Electroweak baryogenesis and dark matter via a gauge singlet scalar

Physics Letters B 323 (1994) 339-346 North-Holland

PHYSICS LETTERS B

Electroweak baryogenesis and dark matter via a gauge singlet scalar John McDonald CFMC-GTAE, Av. Prof. Gama Pinto 2, P-1699 Lisboa, Portugal

Received 26 July 1993 Editor: R. Gatto

We consider the possibility of using a gauge singlet scalar as the source of the additional CP violation needed to allow the observed baryon (B) number asymmetry to be generated during the electroweak phase transition. In addition to extending the standard SU (3) × SU (2) × U ( 1) model by a gauge singlet scalar, it is also necessary to add additional fields, in order to transmit CP violation from the singlet scalar sector to the quarks and so to the B number violating processes in the early Universe. We consider the case of a second Higgs doublet, with Yukawa couplings to the quarks and leptons which are proportional to those of the standard model Higgs in order to avoid large FCNC effects. For reasonable values of the couplings of the gauge singlet scalar it is possible to generate the observed B asymmetry. A specific example is discussed for which we show that the singlet scalar expectation value goes to zero for temperatures below the electroweak phase transition temperature, allowing the model to avoid strong CP violation effects due to spontaneous CP violation and domain wall problems due to discrete symmetry breaking. In this case the singlet scalars are cosmologically stable and can naturally account for the cold dark matter in the Universe. Possible resolutions of the Higgs mass problem of electroweak baryogenesis within the framework of the model are also discussed.

1. Introduction O v e r the past few years there has been m u c h interest in the possibility o f generating the b a r y o n asymmetry o f the U n i v e r s e ( B A U ) during the electroweak phase transition ( E W P T ) [ 1-7 ]. This gives us the exciting possibility o f u n d e r s t a n d i n g m u c h o f the observed U n i v e r s e in terms o f particle physics at a mass scale ~< 1 TeV, in particular nucleosynthesis [ 8 ] a n d the nature o f the d a r k m a t t e r [ 9,10 ] as well as the BAU. In the case o f the s t a n d a r d m o d e l however, it was found that it is not possible to account for the BAU for two reasons: ( i ) there is insufficient C P violation from the quark mass matrices to generate a large enough BAU [ 1 ], and (ii) for a Higgs boson mass larger than the LEP lower bound, the E W P T is not sufficiently strong to ensure that a n o m a l o u s B + L violation is out o f equil i b r i u m at the end o f the phase transition, i.e. the size o f the Higgs expectation value is not large enough to suppress the sphaleron fluctuations [ 1,6,7,11 ]. As a result, even i f a b a r y o n a s y m m e t r y is generated during the EWPT, it will be subsequently washed out b y

the B + L violation which is still in equilibrium. In order to generate the BAU we must therefore consider extensions o f the s t a n d a r d m o d e l which address these problems. Several extensions o f the s t a n d a r d m o d e l which a t t e m p t to o v e r c o m e p r o b l e m s ( i ) a n d ( i i ) have been considered. Two doublet models can allow for additional C P violation and a strong enough E W P T [ 12 ], but it appears that in general the BAU generated is too small [ 13 ]. The m i n i m a l SUSY extension o f the s t a n d a r d m o d e l also allows new C P violating interactions, but it has been f o u n d that this m o d e l can only barely account for the BAU, and any increase in the lower b o u n d on the Higgs mass will rule out this possibility [ 14-16 ], although there m a y exist a small region o f p a r a m e t e r space with a light stop where the Higgs mass lower b o u n d can be significantly larger without ruling out generation o f the BAU [ 17 ]. In general these two doublet models have a difficulty because at the E W P T they behave effectively as m o d e l s with one light Higgs doublet, with the second heavier Higgs doublet decoupling a n d having only a small effect on the phase transition [13]. In this p a p e r we will consider a very simple exten-

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sion of the standard model, namely the addition of a gauge singlet complex scalar. It has already been suggested that adding a gauge singlet scalar which couples to the standard model Higgs can solve problem (ii) [6]. In this paper we concentrate on problem (i) and try to use the singlet scalar as the source of the CP violation responsible for baryon asymmetry generating during the EWPT. We will also discuss possible solutions of problem (ii) and also the possibility of using the singlet scalar to account for the cold dark matter in the Universe.

