Course Exercises

Introduction to Neutrino Physics

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Exercises

Exercise 1.1. How transparent is the Universe to neutrinos?

Estimate the neutrino interaction length in a medium with the mean baryon density of the Universe for two energies:

\[ E_\nu=1~\mathrm{MeV}, \qquad E_\nu=1~\mathrm{PeV}. \tag{1}\]

Use

\[ \lambda_\nu=\frac{1}{n\sigma}, \qquad n\simeq2.5\cdot10^{-7}~\mathrm{cm}^{-3}, \tag{2}\]

\[ \sigma_{\nu e}(1~\mathrm{MeV})\sim10^{-44}~\mathrm{cm}^2, \qquad \sigma_{\nu N}(1~\mathrm{PeV})\sim10^{-33}~\mathrm{cm}^2, \tag{3}\]

and the diameter of the observable Universe \(D_{\mathrm{obs}}\simeq8.8\cdot10^{28}~\mathrm{cm}\).

1. Compute \(\lambda_\nu\) and the ratio \(D_{\mathrm{obs}}/\lambda_\nu\) for both energies.
2. Interpret \(D_{\mathrm{obs}}/\lambda_\nu\) as an optical depth. What is the probability of at least one interaction for \(D_{\mathrm{obs}}\ll\lambda_\nu\)?
3. Why does this estimate require the density of ordinary matter rather than the total energy density of the Universe?
4. Explain why an increase of the cross section by eleven orders of magnitude still does not make a homogeneous Universe opaque to PeV neutrinos.

Exercise 1.2. Solar neutrino flux

The main chain of thermonuclear reactions in the Sun has the overall form

\[ 4p\longrightarrow{}^4\mathrm{He}+2e^++2\nu_e+Q, \qquad Q=26.7~\mathrm{MeV}. \tag{4}\]

Use

\[ L_\odot=3.83\cdot10^{26}~\mathrm{W}, \qquad r_\oplus=1~\mathrm{AU}, \qquad r_J=5.20~\mathrm{AU}, \tag{5}\]

\[ 1~\mathrm{AU}=1.496\cdot10^{11}~\mathrm{m}, \qquad 1~\mathrm{eV}=1.602\cdot10^{-19}~\mathrm{J}. \tag{6}\]

1. Estimate the number of completed reaction chains per second.
2. Find the neutrino production rate \(\dot N_\nu\).
3. Using \(\Phi_\nu=\dot N_\nu/(4\pi r^2)\), compute the flux at the orbits of the Earth and Jupiter.
4. Compare the flux at the Earth with \(\Phi_\nu^{\mathrm{exp}}\simeq6.6\cdot10^{10}~ \mathrm{cm}^{-2}\mathrm{s}^{-1}\).
5. In this estimate the whole energy \(Q\) was identified with the luminosity of the Sun. In which direction will the result change if the energy carried away by neutrinos is taken into account?
6. In the main branch of the pp chain each of the two neutrinos carries away about \(0.27~\mathrm{MeV}\) on average. What fraction of \(Q\) is this, and why can this correction be neglected in an order-of-magnitude estimate of the flux?

Exercise 1.3. How many light neutrinos exist?

The invisible decay width of the \(Z\) boson has been measured as

\[ \Gamma_{\mathrm{inv}}=499.0\pm1.5~\mathrm{MeV}. \tag{7}\]

The Standard Model predicts for a single light neutrino

\[ \Gamma(Z\to\nu\bar\nu)=167.21\pm0.02~\mathrm{MeV}. \tag{8}\]

1. Compute

   \[ N_\nu= \frac{\Gamma_{\mathrm{inv}}} {\Gamma(Z\to\nu\bar\nu)}. \tag{9}\]

2. Estimate the uncertainty of \(N_\nu\), assuming the two errors are independent.

3. Is the result consistent with three neutrino flavors?

4. Why does this measurement constrain the number only of those neutrinos for which the decay \(Z\to\nu\bar\nu\) is kinematically allowed?

5. Can there exist an additional neutrino that is not coupled to the \(Z\) boson? What does the word “sterile” then mean?

Exercise 1.4. Oscillations of atmospheric neutrinos

In the two-flavor approximation

\[ P(\nu_\mu\to\nu_\tau) = \sin^2 2\theta\, \sin^2\!\left( 1.267\, \frac{\Delta m^2[\mathrm{eV}^2]L[\mathrm{km}]} {E_\nu[\mathrm{GeV}]} \right). \tag{10}\]

Set

\[ E_\nu=1~\mathrm{GeV}, \qquad \sin^2 2\theta=1, \qquad \Delta m^2=2.5\cdot10^{-3}~\mathrm{eV}^2. \tag{11}\]

1. Find the distance to the first transition maximum.
2. Compare the probabilities for a neutrino coming from above (\(L=20~\mathrm{km}\)) and one that has passed through the Earth (\(L=12\,742~\mathrm{km}\)).
3. Plot \(P(\nu_\mu\to\nu_\tau)\) as a function of \(x=L/D_\oplus\) for \(0\leq x\leq1\).
4. On the same plot, plot \(P(\nu_\mu\to\nu_\mu)\) and check unitarity.
5. How will the first maximum shift if the energy is increased by a factor of ten?
6. How many full periods of the probability fit into the Earth’s diameter \(D_\oplus=12\,742~\mathrm{km}\) at \(E_\nu=1~\mathrm{GeV}\)?

Exercise 1.5. Minimal masses for the two level orderings

Use the measured mass-squared differences

\[ \Delta m_{21}^2=7.4\cdot10^{-5}~\mathrm{eV}^2, \qquad |\Delta m_{31}^2|=2.5\cdot10^{-3}~\mathrm{eV}^2. \tag{12}\]

1. For the normal ordering, set \(m_1=0\) and find \(m_2\), \(m_3\), and \(\sum_i m_i\).
2. For the inverted ordering, set \(m_3=0\) and find \(m_1\), \(m_2\), and \(\sum_i m_i\).
3. Why do oscillations determine the mass-squared differences but do not fix the mass of the lightest state itself?
4. Repeat the calculation for \(m_{\mathrm{lightest}}=0.05~\mathrm{eV}\). How distinguishable are the two orderings in this case by the mass values themselves?
5. Explain the meaning of the terms “hierarchical” and “quasi-degenerate” spectrum.

Exercise 1.6. Dirac or Majorana neutrino?

The cross sections for neutrino and antineutrino scattering on nucleons differ noticeably. At energies of tens of GeV, for the charged current,

\[ \frac{\sigma_{\nu N}}{E_\nu}\simeq0.68\cdot10^{-38}~ \frac{\mathrm{cm}^2}{\mathrm{GeV}}, \qquad \frac{\sigma_{\bar\nu N}}{E_\nu}\simeq0.34\cdot10^{-38}~ \frac{\mathrm{cm}^2}{\mathrm{GeV}}. \tag{13}\]

The weak charged current contains only the left-handed chiral components of the fields. A state of a particle of mass \(m\) with energy \(E\gg m\) and definite helicity contains a component of the “wrong” chirality with probability

\[ w\simeq\left(\frac{m}{2E}\right)^2. \tag{14}\]

1. Does the difference between the cross sections imply that the neutrino is a Dirac particle?
2. The decay \(\pi^+\to\mu^++\nu_\mu\) produces a neutrino with negative helicity, the decay \(\pi^-\to\mu^-+\bar\nu_\mu\) one with positive helicity. Describe both beams in the language of a Majorana neutrino, for which \(\nu=\bar\nu\). What then distinguishes a “neutrino” from an “antineutrino”?
3. Explain why the left-handed chiral current distinguishes these two states even when \(\nu=\bar\nu\). Why is the difference in cross sections a consequence of helicity rather than of a difference between particle and antiparticle?
4. Estimate \(w\) for \(m=0.1~\mathrm{eV}\) at \(E=1~\mathrm{MeV}\) and \(E=1~\mathrm{GeV}\).
5. Which small parameter suppresses the difference between the predictions for a Dirac and a Majorana neutrino in ordinary scattering experiments? Why does a test of the Majorana nature require a process whose amplitude is proportional to the neutrino mass, for example neutrinoless double \(\beta\) decay?

State the main physical conclusion: what does the difference between the \(\nu\) and \(\bar\nu\) cross sections actually measure?

Exercise 1.7. Where do the cross sections \(10^{-44}\) and \(10^{-33}~\mathrm{cm}^2\) come from?

The lecture used the estimates \(\sigma_{\nu e}(1~\mathrm{MeV})\sim10^{-44}~\mathrm{cm}^2\) and \(\sigma_{\nu N}(1~\mathrm{PeV})\sim10^{-33}~\mathrm{cm}^2\). Obtain their order of magnitude from the Fermi constant. For a point-like four-fermion interaction at \(s\ll M_W^2\)

\[ \sigma\simeq\frac{G_F^2s}{\pi}, \qquad s\simeq2mE_\nu, \tag{15}\]

where \(m\) is the mass of the target at rest. Use

\[ G_F=1.166\cdot10^{-5}~\mathrm{GeV}^{-2}, \qquad (\hbar c)^2=0.3894\cdot10^{-27}~\mathrm{GeV}^2\,\mathrm{cm}^2, \tag{16}\]

\[ m_e=0.511~\mathrm{MeV}, \qquad m_N=0.939~\mathrm{GeV}, \qquad M_W=80.4~\mathrm{GeV}. \tag{17}\]

1. Check that for \(\hbar=c=1\) the quantity \(G_F^2s\) has the dimension of an area. Why is the factor \((\hbar c)^2\) needed when converting to centimeters?
2. Compute \(\sigma\) for neutrino scattering on an electron at \(E_\nu=1~\mathrm{MeV}\) and compare it with \(10^{-44}~\mathrm{cm}^2\).
3. Compute the point-like estimate for a nucleon at \(E_\nu=1~\mathrm{GeV}\) and compare it with the measured \(\sigma_{\nu N}/E_\nu\simeq0.68\cdot10^{-38}~\mathrm{cm}^2/\mathrm{GeV}\). Why is the point-like estimate too large? Take into account that the neutrino scatters on a quark, which carries only a fraction of the nucleon momentum.
4. Extend the linear growth \(\sigma_{\nu N}\propto E_\nu\) to \(1~\mathrm{PeV}\). Compare \(s\) at this energy with \(M_W^2\). Why does the growth of the cross section slow down, and why is the actual value close to \(10^{-33}~\mathrm{cm}^2\)?
5. For \(\sigma=10^{-33}~\mathrm{cm}^2\) and \(n_N=3.3\cdot10^{24}~\mathrm{cm}^{-3}\), find the interaction length and the probability of crossing the Earth along a diameter \(D_\oplus=12\,742~\mathrm{km}\) without interacting, assuming a uniform density.
6. At what energy, in the linear approximation, does the interaction length become equal to \(D_\oplus\)?

State the main physical conclusion: which two factors make neutrino cross sections small, and what limits their growth?

Exercise 1.8. How neutral is the neutrino?

The electric charge of the neutrino is constrained in two independent ways. The first relies on charge conservation in neutron \(\beta\) decay

\[ n\to p+e^-+\bar\nu_e \tag{18}\]

and on experiments on the neutrality of matter: \(|q_n|\lesssim10^{-21}e\) and \(|q_p+q_e|\lesssim10^{-21}e\). The second relies on neutrino scattering on electrons. For an electron recoil energy \(T\ll E_\nu\) the charge and weak-interaction contributions are

\[ \left(\frac{d\sigma}{dT}\right)_q \simeq\frac{2\pi\alpha^2}{m_eT^2}\left(\frac{q_\nu}{e}\right)^2, \qquad \left(\frac{d\sigma}{dT}\right)_{\mathrm{weak}} \sim\frac{G_F^2m_e}{\pi}, \tag{19}\]

where \(\alpha=1/137\), \(m_e=0.511~\mathrm{MeV}\), \(G_F=1.166\cdot10^{-5}~\mathrm{GeV}^{-2}\).

1. From charge conservation, derive \(|q_{\bar\nu}|\le|q_n|+|q_p+q_e|\) and a numerical estimate. What assumption about the charges of \(\nu\) and \(\bar\nu\) is needed to carry it over to the neutrino?
2. Check that both expressions have the same dimension (\(\mathrm{GeV}^{-3}\) for \(\hbar=c=1\)).
3. Find the value of \(|q_\nu|\) at which the charge contribution equals the weak one, for \(T=1~\mathrm{keV}\) and \(T=10~\mathrm{keV}\).
4. How does the sensitivity of an experiment to \(q_\nu\) depend on the threshold \(T\)? Why are low-threshold experiments advantageous?
5. Which threshold corresponds to the bound \(|q_\nu|\lesssim10^{-12}e\) quoted in the lecture?

State the main physical conclusion: why is the indirect bound from the neutrality of matter many orders of magnitude stronger than the direct one, and on which assumption does it rest?

Exercise 1.9. Why are there twice as many atmospheric \(\nu_\mu\) as \(\nu_e\)?

Atmospheric neutrinos are produced in the chains

\[ \pi^+\to\mu^++\nu_\mu, \quad \mu^+\to e^++\nu_e+\bar\nu_\mu, \qquad \pi^-\to\mu^-+\bar\nu_\mu, \quad \mu^-\to e^-+\bar\nu_e+\nu_\mu. \tag{20}\]

Pions are produced and decay at an altitude of about \(h\simeq15~\mathrm{km}\). For the muon \(m_\mu=105.66~\mathrm{MeV}\) and \(c\tau_\mu=658.6~\mathrm{m}\); for \(E_\mu\gg m_\mu\) the decay length is

\[ L_\mu\simeq\frac{E_\mu}{m_\mu}\,c\tau_\mu. \tag{21}\]

To estimate the oscillations, use the two-flavor probability \(P(\nu_\mu\to\nu_\tau)=\sin^22\theta\, \sin^2(1.267\,\Delta m^2[\mathrm{eV}^2]L[\mathrm{km}]/E_\nu[\mathrm{GeV}])\) with \(\sin^22\theta=1\) and \(\Delta m^2=2.5\cdot10^{-3}~\mathrm{eV}^2\).

