Lecture 2
In which the origin of mass is considered and
unsuccessfully measured
The mystery of neutrino
mass
Why are neutrino masses so small?
 Mass in the Standard
Model
In the Standard Model neutrinos are assumed to be massless
If they are not, we need to work out a way of putting them in.
Neutrinos are fermions and obey the Dirac Equation, so there might
be a Dirac Lagrangian term for the neutrino fields
L ν=n(i γ⋅∂−mD )n ⇒ mD n n
Not antineutrino. This is the
adjoint neutrino spinor : n=n†
Dirac mass term
γ0
L mass =m D n n=m D (n L + n R )(n L + n R )=m D (n L n R+ n R n L )
Mass term is the coupling between the left- and right-handed chiral
neutrino states. To conserve gauge invariance we need to introduce the
Higgs field and spontaneous symmetry breaking. The Dirac mass term
explicitly conserves lepton number
 Dirac Mass
nL
x
nR

Higgs mechanism :
< ϕ >∼246 GeV
Dirac Mass
⟨ϕ⟩
mD =G ν
√2
Vacuum
Expectation Value
Hang on, but...
Small m → smaller G
Addition to SM of a sterile nR
that is a distinct state
Majorana Neutrinos
Mass term involves a left- and a right-handed field. For neutrinos we know
that the left-handed field exists. In 1937, Ettore Majorana wondered if you
could make a right-handed field from the existing left-handed one and
form a mass term that way
is a right-handed field ; C = charge
C
T
n L =C n L
conjugation matrix
Can form a Majorana neutrino : n = nL + nLC which is self-conjugate. That
is, the particle is identical to the antiparticle
Clearly this can only happen for neutral particles. A majorana electron, for
example, would violate charge conservation. The neutrino is the only
fermion with potential to be Majorana.
We can also now write down a mass term for Majorana neutrinos
1
1
C
C
C
C
L Maj = mL (n n+ n n )= mL (n L n L + n L nL )
2
2
We
leading to a process
are now coupling neutrinos and antineutrinos,
which violates lepton number by 2
Damn
It turns out that you can't actually form this Majorana term with the lefthanded neutrino field in the Standard Model
nL
I3 = 1/2
Y = -1
C
L
n nL
I3 = 1
Y = -2
To maintain gauge invariance this has to couple to a Higgs-y thing
with Y = +2 and I3 = -1 - that is a Higgs triplet with hypercharge +2.
No such field exists in the Standard Model (although you do get them
if you expand the Higgs sector to include both a scalar doublet and triplet)
We are forced then to consider the existence of an indepedent
right-handed neutrino field even in the case of Majorana neutrinos.
This would be a singlet with I3 = 0 and Y=0 – and can easily form a
standard Dirac term.
The existence of neutrino mass implies physics beyond the Standard
Model,
either from a right-handed state
needed for the standard mass
mechanism, or a Higgs triplet, or a new mass mechanism.
The general mass term
Suppose at the beginning there were 2 Majorana neutrinos. An almost massless
one, and a very heavy one. The mass term looks like
Lmass =m ν ν+ M N N =( ν N )
(
m 0
0 M
)(
ν
N
)
States of definite mass
Suppose the light neutrino mostly is composed of a left-handed chiral
state L and the heavy state is mostly composed of a right-handed chiral
state NR - can write the relationship in terms of some unknown parameter 
ν=cos θ ν L −sin θ N R ; N =sin θ ν L + cos θ N R →
Mass Eigenstates
(Physical particles)
Lmass =( ν L N R )
()(
Chiral Eigenstate
(
c −s
s c
−1
)(
)(
)( )
ν = cos θ −sin θ ν L
N
sin θ cos θ N R
)( )
m 0 c −s ν L
0 M s c NR
off-diagonal mass matrix
The general mass term
The most general mass term combines Dirac and Majorana
masses
From Dirac mass terms
Lmass =( ν
C
L
ν
C
R
)
(
)( )
0 mD νL
mD mR νCR
From Majorana mass terms
Detect these states
which couple to the Weak interaction but these states are not the “particle”
Physical masses are the eigenvalues of the diagonalised
mass matrix (m1,m2).
 =Z
M
−1
 
