Outline. Charged Leptonic Weak Interaction. Charged Weak Interactions of Quarks. Neutral Weak Interaction. Electroweak Unification

Transcription

1 Weak Interactions

2 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

3 The Weak Force

4 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

5 Parity Conservation in QED Consider the QED process e q e q The Feynman rules for QED give: e e Which can be expressed in terms of the electron and quark 4-vector currents: q q with and «Consider what happens to the matrix element under the parity transformation s Spinors transform as s Adjoint spinors transform as s Hence

6 Parity Conservation in QED «Consider the components of the four-vector current 0: k=1,2,3: since since The time-like component remains unchanged and the space-like components change sign Similarly or «Consequently the four-vector scalar product QED Matrix Elements are Parity Invariant Parity Conserved in QED

7 Parity Violation in b-decay «The parity operator corresponds to a discrete transformation «Under the parity transformation: Vectors change sign Axial-Vectors unchanged Note B is an axial vector «1957: C.S. Wu et al. studied beta decay of polarized cobalt-60 nuclei: «Observed electrons emitted preferentially in direction opposite to applied field more e - in c.f. If parity were conserved: expect equal rate for producing e in directions along and opposite to the nuclear spin. «Conclude parity is violated in WEAK INTERACTION that the WEAK interaction vertex is NOT of the form 7

8 Bilinear Covariants «The requirement of Lorentz invariance of the matrix element severely restricts the form of the interaction vertex. QED and QCD are VECTOR interactions: «This combination transforms as a 4-vector «In general, there are only 5 possible combinations of two spinors and the gamma matrices that form Lorentz invariant currents, called bilinear covariants : Type Form Components Boson Spin SCALAR PSEUDOSCALAR VECTOR AXIAL VECTOR TENSOR «Note that in total the sixteen components correspond to the 16 elements of a general 4x4 matrix: decomposition into Lorentz invariant combinations «In QED the factor arose from the sum over polarization states of the virtual photon (2 transverse + 1 longitudinal, 1 scalar) = (2J+1) + 1 «Associate SCALAR and PSEUDOSCALAR interactions with the exchange of a SPIN-0 boson, etc. no spin degrees of freedom

9 V-A Structure of the Weak Interaction «The most general form for the interaction between a fermion and a boson is a linear combination of bilinear covariants «For an interaction corresponding to the exchange of a spin-1 particle the most general form is a linear combination of VECTOR and AXIAL-VECTOR «The form for WEAK interaction is determined from experiment to be VECTOR AXIAL-VECTOR (V A) e n e V A «Can this account for parity violation? «First consider parity transformation of a pure AXIAL-VECTOR current with or

10 V-A Structure of the Weak Interaction The space-like components remain unchanged and the time-like components change sign (the opposite to the parity properties of a vector-current) Now consider the matrix elements For the combination of a two axial-vector currents Consequently parity is conserved for both a pure vector and pure axial-vector interactions However the combination of a vector current and an axial vector current changes sign under parity can give parity violation!

11 V-A Structure of the Weak Interaction «Now consider a general linear combination of VECTOR and AXIAL-VECTOR (note this is relevant for the Z-boson vertex) Consider the parity transformation of this scalar product If either g A or g V is zero, Parity is conserved, i.e. parity conserved in a pure VECTOR or pure AXIAL-VECTOR interaction Relative strength of parity violating part Maximal Parity Violation for V-A (or V+A)

12 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 the spin S is antiparallel or parallel to the direction of motion p Solutions of the Dirac equation: where are helicity eigenstates: For antiparticles the relation is reversed (because v(p) ~ u(-p)):

13 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.

14 Helicity and EM Interaction We can use the projection operators to split the electromagnetic current into 2 pieces: where Since we have and

15 Chiral Fermion States

16 Helicity and EM Interaction Helicity is conserved in the electromagnetic interaction in the high energy (m=0) limit Allowed QED vertices in the high energy limit mirror reflection Helicity is reversed under parity: Equalities are due to parity

17 Chiral Structure of QED «CHIRAL projections operators project out chiral right- and left- handed states «In the ultra-relativistic limit, chiral states correspond to helicity states «Any spinor can be expressed as: The QED vertex in terms of chiral states: conserves chirality, e.g. «In the ultra-relativistic limit only two helicity combinations are non-zero

18 Helicity Structure of the Weak Interaction «The charged current (W ± ) weak vertex is: e n e «Since projects out left-handed chiral particle states: «Writing and from discussion of QED, gives Only the left-handed chiral components of particle spinors and righthanded chiral components of anti-particle spinors participate in charged current weak interactions «At very high energy, the left-handed chiral components are helicity eigenstates : LEFT-HANDED PARTICLES Helicity = -1 RIGHT-HANDED ANTI-PARTICLES Helicity = +1

19 Helicity Structure of the Weak Interaction In the ultra-relativistic limit only left-handed particles and right-handed antiparticles participate in charged current weak interactions e.g. In the relativistic limit, the only possible electron neutrino interactions are: ν e e ν e e + ν e «The helicity dependence of the weak interaction parity violation e.g. RH anti-particle LH particle RH particle LH anti-particle e Valid weak interaction Does not occur

20 Charged Leptonic Weak Interaction The mediators of the weak interactions are intermediate vector bosons, which are extremely heavy: M W = ± GeV/c 2 M Z = ± GeV/c 2 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:

21 Weak Interaction Lorentz condition µ e p = 0 µ

22 Note: also 3 and 4 boson vertices exist Feynman Rules

23 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):

24 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.

