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On the Finite Temperature Effective Potential in

arXiv:hep-ph/9209233v1 14 Sep 1992MPI-Ph/92-67August 1992On the Finite Temperature Effective Potential inScalar QED with N Flavors.M. Carena and C. E. M. WagnerMax Planck Institut f¨ur PhysikWerner Heisenberg InstitutF¨ohringer Ring 6D-8000 M¨unchen 40, Fed.

Rep. of Germany.AbstractThe effective potential of scalar quantum electrodynamics with N flavors ofcomplex scalar fields is studied, by performing a self consistent 1/N expansionup to next to leading order in 1/N.Starting from the broken phase at zerotemperature, the theory exhibits a phase transition to the symmetric phase atsome finite temperature Tc. We work in general covariant gauges and demonstratethe gauge invariance of both, the critical temperature Tc and the minimizationcondition at any finite temperature T. Furthermore, the only minimum of thepotential is at zero scalar vacuum expectation value for any temperature T ≥Tcand varies continuously to nonzero values for temperatures below Tc, implying theexistence of a second order phase transition.1

1IntroductionIn the last years, there has been a great interest in understanding the nature of the finitetemperature symmetry restoration phase transition in the Standard Model (SM) [1] -[12]. Recently, the main motivation arose from the realization that the rate for baryonnumber violation in the SM at finite temperature is much larger than what it was orig-inally assumed [4] - [5].

In fact, once the temperature is much higher than the height ofthe energy barrier between the different topologically distinct vacua [13], the exponentialsuppression of the anomalous baryon number violation processes, present at low temper-ature, disappears [4]-[5]. Thus, at sufficiently high temperatures, the anomalous baryonnumber violation processes could wash out the primordial baryogenesis generated, insome grand unified scenarios, at the very early stages of the universe [14].

In such cases,the generation of baryon number could have occurred through non-equilibrium processesin the transition from the symmetric to the broken SU(2)L × U(1)Y phase [4]. Quitegenerally, models of electroweak baryogenesis [15] fulfill Sakharov’s three requirements[16] for the generation of baryon asymmetry.

Besides the nonconservation of B the un-derlying dynamics must involve nonequilibrium processes and violate CP. The violationof CP is present in the SM and its extensions, although in the SM it is probably too weakin order to generate the required baryon asymmetry [15].

To satisfy the nonequilibriumcondition, these models rely on the presence of a first order phase transition, sufficientlystrong in order to imply the suppression of baryon number violation processes in thebroken phase.The nature of the electroweak phase transition is a fundamental question whichneeds to be carefully explored and requires a detailed study of the effective potential ofgauged scalar theories at finite temperature. A one loop analysis of the abelian theorywas first done by Dolan and Jackiw [2].The one loop effective potential shows theexistence of a first order phase transition, whose strength depends on the values of theelectroweak gauge and the scalar quartic coupling constants [1].

However, the naiveone loop definition of the effective potential is known to have serious infrared problems,such as the appearance of a complex effective potential even at temperatures above2

the critical temperature, Tc, at which the transition from the broken to the symmetricphase takes place. Moreover, it is found that the critical temperature itself is a gaugedependent quantity.

It was realized in Ref. [2] that both - the infrared and the gaugedependence - problems do not appear, if only the dominant terms in the high temperatureexpansion are kept.

Within such an approximation, a definition of a gauge invariantcritical temperature can be achieved by assuming the existence of a second order phasetransition. However, in order to study the nature of the phase transition, higher orderterms in the effective potential need to be computed.Quite recently, an improved analysis of the one loop effective potential has beenconsidered by several authors [7] -[12].The improved one loop effective potential isgenerically obtained by the inclusion of the finite temperature one loop corrections, atzero external momentum, to the boson propagators.

When this process is carefully done,in order to avoid double counting of diagrams, it is equivalent to the inclusion of the socalled ring diagrams in the one loop computation [10]. Such resummation of diagramssolves several problems inherent to the one loop expansion and leads to a first orderphase transition, although weaker than that one found within the one loop expansion[11].

However, in the pure scalar case the ring expansion breaks down at temperaturesclose to the critical one, |T −Tc| ≃˜λTc, with ˜λ the scalar quartic coupling [12]. Hence, itcan not be used to analyze the behavior of the effective potential at T ≃Tc.

In addition,after the inclusion of the gauge fields, for values of the squared gauge coupling g2 of theorder of the quartic coupling it is not possible to conclude that the phase transition isfirst order based solely on the ring expansion. This is due to the fact that, within thering approximation, and for g2 ≃˜λ, the difference between the transition temperatureT0, at which the curvature at the origin vanishes, and the critical temperature Tc isof order g2Tc, but the ring expansion breaks down for |T −Tc| ≤g2Tc [12].

Thus, ananalysis beyond the improved one loop approximation is necessary in order to studythe nature of the phase transition for g2/˜λ ≃O(1), which is the phenomenologicallyinteresting region in which the Higgs mass is of the order of the gauge boson mass.As it has been explained in the seminal work by Dolan and Jackiw [2], for the purescalar case a self consistent resummation of the diagrams contributing to the effective3

potential can be done, by extending the theory to include N flavors of self interactingscalar fields and performing a 1/N study of this model. The 1/N expansion providesinformation which does not appear at any fixed order of perturbation theory and, hence,it is a more appropriate tool to study the nature of the phase transition, which is initself a nonperturbative phenomenon.

Studying the behaviour of the effective squaredmass, by doing a self consistent 1/N expansion, Dolan and Jackiw [2] showed that theinfrared problems inherent to the pure O(N) scalar theory disappear. The explicit formof the effective potential at finite temperature was, however, not derived in this analysis,and it was studied in Refs.

[17] and [18]. A similar resummation of diagrams has beenrecently performed in Ref.

[19], while an analysis of the N component - φ4 theory, by amethod based on average fields, has been considered in Ref. [20].The validity of the finite temperature computations in the scalar O(N) linear modelwas questioned, when it was realized that, in the continuum limit, the effective scalarpotential at zero temperature was either complex, unstable or did not allow the sponta-neous breakdown of the O(N) symmetry [21] - [23].

This issue was clarified by Bardeenand Moshe Moshe [24] , who explained that the problems appearing in the large N com-putations were not associated with the 1/N expansion but with a more fundamentalnonperturbative property of the O(N) scalar model in four dimensions, which is theissue of triviality. In fact, for positive bare quartic coupling, the continuum limit ofthe four dimensional O(N) theory is trivial, in the sense that the renormalized couplinggoes to zero when the cutoffis removed.

This implies that, to work with a nontrivialtheory, one has either to take a negative bare quartic coupling, rendering the theoryunstable at finite temperature [24], or treat the theory as an effective one, by keeping afinite effective cutoff. For a large effective cutoff, the 1/N expansion at zero and finitetemperature can be defined in a consistent way, avoiding the problems mentioned above.Thus, in the following, we shall assume that we are dealing with an effective theory validup to an energy scale of the order of an effective cutoffΛ, at which, quite generally, newphysics should appear.It must be mentioned, that the nature of the phase transition for scalar QED with Nflavors and a scalar Higgs heavier than the gauge boson, have been also studied by using4

a 4 −ǫ expansion [25]-[26]. Within such approximation, the theory may be analyzed byperforming a renormalization group study for small ǫ.

