NEUTRINOS IN THE STANDARD MODEL
Alessandro MIRIZZI
A NEUTRINO TIMELINE
1927 Charles Drummond Ellis (along with James Chadwick and colleagues)
establishes clearly that the beta decay spectrum is really continous, ending all
controversies.
1930 Wolfgang Pauli hypothesizes the existence of neutrinos to account for
the beta decay energy conservation crisis.
1932 Chadwick discovers the neutron.
1933 Enrico Fermi writes down the correct theory for beta decay,
incorporating the neutrino.
1937 Majorana introduced the so-called Majorana neutrino hypothesis in which
neutrinos and antineutrinos are considered the same particle.
1956 Fred Reines and Clyde Cowan discover (electron anti-)neutrinos using a
nuclear reactor.
1956 Discovery of parity violations in beta decay by Chien-Shiung Wu.
1957 Neutrinos found to be left handed by Goldhaber, Grodzins and Sunyar.
1957 Bruno Pontecorvo proposes neutrino-antineutrino oscillations
analogously to K0-antiK0, leading to what is later called oscillations into
sterile states.
1962 Ziro Maki, Masami Nakagawa and Sakata introduce neutrino flavor
mixing and flavor oscillations.
1962 Muon neutrinos are discovered by Leon Lederman, Mel Schwartz, Jack
Steinberger and colleagues at Brookhaven National Laboratories and it is
confirmed that they are different from nu_e.
1964 John Bahcall and Ray Davis propose feasability of measuring neutrinos
from the Sun.
1967 Stewen Weimberg, Sheldon Lee Glashow, Abdus Salam formulate the
Standard Model of Electroweak interactions, in which electromagnetic and
weak forces are unified in sigle fashion.
1968 Ray Davis and colleagues get first radiochemical solar neutrino results
using cleaning fluid in the Homestake Mine in North Dakota, leading to the
observed deficit known as the "solar neutrino problem".
1973 First observation of neutral current of neutrinos in the Gargamelle
experiment at CERN. First indirect confirmation of the Standard Model.
1976 The  lepton is discovered by Martin Perl and colleagues at SLAC in
Stanford, California. After several years, analysis of tau decay modes leads to
the conclusion that tau is accompanied by its own neutrino nu_tau which is
neither nu_e nor nu_μ.
1976 Designs for a new generation neutrino detectors made at Hawaii
workshop, subsequently leading to IMB, HPW and Kamioka detectors.
1980-90 The IMB, the first massive underground nucleon decay search
instrument and neutrino detector is built in a 2000' deep Morton Salt mine
near Cleveland, Ohio. The Kamioka experiment is built in a zinc mine in Japan.
1983 Discovery of the gauge bosons W and Z by the UA1 and UA2
collaborations. First direct confirmation of the Standard Model.
1985 The "atmospheric neutrino anomaly" is observed by IMB and Kamiokande.
The anomaly is at first believed to be an artifact of detector inefficiencies.
1986 Kamiokande group makes first directional counting observation solar of
solar neutrinos and confirms deficit.
1987 The Kamiokande and IMB experiments detect burst of neutrinos from
Supernova 1987A, heralding the birth of neutrino astronomy, and setting
many limits on neutrino properties, such as mass.
1988 Lederman, Schwartz and Steinberger awarded the Nobel Prize for the
discovery of the muon neutrino.
1989 The LEP accelerator experiments in Switzerland and the SLC at SLAC
determine that there are only 3 light neutrino species (electron, muon and
tau).
1991-2 SAGE (in Russia) and GALLEX (in Italy) confirm the solar neutrino
deficit in radiochemical experiments.
1995 Frederick Reines and Martin Perl get the Nobel Prize for discovery of
electron neutrinos (and observation of supernova neutrinos) and the tau
lepton, respectively.
1998 After analyzing more than 500 days of data, the Super-Kamiokande
team reports finding oscillations and, thus, mass in muon neutrinos. After
several years these results are widely accepted and the paper becomes the
top cited experimental particle physics paper ever.
2000 The DONUT Collaboration working at Fermilab announces observation
of tau particles produced by tau neutrinos, making the first direct
observation of the tau neutrino.
2000 Super-K announces that the oscillating partner to the muon neutrino is
not a sterile neutrino, but the tau neutrino.
2001-2 SNO announces observation of neutral currents from solar
neutrinos, along with charged currents and elastic scatters, providing
convincing evidence that neutrino oscillations are the cause of the solar
neutrino problem.
2002 Masatoshi Koshiba and Raymond Davis win Nobel Prize for measuring
solar neutrinos (as well as supernova neutrinos).
2002 KamLAND begins operations in January and in November announces
detection of a deficit of electron anti-neutrinos from reactors at a mean
distance of 175 km in Japan. The results combined with all the earlier solar
neutrino results establish the correct parameters for the solar neutrino
deficit.
2004 Super-Kamiokande and KamLAND present evidence for neutrino
disappearance and reappearance, eliminating non-oscillations models.
2005 KamLAND announces first detection of neutrino flux from the earth
and makes first measurements of radiogenic heat from earth.
[Slide by Eligio Lisi]
THREE FUNDAMENTAL QUESTIONS OF n PHYSICS
HISTORY: BETA DECAY
1914: discovery
(Chadwick)
continuous
energy
spectrum
of
beta
decay
This experimental energy spectrum is
from G.J. Neary, Proc. Phys. Soc.
(London), A175, 71 (1940).
Expected 2-body decay
Problem: nucleus (A,Z) thought to be A protons + (A-Z) electrons
Beta decay: (A,Z) → (A, Z+1) + e- (two body decay, monoenergetic e-)
BOHR EXPLANATION OF THE BETA DECAY SPECTRUM
Niels Bohr:
Energy not conserved
in the quantum domain?
PAULI‟S EXPLANATION OF THE BETA DECAY SPECTRUM
“Neutrino”
(E.Fermi)
“Neutron”
(1930)
Wolfgang Pauli
(1900-1958)
Nobel Prize 1945
THE BIRTH OF THE NEUTRINO
Pauli proposes existence of “neutron” (with spin ½ and mass not more than 0.01
mass of proton) inside nucleus in a famous letter (4 December 1930):
Pauli also left in his diaries: “Today I have done something which no theoretical
physicist should ever do in his life: I have predicted something which shall never
be detected experimentally.“
Chadwick discovers neutron (1932):
– Mass of neutron similar to mass of proton: not Pauli‟s particle!
– Fermi introduces name “neutrino” (ne), which is different to neutron,
and beta decay is decay of neutron:
n  p e ne
-
FOUR-FERMI INTERACTION
Fermi was the first to write down the field theoretical form of an
interaction involving the neutron, the proton, electron and neutrino
fields that would describe weak interactions. The so-called fourFermi interaction Hamiltonian density is written as
H weak
GF