25 Inverse Muon Decay (lowest order diagram) When the amplitude is: Simplifies because M W = 80 GeV much larger than q < (100 MeV).

26 Applying Casimir s trick we find: Inverse Muon Decay trace theorems trace theorems using:

27 Inverse Muon Decay In the CM frame, and neglecting the mass of the electron: where E is the incident electron (or neutrino) energy. The differential scattering cross section is: The total cross section:

28 Decay of the Muon The amplitude: As before:

29 In the muon rest frame: Decay of the Muon Let: Plug in:

30 Decay of the Muon The decay rate given by Fermi s Golden Rule * : where: * a lot of work, since this is a three body decay

31 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:

32 Decay of the Muon Also: The For the integral is trivial. integration, let: and:

33 Decay of the Muon integration: where: The limits of E 2 and E 4 integrals:

34 Using: Decay of the Muon

35 Decay of the Muon

36 Decay of the Muon

37 Decay of the Muon The total decay rate: Lifetime:

38 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: = s

39 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 the electromagnetic fine structure constant!

40 Weak Interactions The weak force is not inherently weak, we just live in a world where typical interactions have q 2 M 2 W Weak force is weak because boson propagator is massive, not because coupling strength is weak Weak Interactions are Parity violating Small CP-violation also observed

41 Decay of the Neutron (the same as in previous case)

42 Decay of the Neutron In the rest frame of the neutron: As before: We can t ignore the mass of the electron. where:

43 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

44 Decay of the Neutron where: and:

45 The range of E 2 integral: Decay of the Neutron E is the electron energy (exact equation)

46 Decay of the Neutron Approximations: Expanding to lowest order:

47 Decay of the Neutron (picture from Griffiths)

48 Decay of the Neutron Putting in the numbers: where:

49 Decay of the Neutron But the proton and neutron are not point-like 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

50 Decay of the Neutron Another correction, the quark vertex carries a factor of is the Cabibbo angle. cosθ C = 0.97 Lifetime:

51 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

52 Decay of the Pion

53 Decay of the Pion

54 Decay of the Pion

55 Decay of the Pion The decay rate: The following ratio could be computed without knowing the decay constant: Experimental value:

56 Decay of the Pion

57 Decay of the Pion Experimentally the ratio is 10-4

58 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:

59 Weak Eigenstates

60 Charged Weak Interactions of Quarks (Cabibbo, 1963) (extra cos or sin in the vertex factor)

61 Cabibbo-GIM Mechanism The decay Amplitude: is allowed by Cabibbo theory., far greater. GIM introduced the fourth quark c (1970). The couplings with s and d : GIM = GLASHOW, ILIOPOULOS, MAIANI

62 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:

63 Cabibbo-Kobayash-Maskawa 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.

64 The CKM Matrix The full matrix: Using the experimental values:

65 Neutral Weak Interactions Neutral weak interaction mediated by the Z 0 boson f stands for any lepton or quark Not allowed:

66 Neutral Weak Interactions Like the photon the Z couples to f and f and NOT to f and f. No flavour change. Unlike the photon the Z couples differently to left and right handed fermions. Like the W the Z coupling too violates parity but unlike the W not necessarily maximally.

67 Neutral Weak Interactions It doesn t matter if we use physical states or Cabibborotated states.

68 Neutral Weak Interactions First attempt to see weak neutral current: D. H. Perkins, Veltman. Failed. Discovery of neutral currents: In an experiment at CERN with a Bubble Chamber. Experiments with μ beams obtained from π decays. First process mediated by Z 0 (Bubble chamber photograph at CERN, 1973)

69 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 higher-order process (which corresponds to a far smaller cross section).

70 Neutral Weak Interactions The coupling to Z 0 : where : ( Weak mixing angle or Weinberg angle )

71 Neutral Weak Interactions Neutral vector and axial vector coupling in GWS model:

72 Neutral Weak Interactions ( Z 0 propagator ) When: the propagator is simply: The masses of the bosons are related by the formula:

73 Elastic Neutrino-Electron Scattering

74 Neutral Weak Interactions Now compute in CM frame and let : (mass of the electron)

75 Neutral Weak Interactions (E is the electron or neutrino energy) Using:

76 Neutral Weak Interactions The total cross section: Compare to: (computed in the earlier) (0.11, experimental)

77 Electron-Positron Scattering Near the Z 0 Pole Most neutral processes are masked by electromagnetic ones.

78 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:

79 Electron-Positron Scattering Near the Z 0 Pole Ignore the mass of quark or lepton (since we are working in the vicinity of 90 GeV) Finally:

80 Electron-Positron Scattering Near the Z 0 Pole problems at Z 0 pole

81 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.

82 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:

83 Electron-Positron Scattering Near the Z 0 Pole

84 Electron-Positron Scattering Near the Z 0 Pole Well below the Z 0 pole: Right on the Z 0 pole:

85 Electron-Positron Scattering Near the Z 0 Pole