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PROGRESS IN THE THEORY OF THE

arXiv:hep-ph/9206259v1 29 Jun 1992SU-ITP-92-18PROGRESS IN THE THEORY OF THEELECTROWEAK PHASE TRANSITIONAndrei Linde 1Department of Physics, Stanford University, Stanford, CA 94305ABSTRACTRecent progress in the theory of the electroweak phase transition isdiscussed. It is shown, that for the Higgs boson mass smaller than the masses ofW and Z bosons, the phase transition is of the first order.

However, its strength isapproximately 2/3 times less than what follows from the one-loop approximation.This rules out baryogenesis in the minimal version of the electroweak theory withlight Higgs bosons. The possibility of the strongly first order phase transition inthe theory with superheavy Higgs bosons is considered.We show that if the Yang-Mills field at high temperature acquires a magneticmass ∼g2T, then the infrared problem and the problem of symmetry behaviorat high temperature effectively decouple from each other, no linear terms appearin the effective potential in all orders of perturbation theory and the symmetryin gauge theories at high temperatures is actually restored.

Even though thelast statement was never questioned by most of the authors, it was extremelydifficult to come to a reliable conclusion about it due to the infrared problem inthermodynamics of non-Abelian gauge fields.The phase transition occurs due to production and expansion of criticalbubbles. A general analytic expression for the probability of the bubble formationis obtained, which may be used for study of tunneling in a wide class of theories.Invited TalkTexas Symposium on Electroweak Baryon Number ViolationYale University, March 19921E-mail: linde@physics.stanford.edu.

On leave from: Lebedev Physical Institute, Moscow1

1IntroductionThis talk is based on the results obtained in our papers with Michael Dine, Patrick Huet,Robert Leigh and Dmitri Linde [1]. It contains also some more recent results on the in-frared problem in the electroweak theory, on the existence of symmetry restoration at a hightemperature and on the possibility of the first order phase transition in the theory withsuperheavy Higgs fields.The existence of the phase transition in the electroweak theories was discovered by DavidKirzhnits twenty years ago [2].

A detailed theory of the phase transition was proposed in1974 by three groups of authors independently (by Weinberg, Dolan and Jackiw and byKirzhnits and Linde [3]), and soon the theory of the electroweak phase transition becameone of the well established ingredients of modern cosmology. Surprisingly enough, the theoryof this phase transition is still incomplete.In the first papers on this problem it was assumed that the phase transition is of thesecond order [2, 3].

Later Kirzhnits and Linde showed [4] that in the gauge theories with manyparticles, and especially with particles which are much more heavy than the Higgs bosonφ, one should take into account corrections to the high temperature approximation used in[2, 3]. These corrections lead to the occurrence of cubic terms ∼g3φ3T in the expressionfor the effective potential V (φ, T).

As a result, at some temperature, V acquires an extraminimum, and the phase transition is first order [4]. Such phase transitions occur throughthe formation and subsequent expansion of bubbles of the scalar field φ inside the symmetricphase φ = 0.

A further investigation of this question has shown that the phase transitionsin grand unified theories are always strongly first order [5]. This realization, as well as themechanism of reheating of the universe during the decay of the supercooled vacuum statesuggested in [4, 6], played an important role in the development of the first versions ofthe inflationary universe scenario [7].

(For a review of the theory of phase transitions andinflationary cosmology see Ref. [8].

)For a long time it did not seem likely that the electroweak phase transition could haveany dramatic consequences. Even though the possibility of a strong baryon number violationduring the electroweak phase transition was pointed out fifteen years ago [9, 10], only afterthe paper by Kuzmin, Rubakov and Shaposhnikov [11] was it realized that such processesdo actually occur and may erase all previously generated baryon asymmetry of the universe.Recently, the possibility that electroweak interactions may not only erase but also producethe cosmic baryon asymmetry has led to renewed interest in the electroweak phase transition.A number of scenarios have been proposed for generating the asymmetry [12] – [19].

All ofthem require that the phase transition should be strongly first order since otherwise thebaryon asymmetry generated during the phase transition subsequently disappears. In all ofthese scenarios the asymmetry is produced near the walls of the bubbles of the scalar field φ.Therefore it is necessary to make a much more thorough analysis of the electroweak phase2

transition than the analysis which is necessary for an approximate calculation of the criticaltemperature.We will say that the phase transition is strongly first order if the ratio of the Higgs fieldφ inside the bubble to the temperature T is larger than one, since otherwise the baryonasymmetry will be washed out by nonperturbative effects.This condition was used in [12, 20] to impose a strong constraint on the Higgs mass inthe minimal version of the electroweak theory, mH<∼42 GeV. This, of course, alreadycontradicts the present experimental limits mH>∼57 GeV [21].

However, more carefulconsideration of various theoretical uncertainties indicated that the constraint might besomewhat weaker, permitting mH up to 55 GeV, or possibly higher [22]. In multi-Higgsmodels [20, 15], the limits are substantially weaker.Before one can discuss details of the process of baryogenesis, it is necessary to checkthat the results of our investigation of the phase transition are reliable.

This is not a trivialissue even in the minimal electroweak theory. Indeed, as stressed in Refs.

[4, 6], each neworder of perturbation theory at finite temperature may bring a new factor of g2T/m ∼gT/φfor the theories with gauge boson masses m ∼gφ. This means that the results of the oneloop calculations may become unreliable at φ <∼gT.

Thus, it became very desirable to gobeyond the one-loop approximation.An example of such an approach is given by the self-consistent approximation elaboratedin [4]. In this approximation, instead of the mass of a particle at zero temperature one usesits temperature-dependent mass, taking into account the contribution from the polarizationoperator.

This method made it possible, in particular, to overcome unphysical difficultiesrelated to imaginary masses of scalar particles at small φ.Recently this approach was reinvented by many authors.Some of the recent resultsobtained by this method were quite surprising. For example, it was claimed that higherorder corrections lead to the appearance of a term in the effective potential ∼−g3φT 3[24, 25].

This term is linear in φ; it is very large at small φ. Depending on its sign, it eithermay remove the local minimum of V (φ, T) created by the cubic term ∼−g3φ3T, or it maymake this minimum much more deep.Our investigation of this problem shows that if one is careful with counting of Feynmandiagrams, neither positive nor negative linear terms ∼g3φT 3 appear in the effective potential[1].

