Exercises: Weak Interaction. Parity Violation

Lecture 02 · Introduction to Neutrino Physics

Author

Dmitry V. Naumov

← Back to the lecture · All course exercises

Exercises

Exercise 2.1. The \(\tau\)–\(\theta\) puzzle

Two charged particles with the same mass and lifetime were observed in the decays

\[ \theta^+\to\pi^++\pi^0, \qquad \tau^+\to\pi^++\pi^++\pi^-. \tag{1}\]

Both particles have spin zero, and the intrinsic parity of the pion is negative: \(P_\pi=-1\). Assume that the final pions are predominantly in a state with zero relative orbital angular momentum.

  1. Find the parity of the two-pion final state.
  2. Find the parity of the three-pion final state.
  3. Why, if spatial parity were conserved, would these decays have to be assigned to two different particles?
  4. Why, on the other hand, did the coincidence of masses and lifetimes point to a single particle?
  5. State how the Lee–Yang hypothesis resolved the paradox.

Exercise 2.2. What does the Wu experiment measure directly?

Let the angular distribution of electrons in the \(\beta\) decay of a polarized nucleus have the form

\[ W(\theta)=1+a\cos\theta, \tag{2}\]

where \(\theta\) is the angle between the electron momentum and the direction of the nuclear polarization.

  1. Indicate which of the quantities \(\mathbf p_e\) and \(\mathbf J\) is a polar vector and which is an axial vector.

  2. Show that under spatial inversion \(\mathbf p_e\to-\mathbf p_e\), \(\mathbf J\to\mathbf J\) and \(\cos\theta\to-\cos\theta\).

  3. Why does a nonzero coefficient \(a\) imply parity violation?

  4. By integrating over the two hemispheres, show that

    \[ N_+\propto1+\frac a2, \qquad N_-\propto1-\frac a2. \tag{3}\]

  5. Express \(a\) in terms of the measured hemispherical asymmetry

    \[ \mathcal A=\frac{N_+-N_-}{N_++N_-}. \tag{4}\]

Exercise 2.3. Can nuclei be polarized with an ordinary magnet?

Consider a nucleus with spin \(1/2\) and magnetic moment \(\mu=4\mu_N\) in a field \(B=10~\mathrm{T}\). Use

\[ \mu_N=3.15\cdot10^{-8}~\mathrm{eV/T}, \qquad k_B=8.617\cdot10^{-5}~\mathrm{eV/K}. \tag{5}\]

The energies of the two orientations are \(E_\pm=\mp\mu B\), and the equilibrium polarization is

\[ \mathcal P= \frac{N_+-N_-}{N_++N_-} =\tanh\!\left(\frac{\mu B}{k_BT}\right). \tag{6}\]

  1. Derive this formula from the Boltzmann populations.
  2. Compute the Zeeman energy \(\mu B\).
  3. Find \(\mathcal P\) for \(T=1~\mathrm{K}\), \(10~\mathrm{mK}\) and \(1~\mathrm{mK}\).
  4. At what temperature is \(\mathcal P=0.90\) reached?
  5. Why does a large magnet by itself not provide strong nuclear polarization?

Exercise 2.4. Adiabatic demagnetization

For \(N\) non-interacting magnetic moments with spin \(1/2\) in a field \(B\), introduce

\[ x=\frac{\mu B}{k_BT}. \tag{7}\]

  1. Show that the partition function is

    \[ Z_N=(2\cosh x)^N. \tag{8}\]

  2. From \(F=-k_BT\ln Z_N\) obtain the entropy

    \[ S=Nk_B\left[\ln(2\cosh x)-x\tanh x\right]. \tag{9}\]

  3. Explain why the ratio \(B/T\) is conserved at \(S=\mathrm{const}\).

  4. The system starts at

    \[ T_i=1.2~\mathrm{K}, \qquad B_i=2.0~\mathrm{T} \tag{10}\]

    and is adiabatically demagnetized to \(B_f=2.0\cdot10^{-2}~\mathrm{T}\). Find \(T_f\).

  5. What real effects prevent the temperature from dropping without limit as \(B_f\to0\)?

  6. Find the limiting forms of the entropy for \(x\ll1\) and \(x\gg1\). Describe the level populations in both limits and explain why \(S\) tends to \(Nk_B\ln2\) and to zero, respectively.

Exercise 2.5. How can we explain to aliens which side is left?

Suppose first that all fundamental interactions conserve spatial parity.

