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A LATTICE MONTE CARLO STUDY OF THE HOT ELECTROWEAK

arXiv:hep-ph/9305345v1 28 May 1993CERN-TH.6901/93A LATTICE MONTE CARLO STUDY OF THE HOT ELECTROWEAKPHASE TRANSITIONK. Kajantiea,b, K. Rummukainena and M. Shaposhnikova1aTheory Division, CERN,CH-1211 Geneva 23, SwitzerlandbDepartment of Theoretical Physics, P.O.Box 9, 00014 University of Helsinki, FinlandAbstractWe study the finite temperature electroweak phase transition with lattice perturba-tion theory and Monte Carlo techniques.

Dimensional reduction is used to approximatethe full four-dimensional SU(2) + a fundamental doublet Higgs theory by an effectivethree-dimensional SU(2) + adjoint Higgs + fundamental Higgs theory with coefficientsdepending on temperature via screening masses and mass counterterms. Fermions con-tribute to the effective theory only via the NF and mtop dependence of the coefficients.For sufficiently small lattices (N3 < 303 for mH = 35 GeV) the study of the one-looplattice effective potential shows the existence of the second order phase transition evenfor the small Higgs masses.

At the same time, a clear signal of a first order phasetransition is seen on the lattice simulations with a transition temperature close tobut less than the value determined from the perturbative calculations. This indicatesthat the dynamics of the first order electroweak phase transition depends strongly onnon-perturbative effects and is not exclusively related to the so-called φ3 term in theeffective potential.CERN-TH.6901/93May 19931On leave of absence from Institute for Nuclear Research of Russian Academy of Sciences, Moscow117312, Russia

1IntroductionThe interest in the study of the high temperature phase transitions in gauge theories,initiated a long time ago in [1], has been revived now in connection with possiblegeneration of the baryonic asymmetry of the universe at the electroweak scale (see, e.g.,reviews [2, 3] and references therein). The common belief, based on the perturbativecalculations of the effective potential for the scalar field, is that the electroweak phasetransition for moderate Higgs boson masses is weakly first order (i.e.temperaturemetastability range is small compared with the critical temperature).

An incompletelist of references containing perturbative analysis of the phase transitions is containedin [4]-[14]. All perturbative calculations, however, work only for sufficiently large valuesof the scalar field, namely φ ≫gT.

This estimate arises as follows. Roughly speaking,there are at least three relevant mass scales at high temperatures.

The first one isrelated to the Debye screening of the gauge charge and is of the order of mD = CDgT(for the electroweak theory with NF = 6 CD =116 ). The second one is associatedwith the non-abelian magnetic sector of gauge theories, mM = CMg2T (CM is somenumber yet to be determined) and the third one with the gauge boson mass inducedby the Higgs mechanism, mW = 12gφ with φ being the condensate of the scalar field.The naive loop expansion for the effective potential works well provided mW ≫mD,i.e.

for φ ≫T. This condition is usually not satisfied for the interesting range offields and temperatures.

The improved loop expansion which takes into account Debyemass effects in all orders of perturbation theory works for a larger interval of the scalarcondensate. However, due to the infrared problem in the high temperature dynamicsof gauge theories [15] any of the loop expansions break down for mW < mM andφ < 2CMgT.

Due to the fact that the coupling constant in electroweak theory is notso small (g ≈2/3) it is important to know the numerical value of the coefficient CM.If it is small, then perturbative calculations of the effective potential are valid in aninteresting range of parameters, while a large value of CM would imply the lack of anyknowledge of the dynamics of the phase transition.The appearance of the magnetic mass is a purely non-perturbative phenomenonand not a lot is known about its value. It is associated with the confinement scaleof 3-dimensional gauge theory derived by dimensional reduction from 4-dimensionaltheory at high temperatures.Different correlation lengths in 3-dimensional SU(2)gauge theory were studied by lattice Monte-Carlo methods in [16].

It was found therethat the 3-dimensional 0++ glueball mass (inverse screening length in this channel)is about ∼2g2T. This indicates that non-perturbative effects are rather large andthat presumably CM ∼2.

If true, a perturbative analysis of the effective potentialin a parameter range relevant for cosmology has low chances to remain if force afternon-perturbative effects are taken into account.In order to clarify this question, one has to use non-perturbative methods for thestudy of the effective potential. They are provided by lattice Monte-Carlo simulations.Some preliminary data derived with the use of 4-dimensional lattices is already available[17]-[19].

