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Baryogenesis at the electroweak scale
I g LllIq IIf_*N"i'.l| f6"! [Qbl "t
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 37A (1994) 117-126 North-Holland
Baryogenesis at the electroweak scale M. Shaposhnikov a" aTheory Division, CERN, CH-1211 Geneva 23, Switzerland
1. I n t r o d u c t i o n
Our universe is charge-asymmetric. The baryonic asymmetry of the universe (BAU) is characterized by the baryon number to the entropy ratio, n__~B= (4 - 10) × 10-11.
(1)
8
The existence of the baryonic asymmetry poses a number of related questions. (i) We would like to know why there is an asymmetry. (ii) We would like to explain its magnitude. (iii) Last, we would like to understand why we have more baryons than antibaryons (sign of the asymmetry). A possible answer to the first question has been known already for about 25 years [1]. Namely, our universe is charge-asymmetric because (i) the baryonic number is not conserved, (ii) the universe is expanding and was very hot in the past, (iii) C and CP parities are not good symmetries of nature. Clearly, the answers to the second and third questions require detailed knowledge of the underlying physics associated with B- and CP-nonconservation. At present, we know two possible sources for B-violation. The first one is connected with Grand Unified Theories (GUTs) of strong, weak and electromagnetic interactions. Here Bnon-conservation is due to the heavy particle (leptoquark) exchange. The mechanism for baryogenesis in GUTs is associated with leptoquark decays at the very early stages of the universe evolution, at temperatures close to the unification "On leave of absence from Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312,
scale (1016 GeV) [1,2]. GUTs provide a plausible explanation of the baryonic asymmetry, but cannot predict its sign and magnitude just because we do not know the details of particle interactions at the unification scale. The other problem with GUT baryogenesis arises in the inflationary universe scenario (the discussion of this point can be found, e.g. in the review [3]). In many models the temperature of the universe after inflation is smaller than the unification scale. This makes the realization of the standard leptoquark decay mechanism impossible (there are no leptoquarks at this stage to decay and produce the baryonic asymmetry]). At the same time, a number of different mechanisms in the framework of GUTs were suggested, which can overcome this difficulty [4]. The second source of B-violation is associated with non-perturbative physics and exists already in the standard electroweak theory [5]. This theory may thus be a proper candidate for an explanation of the baryonic asymmetry of the universe as well [6]. Contrary to the GUT case, we know the parameters of the theory quite well (particle masses and coupling constants). Therefore, one can hope to predict the magnitude and the sign of the baryonic asymmetry, or to predict or constrain the masses of the Higgs boson and top quark, provided the asymmetry depends on them. The firm conclusion that the standard model cannot explain the baryonic asymmetry is also of a great importance, since in this case the existence of BAU will give a clear observational indication on the physics beyond the standard model. In this talk we review some results that have been derived recently in a study of baryogenesis in the minimal standard model [7,8].
Russia.
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00593-K
118
M. Shaposhnikov/Baryogenesis at the electroweakscale
2. B-vlolation at t h e e l e c t r o w e a k scale In spite of the fact that the baryonic number is a good symmetry of the electroweak theory on the perturbative level, it is not conserved due to the anomaly in the fermionic current and complicated vacuum structure of non-abelian gauge theories (for reviews, see e.g. [9,10]). Under usual conditions, the rate of fermionic number non-conservation is suppressed by the semiclassical exponent [5] e x p ( - ~-' 2,r~w)" At high temperatures the rate of B-violation is highly enhanced [6] and proportional to the Boltzmann exponent e x p ( - M s p h ( T ) / T ) . Here Msph(T) is the effective sphaleron [11] mass at high temperatures. At sufficiently small temperatures, T < Te, Msph(T) ~- 3 M w ( T ) / a w , with M w ( T ) being the temperature-dependent W mass. Here Te is the critical temperature of SU(2)xU(1) symmetry restoration [12]. At T > Te, scaling arguments [13,14] show that M, ph(T) -- ~T, where ~¢is some unknown constant. So, one can write for T > Tc and T < Tc correspondingly:
r(T) =
A u ( ~ w T ) 4,
r ( T ) = & \ 4~- ]
• exp
(M,,h(T)) T
\
~
(2)
'
where A, is some factor that cannot be found by semiclassical methods. Real-time numerical simulations [15] give a lower bound A > 0.4, but the values as high as, say, ~ = 20 cannot be excluded at present. For the broken phase Ab ~ NtrNrot, where factors Ntr ~- 26, Nrot ~- 5.3 x 103 are due to the zero-modes normalizations [13]. At T > To, the rate (2) of the anomalous reactions greatly exceeds the rate of the universe expansion, rt71 ~ T2/Mpt, where Mt,t ",, 1019 GeV is the Plank mass. Therefore, anomalous reactions violating the baryon number are in thermal equilibrium untill the moment of the electroweak phase transition. After the phase transition, the vacuum expectation value of the Higgs field is non-zero and the baryon-number violation rate decreases due to the Boltzmann exponential suppression (2). Then, from eq. (2) it follows that
the freezing temperature Ts ~ 100 GeV of the sphaleron processes in the broken phase can be found from equation M, ph (TB) _ 45-~ log(Ppt/TB).
(3)
TB
The equilibrium character of the anomalous fermionic number non-conservation at sufficiently small temperatures influences the primordial baryonic asymmetry, which may be created by GUT interaction. In some cases the primordial asymmetry is washed out by electroweak processes. The analysis of the influence of the anomalous B-non-conserving reactions on the primordial baryonic asymmetry can be found in [16]. 3.
T h e r m a l n o n - e q u i l i b r i u m and t h e Higgs m a s s
bound
on
The baryonic asymmetry of the universe may be generated, provided there are sufficiently strong deviations from thermal equilibrium. It is easy to understand that the departure from the thermal equilibrium coming just from the universe expansion is not enough [6]. Imagine that the electroweak phase transition is of the second order (or weakly first order), so that the temperature dependence of the vacuum expectation value of the Higgs field is continuous:
In this case all deviations from thermal equilibrium come from the universe expansion. Now, the baryonic asymmetry cannot be generated at T > Tn since B-non-conserving reactions are in thermal equilibrium• Therefore, BAU is at most nB/s
~
r(Tn) 6c e
~
TB
~
$cp × 10-18,(5)
which is small compared with observations independently on the magnitude of the CP-violating factor ~cP < 1. The only place for efficient thermodynamical non-equilibrium at the late stages of the universe expansion (T ~ 100 GeV) is the first-order phase transition with the breaking of the SU(2)xU(1) group [17] (see Fig. 1).
M. Shaposhnikov /Baryogenesis at the electroweak scale
0.5
v(¢)
j~
0.2!
