Weak Interaction. Parity Violation

Introduction to Neutrino Physics

Dmitry V. Naumov

JINR

1. Parity as a Symmetry

Space Reflection

before reflection

\(x\) \(y\) \(\mathbf r\)

\(P\) \(\longrightarrow\) \(\mathbf r\mapsto-\mathbf r\)

after reflection

\(x\) \(y\) \(-\mathbf r\)

Spatial parity replaces each point with its mirror image relative to the origin:

\[ P:\quad (x,y,z)\longmapsto(-x,-y,-z). \]

Polar and Axial Vectors

Under space reflection

\[ P:\qquad \mathbf r\to-\mathbf r, \qquad \mathbf p\to-\mathbf p. \]

But angular momentum and spin are axial vectors:

\[ \mathbf L=\mathbf r\times\mathbf p\to\mathbf L, \qquad \mathbf S\to\mathbf S. \]

Therefore the scalar product

\[ \mathbf S\cdot\mathbf p \xrightarrow{P} -\mathbf S\cdot\mathbf p \]

is odd under \(P\). This observation will be the key to the Wu experiment.

Parity of a Quantum State

If the Hamiltonian is invariant under reflection,

\[ [H,P]=0, \]

the states can be chosen to be eigenstates of the parity operator:

\[ P\psi(\mathbf r)=\psi(-\mathbf r)=\eta_P\psi(\mathbf r), \qquad \eta_P=\pm1. \]

Even state

\[ \psi(-\mathbf r)=+\psi(\mathbf r). \]

Odd state

\[ \psi(-\mathbf r)=-\psi(\mathbf r). \]

Why Parity Conservation Was Expected

• In atomic spectroscopy the Laporte rule holds: electric dipole transitions change parity.
• In 1927 Wigner linked the \(\mathbf r\to-\mathbf r\) symmetry with the quantum number of parity.
• Electromagnetic and strong processes confirmed the conservation of \(P\).

By the mid-1950s, parity conservation was perceived almost as a self-evident law of nature.

Eugene Wigner

2. The \(\tau\)–\(\theta\) Puzzle

One Mass — Two Decays

In the early 1950s a positively charged particle with a mass of about \(494~\mathrm{MeV}\) and two decay channels was observed:

\[ \theta^+\to\pi^++\pi^0. \]

\[ \tau^+\to\pi^++\pi^++\pi^-. \]

The masses and lifetimes of \(\theta^+\) and \(\tau^+\) coincided — everything pointed to one and the same particle, the modern \(K^+\).

The Parity Contradiction

The pion has negative intrinsic parity:

\[ \eta_\pi=-1. \]

For decays in the \(S\)-wave:

\[ \theta^+\to2\pi, \qquad \eta_f=(-1)^2=+1. \]

\[ \tau^+\to3\pi, \qquad \eta_f=(-1)^3=-1. \]

One particle cannot have both \(\eta_P=+1\) and \(\eta_P=-1\) if parity is conserved.

Either these are two nearly indistinguishable particles, or the weak interaction violates spatial parity.

3. From a Puzzle to a Testable Hypothesis

An Idea Considered Almost Unthinkable

Murray Gell-Mann

Unable to “prove” the conservation of \(P\), he realized early on: this is not a geometric necessity, but a property of a specific theory.

Richard Feynman

Discussed the hypothesis with Gell-Mann and brought the question to the Rochester Conference.

A Week Spent on an Impossible Proof

Professor Herman Feshbach suggested that students prove parity conservation by a change of coordinates. Gell-Mann spent about a week and discovered that no such proof exists — a coordinate transformation by itself does not dictate the laws of nature.

1955 → Rochester, 1956

Gell-Mann and Feynman discussed the violation of \(P\), but did not connect it with the \(\tau\)–\(\theta\) puzzle. At the conference Feynman passed on the idea of his roommate Marty Block; Yang replied that he and Lee had already considered it — without result.

Moscow, May 1956: the hall responds with hostility

Gell-Mann named the violation of \(P\) as a possible solution of the \(\tau\)–\(\theta\) puzzle. From the crowded hall came objections: “This is a violation of Lorentz invariance!” and “Do the laws depend on rotations?”

Gell-Mann’s answer: the conservation of \(P\) is an empirical fact; there is no fundamental reason for it.

The main barrier was not mathematical. Space reflection was mistakenly perceived as a simple change of coordinates — and therefore as a mandatory symmetry of any law of nature.

Lee and Yang: Testing the Weak Interaction

In the summer of 1956, Tsung-Dao Lee and Chen-Ning Yang systematically studied the existing experiments and noticed:

parity conservation had been well tested for electromagnetic and strong processes, but not for weak ones.

