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The 3d Effective Field Theory

arXiv:hep-ph/9209227v1 11 Sep 1992MPI-Ph/92-72hep-th/9209227August, 1992The 3d Effective Field Theoryof the High Temperature Abelian Higgs ModelVidyut JainMax-Planck-Institut f¨ur PhysikWerner-Heisenberg-InstitutP.O. Box 40 12 12 , D - 8000 Munich 40, GermanyAbstractWe study a weakly coupled 3 dimensional ϕ4 type model consisting of N realscalars ϕi coupled to an abelian gauge field Aa and one extra scalar field ρ. Weargue that, below some scale Λ = O(T), this is the effective field theory of a 3+1dimensional abelian Higgs model at a high temperature T. The effective theory issufficient to study the nature of the phase transition in four dimensions.

By intro-ducing an auxiliary field χ we eliminate the explicit ϕ4 term; the new Lagrangianallows for a simple computation of the dominant corrections to the effective scalarpotential Veff in the large N limit. We study three cases: a) 6g2/λ ≤O(1), b)6g2/λ ∼O(N) and c) 6g2/λ ∼O(N23 ) where g and λ are the 4d gauge and scalarcouplings, respectively.

For case a) which is the most thoroughly studied we findthat the leading O(N)+O(1) result for Veff admits only a second order phase tran-sition. For the other cases we find that b) the leading O(1) result for Veff admits afirst order phase transition whose strength is independent of N and c) the leadingO(N13) result for Veff admits only a second order phase transition – the O(1) cor-rections to this can be interpreted as indicating a first order phase transition whosestrength diminishes as N increases.

1. Introduction.There has been much recent interest in the nature of the electroweak phase transition.In general, for a spontaneously broken 4d φ4 type model, one can attempt to compute thehigh temperature effective scalar potential Veff.

At very high temperature the symmetryis believed to be restored, and at some temperature T = T2 the origin (|φ| = 0) goes frombeing a local minimum to being a local maximum. If at T > T2 the origin is also the globalminimum the theory admits only a second order phase transition which proceeds by aroll–over after the temperature drops below T2.

On the other hand, if at some T1 > T2 thescalar potential has another minimum, degenerate in energy with the one at the origin,the theory can admit a first order phase transition.The determination of a reliable effective potential has proven difficult because, ingeneric models, ordinary perturbation theory in ¯h has infrared divergence problems justat values of φ2 and T where the phase transition occurs. In a φ4 model with N realscalars, Dolan and Jackiw [1] showed over 18 years ago that the one–loop potential hasproblems.

They summed up an infinite class of diagrams, the “superdaisies” to obtain areliable result for the effective mass at the origin.Giving a reliable estimate of Veff in the case of the standard model is much moreinvolved. The typical approach [2,3,4,5] for the corrections from the bosonic sector of thestandard model has been to use not tree–level propogators but propogators corrected byone–loop vacuum polarization effects in a one–loop calculation of the effective potential.This has the effect of summing up an infinite class of diagrams, the so–called “ring” di-agrams, with the correct combinatorics except for the two–loop graphs [4,5,6] which areovercounted.

When only the leading T 2 dependent corrections are used for the vacuumpolarizations the overcounting is not serious, but when field dependent vacuum polariza-tions are used it can result in dangerous T 3|φ| terms in the effective potential [4,5] whichresults in the effective mass blowing up as one approaches the origin.Assuming no such dangerous terms appear, previous studies have concentrated ondetermining the coefficient of the |φ|3 term in their estimates of Veff. For small φ2, theeffective potential can be expanded as (d > 0)Veff = c + aφ2 −b|φ|3 + dφ4 + · · · .

(1)The coefficient a is positive when T > T2 and negative when T < T2.In a general3+1 dimensional scalar and gauge system the coefficients a, b and d can be found inthe ring approximation to Veff as follows [2,3,4,5]. In the background field evaluationof Veff one expands the action about quantum scalar fields ˆφi and quantum gauge fieldsAµ.

Each quantum degree of freedom has a mass depending on the background field φ2.At zero external momentum, the leading vacuum polarization effects give additional T 2contributions to the mass of the quantum scalars and the longitudinal gauge field, but notto the transverse gauge fields. Then, when T is sufficiently bigger than T2, the result ofthe polarization effects is that the scalar and longitudinal gauge boson loops give only φ2and φ4 contributions to eq.

(1). In this limit, the transverse gauge fields still contribute1

a |φ|3 term. It is the existence of this term in the context of the standard model whichled the authors of [2,4] to conclude that the electroweak model with sufficiently smallself–scalar coupling admits a (weak) first order phase transition.The analysis we have just described is however incomplete.

For example, how arethese results modified if one keeps the field dependent contributions to the vacuum polar-izations? As already mentioned, the simple ring diagram procedure can lead to spuriousO(|φ|) contributions to Veff, so obviously some care is needed.

One would also like toinvestigate how the momentum dependence of the vacuum polarizations modifies the |φ3|result. Finally, we would like to know in what way the ring (or “daisy”) diagram sum canbe interpreted as a consistent approximation scheme.To address these questions and problems, we study as a toy model a 3 dimensional ϕ4type model consisting of N real scalars ϕi coupled to an abelian gauge field Aa and anextra scalar ρ, which we argue is the low energy effective field theory for a 3+1 dimensionalAbelian Higgs model at high temperature.

The computations in the effective 3d theory arehowever much easier than in the full four dimensional model, and it is for this reason we feelthis approach will prove very useful in studying the much more complicated electroweakmodel at high temperature. In this paper we denote the 3d scalars ϕ and the 4d scalarsφ.

The 4d gauge and scalar couplings are g and λ, respectively.For 6g2/λ ≤O(1), by introducing an auxiliary field χ we eliminate the explicit ϕ4term and develop a systematic 1/N expansion for Veff and compute the leading andnext–to–leading corrections in this expansion. By matching linearly divergent integrals inthe effective 3d theory with the O(T 2) corrections in the 4d theory we show how to obtainexactly the correct high temperature result of the full 4d model, at least in the g = 0 casewhich is known [6].

The effective 3d model is sufficient to study the nature of the phasetransition which occurs at values of φ2 and χ much less than T 2. In contrast to previousstudies, we find no |φ| term in Veff, nor for 6g2/λ ≪N any significant |φ|3 term in Veff.In fact, to the order we compute, the model admits only a second order phase transition.First order phase transitions are however found for 6g2/λ ∼O(N) and 6g2/λ ∼O(N23).We mention that the high temperature 3+1 dimensional abelian Higgs model has alsobeen investigated in [7] as an effective 3 dimensional theory.

However, in that reference,the ǫ–expansion method is used to study the nature of the phase transition and it doesnot shed any light on the above questions and problems.In section 2 we show how the pure scalar result can be obtained in our approach andfind agreement with an earlier calculation of this author [6]. In section 3 we extend theanalysis to the gauged case for 6g2/λ ≤O(1).

In section 4 we find the leading contributionsto Veff for 6g2/λ ∼O(N) and O(N23). In section 5 we discuss our results and the phasetransition and our conclusions appear in section 6.

The pure scalar example, althoughpreviously studied at finite temperature [6], sets much of the foundation for the analysisin section 3, including in particular the vacuum polarization effects.2

2. φ4 Model to Subleading Order.First consider a model with N real scalars φi with the tree level 3+1 dimensionalLagrangianL[φ] = N2 δij∂µφi∂µφj −Nλ4! (φ2 −v2)2.

(2)λ is a dimensionless coupling and v is the tree level vev of |φ|. The space–time metric isηµν = diag(+, −, −, −).

Furthermore, to avoid problems associated with triviality we willassume λ > 0 and consider the model as an effective low energy model valid below somescale ¯Λ [10].The systematic 1/N expansion allows us to calculate (1) as a perturbation in 1/N nearφ = 0 [8,9]. Root [9] has evaluated the leading corrections and given formal expressionsfor the next–to–leading corrections to the zero temperature scalar potential in 4,3,2 and1 dimensions.