2. The model and BAU generation We first state our model and then explain the reasons for the couplings introduced. We will consider a model with the addition of a gauge singlet S and a second Higgs doublet//2, which will either not develop a VEV or only a very small one. The standard model Higgs doublet is denoted by H1. We wish to show how a time-dependent phase originating in the potential for the scalar singlet can produce a baryon asymmetry. We consider a scalar potential for S, H~ and//2 of the form V= I:1 + V2+ V3+ I:4, where

1:1 = -#2StS+2snStSH~HI+2ss(S?S) 2 ,

(2.1a)

V2 = (m2,$2+ h.c. ) + (2~,S2HTH2 +h.c.) + (2#$4+h.c.),

V3 = - lt2n'[nl + 2 m (H~[H~)2

(2.1b) (2. lc)

and

I:4 =m22H~H2 +2n2(H~H2) 2 +212(HtH1 ) ( H ~ H 2 ) .

(2. ld)

In order to make clear the mechanism for BAU generation in the model, we have retained in Vonly those terms which directly play a role in baryogenesis, rather than including all possible terms. V~ gives an expectation value for S, while the coupling 2si~ gives ( S ) a dependence on ( H ~ ) , which is necessary in order for S to have a time dependent phase during the EWPT. ( S ) 5 0 prior to the EWPT is best since the alternative is for S and H1 to develop an expectation value at more or less the same time, which would in general require some degree of fine-tuning. I:2 is the source of the CP violating phase responsible for the 340

17 March 1994

BAU. In general we can choose the initial phase of ill such that ( H 1 ) is real and positive, and we can choose the phases of H2 and S to make m 2, and 2 , real and positive. The phase of 2#( --- 12#1e i#) is then a physical CP violating phase which cannot be rotated away. We will see that we do not need to introduce explicit CP violation, since ( S ) will spontaneously break CP maximally for a range of parameters. (In fact, a small amount of explicit CP violation (with phase O ( 10- 5) if there is large spontaneous CP violation) will be necessary at the time when S originally develops an expectation value, since otherwise the sign of the B asymmetry will be different in different bubbles of non-zero ( S ) , resulting in a net zero B asymmetry [ 18 ]. In our discussion we are considering a particular value of ( S ) , which can be taken to correspond to the dominant value in the Universe after ( S ) develops). 1:3 and V4 are potentials for the standard model Higgs H~ and the "heavy" Higgs//2 (we consider ran2 to be large compared with the temperature of the EWPT, Tew). The second Higgs is necessary in order that a complex phase coming from ( S ) can interact with the fields of the standard model. In the absence of the second Higgs doublet, the only interactions the scalar can have with the standard model fields are via terms of the form StSH~H~ or S2H~H~ +h.c., which cannot introduce a complex phase into the standard model since the coefficient of H]H~ must be real. (One might ask whether this is true at higher orders in perturbation theory; for example, whether an interaction of the form (aS2+a*S .2) ITVWcould arise at two loops. However, it is straightforward to show directly from considering the possible diagrams that such a term does not in fact arise at two loops. More generally, if such a term could arise at higher orders it would also imply that a term of the form H]H~ ff'Wwould occur in the standard model, as can be seen simply by replacing the S2H~H~ coupling or S*2H] H~ coupling in the diagram responsible by the Higgs self-coupling term (H~[H1)2. In the absence of KM phases, which we can ignore since we are interested in new sources of CP violation, such a term does not arise in the standard model. Thus the CP violating phase cannot be transferred radiatively from the S scalars to the standard model fields when we add only a gauge singlet to the standard model. We must introduce additional fields in order to transfer the CP

