CP-violation and baryogenesis in the low energy minimal supersymmetric standard model

16 October 1997 PHYSICS

BLSEVIER

LETTERS

B

Physics Letters B 411(1997) 301-305

CP-violation and baryogenesis in the low energy minimal supersymmetric standard model Tuomas Multam’ti ‘, Iiro Vilja 2 Department of Physics, University of Turku, FIN-20500 Turku. Finland

Received 5 June 1997; revised 22 July 1997 Editor: P.V. Landshoff

Abstract In the context of the minimal supersymmetric extension of the Standard Model the effect of a realistic wall profile is studied. It has been recently shown that in the presence of light stops the electroweak scale phase transition can be strong enough for baryogenesis. In the presence of non-trivial CP-violating phases of left-handed mixing terms and Higgsino mass, the largest n,/s is created when Higgsino and gaugino mass parameters are degenerate, p= M2. In the present paper we show that realistic wall profiles suppress the generated baryon number of the universe, so that quite a stringent bound lsin&,( 2 0.2 for p-phase & can be inferred. 0 1997 Elsevier Science B.V.

Standard Model (MSSM) has appeared to be one of the most promising candidates to explain the observed baryon asymmetry of the universe nB/s - 10-i’ generated at electroweak scale [l]. Although all requirements are already included in the Standard Model [1,2], the phase transition [3] has appeared to be too weakly first order to preserve the generated baryon asymmetry [4]. Also it has been shown that the CP-violation needed for baryogenesis is too small in the Standard Model [5]. Therefore new physics besides the Standard Model is necessarily needed, provided that the baryon asymmetry is generated during the electroweak phase transition. Because MSSM is one of the most appealing extensions of the Standard Model, it has been worthThe

Minimal

Supersymmetric

’ E-mail: [email protected]. * E-mail: [email protected].

while to study whether it is possible to generate and preserve the baryon asymmetry in it. Indeed, recent analyses shows that there exists a region of the parameter space where the phase transition is strong enough [6,7]. It is required that tan p < 3, the lightest stop is lighter than top quark and the lightest Higss must be detectable by LEP2. The conditions given above may,however, relax due to two and higher -loop effects, which seem to strengthen the phase transition [8,9]. Unlike the Standard Model where the source of CP-violation is solely the CabibboKobayashi-Maskawa matrix, MSSM contains an additional source due to the soft supersymmetry breaking parameters which are related to stop mixing angle. In recent papers Carena et al. [6,10] has analysed the region of supersymmetric parameter space where at the electroweak scale generated baryon asymmetry is consistent with the observed one. The generation and survival of large enough na/s = 4 X IO-” seem

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T. Multamiiki, I. Viua /Physics Letters B 41 I (1997) Sol-30s

302

to require that the lightest stop mass mj is restricted by Z-boson and top masses, Mz < mi < m,. Moreover, the mass of the lightest Higgs boson is bounded by mH < 80 GeV whereas CP-odd boson has mass mA > 150 GeV [6]. The other analysis [ 101 gave dependence of n,/s on Higgsino mass and soft supersymmetry breaking parameters I pl and j A,] and their phases (6, and 4*, respectively, as well as gaugino mass parameters M, and M2. The optimal choice of parameters showed up to be I pi= M,, [sin&,) 2 0.06, and the dependence on M, is weak, so that it can be chosen to be equal to M,. In the latter analysis of Carena et al. it was chosen the left-handed stop mass parameter mQ to be mp = 500 GeV, effective soft supersymmetry breaking parameter i, = A, - ~/tan @= 0 and right-handed stop mass parameter m, = -Kiu < 0 Eiu
2 2 m,u,g3 -