The procedure at finite temperature is very similar.By introducing a dimension two auxiliary field χ we replace the Lagrangian (2):L[φ, χ]=L[φ] + 3N2λ χ −λ6(φ2 −v2)!2=N2 (∂φ)2 + 3N2λ χ2 −N2 (φ2 −v2)χ. (3)The auxiliary field has eliminated the φ4 term; the original form of (3) is easily recoveredby use of the equation of motion for χ.To calculate the effective potential Veff(φ) one proceeds as follows.

First, using thebackground field method one computes the effective potential as a function of backgroundsof φ and χ. Then, the background of χ is eliminated by its equation of motion.The systematic 1/N expansion first requires expanding the Lagrangian L[φ, σ] aboutreal backgrounds φ and χ thus [8,9,11]:φi →φi +ˆφi√N ,χ →χ +ˆχ√N ,(4)and then deleting terms linear in the quantum fields ˆφ, ˆχ.This procedure defines aquantum Lagrangian L[ˆφ, ˆχ], the sum of whose one–particle irreducible (1PI) diagramsgive what we call the effective action.

Since we are interested in the effective potential weassume the backgrounds are space–time constants. We also assume non-negative χ.At finite temperature the model is formally equivalent to a euclidean field theorywith one compact dimension.

When the backgrounds φ2 and χ are much below T 2 itis sufficient to study an effective three dimensional theory, a fact that we exploit below.Thus, at sufficiently high temperature, our results for four dimensions should be similarto those of a three dimensional euclidean field theory with a dimensionful φ4 coupling [7].Indeed, in three dimensions and zero temperature, the leading O(N) potential has long3

been known [8]. It has exactly the same form as the sum of finite temperature superdaisygraphs that were computed by Dolan and Jackiw [1] for a four dimensional φ4 theory.

Animportant point in our approach is that the effect of introducing an auxiliary field χ is toshift the φ mass term and as a result there are no infrared divergences in this formalism.To perform the dimensional reduction from 4 to 3 dimensions, one can follow [7]. Onedecomposes the fields as follows,ˆφi(xµ) =X ˆφin(⃗x)ψn(τ),ˆχ(xµ) =Xˆχn(⃗x)ψn(τ),(5)where ψn(τ) are a complete set of periodic functions on the circle.For what we areinterested in only the zero modes n = 0, for which ψ0(τ) = ψ0(0), are important in theeffective 3d model.

This is because for n ̸= 0 the fields ˆφin pick up nonzero masses ofO(T 2) when the compact dimension is integrated out in the action. Therefore, truncatingthe spectrum to keep only the zero modes and integrating out the compact dimensiongives the effective 3d euclidean Lagrangian (β = 1/T)1βL[ˆφ0, ˆχ0]=−3N2λ χ2 + N2 χ(φ2 −v2)+12ˆφi0[δij(−⃗∂2 + χ)]ˆφj0 +ˆφ20ˆχ02√N + φi ˆφi0ˆχ0 −3N ˆχ202λ .

(6)Recall that φ and χ are space–time constants. Defining the three dimensional quantities,ϕi =qβφi,ˆϕi =qβ ˆφi0,˜λ = λ/β,˜v2 = βv2,(7)we finally obtain the 3d quantum LagrangianL[ ˆϕ, ˆχ]=−3N2˜λ χ2 + N2 χ(ϕ2 −˜v2)+12 ˆϕi[δij(−⃗∂2 + χ)] ˆϕj + ˆϕ2 ˆχ2√N + ϕi ˆϕi ˆχ −3N ˆχ22˜λ .

(8)Here we have dropped the zero mode subscript on ˆχ0.Before proceeding we would like to stress some important points. As shown in [7],if one integrates out the nonzero modes n ̸= 0 at the quantum level rather than justtruncating the spectrum, there is a finite O(T 2) correction to the mass of ˆφi0.Thiscorrection is very important, in fact it is the term that gives symmetry restoration athigh enough temperature.

We will obtain this term another way, in analogy with whatone does in effective low energy theories of the strongly interacting standard model oreffective four dimensional supergravity models inspired by string theory [12]. The threedimensional field theory will be divergent.

Here we will regulate all the divergent integralsby introducing the same scale Λ. In fact, the only divergent integral that will be importantis linearly divergent and we will regulate it by simply using a sharp momentum cutoff.a WeaA different scheme, such as Pauli–Villars, will in general give a different coefficient for the O(Λ)result.

However, the precise coefficient will not be important in what follows [12].4

then give a physical interpretation to Λ, i.e. the scale at which the full four dimensionalphysics becomes important.

Thus, Λ is proportional to T. This is in complete analogywith, for example, the effective 4d theories where the regulating scale is taken to beof order the compactification scale. In our case the identification can be made precisebecause the corresponding 4d correction is well known.

We will see that the identificationΛ = π2T/6 will reproduce exactly the T 2 results from the four dimensional integrals. Inaddition, since the four dimensional model (2) is only valid up to a scale ¯Λ we must requireΛ < ¯Λ, i.e.

that T is sufficiently small. Finally we note that the 4d effective potential canbe obtained from the three dimensional one by dividing by β and using (7).To compute the 3d effective potential we must sum the one–particle irreducible (1PI)diagrams of (8).

For readability, and for later use in the abelian Higgs model, we brieflydescribe the steps that lead to the next–to–leading result for Veff, rather than just statingthe result. Throughout, Tr stands for momentum and internal space traces.Integrating out the ˆφi in this model gives the O(N) corrections to Veff to all ordersin ¯h and defines an effective Lagrangian which only has quantum dependence on ˆχ.

Thepart of this effective Lagrangian which is quadratic in ˆχ includes the tree level part and aone–loop vacuum polarization part. The one–loop ˆχ corrections give the next–to–leadingcorrections in N, also to all orders in ¯h.Because (8) is quadratic in ˆϕ, the O( ˆϕˆχ) mixing term can be removed by making achange of field variables [6].

This has the effect of replacing the mixing term by+ ϕi ˆϕi ˆχ →−ˆχ22ϕ2(−⃗∂2 + χ + ˆχ/√N)→−ˆχ22ϕ2(−⃗∂2 + χ),(9)where the last replacement holds to next–to–leading order in the 1/N expansion.The gaussian integral over ˆϕ is now straightforward and its vacuum polarization con-tribution to the ˆχ propogator at nonzero external momentum is known [8,9] in 3d,Πˆχˆχ(⃗k) =✒✑✓✏=˜λ6Z⃗p1[⃗p2 + χ][(⃗p + ⃗k)2 + χ]=˜λ24π1q⃗k2sin−11q1 + 4χ/⃗k2. (10)The final result for the leading + next–to–leading potential is [8,9,6]Veff = −3N2˜λ χ2 + N2 χ(ϕ2 −˜v2) + N2 Tr ln(−⃗∂2 + χ) + VNtl,(11)where the next–to–leading contribution from the gaussian integral over ˆχ isVNtl = 12Tr ln1 +˜λ24π1q−⃗∂2sin−11q1 −4χ/⃗∂2+˜λϕ2/3−⃗∂2 + χ.

(12)Root [9] has shown, without explicit calculation, that the 3d field theory is renormal-izable to this order in N, as it should be. Although we will give a physical interpretation5

to our regulating scale, renormalizability of the 3d theory places strong constraints onthe type of linearly divergent contributions we can obtain. For example, a contributionof O(Λϕ2) is by itself not renormalizable in our formalism.

How this enters will be seenin the explicit calculations below.Although we did not find a simple expression for VNtl, we found that the dominantcontribution comes from the large external momentum limit of (10), −⃗∂2 ≫χ, not thezero external momentum limit. Although this seems entirely reasonable in the limit ofvanishing χ, why it should be so for sufficiently large χ is not apparent.Indeed, wewould like to stress the importance of vacuum polarization effects at nonzero externalmomentum.

If in (12) we had kept only the zero external momentum part of the ˆχ fieldvacuum polarization, Πˆχˆχ(0) = ˜λ/(48π√χ) then we would have obtained an incorrect (andnonrenormalizable) answer. Πˆχˆχ(0) when used for the whole momentum integral from 0to Λ gives a dangerous O(˜λΛ3/√χ) contribution in the expansion of the log in (12).

