Sphaleron transitions and baryon asymmetry: A numerical, real-time analysis

Nuclear Physics B353 (1991) 346-378 North-Holland

YON ASYMMETRY: NT SITIONS A NU ME CAL, REAL-TIME ANALYSIS J. AMBJORN and T. ASKGAARD The Niels Bohr Institute, Blegdamscej 17, DK-2100 Copenhagen 0, Denmark

H . PORTER Department of'Phl- slcs and Astronomy, Glasgow University, Glasgort G12 8QQ, UK M .E . SHAPOSIINIKOV Institwe for Nuclear Research of the USSR Acadenti- of Sciences, Moscon, 117.312, USSR Received 28 September 199(1

We estimate by numerical simulations the rate l' of sphaleron-like transitions in the electroweak theory in the phase with restored symmetry . There is no suppression and l' = a(a,A,T)4 with a = 0.1-1 .(). We further address the question of a spontaneous CP-breaking in this phase of the electroweak theory . but have not yet found any evidence in this direction.

1 . Introduction 't Hooft was the first to realize that the baryon number is not conserved in the electroweak theory [1]. Baryon number violation is caused by the nontrivial topological winding of the weak SUM gauge fields . The anomaly of the fermionic current relates the winding of the gauge fields and the change in baryon number by Nl dt d3xTrF,,F,,, ., ƒ 167r` 11 JU

t£

(1)

where N,. is the number of families of quarks and leptons. As 't HHooft correctly emphasized the amplitude for such processes is exponentially suppressed as exp(- 27r/a,ti ), a,. = 1130 . This is simply because any gauge field configuration which changes the winding number has an action > 27r/a, Viewed in Minkowski space the change in winding number corresponds to a 0550-3213/91/$03 .5() ,(-, 1991 - Elsevier

SciCnCC

Publishers I3 .V . (North-llolland)

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34 7

tunneling between two neighboring gauge equivalent vacua, and the exponential factor exp( - 2 7r/a W) is a standard WKB tunneling suppression. It is now generally accepted that the situation is quite different at the high temperatures prevailing in the early universe [2]. The analysis that leads to this conclusion was based on the existence of the so-called sphaleron [3]. The sphaleron is a static, finite-energy saddle-point solution to the electroweak field equations, which corresponds to the barrier configuration between topological inequivalent vacua . Its energy is Esph - M w/aw. At nonzero temperatures one can pass over the barrier between the different vacua classically with a probability given by the Boltzmann factor exp(-ßEsph ). For temperatures comparable to Esph this exponential factor goes to one and one would naively expect rapid transitions between topological inequivalent vacua . Such transitions have to be accompanied by a change of the baryon number because of the anomaly and consequently the baryon number should not be a good quantum number at high temperatures . Similar situations occur under other extreme conditions like very high densities, or maybe even under high-energy collisions in the next generation of accelerators . This possibility of generating baryon and lepton-number violating processes induced by instantons in the laboratory is of course extremely interesting, but also less well founded from a theoretical point of view, and it is intensely debated presently. But even the existence of the high-temperature sphaleron transitions has been questioned from time to time and it is indeed not v(-ry clear in detail how the thermal fluctuations manage to induce a classical transition, although it can be analysed in some detail for simple quantum-mechanical systems [4]. If one gets above the temperature for the electroweak transition the symmetry is restored and all weak coupling expansions become unreliable . One cannot expect the sphaleron configuration to be the dominant configuration through which the transition will take place . Other configurations which are not solutions of the classical equations of motion may play an equally important role. Exactly what happens and to what extent one can talk about transitions from one gauge sector to another at temperatures above the electroweak transition is therefore not clear. One purpose of the present work is to analyse by numerical methods what happens above the electroweak transition . Naively one would imagine that the configurations which one has to pass, going from one gauge sector to another, will still be "sphaleron-like'" . The energy of such a configuration will be of the order of T and the extension (a .T) -' in the same way as the energy of a sphaleron in the broken phase is - Mw/a, while its extension is - MW' . Due to the smallness of a,,. the characteristic momenta of the fluctuations forming the "sphaleron-like" configurations arc therefore much smaller than the generic momenta of quantum excitations in the hot plasma which are of order T. Hence the "sphaleron-like" fluctuations decouplc from the quantum fluctuations and we expect that the processes responsible for the change of topological charge are essentially described by classical physics for temperatures so high that we are in the symmetric phase of

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the electroweak theory . This opens an exciting possibility of observing these transitions directly by doing microcanonical simulations of classical statistical mechanics of gauge theories . This will be described in detail in the next section. The second aim of the present work is to investigate some of the implications of the rapid baryon nonconservation in the early universe . It is an old idea that the observed baryon asymmetry of the universe may be explained by a theory incorporating both baryon number and CP nonconservation [5]. These ingredients are present in a number of GUT theories and the possible consequences for the present baryon asymmetry were explored some years ago. Existence of the rapid baryon nonconservation all the way down to a TeV scale implies that any baryon asymmetry generated by GUT interactions will be washed out unless special conditions are satisfied . It is therefore important to explore the possibility of baryon asymmetry generation at the electroweak mass scale. One of the authors (M .E.S.) has suggested such a scenario, where the very small explicit CP-violating terms in the electroweak theory trigger a spontaneous CPbreaking in the hot gauge-Higgs plasma, resulting in a condensation of Chern-Simons (CS) density, which at the electroweak phase transition is converted into baryon number through the anomaly [6]. If n"x denotes the expectation value of the CS-density in the symmetric phase we expect for dimensional reasons n MIX (,s - (aw T)' . This density is now assumed to be converted into baryon number at the electroweak transition, where the vacuum changes into the trivial, perturbative, CPinvariant vacuum. This scenario is able to give a correct order of magnitude of the baryon asymmetry, but it is based on a number of nonperturbative assumptions : The formation of a CS-condensate and the assumption that essentially all of n"X is converted into baryons during the cooling down from the hot phase to the present day perturbative vacuum . In principle we have the possibility of investigating this by numerical methods too. A first attempt to address the question of baryon asymmetry generation in the electroweak theory by Monte Carlo methods was carried out in ref. [7]. However it was impossible to address the question of spontaneous CP-breaking by a condensation of a CS-density simply because it is difficult to include the non-gauge invariant CS-density in the action used for Monte Carlo updating . However, as we shall see, the observation that we can use classical statistical mechanics, makes it possible to incorporate such a CP-breaking term . The second part of this article deals with the attempts to measure a possible CP-breaking, and its consequence for the generation of baryon asymmetry during a first-order phase transition, by means of the classical equations of motion.

