1. Beta Decay history left-handed or right-handed? Four-fermion
One of the puzzles in understanding beta-decay was the emission of particles
(electron, positron, neutrino) that are not present in the atomic nucleus.
1933 quantum theory of radiation developed
1934 Fermi theory of beta decay (based on relativistic formalism). The original Fermi’s idea was that the weak force responsible for beta decay had essentially zero range.
1957 Fall of parity conservation. Fermi theory revisited.
1961 Glashow, introduces neutral intermediate boson of weak interactios
1984 GUT. Georgi and Glashow
1983 W and Z bosons discovered at CERN
The bilinear combinations ("currents") of the fermion fields are Lorentz four-vectors, similarly to the electromagnetic current (coupled to vector four-potential) familiar from QED:
Four-fermion
Lagrangian
an annihilation operator for particle or
a creation operator for antiparticle
Why is it called antineutrino?
left-handed or right-handed?

2. Beta Decay Weak Interaction
Nuclear beta decay is one of the many facets of weak interaction. The basic
reactions involving weak interactions in nuclei may be characterized by the decay of a neutron and a (bound) proton:
A free proton cannot beta decay since a free neutron is more massive ( MeV) than a free proton ( MeV).
Q: what are the other possible channels of proton decay?
There are many other examples of weak decays:
a) semi-leptonic processes (both hadrons and leptons are involved)
b) purely-leptonic processes
The coupling constant (Fermi coupling constant) is:
Force carriers:
interaction range is very short ~10-3 fm
(weak interactions can be considered as zero-range
in nuclear physics!)

3. Beta Decay Energy relations a) b- decay b) b+ decay
atomic mass
nuclear mass
electron
binding energy
P(arent)
D(aughter)
a) b- decay
In the following, we assume that the neutrino mass is ~zero and that the very
small differences in electron binding energy between the parent and daughter atoms can be neglected. This gives:
Consequently, the b- decay process is possible whenever MP>MD
b) b+ decay
Consequently, the b+ decay process has a threshold 2mec2
Atomic electron is captured by a proton. This process leaves the atom in an excited state: a vacancy has been created! The vacancy is quickly filled by producing the characteristic X-ray cascade
c) 
n=K, LI, LII,…

4. Beta Decay examples… mass relationship in electron capture between
the parent and daughter atom
energy relations in various beta decay processes

5. Beta Decay spectrum and lifetime wave function of the parent nucleus
product wave function of the daughter nucleus, electron, and antineutrino
Let us first calculate the density of final states
If we are interested in electrons emitted with an energy between Ee and Ee+dEe,
the variation does not affect the electron observables. Hence one gets:
If we neglect the very small nuclear recoil energy, for constant electron energy
we obtain

6. spectrum and lifetime (cont.)
Beta Decay
spectrum and lifetime (cont.)
The final expression for the density of final states of en electron emitted
with a given energy and momentum (integrated over all angles) is:
Now we need to calculate the interaction matrix element:
The neutrino wave function can be written as:
For the electron, a plane wave approximation is too crude, and one has to consider the distortion of the wave function caused by the interaction with the electromagnetic field of the nucleus. Quantitatively, the main effect is to alter the magnitude of the electron wave function at the origin:
positive (negative) sign used for b- (b+) decay
Fermi function
The Fermi function slightly distorts the beta spectrum shape.
zero-range

7. spectrum and lifetime (cont.)
Beta Decay
spectrum and lifetime (cont.)
Depends on nuclear wave functions

8. Influence of the neutrino mass
Beta Decay
Influence of the neutrino mass
If we assume that the nuclear matrix element is totally independent of pe, and for
vanishing neutrino mass, one gets
Fermi-Kurie plot
The intercept with the energy axis is a convenient way to determine the Q-value!
This procedure applies to allowed transitions. (For forbidden transitions, there is an additional pe dependence of |M’|…
Fermi-Kurie plot for the allowed beta decay in 66Ga
Hypothetical case of beta decay with non-vanishing neutrino mass (3H decay; mnc2=30 eV)
electron scattering within the source
total energy (in mec2)
Deviations around the endpoint due to nonzero neutrino mass…

9. Beta Decay Total half-life f-function electrons positrons
This integral can be expressed as:
where and w0 is the reduced max. electron energy.
If we assume that the matrix element does not depend on w, and after taking out
the strength g of the weak interaction, one obtains:
f-function
electrons
positrons