Infact, the leading field dependent contributions seem to arise from the most ultravioletdivergent field dependent term in the high momentum expansion of Πˆχˆχ. One can checkthis assertion when χ is sufficiently large by the following method.

In this limit, thearcsine term in (12) is small for all values of the momentum, and we expand the ln usingln[1 + x] = x + .... The arcsine possesses two different expansions depending on whether−⃗∂2 is bigger or smaller than 4χ, and accordingly the momentum integral implied by theTr must be broken up into two different regions.

The different contributions can then beevaluated with the result that very little error is made is using only the large momentumlimit of (10) everywhere. This effect will also be very important in the next section.In the large momentum limit, we have(−⃗∂2 + χ)Πˆχˆχ(−⃗∂2) →˜λq−⃗∂248−˜λ√χ12π .

(13)We then write eq. (12) asVNtl ≈12Tr ln"−⃗∂2 + χ +˜λ48q−⃗∂2 +˜λ3 ϕ2 −√χ4π!#−12Tr lnh−⃗∂2 + χi.

(14)It can now be argued that if ˜λ/48 ≪Λ and χ is sufficiently large the O(q−⃗∂2) term inthe log can be dropped [13]. This is what we will assume.We can now give an explicit expression for Veff, up to the approximations we havemade by using the following result for three dimensions (χ ≥0):Tr ln(−⃗∂2 + χ) →Zd3⃗k(2π)3 ln(⃗k2 + χ) = Λχ2π2 −χ326π + const.

(15)A sharp momentum cutoffwas used to evaluate the field dependent divergent part. Addingeverything up we obtain, up to a constant,Veff=−3N2˜λ χ2 + N2 χ(ϕ2 −˜v2) + NΛχ4π2 −χ3212π+ Λ˜λ12π2 ϕ2 −√χ4π!6

−112π χ +˜λ3ϕ2 −˜λ√χ12π! 32−χ32.

(16)To write the effective potential for ϕ2 alone we use the equations of motion for χ toeliminate it. To the order we are working it is consistent to use the leading order equationfor χ to eliminate it [9].

∂Veff/∂χ = 0 from the terms proportional to N gives:χ =˜λ6ϕ2 −˜v2+˜λ6 Λ2π2 −√χ4π!. (17)If we rewrite the O(1) terms in Veff using this equation, and ignore ˜λ/(12π) in com-parison with √χ then we obtain up to a constant,Veff = −3N2˜λ χ2 + N2 χ(ϕ2 −˜v2) + (N + 2) Λχ4π2 −112π χ +˜λ3ϕ2!

32+ (N −1)χ32. (18)This is the leading + next–to–leading result for Veff for a pure scalar theory in 3dwhen √χ is large enough.

To obtain the high temperature four dimensional result, werecall the high temperature 4d leading order mass–gap equation [1,6]:χ = λ6(φ2 −v2) + λ6 112β2 −√χ4πβ!+ . .

. .

(19)Using (7), we see that the identificationΛ = π2T/6(20)in (17) reproduces exactly the four dimensional result (19). In fact, with this identification,(18) reproduces exactly the leading and next–to–leading 4d finite temperature scalarpotential found in [6].

The computations here have been much easier. This potential wasstudied in [6] and admits no first order phase transition for the range of φ2 and T whereit is valid (i.e.

not too close to the origin when the temperature is near T2). Similarconclusions appear in [7,14].

Our result is essentially due to the fact that Veff containsno O(|φ|3) terms for λφ2 < 3χ.The potential (18) is valid when ˜λ/(12π) is ignorable compared to √χ. We will give amore extensive discussion of the phase transition in the more general gauged case, in thelast section.7

3. Abelian Higgs Model, 6g2/λ ≤O(1).Now consider the 3+1 dimensional gauge invariant scalar QED LagrangianL[φ, A] = N2 δij∂µφi∂µφj −Nλ4!

(φ2 −v2)2 −14F µνFµν −gǫij(∂µφi)φjAµ + 12g2φ2A2. (21)Here, g is the gauge coupling, φi, i = 1...N, are real as before and ǫij is antisymmetricwith the nonzero components ǫ12, ǫ21, ǫ34, ǫ43, ... all having magnitude 1.

The field strengthand covariant derivative Dµ are given in terms of the gauge field Aµ byDµφi=∂µφi −gǫijφjAµ,Fµν=∂µAν −∂νAµ. (22)This model is nothing but N/2 copies of the simple scalar QED Lagrangian discussed inmany textbooks, and also in ref.

[1], but with only one gauge vector field.We would like to calculate the leading + next–to–leading corrections to Veff in the1/N expansion at high temperature. As in section 2, we will do this by first writing aneffective 3d theory.

This three dimensional theory will turn out to be an abelian Higgsmodel with an extra scalar. This can be understood intuitively because in 4 dimensionsthe gauge field has three massive components, while in three dimensions it only has two.Hence, to get the same light degrees of freedom we need an additional scalar.To proceed, we introduce an auxiliary field, as in eq.

(3), to obtainL[φ, χ, A] = N2 (∂φ)2+ 3N2λ χ2−N2 (φ2−v2)χ−14F µνFµν −gǫij(∂µφi)φjAµ+ 12g2φ2A2. (23)We then expand this using (4), and delete all terms linear in ˆφ and ˆχ.

Since we areinterested in the scalar potential we need not keep a background for the gauge fields.We also need to gauge fix. We add the gauge fixing termLg.f.

= −12α(∂µAµ + αgǫij ˆφiφj)2. (24)Here, α is an arbitrary parameter.Although the calculations simplify in the Landaugauge α →0, this extra parameter is useful in checking that physical quantities suchas the critical temperature are gauge fixing independent.

The ghost term for this gaugefixing is [1]Lgh = ¯θ(∂2 + αg2φ2)θ,(25)where θ, ¯θ are Grassmanian ghost fields. We assume non-negative g2, α and α ≤O(1) [4].After performing all these operations, the total quantum Lagrangian is given byL[ˆφ, ˆχ, A, θ]=L[ˆφ, ˆχ] + 12Aν∂2Aν + 12(1 −1α)(∂A)2+12g2(φ2 + 2φi ˆφi/√N + ˆφ2/N)A2 −g√N ǫij(∂µ ˆφi)ˆφjAµ−12αg2 ˆφi ˆφkǫijǫklφjφl + ¯θ(∂2 + αg2φ2)θ.

(26)8

We dropped all total divergences and assumed the backgrounds are space–time constants.There is no O(ˆφA) term because of our gauge choice. Finally, L[ˆφ, ˆχ] is the pure scalarpart of the quantum Lagrangian.To write the 3d effective field theory for the high temperature limit of this model wefollow exactly the approach described in section two.

We simply truncate the spectrumto keep the zero modes of the compact dimension of the euclidean theory and then give aphysical interpretation to the scale used to regulate divergent integrals. This truncationmeans ∂0 →0.

Then, with the definitions (7) as well as (a=1,2,3)Aa =qβAa,ρ = iqβA0,ϑ =qβθ,˜g = g/qβ,(27)we write the effective 3d quatum Lagrangian asL[ ˆϕ, ˆχ, A, ρ, ϑ]=L[ ˆϕ, ˆχ] −12Aa⃗∂2Aa −12(1 −1α)(⃗∂A)2 −12ρ⃗∂2ρ+12˜g2(ϕ2 + 2ϕi ˆϕi/√N + ˆϕ2/N)(A2 + ρ2) −˜g√N ǫij(∂a ˆϕi) ˆϕjAa+12α˜g2 ˆϕi ˆϕkǫijǫklϕjϕl + ¯ϑ(−⃗∂2 + α˜g2ϕ2)ϑ. (28)L[ ˆϕ, ˆχ] is given by eq.

(8), and Aa = Aa, ∂a = ∂a.Eq. (28) is nothing but a three dimensional abelian Higgs model with an extra masslessscalar field ρ.

