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Finite Temperature Scalar Potential

arXiv:hep-ph/9205232v1 25 May 1992MPI-Ph/92-41April, 1992Finite Temperature Scalar Potentialfrom a 1/N ExpansionVidyut JainMax-Planck-Institut f¨ur Physik und Astrophysik- Werner-Heisenberg-Institut f¨ur PhysikP.O. Box 40 12 12 , D - 8000 Munich 40, GermanyAbstractWe compute the leading and next–to–leading corrections to the finite temper-ature scalar potential for a 3+1 dimensional φ4 theory using a systematic 1/Nexpansion.

Our approach automatically avoids problems associated with infrareddivergences in ordinary perturbation theory in ¯h. The leading order result does notadmit a first order phase transition.

The subleading result shows that the exacttheory can admit at best only a very weak first order phase transition. For N = 4and weak scalar coupling we find that T1, the temperature at which tunneling fromthe origin may begin in the case of a first order transition, must be less than about0.5 percent larger than T2, the temperature at which the origin changes from beinga local minimum to being a local maximum.

We compare our results to the effectivepotential found from a sum of daisy graphs.

1. Introduction.There has been much recent interest in the nature of the electroweak phase transitionmotivated by the possibility of baryogenesis within the standard model itself.

As notedby Kirzhnits amd Linde [1], Weinberg [2] and Dolan and Jackiw [3] over 18 years ago,a spontaneously broken field theory may have its symmetry restored at high enoughtemperature. For example, in a spontaneously broken φ4 theory, temperature correctionsgive a positive mass–squared contribution at the origin (φ = 0), and at high enoughtemperature this correction results in a global symmetry–unbroken minimum at the origin.In such a model there are two important temperatures as we lower the temperaturefrom a very high value.The first, T1, is the temperature at which a second possibleminimum appears degenerate in energy with the minimum at the origin.

The second, T2,is the temperature at which the effective mass at the origin vanishes, i.e. when the originchanges from being a local minimum to a local maximum.The phase transition from the symmetric to nonsymmetric phase as we cool a systemdescribed by such a model may therefore proceed in two ways, by tunneling when thetemperature is between T1 and T2, or by a rollover when the temperature drops belowT2.

At T = T1, isolated bubbles of the symmetry broken phase will start to be createdby tunneling. If the phase transition completes by bubble nucleation we will call thetransition first order, otherwise if it completes mostly by a rollover after T < T2 then wewill call the transition second order.

If the finite temperature scalar potential possessesonly the local minimum at φ2 = 0 until T = T2 then there is no T1 and the phase transitionis necessarily second order. Even if there is a T1 we will say only that the system admitsa first order transition since if T1 is close enough to T2 we expect the phase transition tocomplete by a rollover.In the context of the standard model, it has been suggested that sufficient baryoge-nesis may occur [4,5,6] if the phase transition is first order.

Unfortunately, an accuratedetermination of the the nature of the phase transition has proven difficult in this case.One needs a reliable determination of the temperature dependent effective potential ofthe scalars near the origin for temperatures between T1 and T2. One–loop calculationssuggest that baryogenesis can only occur in the minimal standard model for Higgs massMH < 45 GeV [7,8], a possibility that appears ruled out by experiment [9].

However, asnoted by both Weinberg [2] and Dolan and Jackiw [3], naive finite temperature pertur-bation theory for even the four dimensional φ4 model suffers from infrared divergences.These divergences invalidate the one–loop approximation. In particular, at temperaturesclose to T2 perturbation theory in ¯h begins to break down.

The reason for this can be eas-ily understood: even the one–loop correction drastically modifies the tree level potential;it is clearly not a small perturbation. Due to such problems, the one–loop result in pureφ4 theory gives a complex potential for small values of φ2 and cannot be used to studythe nature of the phase transition.In the standard model, one may argue that if the phase transition completes “wellbefore” T2 then the one–loop calculation can be used.

However, in the case of the standard1

model the phase transition is expected to be weakly first order, if it is first order at all. Inaddition, it is not a priori clear that resumming some infinite class of diagrams does notrule out a first order transition.

The task then is to extract the leading corrections nearT = T2 to all orders in perturbation theory. For example, Dolan and Jackiw summed aclass of diagrams, the “super–daisies” to circumvent the infrared divergence problem inpure φ4 theory (with N scalars) to get a reliable estimate of T2.

Such diagrams correspondto iterating the daisy graphs✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏♠♠♠♠♠♠♠♠♠· · ·(1)so that each petal has an arbitrary number of petals and so on.Such a summationcorresponds to the leading term in a 1/N expansion and greatly modifies the 1–loopestimate.In this article we will use the systematic 1/N expansion method to calculate the leadingand next–to–leading corrections to the finite temperature effective scalar potential nearthe origin in a weakly coupled φ4 theory in four dimensions. In the case of the standardmodel, one also needs to incorporate corrections to the scalar potential from gauge loops.We do not address this problem here, but will later comment on the applicability ofour methodology to a gauged φ4 theory.