The high temperature theory maybe treated as an effective three dimensional theory and thus, to obtain physical results,the expansion parameter must be taken to be ǫ = 1. In this framework, the critical valuefor the number of flavors, Nc, below which the phase transition becomes first order, wasfound to be surprisingly large, Nc = 183.

Based also on this analysis, Halperin, Lubenskyand Ma [25] predicted a first order phase transition for the smectic-A to nematic phasetransition in liquid crystals. However, when contrasted with experiment, this transitionwas found to be second order [27].

In addition, for ǫ →1, a (2 + ǫ) expansion givesresults which are at variance with those ones coming from the (4−ǫ) expansion [25],[28].In particular, in the case in which the Higgs particle is heavier than the gauge boson,the (2 + ǫ) expansion indicates a much lower critical value of N, if any, for the threedimensional theory [28].This paper is organized as follows: In section 2 we present the model, a scalar theorywith N flavors of complex scalar fields coupled to an abelian gauge field. In section 3we derive the general expression of the effective potential up to next to leading orderin 1/N.

In section 4 we concentrate on the leading order results, and we show that asecond order phase transition takes place in this case. In section 5 we carry out theanalysis of the critical temperature and the nature of the phase transition up to nextto leading order in 1/N.

We start discussing three related aspects of the theory, whichare the renormalization of couplings, the triviality of the theory and the absence oftachyons. Then, we analyze the radiative corrections at finite temperature to finallyperform the minimization of the effective potential up to next to leading order in 1/N.We demonstrate that a gauge invariant expression for the position of the minimum isobtained.

In addition, the position of the minimum is not modified by next to leadingorder corrections and, hence, the phase transition remains second order. A comparison ofour large N results with those previously derived within the improved one loop methodis performed in section 6, where we analyze the source of the discrepancy between theresults obtained in both approaches.

We reserve section 7 for our conclusions.5

2The ModelThe Lagrangian density for a scalar theory with N flavors of complex scalar fields ininteraction with an abelian gauge field is given byL = −14FµνF µν −12α (∂µAµ)2 + 12 (Dµφa)† (Dµφa) −˜λ4!|φ|2 −v22 . (1)In the above expression, a is a flavor index taking values from 1 to N, φa is a complexscalar field, |φ|2 = φ†aφa, Aµ is an abelian gauge field, and Dµ is the covariant derivativeassociated with it.

In addition, we have included a gauge fixing term with α being thegauge fixing parameter (α = 0 is the Landau gauge). Note that, at vanishing gaugecoupling, this model reduces to the O(2N) linear sigma model.Following the method first proposed by Jackiw [29], to compute the effective potentialV (ˆφ) for this theory, we shift the fields φa →ˆφa+φa, where ˆφa is the vacuum expectationvalue of the scalar field and φa stands for its quantum fluctuations.

The Lagrangian mustbe truncated by dropping all terms which are linear in the φa quantum fluctuations. Thenew Lagrangian, L( ˆφa + φa, Aµ), may be decomposed into a term quadratic in the Aµand φa fields and an interaction term.

The effective potential is given by the sum of thetree level potential plus the one loop contributions, obtained from the bilinear part ofthe shifted Lagrangian, plus higher order loop contributions, given by the sum over allconnected one-particle irreducible vacuum graphs for the theory described by the shiftedLagrangian density L( ˆφa + φa, Aµ).The quartic interactions pose a disadvantage for the diagrammatical analysis of thetheory. Therefore, following Refs.

[21] - [22], we shall eliminate the scalar quartic termthrough the introduction of an auxiliary field χ. The modified Lagrangian readsL(φa, Aµ, χ) = L(φa, Aµ) + 32˜λ"χ −˜λ6|φ|2 −v2#2.

(2)The Euler-Lagrange equation for χ is a constraint equation relating χ with the scalarfield φ. In fact, the effective potential for the modified theory V (ˆχ, ˆφa) reduces to theone of the original theory if ˆχ satisfies the requirement∂V∂ˆχ = 0,(3)6

which defines the vacuum expectation value ˆχ as a function of ˆφ.A self consistent 1/N expansion can be defined if the gauge and quartic couplingconstants g and ˜λ, respectively, depend on N so that in the limit of N →∞, g2N and˜λN go to constant values. Hence, for the purpose of our study it is better to definethe new coupling constants e ≡g√N and λ ≡˜λN.

Once the field χ is introduced theLagrangian density of the theory readsL(φa, Aµ, χ)=−14FµνF µν −12α (∂µAµ)2 + 12∂µφa,i∂µφa,i −e√N Aµǫijφa,j∂µφa,i+e22N AµAµφa,iφa,i + 3N2λ χ2 −χ2|φ|2 −v2,(4)where i = 1, 2, φa,1 and φa,2 are the real and imaginary parts of the field φa, respectively,and summation over a and i is understood. Although the introduction of the field χ doesnot alter the dynamics of the full theory, it does lead to a new perturbation series, inwhich the 1/N expansion has a simpler diagrammatic interpretation.

After consideringthe shifted fields, which now include the shift for the auxiliary field χ,φa →√N ˆφa + φa,χ →ˆχ + χ(5)and ignoring the linear terms in the quantum fluctuations χ and φa, the shifted La-grangian readsL=−14FµνF µν −12α (∂µAµ)2 + e22 AµAµ ˆφa,i ˆφa,i −eAµǫij ˆφa,j∂µφa,i−e√N Aµǫijφa,j∂µφa,i + e22N AµAµφa,iφa,i + e2√N AµAµ ˆφa,iφa,i + 12∂µφa,i∂µφa,i+3N2λ χ2 −12 ˆχφa,iφa,i −√N ˆφa,iχφa,i −12φa,iφa,iχ −Vtree,(6)where the tree level potential is given by,Vtree = −3N2λ ˆχ2 + N2 ˆχ|ˆφ|2 −v2. (7)In the above, we have followed Root [22] in rescaling the vacuum expectation valueˆφa and, consistently, v by a factor√N.

This simplifies the 1/N counting. In fact, ifthis rescaling were not done, the vacuum expectation value would be of order N in thenatural scales of the theory, ˆφ2 = NF(µ2, T 2), where F is a function of the bare mass7

µ2 = −˜λv2/6, the temperature T and the couplings of the theory. This extra factor ofN would have to be properly considered while doing the expansion at next to leadingorder.

The physical results, of course, would remain unchanged.The above Lagrangian may be decomposed into a quadratic and an interacting part.The expression of the quadratic Lagrangian readsLquad=12Aµ(∂2 + e2 ˆφ2)gµν −∂µ∂ν1 −1αAν −12φa,i∂2 + ˆχφa,i+3N2λ χ2 −√N ˆφa,iχφa,i −eAµǫij ˆφa,j∂µφa,i(8)from which it is easy to deduce the inverse propagator matrices relevant for the com-putation of the effective potential. The interacting part of the Lagrangian LI, on theother hand, determines the vertices of the boson fields interactions.