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violation from the S scalars to the standard model fields. We note that one could add fields other than the second Higgs doublet which we are considering here. For example, one could add an isosinglet charge 32or - ~ quark field, which would allow the S scalar to have couplings to the standard model quark fields. We are focusing our attention on extending the scalar sector in this paper). In general the minimization of the scalar potential is complicated and depends on the many couplings introduced in (2.1). In order to be able to reach clear conclusions we will restrict our attention to the case where the influence of ( S ) on the first-order phase transition is small. We will thus consider the case where ( H ~ ) is essentially independent of ( S ) and where the effect of mixing between HI and H2 is small. We first consider the evolution of the scalar field expectation values around Tew. (a) T ~ T~w. With ( H 2 ) =0, and defining ( H 1 ) = h~ and ( S ) =pe i°, (2.1a) and (2.1b) can be written as

V~ + V== ( - #2 + 2 suh 2 + 2 m 2, cos 20)p = + [ 2 s s + 2 2 p cos(40+fl) ]p4.

(2.2)

For p to be non-zero at TCw the coefficient of the p2 term should be negative. Then requiring that the p4 term is positive on minimizing the potential with respect to 0, as is necessary for the potential to have a stable minimum, imposes the constraint

).ss > 1 . 22p

(2.3)

From now on we will consider the masses and couplings in V= to be small compared with the corresponding terms in 111. Then minimizing (2.2) with respect to p gives

p2~ # s2- 2 s I ~ h l2 22ss '

(2.4)

where hi is considered independent ofp. Since we will not be concerned with explicit CP violation in the following we can simply choose fl= 0. Minimizing the potential with respect to 0 then gives cos 20= - 2 s s m~, 22p t, ~ , s2- ;~su h2~ ~J

(2.5)

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if Icos 201 < 1, and Icos 201 = 1 if Icos 201 from (2.5) is larger than 1. Note that Icos 201 increases as h 2 increases from zero. Since, as we will show later, the CP violating phase in this model is given by 20, in order to have a non-trivial phase during the electroweak phase transition we must require that Icos 201 < 1 when hi = 0 at Tew. This implies that

2ss #2 22p < m~,-- "

(2.6)

It is important to note that the most natural values for 20 are either 20 ~ nn or 20~ nn/2 (n odd ). Therefore if the CP violating phase from ( S ) is non-trivial, it will tend to have a maximal value. More generally, with fl non-zero, it is easy to show that with hi = 0, the CP violating phase will be significantly different from nzr only if (2.6) is satisfied and that it will then take the value 20~ ( n r c / 2 - fl/2 ). The complex phase from ( S ) is carried to the quarks via the coupling 2~. In order to transfer the complex phase to the quarks, we have to consider Yukawa couplings of Hz to the quarks and leptons of the form

,~uaRJq~QL+~'~aRH2QL+~eaRH2LL+h.c.,

(2.7)

where in order to avoid FCNC effects, 2',, 2~ and 2" must to a good approximation be proportional to the corresponding standard model Yukawa couplings of the quarks and leptons to H1, which we denote by 2 u, 2d and 2e. In the absence of any fundamental understanding of the origin of the Yukawa coupling matrices, this is not obviously any more unnatural than the usual two doublet solution of the FCNC problem, which consists of coupling one Higgs doublet to the u quarks and the other to the d quarks and leptons. Indeed, the solution to the FCNC problem employed in the present model may be regarded as a testable prediction of the model. Since we are considering Tew
,~,~,+~;~.

(2.8)

Assuming that the correction term is small compared with 2t, we can define a corrected t quark Yukawa coupling by 2 t ei#t where 341

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density of t quarks. At finite temperature the free energy of the Universe in the presence of (2.11 ) is minimized for non-zero baryon number. The equilibrium baryon density is then given by [ 3 ]

5

/

l

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/S

72 T 2 . nBeq-- 111 6 ~t-

tL

"

H~

Fig. 1.//2 exchangediagram leadingto a CPviolatingcorrection to the t quark Yukawacoupling.

