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THE EFFECTIVE POTENTIAL AND THE RENORMALISATION GROUP

arXiv:hep-lat/9210033v1 27 Oct 1992LTH 288UM-TH-92-21THE EFFECTIVE POTENTIAL AND THE RENORMALISATION GROUPC. Ford, D.R.T.

Jones, P.W. StephensonDAMTP, University of Liverpool, Liverpool L69 3BX, UKandM.B.

EinhornRandall Laboratory, University of Michigan, Ann Arbor, MI 48109-1120, USAAbstractWe discuss renormalisation group improvement of the effective potential bothin general and in the context of O(N) scalar φ4 and the Standard Model. In thelatter case we find that absolute stability of the electroweak vacuum implies thatmH ≥1.95mt −189 GeV , for α3(MZ) = 0.11.

We point out that the lower boundon mH decreases if α3(MZ) is increased.1

1. Introduction.The effective potential V (φ) plays a crucial role in determining the nature of thevacuum in weakly coupled field theories, as was emphasised in the classic paperof Coleman and Weinberg(CW.

)[1] The loopwise perturbation expansion of V isreliable only for a limited range of φ ; but, as was recognised by CW, it is possibleto extend the range of φ by exploiting the fact that V satisfies a renormalisationgroup (RG ) equation. It is therefore possible to show that in massless λφ4, V (φ)has a local minimum at φ = 0, while massless scalar QED has a local minimumfor φ ̸= 0.Let us review briefly how V is calculated in perturbation theory, using the(functionally derived) elegant method of Jackiw.

[2] In general one shifts scalar fields:φ(x) →φ + φq(x), where φ is x-independent. Then V (φ) is given by the sum ofvacuum graphs with φ-dependent propagators and vertices.

It is not immediatelyobvious from this algorithm what the result for the one loop calculation is; partlyfor this reason, some authors have preferred to consider graphs with one φq-leg,which, it is easy to show, lead to a determination of ∂V/∂φ.All this is veryfamiliar; not so well known, perhaps, is the following point. Jackiw’s algorithmin conjunction with a specific subtraction scheme ( such as MS or MS ) leads toan expression for V (φ) such that V (0) is well defined and calculable: and also, ofcourse generally ignored.

Our point is that unless V (0) is specifically subtracted(or otherwise dealt with,) then V (φ) fails to satisfy a RG equation of the usualform. This fact was noted, for example, in Ref.

3, but has often been overlooked,leading to incorrect “solutions” to the RG equation. This happens because the formof the solution transmogrifies the apparently trivial V (0) term into a φ-dependentquantity.

We will see how this comes about in sections (2) and (3) where we discussvarious strategies for dealing with V (0), and their consequences. We will also arguethat it is in fact simpler to use the RG equation for ∂V/∂φ, since this leads to an“improved” form of V that removes the necessity of considering “improvement” ofV (0).2

In subsequent sections we explore various forms for the RG equation for bothV and ∂V/∂φ for various field theories. We consider in detail scalar φ4 theory,with particular emphasis on the impact of infra-red divergences on the domain ofvalidity of the solution.

We also consider the standard model, where the behaviourof V at large φ is important since it can affect the stability of the electroweakvacuum. Here we improve (in principle) on previous treatments[4,5] by our use ofa correct form of the RG solution, and also by use of a correct form of the 2-loop β-function for the Higgs self coupling[6] ; but, as is easy to anticipate, theanalysis of Ref.

5 should not be materially affected. Interestingly, however, we finddependence on α3(MZ) that differs significantly from that given in Ref.

7.2. The renormalisation group equation for V.In what follows we consider the RG equation in renormalisable field theorieswith a single renormalisation scale µ and couplings λi of dimension δi.Thusthe set λi consists of all masses and coupling constants, both dimensionless anddimensionful.

In general, V is a function V (µ, λi, φa) where φa represents all thescalar fields. In many cases, however, symmetries may be exploited so that V maybe calculated as a function of a single field φ.

This is the case in the standardmodel, for example.In more complicated cases (involving supersymmetry, forinstance) one frequently chooses to explore a specific direction in φ-space.Ofcourse ultimately one must then be able to argue that the absolute minimum ofV is indeed in the chosen direction. (This is not always a trivial matter.

