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COMMENTS ON THE ELECTROWEAK

arXiv:hep-ph/9203201v1 24 Feb 1992SCIPP-92-06SLAC-PUB-5740SU-ITP-92-6February 7, 1992COMMENTS ON THE ELECTROWEAKPHASE TRANSITION∗Michael Dine and Robert G. LeighSanta Cruz Institute for Particle Physics,University of California, Santa Cruz, CA 95064Patrick HuetStanford Linear Accelerator CenterStanford University, Stanford, CA 94309Andrei Linde †Department of Physics, Stanford University, Stanford, CA 94305Dmitri LindeGunn High School, Palo Alto, CA 94305ABSTRACTWe report on an investigation of various problems related to the theoryof the electroweak phase transition. This includes a determination of the∗Research supported by DOE grants 7-443256-22411-3 and DE-AC03-76SF00515, andby NSF grant PHY-8612280.E-mail:LINDE@PHYSICS.STANFORD.EDU; DINE,HUET, LEIGH@SLACVM†On leave from: Lebedev Physical Institute, Moscow.

nature of the phase transition, a discussion of the possible role of higherorder radiative corrections and the theory of the formation and evolution ofthe bubbles of the new phase. We find in particular that no dangerous linearterms appear in the effective potential.

However, the strength of the firstorder phase transition is 2/3 times less than what follows from the one-loopapproximation. This rules out baryogenesis in the minimal version of theelectroweak theory.2

1.With the recognition that baryon number violation is unsuppressedat high temperature in the standard model has come the realization that theelectroweak phase transition might be the origin of the observed asymmetrybetween matter and antimatter [1, 2, 3]. In order to have sufficient departurefrom equilibrium, it is necessary that this transition be rather strongly firstorder.

As a result, there has been renewed interest in understanding un-der what circumstances the transition is first order, and how the transitionproceeds. The minimal standard model almost certainly cannot produce theobserved asymmetry: it has too little CP violation, and, as we will see, itsphase transition is too weakly first order for a Higgs more massive than thepresent experimentalphalimit.

Nevertheless, for considering the features ofthe phase transition, it is a useful prototype, because it is weakly coupledand comparatively simple.Despite its apparent simplicity, understanding the phase transition in thistheory has turned out to be surprisingly complicated, and the literature nowcontains contradictory claims on almost every point. In the first papers onthis subject it was assumed that the phase transition is second order [4].Later it was shown that if the Higgs mass is sufficiently small, the phasetransition becomes first order [5].The problem of bubble formation wasconsidered in some detail; it was argued that at least initially the bubblewalls typically are rather thick [6].

While bubble wall evolution was onlytouched upon in these early efforts [6], simple arguments suggested that, e.g.for a 50 GeV Higgs, the motion of the wall would be non-relativistic.Recently, quite different views have been expressed about all of theseissues. Brahm and Hsu, in an interesting paper [7], have argued that infraredeffects spoil the one-loop analysis and claim to reliably establish that thetransition is second order.

Anderson and Hall [8] have thoroughly considereda number of aspects of the phase transition. Most of their results are inqualitative agreement with the earlier treatments [4, 5] and with our previousresults [9], while providing some quantitative improvement.

However, theyargue that the initial bubbles are thin. Gleiser and Kolb [10], and Tetradis[12] argue that fluctuations are so large that the transition does not proceedthrough the formation of (critical) bubbles.

There has also been controversyas to whether or not bubble wall motion is ultrarelativistic [13, 14].3

In the present note, we will attempt to deal with these various issues.We will focus on the minimal standard model. As we have said, this modelis not realistic, but we expect that the arguments and methods describedhere can be extended to more realistic situations.

We focus on this case onlybecause it is the simplest – and even here, we will frequently have to contentourselves with rather crude calculations. We will be able to give at leastpartial answers to each of the questions raised above.

We will argue that,for sufficiently small coupling, the phase transition can be reliably shown tobe first order. In particular, the linear term in the potential found in Ref.

[7] is not present. On the other hand, we will see that the transition is moreweakly first order than suggested by the one loop analysis.

We will show thatthe bubbles are thick when they form, and the bubble nucleation rate cannotbe reliably computed in the thin wall approximation. We will show that, forthe range of Higgs masses considered here, the transition does in fact proceedby nucleation of critical bubbles.