Even though now the authors of Refs. [24, 25] agree that the linear terms ∼g3φT 3 areabsent, we will repeat our main arguments here, since these arguments may allow us to domuch more than just say that there are no linear terms in order g3.

A generalization of thesearguments allows us to formulate the conditions under which one can show, despite someuncertainties with higher order corrections, that the expectation value of the scalar field φat high temperatures actually disappears, φ = 0. Note, that this would be impossible in thepresence of linear terms of any magnitude and sign.3

However, higher order corrections do lead to a significant modification of the one-loopresults. They lead to a decrease of the cubic term g3φ3T by a factor 2/3 [1].2 This effectdecreases the ratio φ/T at the point of the phase transition by approximately the same factor2/3.

This makes baryogenesis virtually impossible in the context of the minimal standardmodel with mW >∼mH >∼57 GeV.All results discussed above were obtained in the context of the theories with small couplingconstants and light Higgs fields. However, we do not really know whether the Higgs boson islight or very heavy.

If the Higgs boson is superheavy, mH >∼103 GeV, the phase transitionmay become strongly first order. Even though this possibility is extremely speculative, itmay lead to important consequences.

Therefore we will discuss it in this paper.Assuming that one knows the shape of the effective potential at small φ, one should stillwork hard to determine the ratio φ/T at the point of the phase transition. One needs toknow at what temperature the transition actually occurs, and some details of how it occurs.At very high temperatures the effective potential of the Higgs field, V (φ, T), has a uniqueminimum at the symmetric point φ = 0.

As the temperature is lowered, a second minimumappears. At a critical value Tc, this second minimum becomes degenerate with the firstone.

However, the phase transition actually occurs at a somewhat lower temperature, dueto the formation of bubbles of true vacuum which grow and fill the universe. The usualway to study bubble formation is to use the euclidean approach to tunneling at a finitetemperature [27].

One should find high-temperature solutions, which describe the so-calledcritical bubbles.Then one should calculate their action, which leads to an exponentialsuppression of the probability of bubble formation. Typically, these calculations are rathercomplicated, and analytic results can only be obtained in a few cases.

One of these is thethin wall approximation, which is valid (as in the case of transitions at zero temperature)if the difference in depth of the two minima of V (φ, T) is much smaller than the heightof the barrier between them. In this case the radius of the bubble at the moment of itsformation is much larger than the size of the bubble wall, and the properties of the bubblecan be obtained very easily.

However, the thin wall approximation in our case leads to anunderestimate of the tunneling action by a factor of two. Fortunately, we were able to obtaina simple analytic expression which gives the value of the euclidean action for theories withan effective potentials of a rather general type, V (φ, T) = aφ2 −bφ3 +cφ4.

We hope that thisresult will be useful for a future investigation of bubble formation in a wide class of gaugetheories with spontaneous symmetry breaking.On the other hand, validity of the standard assumption that the phase transition occursdue to formation of critical bubbles should be verified as well. Kolb and Gleiser [28] and,more recently, Tetradis [29] have argued that the phase transition may occur by a differentmechanism, the formation of small (subcritical) bubbles.

If this is the case, the transition2A similar result was obtained also by Carrington [26]. However, her original results were different fromours approximately by a factor of two.

Consequently, they lead to an impression that modification of theone-loop results should lead to an increase of the strength of the first order phase transition. At present,there is no disagreement between our results.4

is completed earlier and by a different mechanism than in the conventional picture. Whilethis idea is very interesting, we will argue (see also [30] and the talk of Anderson at thisConference) that it is only relevant in cases where the transition is very weakly first orderand the euclidean action corresponding to critical bubbles is not much larger than one.

Thisis not the case for the strongly first order phase transitions, where the relevant value of theeuclidean action at the moment of the transition is S ∼130 −140.2The Phase TransitionLet us consider the form of the effective potential at finite temperature. Contributions ofparticles of a mass m to V (φ, T) are proportional to m2 T 2, m3 T and m4 ln(m/T).

We willassume that the Higgs boson mass is smaller than the masses of W and Z bosons and thetop quark, mH < mW, mZ, mt. Therefore we will neglect the Higgs boson contribution toV (φ, T).The zero temperature potential, taking into account one-loop corrections, is given by [8]V0 = −µ22 φ2 + λ4φ4 + 2Bv2oφ2 −32Bφ4 + Bφ4 ln(φ2v2o) .

(1)HereB =364π2v4o(2m4W + m4Z −4m4t) ,(2)vo = 246 GeV is the value of the scalar field at the minimum of V0, λ = µ2/v2o, m2H = 2µ2.Note that these relations between λ, µ, vo and the Higgs boson mass mH, which are true atthe classical level, are satisfied even with an account taken of the one-loop corrections. Thisis an advantage of the normalization conditions used in [8].

An expression used in [23] isequivalent to this expression up to an obvious change of variables.At a finite temperature, one should add to this expression the termVT = T 42π26I−(yW) + 3I−(yZ) −6I+(yt),(3)where yi = Miφ/voT, andI∓(y) = ±Z ∞0dx x2 ln(1 ∓e−√x2+y2) . (4)The results of our work are based on numerical calculation of these integrals, without makingany specific approximations [22].

However, in the large temperature limit it is sufficient touse an approximate expression for V (φ, T) [2, 23],V (φ, T) = D(T 2 −T 2o )φ2 −ETφ3 + λT4 φ4 . (5)5

HereD =18v2o(2m2W + m2Z + 2m2t) ,(6)E =14πv3o(2m3W + m3Z) ∼10−2 ,(7)T 2o =12D(µ2 −4Bv2o) = 14D(m2H −8Bv2o) ,(8)λT = λ −316π2v4o 2m4W ln m2WaBT 2 + m4Z ln m2ZaBT 2 −4m4t ln m2taFT 2!,(9)where ln aB = 2 ln 4π −2γ ≃3.91, ln aF = 2 ln π −2γ ≃1.14.It will be useful for our future discussion to identify several ‘critical points’ in the evolutionof V (φ, T).At very high temperatures the only minimum of V (φ, T) is at φ = 0. A second minimumappears at T = T1, whereT 21 =T 2o1 −9E2/8λT1D .