[8]) In anyevent, we will assume for simplicity that it is sufficient to consider the case of asingle φ-field only.It is straightforward to derive the RG equation satisfied by V , but there is onesubtlety. If we calculate V according to the procedure outlined in the previoussection, then the result ˆV (µ, λi, φ) is such that ˆV (µ, λi, 0) is a non trivial func-tion that receives contributions from all orders in perturbation theory.

(In factˆV (µ, λi, 0) may well have an imaginary part if φ = 0 is not a local minimum of3

the tree potential, but let us imagine for the moment that this problem does notarise). Thus we may writeˆV (µ, λi, φ) = ˆV (µ, λi, 0) −∞Xn=11n!φnΓ(n)(pi = 0)(2.1)where Γ(n) represents the 1PI Green’s function with n φ-legs and all external mo-menta set equal to zero.

Then by virtue of the RG equation satisfied by Γ(n), wehaveD ˆV −γφ∂ˆV∂φ = DΩ(2.2)where we have denoted ˆV (µ, λi, φ) by ˆV and ˆV (µ, λi, 0) by Ω. The operator D isD = µ ∂∂µ + βi∂∂λi.

(2.3)Ωis simply a contribution to the vacuum energy on which, outside of gravity, noobservable can depend. Accordingly, we can make a φ-independent shift in V , ie.ˆV →V = ˆV + Ω′(µ, λi) then by choosing Ω′ so thatDΩ′ + DΩ= 0(2.4)we can arrange thatDV −γφ∂V∂φ = 0(2.5)which is the usual RG equation for the effective potential.

Thus the RG equationrestricts the form of the “cosmological constant” Ω+ Ω′ and leads to observableconsequences when we presently consider RG “improvement” of V .On the assumption that we want a potential that satisfies eq. (2.5), then whatis the appropriate choice of Ω′?

The obvious choice is of coursei)Ω′ = −Ω. (2.6)This was advocated, for example, in Ref.

3. Its defect, however, is that as men-tioned above V may have an imaginary part at the origin.

A suitable generalisation4

to the case when the minimum of V lies at non-zero φ is given by[9]ii)Ω′ = −ˆV (φ)φ=v(2.7)where v is the value of φ at its minimum. (If V has more than one local minimumthen any one will give a well defined V satisfying eq.(2.5)).

It is a simple exerciseto show that Ω′ as given by eq. (2.7) satisfies the equationDΩ′ = DΩ−∂ˆV∂φφ=v (γv + Dv)(2.8)so that indeed Ω′ satisfies eq.

(2.4) since by definition∂ˆV∂φφ=v= ∂V∂φφ=v= 0. (2.9)Note that this choice of Ω′ corresponds to setting the cosmological constant to zeroorder by order in perturbation theory.A third possibility which is relevant to some recent work of Kastening .

[10,11] isto chooseiii)Ω′ = Ω′(λi). (2.10)That is, to choose Ω′ to be independent of µ.

To leading order Ω′ is thereforeobtained by solving the equationβi∂Ω′∂λi= −µ∂Ω∂µ =132π2STr M4 φ=0(2.11)where STr is a spin-weighted trace and M2 is the mass matrix for the quantumfields as a function of φ. In section (5) we will construct the solution to eq.

(2.11)for the O(N) scalar case and compare the result with Ref. 11.3.

Solutions to the renormalisation group equation.5

In this section we consider the solution to various forms of the RG equation forV , and show how these solutions can be used to extend the domain of perturbativebelievability (in φ) of the result: or equivalently, sum the leading (and subleading...)logarithms. We suppose that V satisfies the equationDV −γφ∂V∂φ = 0.

(3.1)Straightforward application of the method of characteristics leads to the solutionV (µ, λi, φ) = Vµ(t), λi(t), φ(t)(3.2)whereµ(t) = µet(3.3)φ(t) = φξ(t)(3.4)andξ(t) = exp−tZ0γλi(t′)dt′. (3.5)λi(t) are the usual running couplings and masses, determined by the equationsdλi(t)dt= βiλ(t)(3.6)subject to the boundary conditions λi(0) = λi.

It is sometimes more convenient touse dimensional analysis to recast eq. (3.2) as follows:-¯DV −4¯γV = 0(3.7)6

where¯D = µ ∂∂µ + ¯βi∂∂λi(3.8)and¯βi = (βi + δiλiγ)/(1 + γ)¯γ = γ/(1 + γ). (3.9)Here δi is the dimension of the coupling λi.