In section two we review the 1/N expansionfor a φ4 theory and present our computations. In section three we discuss our results.Qualitatively, we find that the leading order in N correction does not admit a first ordertransition.

The next–to–leading may admit a first order transition, but an extremely weakone. We stress that this approach automatically avoids the infrared divergence problemof ordinary perturbation theory.

In particular, for the range of φ2 and T we are interestedin the potential does not become complex.In the context of the standard model there have been several recent papers on improv-ing the 1–loop estimate [10,11,12]. These attempts mainly involve summing the daisydiagrams.

The simplest procedure [13] has been to use the 1–loop generated temperaturedependent mass to calculate 1–loop corrections. In pure φ4 theory, if properly done, thiscan be interpreted as summing the daisy diagrams.

This procedure has its roots in thework of Weinberg [2]. It is useful to briefly discuss this procedure, and at the same timeintroduce some necessary formulas, before presenting our results.To understand the meaning of such a procedure, it is necessary to recall an importantformula in the background field method.

To be precise we consider a simple O(N) scalarfield theory with action (i = 1 . .

. N)S[φ] = 12Zδij∂µφi∂µφj −λ4!Z(φ2 −v2)2.

(2)2

One calculates [14]Γ[φ]=−i¯h lnZ[D ˆφ] exp i¯h(S[φ + ˆφ] −δS[φ]δφi ˆφi)!=S[φ] −i¯h lnZ[D ˆφ] exp i¯hZ −12ˆφi∆−1ij ˆφj −λ3!φk ˆφk ˆφ2 −λ4!(ˆφ2)2!!. (3)The first term on the RHS of the last line is the classical action.The inverse scalarpropogator is∆−1ij = [δij∂2 + M2ij(φ)].

(4)M2ij is the second derivative of the classical potential w.r.t. the fields φi, M2ij = ∂i∂jV =(λ/6)(δij(φ2 −v2) + 2φiφj).Equation (3) is an effective action in that it incorporates the effects of all one–particle–irreducible (1PI) diagrams and can be given to any order in perturbation theory.

It maybe evaluated in perturbation theory asΓ[φ] = S[φ] + 12i¯hTr ln(δik∆−1kj ) + ¯Γ[φ],(5)The second term is the familiar one–loop contribution and the rest contains all higherloop corrections from 1PI graphs. The one-loop corrected potential is˜V (φ) = V (φ) −12iZp Tr lnhp2δij −M2iji,(6)where the trace is over internal indices and for zero temperature the integration measureover momentum is d4p/(2π)4.

For finite T, in the imaginary time formalism, the integralover four momentum goes over to an integral over three momentum and a discrete sumin the manner given in [3]. We will assume the reader is familiar with finite temperaturefield theory.We now return to problem of understanding what it means to use the one–loop cor-rected mass to calculate one–loop corrections.

If we iterate the one–loop result once weobtain˜˜V (φ) = V (φ) −12iZp Tr lnp2δij −M2ij + 12iZk ∂i∂jTr lnhk2 −M2lki. (7)If we keep only the contributions from (7) when the ∂i∂j both act on the same mass–squared matrix we obtain, by carrying out the differentiation and expanding the log,˜˜V = ˜V + i2∞Xn=1TrZp1n"i[p2 −M2]−1mjZkλ6(δijδkl + δikδjl + δilδjk)[k2 −M2]−1kl#n.

(8)Here the term inside the square brackets is understood as a matrix with two indices; thetrace is over powers of this matrix. It may be verified that the RHS is the tree–level plus3

contributions from the daisy graphs of (1), with the correct combinatorics except for thetwo loop graph (n=1) which is over counted. The correct factor for the two–loop graph ishalf the one appearing from this procedure [14].

In the large N limit if we keep only theleading order in N and account for the two loop difference, this will reproduce the resultof Dolan and Jackiw [3] for the finite temperature daisy sum correction to the effectivemass at the origin:∂˜˜V∂φ2 = ∂˜V∂φ2 + iN2∞Xn=1Zp1n Zk[k2 −λ6(φ2 −v2)]−1!n ∂∂φ2 iNλ6 [p2 −λ6(φ2 −v2)]−1!n. (9)Following [3], one keeps only the most infrared divergent contributions.

This amounts toletting the ∂/∂φ2 act only on [p2 −λ6(φ2 −v2)]−n for n ≥2. For the two loop case it mayact on either [p2 −λ6(φ2 −v2)]−1 or [k2 −λ6(φ2 −v2)]−1 which again explains why thiscase is different.