They proposed specific tests in \(\beta\) decay, pion decays, and muon decays. To test \(\beta\) decay, Lee turned to Chien-Shiung Wu.

4. The Wu Experiment

Chien-Shiung Wu

Chien-Shiung Wu was one of the leading specialists in \(\beta\) decay.

At the end of 1956 her group, together with specialists from the National Bureau of Standards, carried out an experiment with polarized \({}^{60}\mathrm{Co}\) nuclei.

The experiment was to answer a simple question:

do electrons emerge equally often along and against the nuclear spin?

The Idea of the Experiment

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1

\(\mathbf I\) \({}^{60}\mathrm{Co}\)

Polarize the nuclei: align their spins along the magnetic field.

2

Measure the distribution of electrons relative to the spin direction.

3

Any asymmetry between \(\theta\) and \(\pi-\theta\) means parity violation (\(P\) violation).

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Why \({}^{60}\mathrm{Co}\)

The main \(\beta^-\) decay:

\[ {}^{60}_{27}\mathrm{Co}(5^+) \to {}^{60}_{28}\mathrm{Ni}^{*}(4^+)+e^-+\bar\nu_e. \]

After it, the nickel nucleus emits two \(\gamma\) quanta:

\[ {}^{60}\mathrm{Ni}^{*}(4^+) \to {}^{60}\mathrm{Ni}^{*}(2^+)+\gamma \to {}^{60}\mathrm{Ni}(0^+)+2\gamma. \]

• the angular distribution of the electrons tests parity;
• the anisotropy of the \(\gamma\) radiation independently shows the degree of polarization of the nuclei.

Why Millikelvins Are Needed

A magnetic moment in a field has energy

\[ E=-\boldsymbol\mu\cdot\mathbf B, \qquad \frac{n_\uparrow}{n_\downarrow} = \exp\!\left(\frac{2\mu B}{k_BT}\right). \]

\[ \mu_B=\frac{e\hbar}{2m_e} =5.79\cdot10^{-5}\ \mathrm{eV/T}, \]

\[ \frac{\mu_B B}{k_BT}\simeq0.67 \quad (B=1~\mathrm{T},\ T=1~\mathrm K). \]

Electron spins are comparatively easy to polarize.

\[ \mu_N=\frac{e\hbar}{2m_p} =3.15\cdot10^{-8}\ \mathrm{eV/T}, \]

\[ \frac{\mu_N B}{k_BT}\simeq3.7\cdot10^{-4} \quad (B=1~\mathrm{T},\ T=1~\mathrm K). \]

For nuclei, a temperature of order \(10^{-2}~\mathrm K\) is required.

Adiabatic Demagnetization

1

Warm crystal

↗↙→↖↓

\(T_i\simeq1.2~\mathrm K\)

Large spin entropy.

\(B_i\)→

2

Magnetization

↑↑↑↑↑

The crystal is coupled to the helium bath.

Heat flows into the bath.

\(B_i\to B_f\)→

3

Demagnetization

↑↗↑↖↑

Thermal contact is switched off.

Lattice energy flows into the spins.

When the field is decreased adiabatically, the ratio \(B/T\) remains almost constant:

\[ \frac{B_i}{T_i}\simeq\frac{B_f}{T_f}, \qquad T_f\simeq T_i\frac{B_f}{B_i}\sim10^{-2}\ \mathrm K. \]

Experimental Setup

1Cooling1.2 K → 0.01 K

2Counting at B ↑detector points along the field

3Reverse: B ↓the same detector now points against the field

as on the previous slide

Adiabatic
demagnetization

B_cool reduce the field

1.2 K 0.01 K

The spins take energy from the lattice.
The cold sample keeps its polarization.

cooled → measure

electron scintillator position does not change

liquid helium

⁶⁰Co ↑ ↑ ↑

B

γθ ≈ 0

γθ ≈ π/2

two γ counters
anisotropy → polarization

B ↑detector along the field

B ↓detector against the field

equal measurement time

Electrons in the upper detector

N(e⁻)

40

B ↑ along the field

80

B ↓ against the field

N_against > N_along more electrons against the field

independent check γ anisotropy

N(γ)

N_γ(0)

θ ≈ 0 along the axis

N_γ(π/2)

θ ≈ π/2 across the axis

N_γ(0) ≠ N_γ(π/2) anisotropy ⇒ 𝒫 ≈ 0.65

The γ anisotropy independently fixes the nuclear polarization. Then we reverse only the field: the geometry of the electron detector is the same, but the count rate changes — this is the β-decay asymmetry.