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The classical equations and sphalerons

As mentioned above we expect a decoupling of the classical sphaleron-like configurations from quantum fluctuations when the temperature is high. This allows us to use the methods of classical statistical mechanics. To be more specific we will use real-time microcanonical simulation of the high-temperature gauge-Higgs system. This idea was first applied to two-dimensional systems . In ref. [8] the process of kink-antikink pair creation in a Ao4 theory with spontaneous symmetry breaking was investigated . In ref. [9] the "sphaleron" transition in a U(1) abelian gauge-Higgs model was measured by the microcanonical method. The numerical results were in perfect agreement with analytic calculations [10] of the "sphaleron" transition rate. The first numerical study of sphaleron transitions in the four-dimensional electroweak theory was carried out in ref. [7], but the sphaleron transitions were considered in a fictitious Monte Carlo time, and it was not clear how to relate this time to real time and therefore how to get an estimate of the transition rate . As mentioned one of the aims of the present work is to extend the microcanonical methods in ref. [9] to the electroweak theory in 3 + 1 dimensions . In this way it will be possible to perform a quantitative analysis of the sphaleron-like transition rate at temperatures T higher than the critical temperature Tc for symmetry restoration . This is important since any perturbative analytic calculation of the transition rate for T > Tc is impossible due to strong interactions of gauge and Higgs fields at small momentum transfer, k < a w T [11,12]. It can be found from simple scaling arguments [4,13] that the rate T per unit time and unit volume should behave like T-

K(awT )4

with an unknown coefficient K . It is not excluded a priori that this coefficient is so small that the processes with fermion-number nonconservation were never in thermal equilibrium in the early universe . We shall see, however, that such a suppression of the rate is in fact absent at high temperature and the factor K is of order l . The idea of real-time microcanonical simulation is simple and we will now briefly describe it. Every state in the classical theory is characterized by coordinates and momenta . Since we here consider the electroweak theory, but ignore fermions and also ignore the U(1) part of gauge group since it plays no role for the electroweak anomaly, the coordinates are gauge fields A ;' of SU(2)Weak and Higgs fields 0" in the fundamental representation. The temporal gauge where A () = 0 is especially useful since it allows a simple identification of the canonical momentum and the most transparent representation of quantities like the CS-term . We will use the

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350

temporal gauge throughout this work and the conjugate momenta to A ;' will be the electric field E;' = dA°/dt . The conjugate momenta to 0" will be era do"/dt . The probability of having a particular state, characterized by a configuration

is dictated by the Gibbs distribution exp(- H/T) where H is the hamiltonian. Since we consider the classical theory, H is just the classical hamiltonian for the gauge-Higgs system, H = J d3 x

['-,E;'E« + -~ Fi~ F1i +

117.12

+

I Di4l ; + M 2 101 2

+ Alo1 4 1 ,

and in the temporal gauge it should be supplemented by the Gauss constraint in the form Di"" E,1' = ig ((b~ T ' Tr Once we have by some method picked a configuration according to Gibbs distribution we can let it develop according to the classical equations of motion. The energy will be strictly conserved during this classical evolution and if we assume the system is large enough and the equations of motions are sufficiently nonlinear, the last assumption probably being a safe one for the nonabelian gauge-Higgs system, we can appeal to ergodicity and assume that all phase-space configurations with the given energy will be encountered with equal probability during the classical evolution of the system. In other words, the classical evolutior: is creating a microcanonical ensemble of the given energy . In this way we cart simulate the situation in the early universe, assuming that we are interested in physics which can be described satisfactorily by the classical equations of motion : First we heat the system (by some standard metropolis procedure) to the desired tenipeiaiure, next we let the system free to follow the (real) time classical equations of motion. During this evolution we can now measure the quantities of interest. The observable which will have our main interest is the "topological" charge 00

1 327r - ƒ

dt

Of course this quantity as it stands is strictly speaking not a topological quantity. Only if the gauge field A ,1U, 0-r" starts as a pure gauge configuration U - '(x)di U(x ) at time t = () and ends in another pure gauge configuration L' - I (x)di U(x) at time t will Q(t) have a topological meaning and be an integer measuring the difference in winding number between the two gauge transformations U and U. At low tempera-

J. Ambjoin et aL / Sphaleron transitions

35 1

ture the gauge symmetry is spontaneously broken and the sphaleron energy is the barrier between classical vacua with different winding numbers . A typical time evolution picture for Q(t) will depend on the energy. If the energy is smaller than the sphaleron energy Esph, Q(t) will stay bounded, fluctuating around zero, because it is classically forbidden to move to another vacuum sector. If the energy is slightly above Esph, Q(t) will fluctuate for some time around zero, then eventually find the passage (the sphaleron configuration) to the neighboring vacuum sector and fluctuate around plus or minus one, and on a larger time scale one would expect it to perform a random walk between the different vacuum sectors . As the energy goes up or alternatively, the temperature increases, we expect this picture with plateaus and jumps to be more and more blurred and eventually the concept of plateaus should disappear in a rapid change of Q(t). One of the main purposes of this article is to measure the rate of change of Q(t) above Tc . 3. Lattice equations For any kind of numerical investigation we should somehow use discrete space and time. Moreover, classical statistics in continuum space does not exist due to the presence of ultraviolet divergences which are usually referred to as the Rayleigh-Jeans instability. When we go to discrete space (the lattice) we introduce an ultraviolet cutoff which makes the theory properly defined . In the quantum theory this cutoff is provided by the temperature T entering Bose and Fermi distributions of the particles . Hence, we can roughly identify lattice spacing with inverse temperature [9] . We will make this relation explicit later . Of course, there are many ways to perform the discretization ol the equations of motion. The best way is to define a discrete (lattice) version of the gauge-Higgs action from the very beginning and derive equations of motion directly from it. The advantage of this approach is that it allows us to keep all the internal symmetries of the theory (in particular, gauge invariance) automatically during time evolution, provided the discretized action itself respects these symmetries. Therefore, we arrive at a lattice version of the gauge theory which should now be defined in Minkowski space (signature ( - + + + )) rather than euclidean space, as is usually the case when we want to study static properties of quantum field theory . The continuum action corresponding to the hamiltonian (5) is L = - f dt d 3x ['Fr~t, +

ID~10I`

+

M21012

+

h»l 4 ]

(8)

and the most obvious procedure is to take as the corresponding discretized action the standard Wilson action for euclidean lattice gauge theory combined with the lattice Higgs action, and make the obvious change of sign of the spatial parts of

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the gauge and Higgs fields relative to the temporal parts. After that we should take the lattice spacings in the temporal direction to be much smaller than the spacing in the spatial direction. If we denote the lattice spacing in the spatial direction by a and that in the temporal direction a®t we get the following action : t

( E kin -

E pot )

where the kinetic energy term is given by ß1a2 ßG 1 - -, Tr U o ( t )) ~ . F (R - ~ Tr Vx~U ,x+()Vx+()RxRx+~)) ( 2®t ®t2 a, t t o(t)

E kin =

(10) and the potential energy term is t

E pot

- ßG

O(s)

+ßR

(1 - 2 Tr U ~(s)) + 2ßH

x,t

x+t,t

( Rx - 2 T'r

Vx* Ux,x+îVx+îRxRx+î)

2 ( R2 - V' 2 ) .

is the lattice gauge field on the link connecting x and x + /,, Uo denotes the plaquette action for the gauge field and EIM stands for a plaquette in the 0-î plane while o(s) refers to a plaquette in an î-f plane . Further the SUM Higgs doublet 0 is represented as a matrix U,, +A

2 ' 02

-

0 1*

=R-V,

RE=- RVESU(2) .