There is an indication [19] that the strength of the first order phase transitionis enhanced in comparison with perturbative calculations.1

The purpose of this paper is to combine perturbative and non-perturbative methodsfor the study of the finite T electroweak transition. Namely, with the use of dimensionalreduction [20]-[24] one can first derive a 3-dimensional effective action with perturba-tively calculable coefficients, summing up thus the effects of the Debye screening whichare well understood.

Then, one can simulate the phase transition with lattice Monte-Carlo methods.There are a number of advantages of this method in comparison with 4-dimensionallattice simulations. First, it separates the physics in which we are reasonably confident(Debye screening) from the unknown non-perturbative 3-dimensional physics.

In otherwords, any 4-dimensional simulations contain perturbative noise which has nothing todo with non-perturbative effects determining the order of the phase transition. More-over, perturbation theory signals that the first order nature of the electroweak phasetransition is an exclusively 3-dimensional phenomenon, since the so-called φ3 term inthe effective potential (this term induces a jump of the order parameter) arises due toinfrared singularities in loop integrations at high temperatures.

The last, but not leastadvantage is that a lattice simulation of 3-dimensional theories is less time consumingand more transparent from the point of view of scaling behaviour.The paper is organized as follows. In Section 2 we derive the 3-dimensional effectiveaction for our system.

In Section 3 we compute the effective potential in the continuum3-dimensional theory and show how the first order nature of the phase transition arisesin the loop expansion of the effective potential. In Section 4 we study the effectivepotential for the lattice version of the theory which is then compared in Section 5 withMonte Carlo simulations.

Section 6 contains our conclusions.2The effective actionFinite T field theoretic systems are characterised by fields defined over the interval0 < τ < β ≡1/T in imaginary time and extended beyond this region by the con-dition of periodicity (antiperiodicity) with the period β for bosonic (fermionic) fields.If the action is expressed in terms of the Fourier components, the quadratic termsare of the type [(2πnT)2 + k2]|A(n, k)|2, where A(n, k) is a generic bosonic field andn = −∞, . .

. , +∞.

At high T and k < 2πT the nonstatic modes A(n ̸= 0, k) arethus suppressed by the factor (2πnT)2 relative to the static A(0, k) modes. The ideaof dimensional reduction is to exploit this suppression by integrating over the non-static modes in S[A(n = 0, k), A(n ̸= 0, k)] and deriving in this way an effective actionSeff[A(0, k)] for the dominant static modes.

We shall below carry this out for the four-dimensional SU(2) + a fundamental Higgs theory, taking g′ = 0 in the electroweaksector of the standard model. The effective theory then is a three-dimensional SU(2)+ a fundamental Higgs + an adjoint Higgs model with well defined T-dependent coef-ficients.For fermionic fields the square of the inverse propagator is [(2n+1)πT]2 +k2 and allmodes are suppressed at large T. However, the fermionic fields will enter by changingthe coefficients of the effective action of the bosonic static modes.

Their effect can thus2

also be studied in this framework. However, as our calculations so far are only on theone loop level, we can only include a top quark of mass less than 79 GeV, given by oneloop stability [25].The starting point is the action of the 4d SU(2) + fundamental Higgs modelS[Aaµ(τ, x), φi(τ, x)] =Z β0 dτZd3x{ 14F aµνF aµν + (Dµφ)†(Dµφ) + µ2φ†φ + λ(φ†φ)2}, (1)in standard notation.

Possible fermionic terms are not shown explicitly.Including 1-loop corrections and terms not damped by powers of 1/T the dimension-ally reduced effective action becomesSeff[Aai (x), Aa0(x), φi(x)] = 1TZd3x14F aijF aij + 12(DiA0)a(DiA0)a + (Diφ)†(Diφ) ++1223(1 + 14 + NF4 )g2T 2 −(4 + 1)g2TΣcAa0Aa0 +g412π2(1 + 116 −NF8 )(Aa0Aa0)2 ++−12m2H + (18g2 + 116g2 + 12λ + g2m2top8m2W)T 2 −(32g2 + 34g2 + 6λ)TΣcφ†φ + λ(φ†φ)2 ++14g2Aa0Aa0φ†φ,(2)where Σc is the integralΣc =Zd3p(2π)3p2(3)depending linearly on the cutoff. In perturbation theory it cancels against 1-loop diver-gences, as shown explicitly below.

