Baryogenesis through mixing of heavy Majorana neutrinos

26 December 1996

PHYSICS

LETTERS 6

Physics Letters B 389 (1996) 693-699

ELSEWER

Baryogenesis through mixing of heavy Majorana neutrinos Marion Flanz, Emmanuel A. Paschos, Utpal Sarkar

*,

Jan Weiss

Institut jEr Physik, Universitiit Dortmund, D-44221 Dortmund, Germany Received

15 July 1996; revised manuscript received Editor: PV. Landshoff

14 September

1996

Abstract A mechanism is presented in which the mixing of right-handed heavy Majorana neutrinos creates a CP-asymmetric universe. When these Majorana neutrinos subsequently decay more leptons than anti leptons are produced. Due to a resonance phenomenon the lepton asymmetry created by this new mechanism can exceed by a few orders of magnitude any lepton asymmetry originating from direct decays. The asymmetry is finally converted into a baryon asymmetry during the electroweak phase transition.

The generation of the baryon asymmetry in the universe has been discussed in many articles [ 1,2]. Two prominent scenarios are the Grand Unified Theories [ 2,3]and the production of an asymmetry through extended solutions of field theories (sphalerons) [ 4,5]. At this time both scenarios generate asymmetries which are small. This comes about because in the former case CP-violation is produced by higher order effects and in the latter the tunneling rate through potential barriers is small (in addition, if the higgs particles are heavier than 80 GeV, then the baryon asymmetry thus generated will be completely erased) [ 51. In supersymmetric theories there is more freedom and alternative scenarios have been discussed [ 6-81. For instance, a baryon asymmetry is generated in the Affleck-Dine mechanism [6] at an energy where supersymmetty is broken. In a different scheme the sphalerons play a role; they increase the number of superparticles at lower temperatures, which in turn increases the lepton asymmetry in the universe [ 81. Another interesting possibility is a minimal extension of the standard model, which includes heavy Majorana neutrinos. The decays of the neutrinos generate a lepton asymmetry, which later on is converted into the baryon asymmetry. By their very nature Majorana neutrinos posses AL = 2 transitions and in addition they may have couplings which allow them to decay into the standard higgs and leptons, i.e., N + q5t I [ 91. In these models the CP-violation is introduced through the interference of tree-level with one-loop diagrams in the decays of heavy neutrinos, which we shall call El-type effects (direct CP-violation) [9,10]. Contributions from selfenergies are discussed in Refs. [ 11,121 and for supersymmetric theories in [ 131. A new aspect was pointed out [ 121 when it was realised that the heavy physical neutrino states are not CP- or lepton-number eigenstates. Therefore as soon as the physical states are formed there is imprinted on them a CP- and a lepton-asymmetry. This we shall call e-type effects (indirect CP-violation) . It is a property which appears in the eigenstates of the Hamiltonian ’ Permanent address: Theory Group, Physical Research Laboratory, 0370-2963/96/$12.00 Copyright PII .50370-2693(96)01337-S

Ahmedabad-380009,

0 1996 Elsevier Science B.V. All rights reserved.

India.

694

M. Flaw et al. / Physics Letters B 389 (1996) 693-699

ALLa ..

Ni

-.

h ai

-.

--.,

Nf

ALLa ‘.

lIti --=._

$+

- $

Fig. 1. Ni and NF decaying

into 1~~ and (IL~)~.

at an early epoch of thermodynamic equillibrium, even before the temperature of the universe falls down to the masses of the heavy Majorana particles. We demonstrate the phenomenon with a Gedanken experiment. Consider a universe consisting of a large p - ~7 collider which produces s- and S-pairs. The s and S quarks hadronize into Kc and iP mesons whose superpositions are the physical states &cc

[(l+E)KOf(l-E)KO].

The probability of finding a /Kc) is proportional to 11 + (1 - EI* which are not equal. When the particles decay

l/* and

the probability

for a IS)

is proportional

to

the above asymmetry survives as an asymmetry of the detected e”s and e-‘s. The electron-positron asymmetry generated is the same for the KL and KS [ 141. We pointed out that a similar situation arises in the formation and decays of Majorana neutrinos [ 121. In this article we adopt the wave function formalism to calculate the eigenstates of the Hamiltonian and their decay rates. In this formalism we can extend the region of our calculation to the case of very small mass differences. We find that for small mass differences between the two generations the indirect CP-violation (e-type) produces a lepton asymmetry much larger than that of the l ‘-type. We work in an extension of the standard model where we include one heavy right-handed Majorana field per generation of light leptons (Ni, i = 1,2,3). These new fields are singlets with respect to the standard model. The lagrangian now contains a Majorana mass term and the Yukawa interactions of these fields with the light leptons

i +

c n,i

a,i

hzi (eLa)” & (NR~)'