For simplicity, wewill assume in the following thatˆφ1,1 = ˆφ,(9)while all other vacuum expectation values are zero. In fact, it is not important whichstate is chosen as the ground state of the theory.Independent of such a choice theoriginal global U(N) symmetry is spontaneously broken to U(N-1) and 2N-1 Goldstonebosons are generated, one of which is eaten by the gauge field through the usual Higgsmechanism.We obtain two decoupled inverse propagator matrices.

One that mixes the gaugefield Aµ with φ1,2, M(Aµ, φ1,2), and the other one which mixes the auxiliary field χ withφ1,1, M(χ, φ1,1). Thus, in momentum space representation, the nonzero components ofthe inverse propagator matrices read,iD−1µν=−gµνk2 −e2 ˆφ2−kµkν(1 −1α)iD−1a,i;b,j=δabδij(k2 −ˆχ)iD−1χχ=3NλiD−11,1;χ=−ˆφ√NiD−11,2;µ=iekµ ˆφ.

(10)8

Considering the above expressions it is straightforward to compute the determinant ofthe inverse propagator mass matrices, which in Euclidean space are given bydet M(Aµ, φ1,2) = (k2 + ˆχ)(k2 + e2 ˆφ2)3α"k2 + e2 ˆφ2 ˆχα(k2 + ˆχ)#,(11)det M(χ, φ1,1) = −3Nλ"k2 + ˆχ + λ3ˆφ2#. (12)Furthermore, the propagators of the scalars and gauge bosons, which are relevant forthe computation of the effective potential up to next to leading order in 1/N may bethen obtained from Eqs.

(10), (11) and (12), and in Euclidean space are given byiDµν = δµν −kµkνk2!1k2 + e2 ˆφ2 + kµkνk2α (k2 + ˆχ)hk4 + ˆχk2 + αˆχe2 ˆφ2i(13)iDχχ = −(k2 + ˆχ) λ/3Nhk2 + ˆχ + λˆφ2/3i(14)3The Effective PotentialAs we mentioned above, in order to analyze which Feynman diagrams contribute tothe effective potential up to next to leading order in 1/N, it is necessary to consider thebehaviour in 1/N of the propagators and of the interaction vertices. From Eqs.

(10)-(14)we observe that a factor 1/N is associated with Dχχ and a factor 1/√N is associatedwith D1,1;χ. Moreover, a factor of order N is associated with any closed loop of φa fields,due to the summation over all possible internal fields.

From the interaction part of theLagrangian in Eq. (6), we observe that a factor 1/√N is associated with the derivativecoupling of the scalar fields φa to the gauge bosons, while a factor 1/N appears in anyquartic vertex A2φa,iφa,i.

Note that the vertex χφa,iφa,i has, instead, a factor 1 in thecoupling.The only diagram contributing to the potential at leading order in 1/N [17],[18], Vl.o.,is the closed loop involving the 2(N-1) Goldstone bosons which do not mix with eitherAµ or χ,Vl.o. = 2(N −1)2Zk lnk2 + ˆχ.

(15)9

We perform the finite temperature computations in the imaginary time formalism: Aftera Wick rotation to Euclidean space we impose periodic boundary conditions on the timedirection of length L = T −1 ≡β. For simplicity of notation, we have definedZk f(k) ≡1βn=∞Xn=−∞Zd3k(2π)3f(ωn,⃗k),(16)with ωn = 2πn/β.The computation of the effective potential at next to leading order in 1/N involvestwo type of contributions.

There are one loop contributions, involving the two nontrivialpropagator matrices,V 1.l.n.l.o = 12Zk [ln det (M(Aµ, φ1,2)) + ln det (M(χ, φ1,1))] ,(17)and there are those coming from multiloop one particle irreducible graphs, depictedin Fig.1. Observe that any multiloop contribution involving the mixing of χ and Aµ,like those depicted in Fig.2, vanish in the regularized theory.After some work, theresummation of all the multiloop diagrams contributing to next to leading order in 1/NgivesV n−loopn.l.o=−∞Xn=112nZk−λ (k2 + ˆχ)3hk2 + ˆχ + λˆφ2/3inBn(k2, ˆχ)−∞Xn=112nZk Trhie2DµνΠναin ,(18)which can be rewritten asV n−loopn.l.o=12Zk ln1 +λ (k2 + ˆχ)3k2 + ˆχ + λˆφ2/3B(k2, ˆχ)+12Zk Tr lnδαµ −ie2DµνΠνα,(19)whereB(k2, ˆχ) =Zp1(p2 + ˆχ)h(k + p)2 + ˆχi(20)andΠνα(k, ˆχ) =Zp(2p + k)ν (2p + k)α(p2 + ˆχ)h(p + k)2 + ˆχi −2δναZp1p2 + ˆχ,(21)10

is the finite temperature vacuum polarization contribution.Furthermore, using thepropertiesdet M(φ1,2, Aµ) =k2 + ˆχdeth−iD−1µνi(22)anddet M(φ1,1, χ) =k2 + ˆχD−1χχ,(23)we arrive to the final formal expression for the effective potential up to next to leadingorder in 1/N,V (ˆφ, ˆχ)=Vtree + Vl.o. + V 1.l.n.l.o.

+ V n−loopn.l.o.=−3N2λ ˆχ2 + N2 ˆχ|ˆφ|2 −v2+ (2N)2Zk lnk2 + ˆχ+12Zk ln deth−iD−1µν (k, ˆχ, ˆφ) −e2Πµν(k, ˆχ)i+12Zk ln"(k2 + ˆχ) (1 + λB(ˆχ, k2)/3) + λˆφ2/3k2 + ˆχ#. (24)As we discussed above, the expression for the effective potential of the original theorymay be obtained from the above equation by imposing the condition∂V (ˆφ, ˆχ)∂ˆχ= 0,(25)which determines the expression of ˆχ as a function of ˆφ2.

Furthermore, as it was firstnoticed in Ref. [22], in order to obtain the effective potential up to next to leading orderin 1/N, it is sufficient to solve the gap equation, Eq.

(25), up to leading order in 1/N.In fact, calling ˆχ(ˆφ2) the exact solution to the gap equation and ¯χ(ˆφ2) the leading ordersolution, then,ˆχ(ˆφ2) −¯χ(ˆφ2) = O(1/N). (26)Expanding the effective potential around ˆχ(ˆφ2) = ¯χ(ˆφ2), we haveV (ˆφ, ˆχ) = V (ˆφ, ¯χ) + ∂V (ˆφ, ¯χ)∂¯χ(ˆχ −¯χ) + O(1/N).

(27)However, since ¯χ(ˆφ) is the solution to the gap equation at leading order, it follows that∂Vl.o. (ˆφ, ¯χ)/∂¯χ = 0.

Therefore, the second term in the above equation is also of order1/N,V (ˆφ, ˆχ(ˆφ)) = V (ˆφ, ¯χ(ˆφ)) + O(1/N). (28)11

Thus, in order to compute the effective potential up to next to leading order in 1/N wejust need to consider Eq. (24) with ˆχ given by its leading order expression, ˆχ(ˆφ) = ¯χ(ˆφ).Moreover, for the extent of this work we shall concentrate in computing the locationof the extrema of the effective potential, which are derived from the condition,dV (ˆφ, ¯χ)dˆφ2= 0.

