Strong Sphalerons and Electroweak Baryogenesis

Home , Hypercharge

CERN-TH.7080/93

1 2

G. F. Giudice and M. Shap oshnikov

Theory Division, CERN,

CH-1211 Geneva 23, Switzerland

Abstract

We analyze the sp ontaneous baryogenesis and charge transp ort mechanisms sug-

gested by Cohen, Kaplan and Nelson for baryon asymmetry generation in extended

versions of electroweak theory.We nd that accounting for non-p erturbativechirality-

breaking transitions due to strong sphalerons reduces the baryonic asymmetry by the

factor (m = T ) or , provided those pro cesses are in thermal equilibrium.

t W

CERN-TH.7080/93

Novemb er 1993

On leave of absence from INFN, Sezione di Padova.

On leave of absence from the Institute for Nuclear Research of the Russian Academy of Sciences,

Moscow 117312, Russia.

The p opular mechanisms for electroweak baryogenesis discussed in the literature

are the sp ontaneous baryogenesis mechanism and the charge transp ort mechanism,

b oth suggested by Cohen, Kaplan and Nelson [1, 2, 3 ]. It has b een advo cated in

[1, 2, 3] that these mechanisms pro duce an asymmetry parametrically larger that the

mechanism asso ciated with the decay of top ologically non-trivial uctuations of the

gauge and Higgs elds during the electroweak phase transition, considered in refs. [4,

5](for earlier suggestions, see refs. [6, 7 , 8]). In the latter case, the asymmetry pro duced

is prop ortional to the square of the ratio of the top-quark mass to the temp erature

[4], while for the rst two mechanismsitwas argued that this suppression is absent

[1, 2, 3]. The key observation made in [1, 2, 3] is that in the presence of a non-

zero fermionic hyp ercharge density, , the equilibrium value of the baryonic charge,

generally sp eaking, is not equal to zero:

hB i = A ; (1)

where A isanumb er determined by the particle content of the theory and A 6=0 in

the limit m ! 0.

In this pap er, we show that under some circumstances A may b e zero in the massless

quark limit. Moreover, we will argue that this is most probably the case for the two

Higgs doublet theory, considered in [1, 2 ].

In this mo del, in addition to the KM CP-violation, there is an explicit CP-violation

in the Higgs sector. The vevs of the Higgs elds at zero temp erature have the form:

h i =(0;v e ); h i =(0;v ); (2)

1 1 2 2

where is the relative phase, which cannot b e rotated awayby a gauge transformation.

Due to explicit CP-violation in the Higgs p otential this phase is space-dep endent inside

bubble walls forming at the electroweak phase transition (it would b e time-dep endent

for a spino dial decomp osition phase transition). Since quarks are getting their masses

from interactions with the Higgs elds, the most imp ortantinteraction is the top-quark

coupling ,

L = f Q U : (3)

Y t 3 3 1

Here Q are the left fermionic doublets, U ;D are right quark elds, and i is the

i i i

generation index. The b ottom-quark couplings can b e safely ignored provided the

thickness of the domain wall l or the duration of the spino dial decomp osition phase

transition is smaller than the rate of chirality-changing transitions asso ciated with

the b ottom Yukawa coupling; that is, if

2 1

l < ( f T ) ; (4)

0 s

where T is the temp erature. This is always the case for current estimates of the domain

wall thickness, ranging from 40=T [9] to 1=T [10 ]. The same statementisobviously

true for lighter quarks.

We assume that the rst Higgs eld couples only to up-quarks, while the second is coupled only

to down-quarks. This assumption is, however, inessential in what follows. 1

We will clarify our p oint rst within the context of the sp ontaneous baryogenesis

mechanism. Let us take for simplicity the spino dial decomp osition phase transition.