(12)

Finally we note that the summation over spatial points is finite and limited by the lattice 3-volume (periodic boundary conditions) while it is infinite in the time direction. The tree value connection between the lattice parameters in eqs. (9)-(11) and the continuum coupling constants in eq. (8) is as follows M2 ßR ßH

ßR

_

ßH . a 2 gC

-

4 ßG

9

(13)

where a denotes the lattice spacing and we have used the following parametrisation of L.2* + 3 13 1, - 1 . (14) U2- 2 ßR 2ßR

* We keep this unnecessary complicated notation for historical reasons.

J. Ambjorn et al. / Sphaleron transitions

353

The formal continuum limit is obtained by letting the lattice spacing a go to zero and changing from the lattice fields 0 and U to fi --->

a . ~p . V-ll.SH

U -> eiaA

(15)

It requires further a fine tuning of ßR and ßH such that v 2 - a2 as is clear from eqs. (13) and (14). Variation of the action with respect to scalar fields 0x and gauge fields Ux, x+i living on links in spatial directions gives the equations of motion, i .e . connection of fields on the time slice t with the fields on time slices t - ®t and t - 2®t. Variation of the action with respect to gauge fields U,, .,,6 living on the links in the temporal direction gives the Gauss constraint. The lattice equivalent of the temporal gauge condition AO~(x, t) = 0 is U~,x+b(t) = 1 and will be used in the following. As in the continuum formalism, the choice of temporal gauge on the lattice is not a complete gauge fixing. We still have left the invariance under local spatial, but time-independent gauge transformations U(x) and the equations of motion transform gauge covariantly under these residual transformations. Gauss constraints are just the generators of these local, time-independent gauge transformations. Consequently they commute with the hamiltonian, which is invariant under such transformations. In particular this means that the Gauss constraints are consistent with the equations of motions and any initial configuration which obeys Gauss constraints (as it must) will continue to do so at all later times. We introduce the lattice electric field (in the temporal gauge) as follows: _ Ux,x+i(t + Jt)Ux:~a.+ï(t) Ex , x+i --

(16)

and write the matrix E as

E=E° + i7.aEa®t . The electric field in the limit ®t -> 0 is to be identified with Ki(x) -T'E.Q,x+F'

(18)

As is seen from the definition F transforms correctly as an electric field under local, time-independent gauge transformations and further that Ex,-j(t) = Uxlp,x(t)Eá~®i,x(t)Ux- ;,(t) .

(19)

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354

The equation for the gauge field can now be written as Ex_v+î(

t) =

E' . .r+ï(t -At) +

®t 2

j*k

Pil

1®t + 2

~

ßr

JZFk

Tr'rajLly+j,x+j+kQr+j+k,x+kUr+k,x

Tr T t Ur, .r+f~r+j,x+j-kUx+j-k,

Tr

T

a

UX"r+î

(P

.r-kUr-k, .r

(20)

r+ï j)tX

where all variables on the r.h .s . of eq. (20) except E are evaluated at time t . In a similar way we now introduce the canonical momentum P,(t) for the field 0,.(t) by P' (t)at

= (P ,(t) - (,.(t

-At)

(21)

and derive Px(t) =Px(t-dt) +At ~

t

(~x+îUr+î x + ~x-îUr-î .x) -

Ó -}-

4ß It

Pli

(R 2 - 1 ,2 ) 4,

(22)

where again all variables on the r.h.s. of (22) except P are evaluated at time t . Finally the lattice equivalent of the continuum Gauss constraint (6) can be written as G " (X)

=

F'C E TrIT"[E.r . .r+j+E .;. . .r- .l ] +ß II

j

"rr1T"P~r ~,.=0 .

(23)

Using eq. (19) shows that the term involving the E's is precisely the lattice covariant derivative of the electric field . Since time is still associated with a discrete lattice spacing the energy is not defined in a strict sense and we do not really have a lattice version of the hamiltonian (4). In order to derive a hamiltonian defined on the three-dimensional spatial lattice we should strictly speaking take the limit 4t --> 0. Time will then be continuous and we get in a natural way a hamiltonian formalism where the electric field ;~'; appears . However, if At « 1 we expect the approximate conservation of some global lattice quantity which is related to the energy of the discrete system which we would get from (9) in the limit At -> 0 . Therefore, we define an energy functional H from eq. (9) by just performing the sum over one time slice: H = Ekin+ Epc,t ,

In the limit At -> 0 it obviously reduces to the correct hamiltonian.

(24)

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355

We still have to address the question of implementation of the Gauss constraint on the initial configuration. We have now a hamiltonian, given by eq. (24), but if we generate by Standard Metropolis a configuration according to the Gibbs distribution exp(-H/T) it will not satisfy the constraint . The simplest way to impose it in the Metropolis updating procedure is to use the effective hamiltonian

Heff=H+~G2,

(25)

where G denotes the Gauss constraint from eq. (23) : G2=

fGa(X)2

(26)

and ~ is some parameter which governs the accuracy by which the Gauss constraint is satisfied. The Gauss constraint will be exact for 4 --> x . In practice, however, the thermalization of the system with a large value of ~ takes a long time, so a reasonable compromise should be found for the magnitude of 6. From a practical point of view we can afford to spend some time on thermalization since the main part of the computer time anyway will be used on evolving the system according to the iässi iäl cquâtions of motion . In order to be absolutely sure that the Gauss constraint is satisfied we performed at the end of the Metropolis updating an additional "cooling" of the system using the effective action Sef, = G 2 . Such a cooling hardly changes the energy of the transverse modes if the configuration is generated by Metropolis using H from eq . (25) and a reasonable 4, but it kills the longitudinal components responsible for the violation of Gauss law. In this way we could get Gauss law satisfied to any desired accuracy for the initial configuration . No matter whether the initial configuration satisfied Gauss constraint or not and no matter what value we used for ,At, we found that the value of G - was absolutely stable and unchanged from its initial value when we solved numerically the equations of motions by use of eqs . (20)-(22). This invariance was obviously a strong check of the correctness of the numerical code. Let us end this section by briefly discussing an alternative set of equations which we have also used, as an independent check of our results . As mentioned it is quite natural to take the limit It ---) 0 in this action (9) and in this way get a continuous time formalism. Since space is still a lattice, and for our purpose even a finite lattice with periodic boundary conditions, we are in fact dealing with the continuous time development of a classical system with a finite number of degrees of freedom. In order to solve the differential equations describing the time evolution of the system we have again to choose a discrete time step . But we have now a larger freedom in our implementation and one could imagine that it was possible to take advantage of some of the higher-order algorithms for implementing

