Neutrinos
in Physics and Astrophysics
Lecture 1
MEPhI Moscow
Christian Spiering, DESY Zeuthen
Content Lecture 1
1.
Postulate of the neutrino
2.
Fermi theory of weak interactions
3.
Mass of the neutrino
4.
Discovery of the electron (anti)-neutrino
5.
Reminder on Dirac equation and gamma
. matrices
6.
Parity violation in weak interactions
7.
Helicity of the neutrino
2
1.
Postulate of the neutrino
3
Beta decay: the electron spectrum
 Widely accepted assumption in the 1920s:
the nucleus consists of protons and electrons
 Beta decay goes like
nucleus (A,Z)  nucleus (A,Z-1) +
e
 Then, why is the beta spectrum continuous?
Niels Bohr:
Energy conservation only for
an ensemble of interactions,
not for individual interactions ?
(i.e. just another surprise of
quantum mechanics)
Expected for 2-particle final state
4
Answer to question from appetizer lecture
How did Lise Meitner measure the beta spectrum?
1. Momentum selection by magnet
a. Number of electrons by electroscope
b. number of electrons from blackening of films
2. No magnet, electrons to cloud chamber, measure track length
3. Calorimetric measurement
5
Beta decay: spin statistics
 N(A,Z)  N‘ (A,Z-1) +
e
 Nucleus with even number A of nucleons has integernumber spin, nucleus with odd number A has half-integer
spin
 That means, both N and N‘ have the same spin, either
integer or half-integer
 Electron has spin ½
  Angular momentum can be conserved only,
if a second spin ½ particle is emitted.
6
1930: Wolfgang Pauli postulates the neutrino
nucleus A  nucleus A‘ + electron + „neutron“
- neutral
- spin ½
- mass < electron mass
- weakly interacting
Expected for 2-particle final state7
The famous Pauli Letter from Dec. 4, 1930
… to Lise Meitner and a Meeting of German Physicists in Tübingen:
8
2.
Fermi theory of weak
interactions
9
Fermi theory of weak interactions and beta decay
 1932: James Chadwick discovers the neutron, Dmitry Ivanenko and
Werner Heisenberg independently propose that the neutron is a
component of nuclei.
 Pauli: mixed feelings – since this cannot be „his“ neutron.
 1933: Fermi invents the name Neutrino (ital.: the small neutral)
 Fermi‘s theory of -decay
Point-like interaction
10
Fermi Theory
Fermi described the beta decay as
current-current interaction
- A neutron is destroyed, a proton created
- An anti-neutrino is created (equivalent to a
neutrino being destroyed) and an electron created
𝐻~
3
𝑑 𝑥 𝑝 𝑥 𝑛 𝑥 𝑒 (𝑥)(𝑥)
hadronic current leptonic current
But this not was Fermi wrote. He wrote:
𝐻~
𝑑 𝑥 𝑝 𝑥 µ 𝑛 𝑥 𝑒 𝑥 µ(𝑥)
3
(„vector current“, see later on gamma matrices µ)
11
Fermi‘s Golden Rule
𝑑𝑊
2𝜋
2
=
| 𝑓 𝐻 𝑖 | 𝑑𝑛𝑓 /𝑑𝐸
𝑑𝑡
ℎ
 𝑑𝑊
𝑑𝑡
Reaction rate

Matrix element (or the transition operator between
initial and final state
f
H
i
….
 dn f
..
dE
The densitiy of final states (the more available final
states, the faster the decay will go)
12
Value of the Fermi constant G (often „GF“)
 Fermi estimated the value of GF from the decay rate of beta
emitters. That allowed to estimate the neutrino-nucleon cross
section as about 10-38 cm² at 1 GeV.
 Today, the best value for GF is obtained from the decay time of the
muon:
192 𝜋³
𝜏 𝜇 → 𝑒 + 𝑒 +  =
5
𝐺𝐹² 𝑚𝜇
(in units where c=1 and h/2=1)
 From this one gets GF = 1.166  10-5 GeV-2
 Compare this to the coupling constant of electromagnetic
interactions,  = 1/137. Why is GF so small?
13
Why is the weak interaction weak?