If one wants to study the system nonperturbativelywith a finite cutoffit must be included for correct continuum limit [26]. We shall fixthe parameters of the effective action by taking g = 2mW(√2GF)−1/2 = 2/3, mW =80.6 GeV and giving the value of mH.

Then λ/g2 = m2H/(8m2W).The effective theory thus is a 3d SU(2) + fundamental Higgs + adjoint Higgs theorywith coefficients depending on T, NF, the coupling constants and the cutoff. All thethree kinetic terms, the two T = 0 potential terms for φ and the last A0 −φ couplingterm arise from eq.

(1) by naive dimensional reduction (taking fields constant in τ). Thegeneral structure of the 1-loop quadratic potential terms for A0 and φ is c1T 2 −c2T×cutoff, where the first term is the usual 4d 1-loop screening mass and the second termarises from the exclusion of the n = 0 term in the 1-loop integral (since one must onlyintegrate over the nonstatic modes).

Equivalently, this term is the mass countertermof the superrenormalisable 3d theory. The contributions of the various fields to theseterms in eq.

(2) are ordered so that in the coefficient of φ†φ first come the terms withAai in the loop, then A0, then φ and last the fermion loop (only in the T 2 terms). Inthe quadratic and quartic A0 terms the ordering is Aai , φ, fermions in the loop.

The A0loop gives a contribution of order g4 and is neglected here. Also neglected are small1-loop corrections to the quartic φ term.In addition to the mass counterterm eq.

(3) there in 3d actually also is [26] a loga-rithmic 2-loop counterterm ∼λ2T 2/(16π2)R dp/|p|. This is numerically negligible inour case.3

More generally, eq. (2) contains terms with higher powers of fields and their deriva-tives, allowed by the gauge (BRS) invariance of the 3d theory.

For example, the termg6(Aa0Aa0)3/T 2 would appear. If we take the correlation length of the A0 field to be1/(gT) we can estimate from the quadratic term that < AaoAa0 >∼gT 2.

Thus the ratioof terms with consecutive powers of Aa0Aa0 is O(g3) ≪1.3The one-loop effective potential in continuum.To study the theory defined by the action eq. (2) in perturbation theory, we computethe 1-loop correction V1loop(φ, A0) to the tree potential defined by eq.

(2). One findsbefore regularisation thatV1loop(φ, A0) = TZd3p(2π)32 log(p2 + g2A20 + 14g2φ2) + log(p2 + 14g2φ2) ++32 log(p2 + µ2 + λφ2 + 14g2A20) + log(p2 + m2D + 14g2φ2 + λAA20) +(4)+12 log{p4 + p2[µ2 + m2A + (3λ + 14g2)φ2 + (14g2 + 3λA)A20] ++(µ2 + 3λφ2 + 14g2A20)(m2D + 14g2φ2 + 3λAA20) −14g4A20φ2},where φ2 = 2φ†φ and A20 = Aa0Aa0.

The first two terms come from the Ai loop and theremaining ones from the coupled A0, φ loops. UsingZd3p(2π)3 log[p2 + µ2 + m2(φ)] = m2(φ)Σc −16π[µ2 + m2(φ)]3/2(5)one finds that the Σc terms in eq.

(2) and eq. (5) cancel (taking into account the A0loop term −125λATΣcA20 neglected in eq.

(2) for smallness). The finite terms give thequantum correction to the tree potential.

The resulting 1-loop improved potential isthenV (φ, A0)= 12µ2φ2 + 12m2DA20 + 14λ(φ2)2 + 14λA(A20)2 + 18g2A20φ2 −−T6π2g3(14φ2 + A20)3/2 + 18g3φ3 +(6)+(m2D + 14g2φ2 + λAA20)3/2 + 12(m2D + 14g2φ2 + 3λAA20)3/2 ++32(µ2 + λφ2 + 14g2A20)3/2 + 12(µ2 + 3λφ2 + 14g2A20)3/2,whereµ2 = γ(T 2 −T 20 ),γ = 316g2 + 12λ + g2m2top8m2W. (7)We have, for completeness included the small λA term, which will again be neglectedfrom now on.4

Studying the minima of V (φ, A0) one sees that they can be driven to nonzero valuesof A0 only if the negative term is large. This demands that at least g > πq8/3 whichis beyond the domain of validity of this calculation.