As shown in [3], however, the daisy graphs do not exhaust all dominantN contributions.For this one needs to sum the superdaisy graphs.Therefore, evenfor large N, this approximation scheme is not complete. Furthermore, in (7) there areadditional corrections when the ∂i and ∂j both act on different mass–squared matrices.These corrections cannot be written as a sum of 1PI diagrams.Therefore a sensibleapproximation scheme can only be achieved by ignoring these corrections (which shouldin any case be subleading in N).

In the last section we will compare our 1/N expansionresult with the leading O(N) correction from using an effective temperature dependentmass.2. φ4 theory to subleading order.The systematic 1/N expansion allows us to calculate (3) as a perturbation in 1/Nnear φ = 0 [15,16].

Root [16] has evaluated the leading and next–to–leading correctionsto the zero temperature scalar potential in 4,3,2 and 1 dimensions. The procedure atfinite temperature is very similar.At very high temperature, our results for four dimensions should be essentially thoseof a three dimensional euclidean field theory with a dimensionful φ4 coupling.

In threedimensions and zero temperature, the leading O(N) potential has long been known [15].It has exactly the same form as the sum of finite temperature superdaisy graphs that werecomputed by Dolan and Jackiw [3] for a four dimensional φ4 theory. An important pointin our approach is that the effect of introducing an auxiliary field σ is to shift the φ massterm and as a result there are no infrared divergences in this formalism.To proceed, we first set λ = f/N.

The 1/N expansion assumes f is fixed as Nincreases, not λ. Then by introducing a dimension two auxiliary field σ we rewrite the4

action (2):S[φ, σ]=S[φ] + 3N2f σ −f6N (φ2 −v2)!2=12Z(∂φ)2 + 3N2fZσ2 −12Z(φ2 −v2)σ. (10)The auxiliary field has eliminated the φ4 term; the original form of (3) is easily recoveredby use of the equation of motion for σ.To calculate the effective potential V (φ) one proceeds as follows.First, using thebackground field method one computes the effective potential as a function of backgroundsof φ and σ.

Then, the background of σ is eliminated by its equation of motion.The systematic 1/N expansion is performed by expanding the action S[φ, σ] aboutbackgrounds φ and χ thus:φi →√Nφi + ˆφi,σ →χ +ˆσ√N . (11)The factors of√N have been inserted with hindsight.

In particular, the rescaling of thebackground φ is necessary in order to ensure that the effective potential is renormalizableto each order in the 1/N expansion. This will be seen later in the explicit calculations.One now writes a formula similar to (3), with an integral over ˆφ and ˆσ:Γ[φ, χ] = S[√Nφ, χ]−i¯h lnZ[D ˆφ][Dˆσ] exp i¯hZ −12ˆφi∆−1ij ˆφj −ˆφ2ˆσ2√N −φi ˆφiˆσ + 3ˆσ22f!!.

(12)The factor of ¯h is useful in order to compare results in the 1/N expansion to those fromthe usual perturbation theory in ¯h; for practical purposes we will take ¯h = 1. All termslinear in the quantum fields have been discarded, and we have defined∆−1ij = δij(∂2 + χ).

(13)The integral over ˆφ is gaussian and can (formally) be performed exactly to give theleading O(N) contribution to the effective potential. The integral over over ˆφ also pro-duces ˆσ terms which may be expanded in a power series.

It is known that the ˆσ2 termsgive rise to the next–to–leading corrections in the 1/N expansion, which is all we areinterested in. Furthermore, to calculate the effective potential we can take φ and χ asspace–time constants, but not ˆσ.

The integral over ˆφ produces a kinetic term for ˆσ whichmust be carefully determined in order to evaluate the next–to–leading corrections.The ˆφˆσ term in (12) can be rewritten as a term purely in ˆσ by shifting the variable ofintegration ˆφ. This can easily be acheived to all orders in the 1/N expansion, but since5

we are only interested in the leading and next–to–leading terms it suffices to write theargument of the exponential in (12) asi¯hZ −12ˆφi∆−1ij ˆφj −12√Nˆφ2ˆσ + 12 ˆσ2φi∆ijφj + 32f ˆσ2!. (14)Using ˆφi(x) = −i¯hδδJi(x) exp( i¯hR Jk ˆφk) evaluated at Ji = 0, and then rescaling thesource and field ˆφ by ¯h12 we obtain for the integral over ˆφ in (12),Γ[φ, χ, ˆσ]=−i¯h lnZ[D ˆφ] exp i¯hZ −12ˆφi∆−1ij ˆφj −12√Nˆφ2ˆσ!

!=12i¯hTr ln ∆−1ij −i¯h ln"exp i¯hSI[−i¯h12 δδJ ]!exp−i2ZJi∆ijJj#, (15)evaluated at J = 0, whereSI[ˆφ] = −12√NZˆσ ˆφ2. (16)Of course, eq.