The solution of eq. (3.7) isV (µ, λi, φ) = ¯ξ(t)4V (µ(t), ¯λi(t), φ)(3.10)where µ(t) is as in eq.(3.3).

¯ξ(t) and ¯λi(t) are defined as in eq. (3.5) and (3.6) butwith γ →¯γ, β →¯β and λ(t) →¯λ(t).

The absence of a ∂/∂φ from eq. (3.7) accountsfor the fact that φ rather than φ(t) appears on the right-hand side of eq.

(3.10).Either form of the solution may be employed with equivalent results; let us focusfor the moment on eq.(3.10). Let us denote V (µ(t), ¯λi(t), φ) as V (t, φ) for short.Now suppose we wish to calculate V (µ, λi, φ) ( ≡V (0, φ)) for some µ, say, 100 GeV.The key to the usefulness of the RG is that we can choose a value of t such thatthe perturbation series for V (t, φ) converges more rapidly (for certain φ) than theseries for V (0, φ).

Moreover, there is nothing to stop us choosing a different valueof t for each value of φ. Now the perturbation series for V is characterised at largeφ by powers of the parameter λ ln(φ/µ) where λ is some dimensionless coupling.Then clearly perturbation theory is improved if we choose t such that µ(t) ∼φ,as long as λ(t) remains small.

The precise domain of applicability of the solutionfor a given choice of t depends on the details of the theory: in section (5) we willconsider in detail the case of O(N) φ4 theory.Meanwhile, however, let us consider the relevance of the above discussion tothe issue of the subtraction term Ω′(µ, λi) introduced in the previous section. Theimportant point we wish to make here is that whichever procedure we use todefine Ω′, and whether we use the RG solution eq.

(3.2) or eq. (3.10), a choice of tdependent on φ renders Ω′ a function of φ and hence no longer a trivial subtraction.7

This point has been missed in some previous treatments of the RG solution and isimplicit in the treatment of Kastening.It is evident that, with regard to extending the domain of perturbative calcu-lability, one must take into account the behaviour of Ω′µ(t), λi(t)although, sinceit depends on φ only through t, this is unlikely to pose a problem at large φ, forexample. But we can, in fact, finesse this issue altogether by beginning with theRG equation for V ′ ≡∂V/∂φ instead of the one for V (φ), the point being that∂V∂φµ, λi, φ= ∂ˆV∂φµ, λi, φ(3.11)so that the Ω′ term simply does not arise.

The analog to eq. (3.1) isDV ′ −γφ∂V ′∂φ = γV ′(3.12)with solutionV ′(µ, λi, φ) = ξ(t)V ′µ(t), λi(t), φ(t)(3.13)while the analog to eq.

(3.10) is simplyV ′(µ, λi, φ) = ¯ξ(t)4V ′µ(t), ¯λi(t), φ(3.14)since V ′ evidently obeys an RG equation of the same form as eq.(3.7).4. φ4 theory: the N=1 case.In this section we apply the formalism developed in the previous two sectionsto the case of massive λφ4 theory, defined by the LagrangianL = 12(∂µφ)2 −12m2φ2 −λ24φ4.

(4.1)8

ˆV (φ) is given by the loopwise expansionˆV (φ) = ˆV0 + ˆV1 + ˆV2 + ...(4.2)whereˆV0 = 12m2φ2 + λ24φ4(4.3)andˆV1 = κ14H2ln Hµ2 −32. (4.4)In eq.

(4.4), H = m2 + 12λφ2, κ ≡(16π2)−1, and we are using MS as we dothroughout. (The result for ˆV2 may be found in Ref.

12. )At the one loop level the relevant RG functions are given byβ(1)λ= 3λ2κ,β(1)m2 = m2λκ,γ(1) = 0(4.5a, b, c).By virtue of eq.

(4.5c) the two forms of the RG solution are identical, and we haveV (µ, λ, m2, φ) =Ω′µ(t), λ(t), m2(t)+ 12m2(t)φ2 + 124λ(t)φ4+ κ4H2(t)ln H(t)µ2(t) −32+ ...(4.6)where H(t) = m2(t) + 12λ(t)φ2,λ(t) = λ(1 −3λtκ)−1(4.7)andm2(t) = m2(1 −3λtκ)−1/3. (4.8)The function Ω′ depends on the choice made to achieve a V satisfying the RGequation as explained in section (2).