2

'14

( 1 12

to obtain the most optimistic bounds (i.e. maximize 3. With these parameter values the CP-violatn&l ing source is generated essentially by Higgsino and gaugino currents and the right-handed stop contribution is negligible, so that without loss of generality one can set sin( $ + 4AA> = 0. The bound (1) is due to colour non-breaking condition, i.e no colour breaking minimum must not be deeper than the normal electroweak breaking (and colour conserving) minimum. It is defined at zero-temperature, thus ug = 246.22 GeV. Also it was used u, = 0.1 and L, = 25/T. With these conditions large enough baryon asymmetry had been able to create. In the paper [lo] the baryon asymmetry was inferred by calculating first the CP-violating sources and then solving the relevant Boltzmann equations. It was shown that the baryon to entropy ration reads nB -= S

95r,, -g(ki)---, u;s

(2)

where g(ki) is a numerical coefficient depending on the degrees of freedom, 5 the effective diffusion rate, r,, = 6~azT (K = 1) [ll] the weak sphaleron

rate, u, the wall velocity. The entropy density s is given by s=

2n’g,

,vT3

(3) 45 * where g, s is the effective number of relativistic degrees of freedom. The coefficient _& was shown to be a certain integral over the source y(u) = U,,f(ki>a,Jo(U>:

(4) where A, = (u, + \i U: + 4p’o j/(2 Ej> and the wall was defined so, that it begins at u = 0. Here u denotes the co-moving coordinate u = z + u,r supposing that the wall moves in the direction of z-axis. (f(ki) is again a coefficient depending on the number of degrees of freedom present in thermal path and related to the definition of the effective source [ 10,121.) Thus the coefficient ti is dependent on the actual wall shape via

tan/3 = HI/H2 and Hi’s are the real parts of the neutral components of the Higgs doublets. In [IO] the wall shape was, however, taken ad hoc by making a physically acceptable profile. It was assumed to have a simple sinusoidal form where the field H(u) can be given by

+utqu-L,)

and the angle p(u) by

+A@+-L,):

is given by A/3 = p(T,) where A/3 arctan(m,(T,>/m,(T,>), calculated at the temperature where curvature of one-loop effective potential at origin vanishes. (It is thus the angle between the flat direction and vacuum direction.) Inserting these Anzgtse to Eq. (5), we obtain corresponding contribution to the CP-violation

(8) 3 About restrictions

and validity of these results, see [IO].

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T. Muliamiiki, I. Virja / Physics L.&ten B 411 (1997) 301-305

Using this approximation, the results of [lo] was inferred. In the present paper a more realistic prescription of wall shape used. Working at the critical temperature T, and using the one-loop resummed effective potential [ 131 we find numerically the path of smallest gradient y, from ( H, , H2 ) = (0,O) to CH,, Hz) = (v,,v,) = (v,(T,),v,(T,)), which well approximates the true solution. Moreover, using path y,, a upper bound for I is necessarily obtained. The true solution lies necessarily between y, and straight line from (0,O) to (v, ,v,) (which leads to ntg = 0) as can be concluded by studying the Lagrangean 4. Thus 111 along such path is necessarily smaller than along y,. The effective potential for MSSM at finite temperature can be expressed in three parts [13]

and temperature corrected field dependent masses, respectively, n, are the degrees of freedom of each particle (including - -sign for fermions) and

Ji( x2) = (dyy’ln(l

_+ET-~).

( 14)

Note, that we have neglected the b-quark, Ib-squark as wall as other generation contributions as small ones. Nor do the heavy supersymmetric particles contribute. The mass parameters m,, m2, rn,*, are related to p, mA, mH and other parameters of the theory, as given in [ 131. Using these formulas, we can solve y, and define the corresponding CP-violation integral

verf( (PJ) = %( 4) + V,( 4) + Vi.r( 61 + AVr(+),

(9)

where V,, is the tree level zero-temperature potential, V, the renormalized l-loop zero-temperature potential, V,,, the l-loop finite temperature potential and AV, the daisy-resummed part. They are given by

(10)

V&f)