(15) does nothing more than define a perturbative expansion with ascalar propogator ∆ij. The first term in (15) is a one–loop result.

It has been evaluatedmany times before and is given by [3]✒✑✓✏∋i2¯hZx,p Tr lnhp2 −δijχi=−Zx V0 + VT.(17)V0 is the zero temperature result which is divergent and must be regulated. With a sharpmomentum cutoffΛ,V0 =N32π212χ2 ln[χ/Λ2] −14χ2 + χΛ2.

(18)VT is the finite, temperature dependent, result:VT = N−π290β4 +χ24β2 −χ3212πβ −164π2χ2 ln[χβ2] +c64π2χ2. (19)We have dropped O(β2) terms and higher, and c ≈5.41.

This result is valid at highenough temperatures. In the electroweak model, it is known [8] that the high temperatureexpansion is well justified at temperatures relevant for the study of the phase transition.The O(ˆσ2) correction is a 1–loop contribution as well.

Extracting the 1PI part from(15) we obtain to leading order✒✑✓✏∋−i4¯hZx,p1p2 −χˆσ1(p + i∂)2 −χˆσ. (20)6

In deriving this result we took the functional derivatives in (15), fourier transformedthe resulting δ–functions and dropped all total divergences.We further used prop-erties of the translation operators: exp(−ip · x)H(−i∂p) exp(ip · x) = H(x −i∂p) =exp(−i∂x · ∂p)H(x) exp(i∂x · ∂p). Dropping total divergences assumes that the integral iswell regulated (in three dimensions it is finite).

Eq. (20) may be evaluated using Feynmanparameters; the result is✒✑✓✏∋−i4¯hZx ˆσZ 10 dzZp1[p2 −(χ −z(z −1)∂2)]2 ˆσ=¯hZ 10 dzZx125πβ ˆσ1qχ −z(z −1)∂2 ˆσ=¯hZx116πβ ˆσ1√+∂2 sin−11q1 + 4χ/∂2ˆσ,(21)which is 1/β times the corresponding zero temperature result in three dimensions [16].The p integral was evaluated usingRp[p2 −χ]−2 = −∂2/∂χ2 Rp ln[p2 −χ] and only theleading temperature dependent and field dependent correction was kept.

The value of thearcsine is uniquely determined since for ∂2 = 0 the integral over z is unambiguous.The final result for (12) to the required order is thusΓ[φ, χ] = S[√Nφ, χ] + 12i¯hTr ln ∆−1ij + Γ1[φ, χ],(22)where the next–to–leading contribution from the gaussian integral over ˆσ isΓ1[φ, χ] = 12i¯hTr ln1 +¯hf24πβ1√+∂2 sin−11q1 + 4χ/∂2+ f3 φi∆ijφj. (23)We now have all the formalism behind us.

Before presenting explicit calculations it isuseful to rederive the leading order “mass–gap” equation of [3] to see how the infraredproblem is automatically avoided.To leading order in N one drops the ˆσ terms and eq. (12) reduces to¯¯Γ[φ, χ]=ZxN2 (∂φ)2 + 3N2f χ2 −N2 χ(φ2 −v2) + 12i¯hTr ln ∆−1ij=NZx" 32f χ2 −12χ(φ2 −v2) + 12i¯hZp Tr ln[p2 −χ]#.

(24)For convenience, we have rescaled the vev, v2 →Nv2 with respect to the last section.The equation of motion for χ,∂Γ[φ, χ]∂χ= 0(25)7

gives the following equationχ = f6(φ2 −v2) −i¯hf6Zp1p2 −χ. (26)But χ is also just −2∂¯¯Γ/∂φ2, which is the radiatively corrected “effective mass” to leadingorder and is the reason why (26) is called the “mass–gap” equation.

It is exactly the resultderived in [3] by summing superdaisy graphs, each of which is infrared divergent. Thefinite temperature, T = 1/β, result in four dimensions is [3,15,16]χ = f6(φ2 −v2) + ¯hf6 112β2 −√χ4πβ!+ .

. .

,(27)where the “. .

.” refer to(c −12 −ln[Λ2β2]) ¯hf96π2χ + ¯hfΛ296π2 . (28)This result is the sum of the T dependent finite part and the T = 0 divergent part.

Thedivergences may be absorbed by renormalizing v2 and f and by introducing an arbitrarymass scale M [15]. Then for f ∼O(1), and for all reasonable values of Mβ we can forgetthe extra terms at high enough temperatures.To compare, the zero temperature result for the action (2) in three dimensions is [15]χ = f6 (φ2 −v2) −¯hf6√χ4π .

(29)Eq. (27) determines T2, the temperature when the effective mass vanishes at theorigin, to leading order in N. Putting χ = 0 we get (¯h = 1)1β22= 12v2.

