The Weak Interaction

The Weak Interaction Sun sun sunRising sun the creatorMid day blazing sun the destroyer RudraSetting sun the maintainer and continuanceGreatest of allSun sun sun (Gajanan Mishra) 1. Costituents of Matter 2. Fundamental Forces 3. Particle Detection 4. Symmetries and Conservation Laws 5. Relativistic Kinematics 6. The Quark Model 7. The Weak Interaction 8. Introduction to the Standard Model

Simple facts The Weak Nuclear Interactions concerns all Quarks and all Leptons The Weak Interaction takes place whenever some conservation law (isospin, strangeness, charm, beauty, top) forbids Strong or EM to take place In the Weak Interaction leptons appear in doublets: Doublets are characterized by electron, muon, tau numbers (each conserved, except in neutrino oscillations)  whose sum is conserved. …and the relevant anti-leptons. For instance: (see the section on Fundamental Interactions)

Simple facts Weak Nuclear Interaction violates Parity The Parity violation is maximal Discovered first in the Wu experiment. Confirmed in all other experiments on Weak Interactions. The fundamental weak couplings are to fully left-handed fundamental fermions (and fully right-handed fundamental antifermions). Weak Nuclear Interaction violates CP This fact will need to be incorporated in the theory:  a phase in the CKM matrix. CPT is conserved by Weak Interactions Weak Interactions violate P, C, CP, T but not the combination CPT.

Weak Interactions allow for processes otherwise impossible At low energy: Fermi Theory At high (and low) energies: Electroweak Theory The first theory of Weak Interactions was developed by Enrico Fermi in close analogy with Quantum Electrodynamics. The process to be explained was the nuclear beta decay. Nature rejected his paper “because it contained speculations too remote to be of interest to the reader.” ‘Tentativo di una teoria…’ , Ric. Scientifica 4, 491, 1933.

Fermi Theory of the Beta Decay At the fundamental (constituents) level The rate of decay (transizions per unit time) will be: Integration over spins and angles Energy of the final state

F: Fermi transitions. No nuclear spin change ∆J (Nuclear Spin) = 0 Leptonic state: spin singlet ↑↓ │M│2 ≈ 1 GT: transizioni alla Gamow-Teller. Nuclear Spin change ∆J (Nuclear Spin) = +1,-1 Letponic state: spin triplet ↑↑ │M│2 ≈ 3 Several transitions are mixed transitions (F e GT). In the assumption of no interference, one typically has : With weights:

Beta Decay Kinematics dE0 arises from the finite lifetime of the initial state Q-value Final state In the rest frame of the neutron : The recoil kinetic energy of the nucleon Is negligible : Energy carried away by the neutrino :

Number of neutrino and electron available states with electron and neutrino momenta in the ranges p,p+dp e q,q+dq Choosing a normalized volume and integrating over the angles : Neglecting dynamical correlations between p,q… Moreover, there is no free phase space for the proton, since given p,q its momentum is fixed: The phase space is : Now, expressing q as a function of the total available energy and E :

General form Coulombian Correction F(Z,p) Coulombian Correction and non-zero neutrino mass Kurie plot

Beta Decay Spectrum in short The coupling constant enters here

Total decay rate The total decay rate depends on the coupling constant and the phase space. For a fixed coupling constant, the rate is the integral of : over the electron spectrum. This quantity features a sharp dependence on the Q-value E0 This can be quickly appreciated in the (somewhat crude) relativistic electron (E = pc) approximation : Sargent’s rule

Coupling constants : Eelectromagnetic and Weak A reminder : In rationalized and natural units e is adimensional : The Weak Fermi constant The Weak Coupling constant is actually bigger than the fine structure constant. But at low energies it is damped by the W mass into the small GF constant

Weak Decays and Phase Space in the low energy regime According to the Sargent’s rule one has – roughly : The neutron lifetime : And this has a general validity. In fact : The muon lifetime : For a charmed particle :

Electromagnetic Weak High Energy Matrix Element Low Energy Matrix Element

Inverse Beta Decay p is the momentum of the neutron/positron system in their CM This is a mixed (Fermi + Gamow-Teller) transition A very small cross section The cross section increases with E

Neutrino discovery: Principle of the experiment In a nuclear power reactor, antineutrinos come from  decay of radioactive nuclei produced by 235U and 238U fission. And their flux is very high. 1. The antineutrino reacts with a proton and forms n and e+ Inverse Beta Decay 2. The e+ annihilates immediately in gammas Water and cadmium 3. The n gets slowed down and captured by a Cd nucleus with the emission of gammas (after several microseconds delay) Liquid scintillator 4. Gammas are detected by the scintillator: the signature of the event is the delayed gamma signal 1956: Reines and Cowan at the Savannah nuclearpowerreactor

The size of the detector might be important. And this is because of the small cross neutrino section. Not a specific detector. But… the typicalconfiguration of a lowenergy, low background undergound neutrino detector : • Neutrino beam • Massive, instrumented detector • Detector transparent to signalcarriers • Background control! « I went to the general store but they did not sell me anything specific»

Parity violation in Beta Decay 1956: Lee-Yang, studying the decay of charged K mesons hypotesized that Weak Interactions cold not conserve Parity. 1957: esperiment by Wu et al. A sample of Co-60 nuclei at 10 mK in a magnetic field. The Co-60 spin (J=5) get statistically aligned by the magnetic field. The daughter nucleus (Ni*) has spin 4 The experimentally observed distribution for the emitted electron has the form :

This term violates Parity, by correlating the momentum of the electron to the Co-60 spin. This alignment fades away with increasing energy.