It has a simple canonical kinetic term and its tree level coupling to thescalars is simply +˜g2ϕ2ρ2/2. We now compute the 1PI diagrams of this 3d Lagrangian toleading and next–to–leading order in the 1/N expansion.

It is clear that the gauge sectorcontributes only at next–to–leading order; there are no Feynman diagrams involving thegauge fields that contribute at O(N). The correction from the 3d ghosts involves only asimple one–loop calculation.

For the rest, we can either first integrate out the A and ρat the quantum level, or integrate out the ˆϕ. We have done both, and present only thelatter computation.To integrate out the ˆϕ we must first make a field redefinition to eliminate terms in(28) which are linear in ˆϕ.

For ˜g = 0 this is just the replacement (9) to the requiredorder. For ˜g ̸= 0 we have to shift ˆϕ in such a way as to also eliminate the O(˜g2) linearterms in ˆϕ.

When this is done we found no contributions from the O(˜g2 ˆϕi) terms thatwere important at next–to–leading order. Another way of saying this is that there are noFeynman diagrams involving the O(˜g2 ˆϕi) terms that contribute at next–to–leading order.Hence we just forget these terms in the Lagrangian.In the pure scalar case we saw that integrating out the scalars gave the leading orderresult and generated a vacuum polarization term Πˆχˆχ for ˆχ.Here, something similarhappens.

In addition to Πˆχˆχ we will generate a vacuum polarization Πρρ for ρ and avacuum polarization matrix Πab for the 3d gauge fields. This result can be derived verysimply as follows.

The scalar loop integral in (28) contributes a piece12Tr ln"∆−1ij + δijˆχ√N + δij˜g2N (A2 + ρ2) −2 ˜g√N ǫjiAa∂a#(29)9

where the inverse scalar propogator is given by the matrix∆−1ij = δij(−⃗∂2 + χ) + α˜g2ǫii′ǫjj′ϕiϕj′. (30)One can expand (29) as a power series in A; the terms linear in A in this expansion vanishbecause trǫij = 0 and only the terms quadratic a A and ρ contribute at O(1) in the 1/Nexpansion using trǫijǫjk = −N.

At zero external momentum the expansion is easy andgives to O(1)12Tr ln(∆−1ij +δij ˆχ/√N)+ 12˜g2(A2 +ρ2)Tr(−⃗∂2 +χ)−1 + ˜g2AaAbTr∂a∂b/(−⃗∂2 +χ)2. (31)The A2, ρ2 terms are clearly vacuum polarization corrections to the tree level kinetic termsof these fields.

The vacuum polarization for ˆχ is contained in the first term above, andis in fact the same as in the pure scalar case at O(1). Since we have seen that vacuumpolarization effects at nonzero external momentum are very important even in the purescalar case we will also keep them here.

This can be done by carefully accounting forthe 3-space dependence of all the quantum fields in (29) and the result for the full A, ρvacuum polarizations is nothing other than the one–loop result given for scalar QED inmany standard textbooks [15]. Altogether, one hasΠˆχˆχ(⃗k)=✒✑✓✏=˜λ6Z⃗p1[⃗p2 + χ][(⃗p + ⃗k)2 + χ],Πρρ(⃗k)=✒✑✓✏= ˜g2Z⃗p1⃗p2 + χ,Πab(⃗k)=✂✁✂✁✂✁✄✄✄✒✑✓✏✂✁✂✁✂✁✄✄✄+ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✄✄✄✄✄✄✄✒✑✓✏= ˜g2Z⃗p"δab⃗p2 + χ −12(2p + k)a(2p + k)b[⃗p2 + χ][(⃗p + ⃗k)2 + χ]#.

(32)Veff then is given by the tree–level piece, a contribution from the scalars loops,12Tr ln ∆−1ij , a simple one–loop contribution from ghosts ϑ, and the contributions fromquadratic integrals over A, ρ, ˆχ with propogators modified by the vacuum polarizationeffects(32). The result is:Veff=−3N2˜λ χ2 + N2 χ(ϕ2 −˜v2) + 12Tr ln ∆−1ij + VNtl−Tr ln(−⃗∂2 + α˜g2ϕ2) + 12Tr ln(−⃗∂2 + ˜g2ϕ2 + Πρρ) + 12Tr ln(∆−1ab + Πab).

(33)The tree–level gauge kinetic term is given by∆−1ab = −δab⃗∂2 + (1 −α−1)∂a∂b + ˜g2ϕ2δab. (34)The contribution from ˆχ is the same as in the pure scalar case, eq.

(12), because Πˆχˆχ didnot change to O(1) in 1/N.10

A few remarks are in order:• In the Landau gauge α →0, the 3d gauge contributions are given only by the last termin (33). In this gauge, the ρ and Aa contributions are similar to the 4d gauge loop ring sumworked out in refs.

[2,4]. As mentioned in the introduction, their procedure overcountsthe O(¯h2) graphs.For the ˜g dependent O(¯h2) graphs the overcounting is due to anadditional contribution from the scalars because for the scalar loop contribution they usepropogators with ˜g dependent vacuum polarization effects.

Our method of computing Veffautomatically avoids this overcounting. In addition our contribution from (33) includesmuch more than the ring sum in [2,4] because our vacuum polarization effects dependon χ which is a self consistent solution of ∂Veff/∂χ = 0 to all orders in ¯h, whereas thevacuum polarization used in [2,4] is of O(¯h) only.• In a general gauge, α ̸= 0, the contributions to Veff due to nonzero Πab are independentof α.

This is a result due to gauge invariance, i.e. kaΠab(⃗k) = 0 formally holds for (32) –as it also should for a properly regulated Πab.

We have ∆−1ab + Πab = δab(−⃗∂2 + ˜g2ϕ2) +[(1 −α−1)∂a∂b + Πab]. The log of this can formally be expanded as a power series in theterm in square brackets.

Since Πab is transverse, tr[(1 −α−1)∂a∂b + Πab]n factorizes sothat there are no mixed α–dependent and Π dependent terms. Altogether, this meansTr ln[∆−1ab + Πab]=Tr ln ∆−1ab + 2Tr ln(−⃗∂2 + ˜g2ϕ2 + Π) −2Tr ln(−⃗∂2 + ˜g2ϕ2)=Tr ln(−⃗∂2 + α˜g2ϕ2) + 2Tr ln(−⃗∂2 + ˜g2ϕ2 + Π).

(35)Here, Πab = (δab −kakb/⃗k2)Π which is true due to kaΠab = 0 and 3d euclidean invariance.• Renormalizability of the model places strong contraints on the type of linearly divergentcorrections to Veff. The reason is the same as in the pure scalar case and as in section 2we cannot obtain, by itself, a O(˜λΛϕ2) contribution to Veff.• As already described we will give a physical interpretation to the regulating scale fordivergent integrals.

In general, one should introduce arbitrary parameters for the differentdivergent terms [12]. In the pure scalar case we did not do this because it turned outto be unnecessary.

In our computations for this section it turned out that the matchbetween the linearly divergent 3d integrals and the finite T 2 corrections of the full theoryrequired that the linearly divergent part of Πρρ be regulated not with Λ →π2T/6 butwith Λ′ →π2T/3. In general, we should also use unspecified sharp momentum cutoffsfor the linear divergences from the final ρ and Aa loop contributions.

However, the samecutoffΛ →π2T/6 that gave the O(T 2) corrections in the scalar case also worked here.This should be kept in mind in what follows.Many of the traces in (34) are straightforward to compute. Using (15) we find up tofield independent terms,Tr ln ∆−1ij=NΛχ2π2 −χ326π+ α˜g2Λϕ22π2−16πh(χ + α˜g2ϕ2)32 −χ32i,Tr ln(−⃗∂2 + α˜g2ϕ2)=α˜g2Λϕ22π2−α32 ˜g3|ϕ|36π,11

Tr ln(−⃗∂2 + ˜g2ϕ2 + Πρρ)=˜g2Λ(ϕ2 −√χ4π )2π2−˜g3 Λ′2π2 + ϕ2 −√χ4π 326π. (36)Notice that the contribution from the 3d scalar ρ occurs only in the combination ϕ2 −√χ/4π, a fact that is demanded by multiplicative renormalizability of the 3d theory.bWe cannot compute the last term in (33) without first evaluating Πab at nonzeroexternal momentum.