(10)The value of the field φ in this minimum at T = T1 is equal toφ1 = 3ET12λT1. (11)The values of V (φ, T) in the two minima become equal to each other at the temperature Tc,whereT 2c =T 2o1 −E2/λTcD .

(12)At that moment the field φ in the second minimum becomes equal toφc = 2ETcλTc. (13)The minimum of V (φ, T) at φ = 0 disappears at the temperature To, when the field φ in thesecond minimum becomes equal toφo = 3EToλTo.

(14)3Infrared Problems and Reliability of the Perturba-tion ExpansionIn our previous discussion, we have considered only the one loop corrections to the effectivepotential. In this section we discuss the role of higher order corrections.6

It is well known that, in field theories of massless particles, perturbation theory at finitetemperature is subject to severe infrared divergence problems. For small values of the scalarfield, the gauge bosons (and near the phase transition, the Higgs boson) are nearly massless;as a result, as was pointed out in the early work on this subject [4, 6], one cannot reliablycompute the effective potential for very small φ.The problem is that the higher ordercorrections in coupling constants may contain terms of the type of g2 TmN.

As a result,higher order corrections go out of control for m < g2T. For scalar particles this happensnear the critical point only.

Indeed, scalar particles have masses m ∼gT ≫g2T in the hightemperature limit. However, if one takes into account gauge invariance, it can be shown that“magnetic” components of vector particles cannot acquire any contribution to their “masses”larger than g2T.At this point one should be more precise.The Green function of the vector field issingular at k2 ∼g2T 2.

In this sense one may speak about the vector field mass ∼gT.However, the Green function of the vector field at a finite temperature does not have asimple pole singularity. For example, in addition to the singularity at k2 ∼g2T 2, the Greenfunction of a photon has a singularity at k0 = 0,⃗k →0.It is this singularity that isresponsible for all infrared problems in quantum statistics of gauge fields, since the Greenfunctions at k0 = 0 give the leading infrared divergent contribution to thermodynamicalsums [6, 33, 34].

Investigation of the infrared problem in gauge theories without spontaneoussymmetry breaking has shown that the “magnetic mass” (corresponding to the limit k0 =0,⃗k →0) may appear in the non-Abelian theories, but it cannot be larger than O(g2)T.Thus, in the absence of spontaneous symmetry breaking, or at φ <∼gT, when the magneticmass of the vector particles become smaller than g2T, perturbative results may becomeunreliable. One may wonder, therefore, is it possible that some unusual contribution to theeffective potential at φ <∼gT may alter our results.Recently, in a very interesting paper, Brahm and Hsu found that at small φ, higher ordercorrections to the scalar field contribution to the effective potential may produce a largenegative linear term −g3φT 3, which eliminated any trace of a first order transition.On the other hand, Shaposhnikov considered higher order corrections to the vector par-ticle contribution to V (φ, T) and found a large positive term + g3φT 3which made the phasetransition strongly first order (φ/T > 1) even for mH ∼64 GeV [25].We will show that neither positive nor negative linear terms appear in the expressionfor V (φ, T) if one studies higher order corrections paying particular attention to the correctcounting of Feynman diagrams.

[1]. (Additional information on this problem is contained inthe talks by Michael Dine and Robert Leigh at this conference.

)We will consider here for simplicity the contribution of the scalar particles and the Wbosons only; adding the contribution of Z bosons is trivial. As we have already noted, forquestions of infrared behavior, fermions may be ignored.

Coulomb gauge, ⃗∇· ⃗W = 0, is7

particularly convenient for the analysis, though the problem can be analyzed in other gaugesas well. In this gauge, the vector field propagator Dµν after symmetry breaking (and aftera proper diagonalization) splits into two pieces, the Coulomb piece, D00, and the transversepiece, Dij.

For non-zero values of the discrete frequency, ωn = 2πnT, the Coulomb piecemixes with the ’Goldstone’ boson. However, for the infrared problems which concern us here,we are only interested in the propagators at zero frequency.

For these there is no mixing.One has [4]D00(ω = 0,⃗k) =1⃗k2 + m2W(φ)(15)andDij(ω = 0,⃗k) =1⃗k2 + m2W(φ)Pij(⃗k) ,(16)where Pij = δij −kikj⃗k2 . The mass of the vector field W at the classical level is given bymW = gvo/2.

Propagators of the Higgs field φ and of the ’Goldstone’ field χ in this gaugeare given byDφ(⃗k) =1⃗k2 + m2φ,(17)Dχ(⃗k) = 1⃗k2 . (18)Let us review several ways of obtaining the standard one-loop expression for the cubicterm in the effective potential, eq.

(5). The most straightforward is to carefully expand eq.

(3) for the effective potential in yW = mWvoT = gφ2T . Indeed, the contribution of W-bosons tothe effective potential at T > mW(φ) is given byVW(φ, T)=2 × 3 ×−π290T 4 + m2W(φ)24T 2 −m3W(φ)12πT + · · ·=2 × 3 ×−π290T 4 + g2φ296 T 2 −g3φ396π T + · · ·.

(19)Here the expression in brackets coincides with the contribution of a scalar field with mass mW;the factor 2 appears since there are two W-bosons with opposite charges, while the factor 3,which will be particularly important in what follows, corresponds to the two transverse andone longitudinal degrees of freedom with mass mW.Alternatively, we can obtain the cubic term by looking directly at the one-loop Feynmandiagrams. For this purpose, it is only necessary to examine the zero frequency contributions.Certain diagrams containing four external lines of the classical scalar field naively give acontribution proportional to g4φ4; the cubic term arises because the zero frequency integralsdiverge for small mass as T/mW ∼T/gφ.Consider, in particular, the zero frequency part of the expression for the one loop freeenergy in momentum space.

It is simplest to compute the tadpole diagrams for dV/dφ and8

afterwards integrate with respect to φ. The transverse gauge bosons give a contributiondVtrdφ = 2 × g2φT2Zd3k(2π)31⃗k2 + m2W= −2 × g2φ T8πqm2W ,(20)where, by keeping only the zero frequency mode, we have dropped terms which are analyticin m2.

The Coulomb lines give half the result of eq. (20).