-0.25
-0.5
-0.75
-1
'
'
'
¢o
i;o'
'
'
I;o
¢'
'~;c
Figure 1. Typical temperature behaviour of the effective potential for the first-order phase transition.
At T > T+ the effective potential for the Higgs field has just one extremum corresponding to the phase with unbroken symmetry. In the temperature interval Tc < T < T+ the minimum corresponding to the broken phase is metastable, while at T_ < T < Tc the minimum corresponding to the unbroken phase is metastable. So, at T > Tc the symmetry is restored, and the vacuum expectation value of the Higgs field is equal to zero. At T = Tc the free energy of the broken phase is equal to the free energy of the restored phase. However, the phase transition does not occur at this time just because the probability of the tunnelling through the barrier is too small. So the universe waits till the potential barrier separating the broken and unbroken phases is sufficiently small. The temperature T* at which this happens lies somewhere between the critical temperature T¢ and temperature To of absolute instability of the restored phase. The phase transition proceeds via the nucleation of the bubbles of the true phase, which expand with the velocity of the
119
order of the speed of light and fill out the universe. The temperature T" is defined from condition rt~ ,-~ 1, where r is the bubble nucleation rate per unit time and unit volume [17]. The typical size of the bubbles at the end of the phase transition is of the order of 10-4tu ,~ 0.1 cm. It is the motion of the bubble walls through the hot plasma that introduces the deviations from the thermal equilibrium. A constraint on the particle spectrum of the theory can be obtained from Sakharov's out of equilibrium condition [18,19]. Suppose the baryonic excess Ain was generated by some electroweak mechanism at the phase transition that breaks SU(2) xU(1). The source of the baryon excess is not important here in essence. BAU at T* must survive till the present time. The necessary condition for that is freezing out the baryon number non-conserving processes in the phase with broken SU(2) xU(1) at the critical point. Namely, one has TB < T*, or, in particle physics terms
M, nh(T°) T"
> 45.
(6)
The ratio of the sphaleron mass to the temperature at T = T* depends on the Higgs boson mass. In fact, condition (6) implies an upper limit on the Higgs mass. In order to understand why we have an upper bound let us take for simplicity the minimal standard model effective potential in the one-loop approximation at T = T_ 2:
V(¢)T=T_ __ )~¢4
Tg~vCa(2 + 32~r
1 cos30-----~). (7)
This expression is derived using a high temperature approximation and assumes that A << g 2. After the phase transition, the condensate of the field ¢ is non-zero and equals
¢ =
3To 93(2 + - - ) , 1
cos30w
(8)
from which we obtain the upper limit on the Higgs boson mass M~ < Merit -~ 45 GeV. The critical 2The moment of the electroweak phase transition is close to the moment of the absolute instability of the phase with ~b = 0 [17] because t h e t u n n e l l i n g transitions to ~ ~ 0 are suppressed. Using Tc instead of 7"- makes the bound on the Higgs mass somewhat stronger [20].
120
M. Shaposhnikov/Baryogenesis at the electroweakscale
value in this approximation practically does not depend on the value of the top quark mass [21]. This simple consideration, however, is spoiled when one goes beyond one-loop computations. As is well known, perturbation theory in gauge theories at high temperature is intrinsically sick due to the existence of power-like infrared divergences [22]. In particular, the gST4 correction to the free energy, originating first on the 4-loop level, is not computable in principle by perturbative methods. These divergences make the perturbative effective potential divergent at small ¢. The logarithmic divergence appears at the 4-loop level, and at higher loops one gets power singularities. At nonzero ¢, the infrared problem is formally absent, since integrals at small momenta are cut by nonzero gauge boson masses. So, one expects that at sufficiently large ¢ the perturbation theory is applicable. Therefore, perturbation theory seems to be quite suitable for the answer to the following question: 'What is the value of the effective potential at a given temperature and sufficiently large ¢?' In particular, the expectation value of ¢ can be determined with the help of the perturbative effective potential, provided ¢t is large enough. However, the determination of the critical temperature of the phase transition, as well as that of the bubble nucleation temperature T" in cosmology is, strictly speaking, beyond the scope of perturbation theory. Indeed, to find the critical temperature, one must know not only the value of the potential in the broken minimum, but the value of the potential in the phase with ¢ = 0. While the first quantity can be computed in perturbation theory, the second cannot. The same remark applies also to the bubble nucleation temperature T*, since the computation of the rate of bubble nucleation requires the knowledge of the effective action for small enough fields ¢. The determination of these temperatures is impossible in the perturbation theory without additional assumptions on the behaviour of the potential near the origin. In perturbation theory the assumption is: higher-order loop corrections as well as all nonperturbative effects are not important near ~b= 0, i.e. for the determination of Tc and T*. With it the following bounds on the Higgs mass can be
derived in the different approximations: (i) One-loop effective potential for the scalar field and 'tree' sphaleron rate T 4exp(-Msph(T)/T): Merit = 65 GeV [18]. (ii) One-loop effective potential for the scalar field and one-loop sphaleron rate (2): Merit = 45 GeV
[19,21] (iii) One-loop effective potential field with Debye screening taken Merit = 35 GeV [23]. (iv) Two-loop effective potential field with Debye screening taken Merit ~ 40 GeV [24].
for the scalar into account: for the scalar into account:
This assumption, however, is most probably wrong. The first indication comes from the lattice simulations of the 3-dimensional Euclidean pure SU(2) gauge theory, which is a high-temperature limit of the 4-dimensional one. This theory is confining, namely Wilson loops exhibit an area law behaviour (in 2+1 dimensions, this would be the ordinary notion of confinement, so that particle excitations are bound states of SU(2) gluons) 3. The masses of glueballs in 2+1 (or static correlation lengths in our Euclidean theory) were measured on the lattice and appear to be quite large M ,.. 2g2T [25] in comparison with a natural perturbation expansion parameter g2T/2r (2~r's in perturbation theory comes from the integration over momenta). The perturbative computation of the effective potential near the origin uses massless degrees of freedom, while in fact they are massive and unexpectedly heavy. It has been argued in [8] that non-perturbative effects substantially modify the effective potential near the origin. In that paper, a possible model for incorporating some non-perturbative physics to the game was considered. In order to get a total potential incorporating non-perturbative effects, one should add to the perturbative potential the term
Vno,p(O, T) = - A ~ = c T ( g l [ ) 3 f ( g - ~ )
,
(9)
3The confining property of SU(2) in 3d makes the reliable estimate of the critical temperature of the elect roweak phase transition as difficult as the computation of, say, the glueball mass from the first principles of QCD.
