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Baryogenesis Through Lepton Number Violation

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Baryogenesis through lepton number violation 1 Utpal Sarkary yPhysical Research Lab oratory, Ahmedabad - 380 009, INDIA Abstract. The most promising scenarios of baryogenesis seems to b e the one through lepton numb er violation. Lepton numb er violation through a Ma jarana mass of the right-handed neutrinos can generate a lepton asymmetry of the universe when the right- handed neutrinos decay. The left-handed neutrinos get small Ma jorana masses through see-saw mechanism in these mo dels. A triplet higgs scalar violating lepton numb er explicitly through its couplings to two leptons or two higgs doublets can also nat- urally give small Ma jorana masses to the left-handed neutrinos and also generate a lepton asymmetry of the universe. We re- view b oth these mo dels of leptogenesis, where the lepton number asymmetry then gets converted to a baryon asymmetry of the universe b efore the electroweak phase transition. 1. Intro duction To get the baryon asymmetry of the universe [1] starting from a symmetric universe, one requires [2] three conditions (A) Baryon number violation, (B) C and CP violation, and (C) Departurefrom thermal equilibrium. In grand uni ed theories (GUTs) all these conditions are satis ed [3, 4], but the generated asymmetry conserves (B L). It was then realised that the chiral nature of the weak interaction also breaks the global baryon and lepton numb ers in the standard mo del [5]. At nite temp erature these baryon and lepton numb er violating interactions were found to b e very strong in 1 E-mail:[email protected] 1 the presence of some static top ological eld con guration - sphalerons [6]. Although the anomalous sphaleron pro cesses conserves (B L), the GUT (B + L) asymmetry will b e completely washed out by these interactions. Attempts were then made to make use of the baryon numb er violation of the standard mo del to generate a baryon asymmetry of the universe. How- ever, in these mo dels one needs to protect the generated baryon asymmetry after the phase transition, which requires the mass of the standard mo del doublet higgs b oson to be lighter than the present exp erimental limit of 95 GeV. Then the most interesting scenario remains for the understanding of the baryon numb er of the universe is through lepton numb er violation [7]{[14], which is also referred to as leptogenesis. In mo dels of leptogenesis one generates a lepton asymmetry of the uni- verse, which is the same as the (B L) asymmetry of the universe at some high energy. This (B L) asymmetry of the universe then get con- verted to the baryon asymmetry of the universe during the p erio d when the sphaleron elds maintain the baryon numb er violating interactions in equi- librium. Since lepton numb er violation is the source of leptogenesis, they are related to mo dels of neutrino masses. In this article we shall review two scenarios of leptogenesis. In the rst scenario right handed neutrinos are intro duced, which gets a Ma jorana mass and breaks lepton number [7]. The left-handed neutrinos get small Ma jorana masses through see-saw mechanism [15]. In the second scenario only a triplet higgs is intro duced and the fermion content of the standard mo del is unaltered [16, 17, 18]. Unlike earlier treatments, lepton number is now broken explicitly at a very high scale [16,17]. Although the triplet is very heavy, its vev b ecomes of the order of eV to givevery small Ma jorana mass to the neutrinos natu- rally [16]. Decays of the triplet higgs generate a lepton asymmetry of the universe at very high scale. In the next section we shall discuss the electroweak anomalous pro cesses and then how the baryon and lepton numb ers of the universe gets related to the (B L)numb er of the universe. This will imply that if there is vary fast lepton numb er violation in the universe during the p erio d when these pro cesses are in equilibrium, that can also wash out the baryon asymmetry of the universe [19,20]. In the following two sections we present the two scenarios of leptogenesis. 