Weak interaction and Parity β Decay Pion Decay Muon Decay

P318

The electron energy spectrum shows the interaction has “no structure”! ν ν p p ? 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

νμ νe e μ This Fermi theory describes Weak decays very well!

νμ νe e μ νμ νe e μ 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 μ νμ -ig W -

νμ νe e μ νμ νe e μ 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 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 the 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 ..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 左右對稱性 一般預料自然界是遵守鏡射對稱 鏡中世界的物理定律與真實世界形式一樣！

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 left-hander, 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?

當時我也正在重點研究這個問題，曾做過一些嘗試，但未成功。我記得，在1956年4月3－7日羅徹斯特會議結束后的一兩天，即4月8日或9日，我哥倫比亞大學的同事斯坦伯格（J. Steinberger），專程到我的辦公室訪問，討教問題。那時他正在進行不穩定的重粒子的產生和衰變的實驗。在談話的過程中，我忽生靈感，突然很清楚地明了，要解決θ-τ之謎，必須先離開θ-τ系統，必須假定θ-τ以外的粒子也可能發生宇稱不守恆的新現象。我發現，用斯坦伯格實驗中重粒子產生和衰變的幾個動量，便能很簡單地去組織一個新的贗標量。用了這θ-τ以外的贗標量，就可以試驗θ-τ以外的系統宇稱是否不守恆。而這些贗標量，很顯然的，沒有被以前任何實驗測量過。用了這些新的贗標量就可以系統地去研究宇稱是否不守恆那個大問題。θ-τ之謎不再是一個孤立的點，它可以和斯坦伯格正在進行實驗的重粒子連起來，它也可能和其他一切物理整體地連起來。要解開θ-τ之謎，就要去測量弱作用中θ-τ以外的贗標量。我猜想，宇稱不守恆很可能就是一個普遍性的基礎科學原理。這就是宇稱不守恆思想的突破。 據此就可以去測量θ-τ以外的粒子是否也是宇稱不守恆。這是以前別人沒有想到的。這就是我的宇稱不守恆思想的突破。( Lee

“4月底或5月初的一天，我驅車前往哥倫比亞作每周例行的拜訪。我把李政道從他的辦公室接出來，上了車。我們很難找到泊車的空位。后來，我把車泊在百老匯大街和125街的轉角處。那是午飯時分，但附近的飯館尚未開門營業。於是我們就到左近的“白玫瑰”咖啡館，在那裡繼續討論。稍后，我們在“上海餐館”（據我回憶，是這間餐館，但李政道說他記得是“天津餐館”）吃午飯。我們的討論集中在θ-τ之謎上面。在一個節骨眼上，我想到了，應該把產生過程的對稱性同衰變過程分離開來。於是，如果人們假設宇稱隻在強作用中守恆，在弱作用中則不然，那麼，θ和τ是同一粒子且自旋、宇稱為0-（這一點是由強作用推斷出的）的結論就不會遇到困難。這種分離對反應鏈（1）、（2）有特別的意義。李政道先是反對這種觀點。我力圖說服他，因為這種想法可以通過（1）、（2）兩個反應中可能存在的上－下不對稱性而加以檢驗，它就更有吸引力了。后來，他同意了我的意見。那時，李政道還不大熟悉β衰變現象。 Yang

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.

W - W + -ig e νe The weak vertices of leptons coupling with 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 -ig W +

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

W e νe μ νμ -ig

W - e νe W + Feynman Rules of weak interaction, Finally

Look at this weak current It is a projection operator! It project neutrino into the eigenstate of γ5 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 two-component Weyl Equations: Helicity +1 Right-handed Helicity -1 Left-handed 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!

ν Left-handed Neutrino 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.

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

L R W - e νe W + W only couple to the left-handed electron! 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

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

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

The Lord is indeed a weak left-hander.