Sphalerons and baryogenesis: Electroweak CP violation at high temperatures

Nuclear Physics B329 (1990) 493-518 North-Holland

S P H A L E R O N S A N D BARYOGENESIS: ELECTROWEAK CP VIOLATION AT HIGH T E M P E R A T U R E S A.I. BOCHKAREV Institute of Theoretical Pl~vsics, Unieersity of Minnesota. Minneapolis, M N 55455, USA

and Institute for Nuch, ar Research of the USSR A cadency of Sciences, Moscow 117312, USSR*

S.Yu. KHLEBNIKOV and M.E. SHAPOSHNIKOV Institute for Nuclear Research of the USSR Acaden~v of Sciences, Moscow 117312, USSR

Received 10 April 1989

The effective action for the gauge fields in the electroweak plasma of the early Universe is considered. It is argued that in thermal equilibrium the effective action contains a CP-odd term proportional to the Chern-Simons number density provided the exactly conserved fermionic charges L i - B / n i" are nonzero (L i are leptonic generations numbers, B is baryonic charge). In order to obtain the non-equilibrium distributions of fermions in the expanding Universe, the relevant kinetic equations involving both the electroweak CP-violation effects and the anomalous fermion number non-conservation are derived and solved. It is found that the Kobayashi-Maskawa CP-violation is sufficient to ensure the validity of the electroweak (EW) scenario of baryogenesis. We elaborate the program for the determination of the sign of the baryon asymmetry and discuss the interplay between GUT and EW scenarios.

1. Introduction T h e old i d e a that the observed b a r y o n a s y m m e t r y of the universe ( B A U ) m a y be e x p l a i n e d b y the theory i n c o r p o r a t i n g b o t h b a r y o n n u m b e r ( B ) a n d C P n o n - c o n s e r v a t i o n [1], has received new a t t e n t i o n recently. It was realized that the a n o m a l o u s [2] B n o n - c o n s e r v a t i o n [3] in the s t a n d a r d electroweak theory related to the t r a n s i t i o n s between different topological sectors [4-6] is effective at sufficiently high t e m p e r a t u r e s [7]. Together with the explicit C P b r e a k i n g in the q u a r k mixing m a t r i x this o p e n s an exciting possibility to explain the o b s e r v e d B A U in terms of s t a n d a r d m o d e l o n l y and, in particular, to relate the sign of B A U to the sign of a s y m m e t r y in k a o n decays [7,8]. A n i n c o m p l e t e list of references to a n o m a l o u s f e r m i o n i c n u m b e r n o n - c o n s e r v a t i o n is c o n t a i n e d in refs. [7-28]; for a review see ref. [29]. * Permanent address. 0550-3213/90/$03.50 ~ Elsevier Science Publishers B.V. (North-Holland)

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The effect of rapid B non-conservation due to topological transitions was elaborated in more detail in refs. [16-19] and in particular in ref. [22] this effect was derived formally by non-equilibrium statistical mechanics. Moreover, real time numerical simulations of sphaleron transitions [28] completely confirmed quantitative as well as qualitative features of analytical predictions based on statistical physics. The most straightforward cosmological consequence of a high rate of B non-conservation is the considerable dilution of the BAU generated at earlier (say, GUT) scale [7]. The remnant baryon asymmetry is determined by the equilibrium baryon number at the freezing temperature Tf of the anomalous processes [15, 22]: 8ne+4 (B-L)+ B°= 2 2 n f + 13

6 ~

m? ~i ' k i ( t ° ) . ~ "

(1.1)

Here L = E~L i is the full lepton number obtained as a sum over all nf generations. The asymmetry in ( B - L) could be nonzero in GUTs with B - L non-conservation. All A~= B / n , - Li, i = 1 ..... n f, are exactly conserved in the standard theory with massless neutrinos, t o is the decoupling time for GUT processes with Ai non-conservation, m,2 represent the average squared lepton masses for different generations. The BAU predicted by eq. (1.1) can be small compared with the observational one. This happens, for instance, in GUTs with exact B - L conservation like SU(5) where B - L = 0. Moreover, nobody can be sure that GUTs provide a correct description of nature at high energies, so the whole GUT scenario for the BAU production remains questionable. And it is difficult to imagine that it will get a more solid basis in the future due to the lack of experimental support of physics near the Planck scale. Therefore it is important to explore the possibility of the BAU generation at the electroweak mass scale. The scenario of the electroweak BAU generation suggested in ref. [8] is based on the following assumptions: (i) In the high-temperature phase with restored SU(2)× U(1) symmetry gaugeHiggs configurations exist decaying during the first order electroweak phase transition, producing a net topological charge fd4x F,F~* 4= O. (ii) The number of fermions created in the phase transition is obtained by integrating the anomaly equation. (This could not be the case because initial and final configurations are not connected by a large gauge transformation.) (iii) Degeneracy of the equilibrium effective potential for density of ChernSimons charge 1

n c s = 16,rr2eiJkTr( FijA k - ~AiAjAk) ,

(1.2)

which could be discrete (fig. la) or continuous (fig. lb). Recall that Ncs = fd3x ncs

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/ ncs

ncs

I1

s'?

n

-._...j cs

8"

n

cs

b"

Fig. 1. (a, b) Degenerate equilibrium effective potential for the Chern-Simons charge density. (a', b') Breaking of the degeneracy due to CP non-conservation and universe expansion.

is related to B non-conservation through the anomaly 2

O~Jf = n fgw Tr FF*

16rr~ 1

16rr2

Trf'ld4xrF

to

* = Ncs(tl) - Ncs(t0).

(1.3)

(iv) Breaking of the degeneracy by deviations from equilibrium in reactions with C P non-conservation, see fig. la', b'.

We stress that this degeneracy has nothing to do with the motions between different topological sectors which also change Ncs. In fact, if the structure of fig. la, b occurs it will occur near each of the vacua connected by large gauge transformations. However, it will be connected with Ncs[A ] only near the trivial vacuum A = 0. Near some pure large gauge A n it is Ncs[A - A~] that should enter fig. 1 as well as eq. (1.4) below. This explains in particular how the potential for gauge fields introduced above can be a gauge invariant quantity even though Ncs changes by unity under the large gauge transformation (see also ref. [8]). The point is that fig. 1 as well as eq. (1.4) below represent the potential for small vector fields A t ~ g w T and are not applicable for fields involved in the large gauge transformation. So there are only small gauge transformations we should worry about and Ncs is perfectly invariant under them.

