Weak Interactions
OUTLINE CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM Mechanism - Cabibbo-Kobayashi-Maskawa (CKM) Matrix NEUTRAL WEAK INTERACTION - Elastic Neutrino-Electron Scattering - Electron-Positron Scattering Near the Z 0 Pole ELECTROWEAK UNIFICATION - Chiral Fermion States - Weak Isospin and Hypercharge - Electro-Weak Mixing
THE WEAK FORCE
Characteristics of the WEAK FORCE The time scale of the decay is long. Radioactive decays must proceed by the weak force since the timescale ranges from 10-8 s to years E.g. neutron β decay Weak decays often involve neutrinos do not interact by the EM force or the strong force cannot detect in conventional detectors can infer existence from conservation of E, p (Pauli, 1930) Neutrinos would not be directly detected for 25 years: Reines & Cowan, using Savannah River nuclear reactor
DECAY OF THE NEUTRON
HELICITY Helicity is the component of the (spin) angular momentum along momentum vector. For fermions, the value is 1/2 or +1/2, depending on whether spin S is antiparallel or parallel to direction of motion p Solutions of the Dirac equation: where are helicity eigenstates: For antiparticles the relation is reversed (because v(p) ~ u(-p)):
HELICITY with In the limit m=0 (E>>m): obeys Thus the following chirality projection operators are also helicity projection operators for m=0: For m=0, P L projects onto helicity 1/2 fermions but helicity +1/2 anti-fermions.
HELICITY and EM INTERACTION We can use the projection operators to split the electromagnetic current into 2 pieces: where Since we have and
HELICITY and EM INTERACTION Helicity is conserved in the electromagnetic interaction in high energy (m=0) limit Allowed QED vertices in high energy limit mirror reflection Helicity is reversed under parity: Equalities are due to parity
CHARGED LEPTONIC WEAK INTERACTION The mediators of weak interactions are intermediate vector bosons, which are extremely heavy: The propagator for massive spin-1 particles is:, where M is M W or M Z In practice very often: The propagator for W or Z in this case:
WEAK INTERACTION Lorentz condition ε µ p µ = 0
FEYNMAN RULES also 3 and 4 boson vertices exist
CHARGED LEPTONIC WEAK INTERACTION The theory of charged interactions is simpler than that for neutral ones. We start by considering coupling of W s to leptons. The fundamental leptonic vertex is : The Feynman rules are the same as for QED, except for the vertex factor : ( the weak vertex factor ) Weak coupling constant (analogous to g e in QED and g s in QCD) :
CHARGED WEAK INTERACTION The charged weak interaction violates parity maximally By analogy to EM we associate the charged weak interactions with a current, which is purely left-handed: also The charged weak interaction only couples to left-handed leptons (e,µ,τ,ν i ). (Also, only couples to left-handed quarks.) It couples only to right-handed anti-fermions.
Example: Inverse Muon Decay ( lowest order diagram ) When the amplitude is : Simplifies because M W = 80 GeV much larger than q< (100 MeV).
Example: Inverse Muon Decay Applying Casimir s trick we find : trace theorems trace theorems using :
Example: Inverse Muon Decay In CM frame, and neglect the mass of the electron : where E is the incident electron (or neutrino) energy. The differential scattering cross section is : The total cross section :
DECAY OF THE MUON The amplitude : As before :
DECAY OF THE MUON In the muon rest frame : Let : Plug in :
DECAY OF THE MUON The decay rate given by Golden Rule * : where : * a lot of work, since this is a three body decay
DECAY OF THE MUON Perform integral : where : Next we will do the integral. Setting the polar axis along (which is fixed, for the purposes of the integration), we have :
DECAY OF THE MUON Also : The integral is trivial. For the integration, let : and :
DECAY OF THE MUON integration : where : The limits of E 2 and E 4 integrals :
DECAY OF THE MUON Using:
DECAY OF THE MUON
DECAY OF THE MUON
DECAY OF THE MUON
DECAY OF THE MUON The total decay rate : Lifetime :
DECAY OF THE MUON g W and M W do not appear separately, only in the ratio. Let s introduce Fermi coupling constant : The muon lifetime : = 2.2 10-6 s
DECAY OF THE MUON In Fermi s original theory of the beta decay there was no W; the interaction was a direct four-particle coupling. Using the observed muon lifetime and mass : and : Weak fine structure constant : Larger than electromagnetic fine structure constant!