  1. Why are the coordinate systems \((x,y,z)\) and \((-x,y,z)\) physically equivalent in such a world?
  2. Why can a drawing, a written instruction, or a radio signal not by themselves convey the absolute difference between left and right?
  3. Describe a setup with polarized nuclei and \(\beta\) detectors that two civilizations could assemble in the same way.
  4. How would both civilizations, from the sign of the \(\beta\)-decay asymmetry, choose the same orientation of space?
  5. What additional agreement must be communicated so that the procedure does not confuse matter with antimatter?

Exercise 2.6. Why is it easy to polarize electrons but hard to polarize nuclei?

The Bohr magneton and the nuclear magneton are

\[ \mu_B=\frac{e\hbar}{2m_e}=5.79\cdot10^{-5}~\mathrm{eV/T}, \qquad \mu_N=\frac{e\hbar}{2m_p}=3.15\cdot10^{-8}~\mathrm{eV/T}, \qquad k_B=8.617\cdot10^{-5}~\mathrm{eV/K}. \tag{11}\]

For a spin-\(1/2\) moment the equilibrium polarization is \(\mathcal P=\tanh x\), where \(x=\mu B/(k_BT)\).

  1. Find \(\mu_BB\), \(\mu_NB\) and their ratio for \(B=1~\mathrm{T}\). The ratio of which quantities is it equal to?
  2. Compare both energies with \(k_BT\) at \(T=1~\mathrm{K}\): find \(x_e\) and \(x_N\).
  3. Find the polarization of the electron and nuclear moments at \(B=1~\mathrm{T}\) and \(T=1~\mathrm{K}\), taking \(\mu=\mu_B\) and \(\mu=\mu_N\).
  4. To what temperature must the nuclei be cooled to obtain, in the same field, the same \(x\) as for electrons at \(1~\mathrm{K}\)?
  5. The polarized electron shell of an atom creates at the nucleus a hyperfine field far exceeding a laboratory field. What effective field is needed to obtain \(x_N=1\) at \(T=10~\mathrm{mK}\) for \(\mu=\mu_N\)?

State the main physical conclusion: how does the proton-to-electron mass ratio turn into a requirement on the temperature of the Wu experiment, and why do the electron spins of the salt serve as the refrigerator while it is the polarization of the nuclei that is measured?

Exercise 2.7. Why did the Laporte rule make people confident in parity conservation?

The states of the hydrogen atom have the form \(\psi_{nlm}=R_{nl}(r)Y_{lm}(\vartheta,\varphi)\). Under the reflection \(\mathbf r\to-\mathbf r\)

\[ \vartheta\to\pi-\vartheta, \qquad \varphi\to\varphi+\pi, \qquad Y_{lm}\to(-1)^lY_{lm}. \tag{12}\]

The measured lifetimes of the levels are

\[ \tau(2p)=1.60~\mathrm{ns}, \qquad \tau(2s)=0.122~\mathrm{s}. \tag{13}\]

  1. Show that the parity of the state \(\psi_{nlm}\) is \((-1)^l\). Find the parities of \(1s\), \(2s\), \(2p\), \(3d\).
  2. The dipole moment operator \(\mathbf d=-e\mathbf r\) is odd. Assuming \([H,P]=0\) and \(P|i\rangle=\eta_i|i\rangle\), \(P|f\rangle=\eta_f|f\rangle\), show that \(\langle f|\mathbf d|i\rangle=-\eta_f\eta_i\langle f|\mathbf d|i\rangle\). State the Laporte rule.
  3. Which of the transitions \(2p\to1s\), \(2s\to1s\), \(3d\to2p\), \(3d\to1s\), \(3s\to2p\) are allowed for electric dipole radiation?
  4. For \(2s_{1/2}\to1s_{1/2}\) angular momentum conservation does not forbid the emission of a single photon of spin \(1\). Why is this transition nevertheless suppressed, so that the \(2s\) level lives much longer? Find \(\tau(2s)/\tau(2p)\).
  5. Suppose that a parity-violating interaction admixes the \(2p\) state to \(2s\) with a small amplitude \(\varepsilon\). Then the one-photon decay rate is \(\varepsilon^2/\tau(2p)\). For what \(\varepsilon\) would it equal the decay rate \(1/\tau(2s)\)?

State the main physical conclusion: what do parity selection rules in atoms test, and why do they say nothing about the weak interaction?

Exercise 2.8. Why do \({}^{60}\mathrm{Co}\) electrons fly opposite to the nuclear spin?