Expected Angular Distribution

For the electrons of \(\beta\) decay

\[ W(\theta) = 1+A\,\mathcal P\,\frac{v}{c}\,f_{\rm bs}\cos\theta, \]

where

\[ \mathcal P=\frac{\langle I_z\rangle}{I} \]

is the nuclear polarization, \(f_{\rm bs}\) is the backscattering correction, and \(A\) is the asymmetry coefficient.

If parity is conserved, the directions \(\mathbf I\) and \(-\mathbf I\) are physically equivalent, and therefore

\[ A=0. \]

What Reflection Does

The angle is set by the correlation

\[ \cos\theta = \frac{\mathbf I\cdot\mathbf p_e}{I\,p_e}. \]

Under space reflection

\[ \mathbf I\to\mathbf I, \qquad \mathbf p_e\to-\mathbf p_e, \qquad \cos\theta\to-\cos\theta. \]

Consequently,

\[ W(\theta)\xrightarrow{P}W(\pi-\theta). \]

A nonzero term proportional to \(\cos\theta\) means that the mirror process has a different probability.

Result

\(W(\theta)\)

\(\cos\theta\)

Measured asymmetry:

\[ \frac{W(0)-W(\pi)}{W(0)+W(\pi)}\simeq-0.25. \]

For \(\mathcal P\simeq0.65\), \(v/c\simeq0.6\) and \(f_{\rm bs}\simeq2/3\):

\[A\simeq-1.\]

The electrons were emitted predominantly opposite to the nuclear spin. The effect disappeared after the sample was heated and its polarization was lost — which is exactly how a physical asymmetry should behave.

The World and Its Mirror Image

Our world

\(\mathbf I\) \(e^-\) \(\mathbf p_e\)

The electrons are emitted predominantly opposite to the nuclear spin.

Mirror world

\(\mathbf I\) \(e^-\) \(\mathbf p_e\)

The mirrored process predicts the opposite asymmetry.

The weak interaction distinguishes these two worlds: spatial parity is violated.

5. Verification and Physical Meaning

“The Experiment Must Be Repeated”

Wolfgang Pauli

The result was so unexpected that many theorists were at first skeptical of it.

Pauli insisted: the experiment must be repeated and checked by an independent method.

That is exactly what happened almost immediately.

Independent Check with Muons

In the decay

\[ \pi^+\to\mu^++\nu_\mu \]

muons are produced polarized. Their polarization can be measured from the asymmetry of positrons in the subsequent decay

\[ \mu^+\to e^++\nu_e+\bar\nu_\mu. \]

Leon Lederman, Richard Garwin, and Marcel Weinrich prepared the experiment in less than a week and in early 1957 confirmed the violation of parity.

Two different weak processes gave the same answer.

What Remained of Parity

Interaction	Spatial parity
strong	conserved
electromagnetic	conserved
weak	maximally violated

In the modern theory the charged weak current contains the left-handed projector:

\[ J_L^\mu = \bar\psi\gamma^\mu\frac{1-\gamma^5}{2}\psi. \]

The index \(L\) in the gauge group

\[ SU(2)_L\times U(1)_Y \]

is a reminder: the weak interaction fundamentally distinguishes left from right.

How to Explain to Aliens Which Way Is Left?

interplanetary communication session

Earth: the body suggests the picture Mirror Earth: suggests the opposite one We need a physical experiment that does not conserve parity.

A common reference: polarize \({}^{60}\mathrm{Co}\) nuclei and call “left” the direction in which more electrons are emitted.

The 1957 Nobel Prize

Tsung-Dao Lee and Chen-Ning Yang received the Nobel Prize in Physics “for their penetrating investigation of the so-called parity laws, which has led to important discoveries regarding the elementary particles.”

The discovery came a year after their theoretical work.

Mirror symmetry is not a universal law of nature.

Exercises for this lecture

• 2.1. The \(\tau\)–\(\theta\) puzzle
• 2.2. What does the Wu experiment measure directly?
• 2.3. Can nuclei be polarized with an ordinary magnet?
• 2.4. Adiabatic demagnetization
• 2.5. How can we explain to aliens which side is left?

• 2.6. Why is it easy to polarize electrons but hard to polarize nuclei?
• 2.7. Why did the Laporte rule make people confident in parity conservation?
• 2.8. Why do \({}^{60}\mathrm{Co}\) electrons fly opposite to the nuclear spin?
• 2.9. The \(\pi\to\mu\to e\) chain: two parity violations at once
• 2.10. What does “maximal” parity violation mean?

All exercises for lecture 02 → and exercises for the seminar · All course exercises