Integration of the total vectorfield contribution correctly represents the cubic term in (19).A complete gauge boson contribution to the tadpole, including the non-zero frequencymodes, is [4]dVW(φ, T)dφ= 2 × 3 × g2φ48T 2 −3mWTπ+ · · ·= 2 × 3 × g2φ48T 2 −3 gφT2π+ · · ·. (21)One can easily check that integration of this expression with respect to φ gives eq.

(19).With these techniques, we are in a good position to study higher order corrections tothe potential. The authors of Refs.

[24, 25] found a linear contribution to the potential bysubstituting the mass found at one loop back into the one loop calculation. The effectivemasses-squared of both scalar particles and of the Coulomb field contain terms of the form∼g3Tφ, which, upon substitution in (34), give linear terms.

But this procedure is not alwayscorrect. It is well known that the sum of the geometric progression, which appears afterthe insertion of an arbitrary number of polarization operators Π(φ, T) into the propagator(k2 +m2)−1, simply gives (k2 +m2 +Π(φ, T))−1.

Therefore one can actually use propagators(k2 + m2 + Π(T))−1, which contain the effective mass-squared m2 + Π(φ, T) instead of m2.However, this trick with the geometric progression does not work for the closed loop diagramfor the effective potential, which contains ln(k2 + m2). A naive substitution of the effectivemass squared m2 + Π(φ, T) instead of m2 into ln(k2 + m2) corresponds to a wrong countingof higher order corrections.This does not mean that there is no regular way to make this trick for the effectivepotential.

One may add and subtract from the Lagrangian the term −12 φ2 Π(φ(T), T) tothe Lagrangian, and a similar term for the vector field as well.Here Π(φ(T), T) is thepolarization operator with an account taken of all daisy and superdaisy diagrams, φ(T) isa classical field, not an operator. Then the effective mass (at zero momentum) becomesrenormalized, m2 →m2 + Π(φ, T), but one should add some extra diagrams containing theinsertion −12 φ2 Π(φ(T), T).

These diagrams were not considered in [24, 25]. Here one canclearly understand the difference between calculating effective potential and tadpoles.

Ifone uses this method to calculate the tadpole diagrams, insertions of −12 φ2 Π(φ(T), T) self-consistently cancell the diagrams which would generate extraneous polarization operatorcorrections to already corrected effective mass m2(φ(T), T) = m2 + Π(φ, T). This gives usthe standard prescription of simply substituting m2(φ(T), T) = m2 + Π(φ, T) instead of m2in all propagators (k2 + m2)−1.9

Thus, the simplest way to take into account high temperature corrections to masses ofvector and scalar particles without any problems with combinatorics is to compute tadpolediagrams for d Vdφ ; these are then trivially integrated to give the potential. One can easily checkby this method that no linear terms appear in the expression for V (φ, T).

Indeed, at a giventemperature and effective mass, the tadpoles are linear in φ (see e.g. equation (21)).

To takeinto account the mass renormalization in the tadpoles, one should substitute the effectivemass squared m2 + Π(φ, T) into the one-loop expression for the tadpole contribution; as weexplained above (see also ([4])), this is a correct and unambiguous procedure for tadpoles.dVtrdφ = 2 × g2φT2Zd3k(2π)31⃗k2 + m2 + Π(φ, k, T). (22)This expression could lead to a linear term in the effective potential only if the integralwould behave as φ−1 in at small φ →0.

However, in our case this is impossible, since thepolarization operator calculated in [24, 25] at small k and φ is nonnegative, and the integralconverges to a constant. Therefore its subsequent integration with respect to φ, which givesthe correction to the effective potential, is quadratic in φ, i.e.

it does not contain any linearterms.We should emphasize that our approach effectively takes into account all diagrams con-sidered in [24], [25], but with correct combinatorics. Note also, that this approximation workseven if the polarization operator depends on the classical field φ.3 An apparent triviality ofthe investigation with the help of tadpoles is not due to its incompleteness, but due to thepower and simplicity of this method, which was elaborated in [4].

In particular, it was evenunnecessary for us to use a particular expression for the polarization operator, as far as it isnonnegative. We will return to this issue shortly.Even though there are no linear terms ∼g3φT 3, higher order corrections do have adramatic effect on the phase transition.

This effect is a modification of the cubic term.As we have shown above, the cubic term appears due to the contribution of zero modes,ωn = 2πnT = 0. This makes it particularly easy to study its modification by high ordereffects.

Indeed, it is well known that the Coulomb field at zero frequency acquires the Debye‘mass’, m2D = Π00(ωn = 0,⃗k →0) ∼g2 T 2. This leads to an important modification of theCoulomb propagator (15):D00(⃗k →0) =1⃗k2 + m2D + m2W(φ).

(23)For the values of φ of interest to us, m2D ≫m2W(φ). Thus, repeating the calculation ofthe cubic term, the Coulomb contribution disappears.

However, the transverse contribution,3We disagree with the recent claim made in [35] that our approximation works only if the polarizationoperator Π does not depend on φ, and that we do not take into account ‘subleading’ diagrams which giverise to dangerous linear terms obtained in [24], [25]. Indeed, in [1] we did not take into account diagrams ofthe type shown in Fig.

2; see their discussion below. However, these diagrams were not considered in [24],[25] as well, since they do not lead to the linear terms in the order g3.10

which is two times larger than the Coulomb one, is unaffected at this order, due to thevanishing of the ‘magnetic mass’ [6, 33, 34]. As a result, the cubic term does not disappear,but it is diminished by a factor4 of 2/3:E =16πv3o(2m3W + m3Z) .

(24)This small correction proves to be very significant. Indeed, eqs.

(13), (14) show thatthe ratio of the scalar field φ to the temperature at the moment of the phase transition isproportional to E, i.e. to the cubic term.

Actually, the dependence is even slightly stronger,since for smaller E the tunneling occurs earlier.Even before the reduction of the cubicterm was taken into account, the ratio φ/T for mH>∼57 GeV was slightly less than thecritical value φ/T ≈1. The decrease of this quantity by a factor of 2/3 makes it absolutelyimpossible to preserve the baryon asymmetry generated during the phase transition in theminimal model of electroweak interactions with mH >∼57 GeV.Are these results completely reliable?The effective coupling constant of interactionsbetween W bosons and Higgs particles is g/2.