+ c hai(NRilC4 (eLa>c

a.i

(1)

0,i

where 4T = ( :@, 4- ) is the higgs doublet of the standard model, which breaks the electroweak symmetry and gives mass to the fermions; 1~~ are the light leptons, h,i are the complex Yukawa couplings and (Y is the generation index. We have adopted the usual convention for charge conjugation: NC = CF. Without loss of generality we work in a basis in which the Majorana mass matrix is real and diagonal with eigenvalues Mi. The states 1Ni) decay only into leptons, while the states 1NF) decay only into antileptons (Fig. 1). For this reason the states INi) and IN;) have definite lepton numbers and are the appropriate states to describe CP-violation in the leptonic sector. They are analogous to the Ke and ?? states. The idea is as follows: Through the presence of the Yukawa interactions we obtain one loop corrections to the mass matrix (Fig. 2), such that the corresponding mass eigenfunctions are no longer the 1Ni) and 1NF) states, but a mixture of them. It is these physical eigenstates which evolve in time with a definite frequency. If they are shown to be asymmetric linear combinations of the INi) and INf)‘s then we have created a CP-asymmetric universe. By asymmetric linear combinations we mean that the INi) and INF)‘s enter with different complex phases into the decomposition of the eigenfunctions. The subsequent decay of these fields will produce the desired lepton asymmetry.

M. Fhnz et ul. / Physics Letters B 389 (1996) 693-699

4

Fig. 2. One loop contributions

695

to the mass matrix.

As a result even if we start with equal numbers of INi) and IN:), they will evolve according to the asymmetric eigenstates of definite time developement. Since INi) and INi) carry different lepton numbers given by their interactions, this means that a lepton asymmetry is established through mixing before the fields actually decay. Herein lies the main difference between our model and the literature. For the sake of simplicity we consider two generations of Majorana neutrinos, where the indices i and j take the values 1 and 2. We assume the hierarchy M2 > Mt. In the basis ( IN:) IN;‘) INI) IN2) ) the effective Hamiltonian of this model can be written as

Once we include the one loop diagram of Fig. 2, there is an additional contribution to the effective Hamiltonian, which introduces CP-violation in the mass matrix. We treat the one loop contributions as a small perturbation to the tree level Hamiltonian:

(3)

with (4)

(5)

(6)

6%

M. Flanz et al./ Physics Letters B 389 (1996) 693-699

as can be easily read off from Fig. 2. The dispersive renormalization. 1

ab gai,j

The absorbtive

=

G

part &

part g”$j of the loop integrals

can be absorbed in the wave function

ab =

(7)

g

neglecting terms of order 0 (mz/p2), 0 (m$/p2) with p2 2 M?. As we will refer to certain limiting regions o,f the mass difference of the Majorana neutrinos be useful to rewrite the effective Hamiltonian H = g(O) + $?(I) in the following form: 0

0

where M = MI and @I = MT-M,.

Hi;) H$) Hi;) (M+vM) +H$ 0 0 0 0

This Hamiltonian

A:.2 = 4 [A2 + L?2+ 2H;;)fi,(;) S =

The corresponding

is exactly solvable and its squared eigenvalues

are given by

[,a' + B2]+4dB( H;iJ2+ i;rf;j2)

I3= M +qM eigenfunctions

(8)

% &]

[d2- B212+ 4H;;)@;)

A= M+H$,

it turns out to

M+

0 M +'H[;' fip q (M+vM)+H;;)

g=

and mass

is given by

+ H$;?