121
M. Shaposhnikov /Baryogenesis at the electroweak scale
where f(0) = 1, and A~c > 0 is some unknown number. The function f(z) is expected to be exponentially small at large z. Some guess for this function can be derived by computing the contribution of a certain set of non-perturbative calculations to the functional integral: f
= F
2~zvg~(z),
(10)
where K is a modified Bessel function and -
3~2¢ gwT'
(11)
while a rough estimate of A~ac gives A~ac ~ 0.4. A contribution of this type does not contradict any perturbative computation of the effectivepotential. So it is expected that non-perturbative effects dig a pit near the origin, therefore reducing the value of the critical temperature and T*. The lattice simulations of the electroweak phase transition [8] reveal the systematic decrease of the critical temperature with respest to two-loop perturbative computations, thus indicating that nonperturbative effects are really important and that they indeed tend to strengthen the first-order nature of the phase transition. Non-perturbative effects may considerably change the perturbative picture of the electroweak phase transition. For the effective potential containing contributions (9,10) the temperature T" is noticeably smaller than the critical temperature To, so that the vev of the Higgs field at T = T* is considerably larger than expected from perturbative computations4. The estimate of the critical Higgs moves up to a value about 80-100 GeV. At the same time, the bubble wall thickness (Higgs correlation lengths) is about 1-3 T" and is substantially smaller than one would expect from perturbative computations5. It is interesting to note that an upper bound on the Higgs mass implies also an upper bound on 4On the contrary, in refs.
[24,26] it was assumed that
T" =Tc. 5We stress that these numbers axe based on the computations in the framework of the model (9) for nonperturbative effects. The partial test of this model is possible with 3d lattice simulations of the phase transition
[8].
the top-quark mass. From the vacuum stability argument, it follows [27] that m t < 140GeV +
MH -- 75GeV 1.64
(12)
If, say, Merit = 90 GeV, then Mt < 150GeV. One can see, therefore, that cosmology in the combination with particle physics may imply a number of non-trivial bounds on the particle spectrum of the electroweak theory. As one can see, the uncertainties in computations of the critical Higgs mass allow no conclusion as to whether MSM baryogenesis is ruled out or not by direct searches of the Higgs boson at LEP. However, all estimates indicate that the Higgs boson should be quite light in order to make the electroweak baryogenesis possible. If the baryonic asymmetry of the universe is indeed due to MSM interactions, then the Higgs boson may be found on LEP1 or LEP2. Much more theoretical work is needed in order to refine these estimates. 4.
Baryonic standard
asymmetry model
in
the
minimal
It is easy to understand that electroweak baryogenesis has to be associated with the walls of the expanding bubbles of a new (broken) phase. Indeed, outside the bubbles the rate of the fermionic number non-conservation greatly exceeds the rate of the universe expansion, so that sphaleron processes are in thermal equilibrium. On the contrary, inside the bubbles, sphaleron processes are switched off and the fermionic number is effectively conserved. Two types of the mechanisms for electroweak baryogenesis have been considered in the literature. In the first case [28-32], non-trivial configurations of the gauge and/or Higgs fields (sphalerons or thermal fluctuations) interact with the moving domain wall in a CP-violating way. Then, net fermionic number is produced when these field configurations decay inside the bubble wall. In the second class of mechanisms [33], CPnon-invariant interactions between fermions and the bubble wall lead to a separation of some CPodd charge by the bubble wall, w h i c h is then converted to an asymmetry in the baryonic number
122
M. Shaposhnikov /Baryogenesis at the electroweak scale
by sphaleron processes in the unbroken phase. We shall discuss here in some detail the implementation of the last type of mechanism for the minimal standard model, following closely our work with Glennys Farrar [7] (see also [34]). Clearly, the structure of CP-violation is essential. In the minimal standard model CP violation comes from the quark mass-mixing matrix. It can be rotated away provided (i) there is a degeneracy in the up-quark sector, i.e. mu = rne or rnu = me or m t = me, (ii) or there is a degeneracy in the down-quark sector, i.e. md = ms or m g = mb or mb = m , , (iii) or some of the mixing angles between different generations is zero, (iv)or the CP-violating phase is zero. In other words, CP violation vanishes together with the product (known as the Jarlskog determinant [35]) dee = sin(012) sin(023) sin(Ols) sin ~cp -
--
mu)(m
-
-
md)(m.
(13) -
The structure of CP violation makes baryogenesis in the minimal standard model a very nontrivial problem. Indeed, the electroweak phase transition, where strong deviations from thermal equilibrium are expected, happens at rather large temperatures .~ 100 GeV. It seems, therefore, that quark masses (maybe with the exception of the t-quark) can be treated as a perturbation, so that a dimensionless measure of CP violation is just Jcp "" d c p / T 12 "~ 10-20 [19,28]. Of course, this number is too small to account for the observed asymmetry. This argument relies heavily on the applicability of perturbation theory on quark masses (or, which is the same, on Yukawa coupling constants) and uses an a s s u m p t i o n that the typical energy scale essential for the estimate of the asymmetry is the temperature of the phase transition. Imagine now that the relevant scale is m, rather than T: then GIM cancellation does not occur, and what is left from d c p is just the product of mixing angles and CP-violating phase ~ 10 -5. The example of the breaking of perturbation theory is provided by the quark scattering on the bub-
ble walls. Outside the bubble, the electroweak symmetry is unbroken, and the Higgs contribution to the quark mass is zero. Inside the bubble, the vev of the Higgs field is non-zero and quarks are massive. To simplify the discussion, let us forget for a moment the fact that quark propagation through the hot medium is different from that in vacuum. Then, if the energy of the quark in the unbroken phase is smaller than its mass in the broken phase, it will be reflected from the domain wall with unit probability independently on the value of its mass (Yukawa coupling). This is precisely the place where the 'no-go' argument may be wrong. The perturbation theory does not work only in a small fraction of the phase space determined by the quark masses, but the loss in the phase-space factor may be much smaller than the gain in CP-violating amplitude. With these remarks let us see how CP violation can reveal itself in a quark scattering on the bubble walls. Due to CP violation, the reflection amplitude for a quark coming from the unbroken phase is not equal to the reflection amplitude for an antiquark. Hence, there is a net flux of baryon number through the wall and an equal flux of antiquarks reflected from the wall. Now, the fate of the fermions inside and outside the bubble is different. Baryons inside the bubble survive till the present time because the rate of B-violating reactions is highly suppressed in the broken phase, while the antifermions outside the bubble disappear through equilibrium anomalous reactions. The final baryonic asymmetry is just proportional to the difference of reflection coefficients from the domain wall of quarks and antiquarks. The domain wall acts like a separator for baryon number, filling the bubble with fermions and outer space with antifermions, while anomalous reactions destroy the antifermion excess in the unbroken phase. Even an order of magnitude estimate of the baryonic asymmetry in this particular mechanism is a very difficult task, due to a poor knowledge of many high-temperature effects [7]. A computation in the one-loop approximation for the thermal fermionic self-energies has been performed in [7]. Schematically, the baryonic asymmetry can
M. Shaposhnikov /Baryogenesis at the electroweak scale
123
be expressed as '
ns
s
1
-
geII'6,~icro (kinetic factors). x
I
.