Finally, we will make some estimates of thebubble wall thickness and velocity. We will see that the early estimates ofRef.

[6] are only reliable if particle mean free paths are very long. In practice,the relevant mean free paths are short, and the velocity of the bubble wallsomewhat larger than these early estimates suggest.

The problem of bubblewall propagation turns out to be surprisingly complicated, and we mentionsome of the issues which must be considered. We will see, however, that,for the first order transitions being considered here, the motion of the walltends to be non-relativistic.

In this paper, we will outline our treatment ofeach of these issues and describe the major results; details will be given in asubsequent publication [15].2.We first consider the question of the order of the phase transition.The standard approach to this problem consists of computing the effectivepotential to one loop order. At high temperature, the one-loop expressionfor VT is given, to a good approximation, byV (φ, T) = D(T 2 −T 2o )φ2 −ETφ3 + λT4 φ4 .

(1)Here λ = m2H/2v2o, andD =18v2o(2m2W + m2Z + 2m2t) ,E =14πv3o(2m3W + m3Z) ∼10−2 ,(2)4

T 2o =14D(m2H −8Bv2o) ,B =364π2(2m4W + m4Z −4m4t) ,(3)λT = λ −316π2v4o 2m4W ln m2WaBT 2 + m4Z ln m2ZaBT 2 −4m4t ln m2taFT 2!. (4)Here vo = 246 GeV is the value of the scalar field at the minimum of V (φ, 0)and ln aB = 2 ln 4π −2γ ≃3.51, ln aF = 2 ln π −2γ ≃1.14 [4, 9, 8].The behavior of V (φ, T) is reviewed in Refs.

[16, 17].At very hightemperatures the only minimum of V (φ, T) is at φ = 0. A second minimum,φ1, appears at T = T1, whereT 21 =T 2o1 −9E2/8λT1D ,φ1 = 3ET2λT.

(5)The values of V (φ, T) in the two minima become equal to each other at thetemperature Tc, whereT 2c =T 2o1 −E2/λTcD ,φc = 2ETλT. (6)The minimum of V (φ, T) at φ = 0 disappears at the temperature To, whenthe field φ in the second minimum is φo = 3ET/λT.3.The first order character of the transition is due to the φ3 termin the potential.

The appearance of such a term non-analytic in |φ|2 is thesignal of an infrared problem, and raises concerns about the validity of theperturbation expansion. Indeed, these issues were raised in the early workon the subject [5, 18, 19].

The problem of infrared divergences at high tem-peratures in theories with light or massless particles has received extensiveattention in the literature and is discussed in many textbooks [16, 20]. Itis well known that these problems arise from the zero-frequency terms inthe discrete frequency sums which appear in the computation of equilibriumquantities at finite temperature.

The corresponding Feynman diagrams arethose appropriate to a three dimensional field theory. For simplicity, we willconsider here the contribution of W bosons.

In the present case, the problemis most easily analyzed in Coulomb gauge, ⃗∇· ⃗W = 0. The relevant propa-gators are those for the Coulomb lines, D00, and transverse lines, Dij.

For5

zero frequency,D00 = 1⃗k2 ,Dij(⃗k) =1⃗k2 + m2W(φ)Pij(⃗k) ,(7)where mW = gφ/2, and Pij = δij −kikj⃗k2 .The ’Goldstone’ field and thephysical Higgs scalar have standard scalar propagators with mass terms whichare independent of the gauge coupling. The cubic term is readily extractedfrom the zero-frequency piece of the determinant.

2/3 of it arises from thetransverse gauge bosons; the other 1/3 is obtained from the Coulomb line.In this gauge it is rather easy to see how higher orders of perturbationtheory behave. At one loop, it is well-known that the Coulomb line, evenfor φ = 0, acquires a mass m2D = (Ng + Nc)g2T 2/3 [21].

In order to obtaina sensible perturbation theory for small φ, it is necessary to partially resumthe perturbation expansion, i.e. to use in each order of the loop expansiona Coulomb propagator with this effective mass.

In the infra-red, this resum-mation corresponds to integrating out all heavy modes (to one loop order),leaving only the ω = 0 modes, with a Debye mass. In practice, the tree levelgauge boson mass at the minimum of the potential is small compared to mD.As a result, repeating the one loop calculation with this effective mass, thecontribution to the cubic term from the Coulomb line disappears.