 What Fermi considered a point-like interaction is actually the exchange of
a W-boson
„propagator“
 The weakness of weak interactions is due to the high mass of the vector
bosons (illustratively: heavy exchange particles cannot fly far from their emission
point since the energy borrowed from the vacuum must fulfill E t < h/2)
 The dimensionless coupling constant of the weak interaction (the
equivalent of the Sommerfeld constant 1/137) is:
2
8 𝑚𝑊
𝐺𝐹
1
That means, the weak interaction is intrinsically
𝛼𝑊 =
≈
30
stronger than the electro-magnetic interaction!
4 2𝜋
14
Why is the weak interaction weak?
 For a process where the exchange boson carries four-momentum q, the
QED and weak interaction propagators are, respectively:
 𝑃𝑄𝐸𝐷 ~
1
𝑞²
𝑃𝑤𝑒𝑎𝑘 =
1
2
𝑞²− 𝑚𝑊
2
(i. e. for small 𝑞 ) ≈
1
− 2
𝑚𝑊
 Therefore weak interaction decay rates, which are proportional to P², are
supressed by a factor q4/mW4 relative to QED decay rates.
 (neutron)~10³ sec, (pion)~10-8 sec, compared to (e.m.decay) ~ 10-19 sec
 In the high energy limit where |q²| > mW², the electromagnetic and weak
interactions have similar strength.
 Another example for the small cross section: pp  de+e in the Sun.
Density ~ 100g/cm³, lifetime of protons before reaction ~1010 years!
15
The beta spectrum
 Again from the Fermi rule, one can derive
𝑑𝑊
𝑑𝑡
=
2𝜋
ℎ
|𝑓𝐻𝑖|
2 𝑑𝑛𝑓
𝑑𝐸
=
2
𝐺² 𝑀𝑖𝑓
2 𝜋³ 𝑐³ ℎ7
(𝑄 − 𝑇𝑒 )2 𝑝² 𝑑𝑝
16
More precisely (but still assuming m=0):
17
The Kurie-Plot
1/ 2
 N ( p) 
 2

 p F ( Z , Te ) 
3
H  He + e  e
3
-
Q  18.62keV
KEe (MeV)
18
3.
Mass of the neutrino
19
How looks the spectrum if the neutrino has mass?
Fermi
20
How looks the spectrum if the neutrino has mass?
 For a non-zero neutrino mass, the expression has a simple change:
m(e) = 0 eV
m(e) = 20 eV
Electron Energy (keV)
21
Techniques in  spectroscopy
(from G. Drexlin, KIT)
22
History of modern neutrino mass experiments
(from G. Drexlin, KIT)
23
History of modern neutrino mass experiments
(from G. Drexlin, KIT)
24
The MAC-E Principle
Troitsk, Mainz, KATRIN
(from G. Drexlin, KIT)
25
The MAC-E Principle
Troitsk, Mainz, KATRIN
26
The MAC-E Principle
Troitsk, Mainz, KATRIN
27
Troitsk and Mainz experiments
(from G. Drexlin, KIT)
28
KATRIN
29
KATRIN
30
KATRIN: some pictures
31
KATRIN: some pictures
32
KATRIN: some pictures
33
KATRIN mass sensitivity and schedule
(from G. Drexlin, KIT)
34
Why early sensitivity to sterile neutrinos?
 Remember:
 e is not a mass eigenstate:
 From oscillation studies: e consists of ~70% 1, ~30% 2 and ~1% 3
 The spectrum must be understood as composed from decays from all
three mass eigentstates, with contributions proportional to
𝑐2
𝑚1
0.7² 1 −
𝐸0 − 𝐸
2
𝑐2
𝑚2
0.3² 01 −
𝐸0 − 𝐸
2
𝑐2
𝑚3
< 0.001 1 −
𝐸0 − 𝐸
2
35
Why early sensitivity to sterile neutrinos?
 The approximation
1−
𝑚𝑒 𝑐²
2
𝐸0 −𝐸
in
is o.k. if all mi are smaller than the energy resolution.