 p  n e  n e
2
GF = 1.16637(1) x 10-5 GeV-1
The form of the weak Hamiltonian in low energy weak decays has
been confirmed over the year to basically have the same form as in
the four-Fermi prescription, although a great deal more is known
about its Lorentz structure
MODERN FORM OF FOUR-FERMI INTERACTION
The main observation that demand some modification of Fermi‟s
original four-Fermi interaction are as follows:
The fundamental fields entering the Hamiltonian are not hadronic
fields like the proton or the neutron, but rather are fields
describing quarks
The space-time structure of the currents have not only the vector
currents (V), but also axial vector currents (A) [V-A form]
The weak-interactions consist not only of charged current
processes like the beta-decay in which electric charge of leptons
and quarks change, but also neutral current processes where the
electric charge remain unchanged among the leptons and the
hadrons.
The weak interaction Hamiltonian can be written in a compact form as
charged current (CC)
neutral current (NC)
CC:
Chirality projection operators
Fermion fields
The Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix
From the “Review of Particle Physics 2009”
[http://pdg.lbl.gov/2009/reviews]
There is also a phase in the matrix responsible for CP violation
CC interactions give rise to a host of new weak interaction phenomena :
NC:
No term which connects two different species of quark or leptons
No flavor-changing neutral current (FCNC)
PROBLEMS WITH THE FOUR-FERMI INTERACTION
The four-Fermi Lagrangian is not-renormalizable → If one calculates
higher order corrections to any lowest energy process, the contributions
are divergent and the divergence at the L-th loop goes as L2L, L being
the cutoff for the theory.
E.g.: the total cross section for n e scattering is proportional to GF2 s,
being s = (pne + pe)2 . With increasing energy this cross section grows
without limit. Since ne e scattering occurs in s-wave, this process
should obey the s-wave unitarity bound,