.
.
.
I
.
.
.
.
I
.
.
.
.
I
(14) exlO-e
.t
where N~11 ... 100 is the effective number of massless degrees of freedom. The quantity ~micro is given by particle physics:
~miero f dwT(w)A(w), =
(15)
4x 10.4
?~.
q
;~xlO-e
where w is the energy of a quark, 7(w) is a phasespace factor, = E[Iri#l 2 are the reflection coefficients for a process with a quark (antiquark) of the i-th generation incident in the unbroken phase and the reflected quark (antiquark) of the j-th generation. The last factor in (14) takes into account the statistical physics effects such as the magnitude of the deviation from thermal equilibrium, the finiteness of the rate of fermionic number non-conservation in the unbroken phase, etc. An essential part of the analysis is the use of quasi-particle formalism. In the medium, particle excitations are different from bare quarks. In particular, there is non-trivial mixing between quarks of different generations in the broken as well as unbroken phases. Moreover, the quarkmixing matrices are different in different phases, and this difference is not suppressed by Yukawa coupling constants (as happens for bare zerotemperature quarks). The combination of the phenomenon of the total reflection with this fact enhances the asymmetry a lot, in comparison with (13). As is expected from the general discussion, the differential asymmetry A(w) was found to be substantially different from zero only in a small region of phase space, namely when a strange quark is reflected from the domain wall with probability of order 1. Everywhere else the asymmetry is very small due to GIM cancellation. The typical picture for A(w) for quark motion perpendicular to the domain wall is shown in Fig. 2, for experimentally favorable values of mixing angles and mt = 1 5 0 G e V 6 6A p e r t u r b a t i v e a n a l y t i c c o m p u t a t i o n of t h e a s y m m e t r y in a thin-wall limit c a n be f o u n d in [7].
k
ii
,
I
47.9
. . . .
I
48
. . . .
I
48.1
,~(a,v)
. . . .
'
I
,
I
I
48.2
fA A--,-a,a~o4w-.oe~,eV
Figure 2. The asymmetry A in the reflection probabilities of right-chiral quarks and antiquarks incident from the unbroken phase.
The upper pair of peaks occupies the energy range in which the strange quark is totally reflected. The 'notch' in the middle is the region in which the down quark is also totally reflected and GIM cancellation is perfect. The unfamiliar feature that total reflection occurs for a range of energies, rather than for all energies less than some value, results from the unusual properties of the quasi-particle dispersion relation. At lower energy there is another region of a different character, involving transitions between d or s and b quarks. The sign of the predicted asymmetry coincides with observations provided sin(Sop) > 0. Unfortunately, uncertainties in the kinetic factors and in the phase-space volume (coming from the unknown velocity of the domain wall, sphaleron rate in unbroken phase, domain-wall thickness, non-coherent quark scattering, dynamics of the domain-wall propagation through the plasma, non-perturbative effects, etc.) do not allow us
124
M. Shaposhnikov/Baryogenesis at the electroweak scale
to conclude convincingly whether the minimal standard model can be responsible for the baryonic asymmetry of the universe. Combining all factors together, we get [7] ~$ ,~ 10 -1° -10 -lz. A much better understanding of the high-temperature phenomena is required in order to settle this question z. 5. Conclusion While the resolutionof the question of whether the minimal standard model can or cannot produce a sufficientamount of baryons requires the clarificationof a number of very hard dynamical questions, the situation with electroweak baryogenesis in extended versions of the theory is not so difficult.First of all,the extension of the Higgs sector allows us to escape a strong upper bound on the Higgs mass. This happens because in the models with many Higgs fieldsthere is much more freedom, and eq. (6) is now a constraint on some complicated combination of Higgs masses together with coupling constants, rather than on the Higgs mass alone as in the M S M [37]. Moreover, the experimental limit on the Higgs mass is weaker in the extensions of the standard model. The situationwith C P violationis also much simpler: extended versions usually contain extra CPbreaking phases. A great deal of work has been put in the analysisof the baryogenesis in extended versions of the electroweak theory (see [3,38]for a review with references to originalpapers). The general conclusion is quite optimistic- B A U can be a result of anomalous electroweak numberviolating processes. Unfortunately, the direct experimentalcheck of the electroweak character of the baryonic asymmetry is most probably impossible (a gedanken experiment would be the measurement of the leptonic asymmetry L of the universe and checking that B - L = 0). The experimental discovery of any type of new physics below, say, 1 TeV (be"tina recent p r e p r i n t , Gavela et al. [36] a r g u e t h a t d u e to t h e loss of q u a n t u m c o h e r e n c e in the scattering of q u a r k s w i t h g l u o n s , t h e a c t u a l n u m b e r is m u c h smaller. As far a~ we c a n see f r o m their paper they worked in a nonc o n s i s t e n t approximation, throwing away a number of imp o r t a n t contributions to the m s y m m e t r y , so that their conclusions a r e n o t justified by the work r e p o r t e d in [36].