 Adding a fourth neutrino with mass m4 on the eV scale (comparable to
the energy resolution of KATRIN, then this term has to be replaced by
2
(1-|𝑈𝑒4 | ) 1 −
𝑚𝑒 𝑐²
𝐸0 −𝐸
2
2
+ |𝑈𝑒4 |
1−
𝑚4 𝑐²
2
𝐸0 −𝐸
36
Why early sensitivity to sterile neutrinos?
 Then the shape of the  spectrum would be modified
(shown here for ms = 3 eV,  = 45°):
already excluded
 This might be earlier detectable than a sub-eV e mass.
37
4.
Discovery of the electron
(anti)-neutrino
38
The discovery of the neutrino
Remember Appetizer Lecture:
 Peierls and Bethe predict from Fermi theory a cross section for
neutrinos from nuclear reactors
 This is billions times smaller than that of electrons with protons.
Seemed to be hopeless to ever try detecting neutrinos!
 First idea of Fred Reines:
detect
neutrinos from
fission bomb
 Only later:
neutrinos
from a reactor
39
Nuclear reactors as  source
1953: Hanford Reactor
1955: much stronger military
reactor in Savannah River
Clyde Cowan
ON/OFF
Fred Reines
Fred Reines
Clyde Cowan
40
First attempt: Hanford detector
Anti-Electron
Neutrinos
from
𝝂𝐞
Hanford
Nuclear Reactor
n
Cd


p
e+
e−
3 Gammas
in coincidence

41
Savannah River experiment: the principle
photomultipliers
CdCl solved
in water
scintillator
42
Savannah River: Discovery 1956
The detector
Nobel Prize 1995
A neutrino signature
positron neutron
43
Pauli‘s answer to Reines
45
Reines´ cross sections from 1956 and 1959
 1956:
The rate of detected reactions e + p  n + e+ was consistent with the
prediction 6  10-44 cm².
 1957:
Parity violation in weak interactions is discovered (see next chapter).
This increases the expected cross section by a factor of 2.
 1959:
The final result from the Savannnah River reactor is published. The
cross section derived from the rate is now (112.6)  10-44 cm².
Some people raised the question whether the results were „tuned“ to the
expectations. It is often guessed that this was the reason that Reines did
not get the Nobel Prize in the 1960s but only 1995.
46
Savannah River: Discovery 1956
The cross section is
𝜎=
𝑅
cm²
3600 𝑓 𝑛 𝜀𝛽 𝜀𝑛
 R = signal in counts/hour
 n = number of target protons = 8.31028
 f = anti-neutrino flux at the detector, /(cm²sec) = 1.31013,
assuming 6.1 anti-‘s per fission
  , n = detection efficiencies for positron and neutron
- E = 3.53 + E
(in units of electron mass  c²)
-  spectrum from beta source
  = (112.6)  10-44 cm²
47
5.
Reminder on Dirac equation,
gamma matrices etc.
48
Reminder on Dirac equation
For relativistic particles one has to replace the Schrödinger
𝛿
1 𝛿² 
equation i
=+ V by the Dirac equation:
𝛿𝑡
2𝑚 𝛿𝑥²
𝐸 = (𝛼𝒑 + 𝛽𝑚) 
𝑖  = (−𝑖𝛼