(30)Furthermore, at the origin, √χ has a remarkably simple solution for T just above T2 [3]√χ = 2π3 1β −1β2!. (31)In general, we find for √χ the leading result√χ =f48πβ"s1 + 32π2f(12φ2β2 −12v2β2 + 1) −1#.

(32)We have chosen the sign in the solution (32) of (27) so that √χ is positive for T biggerthan T2. Eq.

(32) simplifies in various limits. At φ2 = 0, for T just above T2 we obtain(31).

At 384π2φ2β22 ≫f and T just above T2,√χ ≈sfφ26 . (33)8

Finally, when T/T2 −1 ≫f/(64π2) (i.e. 32π2(1 −12v2β2)/f ≫1) then for all φ we havethe simplificationqφ ≈sf6s112 ((T −T2)2 + 2(T −T2)T2) + φ2.

(34)We now return to the computation of Γ1 in the high temperature limit. Some care mustbe taken to ensure that the final answer contains no temperature dependent divergences[17].

Also, to the order we are working in, it is sufficient to use the solution of (27) for χ[16]. We will drop all field independent divergences.We make the replacement ∂2 →−p2 and the trace becomes an integral over space–timeand momentum.

At finite T, iRp →−β−1 R d3⃗p2/(2π)3 Pn = −[4π2β]−1 R ∞0 (d⃗p2)√⃗p2 Pnwith −p2 →⃗p2+4π2n2T 2 in the integrand. The discrete sum over n runs over all integers.One then obtains,Γ1=−¯h2βZd3⃗p2(2π)3+∞Xn=−∞ln1 +¯hf/(24πβ)√⃗p2 + 4π2n2T 2 sin−11q1 + 4χ/(⃗p2 + 4π2n2T 2)+fφ2/3⃗p2 + 4π2n2T 2 + χ.

(35)The integral is not infrared divergent, but is ultraviolet divergent. The ultravioletdivergences should be the same as those appearing if we did the momentum integral overfour momenta.

[This not quite the same as taking β →∞here, because we use thehigh temperature result for the ˆσ propogator.] We must isolate these divergences beforecalculating the temperature dependent part in the high temperature limit.

Furthermore,even for say the n = 0 contribution we were not able to find an analytic expression forthe integral. We can however evaluate it in some limits.

[For the rest of this section, wedelete overall integrals over space–time. ]Consider the n = 0 contribution.

We then change variables: t =q4χ/⃗p2. Equation(35) becomesΓ1(n = 0) = −(4χ)32(2π)2β¯hZ ∞0dtt4 ln"1 + αt tan−11t+ fφ23χt2t2 + 4#,(36)where we have defined α = ¯hf/(24πβ√4χ) and used the identity sin−1(1 + t2)−1/2 =tan−1 t−1.

At φ2=0, α is small to temperatures just above T2 because from (31) we haveα < 1 as long as 1−β/β2 > f/(32π2) at the origin. In what follows, we will assume smallα and small enough φ2/χ.

In (36), as t ranges from 0 to ∞, t tan−1 t−1 ranges from 0 to1. At t = 1, t tan−1 t−1 = π/4.

Therefore for small α, αt tan−1 t−1 remains small for allvalues of t in the integral. Hence, for small α and small φ2 we can use the simplificationln[1 + r] ≈r to simplify (36) or (35).9

For the moment let us ignore the explicit φ2 term in Γ1. To calculate (35) for smallα we note the following observation for the n ̸= 0 contributions.

If √χ < πT then thearcsine possesses an expansion in inverse powers of the momentum. We have,sin−11√1+4χ/(⃗p2+4π2n2T 2)√⃗p2 + 4π2n2T 2=π/2√⃗p2 + 4π2n2T 2 −∞Xl=0(−1)l(4χ)l+ 12(2l + 1)(⃗p2 + 4π2n2T 2)l+1.

(37)The n = 0 contribution unfortunately does not possess such a straightforward expansionfor all ⃗p2. If ⃗p2 > 4χ then the expansion is as above, otherwise it issin−11√1+4χ/⃗p2√⃗p2=1√χ∞Xl=0(−1)l(⃗p2)l(2l + 1)χl .

(38)We then split up Γ1 into four parts: A, the contribution from n = 0 for ⃗p2 > 4χ; B, thecontribution from n = 0 for ⃗p2 < 4χ; C, the contribution from n ̸= 0 for ⃗p2 > 4χ; D, thecontribution from n ̸= 0 for ⃗p2 < 4χ. B is found using the expansion (38), whereas all therest involve the expansion (37).

We denote the contributions to A, C and D from the firstterm in (37) by a, c and d, respectively. We further define b to be a contribution similar toa but with the momentum integral up to ⃗p2 < 4χ.