sides t-quark and Higgs boson) would mean that the electroweak origin of BAU is very probable. From my point of view, the most important theoretical problem is to understand the relevance of the minimal standard model to baryogenesis. The resolution of this problem will either provide strong evidence on the existence of new physics beyond the standard model, or give constraints on the masses of t-quark or Higgs boson, the only missing blocks of the standard model. REFERENCES
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M. Shaposhnikov/Baryogenesis at the electroweak scale
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125
25. A. Irb~k and C. Peterson, Phys. Left. 174B (1986) 99. G. Koutsoumbas, K. Farakos and S. Sarantakos, Phys. Lett. 189B (1986) 173. M. Teper, Phys. Lett. B289 (1992) 115. 26. P. Arnold and L. Yaffe, Preprint UW-PT-9324, (1993) 27. M. Sher, Preprint WM-93-108, (1993). 28. M.E. Shaposhnikov, Nucl. Phys. B299 (1988) 797. 29. L. McLerran, Phys. Rev. Lett. 62 (1989) 1075. 30. N. Turok and J. Zadrozny, Phys. Rev. Lett. 65 (1990) 2331; Nucl. Phys. B358 (1991) 471. 31. L. McLerran, M.E. Shaposhnikov, N. Turok, and M.B. Voloshin, Phys. Lett. 256B (1991) 451. 32. M. Dine, P. Huet, Ft. Singleton, and L. Susskind, Phys. Lett. 257B (1991) 351. 33. A. Cohen, D. Kaplan and A. Nelson, Phys. Lett. B263 (1991) 86; Nucl. Phys. B373 (1992) 453. 34. M. Shaposhnikov, Phys. Lett. B277 (1992) 324; B282 (1992) 483(E). 35. C. Jarlskog, Phys. Ftev. Lett. 55 (1985) 1039. 36. M. B. Gavela, P. Hernandez, J. Orloff and O. Pene, preprint CERN-TH.7081/93. 37. A.I. Bochkarev, S.V. Kuzmin, and M.E. Shaposhnikov, Phys. Lett. B244 (1990)275; Phys. Ftev. D43 (1991) 369. N. Turok and J. Zadrozny, Nucl. Phys. B369 (1992) 729. G. F. Giudice, Phys. P~ev. D45 (1992) 3177. S. Myint, Phys. Lett. 287B (1992) 325. M. Pietroni, Nucl. Phys. B402 (1993) 27. 38. A. Cohen, D. Kaplan and A. Nelson, Ann. Rev. Nucl. Part. Phys. 43 (1993) 27. M. Dine, talk at this Conference.
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 37A (1994) 117-126 North-Holland
Baryogenesis at the electroweak scale M. Shaposhnikov a" aTheory Division, CERN, CH-1211 Geneva 23, Switzerland
1. I n t r o d u c t i o n
Our universe is charge-asymmetric. The baryonic asymmetry of the universe (BAU) is characterized by the baryon number to the entropy ratio, n__~B= (4 - 10) × 10-11.
(1)
8
The existence of the baryonic asymmetry poses a number of related questions. (i) We would like to know why there is an asymmetry. (ii) We would like to explain its magnitude. (iii) Last, we would like to understand why we have more baryons than antibaryons (sign of the asymmetry). A possible answer to the first question has been known already for about 25 years [1]. Namely, our universe is charge-asymmetric because (i) the baryonic number is not conserved, (ii) the universe is expanding and was very hot in the past, (iii) C and CP parities are not good symmetries of nature. Clearly, the answers to the second and third questions require detailed knowledge of the underlying physics associated with B- and CP-nonconservation. At present, we know two possible sources for B-violation. The first one is connected with Grand Unified Theories (GUTs) of strong, weak and electromagnetic interactions. Here Bnon-conservation is due to the heavy particle (leptoquark) exchange. The mechanism for baryogenesis in GUTs is associated with leptoquark decays at the very early stages of the universe evolution, at temperatures close to the unification "On leave of absence from Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312,
scale (1016 GeV) [1,2]. GUTs provide a plausible explanation of the baryonic asymmetry, but cannot predict its sign and magnitude just because we do not know the details of particle interactions at the unification scale. The other problem with GUT baryogenesis arises in the inflationary universe scenario (the discussion of this point can be found, e.g. in the review [3]). In many models the temperature of the universe after inflation is smaller than the unification scale. This makes the realization of the standard leptoquark decay mechanism impossible (there are no leptoquarks at this stage to decay and produce the baryonic asymmetry]). At the same time, a number of different mechanisms in the framework of GUTs were suggested, which can overcome this difficulty [4]. The second source of B-violation is associated with non-perturbative physics and exists already in the standard electroweak theory [5]. This theory may thus be a proper candidate for an explanation of the baryonic asymmetry of the universe as well [6]. Contrary to the GUT case, we know the parameters of the theory quite well (particle masses and coupling constants). Therefore, one can hope to predict the magnitude and the sign of the baryonic asymmetry, or to predict or constrain the masses of the Higgs boson and top quark, provided the asymmetry depends on them. The firm conclusion that the standard model cannot explain the baryonic asymmetry is also of a great importance, since in this case the existence of BAU will give a clear observational indication on the physics beyond the standard model. In this talk we review some results that have been derived recently in a study of baryogenesis in the minimal standard model [7,8].
Russia.
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00593-K
118
M. Shaposhnikov/Baryogenesis at the electroweakscale
2. B-vlolation at t h e e l e c t r o w e a k scale In spite of the fact that the baryonic number is a good symmetry of the electroweak theory on the perturbative level, it is not conserved due to the anomaly in the fermionic current and complicated vacuum structure of non-abelian gauge theories (for reviews, see e.g. [9,10]). Under usual conditions, the rate of fermionic number non-conservation is suppressed by the semiclassical exponent [5] e x p ( - ~-' 2,r~w)" At high temperatures the rate of B-violation is highly enhanced [6] and proportional to the Boltzmann exponent e x p ( - M s p h ( T ) / T ) . Here Msph(T) is the effective sphaleron [11] mass at high temperatures. At sufficiently small temperatures, T < Te, Msph(T) ~- 3 M w ( T ) / a w , with M w ( T ) being the temperature-dependent W mass. Here Te is the critical temperature of SU(2)xU(1) symmetry restoration [12]. At T > Te, scaling arguments [13,14] show that M, ph(T) -- ~T, where ~¢is some unknown constant. So, one can write for T > Tc and T < Tc correspondingly:
r(T) =
A u ( ~ w T ) 4,
r ( T ) = & \ 4~- ]
• exp
(M,,h(T)) T
\
~
(2)
'
where A, is some factor that cannot be found by semiclassical methods. Real-time numerical simulations [15] give a lower bound A > 0.4, but the values as high as, say, ~ = 20 cannot be excluded at present. For the broken phase Ab ~ NtrNrot, where factors Ntr ~- 26, Nrot ~- 5.3 x 103 are due to the zero-modes normalizations [13]. At T > To, the rate (2) of the anomalous reactions greatly exceeds the rate of the universe expansion, rt71 ~ T2/Mpt, where Mt,t ",, 1019 GeV is the Plank mass. Therefore, anomalous reactions violating the baryon number are in thermal equilibrium untill the moment of the electroweak phase transition. After the phase transition, the vacuum expectation value of the Higgs field is non-zero and the baryon-number violation rate decreases due to the Boltzmann exponential suppression (2). Then, from eq. (2) it follows that
the freezing temperature Ts ~ 100 GeV of the sphaleron processes in the broken phase can be found from equation M, ph (TB) _ 45-~ log(Ppt/TB).
(3)
TB
The equilibrium character of the anomalous fermionic number non-conservation at sufficiently small temperatures influences the primordial baryonic asymmetry, which may be created by GUT interaction. In some cases the primordial asymmetry is washed out by electroweak processes. The analysis of the influence of the anomalous B-non-conserving reactions on the primordial baryonic asymmetry can be found in [16]. 3.