With choice (iii), ie Ω′(µ, λ, m2) = Ω′(λ, m2)9

it is easy to show using eq. (2.11) thatΩ′(λ, m2) = −m42λ + cm4λ−2/3.

(4.9)where c is an arbitrary constant. Notice, that when m2 and λ become t-dependentin accordance with eq.

(4.6)-(4.8) the c-term above remains t-independent andtherefore harmless; so we may set c = 0.Note that this choice of Ω′ has thecurious feature that in the free field limit (λ →0) it corresponds to an infinitevacuum subtraction. We will return later to the consequences of choice (ii) forΩ′; for the time being let us persist with eq.(4.9).

With this Ω′, in fact, eq. (4.6)essentially reproduces the leading logarithms sum of Kastening (eq.

(25) of Ref. 11).

The natural choice of t from the point of view of eq. (4.6) is given by the equationµ2(t) = µ2e2t = m2(t) + 12λ(t)φ2(4.10)since this evidently removes the ln(H/µ2) terms to all orders.

An alternative choicewhich enables us to make contact with Kastening’s work is to choose†µ2(t) = µ2e2t/¯h = m2 + 12λφ2(4.11)which is a less implicit definition of t inasmuch as nowt = ¯h2 ln m2 + 12λφ2µ2. (4.12)Now we show how the various leading logarithm (subleading logarithm....) sumscollected in Kastening’s functions f1, f2 etc.

are in fact subsumed in our solution. (We choose now to work with eq.

(3.2) rather than eq.(3.10).) We need to expand† For the purposes of this discussion we found it convenient to write in the factors of ¯hexplicitly.10

the solution V (µ(t), λ(t), m2(t), φ(t)) in powers of ¯h but retaining all orders in t.Thus, from the expression for βλ incorporating two-loop corrections:dλ(t)dt= 3λ2(t)κ −173 ¯hλ3(t)κ2 + .....(4.13)it is easy to show thatλ(t) = λ(1 −3λtκ)−1 + 179 ¯hλ2κ(1 −3λtκ)−2 ln(1 −3λtκ) + O(¯h2). (4.14)Similarly we can evaluate m2(t), φ(t) and ξ(t) through two loops.

The relevanttwo-loop contributions to the RG functions areβ(2)λ= −173 ¯h2λ3κ2,β(2)m2 = −56m2¯h2λ2κ2,γ(2) = 112¯h2λ2κ2. (4.15a, b, c)Using these results we getm2(t) =m2(1 −3λtκ)−1/3+ ¯hm2(1 −3λtκ)−4/3h1727κλ ln(1 −3λtκ) + 1917λ2tκ2i+ O(¯h2)φ(t) =φ −112¯hλ2tκφ(1 −3λtκ)−2 + O(¯h2)ξ(t) =1 −112¯hλ2tκ(1 −3λtκ)−2 + O(¯h2).

(4.16)Using the formulae for λ(t), m2(t), φ(t) and ξ(t) together with eq. (3.2) or (3.13)one can sum the leading (subleading...) logarithms in V (φ) or V ′(φ) respectively.The sum of the leading logarithms is given by the ¯h0 term in (3.2):L1 = 12m2φ2(1 −3λtκ)−1/3 + 124λφ4(1 −3λtκ)−1 −m42λ (1 −3λtκ)1/3.

(4.17)With t defined as in eq. (4.12) this is identical to the result of Ref.

10. To sum thesubleading logarithms one simply takes the O(¯h) contribution to (3.2).

(Note thatwe would need to calculate the one loop contribution to Ω′).11

We have gone through this exercise to demonstrate how the results of Refs. 10,11 maybe recovered directly from the solution of the RG equation.

The analysis is foundedon choice (iii) for Ω′, which, as we have already indicated, we find somewhat ar-tificial, particularly with regard to the free field limit. In addition, in more com-plicated theories with many couplings the determination of the Ω′(λi) satisfyingeq.

(2.11) becomes onerous. We could choose to adopt choice (ii); it is easy to see,however, that the result will then include terms of the form H′2 ln H′/µ2(t) whereH′ = m2(t) + 12λ(t)⟨φ⟩2.

Although such terms are not dangerous at large φ sincethey do not grow as φ4, they do lead to an unwieldy form of the solution. Witha view to more complicated theories , it appears to us simpler, as we indicatedalready, to work with V ′ = ∂V/∂φ.