(29)In fact, the number and location of the extrema of the effective potential provide suf-ficient information to study the nature of the phase transition. In addition, as we willshow below, a gauge independent expression for the solutions to Eq.

(29) is found. Ob-serve that if Eq.

(29) is not fulfilled for any nontrivial value of the scalar field, the onlyminimum would be at the origin and the global U(N) symmetry, together with the localU(1) symmetry of the theory would be preserved. From Eq.

(24), we find the relation∂Vl.o./∂ˆφ2 = N ¯χ/2, and recalling the fact that ∂Vl.o./∂¯χ = 0, we obtaindVdˆφ2=N ¯χ(ˆφ2)2+ ∂∂ˆφ2 + ∂¯χ∂ˆφ2∂∂¯χ! 12Zk ln deth−iD−1µν (k, ¯χ, ˆφ) −e2Πµν(k, ¯χ)i+12Zk ln"(k2 + ¯χ) (1 + λB(¯χ, k2)/3) + λˆφ2/3k2 + ¯χ#),(30)which is the formal expression from which, after proper integration and renormalizationprocedures, we shall be able to determine the critical temperature and the order of thephase transition up to next to leading order in 1/N.4Critical Temperature and Nature of the PhaseTransition at Leading Order in 1/NIn section 3 we derived the expression of the effective potential up to leading order in1/N to be,V (¯χ, ˆφ) = −3N2λ0¯χ2 + N2 ¯χˆφ2 −v20+ NZk lnk2 + ¯χ,(31)where we have introduced the subscript zero to denote unrenormalized quantities.

Thus,at leading order in 1/N, the gap equation, which determines the expression of ¯χ as a12

function of ˆφ2, reads∂V∂¯χ = −3Nλ0¯χ + N2ˆφ2 −v20+ NZk1k2 + ¯χ = 0. (32)In order to explicitly evaluate the temperature dependent integrals, we shall use therelation [30]Zk f(k0,⃗k) =Zd4k(2π)4f(k) +Z ∞+iǫ−∞+iǫdk02πZd3k(2π)3f(k0,⃗k) + f(−k0,⃗k)(exp(−iβk0) −1),(33)which allows a decomposition in the zero temperature and the finite temperature con-tributions - the first and second terms in the right hand side of the above equation,respectively.

For the particular case of the integral appearing in Eq. (32), we obtain,Zk1k2 + ¯χ =Zd4k(2π)41(k2 + ¯χ) +Zd3k(2π)31q⃗k2 + ¯χexpβq⃗k2 + ¯χ−1,(34)where the k0 dependence in the last integral was evaluated by performing a contourintegration in the complex k0 plane.

At large temperatures, T ≫¯χ, the above integralmay be expanded in a way first derived by Dolan and Jackiw [2]Zd3k(2π)31q⃗k2 + ¯χexpβq⃗k2 + ¯χ−1 =112β2 −√¯χ4πβ +¯χ16π2 ln cT 2¯χ!+ h.o. (β2 ¯χ)(35)where c = 16π2 exp(1 −2γ), with γ ≈0.577.Moreover, the zero temperature contribution is quadratically divergent and needs tobe regularized.

Introducing a momentum cutoffΛ, so thatZd4k(2π)41(k2 + ¯χ) =116π2"Λ2 −¯χ ln Λ2¯χ!#,(36)we then absorb the quadratic and logarithmic divergences by defining the renormalizedquantitiesv2 = v20 −Λ28π2(37)and1λ = 1λ0+148π2 ln" Λ2M2#,(38)13

where M2 is a renormalization scale. Observe that, for any positive value of the unrenor-malized quartic coupling λ0, the renormalized coupling λ vanishes at Λ →∞, and thetheory becomes trivial in the continuum limit.

If the unrenormalized quartic coupling istaken to be negative, the theory becomes unstable [24]. Therefore, the only consistentdefinition of the theory is when it is considered as an effective theory with an effectivefinite cutoffΛ.We can further define a temperature dependent coupling by1λT= 1λ −148π2 ln"cT 2M2#(39)to finally rewrite the gap equation as¯χ −λT6 ˆφ2 −v2 +16β2!+λT12πβ√¯χ = 0.

(40)The above expression gives a quadratic equation in √¯χ, which can be easily solved tofind [2], [18]√¯χ = −λT24πβ +vuut λT24πβ!2+ λT6 ˆφ2 −v2 +16β2!,(41)where the plus sign has been chosen in order to obtain positive values of √¯χ, which isa precondition for the validity of the gap equation derived above.As we have shown in section 3, the minimization condition up to leading order in1/N reads,2dV (ˆφ)dˆφ2≡N ¯χ(ˆφ2) = 0. (42)Considering the explicit expression for ¯χ(ˆφ2), given in Eq.

(41), this yields the relationˆφ2 −v2 +16β2 = 0,(43)which determines the value of ˆφ at the minimum. Eq.

(43) has no solution for temper-atures above the critical valueT 2c ≡1β2c= 6v2,(44)14

and, hence, the minimum is at ˆφmin = 0 in such region of T.1 On the other hand, attemperatures below the critical one, the vacuum expectation value is given byˆφ2min = 16 1β2c−1β2!. (45)Equation (45) shows that the vacuum expectation value varies continuously from zeroto nonzero values, signaling the presence of a second order phase transition within theleading order in 1/N solution of the model.

It is worth to remark that, for values ofˆφ2 at the left of the minimum, ˆφ2 < (v2 −1/6β2), √¯χ takes negative or even complexvalues, showing that, as was first discussed by Coleman, Jackiw and Politzer [21], theeffective potential can not be defined in a sensible way in that region.5Critical Temperature and Phase Transition: Anal-ysis up to Next to Leading Order in 1/N.In terms of the renormalized quantities defined in section 4, and within the frameworkof the high temperature expansion, we can rewrite the effective potential up to next toleading order in 1/N, Eq. (24), asV (ˆφ, ¯χ)=−3N2λT¯χ2 + N2 ¯χˆφ2 −v2+ N ¯χ12β2 −N ¯χ3/26πβ+12Zk ln deth−iD−1µν (k, ¯χ, ˆφ) −e20Πµν(k, ¯χ)i+12Zk ln"(k2 + ¯χ) (1 + λ0B(¯χ, k)/3) + λ0 ˆφ2/3k2 + ¯χ#.

(46)As we shall explicitly show in section 5.1, B(¯χ, k2) and Πµν (¯χ, k2) are logarithmicallydivergent quantities. However, these logarithmic divergences can be naturally absorbedin the renormalization of the gauge and quartic couplings entering in the integrands ofEq.(46).