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Schematically, the sequence of events looks like the following [8]. The initial state of the universe was presumably CP symmetric and the average value of the Chern-Simons condensate was zero. Then, after some time, the system finds itself in the state with maximal possible Chern-Simons density with the sign determined by the phase 3 in the Kobayashi-Maskawa matrix. During the first order phase transition the Chern-Simons condensate is converted into real fermions, thus giving rise to the observed BAU. It is worth noting that the electroweak scenario of the BAU production constrains the masses of particles (top-quark and Higgs), yet to be observed by future accelerators [8,16]. In order to demonstrate the viability of the scenario it is necessary to clarify the status of assumptions (i)-(iv). The first two have been recently proven in ref. [20], the third requires a lot of work connected with lattice simulations of the hightemperature ground state of the gauge;Higgs system (see ref. [20]). The fourth statement, while being more or less evident, was not put on quantitative grounds till now. In ref. [8] it was suggested that the effective action for the nearly static gauge fields (fermions are integrated out) in the expanding universe contains a CP- and CPT-odd correction of the form:

T2 A~ = ~ms~00 NCS

(1.4)

( M 0 - M m, see sect. 3). This is exactly the contribution making the states with maximal possible Chern-Simons density energetically favourable. Moreover, the sign of ~ms determines the sign of the baryon asymmetry. The aim of the present paper is to derive eq. (1.4). We outline a procedure for calculating the parameter ~ms and perform an estimate. We argue that the magnitude of 6ms is sufficient (cf. ref. [8]) to bring the system into the state with maximal possible Ncs - a 3w for the case of a flat potential (fig. lb, b'). Thus if the case in fig. 3 and may coincide l b is realized, then the asymmetry produced is of the order a w with the observational one, provided the Higgs boson mass M n = Men t = 45 GeV or M H = Hcw(m~) (Mcw is the Coleman-Weinberg value [30] of the Higgs mass) [8, 16]. We have not calculated yet the sign of the BAU. However, the problem turns out to be not so formidable as it seemed before [8]. Instead of calculating 14th order (in Yukawa couplings) ill-defined diagrams one should calculate one-loop graphs for annihilation of a quark-antiquark pair into three Higgs bosons. We will turn to the actual calculation in future. Let us give the main idea of our approach. To get an effective action for gauge fields one should integrate out fermionic degrees of freedom with properly specifying the boundary conditions for fermions. For a hot plasma in the expanding universe these are given by the semi-equilibrium density matrix, the universe expansion resulting in deviation of particle concentrations from the equilibrium ones. This is taken into account by introducing chemical potentials for all species of

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fermions and anti-fermions: n F ( E ) = (exp

+ 1

)'

,

nF(E) =

exp

+1

, (1.5)

where E is the energy of particle. Of course, there are also deviations from the Fermi distributions coming from the slightly non-equilibrium character of elastic reactions establishing kinetic equilibrium. However, elastic reactions are much faster than those changing the parameters /~, so eq. (1.5) is a good approximation. The chemical potentials /~ can be determined from the system of Boltzmann-like equations describing the behaviour of fermion densities in the expanding universe. Almost all of them are standard and take into account perturbative reactions: qLq~ --' G H , qLCtL ---' W + W -, etc. (G is gluon, H is Higgs scalar). The quantities B and L i are perturbatively conserved. To terminate the system we add an equation describing the anomalous fermion number non-conservation through sphaleron decays. A crucial property of the solution is the presence of asymmetries in the quark densities/~q -/~c~ :~ 0, even when all strictly conserved numbers B / n r - Li are equal to zero. This ensures CP and C P T non-invariance of the fermionic density matrix and gives rise to the term (1.4) in the effective action for gauge fields. Again we note that (1.4) refers only to sufficiently small vector fields and may be viewed as the first term in the expansion of a yet unknown full expression for A~? near A = 0. The full expression should be invariant with respect to large gauge transformations but not necessarily the lowest term. In fact, this technical issue is very essential to understand why the term (1.4) ever appears in the effective action. The paper is organized as follows. In sect. 2 we calculate the CP and C P T non-invariant term (1,4) treating the chemical potentials/~ as input parameters. At T < T~, Tc being a critical temperature of the phase transition, we include fermion mass corrections which appear to be important. At T > Tc, the case we are mostly interested in, similar corrections are expected from Yukawa contributions to the fermion propagators. These are multiloop graphs in the effective action, and here we give only an order-of-magnitude estimate. Sect. 3 is devoted to the study of non-equilibrium effects. We formulate and solve the kinetic equations for fermionic n u m b e r densities in the expanding universe accounting for CP violation and anomalous reactions. The importance of conformal anomaly is clarified. In sect. 4 we estimate the magnitude of the CP-odd term in the effective action for gauge fields in pure EW theory and discuss its cosmological implications for baryogenesis. In sect. 5 we investigate the influence of G U T s on the EW scenario of the BAU generation. In sect. 6 we summarize the results of the paper. 2. Effective action for gauge fields Since the quantity Ncs is CP and CPT-odd, no term linear in Ncs can appear in the effective action if both the lagrangian and the state of the system are either CP

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or C P T invariant. Our aim here is to investigate the influence of CP and C P T n o n - i n v a r i a n c e coming from non-zero chemical potentials for fermions. This was d o n e in the one-loop approximation for massless fermions in ref. [31]. We will see that mass insertions for T < Tc and multiloop corrections in gauge and Y u k a w a couplings for T > Tc are important for cosmological applications. T h e effective action is the usual grand canonical potential f2 defined as e - ~ = f D P D Q exp

- Her f d r

- Tre -B'o",

(2.1)

where Q and P are canonical coordinates and m o m e n t a , respectively. Her r is the h a m i l t o n i a n modified to include the chemical potentials for all particle species (both bosons (~a b ) and fermions (/~ f )) Her f = H -

S'~/IrNf - E / ~ b N b . f

(2.2)

b

T o include the effect of static background bosonic fields we d e c o m p o s e as usual: A l = A / + a / (l = 1,2, 3) and drop the terms linear in a t. N r and N b a r e free particle n u m b e r s defined generically as: N = E k a t ( k ) a ( k ) . W e develop a perturbation theory in: (i) the n u m b e r of external bosonic legs; (ii) ~ f , b / T ; (iii) coupling constants; (iv) m / T , where m is the particle mass. The entry (i) is the limit of sufficiently small fields already discussed in sect. 1. The leading p a r t of the effective action is related to the M a t z u b a r a two-point function of weak fermionic currents:

as2 = - ½Bf d ~ ((...

d3y((J~(x)Jbm(y)))A~(x)A~(y), (2.3)

55 = T r ( . . . e x p ( - - j S H a r ) ) / T r e - ~ N ° " l A = o ,

where a, b are SU(2) indices. We now proceed to the calculation of (2.3). 2.1. LOW TEMPERATURES: T < T,.

