Neutrino Masses and Flavor Mixing, Part 1

Neutrino Masses & Flavor Mixing
Zhi-zhong Xing
邢 志 忠
(IHEP, Beijing)
＠Schladming Winter School 2010, Styria, Austria
Lecture A
Origin of “Flavor”
2
The term Flavor was coined by Harald Fritzsch and Murray Gell-Mann at a
Baskin-Robbins ice-cream store in Pasadena in 1971 (Browder et al., 09).
Neutrinos are
super-light
flavors:
mass,
flavor mixing
and CP violation
What is Mass?
3
Higgs is the God particle in the SM:
it gives masses to other particles.
A lesson learnt from history:
All known bosons were discovered
in Europe and most fermions were
discovered in America.
The LHC is very likely to see Higgs.
Question: What is mass?
Mass is the inertial energy of a particle when it exists at rest.
 A massless particle has no way to exist at rest. It must always
move at the speed of light.
 A massive fermion (a lepton or a quark) must exist in both the
left- and right-handed states.
Why ’s are massless in the SM?
4
All ’s are massless in the SM, a result of the model’s simple structure:
--- SU(2)_L×U(1)_Y gauge symmetry and Lorentz invariance;
Fundamentals of the model, mandatory for its consistency as a QFT.
--- Economical particle content:
No right-handed neutrinos --- a Dirac mass term is not allowed.
Only one Higgs doublet --- a Majorana mass term is not allowed.
--- Mandatory renormalizability:
No dimension ≥ 5 operators --- a Majorana mass term is forbidden.
In other words, the SM accidently possesses the (B - L) symmetry .
In practice, it is natural for neutrinos to be massless:
--- They are too light to be weighed out & they are almost left-handed;
In principle, it is unnatural for neutrinos to be massless:
--- No fundamental symmetry or conservation law to ensure m_ = 0.
How to give ’s masses?
5
To generate -masses, one of the above 3 constraints must be relaxed.
--- The gauge symmetry and Lorentz invariance cannot be abandoned;
--- The particle content can be modified;
--- The renormalizability can be abandoned.
How many ways?
Outline of Lecture A
In this lecture we shall discuss the following topics:
--- How to write out a Dirac mass term for ’s --- why L is conserving?
--- How to write out a Majorana mass term for ’s --- why L is violated?
--- Preliminary ideas of seesaw mechanisms --- a type-(I+II) example.
--- Electromagnetic dipole moments of massive neutrinos --- overview.
For simplicity, our discussions will try to avoid any model dependence.
Part A
Some Notations
6
Define the left- and right-handed neutrino fields:
Extend the SM’s
particle content
Their charge-conjugate counterparts are defined below and transform
as right- and left-handed fields, respectively:
(can be proved easily)
Representation-independent algebra of the charge-conjugation matrix:
They come from the requirement that the charge-conjugated field must
satisfy the same Dirac equation (
in the Dirac representation).
Part A
Dirac Neutrino Mass Term
A Dirac neutrino is described by a 4-component spinor:
Step 1: the gauge-invariant Dirac mass term and SSB:
Step 2: diagonalization
(basis transformation):
Mass states link to flavor states:
Step 3: physical mass term
and kinetic term:
7
Part A
Dirac Neutrino Flavor Mixing
8
Weak charged-current interactions of charged leptons and neutrinos:
In the flavor basis
In the mass basis
Without loss of generality, one may choose mass states = flavor states
for charged leptons. Then V is just the MNSP matrix of neutrino mixing.
Both the mass and CC terms are invariant with respect to global phase
transformations: lepton number (flavor) conservation (violation).