Then through one loop we have (from eithereq. (3.13) or (3.14)) simplyV ′ = m2(t)φ + 16λ(t)φ3 + κ2λ(t)φH(t)ln H(t)µ2(t) −1+ ...(4.18)We now evaluate V ′ and hence (numerically) V with t defined as in eq.(4.10).

(Notethat since t depends nontrivially on φ, the result for V differs from that obtainedfrom the equivalent RG equation for V itself). For m2 > 0 and sufficiently smallλ, the result differs insignificantly from the tree result for φ < O(µe1/λκ), whichcorresponds to the approach of λ(t) to the Landau pole.

For m2 < 0 there is thefact that for H = m2 + 12λφ2 < 0 the “unimproved” potential develops an imagi-nary part, and there is no solution for t to eq.(4.10). Discussion of the imaginarypart notwithstanding, it is clear that perturbation theory is not to be trusted forH →0, as follows.

If we consider the higher order graphs constructed from thecubic interaction only, then using dimensional analysis these contribute to V (φ)terms of the general form (λφ)4ηL−3 whereη =κλ2φ2m2 + 12λφ2(4.19)and L is the number of loops. Since η →∞as H →0 we clearly have perturbativebreakdown in this region.

This sort of infra-red problem is characteristic of super-renormalisable interactions and is important, of course, in calculations of V at finite12

temperature. Note that in the neighbourhood of the tree minimum, m2+ 16λφ2 ≈0,we have η ∼λ so perturbative calculability requires merely κλ(t) ≪1 as we havealready assumed.Finally let us consider briefly the massless case, m2 = 0.

As originally indicatedby CW , V then remains well defined and perturbatively calculable for φ →0, sothat φ = 0 remains a local minimum (and the global one, modulo the fact that asbefore V can not be calculated in the neighbourhood of the Landau pole).5. O(N) φ4 theory.Here we generalise section (4) to the case of massive O(N) symmetric φ4 theory,defined by the LagrangianL = 12(∂µ⃗φ)2 −12m2⃗φ2 −λ24(⃗φ2)2(5.1)where ⃗φ2 = PNi=1 φiφi.

Including one loop corrections the effective potential isgiven byV (φ) =Ω′ + 12m2φ2 + λ24φ4+ κ4H2ln Hµ2 −32+ κ4(N −1)G2ln Gµ2 −32(5.2)where G = m2 + λφ2/6, and we have exploited the O(N) invariance to write V asa function of a single field φ. Once again the two-loop corrections may be foundin Ref.

12.At the one loop level the relevant RG functions areβ(1)λ= N + 83λ2κ,β(1)m2 = N + 23m2λκ,γ(1) = 0. (5.3a, b, c)As explained in previous sections, we prefer to deal with the RG equation forV ′ but we note for completeness that if we choose to define V `a la Kastening then13

writing Ω′ = m4f(λ) we have from eq. (2.11) thatλ dfdλ + 2N + 2N + 8f =3N2(N + 8)λ(5.4)with solutionf =(3N[2(N −4)λ]−1 + cλ−2(N+2)/(N+8),if N ̸= 4(ln λ)/(2λ) + c/λ,if N = 4(5.5)As in the previous section the c-terms in Ω′ are in fact t-independent in the RGsolution so we may set c = 0.

It is easy to see that for N ̸= 4 eq. (5.5) correspondsto eq.

(15) of Ref. 11 (with t = 0).Reverting now to V ′, we have from eq.

(3.13) thatV ′(µ, m2, λ, φ) =m2(t)φ + 16λ(t)φ3+ κ2λφH(t)ln H(t)µ2(t) −1+ κ6(N −1)λφG(t)ln G(t)µ2(t) −1+ ....(5.6)Evidently there is no choice of t which eliminates the logarithms to all orders: butif our concern is to control the behaviour of V at large φ then any choice such thatµ2(t) ∼φ2 will do. With (say) t = ln(φ/µ), it is a simple matter to compute V ′ asdefined by eq.

(5.6) and hence (numerically) V (µ, m2, λ, φ). For κλ ≪1, the resultdiffers little from the tree approximation out to φ ∼µe1/(κλ) just as in the N = 1case.As in the N = 1 case perturbation theory will break down (for m2 < 0) inthe region H ≈0.

We now, however, have also to consider whether there arealso IR problems at G ≈0: ie at the tree minimum. Evidently for G < 0, Vbecomes complex: but how closely can we approach G = 0 from above and retainperturbative calculability?

