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Baryogenesis and phase transition in the standard model

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Baryogenesis and phase transition in the standard model D.Karczewska 1R.Ma′n ka 2Department of Astrophysics and Cosmology,University of Silesia,Uniwersytecka 4,40-007Katowice,Poland ABSTRACT The sphaleron type solution in the electroweak theory,generalized to include the dilaton ?eld,is examined.The solutions describe both the variations of Higgs and gauge ?elds inside the sphaleron and the shape of the dilaton cloud surrounding the sphaleron.Such a cloud is large and extends far out-side.These phenomena may play an important role during the baryogenesis which probably took place in the Early Universe.1.INTRODUCTION In this paper the electroweak theory will be extended by the inclusion of dilatonic ?eld.The Glashow-Weinberg-Salam dilatonic model with SU L (2)×U Y (1)symmetry is described by the lagrangian L =?14

e 2?(x )/

f B μνB μν+12igW a μσa ?1

2W a μσa are

respectively local gauge ?elds associated with U Y (1)and SU L (2)symmetry

groups.Y denotes the hypercharge.The gauge group is a simple product of U Y (1)and SU L (2)hence we have two gauge couplings g and g ′.The generators of gauge groups are:a unit matrix for U Y (1)and Pauli matrices for SU L (2).In the simplest version of the standard model a doublet of Higgs

?elds is introduced H = H +

H 0 = 012v ,with the Higgs potential

U (H +,H,?)=λ H +H ?1

√

describe the hedgehog structure.This

produces a nontrivial topological charge of the sphaleron.The topological charge is equal to the Chern-Simons number.Such a hedgehog structure determines the asymptotic shape of the sphaleron with gauge ?elds di?erent from zero W a i =?aij n j 1?s (r )2f 2 d ′(r )

2v 20h ′(r )2?1

4r 2v 20(3?s (r ))2h (r )2?

1r 2(3?4s (r )+s (r )2)2+2s ′(r )2 ?12 (5)

Then we switch to dimensionless variables x=M W r=r/r W,where M2W= 1

M W～10?18cm.The resulting Euler-Lagrange equa-tions are following:s(x)function,which describes the gauge?eld in the

electroweak theory and satis?es the equation

s′′(x)+2d(x)′

d(x)2

(3?s(x))

x +

M2W

(d(x)2?h(x)2)h(x)

?8C2x2(s(x)?3)2h(x)=0,(7) where M2H=2λv20determines the Higgs mass.The d(x)function describing the dependence of a dilaton?eld on x in extended electroweak theory obeys the equation:

d′′(x)+2

d(x)

M2H

(s(x)?3)2(s(x)?1)2d(x)3

+2λv40(d(x)2?h(x)2)d(x)3+4

gf )2～10?9.This

practically means that the dilaton?eld is a free?eld.The simplest solutions h(x)=1(shown in Fig.1),s(x)=3(Fig.2),d(x)=1(Fig.3)are global ones corresponding to the vacuum with broken symmetry in the standard model.It is obvious that far from the center of the sphaleron our solutions should describe the normal broken phase which is very well known from the standard model.Knowing the asymptotic solutions we are able to construct a two-parameter family of solutions(for details see[5]):

s(x)=1+2tanh2(tx)(9)

h(x)=tanh(ux)(10)

d(x)=a+(1?a)tanh2(kx)(11) where t,u,a,k are parameters to be determined by the variational procedure. The relevant values of the parameters are those which minimize the energy. For example,with the standard values of M W=80.6GeV,M Z=91.16GeV, M H=350GeV we found the numeric solutions t,u,k,as functions depend-ing on the initial conditions of the dilaton?eld d(0)=a in the center of the sphaleron.Our solutions describe both the behavior of Higgs?eld and gauge ?eld inside the sphaleron and the shape of the dilaton cloud surrounding the sphaleron.Such a cloud is large and extends far outside the sphaleron. The sphalerons are created during the?rst order phase transition in the ex-panding universe as inhomogeneous solutions of the motion equations.These phase transition bubbles,which probably took place in the early universe, break the CP and C symmetry on their walls and can cause the breaking of baryonic symmetry.Detailed consideration of this problem will be the subject of a separate paper.

3.CONCLUSIONS

Numerical solutions suggest that sphaleron possess an‘onionlike’structure. In the small inner core the scalar?eld is decreasing with global gauge symme-try restoration SU(2)×U(1).In the middle layer the gauge?eld undergoes sudden change.The sphaleron coupled to dilaton?eld has also an outer shell, where dilaton?eld changes drastically.The spherically symmetric dilaton so-lutions coupled to the gauge?eld or gravity are interesting in their own rights and may further in?uence the monopole catalysis of baryogenesis induced by sphaleron.

This paper is sponsored by the Grant KBN2P30402206. References

[1]A.Riotto,Phys.Rev.D49,730-738,(1994),Singlet Majoron model with

hidden scale invariance.

[2]R.Manka,D.Karczewska,Z.Phys.C57,417-420,(1993),The neutrino

ball in the standard model.

[3]R.Manka I.Bednarek D.Karczewska,Phys.Scr.52,36-40,(1995),On the

neutrino ball model.

[4]B.Kleihaus,J.Kunz,Y.Brihaye,Phys.Lett.B273,(1991)100.

[5]R.Manka D.Karczewska,Baryogenesis in the dilatonic electroweak the-

ory,preprint U′SL-TH-96-2.

FIGURE CAPTIONS

Figure1.The dependence of the Higgs?eld h(x)on x.

Figure2.The dependence of the gauge?eld s(x)on x.

Figure3.The dependence of the dilaton?eld d(x)on x.