马约拉纳大师中微子质量参数化
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Master Majorana neutrino mass parametrization
I. Cordero-Carrión,
1,*
M. Hirsch,
2,†
and A. Vicente
2,‡
1
Departamento de Matemáticas, Universitat de Val`encia, E-46100 Burjassot, Val`encia, Spai n
2
Instituto de Física Corpuscular (CSIC-Universitat de Val`encia),
Apartado de Correos 22085, E-46071 Valencia, Spain
(Received 17 December 2018; revised manuscript received 25 February 2019; published 15 April 2019)
After introducing a master formula for the Majorana neutrino mass matrix, we present a master
parametrization for the Yukawa matrices automatically in agreement with neutrino oscillation data. This
parametrization can be used for any model that induces Majorana neutrino masses. The application of the
master parametrization is also illustrated in an example model, with special focus on its lepton flavor violating
phenomenology.
DOI: 10.1103/PhysRevD.99.075019
I. INTRODUCTION
The Standard Model (SM) of particle physics stands as
one of the most successful physical theories ever built.
However, despite its tremendous success, it cannot describe
all particle physics phenomena. Neutrino oscillation experi-
ments have firmly established that neutrinos have nonzero
masses and mixings, hence demanding an extension of the
SM that accounts for them.
Many neutrino mass models have been proposed.
A short list, to quote only a few reviews and general
classification papers, includes models with Dirac [1,2] or
Majorana neutrinos [3], with neutrino masses induced at
tree level or radiatively at one-loop/two-loop [4] or three-
loop [5],atlow-[6] or high-energy scales, and by operators
of dimension 5 or higher dimensionalities [7].
Thegoalof this article is twofold. First, wewill introduce a
master formula that unifies all Majorana neutrino mass
models, which can be regarded as particular cases of this
general expression. And second, we will present a master
parametrization for the Yukawa matrices appearing in this
formula. The parametrizationpresented inthisarticle extends
previous results in the literature [8] and can be used for any
model that induces Majorana neutrino masses.
II. THE MASTER FORMULA
With full generality, a Majorana neutrino mass matrix
can be written in the form
m ¼ fðy
T
1
My
2
þ y
T
2
M
T
y
1
Þ: ð1Þ
Here m is the 3 × 3 complex symmetric neutrino mass
matrix,
1
which can be diagonalized as
D
m
¼ diagðm
1
;m
2
;m
3
Þ¼U
T
mU; ð2Þ
with U a 3 × 3 unitary matrix (U
†
U ¼ UU
†
¼ I
3
). The
matrices y
1
and y
2
are general dimensionless n
1
× 3 and
n
2
× 3 complex matrices, respectively, and M is a n
1
× n
2
complex matrix with dimension of mass. Without loss of
generality, we will assume n
1
≥ n
2
. We note that m must
contain at least two nonvanishing eigenvalues in order to
explain neutrino oscillation data. Therefore, in the follow-
ing we consider r
m
¼ rankðmÞ¼2 or 3.
Equation (1) is a master formula valid for all Majorana
neutrino mass models. This can be illustrated with several
examples. The simplest one is based on the popular seesaw
mechanism [9–14], in particular, on the standard type-I
seesaw with three generations of right-handed neutrinos.
The light neutrino mass matrix in this model is given by
m ¼ −hH
0
i
2
y
T
M
−1
R
y, an expression that can be obtained
with the master formula by taking f ¼ −1, y
1
¼ y
2
¼
y=
ffiffiffi
2
p
, and M ¼hH
0
i
2
M
−1
R
.Here,hH
0
i¼v=
ffiffiffi
2
p
is the
SM Higgs vacuum expectation value (VEV) and M
R
the
Majorana mass matrix for the right-handed neutrinos.
Moreover, these matrices are all 3 × 3 and hence n
1
¼ n
2
¼
3 in this model. The mass matrices of more complicated
Majorana neutrino models can also be accommodated with
the master formula. For instance, the inverse seesaw [15]
would correspond to M ¼hH
0
i
2
ðM
T
R
Þ
−1
μM
−1
R
,withμ the
small lepton number violating mass scale in this model,
*
isabel.cordero@uv.es
†
mahirsch@ific.uv.es
‡
avelino.vicente@ific.uv.es
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP
3
.
1
We focus on the case of three generations, because there are
only three active neutrinos. It is straightforward to generalize to a
larger number, if one wants to include, e.g., light sterile neu trinos.
PHYSICAL REVIEW D 99, 075019 (2019)
2470-0010=2019=99(7)=075019(6) 075019-1 Published by the American Physical Society
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