10. Beta Decay Total half-life
From the expression for fT, it is possible to determine the strength g of the beta-decay process, if one knows how to determine the reduced matrix element. As will be discussed later, for superallowed transitions, the matrix element is so the fT values should be identical.
3088.6(2.1) s
or, introducing the dimensionless constant G:

11. Beta Decay Microscopic picture _ e- ne e- p n W- Z0 e- n n
On a more fundamental level, beta decay of hadrons can be viewed as the transformation of one type of quark to another through exchange of charged weak currents (W bosons carry net charges; Z boson is neutral - it is the source of neutral weak current). _ e- ne e- p n W- Z0 e- n n
The flavor of quarks is conserved in strong interactions. However, weak interactions change flavor! For example:

12. Beta Decay Microscopic picture Cabibbo angle
When a quark decays, the new quark does not have a definite flavor. For instance:
Cabibbo angle
However, the observed weak transitions are between quarks of definite flavor. The strong-interaction quark eigenstates
are different from weak interaction eigenstates).
This means that the observed beta-decay strength in reactions is modified by the mixing angle.
Cabibbo _Kobayashi-Maskawa (CKM) matrix
For nuclear beta-decay, we are mainly concerned with the transition between u- and d-quarks. As a result, only the product
enters into the process.

13. Beta Decay Microscopic picture helicity
What are the consequences of parity violation in beta decay?
helicity
The eigenvalue of h is v/c. For a massless particle, the eigenvalues of h can be only +1 or -1. In general, the particle with
h>0 is called “right-handed”
h<0 is called “left-handed”
Experimentally,
All the leptons emitted in beta-decays are left-handed and all antileptons - right-handed!
The operators that are scalars, pseudoscalars and tensors produce leptons of
both helicities under a parity transformation. Only vector operators V and axial vector operators A can accommodate the observed result. Furthermore, since V and A are of different parity, they must appear in a linear combination. This leads to the V-A theory of beta decay. In principle, both V and A parts should be
characterized by different coupling constants, GV and GA, respectively.
The vector current is known to be a conserved quantity (CVC hypothesis)
four-divergence
For the axial vector current, there is not such a relation. The four-divergence
of A (a pseudoscalar!) does not vanish. The pion is a pseudoscalar particle. Hence
the weak interaction is modified in the presence of strong interactions. This leads
to a partially conserved axial-vector current (PCAC) hypothesis:
a constant
the pion field

14. Microscopic picture (cont.)
Beta Decay
Microscopic picture (cont.)
How to relate GV and GA?
pion decay constant
pion-nucleon coupling constant
Goldberger-Trieman
relation
Experimentally, gA=-1.259
This value is close to obtained from the relation above. It is a nice
confirmation of the PCAC
Matrix elements
zero-range
The nuclear operator transforming a neutron into a proton must be one body
in nature. Hence it must involve the isospin raising or lowering operators.
In the non-relativistic limit, the vector part may be represented by the unity operator times and the axial-vector part by a product of and s. (A proper derivation requires manipulation with Dirac 4-component fuctions and g matrices!)
Gamow-Teller decay, carries one unit of angular momentum
Fermi decay, carries zero angular momentum

15. Beta Decay Allowed decays Fermi transitions Gamow-Teller transitions
In reality, isospin is violated by the electromagnetic force, but the violation is weak.
T+ has rank unity!
Gamow-Teller transitions
The matrix element strongly depends on the structure of the wave function!
The absolute values of GT matrix elements are generally smaller than those
for Fermi transitions.
squared matrix elements

16. Forbidden transitions
Beta Decay
Forbidden transitions
Forbidden transitions involve parity change and a spin change of more than
one unit. They come from the higher-order terms in the expansion of electron
and neutrino plane waves into spherical harmonics. Forbidden decays are classified into different groups by the L-value of the spherical harmonics involved. The selection rules for the Lth-order forbidden transitions are:
Experimental log fT
values

17. electron capture processes
Beta Decay
electron capture processes
Electron capture leads to a vacancy being created in one of the strongest bound atomic states, and secondary processes are observed such as the emission of X-rays and Auger electrons. Auger electrons are electrons emitted from one of the outer electron shells and take away some of the remaining energy.
Capture is most likely for a 1s-state electron. The K-electron wave function at the origin is maximal and is given by
The electron capture probability is thus given by:

18. Beta Decay the neutrino inverse beta processes cross section
One of the most pervasive forms of matter in the universe, yet it is also one of the most elusive!
inverse beta processes
Shortly after publication of the Fermi theory of beta decay, Bethe and Peierls
pointed out the possibility of inverse beta decay (neutrino capture):
extremely small
cross sections!
Let us first consider
cross section
neutrino flux
mean-free path
For protons in water n~
This gives the mean-free path of
cm or ~300 light years!
number of nuclei per cm3

19. the neutrino detection
Beta Decay
the neutrino detection
In 1930 Wolfgang Pauli proposed a solution to the missing energy in nuclear beta decays, namely that it was carried by a neutral particle This was in a letter to the Tubingen congress. Enrico Fermi in 1933 named the particle the "neutrino" and formulated a theory for calculating the simultaneous emission of an electron with a neutrino. Pauli received the Nobel Prize in 1945 and Fermi in The problem in detection was that the neutrinos could penetrate several light years depth of ordinary matter before they would be stopped.
In 1951 Fred Reines at Los Alamos thought about doing some real challenging physics problem. In a conversation with Clyde Cowan they decided to work on detecting the neutrino. Their first plans were to detect neutrinos emitted from a nuclear explosion. Realizing that nuclear reactors could provide a neutrino flux of 1013 neutrinos per square centimeter per second, they instead mounted an experiment at the Hanford nuclear reactor in The Hanford experiment had a large background due to cosmic rays even when the reactor was off. The detector was then moved to the new Savannah River nuclear reactor in This had a well shielded location for the experiment, 11 meters from the reactor center and 12 meters underground. The target was water with CdCl_2 dissolved in it. The positron was detected by its slowing down and annihilating with an electron producing two 0.5 MeV gamma rays in opposite directions. The pair of gamma rays was detected in time coincidence in liquid scintillator above and below the water by photomultiplier tubes detecting the scintillation light. The neutron was also slowed by the water and captured by the cadmium microseconds after the positron capture. In the capture several gamma rays were emitted which were also detected in the scintillator as a delayed coincidence after the positron's annihilation gamma ray detection. .

20. the neutrino detection (cont.)
Beta Decay
the neutrino detection (cont.)
The rarity of neutrino capture is shown in their signal rate, which was about three events per hour in the entire detector. The signal to background ratio was about four to one. Thus in 1956 was born the rich and continually exciting field of experimental neutrino physics, as discussed in other articles in this newsletter. This discovery was recognized by honoring Frederick Reines with the Nobel Prize in Clyde Cowan died in 1974….
Original papers:
“Detection of the Free Neutrino: A Confirmation", C.L. Cowan, Jr., F. Reines, F.B. Harrison, H.W. Kruse and A.D. McGuire, Science 124, 103 (1956).
"The Neutrino", Frederick Reines and Clyde L. Cowan, Jr., Nature 178, 446 (1956).
"Neutrino Physics", Frederick Reines and Clyde L. Cowan, Jr., Physics Today 10, no. 8, p.12 (1957).

21. Beta Decay double beta decay Seen in: 48Ca, 76Ge, 82Se, 96Zr,
100Mo, 116Cd, 150Nd,…
second order process
summary of selected double beta-decay results
extraction of the daughter nuclei (Z+2)
from the parent (Z) in an old ore
Typical lifetimes are of the order of 1020 years

22. double beta decay (cont.)
It was suggested that neutrino can be identical or different to its charge conjugate:
Majorana particles appear in a natural way in GUT theories that unify the strong
and electroweak interactions with the possibility that the lepton number is no longer conserved, since now the emitted antineutrino could be absorbed as neutrino:
neutrinoless double beta-decay
2nbb
two-electron spectrum
0nbb
The estimated transition probability for the neutrinoless decay is more than
105 shorter than the “normal” double-beta decay

23. double beta decay (cont.)
The difference between Majorana and Dirac neutrinos were tested by Davis
in The reaction:
has not been observed. This can be understood in light of parity violation
by weak interaction. Indeed, the inverse reaction needs different helicity:
By the same token, the neutrinoless double-beta decay is forbidden, even
if neutrino has a Majorana character! Note, however, that for the massive particle helicity is not a fixed quantum number!