T h e explicit calculation of fig. 2 yields \

7~(3) A~2 --

32vr 2

/x1 + g~

+ E IIG~L2(~I + ~B2 \ a,fl

×

4~r2T 2

(~lmla-,~ 2 +

t,2mz,~)Al-,~2

30mAn3

7~(3) 16~r2T 2

+

amAlo+ A 2 areA2.)),

1o +

+

/

A.L Bochkarev et al. / Ba~ogenesis

499

Fig. 2. Massive one-loop diagram giving the CP-odd contribution to the effective action for gauge fields in an electroweak plasma with non-zero fermionic chemical potentials.

where a, fl run over doublets, g is the half-difference of the chemical potentials of the particle and antiparticle, for instance: m

0

-

(z.5)

0

/~, ~ are the chemical potentials of upper and lower members of doublet, ml, m 2 are the corresponding masses. K,~ is the mixing matrix which equals 8~a if at least one of indices corresponds to the lepton doublet and equals the familiar Kobayashi-Maskawa matrix when both indices correspond to quarks. Thus bilinear parts of Arcs appear, rnl :gm 2 signals SU(2) breaking. 2.2. HIGH TEMPERATURES: T> T~. The flavour asymmetric term in the effective action we are mainly interested in necessarily involves Yukawa couplings. At T < Tc it comes from the one-loop diagram and is proportional to scalar condensate @ ) . Strictly speaking, the latter may be non-zero but parametrically small at high temperatures [8, 36], (q~) - gw T. The infrared domain of such small fields is not controlled by perturbation theory. In general, an analogous statement may hold true for the flavour symmetric term if the condensate of the gauge field is non-zero, ( A ) - gw T. The other possible source of corrections is multi-loop graphs. The two-loop diagrams due to gluon and Higgs exchanges are shown in figs. 3 and 4 respectively. We claim, however, that these graphs give no contribution to the C P T odd part of the correlator (2.3) and hence do not renormalize a coefficient against Ncs-term. To see it one has to recall a theorem proved in refs. [37, 38] which states that in the abelian gauge theory in three dimensions all multi-loop contributions to the polar-

t_

L Fig. 3. Two-loopstrong radiative correctionsto the gauge effectiveaction.

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A.I. Bochkareu et al, / Baryogenesis P,

~'X,A \ Rk / / ~ H ,,L/ tk

Fig. 4. Yukawacorrectionsto the effectiveaction. ization operator behave like O(k2). Only one-loop corrections may produce a term O(k). It is straightforward to generalize this result to the finite temperature effective action of the gauge field at non-zero chemical potential for fermions. In other words, it is possible to prove the non-renormalization theorem for the Ncs term in the abelian gauge theory without massless matter fields at T, /~ ~ 0 claiming that there are no corrections to it beyond one loop. This theorem is not applicable to non-abelian electroweak theory. Potentially possible contributions to Ncs may come from three- and higher-order loop diagrams involving three and four gluon vertices. At this level massless propagators for magnetic components of SU(2) gauge fields as well as the non-abelian character of interaction enter the game. In what follows, we shall assume that there are corrections to the C P T odd part of the effective actions coming from three-loop diagrams a n d / o r scalar and gauge condensates. Generally, they are expected to be of order awf 2 for the flavour asymmetric contribution and - awa S for the symmetric contribution. At high temperatures there is an ambiguity in the introduction of chemical potentials for quarks, because of the degeneracy of quark flavours. This is the problem of finding a basis for quark densities in eq. (2.2). To find the basis we account for quark mixing. The only interaction changing quark flavours is the Yukawa coupling which we choose in the form a(f y =

(2.6)

QLOKMDDRCP + QLOMuUR(P + h.c.

where K is the Kobayashi-Maskawa matrix, O is a matrix specifying the basis (to be determined later) and M D and M u are the diagonal Yukawa couplings: M D -- ~ -g~' Mw diag(md, ms, rob),

gw

M u - V/~ Mwdiag(mu, m

m~, mt),

where rn and M w are zero temperature masses. We will have a correct particle interpretation of the creation and annihilation operators (a t and a) if the thermal averages of the particle density operators a~aj are diagonal in the space of quark flavours (i and j are generation indices), i.e.

)

--= (2 r)

32Eknv(Ek)8,jS(k

-

k'),

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A.I. Bochkarev et al. / Ba(vogenesis

R

Fig. 5. Left quark mixing at finite temperatures.

where E~ is fermionic energy (not necessarily equal to [k] due to finite temperature effects) and n F is the Fermi distribution. We get for left fermions (right fermions are already diagonal if the interaction is chosen in the form (2.6))

(a~a,) oc ai. 1. + dliif(k

(2.7)

),

where f(k) is some function of momenta coming from calculating the Feynman graph of fig. 5, •//¢,j = J / f u + J / D ,

~//u =

0M20I,

~/D = OKM2K*O*.

(2.8)

Therefore, we have to choose the matrix O in such a way that ~ / u is diagonal. Of course, the degeneracy exists in the leptonic sector too. However, it does not cause any trouble due to the absence of right neutrinos and neutrino mixing in standard theory. Even if mixing is present in some extended theory, it is too small to provide any substantial effect. Guided by the result of the low-temperature calculation we write aa2=

E~'L(1 + C~a, + Cy.W,i) + Y ' . ~ (1 + i

Cyff2)

.j

l

+ C r~Ftt-I~-" ( "/Y"i + f e~) ) Ncs

j

(2.9)

Here ffL, -i if'u, - /z~ - and ffH are chemical potentials for left quarks, right U ff~, ff'~, and D quarks, left and right leptons and Higgs bosons respectively. C~, C m C R and C v are yet unknown coefficients of order a w expected from multiloop calculations a n d / o r the non-perturbative scalar condensate; ft are Yukawa couplings of leptons. We introduced here also chemical potentials for right fermions and Higgs bosons.

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The next stage is determining the g's. In equilibrium these can be non-zero only if there exist non-zero densities of conserved quantum numbers. If we assume no mixing between leptons of different generations then there are nt + 2 independent strictly conserved quantities: isospin T3, hypercharge Y, and n r anomaly free combinations of baryon and lepton numbers: B ~i = - -

L i,

i = 1 .....

n t .