This can be explicitly done by following [15]. Gauge invarianceforbids a linearly divergent gauge mass correction and in dimensional regularization thevacuum polarization displays no poles at d = 3c.

Since Πab = (δab −kakb/⃗k2)Π, we simplycompute Π = 12δabΠab. We findΠ = ˜g28π−√χ + 2χ + 12⃗k2q⃗k2sin−11q1 + 4χ/⃗k2.

(37)This has the zero momentum valueΠ = 0,(38)and the high momentum limitΠ ∼−˜g2√χ4π+ ˜g2q⃗k232. (39)In the zero momentum limit Πab = 0d, and if this is wrongly used in the whole momentumintegral in (33) we will obtain a nonrenormalizable answer for the 3d potential.

This ismost easily seen in the Landau gauge for which all linearly divergent corrections fromthe 3d scalar sector come in a combination proportional to ϕ2 −√χ4π .The 3d gaugeloop contributions for Πab = 0 however contribute O(˜g2Λϕ2) corrections which are notrenormalizable. Thus, keeping only the zero momentum limit of Πab is incorrect.We believe that the dominant contributions to Veff from the 3d gauge loop integralarise from the high momentum limit of Πab; we justify this as follows.

For ˜g2/12π ignorablein comparison to √χ it follows that Π(−⃗∂2) is small compared to −⃗∂2 for all values of−⃗∂2. We then expand Tr ln(−⃗∂2 + ˜g2ϕ2 + Π) as a power series in Π.

The momentumintegrals implied by the Tr must be broken up into the regions −⃗∂2 > 4χ and −⃗∂2 < 4χin accordance with the different expansions Π possesses in these two regions. At O(Π) thedifferent contributions can be computed with the result that only a small error is madein using (39) for the full momentum integral [17].bNote that due to our aim of calculating the high temperature Veff, we identify ρ with the longitudinal4d gauge field so that a bare mass for ρ is neither necessary nor allowed.cIn 3d this last statement can be understood as the requirement that there should be no nonlocalultraviolet counterterms [16].dThis agrees with the results given for the transverse components of the photon in [5].12

When Λ ≫ϕ2 −√χ/4π ≫˜g2/(64)2 [13] we find, using eq. (35), a sharp momentumcutoffΛ and ignoring constants,Tr ln(∆−1ab + Πab) = α˜g2Λϕ22π2−α32 ˜g3|φ|36π+˜g2 ϕ2 −√χ4πΛπ2−˜g3 ϕ2 −√χ4π 323π.

(40)Adding everything up, we obtain for the 3d Veff,Veff=−3N2˜λ χ2 + N2 χ(ϕ2 −˜v2) + Λχ4π2(N + 2 + 18˜g2˜λ )−112π 3χ +˜λ3(˜v2 −Λ/2π2)! 32+ (N −1)χ32−(6˜g2/˜λ)3212π χ +˜λ˜v26+˜λ12π2(Λ′ −Λ)!

32+ 2 χ +˜λ˜v26−˜λ12π2Λ! 32+Vα + neglected terms + O(1/N) terms.

(41)Vα is the pure gauge fixing dependent part,Vα = −112πh(χ + α˜g2ϕ2)32 −χ32 −α32 ˜g3|ϕ|3i,(42)and satisfies Vα(χ = 0) = Vα(ϕ2 = 0) = 0. The “neglected terms” in (41) refers to the(small) errors that were made in only keeping the large external momentum limit of thevacuum polarizations, as well as other approximations.

Here, we also used the leadingorder result for χ, eq. (17), to rewrite the next–to–leading terms in Veff – a substitutionthat only changes the O(1/N) terms.To get the high temperature 4d Veff from (41) we must match the Λ, Λ′ terms withthe 4d result for T 2 finite pieces as well as multiplying the 3d potential by T and usingeqs.

(7), (27). For the case ˜g = 0 we already found Λ = π2T/6 reproduces the correct 4dresult.

In fact, this identification also works in the ˜g ̸= 0 case. This can be checked byexamining the O(¯h) (i.e.

one loop in usual perturbation theory) result for which the thirdterm in (41) reduces to [6] Λϕ2(N˜λ + 2˜λ + 18˜g2)/24π2. The identification Λ = π2T/6reproduces the one–loop result for scalar QED given by Dolan and Jackiw [1].

Finally,Λ′ = π2T/3 reproduces the O(T 2) scalar QED longitudinal gauge boson polarization givenin refs. [2,4,5]e. To arrive at (41) we made the assumptions Λ ≫√χ > ˜λ/12π, ˜g2/12π,Λ ≫ϕ2 −√χ/4π ≫˜g2/(64)2 [13].eTo compare we must rescale ϕ2 →ϕ2/N, ˜g2 →˜g2N and set N = 2.13

4. Abelian Higgs Model, 6g2/λ ∼O(N) and O(N23).The case 6g2/λ ∼O(N).

In section 3 we considered λ fixed as N increases. In thiscase we must assume (λN) fixed as N increases.

For a weakly coupled model this meansthe scalar self–coupling is very weak. First let us consider the pure scalar case of section2.

We again work in 3d. To obtain the leading corrections we make the replacements˜λ = ˜λ′/N, χ →χ/N, ˆχ →ˆχ/N in (8) and (11).

Then VNtl is at most of O(1/N); theremainder of (11) is of O(1) and O(1/√N). Keeping only the O(1) and O(1√N) termsthe answer can be exactly computed.

For non-negative χ, (11) gives:Veff = −32N˜λχ2 + 12χ(ϕ2 −˜v2) + Λχ4π2 −χ3212π√N + const. (43)χ should be eliminated by its leading order equation from the the first three terms above.For g ̸= 0 the computation of all O(1/√N) terms is nontrivial.

The replacements ˜λ =˜λ′/N, χ →χ/N in (33) does not give all O(1/√N) terms. The O(1) terms can howeverbe found in this way.

The final expression isVeff=−32N˜λχ2 + 12χ(ϕ2 −˜v2) + Λχ4π2+3Λ˜g24π2 ϕ2 −˜g312πh(Λ′/2π2 + ϕ2)32 + 2|ϕ|3i+ const. (44)χ is given by the same solution as in the ˜g = 0 case.

Eq. (44) is α independent tothis order and is valid for non-negative χ, √χ and Λ ≫χ, ϕ2 and ϕ2 ≫˜g2/(64)2.

Thematching conditions for the linear divergences is the same as in section 3.The case 6g2/λ ∼O(N23). Here we must assume (λN23) fixed as N increases.

We cancompute the leading corrections by using the replacements ˜λ = ˜λ′/N23, χ →χ/N23 andˆχ →ˆχ/N23 in the results of section 2 and 3. For the pure scalar case, VNtl in eq.

(11) givesat most O(N−23) corrections. The rest of eq.

(11) gives O(N13) and O(1) corrections. Forg ̸= 0, the gauge-loop ring sum included in (33) gives O(1) and also more subdominantcorrections.

Thus the O(N13)+O(1) corrections can then be easily found in this way. Thefinal expression isVeff=−3N132N23 ˜λχ2 + N132 χ(ϕ2 −˜v2) + N13Λχ4π2−χ3212π+3Λ˜g24π2 ϕ2 −˜g312πh(Λ′/2π2 + ϕ2)32 + 2|ϕ|3i+ const.

(45)χ is found from its O(N13) equation of motion, and the comments after (44) apply.For negative χ we have to give meaning to eq. (15).

One possibility is to simply useTr ln(−⃗∂2 + χ) = Λχ/2π2+const. This leads to a real potential, but we have no rigorousway of justifying this prescription.14

5. The Phase Transition.We first discuss our results generally before investigating the phase transition.

Asmentioned in the introduction, ordinary 3+1d perturbation theory in ¯h is not reliablefor high temperature studies of spontaneously broken theories because it suffers frominfrared divergence problems just near the value of the temperature at which the phasetransition from the symmetric phase to the broken symmetry phase occurs. One can tryto circumvent this problem either by trying to resum certain infinite classes of Feynmandiagrams to all orders in ¯h or by performing perturbation theory in another parameter, say1/N.