Part B
Majorana Neutrino Mass Term (1)
9
A Majorana neutrino mass term can be obtained at low energy scales by
introducing a Higgs triplet into the SM, writing out the gauge-invariant
Yukawa interactions and Higgs potentials, and then integrating out the
heavy degrees of freedom (type-II seesaw mechanism):
The Majorana mass matrix must
be a symmetric matrix. It can be
diagonalized by a unitary matrix.
Diagonalization:
Physical mass term:
Part B
Majorana Neutrino Mass Term (2)
Kinetic term:
(you may prove
10
.)
A small question: why is there a factor 1/2 in the Majorana mass term?
Answer: it allows us to get the correct Dirac equation of motion for ’s.
Proof: write out the Lagrangian of free massive Majorana neutrinos:
Euler-Lagrange
equation:
Part B
Majorana Neutrino Flavor Mixing
11
Weak charged-current interactions of charged leptons and neutrinos:
In the flavor basis
In the mass basis
The MNSP matrix V contains two extra Majorana CP-violating phases.
Salient feature of massive Majorana neutrinos: lepton number violation
because the mass and CC terms are not simultaneously invariant under
a global phase transformation.
Neutrinoless double-beta decay
Part C
Hybrid Neutrino Mass Term (1)
12
A hybrid neutrino mass term can be written out in terms of the left- and
right-handed neutrino fields and their charge-conjugate counterparts:
type-(I+II) seesaw
Here we have used
Diagonalization by means
of a 66 unitary matrix:
Majorana mass states
It is actually a Majorana mass term!
Part C
Hybrid Neutrino Mass Term (2)
Physical mass term:
Kinetic term:
13
Part C
Non-unitary Neutrino Flavor Mixing 14
Weak charged-current interactions of charged leptons and neutrinos:
In the flavor basis
In the mass basis
V = non-unitary light neutrino mixing (MNSP) matrix
R = light-heavy neutrino mixing (or CC interactions of heavy neutrinos)
Neutrino
V
oscillations
Collider
R
signatures
TeV seesaws might bridge the gap between neutrino & collider physics.
Part D
Seesaw Mechanisms (1)
15
A hybrid neutrino mass Lagrangian contains three distinct mass terms:
(1) Normal Dirac mass term, which is characterized by the electroweak
symmetry breaking scale (i.e., the Higgs’ vev v ~ 174 GeV);
(2) Light Majorana mass term, which violates the SM gauge symmetry
(e.g., from a Higgs triplet) and has a scale much lower than v ;
(3) Heavy Majorana mass term, which is from the SU(2)_L singlets and
has a scale much higher than v (e.g., close to the GUT scale).
A strong hierarchy of 3 mass scales allows us to make approximations
in the diagonalization of the 6×6 neutrino mass matrix:
Part D
Seesaw Mechanisms (2)
16
The above unitary transformation leads to the following relationships:
Then we arrive at the type-(I+II) seesaw formula:
Type-I seesaw limit:
Type-II seesaw limit:
(Fritzsch, Gell-Mann, Minkowski,
1975; Minkowski, 1977; …)
(Konetschny, Kummer, 1977; …)
Part D
History of Seesaw
17
The seesaw idea was originally mentioned in a paper’s footnote.
Seesaw—A Footnote Idea:
H. Fritzsch, M. Gell-Mann,
P. Minkowski, PLB 59 (1975) 256
This idea was very clearly elaborated by Minkowski in his paper
PLB 67 (1977) 421 ---- but it was unjustly forgotten until 2004.
The idea was later on embedded into the GUT
frameworks in 1979 and 1980:
— T. Yanagida 1979
— M. Gell-Mann, P. Ramond, R. Slansky 1979
— S. Glashow 1979
— R. Mohapatra, G. Senjanovic 1980
It was Yanagida who named this mechanism as “seesaw”.
Part E
Electromagnetic Dipole Moments
18
A neutrino does not possess electric charges, but it has electromagnetic
interactions with the photon via quantum loops.
Given the SM interactions, a massive Dirac neutrino
can only have a tiny magnetic dipole moment:
 ~
m
3eGF
 20
m

3

10
B

2
0.1 eV
8 2
A massive Majorana neutrino can not have magnetic
& electric dipole moments, as its antiparticle is itself.
Proof: a Dirac neutrino’s electromagnetic vertex can be parametrized by
Majorana
neutrinos
intrinsic property of Majorana ’s.
Part E
Transition Dipole Moments
19
Both Dirac and Majorana neutrinos can have transition dipole moments
(of a size comparable with _) which may give rise to neutrino decays,
scattering with electrons, interaction with external magnetic fields (red
giant stars, Sun, supernovae), and contributions to neutrino masses.
neutrino decays
scattering
Summary of Lecture A
20
(A) Obvious reasons for neutrinos to be massless in the SM.
(B) The Dirac neutrino mass term & lepton number conservation.
(C) The Majorana neutrino mass term & lepton number violation.
---- the Majorana neutrino mass matrix must be symmetric;
---- how to diagonalize a complex symmetric mass matrix;
---- the factor 1/2 in front of the mass term makes sense.
(D) The hybrid neutrino mass term & seesaw mechanisms.
---- both light and heavy neutrinos are Majorana particles;
---- the 33 light neutrino mixing matrix is non-unitary;
---- light neutrino masses arise from the type-(I+II) seesaw.
(E) Electromagnetic dipole moments of massive neutrinos.
---- Dirac neutrinos have finite magnetic dipole moments;
---- Majorana neutrinos have no electromagnetic moments;
---- Dirac & Majorana neutrinos have transition moments.
References about the electromagnetic properties of massive neutrinos:
---- C. Giunti and A. Studenikin, Phys. Atom. Nucl. 72 (2009) 2089;
---- W. Grimus and P. Stockinger, Phys. Rev. D 57 (1998) 1762.
How to Diagonalize a Symmetric Matrix
Proof: M L  VMˆ LV T
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Thank you for your
attention!