V-A structure of Weak Interactions The helicities of neutrino and electron are : Neutrinos are considered massless ! This property must be part of a consistent theory of Weak Interaction: the description of Dirac-type elementary constituents «Electroweak analogy». What is the structure of the weak current(s) ? Weak Electromagnetic

At low energy Charged weak currents According to the original idea by Fermi : In the earliest days of the parity violation discovery, it was natural to guess that the violation itself might be a special property of neutrinos. The two component neutrino theory: if neutrinos were massless , then they could be polarized only parallel to the direction of motion (positive helicity) or antiparallel to it (negative helicity). But parity violation was seen also in reactions like And was found to be a general property of the Weak Interactions. A theory of the Weak Interactions had to be based on concepts like universality and parity violation.

The two-component theory of the (massless) Neutrino The spin-1/2 pointlike particle wave function obeys the Dirac Equation : Four components : two spin states of particle two spin states of antiparticle • Massive particle: both spin states must be described by the same wavefunction because the spin direction is not Lorentz-invariant. • Massless particle: it always travel at the speed of light, so its spin direction can be defined in a Lorentz-covariant way (parallel or antiparallel to the direction of the momentum, i.e. positive or negative helicity). In the Weyl representation of the Gamma Matrices:

Introducing the bispinors (upper and lower components) : Dirac Equation in the Weyl representation For a massles fermion, the upper and lower components are decoupled : For a massles particle, E= p Left-handed spinor Right-handed spinor

Let us now introduce the Gamma-5 matrix (in the Weyl representation) : One can then build right-handed or left-handed wavefunctions by using the projectors More in general, in the case of massive particles : gives a v/c polarization along the direction of p (+1 when v=c) gives a -v/c polarization along the direction of p (-1 when v=c) Before the Parity violation experiments, there was no reason con consider the right and left-handed spinors as particularly useful. However, detailed evidence was found that only the left-handed spinor occurs in Weak Interactions

Only left-handed spinor particles (and right-handed spinor antiparticles) take part in the Weak Interactions. This has several consequences : • The existence of a two-component massless neutrino theory • Maximal Parity violation • Maximal C violation • T conservation • CP conservation (see before) If we carry out the P operation on the neutrino described by ψL, we obtain a neutrino described by ψR, which is unallowed in the theory. If we carry out the C operation on the neutrino described by ψL, we obtain an antineutrino described by ψL, which is unallowed in the theory. This is because T reverses both spin and linear momentum. (see the lecture on Symmetries and Conservation Laws) There exists – however – tiny violations of CP and T invariance in the Weak Interactions

Notable properties of the projection operators

(an odd number of exchanges with a different matrix) And this is because : The Universal Four-Fermion Matrix Element Propagator and coupling constant The low energy matrix element : Now, which is the form of the current ? We know that it has to be of the form :

To prove It is sufficient to prove :

Now, which is the form of the current ? We know that it has to be of the form : (because of Lorentz invariance requirements) In the case of the WeakInteractions : • Scalar (originates F transitions) • Vector (produces F transitions) • AxialVector (GT transitions) • Pseudoscalar • Tensor (GT transitions)

The Universal Four-Fermion Matrix Element : ..can be constructed with the only non-zero matrix elements (V and A). A general form could be : The fact that a massless neutrino is produced in a pure helicity eigenstate requires CA= - CV giving precisely the helicity projector in the current : In general, this holds for any massive fermion, leading to the general form : Low-E «propagator» Weak Current Weak Current

Corrections to the V-A current structure ? They need to be considered when the Weak Interaction involves Hadrons ! Let us first consider the electric charge of a proton The proton is a complicate object, continually emitting and absorbing quark-antiquark pairs as well as gluons The charge of the proton – however – is equal to the charge of the (elementary) electron ! The electric current (a vector current V) is conserved by the Strong Interaction What about the Weak interaction V-A current ? The general experimental situation indicates that the V part is conserved (Conserved Vector Current, CVC hypotesis. Goldberger-Treiman). The A part of the corrent gets (most or all of) the Strong Interaction corrections :

Pion decay and V-A structure of Weak Interactions Let us compare the decays : H Negative Pion has spin 0  Neutrino and muon must have antiparallel spins (J conserved) Neutrino has -1 helicity  For a massless neutrino helicity is an exact quantum number  Muon MUST have negative helicity (the «wrong» helicity!) If we compare the two processes : From fundamental physics viewpoint, coupling constant, Feynman diagrams, they are essentially the same thing! The main difference is the phase space.

