator of thecomplete ˜ΦR(p) field must be written as:φ2R¯δ4(p) +Pp2 −ω2(φR),(5.5)with ω2(φR) = 8π2φ2R. (At φR = v this corresponds to the Fourier transform of Eq.

(2.2). )The P projection makes no difference except in the symmetric phase (where ω2 and φ2Rvanish): the propagator is then formally P/p2, not the free, massless propagator 1/p2.This form may allow not-entirely-free behaviour, while still being compatible with theconstraints imposed by scale and conformal invariance (see Ref.

[28]).Secondly, recall that we fixed the absolute normalization of Zφ by requiring the physi-cal mass m2h = 12λBv2B to be equal to the second derivative of V 1−loop(φR) at its minimum.This arises from the well-known connection between the derivatives of the effective po-tential and the zero-momentum limit of the 1PI Green’s functions [24]. Here, in the SSBvacuum, the renormalized inverse propagator of the h field is just p2 −m2h, which tendsto −m2h as pµ →0.

This must agree with (minus) the second derivative of Veff(φR) at thevacuum.We can re-state the argument in more physical terms: For self-consistency – especiallyif we believe the h field to be truly a free field – the effective potential near its mini-mum should look like a free-field potential 12m2h(φR −v)2 with the same mass as in thepropagator. (This is extremely intuitive if one first thinks about the 0+1 dimensional,quantum mechanical case.) Furthermore, because fluctuations of h(x) that are finite onthe scale of φB are only infinitesimal on the scale of φR, it is quite reasonable that the h(x)field is only sensitive to the quadratic shape of the renormalized Veffin the infinitesimalneighbourhood of the vacuum.

Therefore, we can understand why h(x) behaves as a freefield, and thus close the circle of our arguments.In fact, we may show directly that h(x) is non-interacting. The bare 3-point and 4-point vertices of the shifted theory are proportional to λBφB and λB, respectively.

Theseare both infinitesimal, of order √ǫ and ǫ, respectively (where ǫ is 1/ ln Λ in cutoffregular-ization, or 4 −d in dimensional regularization). Any diagram with T 3-point vertices, F8

4-point vertices, and L loops can, at most, be of order (λBφB)T (λB)F (1/ǫ)L = ǫT/2+F −L.But it is an identity that T/2 + F −L = n/2 −1, where n is the number of external legs.Thus, all diagrams contributing to the n-point function are suppressed by a factor of atleast ǫn/2−1, and hence vanish for n ≥3 in the continuum limit. Thus, there are no hself-interactions, as a direct consequence of the fact that h(x), unlike the constant φ field,has no infinite re-scaling that can compensate for the infinitesimal strength of λB.

Thus,our results confirm our initial expectation that λΦ4 is a generalized free field theory.6BEYOND ONE-LOOP: EXACTNESS CONJECTURENow, if the h(x) field does not self-interact, then the effective action:Γ[ΦR] =Zd4x(12(∂hR)2 −12(8π2φ2R)h2R −π2φ4R ln φ2Rv2 −12! ),(6.1)that embodies our results (4.12, 4.13), ought to be exact [29].The shifted field h(x)has made a non-trivial contribution to Veffthrough its zero-point energy, but any furthermodifications would have to be due to its self-interactions – and physically it has none!This conclusion would not be immediately obvious diagrammatically, because thereare indeed vacuum diagrams at all orders that give 1/ǫ and finite contributions, as ourargument above (for n = 0) shows.

Similarly, for n = 2, we see that there are diagramsgiving finite contributions to the propagator in all orders. However, since the h(x) field hasno physical interactions, the only reasonable expectation is that the apparent “higher-ordercorrections” to Veffand to m2h either cancel or can be re-absorbed into the renormalizationconstants so as to leave the physical results (4.12, 4.13) unchanged.This conjecture is supported by the observation [10, 11] that exactly the same physicalresults (4.12, 4.13) for Veffand m2h are obtained in the Gaussian approximation.