(2.10)

nf

It is convenient to introduce chemical potentials corresponding to these quantities or, equivalently, decompose 1

~q-

3n r

1

--te~

,

1

gR _

~t~y+ ~u 1

/.tD .-R _ 3 n f G ' / t a ' - 3~Y + ~ ,

(2.11)

etc., where we have put /~T~= 0 as follows from T3 = 0. The usual cosmological case is also Y = 0. However, this does not i m p l y / ~ y = 0 [22]. The decomposition (2.11) separates equilibrium and non-equilibrium contributions: the primed ~"s are zero in equilibrium. 2.2.1. Equilibrium contribution. Substituting the full system (2.11) into eq. (2.9) we see immediately that the leading part of the effective action not proportional to a s or f 2 identically vanishes. As claimed above, the corrections are important. We now have to express the /~'s through A i (using Y = 0) with the same accuracy. To this end we should know the relation between particle number and its chemical potential up to first order in the interaction. This is a two-loop problem (see figs. 5, 6) and will not be addressed here. However, there may arise a suspicion that the corrections in the expression for /~ through Ai will cancel with the corrections in

Fig. 6. Radiative corrections to fermionic number densities.

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eq. (2.9) for £2 through/~, so that the whole effect of these subleading terms will be zero just as it happened for the leading ones. To show that this is generally not the case let us return to low temperatures T < Tc, where corrections are mainly due to fermion masses and can be analyzed explicitly. Expanding the Fermi distribution in t~/T and m 2 / T 2 w e get for a number of particles in a single spin-isospin-colour state:

N v ( # ) = Nr(O ) + I~VT 2

1 1 12

m 2

8 ~2T2

) '

(2.12)

where V is the volume of the system. This cannot cancel with (2.4) since there is no ~-function here*! We are now motivated to conclude that cancellation is unlikely in the high temperature phase as well. The final expression for the effective potential in equilibrium at T > Tc is

ow(

AY2 -- - - ~

)

Ce~(B - L ) + ZCifE2Ai NCS ,

(2.13)

i

where C and C, are some numerical coefficients. Note that due to mixing of quarks, their Yukawa couplings do not contribute to the generation asymmetric part. Of course, Yukawa couplings of quarks and leptons contribute to the symmetric part but this may be neglected compared to the contribution of strong interactions. The reader believing in Grand Unification can easily skip the rest of this section and sects. 3 and 4 and go directly to sect. 5 where the interplay between G U T and EW scenarios of the BAU production is discussed. 2.2.2. Non-equilibrium contribution. The non-equilibrium contribution is much smaller than equilibrium one unless A~ = 0, so we concentrate on this case in what follows. In principle, non-equilibrium effects make both /~, /~r and primed /L'S non-zero. However, the first n f + 1 quantities which are common for all generations of a given species, appear as traces over quark flavours and hence are much smaller than primed /x's (see also sect. 3). In other words, in first approximation there are no asymmetries in the total baryonic and leptonic numbers. However, there are partial asymmetries in different quark species (given by eqs. (3.30) and (3.31)), resulting in a CP-odd contribution to the effective action. Those readers who find these arguments compelling are invited to go directly to sect. 4 where we use the results for quark asymmetries obtained in sect. 3 for the EW scenario of baryogenesis. The others are invited to follow through the arguments of sect. 3, where we proceed to derive the kinetic equations for these quantities. We do not distinguish between /~ and /~' below. * f(3) appears in Nr:(0), see eq. (3.10), but it cancels out when the contribution of antiparticles is included.

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Baryogenesis

3. Kinetic equations for particle number densities in the expanding universe The aim of this section is the derivation of kinetic equations for the evolution of asymmetries in quark and lepton flavours in the expanding universe. The questions we are going to discuss include (i) the importance of conformal anomaly for effective C P T breaking in the early universe, (ii) taking into account fermionic number non-conservation due to anomalous processes. 3.1. GENERAL STRUCTURE OF KINETIC EQUATIONS Consider the general structure of the kinetic equations. (See ref. [32] for an introduction to the physics of particle reactions in the hot universe.) Let n r and ~ be the concentrations of different particles and antiparticles. The kinetic equations can be written symbolically in the form On,. -+ 3 H n r = I~ol +o¢r Ot '

Onr

--

Ot

--r + 3 H ~ =/Col

_1_ ~i~r

(3.1)

where the factor 3Hn describes universe expansion, H is the Hubble constant, I~o1 is the usual Boltzmann collision integral while o¢r is the contribution of anomalous processes. We introduce the following notations: FIr = FIr0

+

=

FIr0 - -

hr.

(3.2)

In linear approximation with respect to/z's, we have 1 0,a~2 n 0r __ n eq r + T~l.tri , -

-

br = ~ , r T 2

(3.3)

gl.tr T --1

(3.4)

for fermions and FI 0r -__- Fierq =

1 . 0,7-2 gl,£r I ,

br =

2

for bosons, where the chemical potentials/~0 and g r were introduced in sect. 2 (see eq. (2.5)). The general structure of the equations for ~t° and ~r is a3 1

n°ra 3) = Ar,~ ° + Br,~t,, 0

3

a3 ~-~(b~a ) = C r , bt° + Drs~s

(3.5)

(3.6)

where a is the scale factor, H = 6 / a , A, B, C, and D are kinetic coefficients. In CP-conserving theory the coefficients B and C are equal to zero; in our case B << A, C << D. Therefore we can neglect the term proportional to ~ in eq. (3.5). So first we should solve eq. (3.5), determining the influence of the universe expansion on the CP-even part of the particle densities and then use the result for the

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calculation of the asymmetries b i through eq. (3.6), where the term proportional to /~0 will play the role of the CP and CPT non-invariant source. The solution to (3.5), (3.6) is /Y(t) = e x p ( A ( t ) ) ~ ( 0 ) + ~ ( t ) ,

fi'= (lu °, ~ r ) ,

(3.7)

where fi'(0) is the initial condition for the chemical potentials, A is the relaxation matrix obtained from the solution of the homogeneous system. After some short time ~- << t u (t U is the age of the universe, t u = M o / T 2, M o = Mpl/1.66 N~f(2, Nef~ is the number of effectively massless bosonic (B) and fermionic (F) degrees of freedom, Ncff = N B + { N F ) , the system "forgets" the initial conditions and fi~(t) ---= ((t). In this asymptotic region, chemical potentials satisfy the system of algebraic (not differential) equations l 8 a30t(nerqa3)=arsll 0 ,

CrsP,°+ Drs~s=O.