The main transition

\[ {}^{60}\mathrm{Co}(5^+)\to{}^{60}\mathrm{Ni}^*(4^+)+e^-+\bar\nu_e, \qquad T_{\max}=0.318~\mathrm{MeV}, \qquad m_e=0.511~\mathrm{MeV}, \tag{14}\]

is allowed: the leptons carry away zero orbital angular momentum. In Fermi transitions the lepton spins form a singlet (\(\Delta J=0\)), in Gamow–Teller transitions a triplet (\(\Delta J=0,\pm1\), except \(0\to0\)); the nuclear parity does not change. The electron from weak decay is longitudinally polarized opposite to its momentum, and the antineutrino along its momentum:

\[ \langle\boldsymbol\sigma_e\rangle=-\frac{v}{c}\,\hat{\mathbf p}_e, \qquad \langle\boldsymbol\sigma_{\bar\nu}\rangle=+\hat{\mathbf p}_{\bar\nu}. \tag{15}\]

  1. Check that the transition is allowed and pure Gamow–Teller.
  2. Let the nucleus be fully polarized: \(M=+5\) along the \(z\) axis. Which values of \(M\) are possible for \({}^{60}\mathrm{Ni}^*(4^+)\)? Show that the leptons must carry away \(J_z=+1\), that is, both spins point along the nuclear spin.
  3. The electron is emitted at an angle \(\theta\) to the \(z\) axis. Using the electron polarization, show that the probability of finding its spin along \(+z\) is \(\tfrac12\left(1-\tfrac vc\cos\theta\right)\). Obtain \(W(\theta)\propto1-\tfrac vc\cos\theta\) and the asymmetry coefficient \(A=-1\).
  4. In which direction does the antineutrino preferentially fly?
  5. Find \(v/c\) for electrons with \(T=T_{\max}\) and \(T=0.1~\mathrm{MeV}\).
  6. For a nuclear polarization \(\mathcal P\simeq0.65\), a velocity \(v/c\simeq0.6\) and a backscattering correction \(f_{\mathrm{bs}}\simeq2/3\), find the expected asymmetry \(A\mathcal P(v/c)f_{\mathrm{bs}}\) and compare it with the value \(-0.25\) measured in the Wu experiment.

State the main physical conclusion: which property of weak-decay electrons determines the sign of the asymmetry in the Wu experiment?

Exercise 2.9. The \(\pi\to\mu\to e\) chain: two parity violations at once

A pion at rest decays as \(\pi^+\to\mu^++\nu_\mu\); the muon stops and decays: \(\mu^+\to e^++\nu_e+\bar\nu_\mu\). Use

\[ m_\pi=139.57~\mathrm{MeV}, \qquad m_\mu=105.66~\mathrm{MeV}, \qquad \tau_\mu=2.197~\mathrm{\mu s}, \qquad \frac{\gamma_\mu}{2\pi}=135.5~\mathrm{MHz/T}, \tag{16}\]

where \(\gamma_\mu\) is the gyromagnetic ratio of the muon. The energy-integrated distribution of positrons from muons with polarization \(\mathcal P_\mu\) is

\[ \frac{dN}{d\cos\theta}\propto1+\frac{\mathcal P_\mu}{3}\cos\theta, \tag{17}\]

where \(\theta\) is the angle between the positron momentum and the muon spin.

  1. Find the momentum, kinetic energy and velocity of the muon in the pion rest frame.
  2. The pion spin is zero, and \(\mu^+\) and \(\nu_\mu\) fly in opposite directions. Show that their helicities are equal. What helicity does the \(\mu^+\) have if the \(\nu_\mu\) is left-handed?
  3. Why does the longitudinal polarization of the muon by itself already mean parity violation in pion decay?
  4. Find the ratio of the numbers of positrons emitted along and opposite to the muon spin for \(\mathcal P_\mu=1\).
  5. In the experiment of Garwin, Lederman and Weinrich the muons were stopped in graphite and their spin was rotated by a magnetic field. Find the precession frequency for \(B=5.0~\mathrm{mT}\) and the angle through which the spin turns during the time \(\tau_\mu\). Write down the time dependence \(N(t)\) of the counting rate of a fixed positron counter.

State the main physical conclusion: which two independent P-odd effects does this chain contain, and why did its result confirm the Wu experiment?

Exercise 2.10. What does “maximal” parity violation mean?