𝛿
𝛿𝑡
𝛿
𝛿𝑥
− 𝑖𝛼
𝛿
𝛿𝑦
− 𝑖𝛼
𝛿
𝛿𝑧
+ β𝑚)
Energy-momentum conservation imply conditions on the form
of the wave function  and the factors  and :
 and  must by 44 matrices and
 a four-component wavefunction,
known as Dirac-spinor
1
2
= 
3
4
49
Reminder on Dirac equation
For relativistic particles one has to replace the Schrödinger
𝛿
1 𝛿² 
equation i
=+ V by the Dirac equation:
𝛿𝑡
2𝑚 𝛿𝑥²
𝐸 = (𝛼𝒑 + 𝛽𝑚) 
𝑖  = (−𝑖𝛼
𝛿
𝛿𝑡
𝛿
𝛿𝑥
− 𝑖𝛼
𝛿
𝛿𝑦
− 𝑖𝛼
𝛿
𝛿𝑧
+ β𝑚)
Matrices  and  are given by
𝐼
𝛽=
0
0
−𝐼
0
and 𝛼𝑖 =
−𝜎𝑖
𝜎𝑖
0
50
Reminder on Dirac equation … and gamma matrices
51
Reminder on Dirac equation … and gamma matrices
52
Reminder on Dirac equation … and gamma matrices
𝐼
𝛽=
0
0
−𝐼
0
and 𝛼𝑖 =
−𝜎𝑖
0
𝛾 ≡𝛽
𝜎𝑖
0
𝑖
𝛾 ≡ 𝛽𝛼𝑖
(i=1,2,3)
53
Dirac equation in covariant formulation
𝑖  = (−𝑖𝛼
𝛿
𝛿𝑡
𝛿
𝑥 𝛿𝑥
𝛿
𝑦 𝛿𝑦
− 𝑖𝛼
− 𝑖𝛼
𝛿
𝑧𝛿𝑧
+ β𝑚)
𝛿
𝑧𝛿𝑧
− 𝑚)
multiply that with  (and note that ² = 1)
𝛿
𝛿𝑡
𝛿
𝑥 𝛿𝑥
(𝑖 +𝑖𝛼
Then, with
𝛿
𝑦𝛿𝑦
+ 𝑖𝛼
0
𝛾 ≡𝛽
and 𝛿µ ≡ 𝜕0 , 𝜕1 , 𝜕2 , 𝜕3 ≡
+ 𝑖𝛼
𝑖
𝛾 ≡ 𝛽𝛼𝑖
𝜕 𝜕 𝜕 𝜕
, , ,
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧
one gets the covariant formulation of the Dirac equation
(𝑖𝛾 𝛿𝜇 −𝑚) = 0
𝜇
54
Solutions of the Dirac equation
𝑖 = 𝑢𝑖 𝐸, 𝒑
𝑖(𝒑∙𝑥
−𝐸𝑡)
𝑒
 Positive energy
 Negative energy
55
6.
Parity violation
in weak interactions
56
Parity Operator
 The Parity Operator P causes a mirror transformation on the coordinate
origin (coordinate inversion): (x,y,z)  (-x,-y,-z):
𝑷  𝒓   (−𝒓)
 Repeated operation: P² = 1 (P is a unitary operator)
 The parity operator for Dirac spinors can be identified as either
P = + 0 or P = - 0
 Eigenvalue (if existent!) is 1
… for instance = cos 𝑥
= sin 𝑥
.
𝑃   cos −𝑥 = cos 𝑥
P = +1
𝑃   sin −𝑥 = − sin 𝑥
P = -1
 = sin 𝑥 + cos 𝑥 𝑃  = −sin 𝑥 + cos 𝑥  
parity not defined !
57
Intrinsic Parity
The intrinsic parity of a particle is defined by the action of the parity
operator 𝑃 = 0 on a spinor for a particle at rest.
1
0
For instance, the u1 spinor for a particle at rest
0
0
is an eigenstate of the parity operator with
 The intrisic parity of a spin ½ particles is opposite to that of an
anti-particle. Per convention one defines the intrinsic parity of
particles as positive and that of ant-particles as negative
58
Intrinsic Parity and parity of composed systems
For example
𝑃 𝑒 − = 𝑃 𝑞 = 𝑃(e) = +1 and 𝑃 𝑒 + = 𝑃 𝑞 = 𝑃(e) = -1
Form QFT it can be shown that the vector bosons responsible for e.m.,
strong and weak interactions have all negative parity
𝑃  = 𝑃 gluon = 𝑃 𝑊 ± = 𝑃 𝑍 = −1
2-particle system:
Parity = product of intrinsic parities and parity of orbital wave function,
which is given by (−1𝑙 ), where 𝑙 is the orbital momentum in the final
state.