M Z=
m 1
0
0
m 2
1
2
2
m̃1,2= m R ± m R +4 m D
2
[
√
]
Seesaw Mechanism
Suppose there are two Majorana neutrinos – a light one and
a heavy one and that they are not chiral eigenstates .
Then the mass states are a super-position of the chiral
states.
m 1=
m
mR
m 2=m R 1
2
D
on the order of an MeV or so – Dirac masses like
other charged leptons
=m
m
2
D
m
2
R
very large – at the GUT scale - 1015 GeV
the physical field m now
naturally has a very small
mass (“our” neutrino)
≈m R =m N
Leptogenesis
Seesaw mechanism requires a GUT scale heavy
Majorana neutrino partner.
In many GUT theories, B-L must be conserved
(baryon number - lepton number)
Suppose there is direct CP violation in the heavy neutrino
decay? This generates a violation of L.
N →l +- + H -+
N → ν+H 0
0
0
Γ ( N → ν + H )≠Γ( N → ν+ H )
+
+
Γ ( N → l + H )≠Γ( N →l + H )
Produces an asymmetry
in the number of leptons
and antileptons
Leptogenesis
To keep B-L conserved one needs to violate B as well.
Could neutrino mass help explain CP violation in the
baryons? In other words, could neutrinos help explain why
there is more matter than antimatter in the universe?
Need to find out whether neutrino is Majorana or not.
Seesaw and GUTs
Electromagnetic, strong
and weak forces have very
different strengths
If supersymmetry is valid
their strengths are the same
at around 1016 GeV
To explain light neutrino
masses through the see-saw
mechanics, we need a heavy
neutrino partner with mass
1016 GeV
Probing
of GUT scale physics
using light neutrinos!
(Attempts at) mass measurements
 decay
Measurement of  mass from kinematics of  decay.
d i
dE

=C p Em e  E 0 −E  E 0− E2−m2 F E   E 0−E−m  
Observable is m2
Requirements
The number of electrons close to the endpoint should
be large
Good (and well-understood) electron energy resolution
No (or minimal) electron energy loss within the source
Minimal atomic and nuclear final state effects, of
excited transitions
3
3
+
­
Gaseous Tritium : H  He e  e
Endpoint is at 18574 eV
No molecular excitation above 18547 eV
Still only 10-9 electrons in this region
Gaseous so you can have a very large source
Mainz Experiment
The current standard for tritium beta decay experiments
2 acceptance
High energy resolution
E
E
~0.03%
Electrostatic
MAC-E Filter
History of Tritium- decay
Present Status
Both experiments have reached the intrinsic limit of their
sensitivity.
KATRIN
Due to start 2013 (hmmmmmm....)
Expected limit : me < 0.2 eV (90% CL)
Discovery potential : me = 0.35 eV at 5 
Katrin on the move
KATRIN on the move
Katrin data
KATRIN Sensitivity
3 year run period
Starts in 2016 (!!!!!!?)
 mass
Easiest way is to use pion decay at rest

+
2

2

2


2

m =m m −2 m  p m
2

m =139.56995±0.00035 MeV
m  =105.658358±0.000005 MeV
2

m =−0.016±0.023 MeV
p =29.792±0.00011 MeV
m ν < 190 keV (90%CL )
2
 mass
+
­
+
e e  
↳
­
+­
0
   5   
m 15.5 MeV 95% CL
s

E=

2
Cosmology
m eV
m eV
Density fluctuations
are affected by
neutrino mass in the
early universe
Highly model
dependent
WMAP,2dF,ACBAR,
CBI
∑ mν <0.27−0.56 eV
i
m eV
m eV
2 Decay
Neutrinoless double beta decay is considered a
golden channel for the measurement of neutrino
mass.
In some nuclei 
decay is forbidden
but double
beta decay is not
­
 Z , A  Z2, A2 e 2  e
2 Decay
−1
[T ]
2
1/ 2
=G Q , Z ∣M
Calculable
phase space
2
2
∣
2
Nuclear
matrix element
Second order process in perturbation theory
Severe test for nuclear matrix element calculation
Nuclear structure effects cause variations in the nuclear
matrix elements of factors of 10
2 Decay
Only occur in 36
known sources
Rarest natural
radioactive decay
extremely long
half-lives
Neutrinoless  Decay
Neutrino must have
mass
(RH chiral 
LH helical) e
e (LH chiral
LH helical)
Neutrino is a Majorana
particle
Violation of lepton
number conservation
m
ν L =ν h=−1+ ν h=+ 1
E
Experiments are crucial
to understanding
helicity states
2
 0  =G0  ∣M 0 ∣
2
2
∣∑ ∣U ∣ m ∣ ⇒T
i
ei
i
27
1 /2
~10 years
0 signal
Experimental Requirements
Types of experiments
Passive Source - NEMO3
Advantage : electron tracking
Disadvantage : less source
material and worse energy
resolution
Active Source Heidelberg-Moscow
11 kg of Ge enriched to 86% of 76Ge in the form of 5 Ge diodes
surrounded by Cu,Pb,Bn shielding
0 electrons detected by Ge detectors themselves
Only sum of electron energy measured
0
Heidelberg-Moscow
m = 0.4 eV
GERDA
GERDA
t1/2 > 2.1 x 1025 yr @ 90% CL
Inconsistent with HdM, but not
definitive (yet)
Future Program
Direct mass measurements
Tritium  decay
1
2 2
i
∑ ∣ ∣ 
2
i
U ei m
< 2.3 eV
Katrin extends sensitivity to 0.2 eV
0 decay
Cosmology
∣
∣
2
U
∑i ei mi
<0.3-1.2 eV
<m>=440 meV from HM
∑i mi0.7 eV
Pion decay
m  190 keV
Tau decay
m   18.2 MeV