In the presence of sphaleron-induced B + L violation which is in thermal equilibrium, the baryon number will increase towards the equilibrium baryon density according to the rate equation

dnB 2-~ m22 sin 20.

(2.9)

This phase will depend on h~ during the EWPT as a result of the dependence o f p on h~. Since only the t quark Yukawa coupling is large enough to produce interactions which are in thermal equilibrium inside the electroweak bubble walls at Tcw [ 3 ], we need only consider the phase for the t quark, Or, when discussing baryogenesis. We follow the procedure of refs. [3,4] in calculating the B asymmetry. We first perform an anomaly-free rotation of the quarks and leptons, in order to eliminate the phase ~t from the corrected t quark Yukawa coupling. We do this by rotating the quarks and leptons by an amount proportional to their hypercharge: tL~tL eic°/6 and tR--~tRe2it°/3, with 09=2~/. Since 09 is spacetime dependent, the kinetic terms for the t quarks will then give rise to a new CP violating term in the Lagrangian,

t-~Z-

~0U~t(47R~)UtR +/-L~UtL) .

(2.10)

The baryon asymmetry produced during the first-order phase transition can then be calculated using the thermodynamic argument of ref. [ 4 ]. ~t can be taken to be time dependent but space independent, since the mean free path associated with the t quarks and the gauge bosons (for the t quark ~ 4 / T ) is small compared with the length scale over which ~t varies during the first-order transition, which corresponds to the thickness of the electroweak bubble walls ( ~ 4 0 / T ) [ 7 ]. Then (2.10) gives an additional term in the Lagrangian: -

316t(4/'RT0tR+ t'LT0tL) ,

(2.1 1 )

which has the form of a chemical potential term for a 342

(2.12)

dt

_ 18 Fsv

~

nB~q.

(2.13)

The sphaleron transition rate Fsv can be approximated by [2,4,5 ] ffsp =K(Otw T) 4 mw
mw>aawT.

(2.14)

a and K are numerical constants reflecting the uncertainty in the estimates; K has been estimated numerically to be between 0.1 and 1 [ 19 ] and ais estimated to be between 2 and 7 [2,5]. Integrating the rate equation then gives for the baryon asymmetry generated during the EWPT nB= 216 Ko~4T3~t,

(2.15)

where ~ t is the change in ~t as the Higgs field increases from h~ = 0 to the value at which the anomalous B + L violation goes out of thermal equilibrium, which from (2.14) is given by h~c- (x/~/g)aotwT¢w. Assuming the change in ~, during the EWPT is mainly due to the change in p2 and using (2.9), we obtain for the final baryon-to-entropy ratio nB 1215 Ka2ot5w s -- lllrt3g(Tew)

~.aASH '~/t T2 × 2ss 2t (--sin 20) m22 ,

(2.16)

where g(Tcw) is the number of degrees of freedom in thermal equilibrium at T~. Numerically (2.16) gives (with g(Tow)= 100)

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ns -- ( 1.5X 10-1°)Ka 2 S

~¢tASH ~tt

×

T2

2~-s 2t ( - s i n 20) m--~m"

(2.17)

To see what values for the couplings are required in order to generate the observed baryon asymmetry, (ns/s)o~= (0.4-1) × 10 -1° [ 10], we use the following values for the unknown parameters. The value of Ka 2, using the above estimates for the values of K and a, is expected to be between 0.4 and 50; we will use K = 0 . 4 and a = 5 , which gives Kt72= 10. Tew can be expressed in terms of the physical Higgs boson mass; for a 120 GeV t quark we find T~w~ 1.6mh, where we have taken the temperature of the EWPT to be approximately equal to the temperature at which the hi = 0 minimum becomes unstable (T2 in ref. [6] ). The largest value of T~w/rnl~ is obtained once mm < O (T~w). Following ref. [ 13 ] we will conservatively consider the largest value of T¢w/mm to be obtained once rnm < nT~w,~ 5mh, in which case mH~ in (2.17) is replaced by IrTew, although it is possible that in the present case mm could be smaller, since ref. [ 13 ] considers the CP violating operators to come from integrating out loop diagrams involving heavy ferrnions rather than the scalar exchange considered here. We will use mm = nT~w in our estimates. (With mh between 60 GeV and 100 GeV the upper bound on mm from m m ~
2 - ~ s >~0.26.