In this case, a general investigation of theinfrared problem in the non-Abelian gauge theories at a finite temperature suggests that theresults which we obtained are reliable for φ>∼g2 T ∼T/3 [6], [33, 34]. Thus, a moredetailed investigation is needed to study behavior of the theories with mH>∼102 GeVnear the critical temperature, since the scalar field, which appears at the moment of thephase transition in these theories, is very small.

However, we expect that our results arereliable for strongly first order phase transitions with φ >∼T/3, which is quite sufficient tostudy (or to rule out) baryogenesis in the electroweak theory. Recent investigation of higherorder corrections to this theory indicates [36] that these results may be reliable even downto φ ∼gT/10.Finally, we would like to address a fundamental question: since the theory for φ ≪gT is infrared divergent, can we definitely establish that the symmetry is restored at hightemperature, or is it possible that φ always has some small, non-zero value?

Indeed, if ourapproximation breaks down at φ <∼gT, how do we know that the symmetry restorationactually takes place, i.e. φ = 0 at T > To?To address this question, we can work far away from the critical point, at T −To ≫To.The best approach, as before, is to study all possible higher order tadpole diagrams, see Figs.1, 2.There are two different classes of diagrams to be considered.

External line of the scalarfield may split either into two lines of the vector field, Fig. 1, or into two lines of vectorfield and one line of scalar field, Fig.

2. All diagrams of the first type can be representedas the trivial one-loop diagram, plus the diagrams with an arbitrary number of polarization4There was also a claim that the cubic term disappears completely [37], but recently this claim waswithdrawn.11

operator insertions. The simplest diagram of that type is shown in Fig.

1. The black circlestands for the exact polarization operator Π(φ, k, T), including all higher order corrections.The sum of all these diagrams gives us the one-loop diagram with an exact Green functionof the vector field instead of the free field propagator.

In other words, as usual, one mustadd polarization operator to the mass squared of the vector field.The behavior of Π(φ, k, T) at k > g2T is known perturbatively. It leads only to high-ordercorrections to V (φ, T), which do not contain any nonanalytic terms, such as the dangerouslinear terms in φ.The only possible source of problems is our absence of knowledge ofΠ(φ, k, T) at φ <∼gT, k0 = 0, |⃗k| <∼g2T.

Indeed, in this domain all higher order correctionsto Π(φ, k, T) are equally important.However, the consequences of this uncertainty mayappear not very significant.Indeed, consider again the most dangerous part of the tadpole diagram, eq. (20), andadd Π(φ, k, T) to m2W(φ):dVtrdφ = 2 × g2φT2Zd3k(2π)31⃗k2 + g2φ24+ Π(φ, k, T).

(25)The part of the integral in the domain of uncertainty, |⃗k| <∼g2T, is given byg2φT2π2Z g2T0k2dkk2 + g2φ24+ Π(φ, k, T). (26)On dimensional grounds, one expects that at gφ ≪g2T and |⃗k| ≪g2T the polarizationoperator has some value of the order g4T 2 [6, 33, 34].

This follows from the fact that themost infrared divergent part of the theory corresponds to the three-dimensional theory, withg2T being the only mass (or coupling constant) scale.One cannot exclude a possibility that the polarization operator is proportional to −g2|⃗k|T,or it is of the order g4T 2, but is negative, and the integral in (26) diverges in the limitφ →0.One should keep this possibility in mind, since it may lead to interesting andunusual consequences, like Bose-condensation or even crystallization of the Yang-Mills fieldat high temperature [33]. Indeed, the tadpole integral has a simple interpretation in termsof integration over the occupation numbers of bose fields.

Large contribution to this integralmay be interpreted as a result of Bose condensation of particles in a state with a nonvanishingmomentum.In our case, this effect may also lead to absence of a complete symmetryrestoration at T > To. Note, that this effect may occur only at φ <∼gT, as we anticipated.However, it is not quite clear that this possibility is physically viable.

Whereas occupationnumbers may be large, they can hardly be negative.The standard (and most conservative) assumption is that 0 ≤Π(φ = 0, k = 0) <∼g4T 2[6, 33, 34]. This corresponds to generation of a magnetic mass 0 ≤m2 <∼g4T 2.

In such case12

we may estimate the corresponding integral at small φ asg2φT2π2Z g2T0k2dkk2 + O(1) g4T 2 ∼g4φT 2 . (27)This term after integration over φ does not give any linear terms in φ.

It gives just a smallcorrection to the quadratic part of the effective potential, ∆V ∼g4φ2T 2. Such correctionsdo not alter our conclusions concerning symmetry restoration at high temperature.

Note,that this term corresponds to the sum of all most dangerous contributions to the tadpolediagrams of the type of Fig.1, to all orders in g2.Now let us consider the diagrams which contain internal lines of scalar field, Fig. 2.

Thesediagrams may be dangerous near the critical point, where the scalar fields are almost massless,but they are much less dangerous than the diagrams of the first class at the temperaturemuch higher than critical. The reason is that at high temperature the scalar field acquires alarge mass m2 ∼g2T 2 ≫g4T 2.

Therefore scalar particles by themselves do not lead to anyinfrared problems outside of a small vicinity of the critical point. The presence of such heavyparticles effectively cuts infrared divergencies in the diagrams with vector particles as well.One can easily check that the diagram with one vector loop, Fig 2a, gives the contributiong4φ2T 2 to the effective potential, the diagram with two vector loops gives g5φ2T 2, the diagramwith three vector loops gives g6φ2T 2.

Starting with this diagrams, infrared problem becomesmanifest in that each diagram of this type with higher number of vector loops gives thecontribution of the same order g6φ2T 2.Thus, the infrared problem in thermodynamics of gauge fields does not permit us tocalculate the effective potential at high temperature to all orders of perturbation theory.The diagrams Fig.2 contain uncertainties at the level of g6φ2T 2; the diagrams Fig.1contain uncertainties at the level of g4φ2T 2. However, neither of these diagrams producelinear terms in φ, unless the Green functions of a massless Yang-Mills field has a pathologicalbehavior at large temperature.