(9)

are found to be

with H(1)

A

dfii;' + BH,(;)

ai= --1?_di + n-c', bi Ai I

Ci=Ci,

2 rT/+li=]

_ 82

_

j;i”‘@ 12

ci

+ lbij2 + Ic;12 + ldi12 (i= 1,2). with mass eigenvalues &At. They are related to each other by a the same physical state. The same holds for the states J?\I) and our calculation we shall only consider 191) and lq2). as

rvi+lc (11)

ry;--11

+ rY,-P

which is a measure of the lepton asymmetry calculated using

rY;+l a

,jT I

where the normalization factor is fi = Iail2 The states 191) and I*;) are eigenstates chiral ys-transformation and correspond to 19;) with eigenvalues fA2. For the rest of The asymmetry parameter can be defined A=c

di=

generated

when the physical states [‘Pi) finally decay. This can be

cn ICikrl-I- dikx212 = c [ICi121ha]12 + ldi121ha212 + 2Re(cfdihz,h,2)] (Y

r* f-1’ m C a

IQhZl+ bihZ212 = C [Iaij2[ha112 + Ibi121ha212 +2Re(aib,*hz,hn2)] a

(12)

M. Flanz et al. / Physics Letters B 389 (1996) 693-699

697

In addition to the CP-violating contribution due to the mixing of the states ]Nj) and IN:), which we call S, there is another contribution l’ coming from the direct CP-violation through the decays of INi) and IN;), (13) which has been discussed in the literature extensively [ 9,101. The new indirect CP-violation 8, which enters through the mass matrix, is given by

8=X5

(14) +

i

where

a

a

Xi = (lci12 f lai12) Clh,11~+ (ldi12h lbi12> Clha212 +2Re

(I

Chithn2(crdi

faibr)

1.

In order to get some simple results we restrict ourselves to the two limiting regions of large and small Majorana mass differences. In the case of a large mass difference, by which we mean Mq =

IM2 - MjI >> ,$.“I

or

lfiy’l.

8 reads

where

From Eq. ( 15) it is clear that this contribution becomes significant when the two mass eigenvalues are close to each other (Fig. 3). It already indicates a resonance like behaviour of the asymmetry if the two mass eigenvalues are nearly degenerate. For very large values of ~17Mthe two contributions E’ and S are of the same order of magnitude. In the case that the two heavy Majorana neutrinos are nearly degenerate in mass we are led to an interesting resonance phenomenon. In this limiting region we have @4 = iA42 - A411 5 ]@~‘I The asymmetry

is then estimated

or

IZft’l.

to be (17)

where we have made use of the additional approximations Ifit)] = JH,:)] and [Hi,” - H:i)] < IHi:) + g$‘]. The above expression vanishes for 7 0 as expected, because there is no detectable mixing between two identical particles (Fig. 4). Striking is the resonance behaviour in the new curve (Fig. 4) where a remarkable enhancement of the lepton asymmetry is evident. The maximum value occurs for

and the corresponding

enhancement

factor is given by [ Re( c,

hz, hn2) ] -’ neglecting

factors of order unity.

698

M. Flunz et ul./ Physics Letters B 389 (1996) 693-699 2.5

2.0

i 5

1.5

72 c ;

1.0

3

0.5

100

0.0

1.2

1.6

1.4

1

I

.s

2.0

1.

r --,

l.WOS

1.0010

r ->

Fig. 3. Comparison of the two CP-violating contributions Iepton asymmetry in units of C, where r = M;/M:.

to the

Fig. 4. Lepton asymmetry difference, where P = c,

in units of C generated for a small mass hz, ha2 and r = h&j/M:.

The dynamical evolution of the lepton number in the universe is governed by the Boltzmann equations. This is the appropriate tool to describe any deviation from thermal equilibrium. The framework and the derivation of the Boltzmann equation is reviewed in [ 21; we adopt the same notation. We make the approximations of kinetic equilibrium and Maxwell-Boltzmann statistics. The out-of-equilibrium conditions occur, when the temperature T drops below the mass scale Me,, where the inverse decay is effectively frozen out. The density of the lepton number asymmetry n.,, = nl - nlc for the left-handed leptons can be shown to evolve in time according to the equation

where 61 is the contribution proportional to l/ C, (h,t 12 in expressions ( 15) and ( 16). The second term on the left side comes from the expansion of the universe, where H is the Hubble constant. I’$‘, is the thermally averaged decay rate of the [tit) state, n,. is the usual photon density and the term (gIu\) describes the thermally averaged cross-section of 1 + 4t +---+ F + 4 scattering. We note that the first term of the right side of this equation describes the creation of lepton number and is proportional to (E’ + Si), while the last two terms are responsible for any depletion of lepton number and are coming from the inverse decay and the scattering respectively. The density of the @i state satisfies the Boltzmann equation, dr

+

3&h = -I$,

(n+, - nz ) .