The sameis not true, however, for the transverse bosons. For zero φ, gauge invarianceforbids a one-loop mass for these fields; the transverse polarization tensor isin fact given by Πij = 15g2T√⃗k232Pij.

As a result, the contribution to the cubicterm from these fields survives, and the net effect is to reduce the coefficientof the cubic term in eq. (1) by 2/3: 1E =16πv3o(2m3W + m3Z) .

(8)This reduction means that, for a given Higgs mass, the phase transitionis significantly less first order than one expects from the one loop analysis.1After we obtained this result, we received a very interesting paper by Carrington [22]where the modification of the cubic term by high order corrections was also considered.Even though the author did not claim that these corrections reduce the cubic term by afactor of 2/3, after some algebra one can check that his result is equivalent to ours.6

It has been pointed out that a minimal requirement of the phase transitionis that the sphaleron rate after the transition be sufficiently small that thebaryon number not be washed out. Using the (unimproved) one loop result,this gives a limit of about 42 −55 GeV [23, 9], at best just barely consistentwith the present experimental constraints [24].

Allowing for the correctionobtained here, the limit on mH is reduced by about 25%, clearly ruling outbaryogenesis in the model.More generally, however, we can ask about the behavior of the pertur-bation expansion, particularly at small φ. Before making general remarks,it is helpful to consider the two loop diagrams involving transverse gaugebosons and scalars (Fig.

1). The zero frequency pieces of these diagrams,separately, give contributions ∼g3|φ|T 3.

If one combines these diagrams,however, being careful about combinatorics, they have the structure of aninsertion of a polarization tensor on the transverse gauge boson line. Be-cause, as mentioned above, this tensor vanishes at zero momentum, the sumof these diagrams is less singular at small φ, and simply gives a correction tothe quadratic term.

We have checked all other diagrams at two loop orderand shown that there are no linear terms in the potential.The authors of Ref. [7] found a linear contribution to the potential bysimply substituting the scalar mass found at one loop back into the one loopcalculation.

Such a procedure is generally reliable when calculating Green’sfunctions or tadpoles. Indeed, it is well known that the sum of the geometricprogression, which appears after the insertion of an arbitrary number ofpolarization operators Π(T) into the propagator (k2 + m2)−1, simply gives(k2 + m2 + Π(T))−1.

However, this trick does not work for the closed loopdiagram for the effective potential, which contains ln(k2 + m2).A naivesubstitution of the effective mass squared m2 + Π(T) instead of m2 intoln(k2 + m2) corresponds to a wrong counting of higher order corrections.The simplest way to avoid the ambiguity is to calculate tadpole diagrams fordVdφ instead of the vacuum loops, and then integrate the result with respect toφ. One can easily check by this method as well, that no linear terms appearin the expression for V (φ, T).The absence of linear terms does not automatically mean that higherorder corrections are completely under control.

A general investigation of7

the infrared problem in the non-Abelian gauge theories at finite temperaturesuggests that the results which we obtained are reliable for φ ∼> g2 T ∼T/3[18, 19]. Thus, a more detailed investigation is needed to study behaviorof the theories with mH ∼> 102 GeV near the critical temperature, since thescalar field, which appears at the moment of the phase transition in thesetheories, is very small (see Fig.

2). However, we expect that our results arereliable for strongly first order phase transitions with φ ∼> T, which is quitesufficient to study (or to rule out) baryogenesis in the electroweak theory.4.We turn now to the problem of bubble formation.

At high tempera-tures, this becomes a problem in classical thermodynamics. One looks for astationary point of the free energy, with the property that φ →0 as r →∞,i.e.

a solution of the classical field equations with potential V0 + VT. Somecare is required in solving this equation, however, since necessarily one isconstructing a saddle point of the action (unstable modes corresponding tobubble growth or collapse). As a result, if one makes a poor approximation,one overestimates the probability of bubble formation.

We believe this is thecase of the analysis of Ref. [8], where formation of bubbles was studied in thethin wall approximation.

This approximation works well if the height of thebarrier between the two minima is much larger than the difference betweenthe values of the effective potential in each of them. This is not the case forthe phase transition with mH ∼< 60 GeV.