Discardingterms that will vanish in the limit Λ →∞, the Gaussian effective potential for the masslesscase can be expressed as [10, 11]:V Gauss = λG4! φ4B + Ω4(φB)64π2 ln Ω2(φB)Λ2−12!,(6.2)with Ω2(φB) = 12λGφ2B and λG = 23λB.

This has the same structure as the one-loop result,Eq. (4.4).

After eliminating Λ in favour of vB, and then re-scaling the bare vacuum fieldby ZGaussφ= 16π2λG= 24π2λB , the two approximations are seen to be physically equivalent.There are differences by factors of 2/3 in the unobservable quantities λB and Zφ, but thesedifferences cancel out in the physically meaningful results (4.12, 4.13) [10, 11].9

Notice that both the 1-loop and Gaussian approximations have a variational character:the one-loop potential is the ground-state energy of that part of the Hamiltonian which isquadratic in the shifted field, while the Gaussian approximation corresponds to minimizingthe expectation value of the full Hamiltonian in the subspace of Gaussian wavefunctionals.The Gaussian approximation can also be described as a re-summation of an infinite subset,a convergent subseries, of ‘daisy’ and ‘superdaisy’ diagrams [30, 31].Other truncations of the diagrammatic series for the effective potential, if they do notcorrespond to any variational procedure, and hence do not enjoy a stability property, maygive rise to spurious difficulties. Apparent contradictions will inevitably arise if one triesto use an approximation method that is inherently incompatible with generalized-free-fieldbehaviour, Eqs.

(2.1–2.4). This is the case with the usual loop expansion beyond one loop.In other words, one should not consider the β and γ functions in Eqs.

(4.6, 4.7) asthe first terms of a power-series expansion in λB which can be systematically improvedorder by order in the loop expansion for the effective potential.Rather, the form ofEqs. (4.6, 4.7) will arise in any approximation scheme, no matter how sophisticated, thatallows generalized free-field behaviour to be a possible solution.

The 2-loop, 3-loop, ...approximations to the effective potential, on the other hand, have a built-in incompatibilitywith generalized free-field behaviour (think of the spurious Zh ̸= 1 effect, starting at the2-loop level, for instance) and represent merely a perturbative construction.This point can best be understood in terms of the effective potential for compositeoperators introduced by Cornwall, Jackiw and Tomboulis [27] (CJT), which is based onthe rigorous definition of the effective potential through the exact relationZd3x Veff(φ) = E[φ, Go(φ)],(6.3)where E[φ, G] = min⟨Ψ|H|Ψ⟩, minimized over all normalized states |Ψ⟩, subject to theconditions ⟨Ψ|Φ|Ψ⟩= φ and ⟨Ψ|Φ(⃗x, t)Φ(⃗y, t)|Ψ⟩= φ2 + G(⃗x, ⃗y), and where Go(φ) isobtained fromδEδG(⃗x, ⃗y)G=Go(φ)= 0. (6.4)Equation (6.4) can be solved exactly for Go in the subspace of Gaussian states and inthat case Eq.

(6.3) leads to the Gaussian effective potential. The problem can also beapproached diagrammatically, using CJT’s result that E[φ, G] has a manifestly covariantexpansion containing the one-loop term and the series of 2-particle-irreducible vacuumgraphs with propagator G and vertices given by the shifted interaction Lagrangian.

How-ever, in an n-loop approximation to E[φ, G] (n ≥2), one cannot exactly solve the resulting10

optimization equation, (6.4). One can only solve it in a perturbative sense.

Since this doesnot provide a true stationary solution for G what one obtains is really not an effectivepotential, except in a perturbative sense. Consequently, one should not attempt to applythe RG equation, (3.3), to the 2-loop, 3-loop, ... approximations to the effective potential.That would be similar to trying to define β through the perturbative 4-point functionat non-zero external momenta and, as we explained in Sect.

2, that cannot produce aconsistent continuum limit.It is possible, in principle, to consider truncations of the CJT construction that improveupon the one-loop or Gaussian approximations, yet still allow the resulting optimizationequation, (6.4), to be solved exactly. For example, one can consider post-Gaussian vari-ational calculations (either Hamiltonian [32] or covariant [33]).In such a calculation,however, we would expect the optimal G to reduce to a free propagator, and our equa-tions (4.12, 4.13) to be unmodified in the continuum limit.