As for the quadratic terms, they can be calculated at leastwith an accuracy up to g3φ2T 2, or maybe even up to g4 ln g φ2T 2. This is quite sufficient tocalculate the critical temperature and to make a conclusion that at the temperature higherthan critical the scalar field φ vanishes.These considerations indicate that the situation with the phase transitions in the non-Abelian gauge theories is probably the same as in the standard case: infrared problems mayprevent a simple description of the phase transition in a small vicinity of the critical point(unless the phase transition is strongly first order), but everywhere outside this region, thesymmetry behavior of gauge theories can be described in a reliable way.13

4First order phase transitions with superheavy Higgs?In our previous investigation we neglected the contribution of Higgs bosons to the one-loopeffective potential. The reason was very simple: We have seen that the increase of the Higgsboson mass decreases the strength of the phase transition.

This can be easily understood byconsidering a model of a single scalar field with the effective potentialV0 = −µ22 φ2 + λ4φ4 ≡−m2H4 φ2 + λ4φ4 . (28)Near the point of the phase transition, where the high-temperature approximation works well,one may investigate the symmetry behavior in this theory in a self-consistent approximationsuggested in [4], where only cactus diagrams should be evaluated.

In this approximationthe effective mass of the scalar field and the first derivative of the effective potential in itsextremum are simply related to each other:m2(T, φ) = 3λφ2 −µ2 + Π(T, m(T, φ)) ,(29)and1φdV 1dφ = Π(T, m(T, φ)) . (30)Here V 1 is the one-loop contribution to the effective potential,1φdV 1dφ = 3λ2π2Z ∞0k2 dkqk2 + m2(T, φ)exp√k2+m2(T,φ)T−1 .

(31)At the minimum of the effective potential with φ(T) ̸= 0dV 1dφ + λφ3 −µ2φ = 0 ,(32)which, together with (29) and (30), gives m2 = 2λφ2(T) and1φdV 1dφ = 3λ2π2Z ∞0k2 dkqk2 + 2λφ2(T)exp√k2+2λφ2(T)T−1 = λT 24 1 −3√2λφ(T)πT+ ...!. (33)On the other hand, the local minimum of V (φ) at φ = 0 disappears when m(T, 0) = 0.

Thisgives the critical temperatureTo = 2v ,(34)where v = µ/√λ = 246 GeV, v > φ(T). For small λ the last term in eq.

(33) is small atT ≈To, which implies that the phase transition is weakly first order. Actually, we cannoteven say from eq.

(33) whether the phase transition is second order or weakly first order,since near the critical point the higher order corrections are large [4].14

Let us nevertheless use eq. (33) to make a bold estimate of conditions under which thephase transitions could be strongly first order.

From eqs. (33), (34) it follows that the jumpof the scalar field at the critical temperature (assuming that To is close to the temperatureof the phase transition) is given by 3√2λT4π.

It is larger than the temperature T (which is thecondition for baryogenesis) if λ >∼9, or, equivalently,mH >∼103 GeV . (35)For obvious reasons, this estimate should not be taken as a serious indication of existenceof strongly first order phase transitions and baryogenesis with superheavy Higgs bosons.However, the stakes are high, and the possibility to have a strongly first order phase transitionand baryogenesis in the strong coupling regime with superheavy Higgs bosons (technicolour?

)should not be overlooked.5Bubble FormationIn the previous section we noted that the two minima of V (φ, T) become of the same depthat the temperature Tc, eq. (12).

However, tunneling with formation of bubbles of the field φcorresponding to the second minimum starts somewhat later, and it goes sufficiently fast tofill the whole universe with the bubbles of the new phase only at some lower temperature Twhen the corresponding euclidean action suppressing the tunneling becomes less than 130 –140 [14, 22, 23]. In [1] (see also [22]) we performed a numerical study of the probability oftunneling.

Before reporting our results, we will remind the reader of some basic concepts ofthe theory of tunneling at a finite temperature.In the euclidean approach to tunneling (at zero temperature) [32], the probability ofbubble formation in quantum field theory is proportional to exp(−S4), where S4 is the four-dimensional Euclidean action corresponding to the tunneling trajectory. In other words, S4is the instanton action, where the instanton is the solution of the euclidean field equations de-scribing tunneling.

A generalization of this method for tunneling at a very high temperature[27] gives the probability of tunneling per unit time per unit volumeP ∼A(T) · exp(−S3T ) . (36)Here A(T) is some subexponential factor roughly of order T 4; S3 is a three-dimensionalinstanton action.

It has the same meaning (and value) as the fluctuation of the free energyF = V (φ(⃗x), T) which is necessary for bubble formation. To find S3, one should first find anO(3)-symmetric solution, φ(r), of the equationd2φdr2 + 2rdφdr = V ′(φ) ,(37)15

with the boundary conditions φ(r = ∞) = 0 and dφ/dr|r=0 = 0. Here r =qx2i ; the xi arethe euclidean coordinates, i = 1,2,3.

Then one should calculate the corresponding actionS3 = 4πZ ∞0r2 dr[12 dφdr!2+ V (φ(r), T)] . (38)Usually it is impossible to find an exact solution of eq.

(37) and to calculate S3 withoutthe help of a computer. A few exceptions to this rule are given in Refs.

[8, 27]. One of theseexceptional cases is realized if the effective potential has two almost degenerate minima, suchthat the difference ε between the values of V (φ, T) at these minima is much smaller than theenergy barrier between them.

In such a case the thickness of the bubble wall at the momentof its formation is much smaller than the radius of the bubble, and the action S3 can becalculated exactly as a function of the bubble radius r, the energy difference ∆V and thebubble wall surface energy S1:S3 = −4π3 r3∆V + 4πr2S1 ,(39)whereS1 =Z ∞0dφq2V (φ, T) . (40)The radius of the critical bubble r can be found by finding an extremum of S3(r).

However,one must be very careful when using these results. Indeed, as can be easily checked, thisextremum is not a minimum of the action, it is a maximum.

Therefore, the action corre-sponding to the true solution of eq. (37) will be higher than the action of any approximatesolution.

As a result, one can strongly overestimate the tunneling probability by calculat-ing it outside the limit of validity of the thin wall approximation. In our case the thin wallapproximation underestimates the tunneling action by a factor of two, i.e.

it gives the proba-bility of tunneling about e−100 where the correct answer is e−200. If the only thing one wishesto know is the time when the tunneling occurs, this error is not very important.