(19)

In order to find a solution to this set of coupled differential equations it turns out to be useful to transform to new variables. We introduce the dimensionless variable x = M+,,/T, a particle density per entropy density F = ni/s and make use of the relation t = x2/2H( x = 1). In addition we define the parameter K = I$, (x = l)/H( x = 1) which is a measure of the deviation from equilibrium. For K << 1 at T x AI,,+, we are far from thermal equilibrium so that the last two terms in Eq. (18) can be safely ignored. With these simplifications and the above redefinitions the Boltzmann equations effectively read

dYL dx

=

(Y,, - $p)(E’+s*,Kx*,

dTh - _(Y$, - ypX2. dx

-

For very large times the solution for YLhas an asymptotic value which is approximately given by

(20)

M. Flanz et al./ Physics Letters B 389 (1996) 693-699

699

(21) where g, denotes the total number of effectively massless degrees of freedom and is of the order of 0( 102) for all usual extensions of the standard model. The lepton asymmetry thus generated will then be converted to the baryon asymmetry of the universe in the presence of the sphalerons during the electroweak phase transition [ 151 and is approximately given by ns M ins. To sum up, we have studied the scenario of baryogenesis in which the baryon asymmetry is generated from a lepton asymmetry. We have demonstrated that the contribution to the lepton asymmetry coming from the mixing of the heavy Majorana neutrinos can dominate any contributions originating from direct decays by several orders of magnitude. This enhancement of the lepton asymmetry was shown to be due to a resonance phenomenon. We thank Dr. A. Pilaftsis for discussions. The financial support of BMBF under contract 056D093P(5) is greatfully acknowledged. One of us (US.) would like to acknowledge a fellowship from the Alexander von Humboldt Foundation and hospitality from the Institut fur Physik, Univ Dortmund during his research stay in Germany and (M.F.) wishes to thank the Deutsche Forschungsgemeinschaft for a scholarship in the Graduiertenkolleg “Production and Decay of Elementary Particles”.

References 1 I] A.D. Sakharov, Pis’ma Zh. Eksp. Teor. Fiz. 5 ( 1967) 32. 12 1 E.W. Kolb and MS. Turner, The Early Universe (Addison-Wesley, Reading, MA, 1989). 131 M. Yoshimura, Phys. Rev. Lett. 41 (1978) 281; 42 (1979) 746 (E). 141 G. ‘t Hooft, Phys. Rev. Lett. 37 ( 1976) 8; E Klinkhamer and N. Manton, Phys. Rev. D 30 (1984) 2212. 15 1 For a review, see V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B 155 ( 1985) 36; V.A. Rubakov and M.E. Shaposhnikov report no. hep-ph/9603208, and references therein. 161 I. Af8eck and M. Dine, Nucl. Phys. B 249 (1985) 361. 171 B. Campbell, S. Davidson and K.A. Olive, Phys. Lett. B 303 (1993) 63; Nucl. Phys. B 399 ( 1993) 111; H. Murayama, H. Suzuki and T. Yanagida, Phys. Rev. Lett. (1993) 19 12; H. Murayama and T. Yanagida, Phys. L.&t. B 322 ( 1994) 349. 18 1 A. Masiero and A. Riotto, Phys. Lett. B 289 ( 1992) 73. 191 M. Fukugita and T. Yanagida, Phys. L&t. B 174 ( 1986) 45. ( 101 M.A. Luty, Phys. Rev. D 45 (1992) 455; P. Langacker, R.D. Peccei and T. Yanagida, Mod. Phys. Lett. A I (1986) 541; A. Acker, H. Kikuchi, E. Ma and U. Sarkar, Phys. Rev. D 48 ( 1993) 5006; P.J. O’Donnell and U. Sarkar, Phys. Rev. D 49 (1994) 2118; M. Pliimacher, DESY report no. DESY-96-052 (April 1996). [ I I I A.Yu. Ignatev, V.A. Kuzmin and M.E. Shaposhnikov, JETP Lett. 30 ( 1979) 688; F.J. Botella and J. Roldan, Phys. Rev. D 44 (1991) 966: J. Liu and G. Segre, Phys. Rev. D 48 ( 1993) 4609. 1121 M. Flanz, E.A. Paschos and U. Sarkar, Phys. Lett. B 345 (1995) 248. [ 131 L. Covi, E. Roulet and E Vissani, SISSA report hep-ph/9605319. [ 141 For a review see, E.A. Paschos and U. Tiirke, Phys. Rep. 178 ( 1989) 147, in particular [ I5 1 J.A. Harvey and M.S. Turner, Phys. Rev. D 42 ( 1990) 3344.

chapters 4 and 6.