As in [1], we assume that the rate of chirality- ipping transitions for the top quark is

larger than the typical inverse-time scale of the variation of the phase during this

transition. Now, following [1], we makea hyp ercharge rotation of the fermionic elds

in sucha way that time-dep endence disapp ears from the Yukawa coupling (3). This

change of variables intro duces the following mo di cation of the e ective lagrangian:

L = L + Y ; (5)

ef f F

where Y is a fermionic hyp ercharge op erator,

1 4 2

Y = : (6) Q Q + U U D D L L 2E E

F i 0 i i 0 i i 0 i i 0 i i 0 i

3 3 3

i=1

Wewanttocheck whether in thermal equilibrium with resp ect to fermionic number

non-conserving pro cesses in a theory with e ective action (5) the baryonic number is

nonzero. To determine the number A de ned in (1), the following standard pro cedure

must b e used (see e.g. ref. [11]). One has to de ne the complete set of conserved

charges X and construct the most general equilibrium density matrix with the help

of those charges, intro ducing chemical p otentials for each of them. The partition

function is

" #

Z = Tr exp (H Y X ) ; (7)

F i i

while the density matrix is

1 1

= exp (H Y X ) : (8)

F i i

Z T

Now, since X are conserved op erators, their average must b e equal to zero. This

requirement xes the chemical p otentials through the system of equations

Z =0: (9)

Then the baryonic numb er is just

hB i = Tr[B]: (10)

Let us de ne the set of conserved numb ers. It is sucient to consider only SU(3)

gauge singlets. We start from purely fermionic currents. In a mo del with massless neu-

trinos (no right-handed neutrinos!), the total numb er of di erent fermionic currents

is 15, the numb er of fermionic degrees of freedom. They are: 3 left quark currents,

Q Q , 6 right quark currents, U U and D D , 3 left leptonic currents, L L

i i i i i i i i

(L is the left leptonic doublet), and 3 right leptonic currents, E E . Not all of these

i i i

currents are conserved. One has to takeinto account the following pro cesses: 2

(i) Perturbativechirality-changing transitions due to Yukawainteractions (3). The

rate of these pro cesses is estimated to b e 30=T [3]. This decreases the number

of conserved currents by one: instead of the pair Q Q and U U ,wehave a con-

3 3 3 3

served linear combination Q Q + U U .

3 3 3 3

(ii) Non-p erturbativechirality-breaking transitions due to strong interactions. It

is well known that the quark axial vector current has an anomaly and therefore is

not conserved. The rate of chirality non-conservation at high temp eratures is

str ong

connected with the rate of top ological transitions in QCD (\strong" sphalerons, [12 ]),

@Q 12 6

= Q ; (11)

str ong 5

@t T

where Q is the axial charge. The factor of 12 comes from the total numb er of quark

chirality states, the factor of 6 from the relation b etween the asymmetry in quark

numb er density and the chemical p otential,

2 3 2

T d k 3m

[n (; ) n (; )] = ); (12) (1

F F

3 2 2

(2 ) 6 2 T

2 2 2

where n (; )=1=(exp ( ) + 1) is the Fermi distribution, = k + m . The rate

of the strong sphaleron transitions is related to the rate of weak sphaleron transitions

through

4 4

= ( ( T ) ; (13) ) =

str ong s sph

3 3

where is the usual parameter characterizing the strength of the electroweak sphaleron

transitions in the unbroken phase. The characteristic time of these transitions is there-

fore

= : (14)

str ong

192 T

Using the conservativelower b ound on derived in lattice simulations [13], >0:5

(see also discussion in ref. [14 ]), we obtain a conservative b ound < 100=T . With

str ong

20, so estimated by a di erent metho d in ref. [15], we obtain 2:5=T .

str ong

Therefore, the rate of strong sphaleron transitions is comparable to or even larger than

the rate of chirality- ip transitions through the Yukawa coupling of the top quark.

Hence, these pro cesses must b e taken into account. This decreases the numb er of con-

served currents by one .

(iii) Anomalous fermion numb er non-conservation must b e taken into account [17].

It decreases the numb er of conserved currents by one.