1. Count the numbers of \(\nu_\mu+\bar\nu_\mu\) and \(\nu_e+\bar\nu_e\) per complete chain and obtain the ratio \(R=(\nu_\mu+\bar\nu_\mu)/(\nu_e+\bar\nu_e)\).
2. At what energy \(E_\mu\) does the decay length of a vertical muon equal \(h\)? How does \(R\) change at higher energies, and why?
3. Find the probability of \(\nu_\mu\to\nu_\tau\) at \(E_\nu=1~\mathrm{GeV}\) for neutrinos coming from above (\(L=20~\mathrm{km}\)). Why, for neutrinos that have crossed the Earth (\(L\gtrsim5000~\mathrm{km}\)), should one take the average \(\langle\sin^2\rangle=1/2\) given a finite energy resolution?
4. Find \(R\) for neutrinos from above and from below, assuming that the \(\nu_e\) flux does not change. Taking roughly half of the events to come from above without oscillations and half from below, estimate the averaged \(R\) and the double ratio \(R_{\mathrm{meas}}/R_{\mathrm{exp}}\).
5. The absolute flux of atmospheric neutrinos is predicted to about 20 %, while the ratio \(R\) is predicted to a few percent. Why, therefore, does one measure the ratio and its dependence on direction rather than the total flux?

State the main physical conclusion: which observation distinguishes oscillations from an error in the flux calculation?

Exercise 1.10. Why does CP violation need three flavors?

The transition probability in vacuum is

\[ P(\nu_\alpha\to\nu_\beta)= \Bigl|\sum_i U_{\alpha i}^*U_{\beta i}\, e^{-im_i^2L/2E}\Bigr|^2, \tag{22}\]

and for antineutrinos \(U\) is replaced by \(U^*\) in this formula.

1. Show that for a real matrix \(U\) \(P(\nu_\alpha\to\nu_\beta)=P(\bar\nu_\alpha\to\bar\nu_\beta)\).
2. How many real parameters does an \(n\times n\) unitary matrix contain? How many of them can be regarded as rotation angles (as for an orthogonal matrix), and how many as phases?
3. The phases of the fields of \(n\) charged leptons and \(n\) Dirac neutrinos can be redefined: \(U\to D_\ell UD_\nu\), where \(D_\ell\) and \(D_\nu\) are diagonal phase matrices. Explain why only \(2n-1\) phases are removed in this way, and show that \((n-1)(n-2)/2\) physical phases remain. How many are there for \(n=2\) and \(n=3\)?
4. In the standard PMNS parametrization the phase enters only in the combination \(s_{13}e^{\pm i\delta_{\mathrm{CP}}}\). Why is the phase \(\delta_{\mathrm{CP}}\) unobservable for \(\theta_{13}=0\)? What do the values \(\delta_{\mathrm{CP}}=0\) and \(\pi\) give?
5. For Majorana neutrinos the phases of the fields \(\nu_i\) cannot be redefined, and two more phases remain in \(U\): \(U\to U\,\mathrm{diag}(1,e^{i\alpha},e^{i\beta})\). Show that they drop out of the oscillation probability.

State the main physical conclusion: which conditions on the number of flavors and on the mixing angles are necessary for CP violation in oscillations, and why do oscillations not distinguish Dirac from Majorana neutrinos?

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Exercises for discussion in the seminar

Seminar 01.1. How many neutrinos pass through a person?

The solar neutrino flux at the Earth is \(\Phi_\nu=6.6\cdot10^{10}~\mathrm{cm}^{-2}\mathrm{s}^{-1}\). Take the cross-sectional area of a person to be \(0.70~\mathrm{m}^2\), the thickness of the body along the beam to be \(0.30~\mathrm{m}\), the electron density of water to be \(n_e=3.3\cdot10^{29}~\mathrm{m}^{-3}\), and the characteristic interaction cross section to be \(\sigma=10^{-48}~\mathrm{m}^2\).

1. How many solar neutrinos pass through a person per second?
2. Estimate the interaction probability for a single neutrino as \(n_e\sigma l\).
3. How often does an interaction occur under these assumptions? Why does the enormous neutrino flux not make them easy to detect?

Seminar 01.2. Charged or neutral current?

For each reaction, determine whether it proceeds through the charged or the neutral weak current:

\[ \nu_e+n\to p+e^-, \qquad \bar\nu_e+p\to n+e^+, \tag{23}\]

\[ \nu_\mu+e^-\to\nu_\mu+e^-, \qquad \nu+N\to\nu+N. \tag{24}\]

Which reactions directly indicate the flavor of the incoming neutrino, and which merely reveal the fact of its interaction? Check the conservation of electric charge in each reaction.

Seminar 01.3. Why do oscillations not measure the absolute mass?

The interference phase of two mass states has the form

\[ \Delta\phi_{ij}=\frac{\Delta m_{ij}^2L}{2E}. \tag{25}\]

1. Show that the oscillation probabilities do not change under the simultaneous replacement \(m_i^2\to m_i^2+C\) for all \(i\).
2. What information about the masses do oscillations nevertheless provide?
3. Which combination of masses is measured from the shape of the endpoint of the \(\beta\) spectrum?

Seminar 01.4. Which experiment answers which question?

Match each observation to a physical question:

• neutrino oscillations;
• the endpoint of the \(\beta\) spectrum;
• the invisible width of the \(Z\) boson;
• the search for neutrinoless double \(\beta\) decay;
• neutrino scattering on matter.

Questions: mass-squared differences and mixing; the absolute mass scale; the number of light active neutrinos; lepton number violation and the Majorana nature; the structure of the weak interaction. Explain why none of these experiments replaces all the others.

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

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

\[ \theta^+\to\pi^++\pi^0, \qquad \tau^+\to\pi^++\pi^++\pi^-. \tag{26}\]

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

\[ W(\theta)=1+a\cos\theta, \tag{27}\]

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

   \[ N_+\propto1+\frac a2, \qquad N_-\propto1-\frac a2. \tag{28}\]

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

   \[ \mathcal A=\frac{N_+-N_-}{N_++N_-}. \tag{29}\]

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

\[ \mu_N=3.15\cdot10^{-8}~\mathrm{eV/T}, \qquad k_B=8.617\cdot10^{-5}~\mathrm{eV/K}. \tag{30}\]

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

\[ \mathcal P= \frac{N_+-N_-}{N_++N_-} =\tanh\!\left(\frac{\mu B}{k_BT}\right). \tag{31}\]

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

\[ x=\frac{\mu B}{k_BT}. \tag{32}\]

1. Show that the partition function is

   \[ Z_N=(2\cosh x)^N. \tag{33}\]

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

   \[ S=Nk_B\left[\ln(2\cosh x)-x\tanh x\right]. \tag{34}\]

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

4. The system starts at

   \[ T_i=1.2~\mathrm{K}, \qquad B_i=2.0~\mathrm{T} \tag{35}\]

   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

\[ \mu_B=\frac{e\hbar}{2m_e}=5.79\cdot10^{-5}~\mathrm{eV/T}, \qquad \mu_N=\frac{e\hbar}{2m_p}=3.15\cdot10^{-8}~\mathrm{eV/T}, \qquad k_B=8.617\cdot10^{-5}~\mathrm{eV/K}. \tag{36}\]

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 \(\mathbf r\to-\mathbf r\)

\[ \vartheta\to\pi-\vartheta, \qquad \varphi\to\varphi+\pi, \qquad Y_{lm}\to(-1)^lY_{lm}. \tag{37}\]

The measured lifetimes of the levels are

\[ \tau(2p)=1.60~\mathrm{ns}, \qquad \tau(2s)=0.122~\mathrm{s}. \tag{38}\]

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

\[ {}^{60}\mathrm{Co}(5^+)\to{}^{60}\mathrm{Ni}^*(4^+)+e^-+\bar\nu_e, \qquad T_{\max}=0.318~\mathrm{MeV}, \qquad m_e=0.511~\mathrm{MeV}, \tag{39}\]

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:

\[ \langle\boldsymbol\sigma_e\rangle=-\frac{v}{c}\,\hat{\mathbf p}_e, \qquad \langle\boldsymbol\sigma_{\bar\nu}\rangle=+\hat{\mathbf p}_{\bar\nu}. \tag{40}\]

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

\[ m_\pi=139.57~\mathrm{MeV}, \qquad m_\mu=105.66~\mathrm{MeV}, \qquad \tau_\mu=2.197~\mathrm{\mu s}, \qquad \frac{\gamma_\mu}{2\pi}=135.5~\mathrm{MHz/T}, \tag{41}\]

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

\[ \frac{dN}{d\cos\theta}\propto1+\frac{\mathcal P_\mu}{3}\cos\theta, \tag{42}\]

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

\[ (\gamma^5)^2=1, \qquad \gamma^5\gamma^\mu=-\gamma^\mu\gamma^5, \qquad P_{L,R}=\frac{1\mp\gamma^5}{2}, \qquad P:\ \psi(t,\mathbf r)\to\gamma^0\psi(t,-\mathbf r). \tag{43}\]

Consider the general coupling of a fermion to a vector boson

\[ \bar\psi\gamma^\mu\left(g_V-g_A\gamma^5\right)\psi. \tag{44}\]

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

   \[ \mathcal A_{LR}=\frac{g_L^2-g_R^2}{g_L^2+g_R^2} =\frac{2g_Vg_A}{g_V^2+g_A^2}, \qquad |\mathcal A_{LR}|\le1. \tag{45}\]

   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

\[ \mathbf r,\quad \mathbf p,\quad \mathbf E,\quad \mathbf J,\quad \mathbf B. \tag{46}\]

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

\[ W(\theta)\propto 1+A_\beta P\beta\cos\theta. \tag{47}\]

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

\[ A^\uparrow=A_{\mathrm{inst}}+A_{\mathrm{phys}}, \qquad A^\downarrow=A_{\mathrm{inst}}-A_{\mathrm{phys}}. \tag{48}\]

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

\[ U=-\boldsymbol\mu\cdot\mathbf B. \tag{49}\]

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?

---

Exercise 3.1. Nuclear recoil in the Goldhaber experiment

In the Goldhaber experiment, electron capture leads to the reaction

\[ {}^{152m}\mathrm{Eu}+e^- \longrightarrow {}^{152}\mathrm{Sm}^*+\nu_e. \tag{50}\]

Take

\[ E_\nu\simeq1~\mathrm{MeV}, \qquad M_{B^*}\simeq152~u, \qquad uc^2=931.5~\mathrm{MeV}. \tag{51}\]

1. Neglecting the momentum of the captured electron, show that \(p_{B^*}\simeq p_\nu\simeq E_\nu/c\).

2. Calculate the recoil energy

   \[ T_{B^*}\simeq\frac{E_\nu^2}{2M_{B^*}c^2} \tag{52}\]

   and the nuclear velocity \(v_{B^*}=p_{B^*}/M_{B^*}\).

3. If the nucleus loses the direction of recoil over a length \(\ell=1\)–\(5~\text{Å}\), estimate \(\tau_{\mathrm{stop}}\sim\ell/v_{B^*}\).

4. Compare the result with the lifetime of the excited state \(\tau(B^*)\simeq0.03~\mathrm{ps}\).

5. Why is it important for the measurement of helicity that the \(\gamma\)-quantum be emitted before the memory of the recoil direction is lost?

Exercise 3.2. Helicity and chirality of a massive neutrino

A neutrino produced by a left-chiral weak current contains, at \(m_\nu\ne0\), a small component of opposite helicity:

\[ P_{\mathrm{opp}}=\frac{1-\beta_\nu}{2}, \qquad \beta_\nu=\frac{p_\nu}{E_\nu}. \tag{53}\]

1. Using \(E_\nu^2=p_\nu^2+m_\nu^2\), show that for \(E_\nu\gg m_\nu\)

   \[ P_{\mathrm{opp}}\simeq \left(\frac{m_\nu}{2E_\nu}\right)^2. \tag{54}\]

2. For \(E_\nu=1~\mathrm{MeV}\) and \(m_\nu<0.45~\mathrm{eV}\) obtain the upper bound \(P_{\mathrm{opp}}\).

3. Why do helicity and chirality practically coincide for an ultrarelativistic neutrino, even though they are different concepts?

4. Which of these characteristics can change sign upon a transition to another inertial frame?

5. How does the mass term of the Dirac equation couple the left and right chiral components?

Exercise 3.3. Spin projections in electron capture

Consider the allowed electron capture

\[ A(J=0)+e^- \longrightarrow B^*(J=1)+\nu_e. \tag{55}\]

Direct the axis \(z\) along the neutrino momentum; neglect the orbital angular momentum of the final state.

1. Write down the conservation law for the projection of the total angular momentum:

   \[ m_e=m_{B^*}+m_\nu. \tag{56}\]

2. List the allowed values of \(m_e=\pm\tfrac12\), \(m_\nu=\pm\tfrac12\) and \(m_{B^*}=-1,0,+1\).

3. Construct a table of the allowed triples \((m_e,m_{B^*},m_\nu)\).

4. For which values of \(m_{B^*}\) does the projection of the nuclear spin unambiguously fix the neutrino helicity? Why is the state \(m_{B^*}=0\) unsuitable as a label?

5. Why does the relation lose its unambiguous meaning if the quantization axis is chosen not along the neutrino momentum?

Exercise 3.4. Doppler selection of \(\gamma\)-quanta

An excited nucleus \(B^*\) moves with velocity \(v\ll c\) and emits, in its own rest frame, a photon of energy \(E_\gamma\). The angle \(\theta\) is measured between the direction of the photon and the velocity of the nucleus.

1. From the transformation

   \[ E_\gamma^{\mathrm{lab}} =\gamma E_\gamma(1+\beta\cos\theta) \tag{57}\]

   obtain, to first order in \(v/c\),

   \[ E_\gamma^{\mathrm{lab}} \simeq E_\gamma \left(1+\frac vc\cos\theta\right). \tag{58}\]

2. Find \(E_{\max}\), \(E_{\min}\) and the width \(E_{\max}-E_{\min}\).

3. Using the velocity from Exercise 03.1, estimate the relative Doppler shift.

4. In both emission and absorption of a photon a free nucleus of mass \(M\) receives the recoil energy

   \[ E_R\simeq\frac{E_\gamma^2}{2Mc^2}, \tag{59}\]

   so the emission and absorption lines are separated by an interval \(2E_R\). Let the nucleus move with the recoil velocity after electron capture, \(v\simeq E_\nu/(Mc)\), with \(E_\nu\simeq E_\gamma\). Show that a photon emitted along the recoil gains an extra energy \(\simeq2E_R\), that is, exactly the amount missing for resonance.

5. Why does resonant absorption select a narrow range of angles?

6. Explain how energy selection turns into selection of the direction of the \(\gamma\)-quantum relative to the nuclear recoil.