Numerical solution of the equationsyields an action typically a few times larger than that obtained from the thinwall approximation.Even though the thin wall approximation fails, one can still study bubbleformation analytically in a wide class of theories. Indeed, we have foundthat in the vicinity of the phase transition one can write, to a very goodapproximation,S3T = 38.8 D3/2E2·∆TT3/2× f2 λo D ∆TE2 T.(9)wheref(α) = 1 + α4h1 +2.41 −α +0.26(1 −α)2i.

(10)Parameter α changes from 0 to 1 in the range of temperatures for which thephase transition is possible. We have found that eq.

(10) is correct with anaccuracy about 1% in the most interesting range 0 ≤α ≤0.95.8

5.In the case that the transition is weakly first order, it is natural to askwhether the transition actually proceeds through formation of bubbles, or ifother sorts of fluctuations might be more important. In most treatments, itis assumed that the transition occurs once the bubble nucleation rate is largeenough that the universe can fill with bubbles.

In practice, because of theextremely slow expansion rate at the time of the transition, this means thatthe barrier is still high enough that the naive calculation of the nucleationrate gives an extremely small result; the three dimensional action is of order130−140. Given that the rate of formation of critical bubbles is so small, onemight expect that other types of fluctuations which might equilibrate the twophases would be extremely rare.

In Refs. [10, 11, 12], however, it has beenargued that this is not the case.

Roughly speaking, these authors arrive atthis conclusion by estimating the mean square fluctuation of the scalar fieldabout the symmetric minimum, φ2rms =< φ2 >, and comparing this with thevalue of the field at the other minimum. A rough estimate leads them tothe conclusion that the < φ2 >∼mT ∼φ2c, so that it is not meaningful toconsider the system as sitting in one vacuum or the other.

Here m is theHiggs field mass near φ = 0. Subcritical bubbles, they argue, equilibrate thetwo phases even before one reaches the temperature Tc.While we believe that for some range of parameters subcritical bubblesmay be important, we do not believe that this is the case for the Higgsmasses under consideration here.

In estimating φrms, one should be carefulto consider only long wavelength modes. Short wavelength modes will beassociated with configurations with large gradient terms, which will collapsein a microscopic time.

Also, one must be very careful with factors of π and 2.A more detailed investigation based on the stochastic approach to tunnelinggives the estimate for the amplitude of relevant fluctuations < φ2 >∼mTπ2[25]. Combining this estimate with our results for the mass m, i.e.

for thecurvature of the effective potential near φ = 0 in the relevant temperatureinterval, To < T < T1, one obtains an estimate φrms ∼.1 T.From this analysis, we see that the typical amplitude of the relevant scalarfield fluctuations is substantially less than the separation of the two minima,unless the phase transition is very weakly first order. Even for mH ∼60GeV, the distance between the two minima remains five times greater thanφrms.

Including fluctuations with k ≫kmax will give larger φrms, but these9

will collapse in a microscopic time, and will not serve to equilibrate the twophases.6.Understanding the motion of the wall is important mainly in the con-text of baryogenesis. We have already seen that no baryons will be generatedin the single Higgs theory, unless the Higgs mass is well below the presentexperimental limit.

We expect, however, that, for mH ∼35 GeV (for whichφ/T > 1), the bubble wall motion in this model will have many features incommon with more realistic theories of baryogenesis, which require the phasetransition to be strongly first order. Even in this simple model, determiningthe wall velocity and shape is a difficult problem, and we will content our-selves with rather crude estimates.

In our analysis, we will assume that thewall achieves a steady state after some time. In particular, there is a frame,which we refer to as the ‘wall frame’, in which the scalar field and the parti-cle distributions are independent of time.

We will assume that the principlesource of damping of the walls motion is elastic scattering of particles fromthe wall; if there are additional sources of damping, they can only slow thewall even further. We will treat the velocity of the wall as a small parameter.We will have to check its validity a posteriori.

With these assumptions, it ishelpful to consider two limiting cases, depending on whether the size of thewall, δ, is large or small compared to the relevant mean free paths for elasticscattering, ζ. As an estimate of these mean free paths, we follow Ref.