(3.8), (3.9)

The first relation demonstrates the importance of conformal anomaly in the whole issue at T > Tc. Indeed, in the scale invariant theory a T = const, and neqoc T 3, so the unique solution to eq. (3.8) is ~'= 0. 3.2 C O N F O R M A L A N O M A L Y

Let us estimate the 1.h.s. of eq. (3.8) for different kinds of particles. We can represent n eq in the form g/eq ~___p

r 3 1 + --ffs(T) q/

+

,

(3.10)

where 0 = 1 for bosons and 3 for fermions. The coefficients a r and br are to be fixed by calculation of the radiative corrections for the particle number densities (see fig. 6). For example, a r = 0 for leptons, scalars, SU(2) and U(1) gauge bosons. We do not include here Yukawa and U(1) couplings because they are much smaller. To calculate the time derivative in eq. (3.8) we should know also ( 0 / 0 t ) a T . It is not zero due to deviations of the equation of state from that for the ultra-relativistic ideal gas p = ~e, where p is pressure and e is the energy density. This problem was solved in the context of asymptotically free SU(5) G U T in ref. [33]. We will apply the general formulas of ref. [33] for SU(3) × SU(2) × U(1) theory. The pressure of the electroweak plasma in equilibrium is equal to [34]

(

os

p= T4 c0+A-- +B rr A = -

8Tr2 3-~-(3 + ~nf),

'

co=

B-

o2 9-0

Neff , 3Trz 36 (2 + 5 n f ) ,

(3.11)

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506

so the energy density is given by 1

2 0

+ - T 4 ( A t ~ ( a s ) + Bfl(aw) ) , q7

(3.12)

where/~(a) are corresponding/~-functions, fi(a) = TOa/OT. With the use of the Einstein equations

// a

- 4 ~ r G ( p + .~e)

=

3

~

(3.13 /

we finally get 1

0

1

~3 0 T ( a T ) 3 = - - ~ T 2 7 ) ( A ~ ( a s ) + B~(aw)).

(3.141

Combining eqs. (3.10) and (3.14) we obtain the 1.h.s. of eq. (3.8) •

A

r rrT

B

(3.15)

In other words, the source term for the equation for/~0 coming from the conformal anomaly is of order HT3~(as) for all particles.

3.3. CP-EVEN DEVIATIONS FROM THERMAL E Q U I L I B R I U M

All inelastic processes give contributions to the r.h.s, of eq. (3.8). They include ordinary kinetic reactions described by the Boltzmann collision integral as well as anomalous ones with fermionic number non-conservation. Let us suppose that the rate of anomalous reactions is small compared with that of the ordinary lowest order perturbative processes. This is perfectly true at sufficiently small temperatures where the rate of B non-conservation is suppressed by the Boltzmann exponent e x p ( - M ~ p h / T ), while the rate of ordinary reactions is not. At high temperatures T > Tc only power suppression of the anomalous rate is expected, VB cc a 4w [18, 22]. As we shall see below the slowest kinetic process determining /~°r has the rate Viin ec a~. Parametrically, this is much larger than VB *. Therefore, neglecting of the anomalous reactions is quite justified. * Real calculations of the rate VB are absent. Therefore we cannot a priori exclude the possibility that the coefficient in front of a~ is large (some estimates can be found in ref. [18]) and VB >> Vkin. This is unlikely to happen but if this is indeed the case then asymmetries in quark flavours are smaller than those found in subsect. 3.6 by a factor Vkin/V,~.

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A.L Bochkare~ et al. / Baryogenesis

The fastest 2 ~ 2 reactions are q L q L ~ G G , qRqa ~ G G , q~C:tR ~ G H , { L { L ~ W W , (LgR ~ W H , ~R#R ~ BB

GW

~ GB

--, W H

~ WB

---, G B

-~ BB

~ BH

~ HH

---, W W

-~ H H

--* BH

~ HH

---, W B ---, BB ~HH together with cross channels (here B is U ( I ) gauge boson). We shall not write the contributions of all these reactions to the r.h.s, of eq. (3.8) since their structure is obvious. F o r example, processes qLCtL--, G G , qLUR--* G H , and qLDR--* G H give for the change of the left quark concentration

E

,,#,

- I2

I"

.o

.u,

j

_

2 0 , 7 ( . o ' - . o + .0D, - . o ) .

(3.16

/

Here o and o,j are cross sections averaged with thermal distributions, i and j are generation indices. However, 2 ---, 2 reactions do not fix the chemical potentials , 0 completely. The reason is simple. These reactions conserve the numbers of particles so that not all the eqs, (3.8) are linearly independent in this approximation. Namely, the sum over r of the collision integrals for 2 ~ 2 processes identically vanishes. Therefore we have to take into account processes with non-conservation of the n u m b e r of particles. The most rapid 2---, 3 reactions contain strongly interaction quark and gluons. They are qLEtL ---, G G G ,

qRqR ~ G G G ,

G G ---, G G G .

0 where ~2A,~0 = 0. The parameter ~0 We introduce the decomposition ,,0. = , o -I- A,~, is an undetermined constant if only 2 --* 2 reactions are taken into account. Since the cross section of strong 2---, 2 reactions is much higher than that for 2 ~ 3 processes, " o >>/3/{,!. To evaluate "o, we sum up all the eqs. (3.8) for quarks, gluons, leptons, W, B and scalars. Only the 2 --, 3 collision integral gives a contribution. We get

o

/~o

°2 ~ 3

Mo

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A. I. Bochkoreu et al. / Burpgenesis

Now it is possible is done result

by solving

to estimate

deviations

the eqs. (3.8) through

of the chemical iterations

potentials

with respect

from pa. This to ApLO,/p,. The

is

where (Y, is connected with the Yukawa coupling constant for the heaviest quark (Y,= a,Tr&‘/Mi, parameter 1) is fixed by condition CAP! = 0. Let us turn to the question of CP violation in different reactions.