Exercise 3.5. Can one overtake a neutrino?

A massive neutrino moves along the axis \(z\) with energy \(E_\nu\) and momentum \(p_\nu\). An observer moves in the same direction with velocity \(u\). Under a longitudinal boost

\[ p_\nu'=\gamma_u \left(p_\nu-\frac{uE_\nu}{c^2}\right). \tag{60}\]

1. Obtain the condition under which \(p_\nu'\) changes sign. Check the same result with the velocity-addition law \(v_\nu'=(v_\nu-u)/(1-uv_\nu/c^2)\).

2. Show that for \(E_\nu\gg m_\nu c^2\)

   \[ 1-\beta_\nu\simeq \frac{m_\nu^2c^4}{2E_\nu^2}. \tag{61}\]

3. For \(E_\nu=1~\mathrm{MeV}\) and \(m_\nu=0.10~\mathrm{eV}/c^2\) estimate by how much the observer’s velocity must differ from \(c\).

4. Estimate the minimum Lorentz factor of such an observer and compare it with \(\gamma_\nu=E_\nu/(m_\nu c^2)\).

5. Why does the helicity change sign after the overtaking? Why is the same thought experiment impossible for a strictly massless particle?

6. Show directly that the spatial inversion \(\mathbf p\to-\mathbf p\), \(\mathbf S\to\mathbf S\) also changes the sign of \(h\). How does this differ from overtaking: which of the two transformations describes the same neutrino seen by another observer, and which turns it into a different state?

Exercise 3.6. From event counts to helicity

After the resonant selection the \(\gamma\)-quanta pass through magnetized iron. In equal times, for the two field directions, the counts are

\[ N_+=0.988\times10^6, \qquad N_-=1.012\times10^6. \tag{62}\]

The count asymmetry

\[ \mathcal A=\frac{N_+-N_-}{N_++N_-} \tag{63}\]

is related to the circular polarization of the photons by

\[ \mathcal A\simeq k\,h_\gamma, \qquad k=f_{\mathrm{res}}P_ea_\gamma, \tag{64}\]

where \(f_{\mathrm{res}}\) is the fraction of resonantly selected photons, \(P_e\) is the polarization of the electrons in the iron, and \(a_\gamma\) is the analyzing power of the setup. The sign of \(k\) is fixed by which field direction is called “\(+\)”; the calibration gives \(k=0.012\pm0.003\).

1. Compute \(\mathcal A\).

2. Treat each detected photon as an independent trial that falls into the count \(N_+\) with probability \(p=N_+/(N_++N_-)\). Show that

   \[ \sigma_{\mathcal A}\simeq \sqrt{\frac{1-\mathcal A^2}{N_++N_-}}, \tag{65}\]

   and compute \(\sigma_{\mathcal A}\).

3. Find \(h_\gamma\) and its statistical uncertainty. Why, in the geometry selected by the resonance, is this the neutrino helicity?

4. Estimate the systematic uncertainty of \(h_\gamma\) caused by the uncertainty of \(k\). Which of the two uncertainties dominates? Compare the scale with the published result of Goldhaber, Grodzins and Sunyar \(h(\nu_e)=-1.0\pm0.3\).

5. Which slow changes of the setup are suppressed by frequent reversal of the field (drift of the source activity, of the detector efficiency)? Which effects change sign together with the field and therefore require a separate check (for example, the action of the stray magnetic field on the photomultiplier)?

State the main physical conclusion: what does such an experiment determine reliably, the sign of the helicity or its magnitude, and why is this enough to conclude that the weak current is left-handed?

Exercise 3.7. Why does a free nucleus not absorb its own photon?

The excited level of \({}^{152}\mathrm{Sm}^*\) lies \(\Delta E=0.96~\mathrm{MeV}\) above the ground state. Take

\[ Mc^2\simeq1.42\times10^5~\mathrm{MeV}, \qquad \tau\simeq0.03~\mathrm{ps}, \qquad \hbar=6.58\times10^{-16}~\mathrm{eV\cdot s}, \qquad k_B=8.62\times10^{-5}~\mathrm{eV/K}. \tag{66}\]

1. From energy and momentum conservation, show that for a nucleus at rest

   \[ E_\gamma^{\mathrm{em}}\simeq\Delta E-\frac{(\Delta E)^2}{2Mc^2}, \qquad E_\gamma^{\mathrm{abs}}\simeq\Delta E+\frac{(\Delta E)^2}{2Mc^2}. \tag{67}\]

   Find the gap \(\delta=E_\gamma^{\mathrm{abs}}-E_\gamma^{\mathrm{em}}\) in electronvolts.

2. Find the natural width of the level \(\Gamma=\hbar/\tau\). By what factor is the gap larger than the width?

3. Thermal motion of the nucleus with velocity \(v_x\) along the photon direction shifts the photon energy by \(\Delta E\,v_x/c\). Show that the Doppler width of the line is

   \[ \sigma_D\simeq\Delta E\sqrt{\frac{k_BT}{Mc^2}}, \tag{68}\]

   and compute it at \(T=300~\mathrm K\). Compare \(\sigma_D\) with \(\Gamma\) and with \(\delta\). At what temperature would \(\sigma_D\) equal half of the gap? Can the gap be closed by heating?

4. After electron capture the nucleus moves with velocity \(v\simeq E_\nu/(Mc)\), where \(E_\nu\simeq\Delta E\). A photon emitted at an angle \(\theta\) to the recoil has the energy \(E_\gamma^{\mathrm{em}}(1+v\cos\theta/c)\). Assuming that resonance occurs as long as the energy differs from \(E_\gamma^{\mathrm{abs}}\) by no more than \(\sigma_D\), estimate the allowed \(1-\cos\theta\) and the opening angle of the cone. What fraction of all emission directions does this cone represent?

5. Which width, \(\Gamma\) or \(\sigma_D\), determines the angular resolution of the resonant filter? Why is the fraction of resonantly scattered photons in the experiment small?

State the main physical conclusion: why is a gap of a few electronvolts closed only by the directed motion of the nucleus after the neutrino is emitted, and why does exactly this make the resonance a direction filter?

Exercise 3.8. Helicity and chirality in the chiral basis

Set \(\hbar=c=1\). In the chiral basis

\[ \gamma^0= \begin{pmatrix}0&I\\I&0\end{pmatrix}, \qquad \gamma^i= \begin{pmatrix}0&\sigma_i\\-\sigma_i&0\end{pmatrix}, \qquad \gamma^5= \begin{pmatrix}-I&0\\0&I\end{pmatrix}. \tag{69}\]

Consider a spinor with momentum \(\mathbf p\), \(p=|\mathbf p|\), and helicity \(h=\pm1\):

\[ u_h(\mathbf p)= \begin{pmatrix} \sqrt{E-hp}\;\xi_h\\[3pt] \sqrt{E+hp}\;\xi_h \end{pmatrix}, \qquad (\boldsymbol\sigma\cdot\widehat{\mathbf p})\,\xi_h=h\,\xi_h, \qquad \xi_h^\dagger\xi_h=1. \tag{70}\]

1. Show that

   \[ \gamma^\mu p_\mu= \begin{pmatrix} 0&E-\boldsymbol\sigma\cdot\mathbf p\\ E+\boldsymbol\sigma\cdot\mathbf p&0 \end{pmatrix}, \tag{71}\]

   and, using \(\sqrt{E-p}\,\sqrt{E+p}=m\), check that \(u_h\) satisfies the equation \((\gamma^\mu p_\mu-m)u_h=0\).

2. Write down the projectors \(P_{L,R}=(1\mp\gamma^5)/2\). Which components of the spinor do they select? Check that \(u_h^\dagger u_h=2E\).

3. Obtain the fractions of the chiral components

   \[ f_{L,R}(h)=\frac{\|P_{L,R}u_h\|^2}{\|u_h\|^2} =\frac{1\mp h\beta}{2}, \qquad \beta=\frac pE. \tag{72}\]

4. Show that the amplitude of the left-handed component in the state \(h=+1\) is \(\sqrt{E-p}=m/\sqrt{E+p}\), that is, proportional to the mass. The muon neutrino from the decay of a pion at rest has \(E\simeq29.8~\mathrm{MeV}\). Estimate \(f_L(+1)\) for \(m=0.05~\mathrm{eV}\).

5. Take the limit \(m\to0\): which chirality do \(u_-\) and \(u_+\) have? Why do the fractions \(f_{L,R}(h)\) not depend on the choice of basis for the Dirac matrices?

State the main physical conclusion: why do helicity and chirality coincide for a massless particle, while for a massive one a state of definite helicity contains both chiralities?

Exercise 3.9. Relic neutrinos: helicity is conserved, chirality is not

Relic neutrinos stopped interacting with the plasma of the early Universe at a temperature of about \(1~\mathrm{MeV}\), when their typical momentum was of order \(3~\mathrm{MeV}\). Today their temperature and mean momentum are

\[ T_\nu\simeq1.95~\mathrm K, \qquad \langle p\rangle\simeq3.15\,k_BT_\nu, \qquad k_B=8.62\times10^{-5}~\mathrm{eV/K}. \tag{73}\]

Take the mass \(m=0.05~\mathrm{eV}\) and set \(\hbar=c=1\).

1. The free Dirac Hamiltonian is \(H=\boldsymbol\alpha\cdot\mathbf p+\beta m\), where \(\alpha_i=\gamma^0\gamma^i\), \(\beta=\gamma^0\). Using the fact that \(\gamma^5\) anticommutes with all \(\gamma^\mu\), show that

   \[ [H,\gamma^5]=2m\,\beta\gamma^5, \tag{74}\]

   whereas the helicity operator \(\boldsymbol\Sigma\cdot\widehat{\mathbf p}\) commutes with \(H\). Which of the two quantities is conserved in the free motion of a massive particle?

2. Estimate the probability of the “wrong” helicity \(h=+1\) at decoupling, \(P\simeq(m/2E)^2\) for \(E\simeq3~\mathrm{MeV}\).

3. As the Universe expands, the momentum of a free neutrino decreases in inverse proportion to the scale factor, while its direction does not change. Why is the helicity conserved in this process? Find today’s \(\langle p\rangle\), \(\beta=p/E\) and the velocity in km/s.

4. For a state with \(h=-1\) the fractions of the chiral components are \(f_{L,R}=(1\pm\beta)/2\). Compute them today. Compare them with the fractions at decoupling.

5. The Solar System moves relative to the cosmic microwave background with a velocity of about \(370~\mathrm{km/s}\); assume that the relic neutrinos are on average at rest in the same frame. Does the Earth overtake a typical relic neutrino? For which momenta (in units of \(k_BT_\nu\)) is the sign of the helicity of a neutrino moving along the Earth’s motion in the Earth frame opposite to its sign in the frame of the relic gas?

State the main physical conclusion: what does the word “left-handed” mean for a nonrelativistic relic neutrino, helicity or chirality, and which of these characteristics does the weak interaction see when the neutrino is detected?

Exercise 3.10. Pion decay: the muon inherits the neutrino helicity

A pion of spin \(0\) at rest decays through the channels

\[ \pi^+\to\mu^++\nu_\mu, \qquad \pi^+\to e^++\nu_e. \tag{75}\]

Take \(m_\pi=139.57~\mathrm{MeV}\), \(m_\mu=105.66~\mathrm{MeV}\), \(m_e=0.511~\mathrm{MeV}\); neglect the neutrino mass and set \(c=1\).

1. For both channels, find the momentum \(p_\ell\), the total energy \(E_\ell\), the kinetic energy and the velocity \(\beta_\ell\) of the charged lepton.

2. The charged lepton and the neutrino fly apart in opposite directions, and the orbital angular momentum along the decay axis is zero. Show that \(h(\ell^+)=h(\nu)\). What is the helicity of the \(\mu^+\) if \(h(\nu_\mu)=-1\)? Why are muons from the decay of pions at rest fully polarized?

3. The left-chiral charged current produces an antilepton in a right-chiral state; for \(m_\ell=0\) this is the state with \(h=+1\). Why would the decay into a massless lepton be forbidden? For a massive lepton the fraction of the right-chiral component in the state \(h=-1\) is \((1-\beta_\ell)/2\). Show that

   \[ 1-\beta_\ell=\frac{m_\ell^2}{m_\pi E_\ell}, \tag{76}\]

   and compute this fraction for the muon and for the positron.

4. A full calculation with the left-handed current gives

   \[ \Gamma(\pi^+\to\ell^+\nu_\ell)\propto m_\ell^2\left(m_\pi^2-m_\ell^2\right)^2. \tag{77}\]

   Show that \(m_\ell^2(m_\pi^2-m_\ell^2)^2=4m_\pi^3\,p_\ell^2\,(E_\ell-p_\ell)\), and identify the helicity-suppression factor in this expression. Compute \(R=\Gamma(\pi\to e\nu)/\Gamma(\pi\to\mu\nu)\) and compare it with the measured value \(R\simeq1.23\times10^{-4}\) (the small difference comes from radiative corrections). Which channel would look preferable if only the momentum of the outgoing lepton were taken into account?

5. The direction of the positron in the decay \(\mu^+\to e^++\nu_e+\bar\nu_\mu\) is correlated with the muon spin. Explain how this chain makes it possible to test the helicity of the \(\nu_\mu\), even though the neutrino itself is not detected.

State the main physical conclusion: why is the decay into the light positron suppressed by almost a factor of \(10^4\), and what does this say about the chiral structure of the charged weak current?

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Exercises for discussion in the seminar

Seminar 03.1. How does helicity transform?

The helicity of a particle is defined as

\[ h=\frac{\mathbf S\cdot\mathbf p}{|\mathbf p|}. \tag{78}\]

Trace the transformation of \(\mathbf S\), \(\mathbf p\) and \(h\) under

1. an ordinary spatial rotation;
2. spatial inversion \(P\);
3. time reversal \(T\).

Why is a preference for one sign of helicity a signature of parity violation?

Seminar 03.2. The logic of the Goldhaber experiment

After electron capture, the excited recoil nucleus and the neutrino fly off in opposite directions. The nucleus then emits a \(\gamma\)-quantum, whose circular polarization stores information about the spin projection of the recoil nucleus.

1. Draw the directions of the momenta of the neutrino, the recoil nucleus and the selected \(\gamma\)-quantum.
2. Explain the chain “neutrino helicity \(\to\) recoil-nucleus polarization \(\to\) circular polarization of the photon” using conservation of angular momentum.
3. Why must the experiment select photons flying along a definite direction of recoil? What would be lost if photons from all directions were registered?