Note that in uence of strong sphalerons on a di erent mechanism for electroweak baryogenesis

has b een discussed also in ref. [16]. 3

Therefore, a complete set of conserved anomaly-free fermionic currents can b e rep-

resented as

Q Q + R L ;i=1;2; (15)

i i L

Q Q + U U + R=2 L ; (16)

3 3 3 3 L

L L L =3;i =1;2; (17)

i i L

E E ;i =1;2;3; (18)

i i

D D R=2;i =1;2;3; (19)

i i

U U U U ; (20)

1 1 2 2

where

3 2

X X

L = L L ;R= U U : (21)

L i i i i

i=1 i=1

In addition to the quantum numb ers asso ciated with these currents, there exists a

conserved op erator containing scalar elds. This is the familiar hyp ercharge

Y = Y + Y ; (22)

F s

where

Y = i[ D (D ) ] i[1 ! 2] (23)

s 0 1 0 1 1

is the scalar eld contribution. Finally, one should add the third comp onent of the weak

isospin op erator T . As in [1, 3 ], we assume that the classical motion of the scalar eld

do es not intro duce any non-zero hyp ercharge or T density. Therefore the hyp ercharge

density is equal to zero for a uniform spino dial decomp osition phase transition.

Now one can check that, in the massless quark approximation, eqs. (7, 8, 9, 15 -

20) imply that the equilibrium value of the baryonic charge is zero, hB i = 0. In order

to get this result, we used, following ref. [18 ], the particle sp ectrum of the unbroken

phase rather than the broken one.

The reader maywonder whywe did not use the electric charge op erator and the

physical particle sp ectrum of the broken phase instead of Y and T . The reason

is that the use of the physical sp ectrum in such a calculation would corresp ond to

a computation of the free energy of the system in the broken phase in the one-lo op

approximation in a unitary gauge. It is known [19, 20 , 21 ], that p erturbation theory in

a unitary gauge is a very delicate thing (e.g., it do es not converge [21]). In particular,

it gives an incorrect result for the critical temp erature of the phase transitions in gauge

theories.

It is clear that for suciently small vacuum exp ectation values of the Higgs eld, all

e ects asso ciated with sp ontaneous symmetry breaking are just the mass corrections

which can b e neglected in the high-temp erature approximation in any renormalizable

gauge. In the worst case, the expansion parameter maybe(v=T ) . In our case, the

vacuum exp ectation value of the Higgs eld v is b ounded from ab oveby the requirement

Actually, the zero result is repro duced also with Q instead of T and Y and the particle sp ectrum

of the broken phase. 4

that the sphaleron pro cesses are suciently fast, so that v

of the particle sp ectrum of the unbroken phase is p erfectly justi ed.

The zero result is also repro duced when fermionic numb er non-conservation is out

of thermal equilibrium. Then one can write an equation for the baryonic number

evolution [18 ] (see also refs. [1, 2, 3])

@B @F

sph sph

= N = 9 ; (24)

@t T @B T

where F (B ) is the free energy of the system characterized by zero values of all conserved

charges but non-zero baryonic density B , is the rate of sphaleron transitions, and

sph

N = 3 is the numb er of fermionic generations. In our case,

= B; (25)

so that B stays zero all the time.

We stress that the inclusion of strong sphalerons was essential for this result. Strong

sphalerons have the physical e ect of mantaining the same chemical p otential for left-

and right-handed baryonic numb ers. If instead one neglects these pro cesses, then the

system of charges (20) has to b e supplied with an extra charge, say R, while the density

matrix will contain an additional chemical p otential. One can check that if one cho oses

this set of conserved charges, the result is indeed non-zero. Which solution has to b e

used? The answer dep ends on the ratio of the typical time scales. If , then

0 str ong

strong sphalerons are irrelevant, and the asymmetry indeed do es not contain Yukawa

coupling suppression provided that is smaller than the typical time scale of the

Y 0

Higgs phase change in the sp ontaneous baryogenesis mechanism. If, on the contrary,

> or , then the asymmetry vanishes in the massless approximation.