In order to extract reliable results such a new perturbation series, if it is calculableto any given order in 1/N, must be renormalizable and avoid infrared divergence problemsto any given order in 1/N.For a weakly coupled 3+1d abelian Higgs model at high T with 6g2/λ ≤O(1) weintroduced an auxiliary field χ in the Lagrangian so as to develop a systematic expansionfor Veff in 1/N, where N is the number of real scalar fields. This was the content of sectionthree.

The O(N) and O(1) corrections in this expansion are renormalizable and avoid theinfrared divergence problems of perturbation theory in ¯h. The result can be expandedto all orders in ¯h and in fact corresponds to summing certain infinite classes of ordinaryFeynman diagrams in ¯h.

The O(N) result for Veff incorporates the sum of all superdaisygraphs of Dolan and Jackiw [1] at O(N); the next–to–leading order result incorporatesmuch more, i.e. the sum of all Feynman diagrams of O(1) as well as O(1) contributionsfrom the superdaisies.

The O(1) gauge contributions are given by a gauge loop ring sum(but not quite the simple one of [4]). The beauty of our systematic approach over trying toadd up some infinite classes of diagrams by hand is that there is no overcounting of O(¯h2)- or any other - Feynman diagrams and in addition the contraint of renormalizability helpsto locate all relevant contributions to Veff in a consistent computation.For weak λ and strong g it is not possible to define an expansion in 1/N, and for aweakly coupled theory it is only possible to set 6g2/λ ∼O(N) if the scalar self-coupling isnot just weak but very weak.

In this case the expansion of section 3 is unreliable becausethe dominant contribution is no longer from the pure scalar fields; this is demonstratedsoon. For a very weakly coupled 4d pure scalar field theory at high temperature withλN ∼O(1) it is possible to write Veff as the sum of O(1) and O(1/√N) terms, withsubdominant terms of O(1/N).

In section 4 these were found to give just the daisy sum(not superdaisy sum) of Dolan and Jackiw [1]. If computed consistently to O(1/√N) thedaisy sum cannot lead to spurious field dependent terms that come from overcounting ofFeynman diagrams in ¯h.

The daisy sum is of course renormalizable and avoids infrareddivergence problems. In the gauged case with very weak λ and 6g2/λ ∼O(N) a calculationof Veff to O(1/√N) is complicated because the contribution from the gauge sector involvesmore than just the gauge–loop ring sum of section 3.

In fact, if one just adds the purescalar daisy sum result to O(1/√N) with the gauge–loop ring sum to O(1/√N) then theresult is multiplicatively nonrenormalizable in the context of the 3d field theory. We didnot find a simple expression for the O(1/√N) corrections involving gauge fields and do notknow if these corrections have infrared divergence problems or not.

The O(1) corrections15

however have no such problem and in addition give a gauge fixing independent effectivepotential to leading order.For 6g2/λ ∼O(N23) it is possible to easily determine the O(N13) and O(1) contribu-tions to Veff, with subdominant contributions of O(N−13). These were found in section4 and have the interpretation of the daisy sum of the pure scalar case plus a gauge–loopring sum taken in the limit of vanishing (field dependent) scalar mass.

The effective po-tential to O(1) has no infrared divergence problems, is renormalizable and is gauge fixingparameter independent.We now study the implications of our results for the phase transition in the weaklycoupled abelian Higgs model. As noted in the introduction, we derived our results byworking with an effective 3d field theory (given by eq (28)) and matching linearly divergentterms in the effective theory with finite O(T 2) corrections in the full 4d model.The“neglected terms” therefore incorporate subleading terms that were dropped due to theapproximations made in calculating the 3d effective potential, as well as finite terms thatthe 3d effective theory cannot account for and that arise when the full 4d model is used.We stress that when χ ≪T 2 and φ2 ≪T 2, the 3d effective theory is sufficient to calculatethe leading (i.e.

T dependent) finite corrections. This is of course the region of interestfor the phase transition.The case 6g2/λ ≤O(1).

Our main result is the leading + next-to-leading order hightemperature Veff(φ2) for a 3+1d abelian Higgs model with N real scalars, computed inthe systematic 1/N expansion. For the tree Lagrangian given by (21) we obtainedVeff=−3N2λ χ2 + N2 χ(φ2 −v2) + T 2χ24 (N + 2 + 18g2λ )−T12π 3χ + λ3(v2 −T 2/12)!

32+ (N −1)χ32−T(6g2/λ)3212π χ + λ6(v2 + T 2/12)! 32+ 2 χ + λ6(v2 −T 2/12)!

32+Vα + neglected terms + O(1/N) terms. (46)where Vα isVα = −T12πh(χ + αg2φ2)32 −χ32 −α32g3|φ|3i,(47)and χ(φ) is the solution of the equation ∂Veff/∂χ = 0.

To next–to–leading order it issufficient [9] to use not the full solution but the O(N) solution of (19),√χ = λT48πvuut1 + 32π2λ 12φ2T 2 −12v2T 2 + 1!−1. (48)There is another solution of √χ which comes with an overall minus sign, but this isunphysical [6,8,9].

Our Veff is valid for T ≫√χ > λT/12π, g2T/12π and T 2 ≫φ2 −16

T√χ/4π ≫g2T 2/(64)2.Also, eq. (48) was used to rewrite the O(1) terms in (46),which adds λ(∂V1/∂χ)2/3N to the O(1/N) terms, where V1 is the O(1) part of (46).

Inthe Landau gauge, and near T =√12v, this particular O(1/N) term is negligible forχ/T 2 > (λ + 9g2)(1 + (6g2/λ)32/2)/3N2.In our approach vacuum polarization effects at large external momentum played animportant role in the same way as they did for the previously studied [6] g = 0 case.It is known [6] that at order N, the potential (46) admits no 1st order phase transition.The O(N) potential is exact in the small χ limit (up to 4d corrections).To O(N),dVeff/dφ2 = ∂Veff/dφ2 = Nχ/2. At φ2 = 0 this vanishes at T 22 = 12v2.

For T >√12vthe origin is a global minimum, and at T = T2 the potential grows away from the origin.For T > T2 the point χ = 0 is away from the origin and this has the interpretation [8,9]as the symmetry breaking minimum below T2. For the case g = 0, Root [9] has shownfor the 3d case that the point χ = 0 remains a local minimum at next–to–leading order.This analysis did not require a computation of Veff.

Root examined dV/dφ2 in the limitof vanishing χ and showed that the leading order gap–equation for χ (in our case eqs. (17) and (19)) was sufficient to show that Veff/dφ2 = 0 still has a solution at χ = 0 atnext–to–leading order.

His analysis can be extended to our gauged case. We will notpresent a detailed analysis here but instead refer the reader to Root and also ref.

[18]where, following Root, an analysis of the small χ limit in the full 4d high temperatureabelian Higgs model has been performed with the result that χ = 0 remains a point ofvanishing dVeff/dφ2 at next–to–leading order.We believe the fact that χ = 0 remains a point of vanishing dVeff/dφ2 is howeverinsufficient to deduce a second order phase transition to this order. As mentioned byRoot, for sufficiently large N the 1/N corrections cannot overwhelm the leading orderconclusion of a second order phase transition.

For N not arbitrarily large we would liketo know the global properties of Veff away from the point χ = 0, and in particular if thereis a point away from the origin at T =√12v where the 1/N corrections can producea new minimum and possibly even result in the breakdown of the 1/N expansion. Inaddition, for T >√12v the point χ = 0 never occurs, so the analysis of [9] is by itselfinsufficient to deduce φ2 = 0 remains a global minimum for N not arbitrarily large.

Ourcomputation of Veff sufficiently away from χ = 0 gives global information that Root’sanalysis cannot give. In addition, (46) appears to characterize the correct behavior in thelimit of vanishing χ.