T h e r m a l n o n - e q u i l i b r i u m and t h e Higgs m a s s
bound
on
The baryonic asymmetry of the universe may be generated, provided there are sufficiently strong deviations from thermal equilibrium. It is easy to understand that the departure from the thermal equilibrium coming just from the universe expansion is not enough [6]. Imagine that the electroweak phase transition is of the second order (or weakly first order), so that the temperature dependence of the vacuum expectation value of the Higgs field is continuous:
In this case all deviations from thermal equilibrium come from the universe expansion. Now, the baryonic asymmetry cannot be generated at T > Tn since B-non-conserving reactions are in thermal equilibrium• Therefore, BAU is at most nB/s
~
r(Tn) 6c e
~
TB
~
$cp × 10-18,(5)
which is small compared with observations independently on the magnitude of the CP-violating factor ~cP < 1. The only place for efficient thermodynamical non-equilibrium at the late stages of the universe expansion (T ~ 100 GeV) is the first-order phase transition with the breaking of the SU(2)xU(1) group [17] (see Fig. 1).
M. Shaposhnikov /Baryogenesis at the electroweak scale
0.5
v(¢)
j~
0.2!
-0.25
-0.5
-0.75
-1
'
'
'
¢o
i;o'
'
'
I;o
¢'
'~;c
Figure 1. Typical temperature behaviour of the effective potential for the first-order phase transition.
At T > T+ the effective potential for the Higgs field has just one extremum corresponding to the phase with unbroken symmetry. In the temperature interval Tc < T < T+ the minimum corresponding to the broken phase is metastable, while at T_ < T < Tc the minimum corresponding to the unbroken phase is metastable. So, at T > Tc the symmetry is restored, and the vacuum expectation value of the Higgs field is equal to zero. At T = Tc the free energy of the broken phase is equal to the free energy of the restored phase. However, the phase transition does not occur at this time just because the probability of the tunnelling through the barrier is too small. So the universe waits till the potential barrier separating the broken and unbroken phases is sufficiently small. The temperature T* at which this happens lies somewhere between the critical temperature T¢ and temperature To of absolute instability of the restored phase. The phase transition proceeds via the nucleation of the bubbles of the true phase, which expand with the velocity of the
119
order of the speed of light and fill out the universe. The temperature T" is defined from condition rt~ ,-~ 1, where r is the bubble nucleation rate per unit time and unit volume [17]. The typical size of the bubbles at the end of the phase transition is of the order of 10-4tu ,~ 0.1 cm. It is the motion of the bubble walls through the hot plasma that introduces the deviations from the thermal equilibrium. A constraint on the particle spectrum of the theory can be obtained from Sakharov's out of equilibrium condition [18,19]. Suppose the baryonic excess Ain was generated by some electroweak mechanism at the phase transition that breaks SU(2) xU(1). The source of the baryon excess is not important here in essence. BAU at T* must survive till the present time. The necessary condition for that is freezing out the baryon number non-conserving processes in the phase with broken SU(2) xU(1) at the critical point. Namely, one has TB < T*, or, in particle physics terms
M, nh(T°) T"
> 45.
(6)
The ratio of the sphaleron mass to the temperature at T = T* depends on the Higgs boson mass. In fact, condition (6) implies an upper limit on the Higgs mass. In order to understand why we have an upper bound let us take for simplicity the minimal standard model effective potential in the one-loop approximation at T = T_ 2:
V(¢)T=T_ __ )~¢4
Tg~vCa(2 + 32~r
1 cos30-----~). (7)
This expression is derived using a high temperature approximation and assumes that A << g 2. After the phase transition, the condensate of the field ¢ is non-zero and equals
¢ =
3To 93(2 + - - ) , 1
cos30w
(8)
from which we obtain the upper limit on the Higgs boson mass M~ < Merit -~ 45 GeV. The critical 2The moment of the electroweak phase transition is close to the moment of the absolute instability of the phase with ~b = 0 [17] because t h e t u n n e l l i n g transitions to ~ ~ 0 are suppressed. Using Tc instead of 7"- makes the bound on the Higgs mass somewhat stronger [20].
120
M. Shaposhnikov/Baryogenesis at the electroweakscale
value in this approximation practically does not depend on the value of the top quark mass [21]. This simple consideration, however, is spoiled when one goes beyond one-loop computations. As is well known, perturbation theory in gauge theories at high temperature is intrinsically sick due to the existence of power-like infrared divergences [22]. In particular, the gST4 correction to the free energy, originating first on the 4-loop level, is not computable in principle by perturbative methods. These divergences make the perturbative effective potential divergent at small ¢. The logarithmic divergence appears at the 4-loop level, and at higher loops one gets power singularities. At nonzero ¢, the infrared problem is formally absent, since integrals at small momenta are cut by nonzero gauge boson masses. So, one expects that at sufficiently large ¢ the perturbation theory is applicable. Therefore, perturbation theory seems to be quite suitable for the answer to the following question: 'What is the value of the effective potential at a given temperature and sufficiently large ¢?' In particular, the expectation value of ¢ can be determined with the help of the perturbative effective potential, provided ¢t is large enough. However, the determination of the critical temperature of the phase transition, as well as that of the bubble nucleation temperature T" in cosmology is, strictly speaking, beyond the scope of perturbation theory. Indeed, to find the critical temperature, one must know not only the value of the potential in the broken minimum, but the value of the potential in the phase with ¢ = 0. While the first quantity can be computed in perturbation theory, the second cannot. The same remark applies also to the bubble nucleation temperature T*, since the computation of the rate of bubble nucleation requires the knowledge of the effective action for small enough fields ¢. The determination of these temperatures is impossible in the perturbation theory without additional assumptions on the behaviour of the potential near the origin. In perturbation theory the assumption is: higher-order loop corrections as well as all nonperturbative effects are not important near ~b= 0, i.e. for the determination of Tc and T*. With it the following bounds on the Higgs mass can be
derived in the different approximations: (i) One-loop effective potential for the scalar field and 'tree' sphaleron rate T 4exp(-Msph(T)/T): Merit = 65 GeV [18]. (ii) One-loop effective potential for the scalar field and one-loop sphaleron rate (2): Merit = 45 GeV
[19,21] (iii) One-loop effective potential field with Debye screening taken Merit = 35 GeV [23]. (iv) Two-loop effective potential field with Debye screening taken Merit ~ 40 GeV [24].