We conclude that the minimum isalways at A0 = 0; no condensate is formed. Basically this is due to the large value ofmD and the small value of the correction, ∼−T/6π.For A0 = 0 and λ ≪g2 the corrections to the effective potential have the formδV = −g316πTφ3 −T4π(m2A + 14g2φ2)32.

(8)It is the first term which gives rise to the first order phase transition in perturbationtheory. Note that the effective 3-dimensional theory correctly takes into account Debyescreening, which decreases the magnitude of the cubic term relative to the naive loopexpansion by the factor 2/3 [9].

The second term corrects the γ in eq. (7) by the term(−3/16π)q2/3g3.

With this input the equationm2H4T 20= g22 316 + 12m2H8m2W+ m2top8m2W−√38π√2g= m2H4T 2c+g432π2m2Wm2H(9)gives the perturbative result for the transition temperature Tc and the lower end T0 < Tcof the metastability range (the high T phase does not exist for T ≤T0). Numericalvalues for these as well as for some other relevant quantities as calculated from 1-loopperturbation theory [7], [10] are given in Table 1.

For g = 2/3 the relevant formulasare ξ(Tc)Tc = 9πmH/mW, Tc/mW(Tc) = 9πm2H/(2m2W), σ/T 3c = 2m5W/(243π3m5H),L/T 4c = 2m4W[1/6 + m2H/(18m2W) −m2W/(18π2m2H)]/(9π2m4H).mHT0TcT+ξ(Tc)m−1W (Tc)σL3590.8995.2495.8312.28/Tc2.67/Tc0.017T 3c0.085T 4c80182.34183.55183.7128.1/Tc13.9/Tc0.00028T 3c0.0044T 4cTable 1: Values of Tc, the lower (T0) and upper (T+) ends of the metastability range,the correlation length ξ for the Higgs field at T = Tc, the gauge field correlationlength 1/mW(Tc), the interface tension σ and the latent heat L as calculated from1-loop perturbation theory for mH = 35 and 80 GeV, for g = 2/3 and for no fermionsincluded.4The lattice action and effective potential.4.1Lattice action.To latticize eq. (2) we go over to the matrix representation A0 = Aa0T a, T a =12σa,φ →Φ = (φ0 + iσiφi)/√2 and rescale the fields byigaA0 →A0,Φ →sTaβH2 Φ,(10)5

where a is the lattice spacing. The lattice action on an N3 lattice then becomesS = βGXxXi

(13) µ2(T) is the coefficient of the φ†φ term in eq. (2) with the continuum Σcreplaced by Σ(N3)/a.

Introducing this relates βH and T for given values of g, mW, mH(and possibly of mtop):m2H4T 2 =g2βG423−1βH+2 m2H8m2WβHβG−92βG(1+ m2H3m2W)Σ(N3)+g22 316+12m2H8m2W+ m2top8m2W,(14)The curve T = T(βH) is plotted in Fig.1 for a few parameter values. The curvesshift to the right (left) with increasing mH and Σ(N3) (βG).

For example, the valuesof βH corresponding to T = T0 and T = ∞areβH(T0) = 13 +(1+ m2H3m2W)Σ(N3)βG+O(β−2G ),βH(T = ∞) = βH(T0)−m2H108βGm2W. (15)6

Eq. (14) takes correctly into account the divergent mass counterterms of the 3d theory.In addition to these, there will be finite renormalisations of the coupling constants gand λ.

Due to infrared divergences getting more and more serious at higher orders[27] these renormalisations – which will depend on the lattice size N – cannot becalculated perturbatively. We shall observe that the numerical results will follow the”constant physics” curve eq.

(14), with N dependent mass renormalisation but Nindependent bare values of g and λ, rather well. However, some more N dependence,clearly attributable to finite renormalisations of the coupling constants, will remain.In any case, all finite size effects will be very difficult to control.