The remaining divergences, arising from the zero temperature contributions tothe integrals, may be absorbed in the renormalization of λ and v2 at next to leadingorder in 1/N.1 Observe that there is a difference in a factor 2 with respect to the results of Refs. [2],[17],[18] dueto the presence of 2N, instead of N, real scalar bosons in the theory.15

From Eq. (30) the minimization condition is given by¯χ=−1N(" ∂∂ˆφ2 + ∂¯χ∂ˆφ2∂∂¯χ#Zk ln deth−iD−1µν (k, ¯χ, ˆφ) −e20Πµν(k, ¯χ)i+Zk ln"(k2 + ¯χ) (1 + λ0B(¯χ, k)/3) + λ0 ˆφ2/3k2 + ¯χ#!)¯χ=0.

(47)Thus, since the minimum is obtained for ¯χ ≃O(1/N), we can safely set ¯χ = 0 in theright hand side of Eq. (47), under the assumption that no infrared divergences arise inthis process.5.1Tachyon Poles and Triviality of the Gauged O(2N) Modelat Zero TAt zero temperature, it is possible to compute the radiative corrections B(p, ¯χ) andΠµν(p, ¯χ) by making use of a gauge invariant regularization scheme, like Pauli - Villars.This gives the resultBT=0(p2, ¯χ(ˆφ))=116π2(ln Λ2¯χ!−2 4¯χ + p24p2!1/2lnp2¯χ 4¯χ + p24p2!1/2+ 122−1(48)whileΠµνT=0(p2, ¯χ(ˆφ2))=(δµνp2 −pµpν)48π2(−ln Λ2¯χ −23(49)+2 4¯χ + p2p2!

 4¯χ + p24p2!1/2lnp2¯χ 4¯χ + p24p2!1/2+ 122−1The divergence associated with B(p2, ¯χ) may be absorbed in the renormalization of λ[22]. In fact, the expressionF(¯χ, p2) ≡p2 + ¯χ 3/λ0 + BT=0(¯χ, p2)+ ˆφ2,(50)appearing in the potential, Eq.

(24), is proportional to the inverse propagator of themassive scalar particle σ ≡φ1,1 and may be rewritten asF(¯χ, p2) =p2 + ¯χ 3/λ + ¯BT=0(¯χ, p2)+ ˆφ2,(51)16

where¯BT=0(p, ¯χ(ˆφ2))=116π2(ln M2¯χ!−2 4¯χ + p24p2!1/2lnp2¯χ 4¯χ + p24p2!1/2+ 122−1(52)and λ is the renormalized quartic coupling at leading order in 1/N, which is given inEq.(38). Observe that, as first noticed in Ref.

[21], the renormalized theory seems to bespoiled by the presence of tachyons. In fact, since we are working in Euclidean space, atachyon pole in the φ1,1 two point function will appear if, for some positive value of p2,F(¯χ, p2) = 0.

Considering the limit ¯χ = ˆφ = 0, this implies an equation for p2, whichreads3λ +116π2 ln M2p2!+18π2 = 0. (53)For a finite value of the renormalized coupling λ, the above equation has a solution atsufficiently large values of p2, which could be interpreted as the presence of a tachyonin the spectrum.

However, rewriting Eq. (53) in terms of the unrenormalized quarticcoupling, we obtain,3λ0+116π2 ln Λ2p2!+18π2 = 0.

(54)Therefore, for any λ0 > 0, the potentially dangerous pole appears at momentum abovethe cutoffscale [24],p2 ≃Λ2 exp48π2/λ0. (55)Hence, it has no physical consequences.

(Observe that at nonvanishing values of ˆφ and¯χ the pole would be at even larger values of p2.) The apparent discrepancy between theresults obtained in terms of the renormalized and unrenormalized couplings is relatedto the issue of triviality.

In fact, while naively analyzing the existence of a tachyon poleusing Eq. (53), one overlooks the fact that the renormalized quartic coupling λ goes tozero in the continuum limit Λ →∞, and for this reason, a finite effective cutoffis neededin order to have a finite renormalized coupling.An analogous situation occurs in the gauge sector of the theory.

In fact, callingiDµν = DT δµν −pµpνp2!+ DLpµpνp2(56)17

andΠµνT=0(¯χ, p2) = δµν −pµpνp2!ΠT=0(¯χ, p2),(57)a tachyon pole in the current - current correlation function would appear ifGp2, ¯χ≡1e20D−1T¯χ, ˆφ, p2−ΠT=0¯χ, p2(58)had a zero in Euclidean space. At ¯χ = ˆφ = 0, the condition G(p2, ¯χ) = 0 leads to theequation1e20+148π2"ln Λ2p2!+ 83#= 0.

(59)Apart from a change from the gauge to the quartic coupling, the above equation isequivalent to that one found in the previous case, Eq.(54). The would be tachyon poleappears at momentum above the physical cutoffof the theoryp2 ≃Λ2 exp(48π2/e20)(60)and, hence, has no physical implications.The renormalization of the gauge coupling proceeds in similar way to that one of thequartic coupling.

The logarithmic divergences in ΠµνT=0 are absorbed in the definition ofthe renormalized gauge coupling e, which is given by1e2 = 1e20+148π2 ln Λ2M2!. (61)We can therefore define the renormalized vacuum polarization, ¯ΠT=0, by replacingln(Λ2/¯χ) by ln(M2/¯χ) in ΠT=0.

Moreover, the expression of the renormalized gaugecoupling, Eq. (61), shows the triviality of the gauged theory in the continuum limit.5.2Radiative Corrections at Finite TemperatureWe shall now analyze the structure of B(p, ¯χ) and Πµν(p, ¯χ) at finite temperature, which,by making use of Eq.

(33), may be decomposed into zero temperature contributions, ¯BT=0and ΠµνT=0, already analyzed in section 5.1, and the corresponding finite temperatureparts to be computed in the following. The radiative correction to the renormalized χ18

propagator, ¯BT(p, ¯χ), has the expression¯BT(p, ¯χ) −¯BT=0(p, ¯χ)=Z ∞+iǫ−∞+iǫdk02πZd3k(2π)3(1[k2 + ¯χ] [(k + p)2 + ¯χ]+1h˜k2 + ¯χi h(˜k + p)2 + ¯χi1(exp(−iβk0) −1)(62)where ˜k = (−k0,⃗k). The k0 integral may be performed by making a contour integrationin the complex k0 plane, leading to¯BT(p, ¯χ)=¯BT=0(p, ¯χ) + 2Zd3k(2π)31q⃗k2 + ¯χ1expβq⃗k2 + ¯χ−1×⃗k + ⃗p2 −⃗k2 + p20(⃗k + ⃗p2 −⃗k2 + p202+ 4p20⃗k2 + ¯χ).

(63)As we have shown above, in order to study the minimization condition up to nextto leading order in 1/N, it is sufficient to study the behaviour of ¯BT for ¯χ ≃0. Fromthe expression of ¯BT(p, ¯χ), doing an expansion in the neighborhood of ¯χ = 0 at finiteexternal momentum, we obtain¯BT (p0, ⃗p, ¯χ) = ¯BT (p0, ⃗p, ¯χ = 0) + K√¯χβp2 + O(¯χ),(64)where K is a coefficient of order one, which can be evaluated by analyzing the infrareddivergences associated with ∂¯BT /∂¯χ as ¯χ →0.