Take, e.g. the 0 meson, decaying like 𝜌0 → 𝜋 + 𝜋 −
The pions are produced with 𝒍= 1
𝑃 𝜌0 = 𝑃 𝜋 + ∙ 𝑃 𝜋 − ∙ (−1)𝑙=1 −1 = −1 ∙ −1 ∙ (−1)
59
Behaviour under parity transformation
Rank
 Scalar
 Vector
Parity
Examples
0
+
temperatur, mass, …
1
-
momentum p
 Axial Vector 1
+
angular momentum L = x  p
(vector product of 2 vectors)
spin
magnetic field
 Pseudoscalar 0
-
helicity ( spin  momentum)
(scalar products of
. axial vector and vector)
60
Parity violation: principle of the Wu experiment
 1956: Lee and Yang conclude that weak interactions violate parity
conservation (i.e. invariance under spatial inversion):
K+ can decay in 2 (even parity: P=+1) ) and 3 (odd parity, P=-1)
 1957: Dedicated test by Madame Wu and co-workers:
 Place 60Co source in strong magnetic field B  60Co is polarized,
.magnetic moment is µ aligned along B
 Measure electrons from the decay 60Co 
60Ni*
+ e- +  e
 Mirror operation: B and µ are axial vectors and do not change their direction,
.momentum p of electrons changes direction
 However, if parity would be conserved, the number of electrons emitted along µ
.should stay the same. This could only be achieved, if the equal number of electrons
.were emitted in forward and backward hemispheres with respect to nuclear spins.
 This what not observed
 parity is violated
.
in weak interactions!
61
Parity violation: details of the Wu experiment
 Measure electrons from Co decay (weak)
 and photons from Ni decays (electromagnetic
 conserve parity)
 The 2 e.m. decays are non-isotropic (gamma
emission probability is function of the angle 
with the field). Therefore one can monitor the
polarisation of the Co sample by measuring its
anisotrop.y
62
Parity violation: details of the Wu experiment
To achieve a sufficient polarization,
 the Co sample must be cooled down
(a few mKelvin!)
 the magnetic field must be large
 embedding Co in paramagnetic crystal
 if crystal in magnetic field, ist electronic magnetic
moments (which are large) become oriented in the
field and generate inside the crystal local fields of
tens of Tesla
Photomultiplier (PM) far from magnetic field
 need long light guide from the an anthracene
scintillator to the PM
Gammas are detected by NaI crystals
63
Parity violation: details of the Wu experiment
 Magnetize, then switch the magnetic field off and
measure anisotropy
of gamma rates
Polarization ~0.6
 Now measure
electron rate for
both B orientations
64
Parity violation: details of the Wu experiment
 If  decay violates parity, the angular distribution of the electrons must
be asymmetric under   - . Therefore the counting rate
is expected to depend on the angle as
(e = velectron/c)

 = 0 if parity is conserved,  = 1 if parity is maximally violated
= +1  V+A structure of the interaction
= -1  V-A structure of the interaction
 Measurement gave  -1 (maximal parity violation).
Weak interaction have V-A structure
65
V-A structure of weak interactions
 Remember the original form of the weak interaction Hamiltonian as
Fermi wrote it:
𝐻~
𝑑 𝑥 𝑝 𝑥 µ 𝑛 𝑥 𝑒 𝑥 µ(𝑥)
3
µ = 𝑝µ 𝑛
𝑗
 So the current has the form
which turns out to be correct
for QED and QCD where parity is not violated. For weak interactions, a
different form is required. The condition of Lorentz invariance leaves only
five combinations of individual gamma matrices that have the correct
Lorentz transformation properties, such that they can be combined in a
Lorentz invariant matrix element
𝑝𝑛
S (scalar)
𝑝 5𝑛 P (pseudoscalar)
𝑝 µ𝑛
V (vector)
𝑝 µ 5𝑛
A (axial vector)
𝑝 (𝛾 µ 𝛾  − 𝛾  𝛾 µ )𝑛
T (tensor)
= 0123
66
V-A structure of weak interactions
 For maximal parity violation,
only two combinations remain
𝑝 µ𝑛
𝑝 µ 5𝑛
V (vector)
A (axial vector)
𝑗µ = 𝑝(µ − µ5)𝑛 = 𝑝 µ (1 − 5)𝑛 (V-A interaction)
𝑗µ = 𝑝(µ + µ5)𝑛 = 𝑝 µ (1 + 5)𝑛 (V+A interaction)
 V+A is excluded by the experiment of Wu.
 
Weak interaction have V-A structure
67
For those who like colored pictures:
Another scheme of
the Wu experiment
68
7.