In fact there is no problem as G →0; this is evidentexplicitly at one and two[12] loops.To extend this result to higher loops, notethat we have in general cubic vertices of the type H3 and HGG but not G3.14

Consider some graph consisting of HGG vertices only: if it is singular as G →0,then it will still be so if we “shrink” every H propagator by the substitution1/(k2 +H) →1/H. But the diagram will then consist of G4 vertices only, with theeffective coupling λ2φ2/H.

Then by dimensional analysis, or simply by noting thatG4 is a renormalisable (not a super-renormalisable) vertex, it is clear that the graphwill not be singular as G →0. The significance of the fact that ∂2V/∂φ2 is singularat G = 0 is not precisely clear to us; at the true minimum, of course, (calculatedconsistently to any order in ¯h) the matrix ∂2V/∂φi∂φj has no singularities and N-1zeroes corresponding to the would-be Goldstones.6.

The standard model.In this section we consider V (φ) in the standard model (SM ) from the RGpoint of view, with emphasis on the question of vacuum stability. As in the O(N)scalar case we can exploit gauge invariance to write V as a function of a single fieldφ.

We must also choose a gauge; the ’t Hooft-Landau gauge is the most convenient.In this gauge the W, Z and γ are transverse, and the associated ghosts are masslessand couple only to the gauge fields; the would be Goldstone bosons G±,G havea common mass deriving from the scalar potential only.Moreover, the gaugeparameter is not renormalised in this gauge so it does not enter the RG equation.Calculating V through one loop yieldsV (φ) =Ω′(µ, m2, h, λ, g, g′) + 12m2φ2 + 124λφ4+ κh14H2ln Hµ2 −32+ 34G2ln Gµ2 −32+ 32W 2ln Wµ2 −56+ 34Z2ln Zµ2 −56−3T 2ln Tµ2 −32i+ ...(6.1)whereH = m2 + 12λφ2,T = 12h2φ2,G = m2 + 16λφ2,15

W = 14g2φ2,Z = 14(g2 + g′2)φ2.Here h is the top quark Yukawa coupling (we neglect other Yukawa couplingsthroughout).The occurrence of the logarithms of H,G,T,W and Z in the perturbation ex-pansion means of course that no choice of t will eliminate the logarithms altogether.As indicated in the O(N) scalar case, however, it is clear that as long as the initialvalues of the dimensionless couplings are small and they remain small on evolu-tion then as long as we choose µ(t) ∼φ, our RG solution eq. (3.14), say, will beperturbatively believable for all φ.The essential feature that distinguishes gauge theories in general from the purescalar cases discussed in the previous two sections is the fact that λ = 0 is no longera fixed point in the evolution of the quartic scalar coupling λ(t).

Evolution of λwith φ may therefore drive λ negative and hence cause V to develop a second localminimum† at large φ; if this minimum is deeper than the (radiatively corrected)tree minimum then it will result in the destabilisation of the electroweak vacuum.Requiring stability (or at least longevity) of the electroweak vacuum results in anupper limit on mt (for a given mH). The existence of this limit and related issueshas been explored in a series of papers by Sher et al[4] (for a clear and comprehensivereview see Ref.

5).Now (as in fact essentially recognised by Sher in Ref. 5) the form of the RG“improved” V used in Ref.

4 is not completely satisfactory, inasmuch as it is not ingeneral a solution of the RG equation for all values of the Higgs (mass)2 parameterm2. In fact, however, because the false minimum, if present, occurs at large t (andhence φ ≫MZ) this should make little difference.

Provided a choice of t is madesuch that µ(t) ∼φ (at large φ), contributions to V from the Ω′ term and subleadinglogarithms neglected in Ref. 5 are very small for values of mt and mH in the range of† If one chooses to identify this “new” minimum with the true electroweak vacuum then itis easy to see that this results in the “Coleman-Weinberg” vacuum with a concomitantexperimentally disfavoured prediction for the Higgs mass.

[5]16

interest. In fact it is easy to convince oneself that in terms of the solution eq.

(3.10),for example, the question of the existence of a false (deep) minimum at some scaleis simply the question of whether λ(t) goes negative as t increases. Even for verysmall negative λ, the fact that this happens at φ/MZ ≫1 means that the tree termλφ4/24 drives V well below the electroweak minimum.