s
tot
16

s
This leads to a contradiction, implying that the four-Fermi
description must breakdown above a certain energy.
THE STANDARD MODEL
The search for a renormalizable weak interaction
culminated in the discovery of the gauge theories.
Lagrangian
Weinberg (1967) and Salam (1968) proposed the spontaneously
broken SU (2) L U (1)Y theory of leptons in its modern version.
The correct extension to include quarks was done in the 70‟s using
an earlier suggestion by Glashow, Iliopoulos and Maiani.
In 1971 „t Hooft proved that all spontaneously broken gauge
theories that include only interactions of spacetime dimension ≤ 4
are renormalizable. Solve the
problem of the four-Fermi
interaction. Open the possibility to calculate radiative corrections.
In 1973 Gross, Wilczeck and Politzer showed that that unbroken
SU(3)C group of strong interactions is asymptotically free.
There was born the Standard Model of Electro-Weak and Strong
Interactions
GAUGE INTERACTIONS IN THE STANDARD MODEL
SU (2) L U (1)Y
g
g‟
Spontaneous simmetry
breaking (SSB)
Coupling constants
tan W 
g'
g
Gauge bosons
U(1)em
Weinberg angle
Relative strength of CC vs NC interactions
=1 is SSB is associated to Higgs doublet
Relation with the Fermi constant
Y
Q  T3 
2
U(1)
SU(2)
U(1)Y
Gauge Generators
em
L
Fermion content
SU(2)L U(1)Y
e  g sin W
Electric charge
charged current (CC)
neutral current (NC)
ne
Charge rising CC
W+
ee-
Charge lowering CC
Wne
Neutral current
f
Z0
f
The introduction of a finite-mass W boson removes the divergence
of ne e scattering