Assuming no pathological behaviour occurs in an exact computationof Veff at next–to–leading order for √χ < λT/12π, g2T/12π we might expect that (46)gives a good description all the way down to √χ = 0 and also φ2 = 0 at √χ = 0.With these points in mind we now investigate if, more generally, (46) can exhibit afirst order phase transition for 6g2/λ ∼O(1).The critical temperature T2 is given by the vanishing of dV/dφ2 at φ2 = 0. Let uswrite Veff = VN + V1 + Vα, where VN is the O(N) potential.

Then since ∂VN/∂χ = 0 we17

obtaindVdφ2="∂V1∂χ + ∂Vα∂χ# ∂χ∂φ2 + ∂Veff∂φ2="∂V1∂χ + ∂Vα∂χ# 6λ +T8π√χ!−1+ Nχ2−T8παg2qχ + αg2φ2 −α32g3|φ|. (49)As indicated by our preliminary remarks, this vanishes at χ = 0 and at the origin thistranslates to the gauge fixing independent resultT 22 = 12v2.

(50)This is in fact the leading order result again (i.e. as if V1 were absent).

The reason is thatin a general model the O(1/N) terms in Veff are important in determining the O(1/N)corrections to T 22 (this is related to the fact that χ = 0 remains a point of vanishingdVeff/dφ2 at O(1)).For T > T2, φ2 = 0 is the local minimum.At φ2 = 0, Vα = 0 so Veff is gaugeparameter independent at this minimum. For there to be a first order phase transition,eq.

(49) must have zeros for T > T2 and, at T = T2, dV/dφ2 should be negative awayfrom φ2 = 0. Let us work in the Landau gauge, α →0 (in an arbitrary gauge α shouldbe not more than O(1) [4]).

Then at T = T2,∂V1∂χ = T 2212 1 + 9g2/λ −√6g4π (6g2/λ)!−T28π√χ3√3 −1 + 2(6g2/λ)32. (51)χ increases with φ2, hence a necessary condition for dVeff/dφ2 to become negative atT = T2 is that (51) becomes negative.

At g = 0, (51) vanishes only when √χ ∼πT2/6by which time the effective theory is no longer valid and the positive contribution todV/dφ2 from the O(N) term in (49) already dominates. Hence for g = 0 no first orderphase transition is possible by T = T2.

For 6g2/λ ∼O(1), g2 ≪2π, the same conclusioncan be reached, namely that within a consistent 1/N approximation the leading+next-to-leading Veff for our weakly coupled abelian Higgs model does not admit a first orderphase transition. (Note the absence of an O(|φ|3) term at T = T2 – see eq.

(48). )If 6g2/λ is of O(N) then (51) can become strongly negative, indicating the possiblebreakdown of the 1/N expansion.

Note also that the “next–to–leading” corrections tothe coefficient of χ in (46) actually compete with the O(N) coefficient and therefore ourresults cannot be used in this case.The case 6g2/λ ∼O(N). Our main result is the high T contribution to Veff whichsurvives in the limit of arbitrarily large N. For the tree Lagrangian given by (21) weobtained (after eliminating χ):Veff=λN4!

φ2(φ2 −2v2 + T 2/6) + g2T 28φ2−g3T12πh(T 2/6 + φ2)32 + 2|φ|3i+ O(1√N) + neglected terms. (52)18

Note that for weak coupling one must keep g and (λN) fixed as N increases. As mentioned,the potential above is α independent to O(1) in the way we computed it.

One can alsocheck that it is gauge fixing parameter independent with the slightly different gaugefixing considered in [4]. Eq.

(52) is strictly valid for T 2 ≫φ2 ≫T 2g2/(64)2. The criticaltemperature T2 determined from (52) is given by12v2 = T 22"1 + 3(6g2/λN) −g√3π√2(6g2/λN)#.

(53)We assume g ≪2π so that T2 here is lower than the leading order critical temperaturefor the 6g2/λ ≤O(N) case,√12v.f At T = T2 the minimum is no longer at the originand occurs at the point|φ|minT2= 12g(6g2/λN),(54)indicating a first order phase transition whose strength grows with the gauge coupling.Since we did not compute subleading corrections to Veff all we can say is that themodel exhibits a first order phase transition for sufficiently large N. This is in contrastto our results in the 6g2/λN ≤O(1) case.The case 6g2/λ ∼O(N23).This case is the most interesting because, unlike thecase 6g2/λ ∼O(N), the next–to–leading corrections are easily determined. The hightemperature potential isVeff=λN4!

φ2(φ2 −2v2 + T 2/6) + g2T 28φ2 −χ3212πθ(χ)−g3T12πh(T 2/6 + φ2)32 + 2|φ|3i+ O(N−13) + neglected terms. (55)Here,χ =λN236(φ2 −v2 + T 2/12).

(56)We note that for weak coupling, λN23 must be held fixed as N increases and that the resultfor Veff is gauge fixing parameter independent to O(1). Finally, note the appearence ofthe step function θ(χ) = 1 for χ > 0 , 0 otherwise, in the expression for Veff.Asdescribed in section 4 there is an uncertainty involved in defining the gaussian integralover the quantum scalar fields when χ is negative; we have chosen an answer that in lightof the results for the cases 6g2/λ ≤O(1) and 6g2/λ ∼O(N) gives a physically appealingfAt the origin and at T = T2 the auxiliary field χ that was eliminated from (44) is negative, and onemay wonder if (44) and (52) are well defined in such a situation.

However, the leading divergent partof Tr(−⃗∂2 + χ)−1 is independent of χ and well defined. Hence, the linearly divergent term in (44) isunambiguous for negative χ.

In fact, to this order the quantum contribution from only the scalars is justthe O(T 2) shift to the tree–level mass of the scalars already derived by a one–loop O(¯h) calculation in[1].19

interpretation. Therefore, our results here are somewhat speculative.

We assume g ≪2πin what follows. Also Veff is strictly valid for T 2 ≫φ2, χ and φ2 ≫g2T 2/(64)2.The leading order critical temperature is the same as in the case 6g2/λ ≤O(1), i.e.T2 =√12v.

The O(N13), potential given by the first term in (55), does not admit a firstorder phase transition. The next–to–leading order T2 is modified.

We find the solutiondVeff/dφ2 at φ2 = 0 occurs at negative χ and it is again given by relation (53) (but notethat it reduces to 12v2 = T 22 for N →∞here). At T = T2 the global minimum is awayfrom the origin and at (54).

For 6g2/λ ∼O(N23) we have|φ|minT2→0 as N →∞,(57)indicating a first order transition that gets weaker and weaker as N increases.We end this section with the following discussion. If we chose a prescription withoutthe step function in (55) the potential is complex at or just above T2 given by (53) andclose enough to the origin.

Since we have all terms to O(1) the imaginary part cannot beremoved at sufficiently large N by subdominant corrections. We do not know the meaningof this result; neither do we know in cases a) and b) if subdominant corrections we havenot computed can lead to the possibility of a complex potential at T just above when thephase transition is supposed to take place.

However, one can check that in the case hereeven without the step function in (55) the expression for T 22 is real to O(N−13) and givenby (53) and also that the value of Veff at the minimum (54) is real to O(1). Our physicalresults, to the order we are working in, are therefore not really so speculative.6.

Conclusions.We have found that for large N the weakly coupled abelian Higgs model can exhibiteither first or second order phase transitions with first order behaviour possible for 6g2/λ ≥O(N). For 6g2/λ ∼O(N23) the model exhibits a weak first order phase transition at largeN.

Intuitively, this only says that the gauge coupling must be large enough so that thegauge contributions can dominate or at least compete with the pure scalar contributionsto Veff. For the three cases a) 6g2/λ ≤O(1), b) 6g2/λ ∼O(N) and c) 6g2/λ ∼O(N23)we estimate that our results are reliable whenever the parameter a) N−1, b) N−12 andc) N−13 is sufficiently small.

An inspection of our results for case a) shows that even forN = 4 the next to leading corrections to the coefficient of φ2 in Veff compete with theleading correction to the coefficient of φ2 in Veff. We do not believe any of our resultscan be directly applied to the N = 4 case; however they can give some indication of whathappens at N = 4 in a very weakly coupled theory with g ≪1, λ ≪1.