Seminar 03.3. Resonance as an angular filter

To first order in the nuclear velocity, the photon energy in the lab frame is

\[ E_{\mathrm{lab}}\simeq E_\gamma(1+\beta_R\cos\theta). \tag{79}\]

The absorber has a resonance energy \(E_{\mathrm{res}}\) and a width \(\Gamma\).

1. Find the direction \(\cos\theta_{\mathrm{res}}\) that falls into resonance.
2. Show that the width of the selected angular interval in \(\cos\theta\) is of order \(\Gamma/(\beta_RE_\gamma)\).
3. Why does the energy resonance simultaneously become a filter of the emission direction?

Seminar 03.4. Overtaking a massive particle

Let a particle have energy \(E=5mc^2\) and move along the axis \(z\).

1. Find its velocity \(\beta\).
2. What minimum Lorentz factor does an observer need in order to move faster than the particle in the same direction?
3. What happens to the direction of the momentum and to the helicity of the particle in the frame of such an observer?
4. At which step does the argument stop working for \(m=0\)?

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Exercise 4.1. Spectrum of \(\beta\) decay

Consider an allowed \(\beta\) decay with a single neutrino mass state. Neglect the nuclear recoil and radiative corrections. Take \(F(Z,E)=1\).

Let \(E\) be the kinetic energy of the electron. Use

\[ \frac{d\Gamma}{dE} = C\,p_e(E+m_e)(E_0-E) \sqrt{(E_0-E)^2-m_\nu^2}\, \Theta(E_0-E-m_\nu), \tag{80}\]

where

\[ p_e=\sqrt{E^2+2m_eE}, \qquad m_e=511~\mathrm{keV}, \qquad E_0=18.575~\mathrm{keV}. \tag{81}\]

1. Write a function that computes the unnormalized spectrum \(d\Gamma/dE\).

2. For \(m_\nu=0\) and \(m_\nu=1~\mathrm{eV}\) make two panels: the entire allowed range and the last \(5~\mathrm{eV}\). Use the same normalization for both masses.

3. Determine the maximum kinetic energy of the electron as a function of \(m_\nu\) and mark both endpoints of the spectrum on the zoomed plot.

4. For \(m_\nu=0\) compute the fraction of all decays falling into the last \(1~\mathrm{eV}\) and \(5~\mathrm{eV}\). Compare the numerical result with the estimate

   \[ \frac{N(E>E_0-\Delta E)}{N_{\mathrm{all}}} \sim\left(\frac{\Delta E}{E_0}\right)^3, \tag{82}\]

   obtained from the \(d\Gamma/dE\propto(E_0-E)^2\) approximation.

5. Convolve the spectrum with a Gaussian resolution function with \(\sigma_E=0.5~\mathrm{eV}\). Show that the reconstructed energy can lie above the physical endpoint \(E_0\), even though the true spectrum is zero there.

Exercise 4.2. Effective mass and the Kurie plot

An electron antineutrino is produced as a mixture of mass states. Near the endpoint of the spectrum, set \(E_\nu=E_0-E\) and use

\[ \frac{d\Gamma}{dE} =C(E)\sum_i|U_{ei}|^2E_\nu \sqrt{E_\nu^2-m_i^2}\,\Theta(E_\nu-m_i). \tag{83}\]

1. For \(E_\nu\gg m_i\) expand the square root to first order in \(m_i^2/E_\nu^2\).

2. Using \(\sum_i|U_{ei}|^2=1\), show that the result has the form of a spectrum with a single effective mass

   \[ m_\beta^2=\sum_i|U_{ei}|^2m_i^2. \tag{84}\]

3. Explain why the probabilities \(|U_{ei}|^2\) add up here, rather than the amplitudes \(U_{ei}\).

4. For a single mass state, construct the Kurie function

   \[ K(E)\equiv \sqrt{\frac{1}{C(E)}\frac{d\Gamma}{dE}} =\sqrt{E_\nu\sqrt{E_\nu^2-m_\nu^2}}. \tag{85}\]

   Compare \(m_\nu=0\) and \(m_\nu=1~\mathrm{eV}\) in the last \(5~\mathrm{eV}\).

5. Why are the individual kinks at \(E_\nu=m_i\) not resolved if the mass differences are much smaller than the energy resolution of the experiment?

Exercise 4.3. Coulomb focusing and the Fermi function

In the point-Coulomb-field approximation, the Fermi function can be written in terms of the Sommerfeld parameter

\[ F(\eta)=\frac{2\pi\eta}{1-e^{-2\pi\eta}}, \qquad \eta=\pm Z_d\alpha\frac{\mathcal E_e}{p_e}, \tag{86}\]

where \(\mathcal E_e=E+m_e\) is the total energy of the charged lepton, the sign “\(+\)” refers to \(\beta^-\) decay, and the sign “\(-\)” to \(\beta^+\) decay. Use \(\alpha=1/137\) and \(m_e=511~\mathrm{keV}\).

1. Show that for \(|\eta|\ll1\)

   \[ F(\eta)=1+\pi\eta+O(\eta^2). \tag{87}\]

   Explain the sign of the first correction for the electron and the positron.

2. Obtain the asymptotics

   \[ F(\eta)\simeq2\pi\eta \quad (\eta\gg1), \qquad F(\eta)\simeq2\pi|\eta|e^{-2\pi|\eta|}\quad(\eta\ll-1). \tag{88}\]

   Relate them, respectively, to Coulomb focusing and defocusing of the wave in the nuclear region.

3. For \(Z_d=29\) compute \(F(+Z_d,\mathcal E_e)\) and \(F(-Z_d,\mathcal E_e)\) at \(E=10~\mathrm{keV}\), \(100~\mathrm{keV}\), and \(1~\mathrm{MeV}\). In which energy range is the plane wave a particularly poor approximation?

4. Prove the identity

   \[ \frac{F(+|\eta|)}{F(-|\eta|)}=e^{2\pi|\eta|} \tag{89}\]

   and explain why it does not imply a difference between the weak matrix elements for \(\beta^-\) and \(\beta^+\).

5. In the nonrelativistic limit \(E\to0\) show that for \(\beta^-\) the product \(F\,p_e\mathcal E_e\) tends to a finite limit, although \(F\) itself diverges as \(1/p_e\). What happens to the same product for \(\beta^+\)?

6. Why does a dependence on \(F(Z_d,\mathcal E_e;A)\) appear in the exact expression, although the formula used contains no \(A\)? Name the physical scale through which the mass number enters.

Exercise 4.4. How the Fermi function changes the \(\beta^-\) and \(\beta^+\) spectra

Consider an illustrative allowed transition with

\[ Z_d=29,\qquad A=64,\qquad E_0=2.0~\mathrm{MeV},\qquad m_\nu=0. \tag{90}\]

Use the point-Coulomb approximation from Exercise 04.3 and the spectra

\[ S_\mp(E)\equiv\frac{d\Gamma_\mp}{dE} \propto F(\pm Z_d,\mathcal E_e)\,p_e\mathcal E_e(E_0-E)^2, \tag{91}\]

as well as the plane-wave result \(S_{\mathrm{pw}}\), obtained for \(F=1\).

1. Plot \(F(+Z_d,\mathcal E_e)\) and \(F(-Z_d,\mathcal E_e)\) as functions of \(E\) from \(0.1~\mathrm{keV}\) to \(E_0\).

2. Plot \(S_{\mathrm{pw}}\), \(S_-\), and \(S_+\), normalizing each curve to unit area. Explain why the Coulomb field enhances the low-energy part of the \(\beta^-\) spectrum and suppresses it for \(\beta^+\).

3. Separately plot the ratios \(S_-/S_{\mathrm{pw}}\) and \(S_+/S_{\mathrm{pw}}\) without renormalization. Why is this exactly the Fermi function?

4. For each spectrum, construct the correctly corrected Kurie function

   \[ K_\mp(E)= \left[ \frac{S_\mp(E)}{F(\pm Z_d,\mathcal E_e)p_e\mathcal E_e} \right]^{1/2}. \tag{92}\]

   Verify that both curves are proportional to \(E_0-E\). Then repeat the construction, mistakenly setting \(F=1\) in the denominator. What curvature does this error produce for \(\beta^-\) and \(\beta^+\)?

5. Why does the Fermi function change the probability distribution over energies but not shift the kinematic boundary \(E_{\max}=E_0-m_\nu\)? Formulate the answer in terms of the Coulomb wave function with an already specified asymptotic energy.

Exercise 4.5. Film, energy losses, and a false neutrino mass

An electron is produced at depth \(z\) in a film and exits at an angle \(\theta\) to the normal. In a simple model

\[ \ell=\frac{z}{\cos\theta}, \qquad \Delta E=a\ell, \qquad E_{\mathrm{det}}=E-\Delta E, \tag{93}\]

where \(a\) is a constant stopping power. Assume that \(z\) is uniformly distributed on \([0,d]\), and that the directions are isotropic within the accepted solid angle \(0\leq\theta\leq\theta_{\max}\). Hence \(u=\cos\theta\) is uniformly distributed on \([c,1]\), where \(c=\cos\theta_{\max}\).

1. Find \(\langle\ell\rangle\) and \(\operatorname{Var}(\ell)\). Show that

   \[ \langle\ell\rangle =\frac d2\frac{\ln(1/c)}{1-c}, \tag{94}\]

   \[ \operatorname{Var}(\ell) =d^2\left[ \frac{1}{3c} -\frac14\left(\frac{\ln(1/c)}{1-c}\right)^2 \right]. \tag{95}\]

   From this, find the mean and variance of the energy loss.

2. Let the true spectrum near the endpoint for \(m_\nu=0\) have the form \(S_0(E)\propto(E_0-E)^2\). Show that the measured spectrum is

   \[ S_{\mathrm{obs}}(E_{\mathrm{det}}) =\int_0^\infty L(\Delta E)S_0(E_{\mathrm{det}}+\Delta E)\,d\Delta E, \tag{96}\]

   where the integrand automatically vanishes for \(E_{\mathrm{det}}+\Delta E>E_0\). Denote \(\varepsilon=E_0-E_{\mathrm{det}}\). In the region where \(\varepsilon\) is larger than the characteristic losses, show that

   \[ S_{\mathrm{obs}}(\varepsilon) \propto (\varepsilon-\overline{\Delta E})^2 +\sigma_{\Delta E}^2. \tag{97}\]

   Which parameter of the endpoint fit absorbs the mean loss \(\overline{\Delta E}\), and why is a correction to \(E_0\) alone insufficient?

3. For \(\varepsilon\gg m_\nu\) the massive spectrum has the expansion

   \[ \varepsilon\sqrt{\varepsilon^2-m_\nu^2} \simeq \varepsilon^2-\frac{m_\nu^2}{2}. \tag{98}\]

   Let the true variance of the losses be \(\sigma_{\mathrm{true}}^2\), while the analysis used the value \(\sigma_{\mathrm{mod}}^2\). After the endpoint fit, show that

   \[ m_{\nu,\mathrm{fit}}^2 \simeq 2\left( \sigma_{\mathrm{mod}}^2- \sigma_{\mathrm{true}}^2 \right). \tag{99}\]

   In which case does a false positive \(m_\nu^2\) arise, and in which a negative best-fit value of \(m_\nu^2\)?

4. Let \(ad=40~\mathrm{eV}\). The true acceptance corresponds to \(\theta_{\max}=60^\circ\), but the model uses \(75^\circ\); the mean loss is freely absorbed by the \(E_0\) fit. Estimate the false \(m_{\nu,\mathrm{fit}}^2\) and \(m_{\nu,\mathrm{fit}}\). Compare the scale with the historical ITEP claim of a mass of about \(30~\mathrm{eV}\).

5. Optional. Carry out a numerical simulation without a moment expansion: generate \(z\) and \(\cos\theta\), convolve the massless spectrum with the resulting loss distribution, and perform a fit with a model that has the wrong \(\theta_{\max}\). Investigate how the result depends on the width of the fitted window near the endpoint of the spectrum.

Exercise 4.6. Kink from a heavy mass state

Suppose that the electron antineutrino contains, in addition to the light states (whose masses can be taken as zero on the scale of this problem), a heavy state with mass \(m_4=5~\mathrm{keV}\) and admixture \(|U_{e4}|^2=0.01\). For tritium \(E_0=18.575~\mathrm{keV}\); denote \(\varepsilon=E_0-E\). The spectrum has the form

\[ \frac{d\Gamma}{dE} =C(E)\,\varepsilon \left[ \bigl(1-|U_{e4}|^2\bigr)\varepsilon +|U_{e4}|^2\sqrt{\varepsilon^2-m_4^2}\,\Theta(\varepsilon-m_4) \right]. \tag{100}\]

1. Why do the contributions of the mass states enter with weights \(|U_{ei}|^2\) rather than through a sum of amplitudes? Show that the ratio to the spectrum without the heavy state,

   \[ \mathcal R(\varepsilon) =1-|U_{e4}|^2 +|U_{e4}|^2\sqrt{1-\frac{m_4^2}{\varepsilon^2}}\;\Theta(\varepsilon-m_4), \tag{101}\]

   does not depend on \(C(E)\) and hence not on the Fermi function.

2. Find \(\mathcal R\) for \(\varepsilon<m_4\) and at \(E=0\), and its asymptotics for \(\varepsilon\gg m_4\). At what electron energy \(E\) is the singularity located?

3. Set \(\varepsilon=m_4+\delta\), \(\delta\ll m_4\), and show that

   \[ \mathcal R-\bigl(1-|U_{e4}|^2\bigr) \simeq|U_{e4}|^2\sqrt{\frac{2\delta}{m_4}}. \tag{102}\]

   Why is this a kink with an infinite derivative rather than a jump?

4. Plot \(\mathcal R(E)\) over the whole spectrum and mark the point \(E=E_0-m_4\). For the massless spectrum with \(F=1\) and \(m_e=511~\mathrm{keV}\), compute numerically the fraction of decays with \(\varepsilon>m_4\) and the ratio of the total decay rates with and without the heavy state.

5. In a real fit the normalization of the spectrum is free. Why is the constant factor \(1-|U_{e4}|^2\) by itself unobservable? Which feature of the shape carries the information on \(m_4\) and \(|U_{e4}|^2\)? Why is a heavy state searched for deep inside the spectrum, where the statistics is huge, while the masses of the light states are searched for only at the very endpoint?