;56

J. Anal'jorn et al. / Sphaleron transitions

differential equations, for instance the leapfrog method or Runge-Kutta algorithm. In this way one could hope that it would be possible to use larger 21t and still get a good approximation to the "continuous time" equations . In practice it turned out not to be a great advantage, however. The price one has to pay for such an approach is that the Gauss law is not automatically satisfied . ®f course it will be satisfied to order -it, since to this order the equations are identical to the ones given above, in which Gauss law was compatible with the equations of motion . But in order to keep the violation small we had to take At fairly small too, even for the higher-order implementations, and nothing was gained. arameters or the real-time simulations Let us discuss first the possible procedures for measuring the rate of topological fluctuations . (i): A straightforward method is the one used in ref. [9]. It is based on the observation that if the volume of the system is not very large the trivial (highmomentum) fluctuations of the topological charge occurring in a given gauge sector are small compared with 1 . Then, if the probability of topological transitions is small enough, a plot of the time dependence of Q(t) will consist of a number of different plateaus characterized by small fluctuations of Q(t) near integer values and rare but rapid transitions between the plateaus as we have already discussed. Here the rate T V will be just the inverse of the average time which the system spends in a given gauge sector. In what follows we shall use this method which appears to be most profitable from the point of view of computer time. (ii): Let us mention two other methods which could in principle be used as partly independent methods. They are both based on the same picture as in (i). This means that the "topological" charge Q as defined by eq. (7) can be considered as an analogue of the coordinate of a brownian particle jumping between the different vacuum sectors . From this analogy we expect that for sufficiently large times [ 13] (27) (Q'(t)) = 2M, where I' is the rate of topological transitions and V is the volume of the system . This equation can be the basis of the rate determination . However, it is more time consuming than the former method. The fluctuations of such a random walk on the real axis are known to be large and in our computer simulations we have not been running for times t sufficiently large to verify (27) in a convincing way. Still another method in the same spirit is to break the symmetry between the different vacua by adding a "topological" Chern-Simons term Ncs to the action . Consider the following continuum hamiltonian : Heff=H+ /u,Ncs,

28

357

J. Ainbjmrrt et al. / Sphaleron transitions

where

Ncs=

f d'xn cs(x),

(29)

1 ncs(x) = 167r2EijkTr(FijAk- 2AiAjAk) .

(30)

As mentioned, this term appears naturally in a theory with fermions and nonzero chemical potential tL for the fermion number. The integration over fermionic degrees of freedom gives exactly (28) [141. The term (29) is gauge invariant under small gauge transformations but changes by integer numbers under large gauge transformations which have nonzero winding numbers . However, equations of motion derived from (9) are perfectly gauge covariant as we will discuss later. On top of the periodic vacuum structure we will get superimposed a constant slope and if we again appeal to the brownian motion picture where Q(t) is jumping between different vacuum sectors [131, the diffusion "downhill" will be favored. For small b, we get l

v

dQ dt )=

(31)

rvw .

We can use this equation for the determination of the rate. Again, in practice we encountered problems using this method for reasons to be discussed later. It is possible to estimate the amplitude of thermal oscillations of the Chern-Simons number in a given topological sector in the continuum quantum theory in a finite volume and at temperature T using lowest-order perturbation theory . We have g_
Eijk (Fi'j A")(x)Elnrrr(FJh,Ah)(1,» , 11

(32)

The thermal average can be calculated using the finite-temperature average of the gauge field defined by ( Ai' (x) Aj'(y )> =

f(

d~ k

'~

)

28uh S ijjI B( k )e ~~k ~~

-ik(x -,-)

(33)

with n B(k) being a Bose distribution . The straightforward calculation gives g 2 ~ 48VT Ncs% 7r` 32~r -

- ~(3))( 34 )

35 8

J. Ambjorn et al. / Sphaleron transitions

where ~ is the Riemann ~-function . The result is valid for T >> Mme, since we have assumed in eq. (33) that the gauge fields are massless. Now we have to relate it to lattice gauge theory . To find the correspondence between temperature and lattice spacing we can, for example, calculate the energy of the free bosonic system with one relativistic degree of freedom in the continuum limit and compare that with the lattice result for the same system : V (2,7r) 3J

d 3 kEk n B(k) =

V ~T,

(35)

where the r.h .s. expresses that every lattice site has energy T due to the equipartition property of classical statistics which we expect to be valid at high temperatures. From eq. (35) we get a relation between the lattice spacing a and the continuum temperature T (aT )-1 = 30/-rr I .

(36)

Finally, using eq. (36) we can relate the continuum expression VT appearing in eq. (34) to the lattice and we get the following expression for the thermal fluctuations : < Ncs> = 10- 3 XN 3 lßG1 where N is the lineal- size of the lattice in the spatial direction. This equation is true if Mw. F1 a << 1 . In the opposite limit Mw .,1 a >> 1 we shall get an additional suppression by the factor (aM w,  ) -4 . In the next section we shall see that eq . (37) is quite well satisfied in our numerical simulations. Let us turn now to the discussion of the sphaleron transition rate in different phases . THE HIGGS PHASE

The Higgs phase is characterized by a larger value of the scalar condensate and an average value of VUV ', which should be close to one. In the Higgs phase we have semiclassical calculations of the transition rate of the sphaleron [4, 10]. The transition rate in a volume V is, in the one-loop approximation l'= 0 .007(a,, .T ) 4 X 7 e `V ,

(38)

where we have assumed that the sphaleron energy is E,pn = 3Mw/a,,

(39)

x = E, n n/T == 3Mw/Ta, .

(40)

and the variable x is defined by

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J. Ambjorn et al. / Sphaleron transitions

This calculation is strictly speaking only valid in the semiclassical approximation where the exponential factor exp(-x) is dominating over the prefactor x', which here means We can convert eq. (38) to the lattice and computer time t (recall that the lattice spacing in the time direction is a ®t) by using a,, = 1 /(M'G ), V = N _3a 3 and (aT ) 3 = 30/-rr 2. The average number of transitions . l It) occurring during computer time t can be written as N

(42) . - i '( t ) = 0.0003 (x'e --') t . F' G

or in lattice units 1 /Mw a, we must at least

I /M w ,

Since the sphaleron radius is demand

1

2<

Mw a

N

(43)

~ 4 '

since we want the spatial variation over more than one lattice unit and the use of periodic boundary conditions means that the sphaleron must be less than half the lattice size. Introducing x and 1ßc; allow us to write eq. (43) as _N 1 .4x ~4~

7r 2 pG

(44)

From the classical relations and

M 2 = 4A
MW = ;gWC

~p

2 ),

(45)

which translated to lattice units by eqs. (13)-(15) are MHa 2 =

40R ßH

c,2

and

wa 2 =

2.8 ßG

, c",

(46)

we have, assuming Mw = M  , 2 a2 Mw

=

Ißti

3ßH - 1

where the restriction that we are in the broken phase 011 > -1

27,8,c;

(1,2

> 0) leads to (48)

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J. Ambjorn et al. / Sphalerott transitions

By combining eqs. (44) and (48) it is indeed possible to find values of ß H and ßc where the transition rate is in the semiclassical region and is not exponentially small for the lattice sizes which are available (N = 8-16). The problem is that all the formulas used in the derivation are tree-level formulas which do not take into account the thermal fluctuations . It turned out that (0 2 ) was quite far from the assumed classical value 1,2 and VUV ' was far from 1 . This means that effectively we were in the unbroken phase for the values of 8E,, ßc relevant for the semiclassical limit for our size of lattices . The only way to get around this problem is to lower the temperature, which means that we have to increase /3., and according to eq. (44) the lattice size . Rough estimates showed that the lattice size N should at least be 32 and the simulation became impossible with the computer resources available. The ideal situation would be one where we confirmed the semiclassical calculations, both in order to check the assumptions used in the calculations and the reliability of our lattice approach. This can indeed be done in two dimensions [9] . It would serve as a convincing "calibration" of our numerical simulation before moving to the more interesting region where T > Mw , but as is clear from the above discussion this is unfortunately not possible with the used lattice sizes . THE HOT PHASE