(2.18)

In obtaining (2.18) we have made a number of assumptions. In particular, we have assumed that the value of h~ during the EWPT is essentially independent of p, in which case the EWPT is essentially the same as in the case of the standard model. This simplifies greatly the calculation of the B asymmetry in our model, and also allows us to use, essentially unaltered, the existing standard model analysis of the EWPT. We must, however, consider the constraints imposed on the model in order for these assumptions to be valid. We must also consider the additional fi-

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nite temperature corrections coming from the new couplings in (2.1). From the point of view of determining the minimum of the potential, the most important corrections are those to the scalar mass terms. (Corrections to the ,,0~4,, terms in the scalar potential are of the order of the T-- 0 radiative corrections and have only a logarithmic dependence on T [20 ].) At T ~ T~I the corrections to the potential for S and hi due to the additional non-standard-model couplings in (2.1) result in effective mass terms for S a n d h~ at finite temperature,

lt2 ~lZ21(T) =lZ21- ~ 2 s n T 2 , Iz2 ~ p 2 ( T ) = # 2 - ~ T 2 ( 8 2 s s + 4 2 s n )

(2.19a)

,

(2.19b)

where the corrections are due to the four real scalars in H1 and the two real scalars in S (we are considering the limit of unbroken SU(2) × U ( 1 ) here). Requiring that p ~ 0 at Tew imposes, from (2.19b), the constraint

2sn, 22ss <~ 2/z2 mh2 •

(2.20)

In order that the EWPT is essentially unaltered from the case of the standard model, we will impose the constraint that corrections to the mass term o f the hx potential contribute less than 0.1/12. From (2.19a), this implies that 0.3m 2 2SH< ~ ~0.1 .

(2.21)

In addition, we require that the contribution to the h~ term from the 2sn term in (2.1a) is small; 2snP2
2ss.

(2.22)

We also require that the mixing between/-/1 and/-/2 does not significantly alter the (negative) mass squared of the smallest eigenvalue of the HI -/-/2 mass matrix in the limit where h~ =0. This requires that 2 2 2 (;t~p) /m m < 0.1/t 2, which gives the constraint 2

2~ <~1.5 ~ 2 ss . /Zst l )

(2.23)

From the above constraints, we find that the tightest 343

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bounds on 2sm Ass and 2,~ are given by (2.22), (2.23) and

Ass < m--'-~h"

(2.24)

Thus we find that if we remain consistent with these constraints, A~sn/Ass has an upper bound to its value, m 2 "_.2..~ A~,As,v h~s Ass <0.08/z4(T) .

(2.25)

Thus, noting that #2 (T)~< #2, we see that a sufficient baryon symmetry can be generated if m h / /Zs> 1.8. So we can consistently generate a large baryon asymmetry if mh ~>#s. This result is quite conservative, since we have not assumed especially large values for Ka 2 or A't/At. As a specific example we will consider a set of values for Ash, Ass and A~ which satisfies all of the above constraints and which generates a large enough baryon asymmetry,

(Ash, Ass, A~) = (0.02, 0.05, 0.7)

(2.26)