Helicity of the neutrino
69
Helicity
 Helicity is defined as the spin, quantized along the direction of
motion of the particle (along its momentum vector). For a spin ½
particle, it can be + ½ or -½ (in units of h/2). Typically it is
normalized, then H = 1, corresponding to positive or negative
.
helicity:
positive
negative
𝒑∙𝒔
𝐻=
|𝑝 ∙ 𝑠|
 H is a pseudoscalar: parity operation
changes the sign of the momentum
but not the sign of the spin
 scalar product changes sign
under mirror transformation
70
The Goldhaber experiment
 Electrons in the Wu experiment are maximally polarized (positive
helicity)
 Can one directly measure the helicity of neutrinos?
 1958: Goldhaber and co-workers investigatde the electron capture
𝑒
−
+
152
Eu 𝐽 = 0 →
152
Sm∗ 𝐽 = 1 + 𝑒
and the subsequent decay
152
∗
Sm
𝐽=1 →
152
Sm 𝐽 = 0 + 
finally measuring the polarisation of the gamma rays
71
The Goldhaber experiment
𝑒
−
+
152
152
Eu 𝐽 = 0 →
∗
Sm 𝐽 = 1 →
152
Sm∗ 𝐽 = 1 + 𝑒
152
Sm 𝐽 = 0 + 
1. Eumeritium source
2. Magnetized iron
3. Samarium oxide for resonance
scattering of gamma rays
4. NaI counter to detect scattered
gamma ray
5. Lead shielding (no direct gammas into
NaI counter)
72
The Goldhaber experiment
𝑒 − + 152Eu 𝐽 = 0 →
152
Sm∗ 𝐽 = 1 →
152
Sm∗ 𝐽 = 1 + 𝑒
152
Sm 𝐽 = 0 + 
Electron capture
1/2
1/2
152Eu
+ e-
Blue: spin
Red: momentum
 +
1/2
or
152Eu
152Eu
1/2
1/2
152Eu

1

152
1
152
Left-handed 
Sm*
1/2
Right-handed 
Sm* + 
73
The Goldhaber experiment
𝑒 − + 152Eu 𝐽 = 0 →
152
Sm∗ 𝐽 = 1 →
152
Sm∗ 𝐽 = 1 + 𝑒
152
Sm 𝐽 = 0 + 
Gamma emission
1
1
152Sm*

152Sm
1
or 152Sm* 
Blue: spin
Red: momentum
+
1
152
Sm + 
Gammas which are emitted along the momentum
of the 152Sm are polarized as the neutrino!
So what remains is to determine the polarization
of the gammas.
74
The Goldhaber experiment
𝑒 − + 152Eu 𝐽 = 0 →
152
Sm∗ 𝐽 = 1 →
152
Sm∗ 𝐽 = 1 + 𝑒
152
Sm 𝐽 = 0 + 
Now measure the resonant scattering of gammas in
the 152Sm2O3 target:
𝛾+
152
Sm →
152
∗
Sm →
152
Sm + 𝛾
This works only if the energy of the gammas is a
little higher than the resonance energy, in order to
balance the recoil energy of the 152Sm nucleus.
This on the other hand is only the case for gammas
emitted in forward direction and being blueshifted
due to the recoil energy of the neutrinos.
How do I measure the photon polarization?
The Goldhaber experiment
𝑒 − + 152Eu 𝐽 = 0 →
152
Sm∗ 𝐽 = 1 →
152
Sm∗ 𝐽 = 1 + 𝑒
152
Sm 𝐽 = 0 + 
Gammas have to pass through magnetized iron. An
iron electron with spin opposite to the gamma spin
can absorb the spin (flipping is own spin). An electron
with spin parallel to the photon spin cannot do that.
Therefore, if the gamma direction is parallel to the
magnetic field, the transmission for lefthanded
gammas is much larger than for lefthanded gammas.
Flip the magnetic field and measure NaI signal!
Result of Goldhaber‘s experiment:
H() = -1
More general: in weak interactions all leptons have
negative helicity, all antileptons have positive helicity. Note
that helicity is not a Lorentz invariant value. Only massless
partices have a well-defined helicity which is conserved
under Lorentz transformations.
… from an old German textbook
77
… from an old German textbook
78
End of first lecture
79