In the hot phase (i.e. the phase with restored SUM symmetry) the average of 02 is close to zero and the magnitude of VUV ' is substantially different from 1 . The analytic evaluation of the rate seems to be a very difficult problem. General scaling arguments give [4,13] F = tc(a w T )4

(49)

which should be compared to eq. (38). As eq . (38) it can be converted into the number of transitions, . I'M, between topological sectors for a given lattice of size N3a 3 in the time t - a N3 0 .045tc ß_4t . 6

(50)

Our choice of parameters during the simulations was as discussed above in connection with the I-Iiggs phase and allowed typically a few classical sphalerons to fit on the lattice . Due to thermal fluctuations we were really in the hot phase where the symmetry was restored . A priori it was difficult to judge how deep in the hot phase the choice of coupling constants would bring us, but the obvious requirement for classifying the system as being well into the hot phase is a correct scaling behavior of the results with respect to N and ß(; .

J. Ambjorn et al. / Sphaleron transitions

361

5. Numerical simulations The Monte Carlo results for different quantities like total energy, etc. can be predicted by analytical weak coupling methods. In particular, we should expect from equipartition that the energy per lattice site should be near 10 (we have 20 physical degrees of freedom) . The kinetic energy of the gauge field per lattice site should be equal to 3 (6 degrees of freedom), and the Gauss term in the action should be 1 .5 per lattice site . We indeed obtain numbers like That after about 100 Metropolis steps . During the real-time evolution we had a conservation of Gauss constraints up to computer errors. When we used the equations of motion (16)-(22) the energy was very well conserved for small ®t - 0.01 . However, even for as large ®t as 0.2 there was no sign of instability although of course the energy fluctuation was larger . The advantage of the formulation (16)-(22) is that Gauss law is strictly satisfied even for large step size. We have also used the "continuous" time equation mentioned above and we get identical results for reasonably small step sizes so we will not distinguish here between the two methods. Typical results for the measurements of the topological charge Q(t) are shown in figs. 1-3 for different values of the coupling constants and lattices with different sizes. The At used in these equations is 0.05 (times the lattice spacing in spatial directions) and the total time of the measurement shown in the figures is 7500. The construction of Q(t) along the classical phase-space trajectory is described in detail in ref. [7,, but let us mention for completeness that the lattice representation

3.0 (0

Z 2.0 C)

v

3000 TIME

6000

Fig . 1 . The measured values of Q(t) for b(; = 9.0, 0 11 = 0.34 and 13R = 0.0016. The lattice size is 8 ;. The timestep is 0.05 and the total time 7500.

J. Ambjorn et al. / Sphaleron transitions

362

0

1500

3000 4500 TIME

6000

Fig. 2. The measured values of Q(t) for b, ; = 10 .0, ßß t{ = 0.34 and 61, = 0.0()14 . The lattice size is 12 ;. The timestep is 0.05 and the total time 7500 .

3 .0

0`

1500

0

3000 TIME

4500

6000

7500

Fig. 3. The measured values of Q(t) for b ( ; = 12 .0, f3 tt = 0.34 and ßtt = 0 .0014. The lattice size is 16 ;. The timestep is 0.05 and the total time 7500 . of (7) used was

Q(t)

=

r

, 1:

2:

16Tr` I =(1 xE I'll tice

Tr F'F~ lattic~ = 1~) tv, 13-- I

!

MI , Aé [ ;

Tr

FFI  «(x, t ) ,

Tr UMr UAE

TrU/i ;, TrU4] (x),

where U~;, are the four plaquettes in the /,v-plane associated with point x.

(51)

(52)

J. Arnbjorn et al. / Sphaleron transitions

363

IYm Q >

mu Z

1 .0

0.5

0 6.000

6.500 TIME

7.000

Fig. 4. An enhanced region of fig . 1 . What we see here is a transition which is not very sharp. but still appears to have an interpretation as a transition .

It is seen that Q(t) behaves essentially as one could have hoped for. In most cases it is possible to identify plateaus and transitions between the plateaus . The details of such transitions are shown in figs. 4-6. It is clear that not all changes of Q(t) appear as well-defined rapid transitions. There is no reason why they should. We do not know the exact form of the space of excitations bringing us from one "vacuum" sector to another when we are in the unbroken phase, but it is certainly larger than just "sphaleron-like" excitations. In fact the pleasant surprise is that for large 8, most changes, which are not just thermal fluctuations, can be viewed as such rapid transitions followed by plateaus . This makes the counting of transitions feasible . Even a graph like the one in fig. I has well-defined plateaus . In order to convince the sceptical reader we have plotted a histogram of Q(t) (fig. 7). It is clearly seen that it spends most time at well-defined plateaus, separated by jumps ®Q = ± I*. One can try to quantify these statements. The problem of counting up the number of transitions within the data by eye is that the jumps between different vacuum sectors tend to be obscured by thermal fluctuations of N(,s . A "smoothing" of the data is unsatisfactory since the jump between different sectors can them* Because we have only used a simple lattice version of Ff the change in Q. even in the case where the gauge field moves from a vacuum configuration with winding number n to another with winding number n + I will not he exactly one . but rather between 1).7 and 1 .0 [7]. Since: we have a finite temperature we will in general not even have vacuum configurations .

J. .4irihjorrt et cil. / Sphalcron transitions

364 20

W~l

k04

0

I.0

L

2 400

I

2 800 TIME

,

3200

Fib; . 5 . An enhanced region of fig . 2 . Here we see a sharp transition .

selves be quite rapid . Consequently we have looked for quantitative features of the data, which distinguish it from purely thermal movements, supporting the idea that there is a tendency to remain in distinct topological sectors. A plateau will be characterized by thermal fluctuations about the mean value of Ncs, nm, of the plateau . We expect that the difference of points from the mean, Qi - m, to be normally distributed about m . We also know that the standard deviation of this distribution, a is related to the overall point-to-point standard deviation of the data set, which we will denote by S such that a = S/ C . If we then consider 2n + I consecutive points as a possible candidate for a plateau then we expect only a few to lie more than say 3 or 3 .5 x a from the mean of the 2n + I points. Given n we can calculate a limit on the number of such points ensuring that we would reject less than 2% of actual plateaus by chance. If this limit is satisfied then we accept the (n + I )th point as a point associated with a physical plateau. Every point in the full data set is considered in this way. This technique shows up a clear difference between the lattice data and a set of data which were generated by a gaussian random walk simulation where Qi + I was equal to Qi + 5Qi, the 5Qi being chosen from a gaussian distribution with mean zero and standard deviation S. The number of points associated with plateaus varies according to the plateau length being imposed; there are many more points being accepted from lattice data than from the random walk. We begin by estimating (Ncs) from the point-to-point correlations in the data ; (Qi - Qi+  )2 versus n . As n increases the curve flattens off as Qi+ is further from Qi. We are seeing the points become uncorrelated as we would expect if we