For example, it isnot known how to include the constant mode (the ni = 0 term in eq. (12)) term inperturbative calculations.4.2Effective potential on the lattice.Monte-Carlo lattice simulations should be compared with the perturbative calculationsof the effective potential on the lattice rather than with continuum expressions.

To geta lattice generalization for the effective potential one can just change the integrationover momenta in eq. (5) to a finite sum over the discrete momenta pi = (2π/aN)ni, ni =0, ..., N −1.

Including only the Ai loop term, which is relevant for the first order natureof the transition, the lattice effective potential becomesVlatt = 12γ(T 2 −T 20 )φ2 + 14λφ4 +3T(aN)3N−1Xni=0log1 + (gaφ/4)2d−(gaφ/4)2d,(16)whered = sin2(πn1/N) + sin2(πn2/N) + sin2(πn3/N). (17)Note that in this sum the term with ni = 0 must not be included not only since wecannot handle it but also by definition since the effective potential is constructed forx-independent φ and thus constant φ configurations must not be integrated over.To study the order of the transition it is convenient to study the zeroes ofdVlattφdφ = γ(T 2 −T 20 ) + λφ2 −3g4128φ2aT 1N3N−1Xni=01d[d + (gaφ/4)2],(18)In the continuum limit a →0, N →∞, aN →∞the last term becomes −(3/16π)g3Tφ.For T = T0 one then is solving λφ2 −(3/16π)g3Tφ = 0, which trivially leads toa second minimum.This also exists for some temperatures above T0, up to T =T+.The case of finite lattices is completely different.It is clear from the latticeexpression for the effective potential that it is an analytic function of φ2 at least forφ2 < (16/g2a2) sin2(π/N) and , therefore, no term ∼−φ can appear.

Nevertheless,a second minimum can exist for sufficiently large lattice sizes. The condition for thissimply is that the equationλ = 3g4128aT 1N3N−1Xni=01d[d + (gaφ/4)2](19)7

have a solution. The right hand side decreases monotonically when φ increases and wecan take φ = 0; if a solution appears it appears first at φ = 0.

Inserting aT = 4/(g2βG)and λ = g2m2H/(8m2W) gives the relation43βGm2Hm2W= 1N3N−1Xni=01d2 ≈0.17N. (20)The approximation is the result of an explicit numerical calculation of the sum.

Forthe value βG = 20 used in our numerical simulations one sees that for mH = 35 GeVa first order transition appears for N > 29 and for mH = 80 GeV for N > 160. Theuse of so large lattice volumes is not possible in practice, and we at this stage confinedourselves to smaller N, N ≤20.

If perturbation theory works well, one should not findany signal of a first order phase transition in the Monte-Carlo simulations with thesesmall lattices.It may be of interest to note that in the limit of large lattices (a = constant, N →∞)the last term in eq. (16) can be written in the form [29]3Ta3Z ∞0dy1y(1 −exp[−(gaφ/4)2y]) −(gaφ/4)2exp(−3y)I30(y),(21)with the aid of the Bessel function I0.

Numerically this is very close to the continuumresult, in particular, the φ →0 limits are the same.In other words, the criticaltemperature as well as the metastability range given in Table 1 are practically thesame for very large N and for the continuum.5Lattice simulationsOur choice of parameters for the simulations is motivated as follows. We have alreadyfixed that g = 2/3 and mW = 80.6 GeV.

Since our aim is to test the validity ofperturbation theory we choose mH = 35 GeV, which makes λ small = 0.0105 but is nottoo close to the vacuum stability limits of somewhat less than 10 GeV. For comparisonwe also choose mH = 80 GeV (λ=0.0547).

We also do not expect fermions (exceptperhaps for the top quark) to qualitatively change the nature of the transition and thuschoose NF = 0. Light fermions could simply be included by changing the numericalvalue of the coefficients as in eq.

(2). Finally, we choose βG = 20, which basically fixesthe lattice spacing a in physical units via the first equation in eq.

(13): a = 0.45/T.As should, this is smaller than the thermal distance scale 1/T, the average distancebetween particles. Equivalently, in order to describe correctly effects associated withmagnetic sector of the 4-dimensional theory one must have a ≪1/mM.

With theuse of CM ≈2 and βG = 20 we get a ≈0.5/mM. For this a the perturbative Higgsfield correlation lengths in Table 1 are 27a (mH = 35) and 42a (mH = 80) while thegauge field correlation lengths are 6a (mH=35) and 14a (mH = 80).