WEAK INTERACTIONS Weak force is weak because boson propagator is massive, not because coupling strength is weak.
DECAY OF THE NEUTRON ( the same as in previous case )
DECAY OF THE NEUTRON In the rest frame of the neutron : As before : We can t ignore the mass of the electron. where :
DECAY OF THE NEUTRON The integral yields : and : Setting the z-axis along (which is fixed, for the purposes of the integral), we have : and :
DECAY OF THE NEUTRON where : and :
DECAY OF THE NEUTRON The range of E 2 integral : E is the electron energy ( exact equation)
DECAY OF THE NEUTRON Approximations : Expanding to lowest order :
DECAY OF THE NEUTRON
DECAY OF THE NEUTRON (picture from Griffiths)
DECAY OF THE NEUTRON Putting in the numbers : where :
DECAY OF THE NEUTRON But the proton and neutron are not point particles. Replacement in the vertex factor: c V is the correction to the vector weak charge c A is the correction to the axial vector weak charge
DECAY OF THE NEUTRON Another correction, the quark vertex carries a factor of is the Cabibbo angle. cosθ C = 0.97 Lifetime :
CHOICE OF WEAK EIGENSTATES
4-Fermion INTERACTION
DECAY OF THE PION The decay of the pion is really a scattering event in which the incident quarks happen to be bound together. We do not know how the W couples to the pion. Use the form factor. form factor
DECAY OF THE PION
DECAY OF THE PION
DECAY OF THE PION
DECAY OF THE PION The decay rate : The following ratio could be computed without knowing the decay constant : Experimental value :
DECAY OF THE PION
CHARGED WEAK INTERACTIONS OF QUARKS For leptons, the coupling to W + and W - takes place strictly within a particular generation : For example : There is no cross-generational coupling as : There are 3 generations of quarks : Coupling within a generation : There exist cross-generational coupling as :
CHARGED WEAK INTERACTIONS OF QUARKS (Cabibbo, 1963) (extra cos or sin in the vertex factor)
Example : Leptonic Decays l is an electron or muon. The quark vertex : Using a previous result : The branching ratio : Corresponding to a Cabibbo angle :
Example : Semileptonic Decays (semileptonic decay) (non-leptonic weak decay)
Example: Semileptonic Decays Neutron decay : Quark process : There are two d quarks in n, and either one could couple to the W. The net amplitude for the process is the sum. Using the quark wave functions The overall coefficient is simply cos, as claimed before. In the decay: But : the quark process is the same we get an extra factor
Example : Semileptonic Decays The decay rate :
GIM MECHANISM
Cabibbo-GIM Mechanism The decay is allowed by Cabibbo theory. Amplitude :, far greater. GIM introduced the fourth quark c (1970). The couplings with s and d : GIM = GLASHOW, ILIOPOULOS, MAIANI
Cabibbo-GIM mechanism Now the diagrams cancel.
Cabibbo-GIM mechanism The Cabibbo-GIM mechanism : Instead of the physical quarks d and s, the correct states to use in the weak interactions are d and s : In matrix form : The W s couple to the Cabibbo-rotated states :
Cabibbo-Kobayashi-Maskawa (CKM) Matrix CKM is a generalization of Cabibbo-GIM for three generations of quarks. The weak interaction quark generations are related to the physical quarks states by Kobayashi-Maskawa (KM) matrix For example : Canonical form of KM matrix depend only on three generalized Cabibbo angles and one phase factor.