I. Ambjorn et al. / Sphaleron transitions

365

2.0

w _~j

Z

0

5000

5 500

TIME

Fig . 6. An enhanced region of fig . 3. Here we see a sharp transition. 40

w

r

Y

Q

VALUE

Fig. 7. The histogram corresponding to fig . 1. The plateaus are clearly seen as peaks in the distributions .

consider thermal fluctuations within a topological sector . We might expect a complete flattening off of this curve in which case the height of the plateau should be equal to 2(N~s ) . However, because we are including all the data, correlations over the edge of two plateaus are being included, which distort the curve . We may apply the plateau selection technique to include only correlations within topologi-

!. Ambjprn et al. / Sphaleron transitions

366

Fig. 8. Curves illustrating the behavior of <(Q, - Q, + )') for (1) : random walk data, (11): the full set of data from the lattice and (III) : for selected points from the lattice data . The points used for (111) are those associated with plateaus of width of 101 points, none of which lie more than 3.0 times the standard deviation from the mean of the plateau .

cal sectors. The increased flattening off is shown in fig. 8, which also shows how the random walk data behaves. In this case there is obviously no flattening off. We interpret the flattening off in the lattice data as indicating that the system prefers to stick in topological sectors within which there are thermal fluctuations . We can define a parameter a a=

(NC's)PC,IN - .

(53)

We make an approximate estimate of the size of a from the intercept of the plateau with the y-axis. The results are given table 1 . The size of (Ncs > extracted in this way agrees quite well with the perturbative estimate of a. The concept of

The measured value of

Tnm .F I

a x 1() 4 (defined by eq. (53)) within a topological sector for the various (3c ; and lattice sizes

/3ci\L3

83

12 3

163

8.() 9 .0 1() .() 12.0

4.1 + 1 .() 4.5+ 1 .0 4.2+ 1 .0 -

-4.8+2.0 4.3+ 1 .5 -

-3.4+ 1 .0 4.0+ 1 .0

J. Ainbjorn et al. / Splhaleron transitions

367

TABLE 2

The measured number of sphaleron transitions between different topological sectors for the various /3 ( ; and lattice sizes (judgment by the eye) 13 c \L3 8 .0 9 .0 10 .0 12 .0

83 25+5 10+3 2± 1 -

12 3

16 3

-

--

22+4 9±2 -

40±5 11 + 1

thermal fluctuations seems therefore to be well founded and agrees with high-temperature perturbation theory. We now turn to the discussion of the number of transitions. The most simpleminded approach is just a subjective judgment by the eye, to single out these transitions from the thermal noise. The numbers obtained in this way have already been given in a brief report [151 and can be found in table 2. It might be hoped that the scheme of plateau identification could allow the transitions to be more easily counted. This is true for cases in which a large number of data points have been recorded, but the method has its intrinsic difficulties . If the system is really in thermal equilibrium we would expect that the distribution of the length of the plateaus is exponential : P(L1 ti ) = exp( -TJti ) .

(54)

In other words if the plateau length size being sought, x, is reduced, the number of plateaus seen should increase . This is illustrated by fig. 9 where we show an example of first the full data set (fig. 9a), then those points picked out by fitting to a plateau of length 101 (fig. 9b) and finally the points picked out by fitting to a plateau of length 51 (fig . 9c). The optimal approach seems to be to vary both x and the number of standard deviations used in the selection criterion in order to get an upper and a lower limit on the number of plateaus . Of course the number of plateaus is not necessarily the number of transitions as the system can jump back and forth between the same few plateaus ; we count a transition as the separation of two plateaus by more than a 0.5 change in the value of N(-s. The overall estimate of the number of transitions defined this way is given in table 3. It is reassuring to compare it with the "naive" table 2. Only for the high temperature data (ßG = 8) there is not agreement. Here thermal and "topological" fluctuations seem to mix and create certain ambiguities. From table 2 it is seen that there is an acceptable scaling behavior of the number of transitions of Q(t) with N and ß(;, in qualitative agreement with eq. (50) . This

J. AmbjOrn et al. / Sphaleron transitions

368

10 [ (a)

N v Z

10

10

~ (c)

s 6 V

Z

4

2

2000

4000 TIME

6000

0

2000

4000

6000

TIME

Fig . 9. (a) shows a typical set of lattice data for Nc , s plotted against time ; (b) shows those points associated with plateaus of width 101 where none of the plateau points lie more than 3 .0 standard deviation from the mean ; (c) shows the points associated with plateaus of width 51 where up to one of the points within the plateau may lie more than 3 .0 times the standard deviation from the mean .

indicates that we are close to continuum physics, and the deviation from (50) goes in the direction one would naively expect : Small volume suppresses the rate more than with a plain volume factor and for large ß,; the rate is more suppressed than what is expected from (50) which is natural since we move to smaller temperature . This means that the "physical" volume of the lattice becomes smaller and again it can be too small to contain many independent fluctuations .

J. Ambjorn et al. / Sphaleron transitions

369

TABLE 3

The measured number of transitions between different topological sectors for the various ßc and lattice sizes using plateau identification 16c

\ L3

8;

8.0

5+ 1

10.0 12.0

2±1 -

9.0

9±2

123

16.3

18±4

30±10 8+3

10±2 -

TABLE 4

The values of rc deduced from table 2 and eq. (50) ßc

\ L3

8.0

9.0

10.0 12.0

8;

123

163

0.6 ± 0.1

0.25±0.05

-

0.4±0 .1

0.1±0.05 --

0.15±0.03

-

0.30±0.04

0.15 + 0 .02

Using eq. (50) we have converted the counting in table 2 to values of tc in table 4. Since we get closer to continuum physics when we move up and when we move right in the table (both ways bring us to large "physical" volumes) we can get the following conservative bound on the coefficient rc in the confinement phase rc>0 .1 .

(55)

We do not see any suppression of the rate, and probably the real value of rc > 0.3. This means, in particular, that if some effective sphaleron exists in the high-temperature phase its mass cannot be substantially bigger than the temperature .