These are so largethat it is realistically not possible to fit an interface between two phases (broken andunbroken) in the lattice.8

5.1The update algorithm.Because the lattice system described by the action eq. (11) has several qualitativelydifferent components, we used a wide mixture of update algorithms to obtain goodperformance.

The SU(2) gauge field was updated as follows: first, we combined theplaquette action (first term in eq. (11)) and the Φ-field hopping term (fifth term)to form a local SU(2) action of the form βeffi (x)Tr Xi(x)Ui(x), X ∈SU(2).Usingthis action, new link matrices were generated with the Kennedy-Pendleton heat bathmethod [28].

The adjoint field hopping action Shopp(U, A0) (second term), which isquadratic in Ui, was then taken into account by accepting or rejecting the new linkmatrices with the Metropolis method – that is, with the condition exp[Shopp(Unew, A0)−Shopp(Uold, A0)] > r, where r is a random number from an uniform distribution between0 and 1. The acceptance rate for the Kennedy-Pendleton heat bath was ∼99.5% andfor the accept/reject step ∼95%.The length of the adjoint field RA = (Aa0Aa0)1/2 was updated with the Metropolismethod, while the colour space direction, which appears only in the hopping term, wasupdated with an SO(3) heat bath.

Similarly, the fundamental Higgs field was dividedinto radial and SU(2) parts: Φ = RV , R > 0, V ∈SU(2). The V -field, which appearsonly in the hopping term (the fifth term in eq.

(11)), was updated with the Kennedy-Pendleton heat bath algorithm, and the length R was updated with the Metropolisalgorithm.The evolution of the gauge and A0-fields in the simulation time was very rapidcompared to the evolution of the Φ-field. It is crucial to make the Higgs field updateas effective as possible.

Because of the large βG the gauge background of the Higgsfield is very flat, and it is plausible that one could construct an effective multigrid orcluster update algorithm.5.2The results.Results of the simulations are shown in Figs. 2-6.

In Fig.2 we present the probabilitydistribution of the order parameter L = Tr V †(x)Ui(x)V (x + i) for mH = 35 GeV anddifferent lattice sizes. The two peak structure characteristic of a first order phase tran-sition is clearly seen on these 83 and 203 lattices, as well as on intermediate 123 and 163lattices (not shown).

The change of the curves with the lattice volume is qualitativelyconsistent with what one would expect from a first order transition, namely the twopeak structure is more distinguished on the large lattices. Note that the peak positionsdo not depend on the lattice size which indicates that the finite volume effects are notsubstantial.The first order nature of the phase transition can be also seen in Fig.

3 where thesimulation time evolution of the order parameter L is shown. Here initially the systemwas confined in the unbroken phase with small value of L, then it jumps to the phasewith broken symmetry and stays there quite a long time.

These jumps then continuein Monte Carlo ”time”.An important criterion for the order of the phase transition is the finite size scaling9

of the second moment of the order parameter L [30].This is shown on Fig. 4 forvarious lattice sizes.

The continuous curves were obtained by combining the individualruns (performed at various βH’s) with the multiple histogram (Ferrenberg-Swendsen)method [31]. We observe that the second moment of L, as a function of βH, developsa narrow peak as the volume is increased.

The height and the location of the peakhave well-defined infinite volume limits; this behaviour is characteristic for systemsexhibiting a first order phase transition.Analogous data for a more heavy Higgs (mH = 80 GeV) is presented on Figs. 5-6.In this case the correlation lengths for the Higgs and W bosons are larger than thelattice size and strong volume dependence is observed (Fig.

6). No definite conclusioncan be made for this case with the present lattice sizes.According to Fig.2 the transition for mH = 35 GeV takes place at about βH =0.34010 roughly independent of the lattice size.

We thus do not observe precise scalingaccording to eq. (14), which would demand that the value corresponding to some fixedtemperature change according to eq.

(15) with lattice size N. We ascribe this to thenonperturbative renormalisations of g and λ discussed above. If we extrapolate to thecurve corresponding to N = ∞, we see that βH = 0.34010 corresponds to Tc ≈85 GeV,somewhat but not much below the continuum perturbative value of Tc = 95 GeV.