2. Sphaleron pro cesses in thermal equilibrium and relation be- tween baryon and lepton numb ers Anomaly breaks any classical symmetry of the lagrangian at the quantum level. So, all lo cal gauge theories should be free of anomalies. However, there may be anomalies corresp onding to any global current. That will 2 simply mean that such global symmetries of the classical lagrangian are broken through quantum e ects. In the standard mo del the chiral nature of the weak interaction makes the baryon and lepton numb er anomalous and give us non vanishing axial current [5] 2 1 5 ~ ~ Y ] j =6[ W W + Y a a (B +L) 8 8 which will break the (B + L) symmetry, while still preserving (B L), during the electroweak phase transition, Z 2 4 ~ (B + L)=2N d xW W =2N g a g a 8 But their rate is very small at zero temp erature, since they are suppressed 2 by quantum tunnelling probability, exp[ ]; where is the Chern- 2 Simmons numb er. At nite temp erature, however, it has b een shown that there exists non- trivial static top ological soliton con guration, called the sphalerons, which enhances the baryon numb er violating transition rate [6] and the suppres- V 0 sion factor is now replaced by the Boltzmann factor exp[ ] where the T p otential or the free energy V is related to the mass of the sphaleron eld, 0 which is ab out TeV. As a result, at temp eratures b etween 12 2 10 GeV >T >10 GeV (1) the sphaleron mediated baryon and lepton numb er violating pro cesses are in equilibrium. For the simplest scenario of = 1, the sphaleron induced pro cesses are B =L= 3, given by, jvac >! [u u d e + c c s + t t b ]: (2) L L L L L L L L L L L L These baryon and lepton numb er violating fast pro cesses will wash out any pre-existing baryon or lepton numb er asymmetry, or will convert any pre- existing (B L) asymmetry of the universe to a baryon asymmetry of the universe, which can b e seen from an analysis of the chemical p otential [21]. We consider all the particles to be ultrarelativistic and ignore small mass corrections. The particle asymmetry, i.e. the di erence b etween the numb er of particles (n ) and the numberofantiparticles (n ) can b e given + in terms of the chemical p otential of the particle sp ecies (for antiparticles 3 gT the chemical p otential is )asn n =n ; where n = 2 for + d d 6 T b osons and n = 1 for fermions. d In the standard mo del there are quarks and leptons q ;u ;d ;l and iL iR iR iL e ; where, i =1;2;3 corresp onds to three generations. In addition, the iR scalar sector consists of the usual Higgs doublet , which breaks the elec- troweak gauge symmetry SU (2) U (1) down to U (1) . In Table 1, we L Y em 3 presented the relevantinteractions and the corresp onding relations b etween the chemical p otentials. In the third column we give the chemical p oten- tial which we eliminate using the given relation. We start with chemical p otentials of all the quarks ( ; ; ; ); leptons ( ; ; , uL dL uR dR aL aL aR where a = e; ; ); gauge b osons ( for W , and 0 for all others); and W the Higgs scalars ( ; ). 0 Table 1. Relations among the chemical p otentials Interactions relations eliminated y D D = + W 0 q q W = + L L dL uL W dL l l W = + L L W iL iL iL y q u = + L R uR 0 uL uR q d = + L R dR 0 dL dR l e = + iL iR 0 iR iR iL The chemical p otentials of the neutrinos always enter as a sum and for that reason we can consider it as one parameter. We can then express all the chemical p otentials in terms of the following indep endentchemical P P p otentials only, = ; ; = ; = = . We 0 W u uL i iL 0 i i can further eliminate one of these four p otentials by making use of the relation given by the sphaleron pro cesses, 3 +2 + =0. We then u W express the baryon numb er, lepton numb ers and the electric charge and the hyp ercharge numb er densities in terms of these indep endentchemical p otentials, B =12 +6 ; L =3+2 u W i W 0 Q =24 + (12 + 2m) (4+2m) ; Q = (10 + m) u 0 W 3 W where m is the numb er of Higgs doublets . At temp eratures ab ove the electroweak phase transition, T >T , b oth c < Q > and < Q > must vanish, while b elow the critical temp erature 3 < Q > should vanish, but since SU (2) is now broken we can consider L = 0 and Q 6=0. These conditions and the sphaleron induced B L 3 0 conserving, B + L violating condition will allow us to write down the baryon asymmetry in terms of the B L numb er density as, 24+4m 32 + 4m B (T >T )= (BL) B(T <T )= (BL): (3) c c 66+13m 98+13m Thus the baryon and lepton numb er asymmetry of the universe after the electroweak phase transition will dep end on the primordial (B L) asym- 4 metry of the universe, while all the primordial (B + L) asymmetry will b e washed out.
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