From the point of view of the phase space, the decay in the electron is largely favored But…in this decay the LEPTON is forced to have an «unnatural» helicity ! H Negative Experimentally, one has the following electron energy spectrum from stopping pions

The Neutrino C, P are violated in Weak Nuclear Interactions Neutrinos takes part only in Weak Nuclear Interactions In the massless neutrino approximation: Experimental evidence indicates that in Weak Interactions : Neutrinos are always left-handed. Antineutrinos are always right-handed ! P C CP To a very good approximation, Weak interactions conserve CP (not C, not P)

The Pion Decay P CP C To a very good approximation, Weak interactions conserve CP (not C, not P)

Introducing the W and the Z0 And the relevant expression for the propagator : Low energy limit Lifetimes ?

The Weak Charged Current and the Weak Neutral Current Statesconnected by a W: • Leptonsonlywithindoublets • Hadron : anytransition States connected by a Z (no flavor change whatsoever) In fact, there is no (flavor changing) tc,tu,bs,bd,cu

Now let us consider – as a meaningful example – the neutrino scattering process in ordinary matter : If the neutrino is an electron neutrino : If the neutrino is a muon neutrino : There is no annihilation diagram possible (the W connects only states within doublets), leaving only the Z possibility (exchange of a Z between the two leptons). Only Neutral Currents

Recalling the discovery of the third leptonic family: the Tau SLAC, 1975, Martin Perl et al., studying the products of e+e- collisions With hindsight : Detection of final states featuring an electron and a muon This indicates intermediate states emitting invisibile leptons (neutrinos). This is because the Lepton Numbers (elettronic, muonic) are violated. Is this the only possible interpretation of an eμ final state?

The important point was that these events took place when the energy was greater than 3.56 GeV : This has to be disentangled from events with two charged particles produced by the process : • Energy threshold of 3740 MeV (as opposed to 3560) • Additional hadronic particles in the final state (K, pions, muons) Featuring the same leptonic final state With the discovery of the Tau (and the Tau Neutrino in 2002) the fundamental leptons are :

A note on neutrino experimental characterization : flavors and currents Key point: different interaction in materials of a neutrino beam. Charged Currents (CC) and Neutral Currents (NC) Muon in the final state (CC event). Muonic neutrino arriving! Electron in the final state (CC) Electron neutrino arriving! No final state lepton Neutral current (NC) ! Neutrino flavor unknown. Tau neutrino tau interactions  Tau lepton decaying in different ways (including muon, electron)

The Weak Charged Currents Weak Vertex Factor The weak charged coupling to can be also considered as having a fundamental vertex with the V-A features: The Weak Coupling Constant : Charged Currents Weak Interactions at low energy: the muon lifetime The Weak Interaction (CC) lowest order Feynman Diagram :

The muon lifetime result is : At low energies, MW and gw always enter in observable quantities as a ratio, which makes it possible to write : The best Weak Coupling Constant determination at low energies

The Weak Charged Currents : Leptons and Quarks W W The coupling of W to leptons takes place strictly within a given generation: W Purely leptonic Charged Current Weak Processes only involve leptons. Their general structure is : W Weak decays of leptons into other leptons Scattering between leptons (observed only if electrons are present to act as suitable targets)

The coupling of W to Quarks : W W W Similar to the Leptorn case, there is coupling within a generation : But cross-generational couplings are also there (6 couplings, since bu and td are not shown) : Charged Current involving Quarks can originate : W W Hadronic processes Semileptonic processes

Charged Current semileptonic processes : They all feature a leptonic and a hadronic charged current W- d u d d u u The neutron decays (and beta decays) The «inversa beta decay» kind of reaction The decay kind of a heavy baryon Beauty and Charm decays

W Charged Current purely hadronic processes : There are weak processed conserving flavor but they are dwarfed by the much stronger Strong Interaction They are possible (and the only possibility) when the flavor is changed. They can connect quarks in the same generation, like in a cs decay : s c W u u b They can connect quarks in different generations, like in a bu decay : W u They of course involve Mesons and Baryons as well :

Weak Charged Currents : the Cabibbo theory of Mixing (1963) Weak Charged Interactions have been characterized with a unique coupling constant (and the phase space). However, the intergenerational processes seemed to take place less often than the decays within the same generation : The charm quark was not known at that time Experiments say that : occurs more frequently than W W Cabibbo proposed for the first time that the quarks entered the Weak Charged Interactions as “rotated” states :

The Weak Interaction Eigenstates at the time of the Cabibbo Theory (no Neutral Currents, yet, no taus, no c, b and t) : Weak Interaction Eigenstates related to Mass Eigenstates by : Mixing determined by the Cabibbo angle The new interaction vertices for Weak Charged currents are : u u W W d s accounting for both the V-A structure and the quark mixing

The Weak Interaction