It leads onlyto a few percent error in calculation of the temperature of the universe at the moment of thephase transition, since the tunneling action is extremely sensitive to even very small changesof the temperature. Thus, one may argue that the thin wall approximation is still useful.

(See also the talk of Anderson at this Conference.) However, it is possible to determine thetime of the phase transition with an accuracy of few percent without any study of tunneling:It is enough to say that the phase transition happens in the middle of the interval betweenTc and To.

In order to obtain a complete description of the phase transition, including acorrect shape of the bubble wall, one should go beyond the thin wall approximation.We would now like to obtain an analytic estimate of the probability of tunneling in theelectroweak theory, which can be used for any particular numerical values of constants D,E and λT. As shown in Ref.

[23], eq. (5) in most interesting cases approximates V (φ, T)with an accuracy of a few percent.

This by itself does not help very much if one must studytunneling anew for each new set of the constants. However, it proves possible to reduce this16

study to the calculation of one function f(α), where α is some ratio of constants D, E andλT. In what follows we will calculate this function for a wide range of values of α.

This willmake it possible to investigate tunneling in the electroweak theory without any further useof computers.First of all, let us represent the effective Lagrangian L(φ, T) near the point of the phasetransition in the following form:L(φ, T) = 12(∂µφ)2 −M2(T)2φ2 + ETφ3 −λo4 φ4 . (41)Here M2(T) = 2D(T 2 −T 2o ) is the effective mass squared of the field φ near the point φ = 0,<∼t is the value of the effective coupling constant λT near the point of the phase transition(i.e.

at T ∼Tt, where Tt is the temperature at the moment of tunneling). With a very goodaccuracy, the constants λt, λT1, λTc, λTo are equal to each other.Defining φ =M22ET Φ, x = X/M, the effective Lagrangian can be written as:L(Φ, T) =M64E2T 2h12(∂µΦ)2 −12Φ2 + 12Φ3 −α8 Φ4i,(42)whereα = λoM22E2T 2 .

(43)The overall factorM64E2T 2 does not affect the Lagrange equationd2ΦdR2 + 2RdΦdR = Φ −32Φ2 + 12αΦ3 . (44)Solving this equation and integrating over d3X = M−3d3x gives the following expression forthe corresponding action:S3T = 4.85 M3E2 T 3× f(α) .

(45)The function f(α) is equal [27] to 1 at α = 0, and blows up when α approaches 1. In thewhole interval from 0 to 1 this function, with an accuracy about 2%, is given by the followingsimple expression:f(α) = 1 + α4h1 +2.41 −α +0.26(1 −α)2i.

(46)In the vicinity of the critical temperature To, i.e. at ∆T ≡T −To ≪To, the action (45)can be written in the following form:S3T = 38.8 D3/2E2·∆TT3/2× f2 λo D ∆TE2 T.(47)Using these results, one can easily get analytical expressions for the tunneling probabilityin a wide class of theories with spontaneous symmetry breaking, including GUTs and theminimal electroweak theory.17

6Subcritical BubblesDespite our semi-optimistic conclusions concerning the infrared problem, it is still desirableto check that the whole picture of the behavior of the scalar field described above is (at least)self-consistent. This means that if the effective potential is actually given by eqs.

(5), (6), (8),(9), (24), then our subsequent description of the phase transition and the bubble formationis correct. Indeed, one would expect that the theory of bubble formation is reliable, since thecorresponding action for tunneling S3/T is very large, S3/T ∼130 −140.

However, recentlyeven the validity of this basic assumption has been questioned. Gleiser and Kolb [28] andTetradis [29] have argued that in many cases phase transitions occur not due to bubbles ofa critical size, which we studied in section 3, but due to smaller, subcritical bubbles.

Webelieve that these authors raise a real issue. However, we will now argue that this problemonly arises if the phase transition is extremely weakly first order.The basic difference between the analysis of Ref.

[28, 29] and the more conventional oneis their assumption that at the time of the phase transition there is a comparable probabilityto find different parts of the universe in either of the two minima of V (φ, T). The mainargument of Ref.

[28, 29] is that if the dispersion of thermal fluctuations of the scalar field< φ2 >∼T 2 is comparable with the distance between the two minima of V (φ, T), then thefield φ “does not know” which minimum is true and which is false. Therefore it spendscomparable time in each of them.According to [28], a kind of equilibrium between thedomains of the two types is achieved due to subcritical bubbles with small action S3/T ifmany such bubbles may appear within a horizon of a radius H−1.In order to investigate this question in a more detailed way, let us re-examine our ownassumptions concerning the distribution of the scalar field φ prior to the moment at which thetemperature drops down to T1, when the second minimum of V (φ, T) appears.

According to(11), the value of the scalar field φ in the second minimum at the moment when it is formedis equal to φ1 = 3ET2λT . For mH ∼60 GeV (and taking into account the coefficient 2/3 in thecubic term) one obtains φ1 ∼0.4 T. Thermal fluctuations of the field φ have the dispersionsquared < φ2 >= T 2/12.

(Note an important factor 1/12, which was absent in the estimatemade in [28].) This gives dispersion of thermal fluctuations √< φ2 > ∼0.3 T, which is notmuch smaller than φ1.However, as the authors of [28] emphasized in their previous work [38] (see also [29]), thetotal dispersion < φ2 >∼T 2/12 is not an adequate quantity to consider since we are notreally interested in infinitesimally small domains containing different values of fluctuatingfield φ.

They argue that the proper measure of thermal fluctuations is the contribution to< φ2 > from fluctuations of the size of the correlation length ξ(T) ∼M−1(T). This leads toan estimate < φ2 >∼T M(T), which also may be quite large [38].

Here again one shouldbe very careful to use the proper coefficients in the estimate. One needs to understand alsowhy this estimate could be relevant.18

In order to make the arguments of Ref. [28, 29] more quantitative and to outline thedomain of their validity, it is helpful to review the stochastic approach to tunneling (see[39] and references therein).