0 str ong 0 str ong

In the original calculation of Cohen, Kaplan and Nelson [1, 2] a non-zero result for

the baryonic asymmetry was obtained, since strong sphaleron transitions have b een

neglected and only B L, Q, and B (B + B )have b een included as conserved

3 1 2

charges. However, in presence of strong sphalerons transitions, their solution corre-

sp onds to non-zero averages for some of the approximately conserved charges de ned

in eqs. (15 - 20).

Is the conclusion that hB i = 0 fatal for the sp ontaneous baryogenesis mechanism?

In fact, this is not the case when quark mass corrections are included. Mass corrections

can b e taken into account in p erfect analogy with the case of the leptonic asymmetries

discussed in refs. [22, 18 ] (for a later discussion see ref. [23]). To make the computation

simpler (and to make clearer whywe previously obtained zero), we observe that in fact

the numb er of indep endentchemical p otentials (14) can b e reduced to a smaller number

(6) with the use of avour symmetry. Namely, our lagrangian has quite a large global

symmetry group SU (2) x SU (3) x SU (2) x SU (3) x SU (3) , where the group

Q D U L E

lab els denote the elds on which the corresp onding group is acting. The conserved

charges invariant under this symmetry are

A = Q Q +2R 2L ; (26)

1 i i L

i=1 5

A = Q Q + U U + R=2 L ; (27)

2 3 3 3 3 L

A = D D 3R=2; (28)

3 i i

i=1

A = E E ; (29)

4 i i

i=1

hyp ercharge Y and weak isospin T .We denote the corresp onding chemical p otentials

by ;i=1,...,4, and . They are to b e found from the equations

i Y T

hA i = hY i = hT i =0: (30)

i 3

Let us rst solve this system neglecting the mass of the top quark. In order to show

that the baryonic density is zero, one can consider only the total baryonic density and

the left-handed leptonic density. The total baryonic densityis

hBi= 4[( + )+2 + ] ; (31)

Y 1 2

while the total left leptonic densityis

: (32) hL i=6[( + )+2 + ]

L Y 1 2

Then from hB L i = 0, it follows hB i =0.For future reference, we give here the

complete solution of eq. (30) in the massless approximation:

4 13 11

_ _ _

= ( + ); = ( + ); = ( + ); (33)

1 Y 2 Y 3 Y

21 21 21

14n

_ _ _

=2(+ ); =0; (+ )= ; (34)

4 Y T Y

9+14n

where n is the numb er of scalar doublets.

The exact prop ortionalitybetween baryonic numb er and left-handed leptonic number

is, however, an artifact of the massless approximation. Using eq. (12), the baryonic

density b ecomes

3m 9n

hB i = = m ; (35)

2 2 2

20 T 10(9 + 14n )

where is the fermionic hyp ercharge density. This is our nal result for the equilib-

rium density of the baryonic numb er in the background of a scalar eld with slowly

changing phase.

If strong sphalerons were not in equilibrium,we should add a new approximately

conserved charge, for instance A R, to our previously de ned set of charges, see

eqs.(26){(29). Following the same pro cedure used to obtain eq.(35), wewould nd:

hB i = T : (36)

6+11n

s 6

m 126

( ) , Therefore the presence of strong sphalerons leads to a suppression factor

185 T

where wehave taken n = 2; this corresp onds numerically to a suppression of ab out

3 10 for f 1, v (T ) g T .

t W

The approximation that anomalous baryon numb er violation is in equilibrium is

unlikely to b e correct. Thus, following ref. [1], we consider the evolution equation for

the baryon asymmetry density, eq. (24). We can compute using the same pro cedure

followed ab ove, adding B to the set of conserved charges and imp osing, b esides eq. (30),

the further constraint hB i =0.We nd, if strong sphalerons are in equilibrium,

9n m

= ; (37)

2 2

4(9 + 14n ) T

and, if strong sphalerons are not in equilibrium,

: (38) =

3(1 + 2n )

The mass suppression present when strong sphalerons are in equilibrium amounts to a

135 m

2 2

factor ( ) 2 10 .

296 T

We can now consider the case of baryogenesis with the charge transp ort mechanism.