Thus although we now haveavailable the two-loop corrections to V[6] for the SM , they will have a negligibleaffect on the outcome. The importance of the evolution of λ to the stability ofthe vacuum was in fact recognised in Ref.

13 and the calculation performed usingthe one loop SM beta functions. The main question we resolve in this section isthe effect of 2 loop corrections on this calculation.

(Previous calculations of thiscorrection are unreliable due to typographical error in the expression for β(2)λgiveninRef 14. )In fact we have also calculated the evolution of m2 through twoloops and hence the improved V as a function of φ but, as anticipated above, therequirement that the electroweak vacuum remains stable turns out to essentiallyidentical to the requirement that λ remains positive.We give the SM β-functions through two loops in an appendix.

It only remainsto discuss boundary conditions. At µ = MZ we use input values for g, g′, α3, λ,h, m2, as follows:g = 0.650g′ = 0.358α3 = 0.10, 0.11, 0.12, 0.13λ = λ0h = h0m2 = m20.

(6.2)In order to translate the results into a limit on mt, mH we use the tree resultsmt = 1√2h0v0m2H = −2m20 = 13λ0v2(6.3)17

where v2 = −6m20/λ0 (= (246 GeV )2). (Of course these relationships are them-selves subject to radiative corrections which we could include in principle).Because the −36h4 term in β(1)λtends to drive λ negative, the result of theevolution is a lower limit on λ0 (and hence mH) for a given h0 (and hence mt).Now the evolution equation for h ( see eq.

(A1)) includes a contribution from α3;increasing the input value of α3 causes h to decrease faster as t increases, and sowe would expect the lower bound on mH to decrease with increasing α3.In fig(1) we display the evolution of λ against t for mt = 120 GeV and threevalues of λ0. For λ0 ≈0.120, λ(t) goes negative but remains small and becomespositive again for t ∼15; but nevertheless because it is negative (albeit small) fort ∼10 this results in a very deep minimum at large φ.

The value λ0 = 0.125 is thecritical value, corresponding to mH = 50.3 GeV .In fig. (2) we display the critical mH as a function of mt for α3 = 0.11, asobtained in the one- and two-loop approximations, respectively.

We see that thetwo-loop corrections are not very large; typically they decrease the lower bound onmH by 2 −4 GeV or so.In fig. (3) we present the critical curve for four input values of α3 .

The depen-dence on α3 is quite marked, and as anticipated above, the lower bound on mHdecreases as α3 increases. This conclusion is at variance to that of Ref.

7, wherethe sensitivity to α3 was indeed noted, but the bound on mH was found to increaseas α3 increases.∗We find, for example that for mt = 130 GeV , the bound on mHis given by 70.1 GeV if α3 = 0.1, but 59.6 GeV if α3 = 0.13.For mt ≥140 GeV the curves are to a very good approximation linear, andstability of the electroweak vacuum corresponds in this region to the relationship( for α3 = 0.11, for example)mH ≥1.95mt −189 GeV. (6.4)∗We thank Marc Sher for confirming that the lower bound on mH indeed decreases as α3(MZ)increases; the result of Ref.

7 was due to a printing error.18

This differs somewhat from the linear approximation given by Sher (Ref. 5 p331),which corresponds to mH ≥1.7mt −160 GeV .

The reason for this discrepancyis that the latter result is based on an extrapolation of the results for lower Higgsmasses.∗∗Let us consider briefly our results in the light of recent predictions[15] for mtand mH based on analysis of LEP data including radiative corrections:mt = 124+26−28 GeV(6.5)andmH = 25+275−19 GeV. (6.6)With mt = 120 GeV , for instance, we have from fig.

(3) that (again with α3 = 0.11)mH ≥50.3 GeV ( 52.6 GeV from a one loop analysis). So with this value ofmt we are already assured of vacuum stability by the direct search limit on mH,mH ≥59GeV .

For mt = 140 GeV , we have from fig. (3) that mH ≥83.2 GeV .Discovery of the Higgs (with this value of mt) in the interval 59 GeV ≤mH ≤83 GeV would strongly suggest the existence of physics beyond the standard model,since the obvious means to rescue electroweak stability would be by new physicsat a scale heavy enough to have negligible impact on the radiative correctionsresponsible for the results eq.

