Gauged B-L Leptogenesis Shaaban Khalil Center for Fundamental Physics Zewail City of Science and Technology
Introduction • The recent observations indicate that the asymmetry between number density of baryon ( ! " ) and of anti-baryon ( ! # " ) of the universe is given • In the SM, the baryon asymmetry is $ " ≈ &' ()* , which is too small to account for the observed baryon asymmetry. • Leptogenesis through the decay of a heavy singlet neutrino is considered as the best scenario for understanding the observed baryon asymmetry of the Universe. • In this mechanism, the lepton asymmetry Y L can be converted to baryon asymmetry Y B via the electroweak sphaleron according to this relation , $ " = , − & $ . * 0 1 23 0 4 / = )) 0 1 2&5 0 4 , and 0 1 is the number of fermions and 0 4 is the number of Higgses.
• The lepton asymmetry is given by • Lepton asymmetry arises due to interference between tree & loop contributions • Thus • Therefore, the necessary condition for leptogenesis is • Due to the unitarity of UMNS, leptogenesis does not depend on low energy phase appears in the leptonic mixing matrix. If the matrices R and M R are real the ! 1 =0.
• Finally, the baryon asymmetry is given by $ $ ) & * ∗ ≈ &- %. ( ) & ! " = $%& ! ' = $%& ( The coffecient ( parametrize the wash out effect due to the inverse decay and the scattering processes. It depends on the ratio / = 0 12& /4 ; ( ( ~ 1/K). • For 6 12& ~ &- &- 789 ⇒ ( ~; & and ! " ≈ &- %. ) & • For 6 12& ≪ 6 12. , 6 12> ;(&- &> ) • Thus, the required baryon asymmetry can be obtained. • In this case, the energy scale involved in a successful application is in the range 10 9 to 10 13 GeV, which renders the idea impossible to verify experimentally
TeV scale Leptogenesis • We now consider TeV scale Seesaw mechanism. In this case, after the TeV scale symmetry (e.g., B−L) breaking, the neutrino mass matrix is given by • With m D = Y ν v 2 , M N = Y N v 1 ʹ. The neutrino masses are • Therefore, if M N ∼ O(1) TeV, the light neutrinos ν ℓ mass can be of order one eV if the Yukawa coupling Y ν ~ 10 −6. • This small coupling is of order the electron Yukawa coupling, so it is not quite unnatural.
In TeV scale type I seesaw, ! " ~ $ %& '( • and ) * ~ $ +,- , the lepton asymmetry can be written as Thus, for /) = ) 1 − ) % ~ $(%& '4 ), the required baryon asymmetry can be • obtained. Based on this fine tuning, a “Resonant Leptogenesis” is defined. •
Inverse Seesaw Mechanism • # and In this class of models, the SM is extended with three right-handed neutrinos, ! " three singlet fermions $ # . • The Lagrangian of neutrino masses, in the flavor basis, is given by: 3 4 + h. c. ) + 9 : ̅ + , - . + 0 1 2 4 3 4 ℒ = (( ) ̅ - .
Leptogenesis with Inverse Seesaw
Gauged B − L Leptogenesis • " # with B − L = 1 implies In supersymmetry, the addition of the singlet superfield ! a fermion " # and a scalar $ " # . After the B−L symmetry breaking by the VEVs < & ',) > = , ',) , a bilinear • ) $ 0 $ 0 is obtained and it is given by coupling - ./ " . " / 5 + 4 " , 3' 7′ ∗ ) = −, 3' 4 " - 1 ) = 0 so that the off-diagonal elements of sneutrino $ " # • Here, we assume that - 1 mass matrix, in the ($ 0 , $ 0∗ ) basis, vanish. " . " . " #∗ are mass eigenstates with mass squared Therefore, $ " # , $ • ∗ < 1 = + > ) + ) BCD2F ' < 1 ? 1 @ < A3 They have lepton numbers G = ∓I respectively. Moreover, if cos 2θ is negative, • " # can be lighter than " # . $ This is the crucial assumption of our proposal. •
The Lagrangian, in flavor eigenstates, relevant for our analysis is given by • The Lagrangian in mass eigenstate is given by • Where • The four-component Majorana spinors are defined as ( , assuming the mass hierarchy Now, we consider leptogenesis by ! " → $ % & • ! ' ) !" ≪ ) !+,- . We assume that only the lightest B − L neutralino $ ≡ $ " and ( are lighter than ! " , and satisfy the relation sneutrino of the third generation & ! - 2 < ) ! " . / 0 + ) & ! - ( carries lepton number, the decay: Due to the fact that 4 ! = 6 , & • ! - ( violates lepton number. ! " → $ & ! -
• ' decay CP asymmetry of ! " → $ % ! & processes is generated by the interference between tree and one- loop level diagrams of vertex and self-energy correction • ' decay processes is generated by the interference CP asymmetry of ! " → $ % ! & between tree and one-loop level diagrams of vertex and self-energy correction • It is defined as • The decay rate ( at one-loop level is given by Where the phase space integral of two-body decay ) * is given by
• ' decay has the structure: We found that the CP asymmetry ! " → $ % ! & where we assumed diagonal right-handed mass matrix. One finds ( "& ~*+, - ≪ " , as 5 !" required by out-of equilibrium condition: ( ( ! " → $ % ' ) < 0(2 = ") , for 2 = ! & 6 . Also ( && ~789 :~" , which leads to a large CP asymmetry. This situation is realized if the mixing matrix ( is almost diagonal. • In our model, baryon asymmetry is obtained through the following procedure: ' decay generates % ' asymmetry ; ∆% 1. ! " → $ % ! & ! & ! . ' decays into (s)lepton by Dirac Yukawa couplings, soft SUSY breaking A-term 2. % ! & and = -term, and resulting (s)lepton asymmetry ; ∆>(∆? @) is obtained by solving the Boltzmann equations. 3. Sphaleron converts total lepton asymmetry ; > = ; ∆> + ; ∆? @ to baryon asymmetry ; A .
Boltzmann Equations The Boltzmann equation describing the evolution of ! " is • ! " is the number of particle in a comoving volume element, which is given by the ratio of # " and the entropy density s .
Enough baryon asymmetry is obtained.
Conclusion We have shown that a successful TeV scale leptogenesis can take place in • gauged B − L supersymmetric model. In this model, if the right-sneutrino bilinear term is absent, then the lightest • $ is lighter than " % and sneutrino is assigned a lepton number. Therefore if ! " # " & is almost diagonal, a large lepton asymmetry can be scalar mass matrix of ! generated by B−L neutralino interactions of O (1) couplings ' ()* and/or + " $ for the decay " % → - ! $ . through the one-loop exchange of " # " # $ is transmitted into asymmetry of lepton and slepton This asymmetry of ! " # • through the Yukawa coupling, trilinear coupling, and μ-term, and sphaleron converts lepton asymmetry to baryon asymmetry.
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