for the scalar into account: for the scalar into account:
This assumption, however, is most probably wrong. The first indication comes from the lattice simulations of the 3-dimensional Euclidean pure SU(2) gauge theory, which is a high-temperature limit of the 4-dimensional one. This theory is confining, namely Wilson loops exhibit an area law behaviour (in 2+1 dimensions, this would be the ordinary notion of confinement, so that particle excitations are bound states of SU(2) gluons) 3. The masses of glueballs in 2+1 (or static correlation lengths in our Euclidean theory) were measured on the lattice and appear to be quite large M ,.. 2g2T [25] in comparison with a natural perturbation expansion parameter g2T/2r (2~r's in perturbation theory comes from the integration over momenta). The perturbative computation of the effective potential near the origin uses massless degrees of freedom, while in fact they are massive and unexpectedly heavy. It has been argued in [8] that non-perturbative effects substantially modify the effective potential near the origin. In that paper, a possible model for incorporating some non-perturbative physics to the game was considered. In order to get a total potential incorporating non-perturbative effects, one should add to the perturbative potential the term
Vno,p(O, T) = - A ~ = c T ( g l [ ) 3 f ( g - ~ )
,
(9)
3The confining property of SU(2) in 3d makes the reliable estimate of the critical temperature of the elect roweak phase transition as difficult as the computation of, say, the glueball mass from the first principles of QCD.
121
M. Shaposhnikov /Baryogenesis at the electroweak scale
where f(0) = 1, and A~c > 0 is some unknown number. The function f(z) is expected to be exponentially small at large z. Some guess for this function can be derived by computing the contribution of a certain set of non-perturbative calculations to the functional integral: f
= F
2~zvg~(z),
(10)
where K is a modified Bessel function and -
3~2¢ gwT'
(11)
while a rough estimate of A~ac gives A~ac ~ 0.4. A contribution of this type does not contradict any perturbative computation of the effectivepotential. So it is expected that non-perturbative effects dig a pit near the origin, therefore reducing the value of the critical temperature and T*. The lattice simulations of the electroweak phase transition [8] reveal the systematic decrease of the critical temperature with respest to two-loop perturbative computations, thus indicating that nonperturbative effects are really important and that they indeed tend to strengthen the first-order nature of the phase transition. Non-perturbative effects may considerably change the perturbative picture of the electroweak phase transition. For the effective potential containing contributions (9,10) the temperature T" is noticeably smaller than the critical temperature To, so that the vev of the Higgs field at T = T* is considerably larger than expected from perturbative computations4. The estimate of the critical Higgs moves up to a value about 80-100 GeV. At the same time, the bubble wall thickness (Higgs correlation lengths) is about 1-3 T" and is substantially smaller than one would expect from perturbative computations5. It is interesting to note that an upper bound on the Higgs mass implies also an upper bound on 4On the contrary, in refs.
[24,26] it was assumed that
T" =Tc. 5We stress that these numbers axe based on the computations in the framework of the model (9) for nonperturbative effects. The partial test of this model is possible with 3d lattice simulations of the phase transition
[8].
the top-quark mass. From the vacuum stability argument, it follows [27] that m t < 140GeV +
MH -- 75GeV 1.64
(12)
If, say, Merit = 90 GeV, then Mt < 150GeV. One can see, therefore, that cosmology in the combination with particle physics may imply a number of non-trivial bounds on the particle spectrum of the electroweak theory. As one can see, the uncertainties in computations of the critical Higgs mass allow no conclusion as to whether MSM baryogenesis is ruled out or not by direct searches of the Higgs boson at LEP. However, all estimates indicate that the Higgs boson should be quite light in order to make the electroweak baryogenesis possible. If the baryonic asymmetry of the universe is indeed due to MSM interactions, then the Higgs boson may be found on LEP1 or LEP2. Much more theoretical work is needed in order to refine these estimates. 4.
Baryonic standard
asymmetry model
in
the
minimal
It is easy to understand that electroweak baryogenesis has to be associated with the walls of the expanding bubbles of a new (broken) phase. Indeed, outside the bubbles the rate of the fermionic number non-conservation greatly exceeds the rate of the universe expansion, so that sphaleron processes are in thermal equilibrium. On the contrary, inside the bubbles, sphaleron processes are switched off and the fermionic number is effectively conserved. Two types of the mechanisms for electroweak baryogenesis have been considered in the literature. In the first case [28-32], non-trivial configurations of the gauge and/or Higgs fields (sphalerons or thermal fluctuations) interact with the moving domain wall in a CP-violating way. Then, net fermionic number is produced when these field configurations decay inside the bubble wall. In the second class of mechanisms [33], CPnon-invariant interactions between fermions and the bubble wall lead to a separation of some CPodd charge by the bubble wall, w h i c h is then converted to an asymmetry in the baryonic number
122
M. Shaposhnikov /Baryogenesis at the electroweak scale
by sphaleron processes in the unbroken phase. We shall discuss here in some detail the implementation of the last type of mechanism for the minimal standard model, following closely our work with Glennys Farrar [7] (see also [34]). Clearly, the structure of CP-violation is essential. In the minimal standard model CP violation comes from the quark mass-mixing matrix. It can be rotated away provided (i) there is a degeneracy in the up-quark sector, i.e. mu = rne or rnu = me or m t = me, (ii) or there is a degeneracy in the down-quark sector, i.e. md = ms or m g = mb or mb = m , , (iii) or some of the mixing angles between different generations is zero, (iv)or the CP-violating phase is zero. In other words, CP violation vanishes together with the product (known as the Jarlskog determinant [35]) dee = sin(012) sin(023) sin(Ols) sin ~cp -
--
mu)(m
-
-
md)(m.