[
[
e-
ne
e-
e-
ne

]
ne
W
e-
ne
=
]
2
F
G s

s
 

s


2
F
G M

2
W
Finite!
NEUTRINOS IN THE STANDARD MODEL
Neutrinos have been used to:
Assess the strength of weak interactions (GF)
Probe the V-A structure of CC
Probe the (T3 – Q s2w) charge of NC
Probe CC+NC interference
PROBING GF IN BETA DECAY AND MUON DECAY
G  (1.16632  0.0002) 10 5 GeV -2
5
G  (1.136  0.003) 10 GeV
-2
G  G cos C
Cabibbo angle
PROBING V-A STRUCTURE IN THE PION DECAY
NEUTRINO-ELECTRON SCATTERING
n
The n e- and n e- processes can only proceed
via a neutral current interactions
n
Z0
e-
q2 <<M2W
e-
n : cA→-cA
The ne e- → ne e- processes offers the intriguing possibility of
studying charged current and neutral current interference
ne
ne
Z0
e-
Fierz
rearranging
ne
ne
W
e-
e-
e-
(- sign: interchange of outgoing leptons)
w.r.t. n e- amplitude: cV → cV +1 cA → cA +1
Hung & Sakurai (1981)
Neutrino-electron
cross
sections form ellispes in
the cv, cA plane
In perfect agreement
with the SM
NUMBER OF LIGHT NEUTRINOS FROM COLLIDER
EXPERIMENTS
The most precise measurements of the number of light neutrino
types, Nn, come from studies of Z production in e+e− collisions. The
invisible partial width, Γinv, is determined by subtracting the measured
visible partial widths, corresponding to Z decays into quarks and
charged leptons, from the total Z width. The invisible width is
assumed to be due to Nn light neutrino species each contributing the
neutrino partial width Γn as given by the Standard Model.
NEUTRINO MASS IN THE STANDARD MODEL
Neutrinos are Left-handed
Neutrinos must be Massless
• All neutrinos left-handed  massless
• If they have mass, can’t go at speed of light.
• Now neutrino right-handed??
 contradiction  can’t have a mass
Anti-Neutrinos are Right-handed
• CPT theorem in
quantum field theory
– C: interchange
particles & antiparticles
– P: parity
– T: time-reversal
• State obtained by CPT
from nL must exist: nR
Other Particles?
• What about other particles? Electron, muon,
up-quark, down-quark, etc
• We say “weak force acts only on left-handed
particles” yet they are massive.
Isn’t this also a contradiction?
No, because we are swimming in a BoseEinstein condensate in Universe
Universe is filled with Higgs
• “Empty” space filled with a BEC: cosmic superconductor
• Particles bump on it, but not photon because it is neutral.
• Can’t go at speed of light (massive), and right-handed and
left-handed particles mix  no contradiction
But neutrinos can‟t
bump because
there isn‟t a righthanded one 
stays massless
Therefore, neutrinos are strictly massless in the Standard
Model of particle physics
WEYL EQUATION FOR MASSLESS NEUTRINOS
Dirac equation
4x4 Dirac matrices
Two component
spinors
Energy-momentum relation: E2=p2
positive and one negative energy solution
Each equation has one
Weyl equation (1929)
Positive energy solution has E = |p|
c describes a left-handed neutrino (helicity l=-1/2) of energy E and
momentum p. The remaining solution has negative energy. Neutrino
solution with –E and –p.
It describes a right-handed antineutrino (helicity l=+1/2) of energy
E and momentum p.
nL and nR
nR and nL
Not-invariant under parity
operation nL → nR
There is no empirical evidence for the existence of nR (and nL)
STATES OF MASSIVE AND MASSLESS PARTICLES
Casimir operators of the restricted Lorentz group
↑
four momentum
covariant spin
Massive particles
orthogonality
Little group is SO(3)
Basis of IR‟s of SO(3)
Helicity
Massless particles
Little group is U(1)
State
for m2 = p2 = 0
l is an additive invariant for massless
particles
For massless particles the two states
are two different IR‟s under
↑
They are connected under the proper Lorentz group
if the parity
is a good simmetry:
i.e. they are two different states of the same particle
But this is not the case for n !
SU (2) L U (1)Y
in one-family approximation
(?)
Experimental properties
Evidence of left-handed n‟s alone
Very small mass, if any, for n‟s
Absence of leptonic mixing (to be discussed later)
On the other hand, the coupling of any right-handed neutral lepton
in SM is:
Most economical solution:
Either nL massless
or nR ruled out
No mass term from SSB
Yukawa coupling of n
Higgs doublet
SSB
Neutrino mass term = 0 either setting
gY= 0 or since nR is ruled out
GAUGE VS MASS EIGENSTATE
From 1 to 2 generations
No leptonic mixing
Cabibbo mixing for quarks
From 2 to 3 generations
Gauge eigenstates
Gauge eigenstates grouped into L-doublets and R-singlets.
What about the mass eigenstates?
The fermionic mass matrices in the basis of gauge eigenstates have
the general form (originating from SSB)
neither symmetric nor hermitian
“No mass term for n‟s !”
M can be diagonalized by a biunitary transformation.
S,T unitary
Md diagonal
Mixing matrix in quark CC coupling
Hadron CC takes form
Kobayashi-Maskawa
2 family theory: one mixing angle (Cabibbo), no phase
3 family theory: three mixing angles (K-M), one phase (CP violating!)
For leptons:
But n‟s are massless, degenerate
n can be assumed as mass
eigenstates
can be assumed
V can be set to be the identity matrix (complete
decoupling)
Trivial leptonic mixing
Neutrino states nL are really weak-interaction eigenstates
n is operationally defined to be the “invisible” partner in ± → e±
decay
In conclusion:
Global symmetry corresponding to the separate conservation of e- lepton number
Forbidden to all the orders
Lepton number conservation
Present experimental evidence and the standard electroweak theory
are consistent with the absolute conservation of three separate
lepton numbers: Le , L , L , except fo the effect of neutrino
mixing associated with the neutrino masses. Searches for
violations are of the following types:
DL=2 for one type of charged lepton. The best limit comes from
the search for neutrinoless double beta decay (Z,A) → (Z+2,A) + e+ e-. The best laboratory limit is t1/2 > 1.9 × 1025 yr (90% C.L.) for
76Ge.
Conversion of one charged lepton type to another. For purely
leptonic processes, the best limits are on → e  and → 3e,
measured as G(→ e /→ all) < 1.2×10-11 and G(→ 3e /→ all) <
1.0×10-12.
Conversion of one type of charged lepton into another type of
charged antilepton. The case most studied is - + (Z,A) → e+ + (Z-2,A),
the strongest limit being G(- Ti→ e+ Ca/ - Ti → all) < 3.6×10-11.
Neutrino oscillations. If neutrinos have mass, then it is expected even
in the standard electroweak theory that the lepton numbers are not
separately conserved, as a consequence of lepton mixing analogous to
Cabibbo quark mixing. However, if the only source of lepton-flavor
violation (LFV) is the mixing of low-mass neutrinos then processes
such as μ → e γ are expected to have extremely small unobservable
probabilities.