State the main physical conclusion: each mass state gives its own threshold \(E_0-m_i\), and the visibility of the threshold is set by the weight \(|U_{ei}|^2\) and by the statistics in the neighborhood of the kink.

Exercise 4.7. What can \(m_\beta\) be?

Oscillations measure only differences of squared masses. Use the rounded values

\[ \sin^2\theta_{12}=0.31, \quad \sin^2\theta_{13}=0.022, \quad \Delta m^2_{21}=7.5\times10^{-5}~\mathrm{eV}^2, \quad |\Delta m^2_{31}|\simeq2.5\times10^{-3}~\mathrm{eV}^2 \tag{103}\]

and

\[ |U_{e1}|^2=c_{12}^2c_{13}^2, \qquad |U_{e2}|^2=s_{12}^2c_{13}^2, \qquad |U_{e3}|^2=s_{13}^2. \tag{104}\]

1. Compute \(|U_{ei}|^2\) and check their sum.

2. In the normal ordering the lightest state is \(m_1\), and \(m_2^2=m_1^2+\Delta m^2_{21}\), \(m_3^2=m_1^2+\Delta m^2_{31}\). Show that

   \[ m_\beta^2=m_1^2+|U_{e2}|^2\Delta m^2_{21}+|U_{e3}|^2\Delta m^2_{31}. \tag{105}\]

   Derive the analogous formula for the inverted ordering, where the lightest state is \(m_3\), \(m_1^2=m_3^2+|\Delta m^2_{31}|\), \(m_2^2=m_1^2+\Delta m^2_{21}\).

3. Find the smallest possible \(m_\beta\) for each ordering.

4. Plot \(m_\beta\) as a function of the mass of the lightest state from \(10^{-4}\) to \(1~\mathrm{eV}\) on a logarithmic scale for both orderings. Above what lightest mass do the two curves differ by less than \(10\%\)? What lightest mass corresponds to \(m_\beta=0.45~\mathrm{eV}\)?

5. For \(m_\beta=0.45~\mathrm{eV}\) estimate \(m_3-m_1\) and \(m_2-m_1\) and compare them with the energy resolution \(\sigma_E=0.5~\mathrm{eV}\). Why is the spectrum described by a single effective mass in this regime?

State the main physical conclusion: what information on the mass spectrum does the kinematic limit on \(m_\beta\) provide, and what would a reliable measurement \(m_\beta<0.04~\mathrm{eV}\) mean?

Exercise 4.8. Nuclear recoil and the endpoint of the tritium spectrum

In the decay \({}^3\mathrm H\to{}^3\mathrm{He}^++e^-+\bar\nu_e\) take \(E_0=18.575~\mathrm{keV}\), \(m_e=511~\mathrm{keV}\),

\[ M({}^3\mathrm{He})c^2\simeq2.81~\mathrm{GeV}, \qquad M({}^3\mathrm{HeT}^+)c^2\simeq5.62~\mathrm{GeV}, \qquad k_B=8.62\times10^{-5}~\mathrm{eV/K}. \tag{106}\]

Set \(c=1\).

1. The recoil momentum equals \(|\mathbf p_e+\mathbf p_\nu|\le p_e+p_\nu\). Show that the largest recoil energy is reached at the endpoint of the spectrum, where \(p_\nu\to0\):

   \[ E_{\mathrm{rec}}^{\max}\simeq\frac{E_0(E_0+2m_e)}{2M}. \tag{107}\]

   Compute it for a free \({}^3\mathrm{He}\) atom and compare it with the scale \(m_\nu\sim0.45~\mathrm{eV}\).

2. In the last \(10~\mathrm{eV}\) of the spectrum \(p_\nu\lesssim10~\mathrm{eV}\). Estimate the spread of the recoil energy due to the angle between \(\mathbf p_e\) and \(\mathbf p_\nu\), and its change when \(E\) is shifted by \(10~\mathrm{eV}\). Why does the recoil near the endpoint only shift the endpoint and get absorbed by the free parameter \(E_0\)?

3. In a molecule the recoil momentum is received instantaneously by the helium nucleus. Show that the center-of-mass motion of \({}^3\mathrm{HeT}^+\) carries away the energy \(p^2/(2M_{\mathrm{mol}})\), while the remainder \(p^2/(2M_{\mathrm{He}})-p^2/(2M_{\mathrm{mol}})\) goes into the internal motion of the molecule. Estimate this remainder at the endpoint of the spectrum. Why does the decay of a molecule inevitably leave the daughter ion in a rotationally and vibrationally excited state?

4. The parent \(\mathrm T_2\) molecules (mass about \(6.03~u\), \(uc^2=931.5~\mathrm{MeV}\)) are in thermal motion at \(T=30~\mathrm K\). The electron energy in the laboratory is shifted by approximately \(p_ev_x\). Show that the Doppler broadening equals \(\sigma\simeq p_e\sqrt{k_BT/M}\), and compute it. Using the result of Exercise 4.5 for an unaccounted Gaussian variance, estimate the shift of \(m^2_{\nu,\mathrm{fit}}\) if this broadening is not included in the model.

State the main physical conclusion: which recoil effects only shift the endpoint, and which change the shape of the spectrum and must therefore enter the model?

Exercise 4.9. How many decays are needed to see the mass?

For tritium \(T_{1/2}=12.32\) years, the molar mass is \(3.016~\text{g/mol}\), \(E_0=18\,575~\mathrm{eV}\). Assume ideal resolution, zero background and a known \(E_0\).

1. Compute the specific activity of tritium in becquerels per gram.

2. Use the estimate of the fraction of decays in the last interval \(f(\Delta E)\simeq(\Delta E/E_0)^3\) (an exact calculation with the slowly varying factor \(C(E)\) gives about twice as much; this does not matter for the estimate). Let the spectrometer collect electrons from an effective activity \(A_{\mathrm{eff}}=10^{11}~\text{Bq}\). What mass of tritium does this correspond to? Find the count rate in the last \(1\) and \(2~\mathrm{eV}\).

3. Show that the number of events in the window \(0<\varepsilon<\Delta\) for mass \(m_\nu\) relates to the massless one as

   \[ \frac{N(\Delta;m_\nu)}{N(\Delta;0)} =\left(1-\frac{m_\nu^2}{\Delta^2}\right)^{3/2} \simeq1-\frac32\frac{m_\nu^2}{\Delta^2}. \tag{108}\]

4. Require that the deficit of events exceed three standard deviations of the Poisson count. Show that \(N\gtrsim(2\Delta^2/m_\nu^2)^2\) events in the window are needed. Find \(N\) and the measurement time for \(m_\nu=0.45~\mathrm{eV}\) and \(\Delta=2~\mathrm{eV}\). By what factor must the statistics be increased to improve the sensitivity to \(m_\nu\) by a factor of \(10\)?

5. Show that in this model the measurement time grows as \(\Delta/m_\nu^4\), that is, a narrow window is more favorable. Why can a real experiment not narrow the window to fractions of an electronvolt, and why is the time obtained a strongly optimistic estimate? Name at least three reasons.

State the main physical conclusion: why does the sensitivity to the mass improve only as the fourth root of the statistics, and why is the main resource of the experiment not only the activity of the source but also control of the shape of the spectrum near the endpoint?

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Exercises for discussion at the seminar

Seminar 04.1. Why is the \(\beta\) spectrum continuous?

Let the initial nucleus be at rest, and the decay energy \(Q\) be fixed.

1. What would the electron energy be in a hypothetical two-body decay \(A\to B+e^-\)?
2. Why can the electron have a continuous set of energies in the three-body decay \(A\to B+e^-+\bar\nu_e\)?
3. Where is the endpoint of the spectrum if the nuclear recoil is neglected but the neutrino mass is retained?
4. What will qualitatively change when the nuclear recoil is taken into account?

Seminar 04.2. The price of approaching the endpoint

Denote \(\varepsilon=E_0-E\). For a massless neutrino near the endpoint of the allowed spectrum,

\[ \frac{d\Gamma}{dE}\propto\varepsilon^2. \tag{109}\]

1. Show that the number of events in the last interval of width \(\Delta\) grows as \(\Delta^3\).
2. By what factor does the statistics increase when the window is widened from \(1\) to \(5~\mathrm{eV}\)?
3. Using \(\varepsilon\sqrt{\varepsilon^2-m_\nu^2} \simeq\varepsilon^2-m_\nu^2/2\), estimate how the relative effect of the mass depends on \(\varepsilon\).
4. Formulate the trade-off between statistics and sensitivity to the mass.

Seminar 04.3. What is visible on the Kurie plot?

For an allowed transition, the function

\[ K(E)= \left[ \frac{d\Gamma/dE}{F(Z,E)p_eE_e} \right]^{1/2}. \tag{110}\]

is defined.

1. Show that for \(m_\nu=0\) the plot of \(K(E)\) is a straight line \(K(E)=E_0-E\).
2. How does a nonzero mass change the position of the endpoint and the shape of the last points?
3. How does this effect differ from an overall shift of the energy scale?
4. How can electron energy losses in the source mimic a change in the shape of the spectrum near the endpoint?

Seminar 04.4. The Fermi function and causality

An electron in the Coulomb field of the daughter nucleus is described not by a plane wave but by a scattering state with a given asymptotic energy \(\mathcal E_e\). The Fermi function \(F(Z,\mathcal E_e)\) compares the density of this wave function in the nuclear region with the density of a plane wave.

1. Why, for \(\beta^-\) decay, does the Coulomb attraction enhance the spectrum at low energies, while for \(\beta^+\) decay the repulsion suppresses it?
2. Why is it incorrect to describe the effect as the emission of an electron with one energy and its subsequent transformation into another?
3. Explain why the neutrino energy and the asymptotic electron energy are fixed by a single energy conservation law, so that no contradiction with causality arises.

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Exercise 5.1. Time of flight along a helical trajectory

An electron with kinetic energy \(E\) is born at \(z=0\) in a uniform field \(\mathbf B=B\mathbf e_z\). The angle between the momentum and the direction of \(+\mathbf B\) is \(\alpha\). The detector is located in the plane \(z=L\).

1. Using the constancy of \(p_\parallel\), obtain \(z(t)\), the time to reach the detector \(t_L\), and the total trajectory length \(\ell=vt_L\).

2. Express the velocity relativistically:

   \[ \gamma=1+\frac{E}{m_ec^2}, \qquad v=c\sqrt{1-\gamma^{-2}}. \tag{111}\]

3. For \(E=18.6~\mathrm{keV}\), \(B=2.0~\mathrm{mT}\) and \(L=5.0~\mathrm{m}\) compute, at \(\alpha=15^\circ\), \(45^\circ\) and \(75^\circ\):

   \[ t_L,\qquad \ell,\qquad T_c=\frac{2\pi\gamma m_e}{eB}, \qquad N=\frac{t_L}{T_c}. \tag{112}\]

   Use \(m_ec^2=511~\mathrm{keV}\), \(c=2.998\cdot10^8~\mathrm{m/s}\) and \(e/m_e=1.759\cdot10^{11}~\mathrm{C/kg}\).

4. What happens to \(t_L\), \(\ell\) and \(N\) as \(\alpha\to90^\circ\)? Which electrons will reach the plane \(z=L>0\) at all?

5. Which of the quantities found depend on \(B\), and which do not? Explain the result physically.

Exercise 5.2. Electrostatic threshold and angular acceptance

In the non-relativistic approximation, an electron with kinetic energy \(E\) and angle \(\alpha\) to the spectrometer axis has

\[ E_\parallel=E\cos^2\alpha, \qquad E_\perp=E\sin^2\alpha. \tag{113}\]

A retarding potential \(U\) is set up in front of the detector. The electron passes if \(E_\parallel>e|U|\).

1. For given \(E\) and \(U\), find the maximum transmission angle \(\alpha_{\max}\).

2. Consider isotropic emission into the forward hemisphere. Show that the fraction of transmitted directions is equal to

   \[ f_\Omega=1-\cos\alpha_{\max} =1-\sqrt{\frac{e|U|}{E}}. \tag{114}\]

3. Compute \(\alpha_{\max}\) and \(f_\Omega\) for \(E=18.6~\mathrm{keV}\) and \(e|U|=18.0~\mathrm{keV}\).

4. Why do electrons of the same total energy but with different \(\alpha\) pass different electrostatic thresholds?

5. Explain why a uniform magnetic field does not fix this problem and why collimation is needed in front of the filter.

Exercise 5.3. Guiding center and the sign of the charge

Let

\[ \mathbf B=B\mathbf e_z, \qquad \Omega=\frac{qB}{m}, \tag{115}\]

and at the initial moment

\[ x_0=y_0=0, \qquad \mathbf v_0=(v_\perp,0,v_\parallel). \tag{116}\]

1. From the equations

   \[ \dot v_x=\Omega v_y, \qquad \dot v_y=-\Omega v_x \tag{117}\]

   show that

   \[ X=x+\frac{v_y}{\Omega}, \qquad Y=y-\frac{v_x}{\Omega} \tag{118}\]

   are constant.

2. Find \((X,Y)\) and the orbit radius for an electron \(q=-e\).

3. Determine the direction of rotation when viewed along \(+\mathbf B\). Repeat the answer for a positive charge.

4. How does the sign of \(v_\parallel\) determine the motion of the guiding center along the field? Does this answer depend on the sign of the charge?

5. Is there a transverse drift of the guiding center in a uniform magnetic field without an electric field?

Exercise 5.4. From the phase ellipse to the variables \((\theta,J)\)

For transverse cyclotron motion the following are introduced:

\[ Q=x-X=-\frac{v_y}{\Omega}, \qquad P=m\dot Q=mv_x. \tag{119}\]

1. Show that

   \[ H_\perp(Q,P) =\frac{P^2}{2m} +\frac{m\omega_c^2Q^2}{2} =E_\perp, \qquad \omega_c=|\Omega|. \tag{120}\]

2. Find the semi-axes of the phase ellipse and compute

   \[ J=\frac{1}{2\pi}\oint P\,dQ =\frac{E_\perp}{\omega_c}. \tag{121}\]

3. Introduce

   \[ \theta= \operatorname{atan2}(m\omega_cQ,P) \tag{122}\]

   and obtain the inverse transformation

   \[ Q=\sqrt{\frac{2J}{m\omega_c}}\sin\theta, \qquad P=\sqrt{2m\omega_cJ}\cos\theta. \tag{123}\]

4. By direct computation, verify the Poisson bracket \(\{\theta,J\}_{Q,P}=1\).

5. Write \(H_\perp\) in terms of \(J\) and obtain from Hamilton’s equations \(\dot J\) and \(\dot\theta\). Why is the angle \(\theta\) undefined at \(J=0\)?