From Eq. (63) it is possible to prove that∂¯BT (p, ¯χ)∂√¯χ→−12πβ (p20 + ⃗p2)(65)as ¯χ →0 and, hence, K = −1/2π.

Furthermore, the last term in the Eq. (64) involveshigher order contributions in ¯χ which are not relevant to solve the minimization condi-tion.

This is due to the fact that, from the gap equation up to leading order in 1/N,Eq. (40),∂¯χ(ˆφ2)∂ˆφ2=4πβ√¯χ1 + 24πβ√¯χ/λT¯χ→0→4πβ√¯χ.

(66)Therefore, the derivative operator involved in the minimization condition, Eq. (47), onlyreceives contributions from the infrared dominant part of ∂Vn.l.o/∂¯χ.19

An analogous procedure may be used to analyze the behaviour of the vacuum polar-ization contribution ΠµνT .ΠµνT −ΠµνT=0=Z ∞+iǫ−∞+iǫdk02πZd3k(2π)3( (2k + p)µ(2k + p)ν[k2 + ¯χ] [(k + p)2 + ¯χ]+(2˜k + p)µ(2˜k + p)νh˜k2 + ¯χi h(˜k + p)2 + ¯χi −4δµνk2 + ¯χ1(exp(−iβk0) −1)(67)At finite temperature it is easy to demonstrate the transversality of ΠµνT by making useof the p0 quantization, p0 ≡ωn = 2πn/β0. However, general covariance is explicitly lostand, hence, the vacuum polarization takes the general form[30]ΠµνT (p, ¯χ) = ΠT1 (p, ¯χ) δµν −pµpνp2!+ ΠT2 (p, ¯χ) δij −pipj⃗p2!.

(68)Recalling the general form of the gauge boson propagator, Eq. (56), it is straightforwardto show thatdet−iD−1µν −e20ΠTµν="1DT (p2, ˆφ)−e20ΠT1 (p, ¯χ) + ΠT2 (p, ¯χ)#2"1DT (p2, ˆφ)−e20ΠT1 (p, ¯χ)#1DL(p2, ¯χ, ˆφ),(69)where, from Eq.

(13), we have,1DT (k2, ˆφ)=k2 + e20 ˆφ21DL(k2, ¯χ, ˆφ)=hk2 (k2 + ¯χ) + αe20 ˆφ2¯χiα (k2 + ¯χ). (70)Therefore,det−iD−1µν −e20ΠTµν=hk2 + e20 ˆφ2 −e20ΠT1 (k, ¯χ) + ΠT2 (k, ¯χ)i2hk2 + e20 ˆφ2 −e20ΠT1 (k, ¯χ)i 1DL.

(71)This expression is equivalent to that one found by Carrington [10] in the limit of smallexternal momentum (p0 = 0, ⃗p ≃0), while computing the ring diagrams contributionto the effective potential in the Landau gauge.Observe that, at zero temperature,ΠT=01= ΠT=0, while ΠT=02= 0 and all logarithmic divergences may be absorbed in the20

definition of the renormalized gauge coupling, which implies that ΠT=01= ΠT=0, turnsinto its renormalized expression ¯ΠT=01= ¯ΠT=0. Consequently, defining ¯ΠT1 as the finitetemperature renormalized quantity, the logarithm of the determinant takes the formln det[−iD−1µν (k, ˆφ, ¯χ)−e20ΠTµν(k, ¯χ)] = 2 lnhk2 + e2 ˆφ2 −e2 ¯ΠT1 + ΠT2i+lnhk2 + e2 ˆφ2 −e2 ¯ΠT1i+ ln(k2 (k2 + ¯χ) + αe20 ˆφ2 ¯χk2 + ¯χ).

(72)The first two terms in the right hand side of the equation above are gauge independentand, for ΠTi = 0, they give the well known one loop contribution to the effective potential.The last term contains all the gauge dependence and its contribution to the effectivepotential only vanishes in the Landau gauge, α = 0. We shall return to the issue of thegauge dependence in section 5.3.Since ΠTµν, Eq.

(68), has only two independent components, we can obtain the func-tions ¯ΠTi by computing the expressions of ΠT00 and ΠTµµ. In fact,ΠT00 −ΠT=000=¯ΠT1 −¯ΠT=01 ⃗p2p2,ΠµµT −ΠµµT=0=3¯ΠT1 −¯ΠT=01+ 2ΠT2 .

(73)Moreover, using Eqs. (49) and (67), we haveΠ00T (k, ¯χ) −Π00T=0(k, ¯χ)=I1(k, ¯χ) −2I2(k, ¯χ)(74)ΠµµT (k, ¯χ) −ΠµµT=0(k, ¯χ)=−4I2(k, ¯χ) −k2 + 4¯χ ¯BT(k, ¯χ, ˆφ) −¯BT=0(k, ¯χ, ˆφ),withI1(p, ¯χ)=Z ∞+iǫ−∞+iǫdk02πZd3k(2π)3(2k0 + p0)2(k2 + ¯χ)h(k + p)2 + ¯χi+(2k0 −p0)2˜k2 + ¯χ ˜k + p2 + ¯χ1(exp(−iβk0) −1)(75)andI2(p, ¯χ) = 2Z ∞+iǫ−∞+iǫdk02πZd3k(2π)31(k2 + ¯χ)1(exp(−iβk0) −1).

(76)21

The integrals Ii may be computed by performing a contour integration in the complexk0 plane. We obtain,I1=2Zd3k(2π)31q⃗k2 + ¯χexpβq⃗k2 + ¯χ−1×hp20 −4⃗k2 + ¯χi p20 +⃗k + ⃗p2 −⃗k2+ 8p20⃗k2 + ¯χp20 +⃗k + ⃗p2 −⃗k22+ 4p20⃗k2 + ¯χ(77)andI2 =Zd3k(2π)31q⃗k2 + ¯χexpβq⃗k2 + ¯χ−1.

(78)It is first interesting to analyze the behaviour of Ii(p0, ⃗p) at ¯χ = 0, p0 = 0 and |⃗p| ≪T.The integral I2 has already been considered in section 4, and its high temperatureexpansion result, Eq. (35), in the limit we are studying, gives I2(¯χ = 0) = 1/12β2.Furthermore, from Eq.

(77), it is straightforward to show thatI1(¯χ = 0, p0 = 0, ⃗p ≃0) = −1π2Z ∞0d|⃗k||⃗k|exp(β|⃗k|) −1 = −16β2. (79)The above results imply that, in this momentum regime,¯ΠT1 −¯ΠT=01= −ΠT2 = −13β2(80)Observe that this contribution gives a static mass term to the abelian gauge field.

Infact, recalling Eq. (71) we obtain that, since ¯ΠT1 and ΠT2 come with opposite signs, onlythe longitudinal mode receives a contribution to the mass term equal toM2D = e23β2,(81)which is the well known Debye screening mass term [30].

It has been recently pointedout in the literature [10]-[12], that this term is important in order to determine thestrength of the phase transition within the framework of the improved one loop effectivepotential. Within such approximation, the phase transition appears to be first orderbut, due to the Debye screening suppression of the longitudinal mode, its strength isabout two thirds of the one obtained using the naive one loop approach.

