Weak interaction and Parity Decay Pion Decay Muon

Weak interaction and Parity β Decay Pion Decay Muon Decay

P 318

The electron energy spectrum shows the interaction has “no structure”! e p ν ? n n The spectrum can be produced by phase space alone! M only need to be a constant!

Muon decay

Positron energy spectrum in muon decay

This Fermi theory describes Weak decays very well!

W

We couple A with the electron current, not axial current because of parity conservation of QED.

The weak vertices of leptons coupling with W W-

W

This is almost right but not exactly true.

Parity Symmetry 鏡射對稱性

Parity non-conservation Vectors are odd under parity Axial Vectors are even under parity

In quantum mechanics: With parity symmetry, particles must be eigenstates of parity: A particle has a intrinsic parity.

And more and more … until they don’t feel fundamental at all…………

Tau-Theta puzzle Tau and Theta have identical masses and charge. Look identical in every aspect. But Theta decays into a parity even state. Tau decays into a parity odd state. How could two different particles look so similar?

Rochester Conference

Could it be that theta and tau are different parity states of the same particle which has no definite parity, that parity is not conserved. Feynman at Rochester 1956

. . many fine Chinese restaurants. Finding they are not open yet, they retired to the nearby White Rose Café. They discussed the possibility that parity is conserved in strong interaction that produced tau and theta, yet violated in the weak interaction by which they decay. Approximate Symmetry, Partial Symmetry

Theta and tau are just K.

Parity symmetry is violated (broken)

How to check Parity violation? Polarize the nucleus, and observe if there is any anisotropy in beta decays. In beta decay of Cobalt 60, most electrons are emitted in direction opposite to nuclear spin. In mirror world, most are parallel.

C. S. Wu

There is an easier way. Helicity:

If neutrino from pion decay is more left-handed than right-handed, the muon will be polarized. The e’s from muon decay will have anisotropy, just like in Wu’s experiment!

I don’t believe that the Lord is a weak lefthander, and I am ready to bet a very large sum that the experiments will give symmetric results. Now after the first shock is over, I begin to collect myself. It’s good that I didn’t make a bet. What shocks me is not the fact that “god is just left-handed” but the fact that in spite of this he exhibits Himself as left/right symmetric when he expresses Himself strongly? How can the strength of an interaction produce or create symmetry?

In fact all bilinears can be classified according to their behavior under Lorentz Transformation:

How does a Dirac Bispinor transform under parity? We can assign: Dirac Equation will be invariant under this assignment with space transforms in the way we expect.

In fact all bilinears can be classified according to their behavior under Lorentz Transformation: True vector Axial vector

We couple A with the electron current, not axial current because of parity conservation of QED.

The weak vertices of leptons coupling with W W+ Now that parity is violated in weak interaction, both vector current (V) and axial vector current (A) could possibly couple to W.

1957 -58 a lot of efforts are devoted to determine g’s. It turned to be much simpler than expected: V-A theory

The weak vertices of leptons coupling with W W+

The weak vertices of leptons coupling with W W+ W- W+

W+ Feynman Rules of weak interaction, Finally

Look at this weak current It is a projection operator! It project neutrino into the eigenstate of γ 5 with eigenvalue of -1. Part of the neutrino and the electron does not participate in weak interaction. This is called Chirality. Only Fermions with Chirality -1 feels weak interaction! What does it mean?

The greek name kheir means "hand. “ So chirality indeed means "handedness. " 手徵 The phenomenon, in chemistry, physics and mathematics, in which an object differs from its mirror image Lord Kelvin’s definition (1904): …I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide by itself. . . Parity change the chirality of a state:

Chirality in Chemistry

Left and right coiling Euhadra snails

What exactly is chirality in weak interacting particles? Anything that breaks left-right symmetry? Right-handed neutrino is not balanced with left-handed neutrino in muon decays.

A 1958 ex. by Goldhaber et al measured the helicity of neutrino! Maybe: Chirality -1 neutrino means left-handed neutrino! Chirality +1 neutrino doesn’t exist!

Chirality means helicity, is this true? The idea f Dirac Eq. comes from factoring the energy momentum relation: The above could be done for 2 by 2 matrices if there is no mass. Massless fermion contains only half the degrees of freedom.

Factoring the energy momentum relation: Expand the right hand side: We get and we need:

交叉項抵銷 We can get two-component Weyl Equations: Helicity +1 Right-handed Helicity -1 Lefthanded These are the Weyl Equations for massless fermions.

Put two Weyl spinors into one four component Dirac bisipnor: We can combine the two Weyl Equations into a Dirac Equation with the following gamma matrices: All gamma’s have non-zero at the same blocks! All sets of Gamma’s are equivalent! Dirac Equation decomposes into two Weyl Equations when mass is zero.

Now Chirality In this representation: Right-handed ψR is the Chirality eigenstate with eigenvalue 1. Left-handed ψL is the Chirality eigenstate with eigenvalue -1. For massless particle, helicity is chirality!

Left-handed Neutrino ν Neutrino is projected into the Left-handed version. Only left-handed neutrino exists anyway!

How about massive fermion like electron? Adopt neutrino handed-ness to denote electron chirality We can combine the two Weyl Equations into a Dirac Equation with the following gamma matrices: All gamma’s have non-zero at the same blocks! All sets of Gamma’s are equivalent! Mass is the factor that mixes the right-handed and the left-handed.

Two bases of Dirac Matrices Dirac Basis As This basis is good for the slow electrons. Weyl Basis As This basis is good for the fast electrons.

Left-handed and Right-handed electrons can not be separated. A real electron has to come with two parts! R Lorentz Transformation Handed electrons are not mass (energy) eigenstates. Mass eigenstate doesn’t have definite Chirality (handedness). L

W only couple to the left-handed electron! W+ L R W Only left-handed fermions participate in weak interaction. This is the essence of parity violation.

Sum up, for massive fermions Chirality Eigenstates Helicity Eigenstates Weak interaction W couple to Dirac Equation solutions Gauge Eigenstate Free propagating fermions Mass Eigenstate

The vertices of weak interaction: leptons coupling with W W+ W+ WRight-handed leptons have no weak interaction!

The vertices of weak interaction: quarks coupling with W W+ W- W+ Right-handed quarks have no weak interaction!

The Lord is indeed a weak left-hander.