Exercise 5.5. Four physical meanings of the action

For a non-relativistic cyclotron orbit use

\[ \rho_L=\frac{p_\perp}{eB}, \qquad J=\frac{p_\perp^2}{2eB}, \qquad L_{\mathrm{orb}}=\rho_Lp_\perp, \tag{124}\]

\[ \mu=\frac{p_\perp^2}{2m_eB}, \qquad \Phi=\pi\rho_L^2B. \tag{125}\]

1. Prove the chain of equalities

   \[ \boxed{ J=\frac{L_{\mathrm{orb}}}{2} =\frac{m_e}{e}\mu =\frac{e}{2\pi}\Phi }. \tag{126}\]

2. Check the dimensions of all four expressions for \(J\).

3. For

   \[ E=1.00~\mathrm{keV}, \qquad \alpha=60^\circ, \qquad B=0.100~\mathrm{T} \tag{127}\]

   in the non-relativistic approximation compute \(\rho_L\), \(\omega_c\), \(J\), \(\mu\) and \(\Phi\). Use

   \[ e=1.602\cdot10^{-19}~\mathrm{C}, \qquad m_e=9.109\cdot10^{-31}~\mathrm{kg}. \tag{128}\]

4. Numerically verify the equalities from part 1.

5. Find \(J/\hbar\) for \(\hbar=1.055\cdot10^{-34}~\mathrm{J\,s}\). What does a large value of this ratio tell us about the applicability of the classical description of the orbit?

Exercise 5.6. Relativistic transverse energy

The formula \(E_\parallel=E\cos^2\alpha\) from Exercise 5.2 is non-relativistic. Let us check how accurate it is for an electron near the endpoint of the tritium spectrum, where the neutrino mass is sought at the electronvolt level.

An electron with kinetic energy \(E\) is emitted at an angle \(\alpha\) to the \(z\) axis and moves in an electrostatic potential \(\Phi(z)\) that depends only on \(z\). At the starting point \(\Phi=0\); at the top of the barrier the potential energy of the electron equals \(e|U|\).

1. Show that \(p_\perp\) is conserved and that the total energy satisfies

   \[ \sqrt{p^2c^2+m_e^2c^4}+q\Phi(z)=\mathrm{const}. \tag{129}\]

2. The electron passes the barrier if \(p_\parallel^2>0\) at its top. Show that the exact transmission condition has the form

   \[ E-e|U|>E_\perp, \qquad E_\perp\equiv\sqrt{p_\perp^2c^2+m_e^2c^4}-m_ec^2, \qquad p_\perp c=\sin\alpha\sqrt{E^2+2Em_ec^2}. \tag{130}\]

3. Expand \(E_\perp\) to second order in \(E/(m_ec^2)\) and obtain

   \[ E_\perp\simeq E\sin^2\alpha \left[1+\frac{E}{2m_ec^2}\cos^2\alpha\right]. \tag{131}\]

   Why does the correction vanish at \(\alpha=0\) and at \(\alpha=90^\circ\)?

4. For \(E=18.6~\mathrm{keV}\) and \(m_ec^2=511~\mathrm{keV}\) compute the exact \(E_\perp\) and the non-relativistic \(E\sin^2\alpha\) at \(\alpha=10^\circ\), \(30^\circ\) and \(60^\circ\). Express the difference in electronvolts.

5. Recompute exactly \(\alpha_{\max}\) and \(f_\Omega\) from Exercise 5.2 for \(e|U|=18.0~\mathrm{keV}\). Is the difference in the angular acceptance noticeable?

State the main physical conclusion: what accuracy can be expected from the non-relativistic split \(E=E_\parallel+E_\perp\), and why does the analysis of the endpoint of the spectrum require relativistic kinematics?

Exercise 5.7. Period from area: a box and a bouncing ball

For one periodic degree of freedom

\[ J(E)=\frac{1}{2\pi}\oint p\,dq, \qquad T=2\pi\frac{dJ}{dE}. \tag{132}\]

Let us test this relation on two systems whose phase orbits are not ellipses.

1. A particle of mass \(m\) moves between elastic walls at \(q=0\) and \(q=a\). Draw the phase orbit at energy \(E\), find \(J(E)\) and obtain the period from it. Compare with \(2a/v\).

2. A ball of mass \(m\) bounces elastically above a floor:

   \[ H=\frac{p^2}{2m}+mgq, \qquad q\ge0. \tag{133}\]

   Draw the phase orbit and show that

   \[ J(E)=\frac{2\sqrt2}{3\pi}\,\frac{E^{3/2}}{g\sqrt m}. \tag{134}\]

   Obtain from \(J(E)\) the period \(T=2v_0/g\), where \(v_0\) is the speed at the floor. Check the answer with elementary kinematics.

3. For the box, the oscillator and the ball \(J\propto E^k\). Find \(k\) in each case and show that \(T\propto E^{k-1}\). Why does the period of the oscillator not depend on the energy, and why is this an exception?

4. A tennis ball of mass \(m=58~\mathrm{g}\) bounces to a height \(h=1.0~\mathrm{m}\). Find \(J\) and \(J/\hbar\) for \(g=9.81~\mathrm{m/s^2}\) and \(\hbar=1.055\cdot10^{-34}~\mathrm{J\,s}\).

5. For a neutron (\(m_n=1.675\cdot10^{-27}~\mathrm{kg}\)) find the bounce height at which \(J=\hbar\). What does this result mean for the classical description of the motion of ultracold neutrons above a mirror?

State the main physical conclusion: what does the area of a phase orbit know about the period of the motion, even when the orbit is not an ellipse?

Exercise 5.8. Which changes of variables are canonical?

For one degree of freedom a change \((q,p)\to(Q,P)\) is canonical if

\[ \{Q,P\}_{q,p} =\frac{\partial Q}{\partial q}\frac{\partial P}{\partial p} -\frac{\partial Q}{\partial p}\frac{\partial P}{\partial q} =1. \tag{135}\]

The same expression equals the Jacobian of the change: \(dQ\,dP=\{Q,P\}_{q,p}\,dq\,dp\).

1. Check whether the following changes are canonical:

   1. \(Q=aq\), \(P=p/a\), \(a>0\);
   2. \(Q=p\), \(P=-q\);
   3. \(Q=q\), \(P=p+kq^2\);
   4. \(Q=q^2\), \(P=p\);
   5. \(Q=q^2\), \(P=p/(2q)\), \(q>0\).

2. Draw the image of the square \(1\le q\le2\), \(0\le p\le1\) for each change and find its area.

3. For a free particle \(H=p^2/(2m)\) write \(H(Q,P)\) in cases (d) and (e). Show that in case (d) \(\dot Q\ne\partial H/\partial P\), while in case

   5. both Hamilton’s equations hold.

4. For the oscillator \(H=p^2/(2m)+m\omega^2q^2/2\) choose \(a\) in change (a) so that the phase orbit becomes a circle. Find its radius and show that the area equals \(2\pi E/\omega\) for any \(a\).

5. Change (b) rotates the phase plane by \(90^\circ\). Apply it to the oscillator in the variables of part 4. How has the Hamiltonian changed? What does this say about the distinction between “coordinate” and “momentum”?

State the main physical conclusion: why is the action \(J\) a property of the orbit itself rather than of the chosen variables?

Exercise 5.9. The pendulum: from oscillation to rotation

The Hamiltonian of the simple pendulum is

\[ H(\varphi,p_\varphi) =\frac{p_\varphi^2}{2I}-I\omega_0^2\cos\varphi, \qquad I=ml^2, \qquad \omega_0^2=\frac{g}{l}. \tag{136}\]

Use units \(I=\omega_0=1\); then \(E\ge-1\).

1. Draw the phase portrait and mark the three regimes: oscillation \(-1<E<1\), the separatrix \(E=1\) and rotation \(E>1\). Find the turning point \(\varphi_0(E)\) for oscillation.

2. Show that for oscillation

   \[ J(E)=\frac{2}{\pi}\int_0^{\varphi_0}\sqrt{2(E+\cos\varphi)}\,d\varphi, \qquad \cos\varphi_0=-E, \tag{137}\]

   and that at small amplitude \(J\simeq(E+1)/\omega_0\), as for an oscillator.

3. Numerically (Python) compute \(J\) and \(T=2\pi\,dJ/dE\) for the amplitudes \(\varphi_0=10^\circ\), \(90^\circ\), \(150^\circ\) and \(179^\circ\). Compare the result with the exact formula

   \[ T=\frac{4}{\omega_0}K\!\left(\sin\frac{\varphi_0}{2}\right), \tag{138}\]

   where \(K(k)\) is the complete elliptic integral of the first kind. In scipy.special.ellipk(m) the argument is \(m=k^2\).

4. Show that on the separatrix \(J=8/\pi\), while the period diverges. How is this seen on the plot of \(J(E)\)?

5. For rotation \(J(E)=(2\pi)^{-1}\int_0^{2\pi}p_\varphi\,d\varphi\). Show that for \(E\gg1\) one gets \(J\to\sqrt{2E}\), that is, the angular momentum of the rotor. Find \(J\) for \(E\to1^+\) and compare it with \(J\) of oscillation on the separatrix. Where does the factor \(2\) come from?

State the main physical conclusion: what do the actions of the oscillator, the pendulum and the rotor have in common, and what changes on crossing the separatrix?

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Exercises for discussion in the seminar

Seminar 05.1. The first bend of the trajectory

Let \(\mathbf B=B\mathbf e_z\), and let the initial velocity of the particle be

\[ \mathbf v=(v_\perp,0,v_\parallel). \tag{139}\]

1. Find the direction of the Lorentz force \(q\mathbf v\times\mathbf B\) for an electron and a positron.
2. Which velocity component starts to change at the first moment?
3. Show directly that the magnetic field does no work.
4. Sketch the initial segment of the trajectory for both signs of the charge.

Seminar 05.2. Two electrons with the same energy

Two electrons with the same energies enter the same uniform field at angles \(\alpha_1=30^\circ\) and \(\alpha_2=60^\circ\) to the field.

1. Compare their cyclotron periods.
2. Find the ratios of the Larmor radii \(\rho_2/\rho_1\) and of the pitches of the helix \(h_2/h_1\).
3. Find the ratio of the times in which the electrons cover the same distance along the field.
4. Which of these differences are visible in the animation, and which require time measurement?

Seminar 05.3. Why does a uniform field not collimate?

In a uniform constant magnetic field, \(p_\parallel\) and \(p_\perp\) are conserved.

1. Show that the pitch angle \(\tan\alpha=p_\perp/p_\parallel\) remains constant.
2. What does the field do to isotropically emitted electrons: does it keep them near a field line, turn them along the field, or both?
3. Why is a uniform field alone insufficient if a small transverse momentum is required in front of the electrostatic filter?
4. Which property of the field will have to be changed in the next exercise of the course?

Seminar 05.4. The action for two periodic motions

For a harmonic oscillator

\[ H=\frac{P^2}{2m}+\frac{m\omega^2Q^2}{2}=E \tag{140}\]

and for a planar rotor

\[ H=\frac{L^2}{2I} \tag{141}\]

compute the action \(J=(2\pi)^{-1}\oint p\,dq\).

1. Find the area of the oscillator ellipse in the \((Q,P)\) plane and show that \(J=E/\omega\).
2. Show that for the rotor \(J=L\).
3. A cyclotron orbit is equivalent to an oscillator with \(E=E_\perp\) and \(\omega=\omega_c\). Obtain \(J=E_\perp/\omega_c=p_\perp^2/(2|q|B)\).
4. Why for a cyclotron orbit \(J=|L_{\mathrm{mech}}|/2\), whereas for a rotor \(J=L\)? What plays the role of the periodic coordinate in these two exercises?

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Exercise 6.1. MAC-E spectrometer numbers

Let the source have \(B_{\mathrm{src}}=3.6~\mathrm{T}\), the maximum field be \(B_{\max}=6.0~\mathrm{T}\), and in the analyzing plane \(B_{\mathrm A}=0.35~\mathrm{mT}\).

1. From the conservation of \(p_\perp^2/B\) obtain the condition for passing the magnetic mirror and the maximum initial angle

   \[ \sin^2\vartheta_{\mathrm{acc}}=\frac{B_{\mathrm{src}}}{B_{\max}}. \tag{142}\]

   Calculate \(\vartheta_{\mathrm{acc}}\).

2. What fraction of isotropically emitted electrons in the forward hemisphere satisfies the condition \(\vartheta<\vartheta_{\mathrm{acc}}\)?

3. For an electron with \(E=18.6~\mathrm{keV}\) estimate the maximum residual transverse energy in the analyzing plane and the resolution

   \[ \Delta E\simeq E\frac{B_{\mathrm A}}{B_{\max}}. \tag{143}\]

4. Explain in one sentence the role of magnetic adiabatic collimation and of the electrostatic filter.

Exercise 6.2. Two adiabaticity conditions

An electron moves in a stationary inhomogeneous magnetic field. On the chosen section the following are known

\[ B=0.10~\mathrm{T}, \qquad L_\perp=0.50~\mathrm{m}, \qquad L_\parallel=2.0~\mathrm{m}, \tag{144}\]

and its kinetic energy and angle to the field are

\[ E=10~\mathrm{keV}, \qquad \alpha=45^\circ. \tag{145}\]

1. Find \(p_\perp\), \(v_\parallel\), the Larmor radius \(\rho_L\) and the pitch \(h=v_\parallel T_c\) with the relativistic correction to the cyclotron frequency.
2. Calculate \(\rho_L/L_\perp\) and \(h/L_\parallel\). In which direction is the inhomogeneity more dangerous for the adiabatic approximation?
3. Explain why \(dB/dt\) along the trajectory can be a nonzero quantity, although \(\partial B/\partial t=0\) throughout the whole setup.
4. What changes in the argument about the energy if the current in the magnet starts to be varied over a time comparable to \(T_c\)?