((2.25) then requires mh/lls> 1.9, while (2.24) requires mh/MS<4.5). With these values for the couplings we find ns/s= 5 × 10-11. This shows that with quite reasonable values for the new scalar couplings (up to a rather large value for A~) we can generate the BAU without significantly altering the analysis of the EWPT from the case of the standard model. (b) T = 0. So far we have discussed baryogenesis at the electroweak phase transition. Once T<< Tew, the Higgs expectation value will be equal to its present ( T = 0 ) value, hl=v,.~ 175 GeV. In general, if is non-zero at T = 0, there will be significant mixing of the scalars from S, H1 and/-/2. In this case the LEP limits on the Higgs boson mass, and the relationship between the strength of the first-order EWPT and the Higgs boson mass, will be altered from the case of the standard model. (For some recent discussions of this in other models with light scalars and scalar mixing see ref. [ 21 ].) However, a particularly simple and interesting case arises if Ashy2> M~. In this case at T = 0 we have < S> = 0. As a result, there is no mixing of the scalars at T = 0 , only is non-zero and the Higgs mass is given by the standard model result. The case with = 0 is interesting for a number of reasons. When =0, because (2.1) is symmetric under S,--,-S, the lightest scalar from S is cos344

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mologically stable and may account for cold dark matter. For the example (2.26), and assuming for example ps=~mh, we find that < S > = 0 if mh<74 GeV. If we consider a Higgs mass rnh= 65 GeV (which is larger than the LEP lower bound of 57 GeV [ 22 ] ), then the physical S scalar mass is given by ms..~ (AsnVA-#2)l/2"~12 GeV. The resulting cosmological density of S scalars can then be calculated using the method of Lee and Weinberg [23]. This gives Qsh2~ 1.0X 10_11

1

Xfs

( 1 - ~xfs)

(2.27)

(1 -- ½Xfs) "

Here t2s=Ps/pc, where Ps is the mass density of the S scalars and anti-scalars and Pc is the critical density needed to close the Universe. is the thermal average of the product of the SS t annihilation crosssection and the relative velocity, h = 0.5-1 gives the uncertainty in the Hubble constant at present, and xfs-Tf/ms, where Tf~ is the freeze-out temperature of the S scalars, xfs is given by msX~s < O'Vrcl>

= I n ( 7 X 1017

\

(1 - ~X~s) ( 2 ~ ) 3 / 2 1

(2.28)

For a 12 GeV S scalar the main decay mode is the annihilation to b quark pairs via s-channel h ° exchange (fig. 2), which gives

m222n ( , _ m2"~3/~ =2zc(4m2 m2)2 \ - ~ s ] '

(2.29)

where mb is the b quark mass. From this we find x g I ~ 15 and I2sh 2 ~ 1.5 for the example with ms= 12 GeV. Thus we see that it is quite natural for the cold dark matter density due to S scalars to have a value around the critical density in this model. Another possible advantage of having < S > ~ 0 as ~x

"> S/

h° ~.~ bL

Fig. 2. SS "tannihilation to b quark pairs via h ° exchange.