FormH = 80 GeV the results are less conclusive, but for the largest lattice studied thetransition takes place at βH = 0.3418. The mH dependence of eq.

(15) is rather wellreproduced, but so far it is impossible to make a definite conclusion concerning Ndependence. Anyway it is suggested by Fig.1 that again the Tc observed is less thanthe perturbative value of 184 GeV.6ConclusionsWe have performed a combined analytical and numerical study of the finite T elec-troweak phase transition.First those degrees of freedom which – with reasonabledegree of confidence – can be treated perturbatively were analytically integrated overand an effective action in the remaining degrees of freedom was derived.

This effectivetheory is a T = 0 3d SU(2) gauge field + adjoint Higgs + fundamental Higgs theorywith known coefficients depending on T and, quite essentially, on the cutoffof thetheory. This bare effective theory was then latticized with 1/a, a = lattice spacing, asthe cutoffand the nonperturbative degrees of freedom were treated numerically withMonte Carlo techniques.In this first numerical application our aim was to study the structure of the theoryand a fairly small Higgs mass, mH = 35 GeV was mainly discussed, with some resultsgiven also for mH = 80 GeV.

A first order transition was clearly seen at least for thesmaller Higgs mass and the numerical value of Tc is rather close to (but less than)the value obtained from 1-loop perturbation theory. For this numerical agreement thecomputable cutoffdependence of the bare effective theory was crucial.

With increasingmH the various characteristic lengths in the problem increase rapidly and the problembecomes numerically more and more difficult.10

Our lattice Monte Carlo results thus clearly indicate that there is a first order phasetransition, at least for small values of the Higgs mass. A very intriguing part of theresult is that we see a first order phase transition also on small lattices where, accordingto the perturbative calculations in Section 4.2, the phase transition must be of thesecond order.

A number of tests including the lattice volume dependence of the orderparameter, temperature metastability range, etc. indicate that the first order characterof the phase transition is not a lattice artifact.

These results indicate that so-calledφ3 term is not the only source for the first order character of the electroweak phasetransition and that non-perturbative effects are important as well. The physical natureof these non-perturbative effects should be presumably related to the confinement in3-dimensional gauge theory.

Unfortunately, the complete solution of the problem ofthe electroweak phase transition by non-perturbative lattice methods requires hugelattices.AcknowledgmentThe authors are grateful to K. Farakos for the collaboration at the initial stage of thiswork and providing us his code for 3-dimensional simulations of 3-dimensional gaugetheory with a Higgs doublet.References[1] D. A. Kirzhnitz, JETP Lett. 15 (1972) 529;D. A. Kirzhnitz and A. D. Linde, Phys.

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0.34000.34100.34200.3430βH507090110130150170190T(βH) GeVmH= 80 GeVmH = 35 GeVN = 8121624 oomH= 80 GeVmH = 35 GeVN = 8121624 ΖFigure 1: The relation between T and βH as given by eq. (14) for mH = 35 and 80GeV, βG = 20 and NF = 0.

The arrows denote the perturbative values of Tc as givenin Table 1.0.02.04.06.08.0P 0.33970.340150.34020.10.30.50.70.9L0.010.020.030.0P0.340060.340090.3401083203Figure 2: The distribution of L = Tr V †(x)Ui(x)V (x + i) for mH = 35 GeV for 83 and203 lattices for βG = 20 and different values of βH.13

050000100000Iteration0.200.400.600.801.00203 lattice, βH = 0.34009Figure 3: The Monte Carlo time history of the lattice simulations for mH = 35 GeVon a 203 lattice near critical βH.0.339950.340050.340150.34025βH0.000.010.020.030.040.050.06<(L - )2>83122162203mH = 35 GeVFigure 4: The dependence of the second moment of the order parameter L on βH fordifferent lattice sizes and mH = 35 GeV.14

0.250.350.450.550.65L0.05.010.015.020.0P0.34140.34160.34180.34200.3422163Figure 5: The distribution of L = Tr V †(x)Ui(x)V (x + i) for mH = 80 GeV and a 163lattice for different values of βH.0.34100.34150.34200.3425βH0.0000.0050.0100.0150.0200.025<(L - )2>83123163mH = 80 GeVFigure 6: The dependence of the second moment of the order parameter L on βH fordifferent lattice sizes and mH = 80 GeV.15

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