Any other result would providestrong evidence that (λΦ4)4 is not, in fact, a generalized free field theory. Therefore, if“triviality” is true, as we believe, then Eqs.

(4.12, 4.13) should be considered exact.7FINITE TEMPERATUREWe briefly discuss finite-temperature effects in order to make two points: (i) our λΦ4theory, even though it has no particle scattering, is not entirely trivial; it has non-trivialcollective effects, and (ii) our renormalized v has real physical significance.Since the h(x) field is a non-interacting boson field with mass ω2 = 12λBφ2B = 8π2φ2R,the only finite-temperature correction to Veffwill be the term [30]:Iβ1 = 1βZd3p(2π)3 ln(1 −exp{−β(p2 + ω2)1/2}),(7.1)where β = 1/T, and T is the temperature in units where Boltzmann’s constant is unity.This term, since it modifies a non-trivial effective potential, produces non-trivial effects.In particular, one finds that the theory undergoes a symmetry-restoring phase transitionat a critical temperature Tc of order v. Note that it is the renormalized v, not vB, thatsets the scale for the physical observable Tc. This is because the depth of the SSB vacuumrelative to the symmetric vacuum (which is invariant under re-scalings of φ) is of order v4.The Gaussian approximation has a very simple generalization to the finite-temperaturecase, and after renormalization it yields exactly the same result as above, because otherwould-be contributions are infinitesimal [18].

Thus, we may take over the results fromRef. [18], converting the notation appropriately [34].

The critical temperature is Tc =11

0.51394 (8π2/e)1/2 v = 2.7699 v. The transition is first order with a barrier height thatis about a quarter of the original depth of the SSB vacua relative to the origin. Beforethe phase transition the SSB minima of the finite-temperature effective potential moveslightly inward from their zero-temperature positions at ±v.

Because of this there is aslight (5%) decrease in the particle mass between T = 0 and T = Tc, with most of thisdecrease occuring close to Tc [18].8REMARKS ON THE PERTURBATIVE PICTUREWe have insisted on being able to take the continuum limit. However, if one is preparedto keep the ultraviolet cutofffinite, so that a finite “λR” exists, then one can proceedperturbatively, and here we make contact with the classic analysis of Coleman and Wein-berg [24].

Their perturbative renormalization λB = λR(1 + O(λR)) leads to perturbativeexpressions for β and γ [24]:βpert = + 3λ2B16π2andγpert = O(λ2B),(8.1)which allow a perturbative solution of the RG equation (3.3), for finite ultraviolet cutoff,valid only up to higher order terms in λR. At one-loop order there is then no wavefunctionrenormalization, so that hB = hR and φB = φR.

Making an “RG improvement” usingβpert produces a re-summation of the “leading log” series in x, defined asx = 3λB32π2 ln 2Λ2λBφ2B,(8.2)yielding a perturbative, running coupling constant:λR(φ2R) =λB1 + 3λB32π2 ln Λ2φ2R. (8.3)(At this level of approximation one may drop the λB/2 factor in the logarithm; it makesonly a sub-leading-log difference.) The resulting “leading-log effective potential” has noSSB minima.

However, this leading-log re-summation procedure is questionable and byno means unique. The “leading-log” series has the form 1−x+x2 −x3 +..., and convergesto11+x only for |x| < 1.

However, the SSB minimum of the original one-loop effectivepotential was precisely at x = 1, so it is not surprising that it can spuriously be made todisappear if one extends the re-summed expression into the region x ≥1.Still, one could argue that the re-summation is trustworthy in the region where φBis not too small and define the “leading-log effective potential” over the whole range of12

φB by analytical continuation. One would start with a very small bare coupling constantλB16π2 ≪1 and a very large cutoffΛ such that x ∼1 but λBx ≪1 so that the importanteffects are restricted to the leading-log sector.