3.4. CP-VIOLATING

EFFECTS

In this subsection we will estimate the CP-violating term in our system (3.9). Generally, the coefficients Cji are proportional to the CP-odd part of the relevant cross section, (3.19) where the bar denotes antiparticles, Imng, stands for the imaginary part of a product of coupling constants entering the tree and loop diagrams describing the reaction, and Im D is the imaginary part of a loop diagram (here coupling constants are omitted) which is a radiative correction to the chosen process (see, e.g. ref. [35]). At high temperatures the only source of CP violation is the complex Yukawa coupling constants. So we expect that the largest contribution to CP non-conservation is provided by quark-Higgs reactions like qq + HH, qS + HHH etc. Let us find the lowest order CP-violating product of Yukawa coupling constants. We have two different building blocks for left fermions: JZ?‘” and Jr,. Note that here we should account also for deviations (3.18) of the chemical potentials py from p,,. M, and M, appearing there for right quarks are converted to JZu and ~4’~ through the identities OMvOt =&!;i and OMpOt =A&. So we look for the imaginary part in the diagonal elements of a matrix being an arbitrary product of Au, An. Using the fact that J+‘” +A,, is a diagonal matrix we find the following

A . I . B o c h k a r e v et al. /

509

Baryogenesis

non-trivial matrix elements:

and [ 'D.//guJg ]ii.

(3.20) In other words, CP violation first appears in the eighth order in Yukawa couplings. Denoting by e,, the generic magnitude of eq. (3.20) we can give an estimate E h (3(.

" m 2t m 4bm 2sS12 s2s3sln ~,

e2L2oc ( - - vvt.2t vvt. 4b vvt. 2s+ m c m2 b m4 ~ l2s l s 22 s 3 s l n" a 2

4

2 2

17L3 Of. - - m c m b m s S 1 S 2 S 3 S l

n

(3.21)

~,

so that Tr e,j = 0. Here s, = sin 0i, 01 is the Cabibbo angle, 6 is the CP-violating phase. Analogous estimates show that CP non-conservation in the processes with right particles reveals itself only in the tenth order in Yukawa coupling constants. For U-type right quarks we have 2L

2L

2L

muEll,

mc~22,

mt833

while for D-type quarks 2L 8D C( mse22,

e~x

2 L mbe33.

So we neglect CP effects in the right quark sector in what follows. Now we will show that the processes qLYqR~ 3H give the largest contribution to the CP-odd part of eq. (3.9). Consider first 2 --+ 2 reactions. On tree level processes qLqL--' HH, qRqR ~ H H are necessary CP symmetric due to the CP invariant character of the initial and final state. Indeed, our choice of basis permits the reactions of this type to take place only inside the same generation. On one-loop level the processes like qL,qLj ~ H H do exist for i 4=j, so that the CP non-conserving source for instance for left quarks in eq. (3.9) is of order 4

E,

L 0

--

" 12M

L- o

Eii / l ~ H

.

(3.22)

Note that /% does not enter (3.22) because 2 ~ 2 reactions conserve the number of particles. Processes qLFqR---}G H do not produce any CP-violating effects in the one-loop approximation because the number of Yukawa couplings (at most six: four come from diagrams and two from A/x°) is not sufficient. In two-loop order for this process we use six Yukawa couplings from matrix elements and two present in

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A.L Bochkarev et al. / Baryogenesis

'",4 .....

t I

',

kt"

F:::

1 ....

Fig. 7. Diagrams for the process qL~tR~ 3H together with one-loop radiative corrections.

A~D ,, SO the estimate of CP effects appears to be the same as in eq. (3.22). An amplification of CP violation occurs in 2 ~ 3 processes because here non-conservation of the number of particles enters the game. This implies CP asymmetry of order L 0 Y'~

_ ~r 1 2 M ~

e~/a°

(3.23)

in reactions qLCtR --' 3H (see fig. 7). It is larger than (3.22) provided that/~0 >> A/~°. It is this estimate which will be used in the determination of asymmetries in quark flavours. 3.5. KINETICS OF FERMIONIC NUMBER NON-CONSERVATION In this subsection we shall find the contribution of anomalous processes to the collision integral J , thus completing the system of kinetic equations. To this end we recall the selection rules for fermionic number non-conservation. Let us first neglect the Yukawa coupling constants. Then only left fermions take part in these processes. Moreover, all the fermionic generations enter on the same footing: fermionic levels for every SU(2) doublet cross zero during the sphaleron-like transition (see fig. 8). Therefore, in this approximation the contribution of B non-conservation is identical for all left fermions and reads J ( l e f t ) = - g e e (g'L + ~/e) where ~'L and ~ respectively.

(3.24)

stand for the chemical potentials for left quarks and leptons

3

nfqL ~

nflL

Fig. 8. Anomalous sphaleron-like transition. 3n r left-handed quarks and nf left-handed leptons take part in the process.

A.L Bochkarev et al. / Bao, ogenesis

511

Fig. 9. E m i s s i o n of a Higgs boson in an a n o m a l o u s reaction.

Now let us include Yukawa corrections. This results in the renormalization of the rate VB found in the "tree" approximation and in the modification of the anomalous collision integral due to the emission of right fermions and Higgs bosons (see fig. 9). The corrections of the first type are not essential because they do not spoil the structure of eq. (3.24). It is not difficult to get non-trivial contributions originating from left-right transitions due to the scalar field. We can write for left quarks AJn(left) --

VR~2 ( ~[(OMu)ijl2(~(~+ ~ ) - (fiic - fiG+~H)I J

\k

ji

j

]

\

]

k

where ~% and ~ are the chemical potentials for the left and right leptons, M E is the leptonic mass matrix. Analogously, we get for right quarks

zaoc"(u)--V~w~w2

. I(OMu),,,I z

(g[+fi~)-(gL-fi;+~H)

,

O~w

AJ"(D)--VB~w(~i I(OKMo)i,,12(~(Ft~+ft~)-(Ft~-~'~-~n))).

(3.26)

The collision integral for leptons has the same structure and we do not write it here.

512

A.I. Bochkarev et al. / Baryogenesis

Finally, summing up all the equations for quarks and leptons we get an evolution of the quantity (B + L) = F: 1 OF -

V Ot

(~i "

°~w( j

M,~

+ ~I(OKMI)),.jl2(g~-~)-gn)+j ~. I(ML) jjj 12(g¢-- ~ -- g H ) ) )

(3.27)

Only anomalous processes are relevant for its changes because F is exactly conserved on perturbative level. Note also that we do not include any CP effects proportional to ~t°. They should be proportional to the eighth power of Yukawa coupling constants as well as to the rate of B non-conservation and hence are small compared with terms found in subsect. 3.4. 3.6. SOLUTION OF THE KINETIC EQUATIONS

N o w we are in a position to specify the system (3.9) for fermions and solve it. The fastest ordinary kinetic reactions changing fermionic chemical potentials are qLV:IR, G H (see fig. 10). Collision integrals for these processes have the form (~(; = 0 from SU(3) symmetry): I~o,(left)

OlsO/w(~] .