[3]and take the longitudinal and transverse gluon propagators to include a massproportional to mD. For top quarks, this yields σt = 16 πα2s/3m2el.

Multiply-ing by the flux one obtains ζ ∼4 T −1 for quarks. For W’s and Z’s the resultis about three times larger.

These numbers are consistent with results whichhave been obtained for the stopping power [26]. To get some feeling for thewall size, consider the system at temperature Tc.

At this temperature, thetwo phases can coexist, separated by a static domain wall. The φ field inthis domain wall is readily obtained by quadrature; for a 35 GeV Higgs, onefinds δ ∼40 T −1.This suggests that the thin wall limit may not be a good one, but it isinstructive to consider it in any case.

In this limit, a typical particle passesthrough the wall, or is reflected from it, without scattering. An estimate inthis limit was given in Ref.

[6]. If the problem is treated semiclassically,it is straightforward to calculate the extra, velocity-dependent force on the10

wall, assuming that all particles approaching the wall from either side aredescribed by an equilibrium distribution at some temperature. (Note thatparticles moving away from the wall are not described by an equilibriumdistribution in this case; these particles are assumed to be equilibrated farfrom the wall, at some possibly different temperature and velocity).

In thismodel it is straightforward to calculate the force on the wall to linear orderin v. For bosons, the leading term is of order m3T, while for fermions, it isof order m4. Equating this force to the pressure difference on the two sidesof the wall gives the velocity.

Definingǫ = Tc −TTc −To,(11)we can obtain an approximate equation for the velocity, valid for small ǫ:v ∼π6ǫ(1 + ǫ) C(mt, λ) ,(12)where the correction C comes predominantly from top quarks and is roughlyof order one. For mt = 120 GeV and mH = 35 GeV (for which ǫ ∼1/4), wefind v ∼0.05.In this discussion we have made a variety of oversimplifications.Onewhich is potentially important is our assumption of equilibrium densities onboth sides of the wall.

Some fraction of incoming particles, however, arereflected from the wall, and this will tend to lead to an enhancement of theparticle density in front of the wall. For definiteness, consider top quarks.As a crude estimate, suppose the fraction of reflected particles is f (of orderm/T), and suppose that the mean free path for processes which can changetop quark number is τ, with ζ ≪τ.

Then the density in front of the wallis enhanced by an amount of order fvnq τζ . For the thin wall case, this islikely to reduce the velocity of the wall somewhat.

In the thicker wall casesconsidered below, however, this is likely to be more important.Consider now the case δ ≫ζ. In this case, to model the deviations fromequilibrium due to the finite velocity of the wall, we can break the wall intosegments of length ζ, and repeat the thin wall analysis for each of thesesegments.

In particular, we assume that the distributions on either side ofeach segment are at equilibrium. We also assume that the temperature and11

velocity of the particle distributions are constant across the wall. This shouldnot be a bad approximation when there are many light species of particlesin the system.

One obtains for the v-dependent force on the wall, a resultroughly suppressed by a factor of orderqζ/δ. For a Higgs mass of 35 GeVwe obtain a velocity ∼0.2.In this thick wall case, the density enhancement described earlier could beextremely important.

Here one expects the extra particles to be distributedmore or less uniformly over the wall. This leads to an extra force ∆F ∼fnvτ ∆ρ /δ where ∆ρ denotes the internal energy difference on the two sidesof the wall.

As an estimate of f, we can take the ratio of the equilibriumdensities on the two sides of the wall; for top quarks, this gives a numberof order 5%. This effect seems to be comparable to that of the precedingparagraph.7.In this paper we have outlined a program for treating the phasetransition in weakly coupled theories with scalar fields.

From the standpointof electroweak baryogenesis, it is important to find models consistent withcurrent experimental bounds in which the phase transition is strongly firstorder. It would be interesting to find models where the bubble wall is thin;in such theories [3] electroweak baryogenesis can be very efficient.

However,our work suggests that in many models, the wall will be slow and thick, andadiabatic analyses of the type of Ref. [2] will be relevant.

In these cases,one may just barely be able to produce the observed asymmetry. Further im-provements in both the theory of the phase transition and that of electroweakbaryogenesis will be necessary to completely settle these questions.We are grateful to L. Susskind, L. McLerran, R. Kallosh, N. Turok andM.

Gleiser for discussions.12

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