Then, employing βpert and γpert one wouldfind that all the leading logs are contained in the running coupling constant λR(φ2R).Consequently, one would end up with a perturbatively renormalized “leading log effectiveaction” that is just the shifted (Φ →φ+h(x)) classical action, with λB replaced by λR(φ2R).However, if one tried to take the cutoffto infinity with βpert governing λB as a functionof Λ, one would inevitably be drawn into a region where λB is not small. Then λR(φ2R)would be driven to zero at any finite φ2R.

Alternatively, we can say that the only regionwith non-vanishing λR(φ2R) in which we can trust the leading-log re-summation is at largeφ2R, of order Λ2. In any finite range of φ2R one cannot justify neglecting “sub-leading”logarithms.The crux of the matter is the qualitative conflict between the one-loop effective poten-tial itself and its “re-summed” or “RG-improved” form.

This unhappy situation can beavoided by going to theories such as scalar electrodynamics. There, as Coleman and Wein-berg argue, there is no doubt that the leading-log re-summation is valid, for sufficientlysmall λ and gauge coupling e. Our type of analysis, though seemingly very different, actu-ally leads to the same physical consequences in this perturbative region [12, 13].

However,outside the perturbative region, the relation between e2B and λB necessary for renormaliz-ability ceases to be of the form e4B proportional to λB. In fact, e2B goes back to zero as λBapproaches the pure-λΦ4-theory value, Eq.

(4.9). Thus, for small eB our approach yieldstwo solutions; one perturbative, and one close to pure λΦ4 [12].

The perturbative solutionyields the 10 GeV Higgs of Coleman and Weinberg, now excluded by experiment.9IMPLICATIONS FOR ELECTROWEAK THEORYWe now consider the implications of this picture for the standard model of electroweakinteractions. This uses four scalar fields in a complex isodoublet:K(x) = 1√2(χ1(x) + iχ2(x), v + h(x) + iχ4(x)),(9.1)where one field χ3(x) = v + h(x) has a non-zero vacuum expectation value.

When thisform is substituted into the tree-level scalar-vector couplings, it generates mass terms forthe gauge fields. Hence, there is a direct relation between v and the Fermi constant GF ;namely, v ∼(√2GF )−1/2 ∼246 GeV.

Thus, v is a physical, measurable quantity, andrepresents the phenomenological vacuum value of the scalar field. The physical origin of13

its non-zero value, however, is ascribed to a presently untested part of the theory, namelythe “Higgs potential”. In textbook treatments the Higgs potential is treated classically andthere is no ‘bare/renormalized’ distiction.

However, in our approach the SSB arises onlyafter the full quantum-dynamical content of the scalar sector has been taken into account.Therefore, in our approach, K(x) represents the O(4) extension of our renormalized fieldΦR(x) (i.e. the Fourier transform of ˜ΦR(p) introduced in Sect.

5). This equals vR + h(x)when evaluated at the minimum of the effective potential.

The v in Eq. (9.1) is thusto be identified with vR, the renormalized vacuum expectation value, i.e., the one whichincludes the full dynamical content of the scalar sector.With v = vR identified with(√2GF )−1/2 ≈246 GeV our result m2h = 8π2v2 gives a Higgs mass prediction of 2.19 TeV.Although the scalar sector must be analyzed and renormalized nonperturbatively, onemay continue to treat the gauge and Yukawa sectors perturbatively; that is gR = gB(1 +O(g2B)).Hence, one obtains (up to small corrections from µ-decay) the renormalizedrelation M2W = 14g2Rv2R, where gR is the running SU(2) coupling constant evaluated at theW mass scale.

In the presence of the gauge and Yukawa couplings, there will be radiativecorrections to the effective potential and to the Higgs mass, but these are small providedthe top mass is less than 200 GeV [10, 11]. [Note that this attitude is not quite the same as in Refs.

[12, 13]. In those papers,which deal only with the effective potential, the gauge coupling was also renormalized ina nonperturbative manner.