6. Simulations including a Chern-Simons density As mentioned in sect . 1 one of the implications of rapid sphaleron-like transitions is that the baryon number need not be a good quantum number in the hot early universe . We will now use the microcanonical simulation to address the question of spontaneous CP-breaking in the early universe triggered by the small CP-breaking term present in the electroweak theory (explicitly manifest in the complex phase of the Kobayashi-Maskawa mass matrix). The simplest way to observe a possible spontaneous CP-breaking is by investigating the effective potential for the CS-density. If the effective potential for the CS-density is flat in

370

J. Ambj0rn et aL / Sphaleron transitioiis

some range we can get a spontaneous breaking of CP-invariance* . An attempt to measure this potential was done in ref. [7] . The results were not incompatible with a flat potential, but cannot really be considered as conclusive . In the old approach a lattice simulation of a (infinitely, by dimensional reduction) hot gauge-Higgs system was performed using a standard Metropolis algorithm. After thermal equilibrium a first-order phase transition to the broken phase was induced and a relaxation equation cooled the gauge-Higgs configuration afterwards . During the rolling down of the gauge configuration to a classical vacuum the change in CS-density was measured . In this way the effective potential for change in CS-density during the first-order transition could be constructed [7]. As will be discussed there is a certain ambiguity associated with this approach. Here we will try an alternative way of observing the spontaneous condensation of CS-density, namely by adding directly the relevant CP-breaking term H~s =1 J d'xncs(x) , it

(56)

where the CS-density is given by n(,s(x) -°

1

;jAk - A ;A i A k )(x) . 16-rr2E'~kTr(F

(57)

Adding such a term was impossible in the old approach [7], where we used the action in the Metropolis updating procedure. Since ncs (x) is not invariant under gauge transformations (although of course the integral Ncs is invariant under all local gauge transformations) there exists no simple convenient lattice version of ncs (x). However, the classical equations derived from an effective hamiltonian H,rr _Ho+Hcs

(58)

do not suffer from this ambiguity. They are in fact covariant and one has d A i' dt

dH,~ I.f.

() LI

r

'

d Ei' ® ® lI N(A f dH, ) _ gB dAi' dt dAi'

(59)

(60)

* For discussions of the effective potential for CS-density, including the problem of gauge invariance, see ref. [6].

J. Ambjorn et al. / Sphaleron transitions

371

where Ba denotes the "magnetic" field associated with the field tensor Fib

Ba=

2 Eijk FÍk .

(61)

This modification of the classical equations of motion is not in conflict with Gauss constraint, since the Bianchi identities ensure us that DabBh = 0 .

(62)

We can transform eqs. (59), (60) to the lattice and perform the same numerical simulations as we did when we studied the sphaleron transitions. During the time evolution we can again measure the "topological" charge Q(t) as we did when t, = 0 . Further we can make successive copies of the gauge-Higgs configurations at times t,, . . . , t . Each of these configurations can be considered as a representative for a hot configuration at a certain stage of the evolution of the universe . It is possible to induce a first-order phase transition on such a configuration, the same way as was done in ref. [7] . It is done by changing by hand the coupling constants such that after the change we will be deep into the broken phase. Such a discontinuous change of the mass excitations gives a good phenomenological representation of a first-order phase transition [11] . After the change in coupling constants we cool the configuration down to a classical ground state by a relaxation equation . We refer to ref. [7] for a careful discussion of this point, here it will be sufficient to mention that this procedure is intended to represent the evolution (cooling) of a typical gauge-Higgs configuration after the electroweak transition until it settles down in a classical vacuum. During this cooling we can now measure the change 4Q, in CS-number, a change which by the anomaly will be related to the number of generated baryons. The index ti denotes the time at which we take a copy of the gauge-Higgs configuration and induce the first-order phase transition on the copy. By repeating the above-mentioned procedure a number of times we can get a probability distribution of the charge Q(t) as a function of time, in the hot phase and we can get a probability distribution of the change ®Q, at t,, . . . , t . Let us discuss shortly what we expect to observe. At least for small values of the chemical potential j, we have the following picture: If /, is zero we have already argued that Q(t) should essentially perform a random walk between the different classical vacua . The classical energy functional is a periodic functional with respect to large gauge transformations which have a winding number. If /, is different from zero, but small, this periodic potential will be slightly tilted because of the additional term ® H = I, N«. This means that there will be a small drift term in the random walk and diffusion "downhill" will be favored. We should therefore see a linear growth in Q(t) for small j, and moderate time t . One of the important questions we will try to answer is whether this grows in Q(t) (which we will actually

J. Amhjorn et al. / Sphaleron transitions

;72

observe) has any influence on the probability distributions of ®Q,,, i = 1, . . . , n, that is on the asymmetry in the anomalous baryon production during the electroweak phase transition. If the picture of anomalous baryon production suggested by one of the authors (1VI .E.S. [6]) is correct, we should see a gradual shift in ®Q,, as a function ti , the distribution changing from being symmetric for small ti (small Q(ti)), to being increasing asymmetric when ti is growing, until a maximum asymmetry is attained, after which only small further changes should be observed . Such a behavior should correspond to an effective potential for n cs which is flat out to some maximum value which for dimensional reasons should be

ncsX ,., (aw T ) 3 .

(63)

Again we refer to ref. [6] for a more careful discussion of the effective' potential V(n cs ) for the CS-density . Let us return to the question of how to transfer the equations of motion (60) to the lattice . This is seemingly straightforward since we can represent Fi'(x) - '-, Tr-r"Uo(,i,j)'

(64)

Defined in this way the lattice quantity will transform covariantly, °xactly as Fi" j . Consequently we simply add to the equations of motion (20) this additional term and we get the following change for the matrix E compared to eo,. ;20)

Eq. (65) reduces in the formal continuum limit (where the lattice spacing a goes to zero) to the correct form, but although eq. (65) transforms covariantly the time evolution will no longer be compatible with the Gauss constraint . The reason is that the Bianchi identities on the lattice are much more complicated than in the continuum . It was possible to reduce the violation by an order of magnitude by using a more complicated lattice implementation of the continuum object Bti`(x ) _ ;Eijti Fi~(x) than (65) Bti`( x)

~

Eiik Tr_

jTct

( U O .r :i .Í

+

Uo( .r-î :i .j)

+

Uo( .r-î-j :i .j)

+ U~( .r-Ï :i,j)~ )

'

(66)

The meaning of eq . (66) is that a symmetric average is taken of all the plaquettes which touch the lattice site x and are orthogonal to the link (x, x + 1F ) . Although not stated explicitly in eq . (66) the product of links in the four plaquettes should always start and end at x, in order that the lattice version of Bk(x) transforms covariantly.

J. Ambjorn ei al. / Sphaleron transitions

373

However, even with this symmetric, improved lattice representation of Fib there was still a small violation of the Gauss constraint . The quantity G2 defined by eq. (26) and being the lattice equivalent to f d 3x(Di Ei )2 was growing approximately linear with the computer time used. Since these excitations represent unphysical modes they are a serious problem for the simulation and we decided to kill them by the cooling procedure also used after the Metropolis heating. Whenever G 2 exceeded a certain limit we relaxed the gauge-Higgs configuration, using the relaxation equation in a fictitious time T d0(x, T)

8G 2 [0]

80( x, d7-

T) '

were 0 =-- U, E, 0, P .

(67)

This cooling was very efficient and in this way we kept the Gauss constraint satisfied to any desired precision. Although a single cooling results only in very small changes in the energy of the system, we have to admit that the accumulative effect of many coolings in long runs could modify the dynamics of the systems. However, we have found no better way to repair the violation of the Gauss constraint . We now repeated the numerical simulations done for the chemical potential t. = 0 for various values of the A . If we want values of t, so small that the CS-term can be viewed as a perturbation to the classical periodic vacuum structure we get btizlttice « ßcx/ 32 ir 2 ,

(68)

where x, defined by eq . (40), essentially is the sphaleron energy in lattice units. Semiclassically we expect x - 10, but since we are in the hot phase x may actually be less and we get (for the range of ß(;'s used in the simulations) A lattice « 0 .1 .