Model dependent
Seesaw and GUTs
Electromagnetic, strong
and weak forces have very
different strengths
If supersymmetry is valid
their strengths are the same
at around 1016 GeV
To explain light neutrino
masses through the see-saw
mechanics, we need a heavy
neutrino partner with mass
1016 GeV
Probing
of GUT scale physics
using light neutrinos!
SN1987A
Neutrinos detected
Four neutrino detectors operating at the time
Kamiokande II, IMB, BST, Mont Blanc
Mass from Velocity
The neutrinos had travelled 150,000 light years – enough
for small mass differences to show up as a difference in
arrival times
L
L E
L

2 c
4
t F =t −t 0= =
c~ 1m E
2
v c p
c
2
 t=t j −t i= t 0 
L m
2c
2

 
1
E
2
j
−
1
E 2i
Estimate dependent on models of supernova process
(emission intervals, size of the neutrino shell etc)
m 5.7 eV
95 CL
e
The General Mass Term
If we are resigned to the existence of a sterile right-handed
state, then we can construct a general mass term with Dirac
and Majorana masses
Lmass =( n
C
L
n
C
R
)
(
m L m D nL
m D m R nCR
)( )
(
nL
1 C
m L mD
C
n≡ C → L mass=− [n M n+ n M n ] with M =
2
mD mR
nR
()
)
Observable masses are the eigenvalues of the diagonalised
mass matrix (m1,m2)
 =Z
M
−1
 
M Z=
Mixing
matrix
m 1
0
0
m 2
m1,2=
1
[2

2
2
 m L mR ±  mL −m R  4 mD
]
Passive Source - NEMO3
Advantage : electron tracking
Disadvantage : less source
material and worse energy
resolution
Cuoricino/Cuore
Heat sink
Thermal coupling
Thermometer
Double beta decay
Crystal absorber
example: 750 g of TeO2 @ 10 mK
C ~ T 3 (Debye)  C ~ 2×10-9 J/K
1 MeV -ray

T ~ 80 K
 U ~10 eV
Cuoricino/Cuore
Heat sink
Thermal coupling
Thermometer
Double beta decay
Crystal absorber
example: 750 g of TeO2 @ 10 mK
C ~ T 3 (Debye)  C ~ 2×10-9 J/K
1 MeV -ray

T ~ 80 K
 U ~10 eV
Cuoricino Results
60
Co
0
2529 keV
0
24
T 1/2 3.0×10 years ⇒ ⟨ m ⟩0.68eV
SNO+
150
Nd loaded - m < 80 meV
The General Mass Term
If we are resigned to the existence of a sterile right-handed
state, then we can construct a general mass term with Dirac
and Majorana masses
Lmass =( n
C
L
n
C
R
)
(
m L m D nL
m D m R nCR
)( )
(
nL
1 C
m L mD
C
n≡ C → L mass=− [n M n+ n M n ] with M =
2
mD mR
nR
()
)
Observable masses are the eigenvalues of the diagonalised
mass matrix (m1,m2)
 =Z
M
−1
 
M Z=
Mixing
matrix
m 1
0
0
m 2
m1,2=
1
[2

2
2
 m L mR ±  mL −m R  4 mD
]
Two ways to go
Dirac neutrinos
There are new
particles (right handed
neutrinos) after all
Why haven't we seen
them?
They must only exist
to give neutrinos mass
Still have to solve the
question of their very
very weak coupling
Two ways to go
Majorana neutrinos
There are new particles
(right handed neutrinos)
after all
If I pass a neutrino and
look back I will see a
right-handed thing
Must be a right-handed
anti-neutrino
No fundamental
difference between
neutrinos and anti neutrinos
(Theorists Favourite!)
Power spectra
“Wavelength” of
density fluctuation
m = 0 eV
m = 1 eV
Tegmark, M. et al. 2004. PhRvD, 69, (10), 103501.