Relating leptogenesis to low energy CP violation Page: 2 of 4
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interactions responsible for generating the lepton masses are,
- = IR(Pij'+ RijI)Lj,L ifij(L LS62ALLj,L + L RCT2ARLj,R )+h.c., (1)
where the matrices Pij, Rij and fij are real due to the assumption of SCPV. The complete
Lagrangian of the model is invariant under the unitary transformation, under which the
matter fields transform as yL -> ULy/L and y/R -> URy/R where yiL,R are left-handed
(right-handed) fermions, and the scalar fields transform according to ( -> UR(U ,
AL -, ULALU , and AR -1 URARUR, with the unitary transformations UL and UR being
UL - eI i'L OYU e RN
UL=(e e-igL, UR=( 0 e-iYR (2)
Under these unitary transformations, the VEV's transform as K -> Ke i(KYR)K -K
K'ei(YL R), VL - vLe 2irL and vR - vRe 2iR Thus by re-defining the phases of matter
fields with the choice of yR = aR/2 and yL = aK + aR/2 in the unitary matrices UL and
UR, we can rotate away two of the complex phases in the VEV's of the scalar fields and
are left with only two genuine CP violating phases, aK' and aL,
G< >(0 Kea, < AL >= vLaL , <AR >= R 0 3
The quark Yukawa interactions give rise to quark masses after the bi-doublet acquires
Mu= KFij + Ke ia,'Gij, Md = K'eia,'Fi + KGij (4)
Thus the relative phase in the two VEV's in the SU(2) bi-doublet, aK, gives rise to
the CP violating phase in the CKM matrix. To obtain realistic quark masses and CKM
matrix elements, it has been shown that the VEV's of the bi-doublet have to satisfy
K/K' ~ mt/mb 1. When the triplets and the bi-doublet acquire VEV's, we obtain the
following mass terms for the leptons
Me = K'eiax'Pij + KRij, Mvirac _ KPj - KeiaK'Rij (5)
MyR vRfij, MLL =vLeiaLj. (6)
The effective neutrino mass matrix, M", which arises from the Type-II seesaw mecha-
nism, is thus given by M" = MvlL - Mv. Assuming the charged lepton mass matrix is
diagonal, the Yukawa couplings Rij can be determined by the charged lepton masses. In
the limit K K', the conventional type-I see-saw term is dominated by the term propor-
tional to K,
MV =(Ma,)T (M<R 1(M ) ~ PTf-P - PT -f P (7)
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Chen, Mu-Chun; /Fermilab /UC, Irvine; Mahanthappa, K.T. & U., /Colorado. Relating leptogenesis to low energy CP violation, article, October 1, 2006; Batavia, Illinois. (digital.library.unt.edu/ark:/67531/metadc880060/m1/2/: accessed January 19, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.