Exercise 6.3. Loss cone in a dipole field

Consider the dipole field line \(L=4\) and a conventional boundary of the atmosphere at altitude \(h_{\mathrm{atm}}=100~\mathrm{km}\). Use \(R_\oplus=6371~\mathrm{km}\) and the formulas

\[ r(\lambda)=LR_\oplus\cos^2\lambda, \qquad \frac{B(L,\lambda)}{B_{\mathrm{eq}}(L)} =\frac{\sqrt{1+3\sin^2\lambda}}{\cos^6\lambda}. \tag{146}\]

1. To what altitude above the equator does \(L=4\) correspond?

2. Find the magnetic latitude \(\lambda_{\mathrm{atm}}\) at which the line crosses the boundary of the atmosphere.

3. Calculate the ratio \(B_{\mathrm{atm}}(L)/B_{\mathrm{eq}}(L)\) and the loss-cone angle

   \[ \alpha_{\mathrm{loss}} =\arcsin\sqrt{\frac{B_{\mathrm{eq}}(L)} {B_{\mathrm{atm}}(L)}}. \tag{147}\]

4. Compare particles with equatorial angles \(5^\circ\), \(20^\circ\) and \(60^\circ\). Which one will be reflected above the atmosphere, and which one will be lost?

5. Draw the sphere of velocity directions and the two loss cones. Why is the loss region not a ring or a flat sector?

Exercise 6.4. Stationary mirror and Fermi acceleration

A particle is elastically reflected from a flat magnetic inhomogeneity moving toward it with speed \(u\ll v\). Consider the motion along the normal to the inhomogeneity and use the nonrelativistic addition of velocities.

1. Go to the reference frame of the inhomogeneity, change the sign of the particle’s velocity upon reflection, and return to the laboratory frame.
2. Show that for a head-on collision the change in velocity is \(\Delta v\simeq2u\).
3. Estimate the relative change in energy for \(u\ll v\).
4. Why is the energy before and after reflection the same in a stationary magnetic mirror, while in this exercise it is not?
5. What energy does the system lose when the particle is accelerated?

Exercise 6.5. Slow or fast: a numerical test of the invariant

Transverse cyclotron motion in a slowly varying field is an oscillator with a variable frequency. Let us check numerically when the action is conserved. Set \(m=1\) and consider

\[ \ddot Q=-\omega^2(t)\,Q, \qquad J=\frac{1}{\omega}\left(\frac{\dot Q^2}{2}+\frac{\omega^2Q^2}{2}\right). \tag{148}\]

The frequency decreases from \(\omega_0=1\) to \(\omega_1=1/4\), as when the field decreases by a factor of four, over a time \(\tau\) (in units of \(1/\omega_0\)).

1. Sudden change, \(\tau\to0\): \(Q\) and \(P=\dot Q\) have no time to change. For the initial state \(Q=\sqrt{2J_0/\omega_0}\sin\theta_0\), \(P=\sqrt{2\omega_0J_0}\cos\theta_0\) show that

   \[ \frac{J_1}{J_0} =\frac{\omega_0}{\omega_1}\cos^2\theta_0 +\frac{\omega_1}{\omega_0}\sin^2\theta_0. \tag{149}\]

   Find the smallest, the largest and the phase-averaged values.

2. Numerically (Python) integrate the equation for the linear profile \(\omega(t)=\omega_0+(\omega_1-\omega_0)t/\tau\), \(0\le t\le\tau\), for \(\tau=1\), \(10\) and \(100\). For \(64\) uniformly distributed initial phases find the smallest and the largest \(J_1/J_0\).

3. Repeat the calculation for the smooth profile

   \[ \omega(t)=\omega_0+(\omega_1-\omega_0)\, \frac{1+\tanh(t/\tau)}{2}, \qquad -8\tau\le t\le8\tau, \tag{150}\]

   for \(\tau=1\), \(3\) and \(10\). Why does the smooth profile conserve \(J\) much better than the linear one? Where is slowness violated in the linear profile?

4. For the linear profile compute the parameter \(|\dot\omega|/\omega^2\) at the end of the frequency change. Is it consistent with the spread of \(J_1/J_0\) you found?

5. What is the energy \(E_1=\omega_1J_1\) in the adiabatic limit? What does this mean for the transverse energy of an electron if the field along its path decreases by a factor of four? Where does the energy difference go in a stationary magnetic field?

State the main physical conclusion: what exactly must be small for \(J\) to be conserved, and why is the conservation of \(J\) independent of the phase only in the adiabatic limit?

Exercise 6.6. Magnetic bottle: oscillations between mirrors

Near the minimum on the trap axis the field is

\[ B(s)=B_0\left(1+\frac{s^2}{\ell^2}\right). \tag{151}\]

A non-relativistic electron with energy \(E\) passes the point \(s=0\) at an angle \(\alpha_0\) to the field.

1. Using the guiding-center Hamiltonian \(\mathcal H_{\mathrm{gc}}=p_\parallel^2/(2m)+\mu B(s)\), show that the guiding center performs harmonic oscillations along the field with frequency

   \[ \Omega_b=\frac{v_{\perp0}}{\ell}, \tag{152}\]

   and amplitude \(s_m=\ell\cot\alpha_0\). Check that at the point \(s_m\) the field equals \(B_0/\sin^2\alpha_0\).

2. Why is the oscillation frequency determined by \(v_{\perp0}\) rather than by the total speed? Does the position of the mirror point depend on the electron energy?

3. For \(E=1.0~\mathrm{keV}\), \(B_0=0.10~\mathrm{T}\), \(\ell=0.50~\mathrm{m}\) and \(\alpha_0=30^\circ\), \(60^\circ\) compute \(\rho_L\), \(T_c\), \(T_b=2\pi/\Omega_b\) and \(s_m\). Is the condition \(T_c\ll T_b\) satisfied?

4. A real trap is finite: at its ends the field reaches \(B_{\max}=RB_0\). Show that a particle from the center escapes through an end if \(\sin^2\alpha_0<1/R\), and that for an isotropic distribution the fraction of escaping particles is

   \[ f_{\mathrm{loss}}=1-\sqrt{1-\frac1R}. \tag{153}\]

   Compute \(f_{\mathrm{loss}}\) for \(R=2\) and \(R=10\).

5. In a tokamak the field is \(B\propto1/R_{\mathrm{maj}}\), where \(R_{\mathrm{maj}}\) is the distance to the axis of the torus. On a magnetic surface of minor radius \(r\) around a circle of radius \(R_0\) the ratio of the largest and smallest fields equals \((1+\varepsilon)/(1-\varepsilon)\), \(\varepsilon=r/R_0\). Show that for an isotropic distribution on the outer side of the torus the fraction of trapped particles is

   \[ f_{\mathrm{trap}}=\sqrt{\frac{2\varepsilon}{1+\varepsilon}}. \tag{154}\]

   Compute it for \(\varepsilon=0.1\) and \(0.3\). Why are there so many trapped particles although the mirror ratio is close to unity?

State the main physical conclusion: what determines whether a magnetic mirror confines a particle, and what does not?

Exercise 6.7. Transmission function of an ideal MAC-E filter

Monoenergetic electrons with kinetic energy \(E=18.6~\mathrm{keV}\) are emitted isotropically in the source. Use the non-relativistic relation \(E_\perp=E\sin^2\alpha\), the conservation of \(p_\perp^2/B\), and

\[ B_{\mathrm{src}}=3.6~\mathrm{T}, \qquad B_{\max}=6.0~\mathrm{T}, \qquad B_{\mathrm A}=0.30~\mathrm{mT}. \tag{155}\]

1. Show that in the analyzing plane \(E_{\perp,\mathrm A}=E\sin^2\alpha_{\mathrm{src}}\,B_{\mathrm A}/B_{\mathrm{src}}\) and that the electron passes the barrier if \(\sin^2\alpha_{\mathrm{src}}<\sin^2\alpha_T\), where

   \[ \sin^2\alpha_T =\frac{E-e|U|}{E}\,\frac{B_{\mathrm{src}}}{B_{\mathrm A}}. \tag{156}\]

2. Because of the magnetic mirror only angles \(\alpha_{\mathrm{src}}<\alpha_{\mathrm{acc}}\) are accepted. Obtain the transmitted fraction of the accepted electrons

   \[ T(E,U)=\frac{1-\cos\alpha_T}{1-\cos\alpha_{\mathrm{acc}}}, \qquad e|U|\le E\le e|U|+\Delta E, \tag{157}\]

   with \(T=0\) below and \(T=1\) above this interval. Show that \(\Delta E=E\,B_{\mathrm A}/B_{\max}\) and that \(T\) is continuous at both ends of the interval.

3. Introduce \(u=(E-e|U|)/\Delta E\) and write \(T(u)\) in terms of the ratio \(B_{\mathrm{src}}/B_{\max}\). Compute \(T\) at \(u=0.25\), \(0.5\), \(0.75\) and the slopes \(dT/du\) for \(u\to0\) and \(u\to1\). Plot it. Is the transmission edge linear?

4. If the integral spectrum is almost linear over the width \(\Delta E\), the smooth edge acts as a step shifted by

   \[ \delta=\Delta E\int_0^1\bigl[1-T(u)\bigr]\,du. \tag{158}\]

   Prove this and compute \(\delta/\Delta E\).

5. Include the relativistic correction: \(\Delta E\) is multiplied by \((\gamma+1)/2\). Assuming the shape of \(T(u)\) unchanged, find \(\Delta E\) and \(\delta\) in electronvolts.

State the main physical conclusion: why does an edge about \(1~\mathrm{eV}\) wide not prevent a search for a neutrino mass below \(1~\mathrm{eV}\), provided the shape of the edge is known?

Exercise 6.8. Why the main spectrometer is the size of a house

The electrons accepted by KATRIN fill a magnetic tube with flux \(\Phi_{\mathrm{tube}}=191~\mathrm{T\,cm^2}\). Use \(E=18.6~\mathrm{keV}\), \(\gamma=1.036\), \(B_{\mathrm{src}}=3.6~\mathrm{T}\), \(B_{\max}=6.0~\mathrm{T}\), \(B_{\mathrm A}=0.30~\mathrm{mT}\). The main vessel has a length of \(23.3~\mathrm{m}\) and an inner diameter of \(9.8~\mathrm{m}\).

1. From flux conservation find the diameter of the tube in the source, at the field maximum and in the analyzing plane.

2. Show that the largest Larmor radius in the analyzing plane is

   \[ \rho_{L,\mathrm A}^{\max}=\frac{p}{e\sqrt{B_{\mathrm A}B_{\max}}}, \tag{159}\]

   and compute it. What sets the diameter of the vessel: the size of an individual orbit or the cross section of the tube?

3. Compute the pitch \(h=v_\parallel T_c\) for an electron moving almost along the field without retardation (\(v_\parallel\simeq v\)) at \(B=6.0~\mathrm{T}\), \(0.10~\mathrm{T}\), \(1.0~\mathrm{mT}\) and \(0.30~\mathrm{mT}\). Where is adiabaticity at risk?

4. Require \(h/L_\parallel\le\varepsilon\) along the whole path. Show that this condition has the form

   \[ \left|\frac{d}{ds}\frac{1}{B}\right| \le\frac{\varepsilon e}{2\pi\gamma m_ev}, \tag{160}\]

   and that the distance from the field maximum to the analyzing plane is at least \(s_{\min}\simeq h(B_{\mathrm A})/\varepsilon\). Estimate \(s_{\min}\) in order of magnitude for \(\varepsilon\sim1\) and compare it with half the length of the vessel.

5. Near the analyzing plane the electron is retarded by the potential. Find the pitch at \(B_{\mathrm A}\) for a longitudinal energy of \(0.1\) and \(1.0~\mathrm{eV}\) (here \(\gamma\simeq1\)). How does the retardation relax the estimate of part 4?

State the main physical conclusion: what two prices does the spectrometer pay for a small ratio \(B_{\mathrm A}/B_{\max}\)?

Exercise 6.9. Where is the electron’s fate decided?

Near the analyzing plane on the spectrometer axis use the model

\[ e|U(s)|=e|U_0|\left[1-\frac{(s-\delta)^2}{a^2}\right], \qquad B(s)=B_{\mathrm A}\left(1+\frac{s^2}{b^2}\right), \tag{161}\]

where \(e|U_0|=18.6~\mathrm{keV}\), \(a=5.0~\mathrm{m}\), \(b=3.0~\mathrm{m}\), and \(\delta\) is the offset of the potential maximum relative to the field minimum. In the source the potential is zero and the kinetic energy of the electron is \(E\). For the accepted electrons \(\mu B_{\mathrm A}\le\Delta E=0.95~\mathrm{eV}\).

1. From the conservation of \(\mathcal E=p_\parallel^2/(2m)+\mu B+q\Phi\) and of \(\mu\) obtain, in non-relativistic form,

   \[ E_\parallel(s)=E-e|U(s)|-\mu B(s). \tag{162}\]

   Explain why the electron passes the filter only if \(\min_sE_\parallel(s)>0\).

2. Let \(\delta=0\). Show that the minimum of \(E_\parallel\) is at \(s=0\) if \(e|U_0|/a^2>\mu B_{\mathrm A}/b^2\), and estimate the ratio of the two coefficients. What is the transmission threshold?

3. Let \(\delta\ne0\). Find the position of the minimum \(s^*\) and show that to leading order

   \[ \min_sE_\parallel \simeq E-e|U_0|-\mu B_{\mathrm A}\left(1+\frac{\delta^2}{b^2}\right). \tag{163}\]

4. Find the width of the threshold for \(\delta=0.5\), \(1.0\) and \(2.0~\mathrm{m}\). For which electrons does the offset not matter?

5. A “paradox”: for \(\delta=1.0~\mathrm{m}\) an electron with energy slightly above \(e|U_0|+\mu B_{\mathrm A}\) has a longitudinal energy of about \(0.7~\mathrm{keV}\) at the point \(s=0\), but can still be reflected. Find this energy and explain where the analyzing plane physically is.

State the main physical conclusion: why must the maximum of the retarding potential coincide with the minimum of the magnetic field?

Exercise 6.10. Whom does the Earth’s magnetic field confine?

In the plane of the magnetic equator the dipole field is

\[ B_{\mathrm{eq}}(r)=B_{\mathrm{eq},\oplus}\left(\frac{R_\oplus}{r}\right)^3, \qquad B_{\mathrm{eq},\oplus}=3.1\cdot10^{-5}~\mathrm{T}, \qquad R_\oplus=6371~\mathrm{km}. \tag{164}\]

Use \(m_ec^2=0.511~\mathrm{MeV}\), \(m_pc^2=938.3~\mathrm{MeV}\) and the relativistic Larmor radius \(\rho_L=p_\perp/(|q|B)\); for the estimate set \(p_\perp\simeq p\).

