Texture zeros in neutrino mass matrix
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ELEMENTARY PARTICLES AND FIELDS Theory
Texture Zeros in Neutrino Mass Matrix∗ B. Dziewit1)** , J. Holeczek1)*** , M. Richter1)**** , S. Zajac2)***** , and M. Zralek1)****** Received May 18, 2016
Abstract—The Standard Model does not explain the hierarchy problem. Before the discovery of nonzero lepton mixing angle θ13 high hopes in explanation of the shape of the lepton mixing matrix were combined with non-Abelian symmetries. Nowadays, assuming one Higgs doublet, it is unlikely that this is still valid. Texture zeroes, that are combined with abelian symmetries, are intensively studied. The neutrino mass matrix is a natural way to study such symmetries. DOI: 10.1134/S1063778817020144
1. INTRODUCTION In the Standard Model (SM) neutrinos interact with charged leptons. Interactions with the Higgs particle and interactions by neutral current with Z particle are not important for our purpose. In the flavor basis the Majorana neutrino Lagrangian effectively takes the form: e ¯l γ μ ναL W − + h.c. L = −√ αL μ 2 sin θW α=e,μ,τ l ¯ l∗ ¯ Mα,β lαR lβL + Mα,β lβL lαR +
As usually, let us introduce the physical basis in which the mass matrices become diagonal: ν Uαi νiL ναL = i=1,2,3
⇒
+
1 2
α,β=e,μ,τ
The neutrino mass matrix Mν is an arbitrary symmetric three dimensional complex matrix. It is diagonalized by the orthogonal transformation with a complex matrix: = δij mνi , mνi ≥ 0. (2) U νT Mν U ν
and also:
αβ
∗
The text was submitted by the authors in English. Institute of Physics, University of Silesia, Katowice, Poland. 2) Faculty of Mathematics and Natural Studies, Cardinal Ste´ fan Wyszynski University in Warsaw, Warszawa, Poland. ** E-mail: [email protected] *** E-mail: [email protected] **** E-mail: [email protected] ***** E-mail: [email protected] ****** E-mail: [email protected] 1)
ULl
= lαL
(4)
and
lαR =
lβL
αβ
β=e,μ,τ
URl
αβ
β=e,μ,τ
lβR .
(5)
After such transformation, UPMNS matrix (Pontecorvo–Maki–Nakagawa–Sakata mixing matrix) appears in the charged current interactions: e LCC = − √ 2 sin θW ¯lαL γ μ (UPMNS ) νiL W − + h.c., × (6) μ
αi
α=e,μ,τ i=1,2,3
ij
The charged leptons’ mass matrix M l is an arbitrary three-dimensional complex matrix diagonalized by the biunitary transformation: = δαβ mlα , mlα > 0. (3) URl† M l ULl
∗ν Uαi νiR ,
i=1,2,3
α,β=e,μ,τ
∗ C C νβL + Mνα,β ν¯βL ναR Mνα,β ν¯αR . (1)
ναR =
(UPMNS )αi =
† (U ν )βi , ULl β=e,μ,τ
αβ
(7)
One should mention that the mixing matrix (7) is expressed in the mass eigenstates basis both for the charged leptons (e, μ, τ ) and for the neutrinos (ν1 , ν2 , ν3 ), so we have the freedom to choose the model. We can choose the basis in which at the beginning M l or Mν are diagonal. Usually the first alternative is considered, and we do the same. In the basis in which M l is diagonal we can rewrite relation (2) as: U T Mν U = mdiag , 353
(8)
354
DZIEWIT et al.
where we introduce the notation: UPMNS = U ν ≡ U.
(9)
The mixing matrix U can be decomp
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