Use

\[ (\gamma^5)^2=1, \qquad \gamma^5\gamma^\mu=-\gamma^\mu\gamma^5, \qquad P_{L,R}=\frac{1\mp\gamma^5}{2}, \qquad P:\ \psi(t,\mathbf r)\to\gamma^0\psi(t,-\mathbf r). \tag{18}\]

Consider the general coupling of a fermion to a vector boson

\[ \bar\psi\gamma^\mu\left(g_V-g_A\gamma^5\right)\psi. \tag{19}\]

  1. Show that \(P_L^2=P_L\), \(P_R^2=P_R\), \(P_LP_R=0\), \(P_L+P_R=1\).

  2. Show that \(P_L\gamma^0=\gamma^0P_R\). Why does reflection turn the left-handed component of the field into the right-handed one?

  3. Write the coupling as \(\gamma^\mu(g_LP_L+g_RP_R)\) and express \(g_L\) and \(g_R\) in terms of \(g_V\) and \(g_A\).

  4. For ultrarelativistic fermions the left- and right-handed components interact with strengths \(g_L^2\) and \(g_R^2\). Show that the asymmetry is

    \[ \mathcal A_{LR}=\frac{g_L^2-g_R^2}{g_L^2+g_R^2} =\frac{2g_Vg_A}{g_V^2+g_A^2}, \qquad |\mathcal A_{LR}|\le1. \tag{20}\]

    Why does a purely vector or purely axial coupling conserve parity, while P-odd effects arise from the interference of \(V\) and \(A\)?

  5. Find \(g_V\), \(g_A\) and \(\mathcal A_{LR}\) for the charged current \(\bar\psi\gamma^\mu P_L\psi\).

  6. For the coupling of the electron to the \(Z\) boson \(g_V=-\tfrac12+2\sin^2\theta_W\), \(g_A=-\tfrac12\), \(\sin^2\theta_W=0.231\). Find \(\mathcal A_{LR}\).

State the main physical conclusion: in what sense is parity violation in the charged current maximal, and why does this statement not carry over automatically to the neutral current?

Exercises for discussion at the seminar

Seminar 02.1. What changes under reflection?

Fill in the table of transformations under spatial inversion \(\mathbf r\to-\mathbf r\) for the quantities

\[ \mathbf r,\quad \mathbf p,\quad \mathbf E,\quad \mathbf J,\quad \mathbf B. \tag{21}\]

Which of them are polar and which are axial vectors? Determine the parity of the quantities \(\mathbf p\cdot\mathbf J\), \(\mathbf E\cdot\mathbf p\) and \(\mathbf J\cdot\mathbf B\). Which of them can serve as an observable signature of parity violation?

Seminar 02.2. Asymmetry from event counts

The angular distribution of electrons from polarized nuclei has the form

\[ W(\theta)\propto 1+A_\beta P\beta\cos\theta. \tag{22}\]

Two identical detectors covering opposite hemispheres registered \(N_+=11\,200\) and \(N_-=8\,800\) events. The nuclear polarization is \(P=0.80\), and the electron velocity is \(\beta=0.75\).

  1. By integrating over the hemispheres, show that \((N_+-N_-)/(N_++N_-)=A_\beta P\beta/2\).
  2. Find \(A_\beta\).
  3. How will the result change if the nuclear polarization is reversed?

Seminar 02.3. How can a physical asymmetry be separated from an instrumental one?

Suppose that the asymmetries measured for the two polarization directions are

\[ A^\uparrow=A_{\mathrm{inst}}+A_{\mathrm{phys}}, \qquad A^\downarrow=A_{\mathrm{inst}}-A_{\mathrm{phys}}. \tag{23}\]

  1. Express \(A_{\mathrm{phys}}\) and \(A_{\mathrm{inst}}\) in terms of the two measured quantities.
  2. Find them for \(A^\uparrow=0.13\) and \(A^\downarrow=-0.09\).
  3. Why is reversing the polarization a more reliable check than comparing two detectors for only one field direction?

Seminar 02.4. Does a magnetic field violate parity?

The orientation of the nuclei is produced by the interaction

\[ U=-\boldsymbol\mu\cdot\mathbf B. \tag{24}\]

Show that \(\boldsymbol\mu\) and \(\mathbf B\) are axial vectors, and that the energy \(U\) does not change under spatial inversion. Why is the very preparation of a polarized target with a magnetic field not a violation of parity, whereas the correlation \(\mathbf J\cdot\mathbf p_e\) in the decay can reveal it?