Kobayashi-Maskawa (KM) matrix The full matrix : Using the experimental values :
THE CKM MATRIX
NEUTRAL WEAK INTERACTIONS Neutral weak interaction mediated by the Z 0 boson f stands for any lepton or quark Not allowed :
NEUTRAL WEAK INTERACTIONS It doesn t matter if we use physical states or Cabibbo rotated states.
NEUTRAL WEAK INTERACTIONS First process mediated by Z 0 (Bubble chamber photograph at CERN, 1973)
NEUTRAL WEAK INTERACTIONS In the same series of experiments : Neutrino-quark process in the form of inclusive scattering The cross sections were three times smaller than the correspondent charged events : Indication of a new kind of interaction, and not simply a high order process. (which correspond to a far smaller cross section)
NEUTRAL WEAK INTERACTIONS The coupling to Z 0 : where : ( Weak mixing angle or Weinberg angle )
NEUTRAL WEAK INTERACTIONS Neutral vector and axial vector coupling in GWS model :
NEUTRAL WEAK INTERACTIONS ( Z 0 propagator ) When : the propagator is simply : The masses of the bosons are related by the formula :
Example : Elastic Neutrino-Electron Scattering
Example : Elastic Neutrino-Electron Scattering Now compute in CM frame and let : (mass of the electron)
Example : Elastic Neutrino-Electron Scattering ( E is the electron or neutrino energy ) Using :
Example : Elastic Neutrino-Electron Scattering The total cross section : Compare to : (computed in the earlier) (0.08, experimental)
Example : Elastic Neutrino-Electron Scattering Most neutral processes are masked by electromagnetic ones.
Example : Electron-Positron Scattering Near the Z 0 Pole f is any quark or lepton (except electron we must include one more diagram) We are interested in the regime : The amplitude : where :
Example : Electron-Positron Scattering Near the Z 0 Pole ( since we are working in the vicinity of 90 GeV ) Ignore the mass of quark or lepton Finally :
Example : Electron-Positron Scattering Near the Z 0 Pole problems at Z 0 pole
Example : Electron-Positron Scattering Near the Z 0 Pole Z 0 is not a stable particle. Its lifetime has the effect of smearing out the mass. Replacement in the propagator : = decay rate The cross section : Because : the above correction is negligible outside Z 0 pole.
Example : Electron-Positron Scattering Near the Z 0 Pole Cross section for the same process, mediated by a photon : ( Q f is the charge of f in units of e ) The ratio :
Example : Electron-Positron Scattering Near the Z 0 Pole
Example : Electron-Positron Scattering Near the Z 0 Pole Well below the Z 0 pole : Right on the Z 0 pole :
Example : Electron-Positron Scattering Near the Z 0 Pole
CHIRAL FERMION STATES To unify the weak and electromagnetic interaction, let s move the matrix into the particle spinor. ( L stands for left-handed ) (But u L is not, in general, a helicity eigenstate)
CHIRAL FERMION STATES If the particle is massless : helicity where : projection operator Using : we can compute the following table è
CHIRAL FERMION STATES
CHIRAL FERMION STATES (inverse beta decay) The contribution to the amplitude from this vertex : Negatively charged weak current and e stand for the particle spinors. Rewrite as : (coupling between left-handed particles only) Note that : Electromagnetic current :
WEAK ISOSPIN AND HYPERCHARGE Negatively charged weak current Positively charged weak current
WEAK ISOSPIN AND HYPERCHARGE We can express both by introducing the left handed doublet : and the matrices :
WEAK ISOSPIN AND HYPERCHARGE We could have a full weak isospin symmetry if only there is a third weak current, neutral weak current. Weak analog of hypercharge (Y) in the Gell-Mann Nishijima formula : Weak hypercharge :
WEAK ISOSPIN AND HYPERCHARGE Everything could be extended to the other leptons and quarks : Weak isospin currents : Weak hypercharge current : where :
ELECTRO-WEAK MIXING GWS model asserts that the three weak isospin currents couple to a weak Isotriplet of intermediate vector bosons, whereas the weak hypercharge current couples to an isosinglet intermediate vector boson. wave functions representing the particles.