+

+ EI(OKMD).jl2(fL"L--M,--ft.)), J

I~°'(C)

~rM~ . OMu)i.I (#L

1" - c o ,tD~ ,,

asa'~ ~rM~~ .[

/~U+~H),

(OKMD)*"I2(t~--~D -

- "

~)"

(3.28)

We should also account for the contribution of anomalous processes (3.25) and

q

i

L

J qL Fig. 10. Fastest reaction establishing the chemical equilibrium between quark generations.

A.L Bochkarev et aL / Baryogenesis

513

(3.26). However, we will show a posteriori that this contribution is inessential in the leading order in Yukawa couplings. Moreover, as was found in subsect. 3.4 the CP non-conservation is substantial for left quarks only. Under these conditions the system (3.9) takes the form 2

-i

-n

Z I(OMu),.I (~L-- ~U + ~H) =0 , i

EI(OKMI))i,,I (t~L-~D 2

--i

--n

_

_

fiH)=o,

i

fi:-~.-fi~=o, O~sO/w

Z/ ( I(Oiu).,I 2(ilL-f,6 . +~H) +I(OKMD)~,I (t~ ~--U*H)) = 1

(2)4 .w

(3.29a)

Consider the implications of eq. (3.29a) for the anomalous collision integrals (3.25) and (3.26). Owing to the similar flavour structure of (3.29a), and (3.25) and (3.26) we find that A J " is proportional to Zi(~'L + g~e) as in the zeroth order (3.24). (For left quarks this is true up to the term proportional to Tr e L -- 0.) Thus, Yukawa corrections spoiling the "tree" structure (3.24) start only from fourth order. Now the anomalous kinetic equation (3.27) yields for semi-equilibrium (OF/Ot = 0):

Y'. ( g~ + ~ ) = O.

(3.29b)

i

Eq. (3.29b) is important to prove the self-consistency of our previous manipulations. Indeed, if we return to the original system (3.28) to restore the anomalous contribution but account for (3.29b) we obtain the same equations (3.29a) as before. This is again due to the similar structure of the collision integrals for ordinary and anomalous processes in the second order in Yukawa couplings. The solution to eq. (3.29a) contains five arbitrary constants /~e, - j Z ~ and ~r~ corresponding to five on this level exactly conserved charges B, L i and hypercharge y*: --1

1

g'R = ~'~- g . ,

1

g'u = g'~ + g . ,

--i

~b = g'L - g . ,

(3.30)

* Obtaining this solution we have taken into account that the t quark is much heavier than the others.

A . L Bochkarev et al. / Baryogenesis

514

where Aft is connected with CP violation and given by the expression 1

1

4

A f t - 7r2 ~C~wl 2M 2

Mw

(](gMD)1212+[(gMD)1312)~lt~o.

(3.31)

We omit the matrix O because its deviations from unity are small, O - 1 O(SlS2(S 2 + s3)m2~/m 2) ~ 10 -2. Eq. (3.29b) ensures in fact the equilibrium character of anomalous processes and fixes one of the arbitrary constants. To fix the others we should add also conditions which guarantee the neutrality of our system with respect to conserved charges A i and hypercharge: A i = 0 and Y = 0. Taking into account radiative corrections in the calculation of the conserved numbers we get

ft;- ft.- Eft'L- o(f )aft,

(3.32)

where f is the generic Yukawa coupling. 4. E l e e t r o w e a k

CP v i o l a t i o n a n d e f f e c t i v e a c t i o n

Thus, we have fixed the CP and CPT non-conserving part in the fermionic density matrix. With the use of the results of sect. 2 it is possible to find the effective action for gauge fields. As seen from (3.32), the knowledge of only Aft is necessary, because all other contributions are suppressed by the powers of Yukawa coupling constants. We get for the Chern-Simons density T 2

2 2 O~w m t . _

AQ=SmsM00 N c s - ~-/1/~ A/'tNcs"

(4.1)

An order-of-magnitude estimate of Aft (eq. (3.31)) gives for the coefficient in front of Ncs _

3ms

C~w/~w]4

4 4 2 mtmbmsS2S3Sln3 --10 is 2 2 +s2mb) 2 2 - - ~ M 2 ] oq(ms

(4.2)

This value is sufficient to bring the system into the state with maximal possible Chern-Simons density if the effective potential for Arcs is flat. Recall [8], that this happens if the quantity Z = 4~raZ3msMo/Tc >_ 1. From (4.2), Z - 1. An important comment is now in order. CP-violating effects found in this paper are much larger than those found in ref. [8]. In fact, they are only sixth order in coupling constants. In ref. [8] it was claimed that CP violation in the effective action for gauge fields reveals itself only in 14th order in Yukawa couplings. The arguments were based on looking for the lowest order CP non-conserving trace of Yukawa constants. It was found that ImTr J g 3 j g 2 ~ ' v ~ ' D ~ 0 (in 12th order there

A.L Bochkarevet a L / Ba(vogenesis

515

9 2 is a compensation of traces Tr.ffg~.ffgD.ffguJg D and T r J g Z , g ~ g D . A ' u ) . The key observation made in this paper is the explicit CP and C P T non-invariance of the fermionic density matrix due to CP breaking and deviations from equilibrium in ordinary kinetic processes. This greatly increases CP effects. Technically that happens because (i) there is no need to calculate traces over fermionic generations (see eq. (3.21)) and (ii) cross sections of the transitions between generations enter the denominator (see (3.31)). The renormalization of the Chern-Simons term even in the case of anomaly-free fermionic densities (like B - L, Ai) discussed in sect. 2 is of great importance too*. It may be shown that if radiative corrections to the CPT-odd part of the effective action of the gauge fields are absent in thermal equilibrium, then CP violation and universe expansion come into play only in 14th order in Yukawa couplings, 6m.~ ~ f14.

5. GUTs and eleetroweak baryon asymmetry In this section we scenarios of the BAU renormalization of the G U T asymmetries in effective lagrangian is

will discuss the interplay between G U T and electroweak generation. The interference does exist because there is a Ncs term in the effective action for the gauge fields due to A, provided they are nonzero. The CP-odd part of the

/ T2 A ~ = t6m%UTT-~-~mEsW~0 Arcs.