We shall now22

demonstrate that, when using the self consistent 1/N expansion up to next to leadingorder in 1/N, and computing the value of ΠµνT at finite external momentum, extra termsappear in the effective potential, which change not only the strength but also the orderof the phase transition.5.3Minimization of the Effective PotentialThe full effective potential up to next to leading order in 1/N, Eq. (24) may be rewrittenasV (¯χ, ˆφ)=Vtree + Vl.o.

+ 12Zk ln(k2 + ¯χ)1 + λ ¯BT(¯χ, k)/3+ λˆφ2/3k2 + ¯χ+12Zkn2 lnhk2 + e2 ˆφ2 −e2 ¯ΠT1 + ΠT2i+lnhk2 + e2 ˆφ2 −e2 ¯ΠT1io+ Vg.d.,(82)where e and λ are the renormalized gauge and quartic couplings, respectively, and ¯BTand ¯ΠTi are the full, temperature dependent radiative corrections contributions with theirlogarithmic divergences removed through the definition of the renormalized couplings,in the way explained above. Moreover, we can rewrite the gauge dependent term, Vg.d.,coming from Eq.

(72) likeVg.d. = 12Zkhlnk2 + R21+ lnk2 + R22−lnk2 + ¯χi(83)whereR21,2 = ¯χ ±q¯χ2 −4αe20 ˆφ2 ¯χ2.

(84)Similarly to the computation of ¯BT, from Eq. (47) it follows that, in order to evaluatethe minimum condition up to next to leading order in 1/N, we only need to studythe behaviour of ¯ΠTi for ¯χ ≃0.

As we have just remarked, the computation of thevacuum polarization expression for finite external momentum is crucial in obtaining thecorrect behaviour of the effective potential at small values of ¯χ. Recalling Eqs.

(49),(67),(74),(77) and (78), it is straightforward to find that, for ¯χ →0, pµ ̸= 0,ΠµµT (p0, ⃗p, ¯χ) = ΠµµT (p0, ⃗p, ¯χ = 0) + 3√¯χ2πβ + h.o. (¯χ),(85)23

andΠ00T (p0, ⃗p, ¯χ) = Π00T (p0, ⃗p, ¯χ = 0) +√¯χ2πβ⃗p2(p20 + ⃗p2) + h.o.(¯χ). (86)Note that, as implied by transversality, the expression of ΠT00 vanishes in the limit ⃗p →0for p0 ̸= 0.

From the above expressions it follows that, at finite momentum transfer andin the small ¯χ limit, the vacuum polarization components read,¯ΠT1 (p0, ⃗p, ¯χ)=¯ΠT1 (p0, ⃗p, ¯χ = 0) +√χ2πβ + h.o. (¯χ)ΠT2 (p0, ⃗p, ¯χ)=ΠT2 (p0, ⃗p, ¯χ = 0) + h.o.(¯χ).

(87)It is important to emphasize that, this is the correct behaviour for ¯χ →0 at anyfinite |p| =qp20 + ⃗p2, which is the relevant regime for the computation of the integralscontributing to the minimum of the effective potential. Using the above expressions,Eqs.

(64),(87), we can rewrite the minimization condition as¯χ=−1N" ∂∂ˆφ2 + ∂¯χ∂ˆφ2∂∂¯χ#{Vg.d.+Zk ln"k2 + e2 ˆφ2 −e2 ¯ΠT1 (k, ¯χ = 0) +√¯χ2πβ!#+2Zk ln"k2 + e2 ˆφ2 −e2 ¯ΠT1 (k, ¯χ = 0) + ΠT2 (k, ¯χ = 0) +√¯χ2πβ!#+Zk ln(k2 + ¯χ)3/λ + ¯BT(k, ¯χ = 0) −√¯χ/(2πβk2)+ ˆφ2k2 + ¯χ¯χ=0(88)The first term in Eq. (88) involves all the possible gauge dependent contributions to theminimum of the effective potential.

However, explicitly applying the total derivative tothe logarithmic expression of Vg.d., ∂∂ˆφ2 + ∂¯χ∂ˆφ2∂∂¯χ!Vg.d.¯χ=0=12(Zk1k2" ∂∂ˆφ2+∂¯χ∂ˆφ2∂∂¯χ# R21 + R22 −¯χ)¯χ=0(89)and using the fact that (R21 + R22 −¯χ) = 0, we observe that Vg.d. does not give anycontribution to the minimization condition up to this order in 1/N.

( Observe that, sincewe are working with regularized integrals with a finite cutoff, we can safely interchangederivatives by integrals in the expression above.) Therefore, although the expression24

of the effective potential is gauge dependent, the minimization condition gives a gaugeindependent result.Furthermore, from Eq. (88), it comes to notice that, as we discussed above, for theminimization of the effective potential, the only relevant contributions from the finitetemperature radiative corrections come from the infrared dominant terms of ¯BT(p, ¯χ)and ¯ΠTi (p, ¯χ).

In addition, recalling the fact that ∂¯χ/∂ˆφ2 →√¯χ4πβ as ¯χ →0, it readilyfollows that the minimization condition reads,¯χ=−1N2e2Zk1 −2√¯χ∂(√¯χ)/∂¯χhk2 + e2 ˆφ2 −e2¯ΠT1 (k, ¯χ = 0) + ΠT2 (k, ¯χ = 0)i+e2Zk1 −2√¯χ∂(√¯χ)/∂¯χhk2 + e2 ˆφ2 −e2¯ΠT1 (k, ¯χ = 0)i+Zk1 −2√¯χ∂(√¯χ)/∂¯χhk23/λ + ¯BT(k, ¯χ = 0)+ ˆφ2i¯χ=0(90)The above equation implies that, at next to leading order in 1/N the only minimum isat¯χmin(ˆφ) = 0. (91)Thus, the value of ¯χ(ˆφ) at the minimum remains unchanged after the inclusion of the nextto leading order contributions.

As we discuss in section 4, Eq. (91) leads to a nontrivialsolution for the vacuum expectation value of the scalar field, ˆφ2min = v2 −1/(6β2), fortemperatures below the critical temperature, T 2c = 6v2.

Hence, the vacuum expectationvalue of the scalar field varies continuously at the critical temperature, characterizing asecond order phase transition.6Comparison with the One Loop ApproximationThe results of section 5 show that, in the presence of a large number of scalars and inthe region of parameters e2/λ ≪N, in which the scalar loop contributions are enhancedin comparison to the gauge loop ones, the phase transition is second order. In spiteof that, the gauge contributions to the effective potential at next to leading order in1/N resemble those ones contributing to the effective potential in the improved one25

loop approximation. In fact, for small ˆφ, and at T = T0, at which the quadratic termin ˆφ vanishes, one would expect that due to the presence of the gauge contributions,a dominant negative cubic term would appear, leading to a nontrivial minimum in theeffective potential and, hence, to a first order phase transition.