Then it can be seen a+b+c+d is fieldindependent and can be dropped, i.e. we evaluate Γ1 = (A−a)+B +(C −c)+(D−d)−b.For B we getB = −(4χ)32(2π)2β ¯hZ ∞1dtt3 α tan−11t≈−f¯h2χ84π3β2.

(39)For b we getb = −¯h2f48πβ2Z ⃗p2=4χ0d3⃗p(2π)3π2√⃗p2 = −¯h2fχ96π2β2. (40)We also get(A −a) + (C −c)=¯h2f48πβ∞Xl=0Il+1(−1)l(4χ)l+ 12(2l + 1)≈¯h2f√χ24πβ Λ216π2 +112β2!−3¯h2fχ56π3β2,(41)where Il is given in the appendix (for ǫ = √4χ, χ = 0).

Furthermore it is straightforwardto show that D −d is subleading in temperature and may be dropped.The φ2 term from the expansion of the log is more easily evaluated and contributesΓ1 ∋−¯hf6 φ2I1 = −¯hf6 φ2 Λ216π2 +112β2 −√χ4πβ!,(42)where I1 is taken from the appendix (ǫ = 0, χ ̸= 0). If we keep all powers of φ2, butignore the arcsine term, we must also includeΓ1 ∋¯hχ324πβ∞Xl=2(−fφ2/6χ)l(2l −5)!!l!.

(43)10

This series converges for all values of φ2/χ we will be interested in.Each of the O(Λ2) divergent terms in the above equations is not separately renormal-izable, however their sum is. This is because for the next–to–leading terms we can use eq.

(27) to express φ2 −¯h√χ/4πβ = 6χ/f + v2 −¯h(12β2)−1 + O(1/N). Since the tree-levelpotential contains a term proportional to v2χ, we see that the above divergences in Γ1can be absorbed by renormalizing v2 (throughout we have neglected field independentconstants).

The dominant contributions to Γ1 are found by summing the above results;we obtainΓ1≈−¯hχ Λ216π2 +112β2!+ ¯hfφ2√χ24πβ+ ¯hχ324πβ∞Xl=2(−fφ2/6χ)l(2l −5)! !/l!,≈−¯hχ Λ216π2 +112β2!+ ¯hχ3212πβh(1 + fφ2/3χ)32 −1i.

(44)Altogether, summing the various partial results we find for the dominant high temper-ature effective scalar potential from (12), V (φ) = −Γ[φ, χ(φ)], the following renormalizedresult up to a constant (¯h = 1):V (φ) = N"12χ(φ2 −v2) −32f χ2#+(N +2) χ24β2 −χ3212πβh(1 + fφ2/3χ)32 + N −1i. (45)where χ is a solution of ∂V/∂χ = 0.

We do not give details of the renormalization whichfor zero temperature and in 3 and 4 dimensions may be found in [15,16].We will use these results in the last section of this paper, where we also discuss therange of φ2 and T for which these results are the most dominant.3. Discussion.Our main result is the leading order in N and next–to–leading order expression forthe high temperature scalar potential, eq.

(45). The tree level potential, with the nor-malizations of the last section isV (φ) = Nf4!

(φ2 −v2)2.(46)Eq. (45) includes loop corrections to this.

We assumed β√χ ≪1 and β√χ ≫f/48π ≈f/150 in obtaining the next–to–leading result. At T = T2 for the leading order result, thisgives approximately 100v2/f ≫φ2 ≫fv2/100.

At φ2 = 0 we require T −T2 ≫fT2/300.To study the nature of the phase transition we may look for zeros of dV/dφi. Thereis always one zero at the origin, φi = 0 for all i.

Away from the origin we can look forzeros of dV/dφ2 = (∂V/∂χ)(∂χ/∂φ2) + ∂V/∂φ2 = ∂V/∂φ2. Since to O(N), dV/dφ2 ∝χ,11

the leading order result has one such zero away from the origin when T < T2 (which isthe local minimum at temperatures below T2). Thus, the leading order result does notadmit a first order phase transition, for which we need zeros away from the origin fortemperatures T ≥T2.At next–to–leading order, the critical temperature T2 is modified from the leadingorder result.

∂V/∂φ2 = 0 for V given by (45) still has a solution when χ = 0. Writingout ∂V/∂χ at φ2 = 0 and setting χ = 0 immediately gives for the temperature T2 ,1 + 2NT 22 = 12v2.

(47)T2 has been reduced from its leading order value. ∂V/∂φ2 = 0 has one further solution atthe origin.

∂V/∂φ2 = 0 givesN2√χ =f24πβ(1 + fφ2/3χ)32,(48)when √χ ̸= 0. At φ2 = 0 this has the solution √χ = f/(12Nπβ).

Writing out ∂V/∂χ = 0and inserting the above expression for √χ gives a second (slightly higher) temperaturewhen the effective mass at the origin vanishes. To O(1/N) it is given by 1 + 2 −f/4π2N!T 2 = 12v2.