=f

(12)

and AV,( H) = -&~n,[m~(HjT-m~(H)], 1 (13)

where mi( H) and m,( H,T) are the zero temperature

4 This is true providing that the field is not strongly oscillating within the wall, but is a smooth configuration.

at the critical temperature along the path y, . It shows up also that the form of profiles are with good acccuracy the form of kink. Indeed, if we reparametrise the field (H, , H,) to component pointing towards (u,, u,>, H,, and component orthogonal to that, HI , appears that the ratio of maximum value of HI to u = d2v, + u2 is in any case smaller that 0.004. Thus the bending of the path, i.e. the deviation from straight line is small. (Note, that here v is the value of vacuum at T,, thus not equal to v,, = 247 GeV.) The ratio Z,,/I, gives immediately the supression of n,/s with respect the results of Carena et al. [lo]. Hence the values needed for soft supersymmetry mixing phase sin$ is increased by factor 1,/i,. In the Fig. 1. we have presented the value of integral Z, as a function of p for several values of mA. From the figure it can be read out that increasing p decreases the (absolute) value of Z, so that for /J > 250 GeV holds IZYl< 1. Also if can be found that increasing mA with factor r decreases IYroughly by factor l/r within parameter range studied. This behaviour is likely be more general that just restricted to analysis of [lo], because the amount of CP-violation is in general proportional to the change

T. Multamiiki, I. Vilja / Physics Letters B 41 I (1997) 301-305

304

El

m,=lSOGcV

In,=175 Gev - - - m,=ZOOGeV -m=225GeV --.’

Fig. I. Values of the integral I,= {z(H’@‘Ye- *+’ (where comma stands for u -derivative) p=M,=MZ with u,=O.l, m,=5OOGeV, fi,=Ap, tanp=2and L,=O.

as function of soft supersummetry

parameter

r

o.ol 0.0

200.0

100.0

p/GeV Fig. 2. Values of the ratio 1,/l, width L,. = 25/T is used.

as function of /.L= M, = M2 with u, = 0.1, me = 500 GeV, 6,

= rT’z’$‘,tanp = 2 and i, = 0. For I, wall

T. Multamiiki, 1. Vilja /Physics

of p on the path from origin to the non-trivial vacuum. In the Fig. 2. a comparison to the result of Carena et al. [lo] is made. We have plotted the ratio Z,/Zs also as function of p with several values of mA. There apears a clear tendency of additional suppression: for very small CL, 1,/Z, = 0.4, whereas for I_L2: 250 GeV, Z,/Z, = 0.1. For optimal values of /.L, 150 GeV I p I 250 GeV the supression factor is at least 0.3. It appers that the dependence of this supression factor on CP-odd boson mass mA is relatively weak. Taking in the account that according to analysis of [lo] it is required that (sin&] 2 0.06, this is now converted to more strignent bound jsin$lk 0.2 which remarkably weakens the possibility of baryogenesis in MSSM. However, for /.L= 250 GeV the bound rises to Isin&\ 2 0.6 which possibly is already too large. In the present paper we have calculated the amount of CP-violation in the bubble wall of minimal supersymmetric extension of the Standard Model at electroweak scale phase transition. It has been shown that the previous estimates tend to be too optimistic: an extra suppression of (at least) 0.3 is present. This tends to make the electroweak baryogenesis in MSSM more difficult and less likely. However, more analysis on the model is needed, in particular to make clear how higher corrections (more scalar insertions) to the CP-violating source behave. If the higher corrections to the source expands in the powers of (H,dH,- H,aH,)no help from them are expected. If they, however, order by some other expansion parameter, their contribution to the source may be remarkably large. Unfortunately this may also lead to the situation where non-perturbative effects are important. Moreover, the effect of higher order corrected effective potential remains to be studied, because the two-loop corrections tend to strenghten the phase transition and thus relaxes the bounds [8,9]. The two-loop contributions may, however, affect directly the value of the integral I needed in the calculation of CP-violating source, too. On the other hand, also the sphaleron rate r,, is under discussion [14] and changes on that may change significantly the conclusions made about baryogenesis. The authors thank A. Riotto for discussions and I. Sirkka for technical advice.

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