As shown in ref. [2], the CP-violating interactions of the fermions with the moving thin

bubble wall lead to a non-vanishing ux of particles carrying some quantum number X .

If, as in the original work of ref. [2], we identify X with the hyp ercharge, then it is easy

to realize that hB i = 0, when strong sphalerons are in equilibrium and all fermions are

massless. The calculation is analogous to the one for sp ontaneous baryogenesis, with

the only di erence that, instead of the source wehave to consider a non-vanishing

Y density.However, as shown by Khlebnikov [24 ], the Debye screening prevents the

transp ort of the gauged charge Y over distances larger than ab out 2=T .

Therefore the recip e prop osed in ref. [25 ] is to identify as the transp orted charges X

only global charges which are orthogonal to gauge charges. We then de ne

0 0

B = B + Y; A = A + Y; i =1; :::; 4; (39)

B i i

0 0

with and chosen such that B and A are orthogonal to Y , i.e., in a massless

B i

quark approximation:

1 1 2 2 1

= = = = = : (40)

B 1 2 3 4

8 4 9 3 10 + n

By intro ducing chemical p otentials for the charges (26){(29), Y , and B ,we can write

the asymmetry density for each sp ecies of particles as:

= q + q + q : (41)

Y Y B B A i

The chemical p otentials , , and are then computed from the equations

Y B i

hAi hBi

= =n 6=0; (42) hY i =0;

B i 7

which corresp ond to a situation in which the gauged charge Y is screened, but the Y

comp onents of the global charges can carry a non-vanishing density n . The solution of

eq. (42) yields = 0 and therefore anomalous baryon numb er violating interactions

are unable to generate any baryon asymmetry. One can also check that 6=0,if

strong sphalerons are not in equilibrium.

The cancellation in the case of charge transp ort baryogenesis may seem more severe

than in the case of sp ontaneous baryogenesis since the relevant pro cesses o ccur in

the unbroken phase, where the Higgs eld condensate is absent. There are twotyp es

of corrections whichsave the situation. Just as in the leptonic case considered in

[18], Yukawa and gauge radiative corrections are imp ortant. The equation for the

asymmetry numb er density of eachchiral fermion in the unbroken phase, including

radiative corrections, can still b e written as in eq.(12), with the replacement:

" #

2 2 2

Y f T

2 2 2 2

m ! C g +(C + sin )g + ; (43)

s W W

s W

4 2 4

4 3

where C is for quarks and 0 for leptons, C is for weak doublets and 0 for

s W

3 4

singlets, Y is the hyp ercharge quantum numb er, and f is the Yukawa coupling, taken

to b e non-vanishing only for the top quark. The solution of eq. (42) now gives:

9 n

2 2 2

= (3f +2g sin ) ; (44)

B W

t W

2 2

64 (9+14n ) T

which, as noticed ab ove, vanishes in the limit of negligible radiative corrections. The

result of eq. (42) would corresp ond to a reduction of the baryonic asymmetry in

27(1+2n )

2 3

comparison with [2, 3] by the factor f 3 10 , if other e ects are

128 (9+14n )

absent. These more imp ortant e ects are asso ciated with electroweak corrections which

break the degeneracy b etween left and right top quarks in the unbroken phase resulting

in a net baryonic current J (see refs. [26, 14 ]) related to the hyp ercharge current J

B Y

through

2 p p

L R

J = J ; (45)

B Y

3 p + p

L R

where p and p are the momenta of incident left and right quarks corresp onding to

L R

the same energy.Now, p and p are di erent since left quarks interact with SU(2)

L R

gauge elds while right quarks do not. With the use of eq. (43) we get a larger baryonic

chemical p otential,

W x

' ; (46)

16 p

wherep>m is a typical transverse momentum of the quarks contributing to charge

transp ort. So the nal estimate of the suppression of the asymmetry in this case

is a factor of ab out 10 . Therefore, the charge transp ort mechanism (as well as

the \top ological" mechanism [4, 5]) can still explain the baryonic asymmetry since,

according to the estimate of ref. [2], n =s can b e as large as 10 . With strong

sphalerons taken into account, this numb er should b e converted to 10 , which is still

larger than observation. 8

To conclude, wehave shown that the baryonic asymmetry in the two Higgs doublet

mo del pro duced byany of the mechanisms considered in [1, 2, 3] contains a parametric

suppression asso ciated with the top Yukawa coupling or electroweak coupling due to

the existence of strong sphalerons. Strong sphalerons have not b een taken into account

inanumb er of pap ers on electroweak baryogenesis and maychange their conclusions.