However, the fundamentaldifference comes from considering the finite external momentum contributions to theradiative corrections at finite temperature.Quite generally, as we have already said, the integralIT = 12Zk ln(k2 + m2)(92)may be evaluated in the high temperature regime to give,IT = IT=0 + m224β2 −m3/212πβ + h.o.(m2β2). (93)Therefore, since the zero temperature gauge boson mass is equal to m2g = e2 ˆφ2 one wouldnaively expect that the inclusion of the gauge field would induce the generation of aquadratic term in ˆφ, which would modify the definition of the transition temperature, aswell as the appearance of a cubic term, which would lead to a first order phase transition.In the next to leading order in 1/N analysis, these two effects are not present, and,hence, some sort of finite temperature screening, which modifies the nature of the phasetransition, should occur.

One type of screening, which is already contained within theimproved one loop approximation, is that one induced by the generation of an effectivetemperature dependent mass for the longitudinal gauge boson mode. As we alreadydiscussed above, when compared with the naive one loop result, the Debye screeningcauses the reduction in two thirds of the value of the coefficient of the cubic term.

Inour analysis, however, the preservation of the value of the critical temperature and ofthe nature of the phase transition after considering the next to leading order in 1/Neffects is a result that would be obtained even in the case of a cancellation of the ¯χ(ˆφ)-independent radiative corrections contributions to the photon and scalar propagators.In fact, within the 1/N expansion, there is a different source of screening, which canonly be obtained by considering the ˆφ dependent, finite momentum contributions to theradiative corrections.26

For values of ˆφ close to the minimum of the effective potential (¯χ ≃0), and atmomentum p2 ≫¯χ, the evaluation of the finite temperature radiative corrections tothe photon and scalar propagators have as outcome the replacement of the functionaldependence on ˆφ2 by a dependence on the combination [ˆφ2 −√¯χ/(2πβ)]. In fact, as it isclearly seen from Eqs.

(82), (85)-(87), in this momentum regime, the finite temperatureeffective mass of the transverse gauge boson mode is given bym2T = e2 φ2 −√¯χ/(2πβ)+ h.o. (¯χ)(94)To easily understand the relevance of this effect, let us first analyze the behaviour of ¯χat T = Tc,√¯χ = −λTc24πβc+vuut λTc24πβc!2+ λTc ˆφ26.

(95)Expanding Eq. (95) in the neighborhood of the origin we obtain√¯χ = 2πβc ˆφ21 −24π2λTc φTc!2.

(96)From Eq. (96) it follows that at T = Tc and for the momentum regime under considera-tion, the effective mass of the photon transverse mode in the neighborhood of the originis given bym2T = ˆφ224π2e2λTc ˆφTc!2+ h.o.

(¯χ)(97)Thus, since for the same range of parameters,¯χ ≃4π2 ˆφ2 ˆφTc!2,(98)the effective squared mass of the photon transverse mode behaves like ˆφ4/T 2 ratherthan the expected ˆφ2 dependence. Within the region of parameters considered above,the radiatively corrected squared mass of the σ particle also behaves as ˆφ4/T 2.The dependence of m2T in the neighborhood of the origin and at T = Tc has twoeffects.

First, it explains the cancellation of the quadratic term in ˆφ at T = Tc andhence the preservation of the value of the critical temperature at next to leading orderin 1/N. Second, it implies that, in the neighborhood of the origin, and independently27

of the existence of ˆφ - independent radiative corrections contributions, no cubic term isinduced and thus, at T = Tc the finite temperature effective potential is well describedby a positive, quartic term.It is important to remark that the expressions we obtain for the radiatively correctedphoton and scalar propagators are only valid for T, k2 ≫¯χ. Moreover, the cubic termarising in the integral, Eq.

(92), receives the most relevant contributions from the mo-mentum regime k0 = 0, ⃗k2 = O(m2). Thus, the present analysis would fail if in theneighborhood of the origin the relation ˆφ2 ≫¯χ were not fulfilled.

However, as readilyseen from Eq. (98), ¯χ ≪φ2 and the effective potential close to the origin is well describedby our approach.7ConclusionsIn this article we have analyzed the finite temperature phase transition of scalar elec-trodynamics with N flavors of complex scalar fields, by means of a large N expansion,up to next to leading order in 1/N.

We have shown that the effective potential takesa compact and simple form, which, although gauge dependent, leads to gauge invariantresults for the critical temperature as well as for the extrema of the potential, which arewell defined physical quantities. At leading order in 1/N, the effective potential coin-cides with the one of the O(2N) vector model, already studied in the literature.

At thecritical temperature Tc = 6v2, the system develops a symmetry restoration phase tran-sition, with an order parameter which varies continuously from zero to nonzero values.The phase transition is hence second order.At next to leading order in the 1/N expansion, the effective potential receives con-tributions from the radiatively corrected σ and gauge field propagators. The gauge fieldcontribution is essentially that one found at one loop, but with the photon corrected byvacuum polarization effects.

The ˆφ dependence of the vacuum polarization effects is cru-cial in order to define a correct 1/N expansion. We have shown that, when the radiativecorrections are considered at finite external momentum, their effects lead to an effectivescreening of the gauge boson mass, which change the nature of the gauge boson loop28

contributions to the effective potential. Due to this screening effect, no extra ˆφ2 term isinduced in the neighborhood of the origin at the leading order critical temperature and,therefore, the critical temperature remains the same as in the leading order analysis.More generally, the position of the minima of the effective potential is not modified bynext to leading order effects and, in addition, no new minimum of the effective potentialappears at next to leading order.

Hence, the order parameter has the same behaviouras in leading order in 1/N and the phase transition remains second order.The results of this work show that in the presence of a large number of flavors ofcomplex scalar fields, the symmetry restoration phase transition of scalar QED becomessecond order. Within the context of this study, however, we can not rule out the presenceof a critical value of N, below which the transition is first order.

In general, we expectthe range of validity of the large N approximation to depend on the relation between thegauge and quartic couplings. More specifically, the expansion considered in the presentwork assumes the dominance of the scalar loops contributions to the effective potential.Therefore, it is rigorously valid for values of e2/λ ≪N.

Thus, for low values of N, theexpansion becomes more reliable for a scalar Higgs mass, m2σ = λˆφ2/3 larger than thegauge boson mass, m2g = e2 ˆφ2, m2σ/m2g ≥1, for which the fluctuations of the scalar fieldsare enhanced.ACKNOWLEDGEMENTSWe would like to thank W. A. Bardeen, R. D. Peccei, R. D. Pisarski and P. Weisz forvery enjoyable discussions and helpful comments. We would also like to thank V. Jain forpleasant and useful discussions and P. Ramond and S. Hsu for interesting conversations.Parts of this work were carried out at the Aspen Center for Physics, at Fermilab NationalLaboratory and at Brookhaven National Laboratory, to which we are grateful.29

FIGURE CAPTIONSFig.1. Multiloop diagrams which contribute to the effective potential at next to leadingorder in 1/N.

Solid and dashed lines denote the propagators of the scalar fields, φa,i andχ, respectively, while the curved lines denote the gauge field propagator.Fig.2. Same as in Fig.

1, but considering those diagrams which involve a mixing ofχ and Aµ.30

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