(6.5) and (6.6). It is also clear that refinement of thevalue of α3(MZ) would be helpful in reducing the uncertainty in the critical curve.∗∗Once again we thank Marc Sher for confirming this.19

7. Conclusions.The renormalisation group expresses the simple fact that observables are in-dependent of the renormalisation scale µ. Consequently, adroit choice of µ leadsto improved perturbation theory by removing large logarithms in processes char-acterised by a single momentum scale.†Application of the RG to the effectivepotential is quite analagous, except now it is the region of large (or small) φ thatbecomes accessible.

In this paper we hope we have elucidated the issues that arise;in particular the relationship between the usual RG approach and the analysis ofRef. 10, 11.

We have also reconsidered the RG improvement of the SM poten-tial, and give a result for the electroweak stability bound on mH based on a fulltwo loop RG analysis. In particular we highlighted the dependence on α3(MZ),showing that the lower bound on mH decreases with increasing α3(MZ).

With thediscovery of the top quark generally expected to be imminent, it will be interestingto see whether the direct search limit on mH leaves a “window of instability”, asdiscussed in section 6.Among further applications of the RG to the effective potential, we might con-sider extension to the supersymmetric SM , and also whether the RG improvedpotential has any bearing on the issue of triviality of non-asymptotically free the-ories.Acknowledgements.While part of this work was done, one of us (DRTJ) enjoyed the hospitality ofthe Institute for Theoretical Physics at Santa Barbara and thanks Jim Langer forhis part in making the visit possible. C.F is grateful to the S.E.R.C for financialsupport; this research was also supported by the National Science Foundationunder grant no.

PHY89-04035, and by a NATO collaboration research grant. The† Processes with several scales may benefit from a multiscale RG approach: see Ref 16.20

work of one of us (MBE) was supported in part by the U.S. Department of Energyand by the Institute for Theoretical Physics at Santa Barbara.21

AppendixWe list the RG functions for the SM (see section 6 for notation and conven-tions) through two loops.The one-loop RG functions areκ−1γ(1) =3h2 −94g2 −34g′2κ−1β(1)λ=4λ2 + 12λh2 −36h4 −9λg2 −3λg′2+ 94g′4 + 92g2g′2 + 274 g4κ−1β(1)h=92h3 −8g23h −94g2h −1712g′2hκ−1β(1)g= −196 g3κ−1β(1)g′ =416 g′3κ−1β(1)g3 = −7g33κ−1β(1)m2 =m2(2λ + 6h2 −92g2 −32g′2). (A1)The two-loop contributions to the RG functions are given by22

κ−2γ(2) =16λ2 −274 h4 + 20g23h2 + 458 g2h2 + 8524g′2h2−27132 g4 + 916g2g′2 + 43196 g′4κ−2β(2)λ= −263 λ3 −24λ2h2 + 6λ2(3g2 + g′2) −3λh4 + 80λg23h2+ 452 λg2h2 + 856 λg′2h2 −738 λg4 + 394 λg2g′2 + 62924 λg′4+ 180h6 −192h4g23 −16h4g′2 −272 h2g4 + 63h2g2g′2−572 h2g′4 + 9158 g6 −2898 g4g′2 −5598 g2g′4 −3798 g′6κ−2β(2)h=h−12h4 + h2(13116 g′2 + 22516 g2 + 36g23 −2λ) + 1187216 g′4−34g2g′2 + 199 g′2g23 −234 g4 + 9g2g23 −108g43 + 16λ2κ−2β(2)g=g3(32g′2 + 356 g2 + 12g23 −32h2)κ−2β(2)g′ =g′3(19918 g′2 + 92g2 + 443 g23 −176 h2)κ−2β(2)g3 =g33(116 g′2 + 92g2 −26g23 −2h2)κ−2β(2)m2 =2m2−56λ2 −6λh2 + 2λ(3g2 + g′2) −274 h4 + 20g23h2+ 458 g2h2 + 8524g′2h2 −14532 g4 + 1516g2g′2 + 15796 g′4. (A2)23

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FIGURE CAPTIONSFig.1Plot of the running coupling λ(t) for mt = 120 GeV and λ0 just above, at andjust below its critical value (0.125).Fig.2Plot of the critical value of mH for vacuum stability against mt, for α3(MZ) =0.11, showing one- and two-loop approximations.Fig.3Plot of the critical value of mH for vacuum stability against mt, for α3(MZ) =0.1, 0.11, 0.12 and 0.13.25

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