This approach is not as precise as the euclidean approach (intheories where the euclidean approach is applicable). However, it is much simpler and moreintuitive, and it may help us to look from a different point of view on the results we obtainedin the previous section and on the approach suggested in [28, 29].The main idea of the stochastic approach can be illustrated by an example of tunnelingwith bubble formation from the point φ = 0 in the theory (41) with the effective potentialV (φ, T) = M2(T)2φ2 −ETφ3 + λo4 φ4.

(48)For simplicity, we will study here the limiting case λo →0.At the moment of its formation, the bubble wall does not move. In the limit of smallbubble velocity, the equation of motion of the field φ at finite temperature is simply,¨φ = d2φ/dr2 + (2/r)dφ/dr −V ′(φ) .

(49)The bubble starts growing if ¨φ > 0, which requires that|d2φ/dr2 + (2/r)dφ/dr| < −V ′(φ) . (50)A bubble of a classical field is formed only if it contains a sufficiently big field φ.

It shouldbe over the barrier, so that dV/dφ < 0, and the effective potential there should be negativesince otherwise formation of a bubble will be energetically unfavorable. The last conditionmeans that the field φ inside the critical bubble should be somewhat larger than φ∗, whereV (φ∗, T) = V (0, T).

In the theory (48) with <∼o→0, one has φ∗= M2/2ET. As a simplest(but educated) guess, let us take φ ∼2φ∗= M2/ET.

Another important condition is thatthe size of the bubble should be sufficiently large. If the size of the bubble is too small, thegradient terms are bigger than the term |V ′(φ)|, and the field φ inside the bubble does notgrow.

Typically, the second term in (50) somewhat compensates the first one. To make avery rough estimate, one may write the condition (50) in the form12r−2 ∼12k2 < 12k2max ∼φ−1|V ′(φ)| ∼2M2.

(51)Let us estimate the probability of an event in which thermal fluctuations with T ≫Mbuild up a configuration of the field satisfying this condition. The dispersion of thermalfluctuations of the field φ with k < kmax is given by< φ2 >k

(52)19

Note that the main contribution to the integral is given by k2 ∼k2max ∼4M2. This meansthat one can get a reasonably good estimate of < φ2 >k

The result we get is< φ2 >k

We can use it to evaluate the probabilitythat these fluctuations build up a bubble of the field φ of a radius r > k−1max. This can bedone with the help of the Gaussian distribution5P(φ) ∼exp(−φ22 < φ2 >k

(54)Note that the factor in the exponent in (54) to within a factor of C2 = 1.02 coincides with theexact result for the tunneling probability in this theory obtained by the euclidean approach[27] (see eq. (46)):P ∼exp(−4.85M3E2T 3 ) .

(55)Taking into account the very rough method we used to calculate the dispersion of the per-turbations responsible for tunneling, the coincidence is rather impressive.As was shown in [39], most of the results concerning tunneling at zero temperature, at afinite temperature and even in the inflationary universe, which were obtained by euclideanmethods, can easily be reproduced (with an accuracy of the coefficient C2 = O(1) in theexponent) by this simple method.Now let us return to the issue of subcritical bubbles. As we have seen, dispersion ofthe long-wave perturbations of the scalar field, < φ2 >k

Its calculation provides a simple and intuitive way to get the sameresults as we obtained earlier by the euclidean approach [39]. To get a good estimate of theprobability of formation of a critical bubble in our simple model one should calculate thisdispersion for kmax ∼2M(T), which gives < φ2 >k

Note, that this estimateis much smaller than the naive estimate < φ2 >∼TM.The crucial test of our basic assumptions is a comparison of this dispersion and the valueof the field φ at the moment T = T1, when the minimum at φ = φ1 ̸= 0 first appears. Using5The probability distribution is approximately Gaussian even though the effective potential is not purelyquadratic.

The reason is that we were able to neglect the curvature of the effective potential m2 = V ′′ whilecalculating < φ2 >k

eqs. (5), (11), one can easily check that the mass of the scalar field at T = T1, φ = 0 is givenbym = 3ET2√λT.

(56)This yieldsq< φ2 >kk

(58)Thus, even with account taken of the factor 2/3 in the expression for E, the dispersion oflong-wave fluctuations of the scalar field is much smaller than the distance between the twominima. Therefore, the field φ on a scale equal to its correlation length ∼M−1 is not equallydistributed between the two minima of the effective potential.

It just fluctuates with a verysmall amplitude near the point φ = 0. The fraction of the volume of the universe filled bythe field φ1 due to these fluctuations (i.e.

due to subcritical bubbles) for mH ∼60 GeV isnegligible,P(φ1) ∼exp −φ22 < φ2 >k

The answer remains rather small even for mH ∼100 GeV,when the phase transition is very weakly first order.Moreover, even these long-wave fluctuations do not lead to formation of stable domainsof space filled with the field φ ̸= 0, until the temperature is below Tc and critical bubblesappear. One expects a typical subcritical bubble to collapse in a time τ ∼k−1max; this isabout thirteen orders of magnitude smaller than the total duration of the phase transition,∆t ∼10−2H−1 ∼10−4Mp T −2.

We do not see any mechanism which might increase τ bysuch a large factor.Despite all these comments, we think that subcritical bubbles deserve further investi-gation. They may lead to interesting effects during phase transitions in GUTs, since thedifference between T −1 and the duration of the GUT phase transitions is not as great as inthe electroweak case.

They may play an important role in the description of the electroweakphase transition as well, in models where the phase transition occurs during a time not muchlonger than T −1. This may prove to be the case for very weakly first order phase transitionswith 103 GeV ≫mH >∼102 GeV, when the distance between the two minima of V (φ, T)at T ∼T1 is smaller than the dispersion √< φ2 >k

7ConclusionsOne of the main consequences of our work [1] is that it is very difficult to generate baryonasymmetry in the standard model without expanding its Higgs sector. One the other hand,now we better understand what is necessary for the electroweak baryogenesis to work andhow to calculate relevant quantities.

Hopefully, this will help us to find a realistic theory ofelementary particles where electroweak baryogenesis is possible.Acknowledgements This work could not be done without collaboration with MichaelDine, Patrick Huet, Robert Leigh and Dmitri Linde. I appreciate very much fruitful dis-cussions with Greg Anderson, Renata Kallosh, Larry McLerran and Lenny Susskind.

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