(5.1)

It is clear that if 6(mUT> 6~ w -- 10 -15, then electroweak CP violation plays no role in the whole scenario because the G U T contribution dominates at all temperatures. Moreover, most rapid changes in the Chern-Simons number in the EW mechanism of the BAU production occur near the critical temperature. So we can neglect 6~ w if the much weaker inequality is fulfilled rc

10 -32

(5.2)

In other words, if a very small baryon asymmetry (even non-sufficient for explaining the observed BAU) is produced by GUTs, then it still determines the CP violation in the anomalous electroweak physics. Hence the sign of the BAU is indeed connected with the sign of CP violation in K ° decays only if G U T asymmetries are in fact absent (see (5.2)). Suppose that the G U T asymmetry is large enough. What is the total contribution to the BAU, from these two sources? The answer depends on the form of the effective potential for Ncs. Consider three possible cases. * Note that quark asymmetries coming from CP violation in the leading order in coupling constants (see eq. (3.30)) introduce a source term for anomaly-free charge, counting the number of quarks in second and third generations minus the number of quarks of first quark family: ~ = ~[ = - ~L-

A.I. Bochkarev et al. / Ba~'ogenesis

516

(i) If the potential is trivial (no degeneracy) the contribution to the BAU coming from SU(2) interactions is small compared with the G U T asymmetry. Using the results of ref. [8] it is easy to see that at least Asu{2) < a 3w A GUT" (ii) In the case of a flat potential, the G U T contribution bends the plateau (see fig. l b ' ) and in addition to the G U T asymmetry we get an SU(2) asymmetry which is maximal ( - a3,) if 6,~u'r > ( T c / a w M o ) 2 ~ 10 -3°. To reduce this asymmetry to the observed value, the Higgs boson mass should be M H = M c w ( m t ) or M H --- Merit = 45 GeV if AGu v < Aob~. On the other hand, M H > Men t if A t , v- r = AobS. The sign of the BAU is determined by G U T physics. Here G U T effects just trigger the SU(2) BAU generation. (iii) If the potential has a double well form (see fig. la), the G U T contribution, being sufficiently large, could purify the initial mixed state of the universe consisting of an equal number of domains with opposite C P parity. (See for a detailed discussion ref. [8].) Following ref. [8], we can write the equations describing the eating of less energetically favourable domains by other ones: 3V + Ot

3V- HV++

c,

-

HV--

(5.3)

c.

Ot

Here c is the velocity of the domain wall, separating fractions of space with opposite parity and volumes V + and V . A simple power counting estimate of c gives c -- 6,c,;~v (cf. ref. [8]). The solution shows that domain structure disappears to the temperature (5.4)

T* = C a w M o .

Therefore, SU(2) asymmetry will be maximal i f ~2y T > T c / / O c w m 0 - 1 0 - 1 5 . Again, if C,UT < A ob~ the main contribution to the BAU comes from the electroweak phase transition. There is no connection between the sign of A and K ° decays. The mass of the Higgs boson must be greater than M e r i t if AGm- = AobS and should be equal to Merit or M e w if AGu r < Otobs. It is possible to have Mew < M < Mere if the domain structure still exists at the moment of the electroweak phase transition, but the relation between different C P phases exactly ensures the correct baryon asymmetry: V+ V

= 1 + Caw

M0 T '

Caw

M0 T

za obs Ama x '

ZXm~

-

3 Otw"

(5.5)

6. Conclusion In this paper we found a way of calculating C P effects in the effective action for gauge fields in the expanding universe. Let us summarize the implications of our results for the EW scenario of baryogenesis.

A.L Bochkarev et al. / Ba(vogenesis

517

T h e C P effects in pure EW theory appear to be sufficiently large to bring the system to the state with maximal possible C h e r n - S i m o n s n u m b e r in the case of infinite degeneracy with respect to Arcs. Baryon asymmetry production in the first 3 The Higgs mass order electroweak phase transition is then maximal, n B / n ~ _ a w. should lie very near M~r~t-~ 45 GeV or the C o l e m a n - W e i n b e r g value of the Higgs mass (--- 9 G e V if m t < Mw). Actual computation of the sign of b a r y o n asymmetry and its relation to the sign of C P violation in K ° decays is possible. It should involve (i) calculation of the C P T - o d d piece of the effective action for gauge fields at high temperatures and small fermionic chemical potentials, (ii) calculation of lowest order strong radiative corrections to particle n u m b e r densities, (iii) estimate of C P violation in the processes qLYqR--* 3H on the one-loop level. We plan to return to these problems in future. W e f o u n d that a G U T asymmetry, even if being too small to explain the b a r y o n a s y m m e t r y of the universe, could ensure the working of the electroweak scenario of the B A U generation. In particular, it brings the universe to the pure state with n o n - z e r o Ncs even in the case of finite degeneracy of the potential for the C h e r n - S i m o n s density corresponding to spontaneous C P violation in the hot electroweak plasma. The crucial question to be solved to clarify the status of the whole scenario is the f o r m of potential for Ncs. Its solution requires Monte Carlo simulations of electroweak theory at large lattices, see ref. [20]. T h e authors are grateful to V.A. Kuzmin, V.A. Matveev, L. McLerran, V.A. R u b a k o v and A.N. Tavkhelidze for discussions related to this work. One of us (A.B.) is indebted to Larry McLerran for kind hospitality at the Theoretical Physics Institute, University of Minnesota, where this work was completed. M.S. would like to thank G. Semenoff for important discussions on non-renormalization theorems and for his kind hospitality at the University of British Columbia (Vancouver).

References [1] A.D. Sakharov, Pisma ZhETF 5 (1967) 32: V.A. Kuzmin, Pisma ZhETF 13 (1970) 335; A.Yu. Ignatiev, N.V. Krasnikov, V.A. Kuzmin and A.N. Tavkhelidze, Proc. Int. Conf. Neutrino-77, vol. 2 (Nauka, Moscow, 1978) p. 293; Phys. Lett. B76 (1978) 436; M. Yoshimura, Phys. Rev. Lett. 41 (1978) 281; 42 (1979) 476; S. Weinberg, Phys. Rev. Lett. 42 (1979) 850; A.Yu. Ignatiev, V.A. Kuzmin and M.E, Shaposhnikov, Phys. Lett. B87 (1979) 114 [2] S. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 60A (1969) 47 [3] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D14 (1976) 3432 [4] R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172 [5] C.G. Callan, R.F. Dashen and D.J. Gross, Phys. Lett. B63 (1976) 334

518 [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16] [17]

[181 [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

[30] [31]

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