We believe that there is ultimately a duality between the twoapproaches. However, for practical purposes – since we know that perturbation theoryworks well in QED and weak interactions – the present attitude is more useful.

]Our mass prediction m2h = 8π2v2 ≈2.19 TeV comes from considering λΦ4 theory fora single scalar field, whereas the standard model involves the O(4) generalization. In theO(4) λΦ4 case the one-loop and Gaussian results are not quite identical: one-loop givesmh = 1.89 TeV while the Gaussian approximation gives mh = 2.05 TeV [12, 13].

However,this difference is probably attributable to an inexactness inherent in using “Cartesian-coordinate” fields. Really the Goldstone fields should be described by angular, “polar-coordinate” fields.

One can argue that the exact result in the O(N) case should be justthe N = 1 result [10, 11]; i.e., that only the radial field affects the shape of the effectivepotential. This idea is motivated by the approach of Ref.

[35] where the functional integralis expressed in polar coordinates; the angular integration gives only a constant term andthe non-trivial Jacobian is handled by ghost fields [36].In any case, one predicts a Higgs mass mh ∼2 TeV in the Standard Model. More-14

over, despite its large mass, the Higgs would be narrow, decaying principally to t¯t, andthere would be no strong interactions among longitudinal gauge bosons. This follows,by the “equivalence theorem” (see below), from the fact that the scalar sector is non-interacting in the absence of gauge couplings.

Now, it might seem at first that “triviality”for broken-phase O(N) theory would be at odds with current algebra, which prescribesdefinite derivative couplings for Goldstone bosons. However, this conflict can be recon-ciled by the fact that vB is infinite, which causes the current-algebra interactions in O(N)λΦ4 theory to be infinitely suppressed.

Thus, both sides of an “Adler-Weisberger” sumrule would be infinitesimal: the scattering cross sections are infinitesimal, reflecting “triv-iality”, while the other side of the equation is supressed by a 1/v2B factor. The scalarself-interactions disappear because vB is infinite, but it is the finite vR that sets the scaleof the symmetry breaking, and governs Tc and the particle masses.Our picture is perfectly compatible with the “equivalence theorem” [37, 38], whosephysical content, paraphrasing Sect.

II of Ref. [38], is the following: For zero gauge cou-pling(s), g, the Goldstone bosons are physical particles while the longitudinal componentsof the W’s are free, unphysical degrees of freedom, included just to maintain manifestLorentz covariance.We call this the “g = 0 theory”.On the other hand, for g ̸= 0,however small, the longitudinal W’s are physical while the Goldstone bosons are now un-physical particles, included just to preserve renormalizability.

This situation we call the“g →0 theory”. The requirement that the physical observables of the two theories arethe same in the limit g →0 implies an equivalence between the physical longitudinal W’sof the “g →0 theory” and the physical Goldstone bosons of the “g = 0 theory.” Thisstatement is made precise by the theorem, which is valid to lowest non-trivial order ing and to all orders in the scalar self-interaction [37, 38].

Thus, presumably the theoremremains valid for a nonperturbative scalar sector. In our picture the g = 0 theory hasnon-interacting Higgs and Goldstone particles, so in the small-g theory we expect onlyweakly interacting Higgs and longitudinal gauge bosons.

So, at low energy scales, the onlytrace of the Higgs particle is represented by the logarithmic one-loop correction to the ρparameter discovered by Veltman [39].In our picture there is just no such thing as a “renormalized λ”, and the quasi-classicalrelation “m2h ∝λRv2”, implying a proportionality between the Higgs mass and its self-coupling, is completely misleading. As has been pointed out by Huang [23, 22], the ratiom2h/v2 is not a measure of the effective scalar coupling strength.ACKNOWLEDGEMENTS15

One of us (M.C.) would like to thank A. Agodi, R. Akhoury, M. Einhorn, P. Federbush,K.

Huang, R. Jackiw, G. Kane, Y. Tomozawa, M. Veltman and Y. Yao for very usefuldiscussions. This work was supported in part by the U.S. Department of Energy underGrant No.

DE-FG05-92ER40717.16

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