(49)For N = 4, f = 1 this is within 0.5 percent of T2. Between this temperature and T2 theeffective mass at the origin is negative!

However, since our approximations break downat β√χ less than about f/48π such a phenomenon is not necessarily physical; here it isan artifact of our approximations.To ascertain the nature of the phase transition we have to study the shape of thepotential near T = T2. If a second minimum degenerate in energy with the one at theorigin does not appear by temperature T1 > (1 + 5f × 10−3)T2 then we cannot reliablydetermine if the theory admits a first order phase transition.

However, in this case, if theexact theory does admit a first order phase transition then it will be only a very weakone.The general solution of χ(φ) is complicated. As noted by Root [16], one can use theleading order result for χ(φ) in the next–to–leading potential (45) to find V (φ) to O(1)in the 1/N expansion.

However, (32) can develop an imaginary part for temperaturesbelow the leading order value for T2. Therefore, it is preferable to use the next–to–leadingorder solution for χ(φ).

Our results in what follows are correct to O(1/N), barring theother approximations we made. At T near T2 given by (47) we once again have (33) whenφ2 ≫v2/300.

To see this, we write out the equation for χ(φ),6χf=φ2 −v2 + 1 + 2/N12β2−√χ4πβ−√χ4πβNh(1 + fφ2/3χ)32 −1i+fφ212πβ√χN (1 + fφ2/3χ)12. (50)12

In this put β = β2 and √χ =qfφ2/6 + δ.Assuming that δ is small one obtainsδ ≈−f/48πβ[1 + (√3 −1)/N] which can be ignored when δ ≪qfφ2/6. We can improveon this estimate for √χ by noting that when √χ ≈qfφ2/6 the last two terms in (50)add to an amount which for large enough N (say N > 3) are much smaller than the thirdlast term.

Hence we have the approximate solution√χ =f48πβ"s1 + 32π2f(12φ2β2 −12v2β2 + 1 + 2/N) −1#. (51)This should be valid at temperatures near T2 and φ2 sufficiently far from the origin.To see if a first order phase transition is possible we must find the zeros of dV/dφ2 =∂V/∂φ2.

There is one solution at √χ = 0 and another given by solving (48), which atT = T2 and χ ≈fφ2/6 has a solution at √φ2 ≈v/q3f/[2N(N + 2)]. This result holdsif N is not too large (√φ2 is not too small).

At N = 4 we have √φ2 ≈√fv/4. We cando better numerically.

We studied the case N = 4, f = 1. The approximate solution(51) suggests T1 = T2(1 + 3.5 × 10−3), with √φ2 ≈0.08v at the second minimum atT = T1.

The height of the barrier separating the two minima is at most of O(2 ×10−7v4).At T = T2 the minimum is at √φ2 ≈0.15v, which is slightly smaller than our cruderestimate above.All the numbers above occur as our approximations break down so we cannot determinethe exact nature of the phase transition. However, as argued above we may deduce fromthem that the exact model can admit only a very weak first order phase transition.

Toget a more accurate picture one should evaluate (35) in the limit β√χ ≪f/48π.To compare these results with the effective mass insertion procedure discussed inthe introduction, we note that the leading O(N) one–loop corrections to (46) give the(renormalized) potential [3]V = Nf4! (φ2 −v2)2 + f(φ2 −v2)/624β2−(f(φ2 −v2)/6)3212πβ.

(52)As indicated in the introduction, this is complex for φ2 < v2. The leading O(N) one–loopcorrected mass isNf6 φ2 −v2 +112β2!.

(53)Using this to perform the one–loop corrections, one getsV = Nf4! (φ2 −v2)2 + f(φ2 −v2 + 1/12β2)/624β2−(f(φ2 −v2 + 1/12β2)/6)3212πβ.

(54)The effective mass at the origin vanishes at T 22 = 12v2, as for the leading O(N) resultfrom the 1/N expansion. However, there is a higher temperature at which this also occurs.13

Namely, if we put T = T2+t then for small t the appropriate equation is easy to solve. Theresult is that the effective mass at the origin also vanishes at T ≈(1+f/144)T2.

Betweenthis temperature and T2 the effective mass at the origin is negative. Thus compared toour results from the 1/N expansion, we see that the effective mass insertion technique isnot reliable at temperatures less than about 1 percent above T2 even in the large N limit.In conclusion, for large enough N the model can admit only a very weak first orderphase transition.

For N = 4 it would be interesting to see how O(1/N) contributionsmodify the results above. At N = 4 the next–to–leading corrections are certainly smallerthan the O(N) corrections and we do not expect O(1/N) corrections to compete withthe O(N) or O(1) contributions.