(13) -
The structure of CP violation makes baryogenesis in the minimal standard model a very nontrivial problem. Indeed, the electroweak phase transition, where strong deviations from thermal equilibrium are expected, happens at rather large temperatures .~ 100 GeV. It seems, therefore, that quark masses (maybe with the exception of the t-quark) can be treated as a perturbation, so that a dimensionless measure of CP violation is just Jcp "" d c p / T 12 "~ 10-20 [19,28]. Of course, this number is too small to account for the observed asymmetry. This argument relies heavily on the applicability of perturbation theory on quark masses (or, which is the same, on Yukawa coupling constants) and uses an a s s u m p t i o n that the typical energy scale essential for the estimate of the asymmetry is the temperature of the phase transition. Imagine now that the relevant scale is m, rather than T: then GIM cancellation does not occur, and what is left from d c p is just the product of mixing angles and CP-violating phase ~ 10 -5. The example of the breaking of perturbation theory is provided by the quark scattering on the bub-
ble walls. Outside the bubble, the electroweak symmetry is unbroken, and the Higgs contribution to the quark mass is zero. Inside the bubble, the vev of the Higgs field is non-zero and quarks are massive. To simplify the discussion, let us forget for a moment the fact that quark propagation through the hot medium is different from that in vacuum. Then, if the energy of the quark in the unbroken phase is smaller than its mass in the broken phase, it will be reflected from the domain wall with unit probability independently on the value of its mass (Yukawa coupling). This is precisely the place where the 'no-go' argument may be wrong. The perturbation theory does not work only in a small fraction of the phase space determined by the quark masses, but the loss in the phase-space factor may be much smaller than the gain in CP-violating amplitude. With these remarks let us see how CP violation can reveal itself in a quark scattering on the bubble walls. Due to CP violation, the reflection amplitude for a quark coming from the unbroken phase is not equal to the reflection amplitude for an antiquark. Hence, there is a net flux of baryon number through the wall and an equal flux of antiquarks reflected from the wall. Now, the fate of the fermions inside and outside the bubble is different. Baryons inside the bubble survive till the present time because the rate of B-violating reactions is highly suppressed in the broken phase, while the antifermions outside the bubble disappear through equilibrium anomalous reactions. The final baryonic asymmetry is just proportional to the difference of reflection coefficients from the domain wall of quarks and antiquarks. The domain wall acts like a separator for baryon number, filling the bubble with fermions and outer space with antifermions, while anomalous reactions destroy the antifermion excess in the unbroken phase. Even an order of magnitude estimate of the baryonic asymmetry in this particular mechanism is a very difficult task, due to a poor knowledge of many high-temperature effects [7]. A computation in the one-loop approximation for the thermal fermionic self-energies has been performed in [7]. Schematically, the baryonic asymmetry can
M. Shaposhnikov /Baryogenesis at the electroweak scale
123
be expressed as '
ns
s
1
-
geII'6,~icro (kinetic factors). x
I
.
.
.
.
I
.
.
.
.
I
.
.
.
.
I
(14) exlO-e
.t
where N~11 ... 100 is the effective number of massless degrees of freedom. The quantity ~micro is given by particle physics:
~miero f dwT(w)A(w), =
(15)
4x 10.4
?~.
q
;~xlO-e
where w is the energy of a quark, 7(w) is a phasespace factor, = E[Iri#l 2 are the reflection coefficients for a process with a quark (antiquark) of the i-th generation incident in the unbroken phase and the reflected quark (antiquark) of the j-th generation. The last factor in (14) takes into account the statistical physics effects such as the magnitude of the deviation from thermal equilibrium, the finiteness of the rate of fermionic number non-conservation in the unbroken phase, etc. An essential part of the analysis is the use of quasi-particle formalism. In the medium, particle excitations are different from bare quarks. In particular, there is non-trivial mixing between quarks of different generations in the broken as well as unbroken phases. Moreover, the quarkmixing matrices are different in different phases, and this difference is not suppressed by Yukawa coupling constants (as happens for bare zerotemperature quarks). The combination of the phenomenon of the total reflection with this fact enhances the asymmetry a lot, in comparison with (13). As is expected from the general discussion, the differential asymmetry A(w) was found to be substantially different from zero only in a small region of phase space, namely when a strange quark is reflected from the domain wall with probability of order 1. Everywhere else the asymmetry is very small due to GIM cancellation. The typical picture for A(w) for quark motion perpendicular to the domain wall is shown in Fig. 2, for experimentally favorable values of mixing angles and mt = 1 5 0 G e V 6 6A p e r t u r b a t i v e a n a l y t i c c o m p u t a t i o n of t h e a s y m m e t r y in a thin-wall limit c a n be f o u n d in [7].
k
ii
,
I
47.9
. . . .
I
48
. . . .
I
48.1
,~(a,v)
. . . .
'
I
,
I
I
48.2
fA A--,-a,a~o4w-.oe~,eV
Figure 2. The asymmetry A in the reflection probabilities of right-chiral quarks and antiquarks incident from the unbroken phase.
The upper pair of peaks occupies the energy range in which the strange quark is totally reflected. The 'notch' in the middle is the region in which the down quark is also totally reflected and GIM cancellation is perfect. The unfamiliar feature that total reflection occurs for a range of energies, rather than for all energies less than some value, results from the unusual properties of the quasi-particle dispersion relation. At lower energy there is another region of a different character, involving transitions between d or s and b quarks. The sign of the predicted asymmetry coincides with observations provided sin(Sop) > 0. Unfortunately, uncertainties in the kinetic factors and in the phase-space volume (coming from the unknown velocity of the domain wall, sphaleron rate in unbroken phase, domain-wall thickness, non-coherent quark scattering, dynamics of the domain-wall propagation through the plasma, non-perturbative effects, etc.) do not allow us
124
M. Shaposhnikov/Baryogenesis at the electroweak scale
to conclude convincingly whether the minimal standard model can be responsible for the baryonic asymmetry of the universe. Combining all factors together, we get [7] ~$ ,~ 10 -1° -10 -lz. A much better understanding of the high-temperature phenomena is required in order to settle this question z. 5. Conclusion While the resolutionof the question of whether the minimal standard model can or cannot produce a sufficientamount of baryons requires the clarificationof a number of very hard dynamical questions, the situation with electroweak baryogenesis in extended versions of the theory is not so difficult.First of all,the extension of the Higgs sector allows us to escape a strong upper bound on the Higgs mass. This happens because in the models with many Higgs fieldsthere is much more freedom, and eq. (6) is now a constraint on some complicated combination of Higgs masses together with coupling constants, rather than on the Higgs mass alone as in the M S M [37]. Moreover, the experimental limit on the Higgs mass is weaker in the extensions of the standard model. The situationwith C P violationis also much simpler: extended versions usually contain extra CPbreaking phases. A great deal of work has been put in the analysisof the baryogenesis in extended versions of the electroweak theory (see [3,38]for a review with references to originalpapers). The general conclusion is quite optimistic- B A U can be a result of anomalous electroweak numberviolating processes. Unfortunately, the direct experimentalcheck of the electroweak character of the baryonic asymmetry is most probably impossible (a gedanken experiment would be the measurement of the leptonic asymmetry L of the universe and checking that B - L = 0). The experimental discovery of any type of new physics below, say, 1 TeV (be"tina recent p r e p r i n t , Gavela et al. [36] a r g u e t h a t d u e to t h e loss of q u a n t u m c o h e r e n c e in the scattering of q u a r k s w i t h g l u o n s , t h e a c t u a l n u m b e r is m u c h smaller. As far a~ we c a n see f r o m their paper they worked in a nonc o n s i s t e n t approximation, throwing away a number of imp o r t a n t contributions to the m s y m m e t r y , so that their conclusions a r e n o t justified by the work r e p o r t e d in [36].
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