In this limit,cases b) and c) admit only a very weak first order phase transition. For case a) it can beargued on the basis of the 3d field theory (and dimensionality of the 3d couplings) thatthe coefficient of φ2 in Veff to lowest order in the couplings and to all orders in 1/N isgiven by the one–loop O(¯h) graphs.

This yields a critical temperature given by [1]T 22 (1 + 2/N + 18g2/λN) = 12v2. (58)20

For N = 4 and 6g2/λ ∼1 this gives a critical temperature T2 much below√12v anddemonstrates that our analysis of case a) is insufficient to rule out a significant first orderphase transition. Assuming that O(1/N) corrections to (46) are negligible it is possibleto use (46) to study the phase transition to temperatures below T =√12v.

The leadingorder solution √χ of (48) becomes negative and eventually complex near the origin aswe lower the temperature below T =√12v. To avoid this problem one can use the fullsolution for χ from (46), not just the leading order one.

This approach is not entirelyconsistent but the result of the analysis can be shown to be again a second order phasetransition to T ≥T2 given by (58). This only demonstrates that any first order behaviouris contained in O(1/N) corrections to Veff which we can expect to be small even at N = 4.Hence our result for all cases is that for a very weakly coupled model the phase transitionof the exact model, even at N = 4, will be for all practical purposes second order.For g ∼1 and N = 4, we believe that a reliable determination of the nature of thephase transition cannot be given by the O(N) + O(1) corrections in case a), nor by thedaisy and ring sum results of cases b) and c).

In all cases more subleading contributionsmust be computed than those that we have computed or currently exist in the literature.These extra contributions may be negligible even for g ∼1 and N = 4 but this is not apriori clear. However if the exact g = 0 model admits only a second order phase transitionthen we can expect that for g ̸= 0 and 6g2/λ ≤O(1) the exact model will admit at bestquite weak first order behaviour.Our physical results are in all cases gauge fixing parameter independent, they do notovercount any Feynman diagrams and do not suffer from any infrared problems.

We foundno terms of O(T 3|φ|) in Veff in any of the cases. One can argue on the basis of the 3dfield theory, dimensionality of the 3d couplings and renormalizability that any such termsmust be very small in a very weakly coupled model.

More generally, any such terms mustbe vanishingly small in the limit of large N.Acknowledgements. I am very grateful to P. Weisz with whom I had many fruitfuldiscussions and for a critical reading of a draft of this paper and to M. K. Gaillard, M.Moshe and F. Zwirner for various useful discussions.

I would also like to thank M. Carenaand C. Wagner for useful discussions and in particular for pointing out that nonzeroexternal momentum vacuum polarization effects in the gauged case, as in the pure scalarcase, can play an important role.Finally, I wish to thank K. Sibold for a clarifyingdiscussion about super-renormalizability.Note Added. As we completed this work we recieved a paper by J. March–Russell,LBL–32540, PUPT–92–1328, which also considers 2nd order phase transitions in gaugetheories, but using ǫ–expansion techniques.21

References[1] L. Dolan and R. Jackiw, Phys. Rev.

D9: 3320 (1974). See also S. Weinberg, Phys.

Rev. D9: 3357 (1974), and for anearlier work on symmetry restoration see D. A. Kirzhnits and A. D. Linde, Phys.

Lett. B42: 471 (1972).

[2] M. E. Carrington, TPI-MINN-91/48-T.[3] M. Dine, R. G. Leigh, P. Huet, A. Linde and D. Linde, SLAC-PUB-5741, SCIPP-92-07, SU-ITP-92-7. [4] P. Arnold, EW/PT–92–06, NUHEP–TH–92–06.

[5] C. G. Boyd, D. E. Brahm and S. D.H. Hsu, CALT-68-1795, HUTP-92-A027, EFI–92–22. [6] V. Jain, MPI–Ph/92-41.

[7] P. Ginsparg, Nucl. Phys.

B170: 388 (1980). [8] S. Coleman, R. Jackiw and H. D. Politzer, Phys.

Rev. D10: 2491 (1974).

[9] R. G. Root, Phys. Rev.

D10: 3322 (1974). Our vacuum polarization Πˆχˆχ(⃗k2) is just minus the 3d B(⃗k2) given here.We pick Πˆχˆχ so that the ˆϕ one–loop contribution to the effective action contains a term −3ˆχΠˆχˆχ(−⃗∂2)ˆχ/(2˜λ).

Πˆχˆχis unambiguous because its value in the limit of zero external momentum is fixed. [10] W. A. Bardeen and M. Moshe, Phys.

Rev. D28: 1372 (1983) and D34: 1229 (1986).

[11] M. L¨uscher, unpublished notes. There is a technical point here.

We used the form of the auxiliary field addition in(3) appropriate for the Minkowski space Lagrangian. If one works in euclidean space then the 1/N expansion can bejustified at next–to–leading order by taking an imaginary background χ and a real fluctuation ˆχ.

Then, the gaussianintegral over ˆχ is convergent for values of χ which we will be interested in. With our form (3) we consider the backgroundreal and the fluctuations imaginary.

[12] M. K. Gaillard, in Particle Physics: Carg`ese 1987 (NATO ASI, Series B: Physics, Vol. 173), ed.

by M. Levy, et.al., Plenum, New York, 1988. See also M. K. Gaillard and T. R. Taylor, LBL–31464 (1992).

The precise answer forthe divergent terms in the effective theory depends on the regularization prescription used. Since we identify theultraviolet regularization scale with the scale at which the full theory becomes important this ambiguity only saysthat the effective theory cannot account for the precise way in which full theory comes into play at high energy; i.e.the effective theory could be the low energy limit of several different models.

Thus, each of the divergent terms shouldbe given arbitrary coefficients (consistent with all symmetries). This arbitrariness can be removed by matching thedivergent terms in the effective theory with the results of known calculations in the full model.

In our case, the lineardivergences in the effective 3d theory are related to quadratic divergences – and hence O(T 2) corrections – in the fulltheory. These are well known and so the match is easy.

In hindsight we found that for the pure scalar case that thesame matching condition worked for all the linear divergences, while in the gauged case we needed two conditions. [13] As pointed out by P. Weisz, Tr ln(−⃗∂2 + ap−⃗∂2 + b) can be evaluated exactly in 3d.

With a sharp momentum cutoffΛ, and the limits Λ2 ≫b ≫a2 one obtains the approximate solution Tr ln(−⃗∂2 + ap−⃗∂2 + b) ≈Tr ln(−⃗∂2 + b) −ab ln(Λ2/b)/4π2. The log divergent term can be ignored in comparison with the linearly divergent term whenever(a/Λ) ln(Λ2/b) ≪1, which is clearly true with our assumptions.

The log divergent piece can be ignored in comparisonto the finite piece from Tr ln(−⃗∂2 + b) whenever (a/√b) ln(Λ2/b) ≪1, which is an extra assumption we will make. [14] J. R. Espinosa, M. Quiros and F. Zwirner, CERN–TH–6451/92.

[15] Quantum Field Theory, C. Itzykson, J.-B. Zuber, McGraw–Hill 1985, page 378.

[16] P. Weisz, private communication. [17] For −⃗∂2 < 4χ, Π is well approximated by zero.

The largest error incurred by using (39) everywhere is proportional toδ = Tr ln(−⃗∂2 + ˜g2ϕ2) −Tr ln(−⃗∂2 + ˜g2ϕ2 −˜g2√χ/4π) with the ⃗k2 integral from 0 to 4χ. For ϕ ̸= 0, χ ≈0, δ goeslike χ32 , i.e its contribution to the effective mass vanishes as χ →0.

In addition, δ is negligible for ϕ2 ≫√χ/4π andϕ2 ≈√χ/4π. We will drop δ as a subdominant term, however the fact that it goes like χ32 for small χ is an implicitassumtion made in the analysis in section 5.

[18] M. Carena and C. Wagner, MPI preprint (1992).22

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