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T--, 0 is that the CP violating phase due to ( S ) , which will introduce large strong CP violation [ 24 ], also goes to zero. This would allow the model to be incorporated into models which seek to eliminate strong CP violation by imposing CP as a symmetry at treelevel, in contrast with dynamical (axion) solutions [ 24 ]. In addition, this also eliminates the cosmological domain wall problem [ 10 ] which would arise in the case where S,--,-S is an exact discrete symmetry, as is true when S is stable. We finally consider the question of the Higgs mass problem of electroweak baryogenesis. In general, as noted above, if ( S ) S 0 at T = 0 , then the relationship between the lightest Higgs boson mass and the strength of the first-order phase transition will be altered because of the mixing between the singlet and doublet scalars [ 21 ]. In addition, this will also alter the LEP lower bound on the Higgs boson mass, which applies only for the case of the pure SU (2) doublet Higgs scalar of the standard model [22 ]. (In general, mixing with a singlet will weaken the lower bound on the Higgs mass [ 21 ]. ) Thus we expect that in general the singlet extension will be able to evade the LEP lower bound via mixing, although it is beyond the scope of the present paper to analyse in detail the case where S, H1 and HE all develop expectation values. However, for the interesting case ( S ) = 0, there is no mixing and so at tree-level the physical Higgs boson eigenstate and its mass are unchanged from the case of the standard model. In this case the model has at least two possible solutions of the Higgs mass problem. One possibility is that finite temperature corrections to g2 due to the additional couplings in (2.1) could cause the effective value of/z 2 at Tew, (2.19a), to be smaller than in the case of the standard model, resulting in a stronger first-order transition for a given physical Higgs mass mh. However, this requires from (2.21) that 2sn>0.1, which in turn requires, from (2.20) and (2.22), that 21z2(Tew)/m 2
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mass and the strength of the first-order phase transition. This possibility has been considered in ref. [6 ], where radiative corrections due to a heavy gauge singlet scalar were considered, and it was shown that if the coupling of the singlet to the Higgs doublet was large enough (corresponding to 2sn>~l in (2.1)), then it would be possible to have a Higgs mass larger than the LEP bound and still have a sufficiently strong first-order phase transition to produce a baryon asymmetry. In the model of interest, we cannot consider such a large value for 2sn if we wish to satisfy the constraint (2.21). However, radiative corrections due to the heavy Higgs doublet//2 can have the same effect. The coupling 212 in (2.1 d) leads to a shift in the upper bound on the Higgs mass coming from the requirement of a sufficiently strong first-order phase transition,

3235(

m~¢(212) =m~o(212 = 0 ) + 3--~n2~,~-~-~j,

(2.30)

where mho(212) is the critical Higgs mass for a given value of 212, which gives the upper bound on the Higgs mass if the phase transition is to be strong enough. The limit on the Higgs mass in the case of the standard model from requiring a sufficiently strong firstorder EWPT is given by mho (212=0) ~ 4 0 GeV [7]. Therefore in order to have mm (212) >t 60 GeV, in order to evade the LEP lower bound (mh> 57 GeV), we require that /

\2/3

(___mH2 ~ 212 > 1.5 \ 2 5 0 GeV, /

•

(2.31)

Although 212 must be large, this at least shows explicitly that it is possible to evade the LEP constraint in the context of the present model. Indeed, this particular solution to the Higgs mass problem should be testable at the next generation of hadron colliders (LHC or SSC).

3. Conclusions We have discussed the possibility of using the phase of a gauge singlet scalar to introduce the CP violation necessary in order to generate the observed baryon asymmetry. In a minimal extension of the standard model it was also necessary to introduce a second 345

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Higgs doublet, not much heavier than 5ran, with Yukawa couplings to the quarks a n d leptons proportional to those of the standard model Higgs in order to avoid F C N C effects. For a particular reasonable set of couplings involving the singlet scalar we found that the observed baryon asymmetry could indeed be generated, a n d that in addition the gauge singlet scalar could naturally account for the cold dark matter in the Universe. Also, the spontaneous CP violation due to the S scalar vanishes as T--, 0, so preventing the i n t r o d u c t i o n of strong CP violation into the standard model from the S scalar sector, as well as avoiding a cosmological d o m a i n wall problem associated with discrete symmetry breaking. Two features of the model are especially noteworthy; (i) the CP violating phase of ( S ) is naturally maximal, being due to spontaneous CP violation, a n d (ii) the same gauge singlet scalar which introduces CP violation to drive electroweak baryogenesis also accounts for cold dark matter. All the features of the model (which introduces only a light gauge singlet scalar a n d a second Higgs doublet of mass mH2 typically less than 500 GeV) should be testable at the next generation of hadron colliders (LHC or SSC).

Acknowledgement This research was supported by the G r u p o Teorico de Altas Energias, Portugal.

References [ 1] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov,Phys. Lett. B 155 (1985) 36; M.A. Shaposhnikov,Nucl. Phys. B 287 ( 1987) 757. [2] P. Arnold and L. McLerran, Phys. Rev. D 36 (1987) 581. [3] A.G. Cohen, D.B. Kaplan and A.E. Nelson, Phys. Lett. B 236 (1991) 86.

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