(69)

In fig . 10 we have shown typical time evolution pictures of the "topological" charge Q(r) in the hot phase . In agreement with the estimate (69) there is a qualitative change in the behavior of Q(t) for j. = 0.04. For smaller t, we have pictures where fluctuations between plateaus are superimposed on a small systematic drift in a direction determined by the sign of A. For larger tL we see larger changes in Q(t) and the systematic drift dominates . However, for these larger values of /, the violation of Gauss constraints is not small and the relaxation had to be used frequently . We cannot rule out the possibility that for large values of t, the excitation of unphysical modes contributes in an important way to the observed value of AQ. Although these modes were explicitly removed by the relaxation procedure (67) they could implicitly affect the physics by a too frequent use of eq. (67) .

J. Ambjorri et al. / Sphaléron transitions

374 11

W J

z

-2 -3 -a -5 -6 < (a)

-7 0

1000

0

1000

I 2000 TIME

1 4000

, 5000

11 3000 4000

i 5000

3000

W J m u

Z

2000 TIME

Fig. 10. The measured values of Q(t) for b G = 10.0, Iß H = 0.34 and ß R = 0.0014 and for various values of j,. The lattice size is 12;. The timestep is 0.05 and the total time 5000. (a) t,L. = 0.02 ; (b) p, = -0.02 ; (c)/,=0 .04 ; (d)/,= -0.04.

The question we want to answer is whether the observed change in n «(x) in the real-time simulation with I.L * 0 can be related to an asymmetry in the observed ®Q,, during the induced first-order transition . Since the "topological" charge Q(t) in the hot phase changes in a systematic way during the time evolution, as is apparent from fig . 10, it would be natural if this change was related to a similar asymmetry in the distribution of ®Q, as a function of the time t; the system spends in the hot phase with a chemical potential u 0 0. Quite disappointingly we

375

J. Ambjorn et al. / Sphaleron transitions

-20

w

Z -100

0

1000

2000

1000

2000

TIME

3000

4000

3000

4000

,

5000

120

90 w J 60 z

30

0

TIME

5000

Fig. 10 (continued).

observed nothing which allowed for this interpretation . For the small values of tL where the qualitative random walk picture might have some validity, the measured distributions of 4Qr~ were, within the statistics available, independent of the starting time tl for cooling of the configurations and identical to the distribution we got if tu, = 0. For growing iu, we saw a gradual change in the distribution of SQ, but it did not appear as a shift in the distribution corresponding to A = 0, cven if the final value of Q(t) in the hot phase could be very large. Three distributions of ®Q are shown in fig. 11 one for A = 0 and one for A = ±0.04. The "periodic"

J. Arnbjorn et al. / Sphaleron transitions

376

90 F (a)

m

45

Z

0 ' .. n .. -2

r

I

-I

r0 Q VALUE

36

36

r

m 20 Z

U 0 1---3

nnnln~

-2

11111 n n

-I

0 0 VALUE

I

I

2

3

0 L-3

«Il

-2

I

-I

0

0 VALUE

I

2

3

Fig . 11 . The distribution of the measured change IQ of the "topological" winding number during the relaxation from the hot phase to the classical ground state . Lattice size is 12 ; , the coupling constants in the hot phase are P c , = 1() .U, /3,, = (1 .34 and i3 1{ = 0 .00 14 . (a) /-t = (1 .() ; (b) A = () .()4 ; (c) u = -(l .()4 .

structure of the distribution of AQ for ,u = U agrees with the observations in ref. [7], while we have no interesting interpretation of the two other distributions. We conclude that we have not observed any indications of a spontaneous breaking of CP-invariance in the hot gauge-Higgs vacuum .

7.

iscussion

We had two objectives with the computer simulations: a nonperturbative investigation of sphaleron-like transitions in the hot phase of the electroweak theory and an attempt to verify the hypothesis that CP-invariance can be spontaneously broken in the same phase . The first part was straightforward . Unfortunately the semiclassical formulas for sphaleron transitions in the broken phase of the electoweak theory could not be

J. Ambjorn et al. / Sphaleron transitions

377

verified with the lattice size used, but in the hot phase we found results supporting the idea that there is no suppression of the transitions . As explained above this has important implications for any attempt to explain the observed baryon asymmetry of the universe . The second part is related to specific attempts to explain the baryon asymmetry entirely within the standard electroweak theory . An important ingredient in such explanations is the formation of a CS-condensate in the hot phase of the universe due to the presence of a small chemical potential j,, and the consequent asymmetry of ®Q during the electroweak phase transition . The inclusion of the CS-term on the lattice turned out to be problematic . To the extent we were able to put in on the lattice we saw no evidence of an asymmetry of ®Q during the phase transition . However, we feel one should not draw too firm conclusions based on the above lattice results for reasons to be discussed now. The first point to worry about has already been mentioned and is the violation of Gauss constraints caused by the presence of the CS-term in the effective hamiltonian . The second point has to do with the cooling of the gauge-Higgs configuration to the classical vacuum after the induced first-order phase transition. Since the configuration has to end in one of the classical vacua (and it does) and since AQ only depends on the initial and final configuration one would naively expect that the details of the relaxation equations used to cool the gauge-Higgs configuration to one of the classical vacua should not matter . This seems to be wrong. Even for j, = U we can get rather different distributions depending on the details of the relaxation equations . Although the qualitative features of the distributions are all the same, it is indeed worrisome that we can observe quantitative differences in the distributions depending on the relaxation equations . One gets the suspicion that lattice artifacts could be important . We analyzed this in some detail in ref. [71 and came to the conclusion that level crossings of the Dirac propagator during the change in Q seem to take place, as expected from continuum physics and needed for the anomaly. 'Nevertheless, it would be highly desirable if one could invent a more robust method of measuring the change ®Q. Let us finally remark that it now appears possible to avoid the trouble of introducing the gauge noninvariant CS-density and still address the question of say spontaneous parity breaking . Integrating out the fermions in a gauge-Higgs system will not only introduce a CS-term, but also parity-odd terms like ` Fi~ ( t7 11 Dkfl (70) P, 1 "Eijk H

fi

H

Î ~. = l Ei k E abccP t Tu D i cP cP t T hD~ cp cP T Dk cP .

(71)

Such terms are explicit gauge invariant and might more easily be included in a lattice approach . Work in this direction is in progress .

378

,J Anrhiorn et at / Sphaleroti transitions

M .S. is indebted to the Niels Bohr Institute, Copenhagen, where this work was done, for the kind hospitality. J.A . thanks the Danish research council for computer time on the Amdahl VP1100. H.P. acknowledges very helpful discussions with C.D. Froggart and D.G. Sutherland and financial support and the provision of Cray computing time from S.E.R .C. Finally we are grateful to Benny Lautrup for generously letting us use his Apollo 10000 workstation . eferences [1] G. 't Hooft, Phys. Rev . Lett. 37 (1976) 8; Phys. Rev. D14 (1976) 3432 [2] [3j [4] [5]

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