One of us (M.S.) is indebted to the High Energy Physics Group at Rutgers University,

where part of this this work has b een done, for kind hospitality.We thank G. Farrar,

D. Kaplan, S. Khlebnikov, A. Nelson and N. Turok for useful discussions.

References

[1] A. Cohen, D. Kaplan, and A. Nelson, Phys. Lett. B263 (1991) 86.

[2] A. Cohen, D. Kaplan, and A. Nelson, Nucl. Phys. B373 (1992) 453.

[3] A. Cohen, D. Kaplan, and A. Nelson, Preprint UCSD-PTH-93-02 (1993).

[4] L. McLerran, M.E. Shap oshnikov, N. Turok, and M.B. Voloshin, Phys. Lett. 256B

(1991) 451.

[5] D. Grigorev, M. Shap oshnikov, and N. Turok, Phys. Lett. B275 (1992) 395.

[6] M.E. Shap oshnikov, Nucl. Phys. B299 (1988) 797.

[7] J. Amb jrn, M.L. Laursen, M.E. Shap oshnikov, Nucl.Phys. B316 (1989) 483.

[8] N. Turok and J. Zadrozny,Phys. Rev. Lett. 65 (1990) 2331; N. Turok and J.

Zadrozny, Nucl. Phys. B358 (1991) 471; N. Turok, Persp ectives on Higgs Physics:

300 (1992).

[9] M. Dine, R. Leigh, P. Huet, A. Linde, and D. Linde, Phys. Rev. D46 (1992) 550.

[10] M.E. Shap oshnikov, Phys. Lett. 316B (1993) 112.

[11] L.D. Landau and E.M. Lifshitz, Statistical Physics, Nauka, Moscow, 1974.

[12] L. McLerran, E. Mottola, and M. Shap oshnikov, Phys. Rev. D43 (1991) 2027.

[13] J. Ambrn, T. Askgaard, H. Porter, and M.E. Shap oshnikov, Nucl. Phys. B353

(1991) 346.

[14] G.R. Farrar and M.E. Shap oshnikov, PreprintRU-93-11, CERN-TH.6734/93,

1993.

[15] P. Arnold and L. McLerran, Phys. Rev. D36 (1987) 581.

[16] R.N. Mohapatra and Xinmin Zhang, Phys. Rev. D45 (1992) 2699; Phys. Rev. D46

(1992) 5331. 9

[17] V.A. Kuzmin, V.A. Rubakov, and M.E. Shap oshnikov, Phys. Lett. 155B (1985)

36.

[18] S.Yu. Khlebnikov and M.E. Shap oshnikov, Nucl. Phys. B308 (1988) 885.

[19] L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 3320.

[20] S. Weinb erg, Phys. Rev. D9 (1974) 3357.

[21] P. Arnold, E. Braaten, and S. Vokos, Phys. Rev. D46 (1992) 3576.

[22] V.A. Kuzmin, V.A. Rubakov, and M.E. Shap oshnikov, Phys.Lett. 191B (1987)

171.

[23] H. Dreiner and G.G. Ross, Preprint OUTP-92-08P (1992).

[24] S. Khlebnikov, Phys. Lett. B300 (1993) 376.

[25] A. Cohen, D. Kaplan, and A. Nelson, Phys. Lett. B294 (1992) 57.

[26] M. Shap oshnikov, Phys. Lett. 277B (1992) 324; 282B (1982) 483 (E). 10