However, it would be necessary to perform an explicitcalculation to check this belief. Finally, in the context of the standard model, one needsto add gauge fields A to this model.In fact, the contributions from gauge loops areexpected to dominate over those from scalar loops in the weak scalar self–coupling limit.The main problem in the case of a gauged model is the gauge fixing dependence of thecomputations.

Otherwise, in principle, is should be possible to determine the dominantcontributions to the finite temperature scalar potential. For example, for SU(2), the gaugekinetic term has two types of A4 interaction ([AiµAµi ]2 and [AiνAµi ]2, where i = 1, 2, 3) aswell as an A3 interaction.

Loops from the second A4 interaction and loops from the A3interaction are subleading to those from the first A4 interaction. If we drop these last twointeractions then what remains can be treated very similarly to the pure φ4 theory.

Onemust then determine what gauge fixing invariant quantities can be extracted from sucha computation. It is expected that, if carefully done, physically measurable quantitiesshould be gauge fixing independent.

Work along these lines is in progress.Acknowledgements.I am greatly indebted to G. Anderson, M.K. Gaillard, S. Naik and especially P. Weisz,who introduced me to the 1/N expansion.

I also want to thank him, M. Carena and C.Wagner for useful comments on the manuscript, and H. Ewen, G. Lavrelashvili, W. M¨osleand O. Ogievetsky for other useful discussions.14

Appendix.The finite temperature quantities we need to evaluate in this paper are all of the formIl = 1β+∞Xn=−∞Z ∞⃗p2=ǫ2d3⃗p(2π)3[⃗p2 + 4π2n2T 2 + χ]−l(55)for l a positive integer. The discrete sum runs over all integer n. We need only the casesǫ = 0 and χ ̸= 0 or ǫ ̸= 0 and χ = 0.

We assume β√χ ≪1, but β√χ ≫f/(48π) ≈f/150.For l > 1 we can find the answer by differentiating the l = 1 result l −1 times w.r.t.χ. To evaluate the expressions we follow [3].

The sum over n is explicitly performed togiveI1 =Z ∞⃗p2=ǫ2d3⃗p(2π)312√⃗p2 + χ +1√⃗p2 + χexp(β√⃗p2 + χ) −1. (56)The first term in this expression is the T = 0 piece and contains all the ultravioletdivergences; the second term is the T dependent piece and is finite.

Changing variables,the T dependent piece isI1(T ̸= 0) =12π2β2Z ∞βǫz2dz√z2 + β2χexp(√z2 + β2χ) −1. (57)In the high temperature limit, and for ǫ = 0, we have the leading result of [3],I1(ǫ = 0) =Λ216π2 + χ8π2 ln"√χΛ#+112β2 −√χ4πβ +χ16π2c −12 −2 ln[β√χ].

(58)The log pieces combine to give a single term proportional to χ ln[βΛ]. Such log–divergentterms in the final answer must be renormalizable.The coefficient of this term is ofO(β√χ/(2π)) down from the preceeding O(T) term, and in our approximation all O(χ)terms can be neglected in comparison to O(√χ/β) terms.Furthermore, to properlyrenormalize such terms in the effective potential we should keep subleading corrections tothe ˆσ propogator, eq.

(21), which we discarded. Therefore it is consistent to drop all theterms proportional to χ.

This is what we do in the following.For ǫ ̸= 0 we subtract from the above the temperature dependent piece12π2β2Z βǫ0z2dzz2 + β2χ"√z2 + β2χexp(√z2 + β2χ) −1#≈12π2β2Z βǫ0z2dzz2 + β2χ1 −12qz2 + β2χ≈ǫ2π2β −√χ2π2β tan−1" ǫ√χ#−ǫ√ǫ2 + χ8π2+ χ8π2 ln"√ǫ2 + χ + ǫ√χ#,(59)and the zero temperature piece,14π2Z ǫ0z2dz√z2 + χ = ǫ√ǫ2 + χ8π2−χ8π2 ln"√ǫ2 + χ + ǫ√χ#,(60)15

to arrive atI1(ǫ ̸= 0) =Λ216π2 +112β2 −ǫ2π2β +√χ2π2β tan−1" ǫ√χ#−π2!. (61)In the limit χ →0 this expression has an expansion in positive integer powers of χ.Therefore, its lth derivative w.r.t.

χ evaluated at χ = 0 is well defined. By expandingthe arctan in this limit we obtain for l a positive integer and χ = 0,Il(ǫ ̸= 0) =" Λ216π2 +112β2#δl−1 +ǫ3−2l2π2β(2l −3).

(62)Il for ǫ = 0 and χ ̸= 0 is simply found by differentiating (61) w.r.t. χ the appropriatenumber of times.

The answer is, for l > 1,Il(χ ̸= 0) = χ32−l4πβ(2l −5)! !2l−1(l −1)!.

(63)16

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