Towards unification of quark and lepton flavors in A₄ modular invariance · arXivDesk
1905.13421May 31, 201919 pages, 6 tables, 9 figures; the quark and lepton sectors have been significantly improved, and final results have been also changed
Towards unification of quark and lepton flavors in A4 modular invariance
We study quark and lepton mass matrices in the A4 modular symmetry towards the unification of the quark and lepton flavors. We adopt modular forms of weights 2 and 6 for quarks and charged leptons, while we use modular forms of weight 4 for the neutrino mass matrix which is generated by the Weinberg operator. We obtain the successful quark mass matrices, in which the down-type quark mass matrix is constructed by modular forms of weight 2
Nearby in the stack
, but the up-type quark mass matrix is constructed by modular forms of weight
6
. Two regions of
τ
are consistent with observed CKM matrix elements. The one is close to
τ=i
and the other is in the larger
Im[τ]
. On the other hand, lepton mass matrices work well only at nearby
τ=i
, which overlaps with the one of the quark sector, for the normal hierarchy of neutrino masses. In the common
τ
region for quarks and leptons, the predicted sum of neutrino masses is
87
--
120
meV taking account of its cosmological bound. Since both the Dirac CP phase
δCPℓ
and
sin2θ23
are correlated with the sum of neutrino masses, improving its cosmological bound provides crucial tests for our scheme as well as the precise measurement of
sin2θ23
and
δCPℓ
. The effective neutrino mass of the
0νββ
decay is
⟨mee⟩=15
--
31
meV. It is remarked that the modulus
τ
is fixed at nearby
τ=i
in the fundamental domain of SL
(2,Z)
, which suggests the residual symmetry
Z2
in the quark and lepton mass matrices. The inverted hierarchy of neutrino masses is excluded by the cosmological bound of the sum of neutrino masses.