1. Show that at the equator the scale of the inhomogeneity across the field is \(L_\perp=r/3\).
2. Compute \(\rho_L\) on the lines \(L=1.5\) and \(L=4\) for a \(1~\mathrm{MeV}\) electron, a \(1~\mathrm{MeV}\) proton and a \(100~\mathrm{MeV}\) proton.
3. Find \(\rho_L/L_\perp\) and show that at fixed momentum this ratio grows as \(L^4\). For which particles and lines is the adiabatic approximation reliable?
4. For a \(100~\mathrm{MeV}\) proton on the line \(L=4\) compare the cyclotron period with the transit time \(LR_\oplus/v\). Can one speak of the conservation of \(\mu\)?
5. Protons with energies up to hundreds of megaelectronvolts are observed in the inner belt, while the outer belt consists mainly of electrons. What does your estimate say about where energetic protons can be stably confined at all?

State the main physical conclusion: why is the same magnetic field a good trap for electrons and a poor one for energetic protons at large \(L\)?

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Exercise 7.1. Final states and scattering in the source

After the molecular decay, the daughter system is left in the state \(j\) with excitation energy \(V_j\) and probability \(P_j\).

1. Write the observed spectrum as a sum of spectra with endpoints \(E_0-V_j\). Check that \(\sum_jP_j=1\) conserves the total number of decays.

2. Let the mean number of inelastic collisions of an electron in the full gas column be \(\tau=1.4\). Compute

   \[ P_s=e^{-\tau}\frac{\tau^s}{s!} \tag{165}\]

   for \(s=0,1,2\) and the probability of at least one collision.

3. The decays are uniformly distributed along the source. If the optical depth \(u\) remaining after production is uniform on \([0,\tau]\), show that

   \[ \langle P_0\rangle =\frac{1}{\tau}\int_0^\tau e^{-u}\,du =\frac{1-e^{-\tau}}{\tau}. \tag{166}\]

   Compute this number and explain its physical meaning.

4. Let \(g(\Delta E)\) be the energy-loss distribution in a single collision. Show that the loss function has the form

   \[ R(\Delta E)=\sum_{s=0}^{\infty}P_s\,g^{*s}(\Delta E), \qquad g^{*0}(\Delta E)=\delta(\Delta E), \tag{167}\]

   where \(g^{*s}\) denotes the \(s\)-fold convolution.

Exercise 7.2. What KATRIN measures

In the ideal approximation, treat the MAC-E filter as a step function:

\[ \mathcal T(E,qU)=\Theta(E-qU). \tag{168}\]

Then the expected count rate at retarding energy \(qU\) is

\[ R(qU)=A\int_{qU}^{E_0}\frac{d\Gamma}{dE}\,dE+b. \tag{169}\]

1. Introduce \(x=E_0-E\) and \(w=E_0-qU\). For a massless neutrino and \(d\Gamma/dE=Cx^2\), obtain

   \[ R(w)-b=\frac{AC}{3}w^3. \tag{170}\]

2. Show that in the ideal case

   \[ -\frac{dR}{d(qU)} =A\left.\frac{d\Gamma}{dE}\right|_{E=qU}. \tag{171}\]

   Explain why scanning the threshold recovers information about the shape of the differential spectrum.

3. Plot on a single graph \(R(qU)\) for \(m_\beta^2=0\) and \(m_\beta^2=1~\mathrm{eV}^2\) over the last \(10~\mathrm{eV}\). Use the numerical integration of the spectrum from Exercise 04.1.

4. Replace the step function with the known transmission function \(\mathcal T(E,qU)\) and write the general integral for \(R(qU)\). Then include the loss function \(R_{\mathrm{loss}}(\Delta E)\) from Exercise 07.1 as one more convolution.

5. Explain why a single point \(qU\) does not allow one to simultaneously determine \(m_\beta^2\), \(E_0\), the normalization \(A\) and the background \(b\), whereas a set of thresholds allows them to be separated by their different dependence on \(qU\).

Exercise 7.3. Why a fit can give \(m_\beta^2<0\)

This exercise links the physics of the spectrum endpoint with the course on statistical analysis. Use a deliberately simple model: \(E_0\), the normalization and the background are assumed known, and the only fit parameter is \(m_\beta^2\).

Divide the interval \(1<x<10~\mathrm{eV}\), where \(x=E_0-E\), into bins of width \(\Delta x=0.5~\mathrm{eV}\). For the center \(x_i\) set

\[ \mu_i(m_\beta^2) =A\Delta x\, \left(x_i^2-\frac{m_\beta^2}{2}\right)+b, \qquad A=3~\mathrm{eV}^{-3},\quad b=1. \tag{172}\]

This is a linearized model of the spectrum endpoint, not the full KATRIN fit. The chosen ranges of \(x\) and \(m_\beta^2\) preserve \(\mu_i>0\) at all scan points.

1. Generate independent event counts \(n_i\sim\operatorname{Poisson}[\mu_i(0)]\).

2. Scan \(m_\beta^2\) from \(-4\) to \(4~\mathrm{eV}^2\) with a step no larger than \(0.02~\mathrm{eV}^2\). For each point compute the Poisson deviance

   \[ D(m_\beta^2)=2\sum_i \left[\mu_i-n_i+n_i\ln\frac{n_i}{\mu_i}\right], \tag{173}\]

   taking the \(n_i\ln(n_i/\mu_i)\) term to be zero for \(n_i=0\).

3. Find the minimum of the grid scan. Repeat steps 1–2 at least 1000 times and construct the distribution of \(\widehat m_\beta^2\). Find the mean, the standard deviation and the fraction of results with \(\widehat m_\beta^2<0\).

4. Repeat the analysis, artificially forbidding \(m_\beta^2<0\). Compare the mean value of the estimate in the two cases and explain the bias that arises at the physical boundary.

5. Now generate pseudo-data with an unaccounted resolution, replacing the true expectation by

   \[ \mu_i^{\mathrm{true}} =A\Delta x\,(x_i^2+\sigma_E^2)+b, \qquad \sigma_E=0.5~\mathrm{eV}, \tag{174}\]

   but fit with the previous model without \(\sigma_E\). Show numerically that the center of the distribution shifts by approximately

   \[ \Delta m_\beta^2\simeq-2\sigma_E^2. \tag{175}\]

6. Explain the result: why does a negative minimum of the likelihood function not imply a tachyonic neutrino, and why does the experiment allow \(m_\beta^2\) to take negative values at the fitting stage?

Exercise 7.4. From calibration to the spectrum

1. The electron gun passes through the full gas column with optical depth \(\tau=1.4\). Find the fraction of electrons on the plateau without losses. Compare it with the fraction of β electrons produced inside the same column and moving along the axis. Why do the two fractions differ?
2. For \(E_{\mathrm{gun}}=18\,600~\mathrm{eV}\) and \(e|U|=18\,585~\mathrm{eV}\), find the excess energy \(E_s\). Which electrons will pass the ideal filter at loss \(0\), \(12.8\) and \(20~\mathrm{eV}\)? How does changing \(U\) help to recover the loss function?
3. For a divider with \(M=1972\), find the voltage on the voltmeter at \(U=-18.6~\mathrm{kV}\). What absolute stability of the high voltage corresponds to the relative requirement \(3\cdot10^{-6}\)?
4. Suppose that, because of the source potential, the electron energies are additionally distributed with zero mean and rms spread \(\sigma_E=0.2~\mathrm{eV}\). Use the estimate from Exercise 7.3 for the shift \(m_\beta^2\). What measurement inside the gas can constrain this spread?
5. Explain why the energy selection of pulses in the silicon detector does not remove all electrons produced after the analyzing barrier.

Exercise 7.5. What one needs to know about the source

The windowless source is a tube \(10~\mathrm{m}\) long and \(90~\mathrm{mm}\) in diameter with a molecular column density \(\rho d=4\cdot10^{17}~\mathrm{cm}^{-2}\). The fraction of tritium atoms among all hydrogen atoms in the gas is \(\varepsilon_T=0.98\), and the half-life of tritium is \(T_{1/2}=12.32\) years. An electron near the endpoint of the spectrum has kinetic energy \(E=18.6~\mathrm{keV}\), \(m_ec^2=511~\mathrm{keV}\), \(1~\mathrm{u}=931.5~\mathrm{MeV}\).

1. Find the number of tritium atoms in the gas column above the cross-section of the tube and the activity of the source. Compare with the characteristic activity of \(\sim10^{11}\) decays per second. Why does an instability of \(\rho d\) change not only the normalization of the spectrum but also its shape?

2. At the very endpoint of the spectrum the neutrino carries away almost no momentum. Show that the daughter molecule of mass \(M\) receives the recoil energy

   \[ E_{\mathrm{recoil}}\simeq\frac{p_e^2}{2M}, \qquad p_e^2c^2=E\,(E+2m_ec^2). \tag{176}\]

   Compute \(E_{\mathrm{recoil}}\) for \(\mathrm T_2\), \(\mathrm{DT}\) and \(\mathrm{HT}\) (\(M\simeq6.03\), \(5.03\) and \(4.02~\mathrm{u}\)). By how much do the endpoints of the decay spectra of \(\mathrm T_2\) and \(\mathrm{HT}\) differ?

3. The decaying molecule moves with a thermal velocity. The projection \(u_\parallel\) of its velocity on the direction of the emitted electron is Gaussian distributed with variance \(k_BT/M\). Show that in the laboratory frame the electron energy is shifted by \(\delta E\simeq p_eu_\parallel\), and obtain

   \[ \sigma_E=p_ec\,\sqrt{\frac{k_BT}{Mc^2}}. \tag{177}\]

   Compute \(\sigma_E\) for \(\mathrm T_2\) at \(T=30~\mathrm{K}\) and \(T=80~\mathrm{K}\).

4. Use the estimate \(\Delta m_\beta^2\simeq-2\sigma_E^2\) from Exercise 7.3. What shift would the Doppler broadening at \(80~\mathrm{K}\) produce if it were left out of the model? Why does a known broadening that is included in the model not produce such a shift?

5. The gas composition is measured from the Raman lines of molecular vibrations. In the harmonic approximation the vibrational frequency is proportional to \(\mu^{-1/2}\), where \(\mu\) is the reduced mass of the nuclei. For \(\mathrm T_2\) the line lies at \(2464~\mathrm{cm}^{-1}\). Predict the positions of the \(\mathrm{DT}\) and \(\mathrm{HT}\) lines (\(\mu\simeq1.51\), \(1.21\) and \(0.755~\mathrm{u}\) for \(\mathrm T_2\), \(\mathrm{DT}\), \(\mathrm{HT}\)) and compare with the measured values \(\simeq2743\) and \(\simeq3435~\mathrm{cm}^{-1}\). In which direction do the estimate and the measurement differ, and why is the discrepancy larger for the lighter molecule? Find the wavelength of the \(\mathrm T_2\) Stokes line for a laser with \(\lambda=532~\mathrm{nm}\).

State the main conclusion: which properties of the source change only the number of counts, and which change the shape of the spectrum near the endpoint and therefore enter \(m_\beta^2\) directly?

Exercise 7.6. The krypton line as an energy standard

In the \(^{83\mathrm m}\mathrm{Kr}\) nucleus the transition energy \(\Delta E_{\mathrm{nucl}}=32.152~\mathrm{keV}\) is transferred to an electron of the atomic shell. The binding energies are \(14.326~\mathrm{keV}\) for the \(K\) shell and \(1.678~\mathrm{keV}\) for the \(L_3\) subshell. The atomic mass is \(M\simeq82.91~\mathrm{u}\), and the natural width of the \(K\)-32 line is \(\Gamma=2.7~\mathrm{eV}\).

1. Find the energies of the \(K\)-32 and \(L_3\)-32 conversion electrons neglecting recoil. Then estimate the recoil energy of the atom \(E_{\mathrm{recoil}}=p_e^2/2M\) for both lines. Compare it with the requirement \(|\delta E|\lesssim56~\mathrm{meV}\) on the energy scale.

2. The edges of the two lines are observed at voltmeter readings \(u_K\) and \(u_L\) behind a divider with ratio \(M_{\mathrm{div}}\). Show that

   \[ M_{\mathrm{div}} =\frac{E_{L_3}-E_K}{e\,\bigl(|u_L|-|u_K|\bigr)}, \tag{178}\]

   and that both the nuclear transition energy and the shift due to the potentials, common to the two lines, cancel in the difference. How precisely must the energy difference be known to obtain \(M_{\mathrm{div}}\) with a relative precision of \(5\cdot10^{-6}\)? Can the difference of the recoil energies then be neglected?

3. The line has a Lorentzian shape

   \[ L(E)=\frac{\Gamma/(2\pi)}{(E-E_K)^2+\Gamma^2/4}. \tag{179}\]

   For an ideal step filter, show that the fraction of transmitted electrons is

   \[ \frac{R(qU)}{R_{\max}} =\frac12-\frac1\pi\arctan\frac{2(qU-E_K)}{\Gamma}. \tag{180}\]

   Find the width of the edge between the \(90\%\) and \(10\%\) levels.

4. Numerical experiment. Draw \(N\) energies from the Lorentzian with \(\Gamma=2.7~\mathrm{eV}\) for \(N=10^2\), \(10^3\), \(10^4\), \(10^5\) and estimate the center of the line in two ways: by the arithmetic mean and by the median. Repeat the sampling at least \(300\) times and compare the spread of the estimates. Why does the mean not improve as \(N\) grows? Using the asymptotic formula \(\sigma_{\mathrm{med}}\simeq\pi\Gamma/(4\sqrt N)\), find how many electrons are needed to determine the center to within \(50~\mathrm{meV}\).

5. The monitor spectrometer sees that the line of a stable source has shifted by \(30~\mathrm{meV}\). What relative drift of the \(18.6~\mathrm{kV}\) high voltage does this correspond to? Does it meet the requirement \(3\cdot10^{-6}\)?

State the main conclusion: why does a line \(2.7~\mathrm{eV}\) wide calibrate the scale at the level of tens of millielectronvolts?

Exercise 7.7. Where measuring pays off most

Take a simplified model of the integral spectrum near the endpoint: a sharp filter, a single final state, all parameters except \(m_\beta^2\) known, and \(w=E_0-qU\):

\[ R(w)=\kappa\left(w^3-\frac32m_\beta^2w\right)+b, \qquad \kappa=10^{-3}~\mathrm{s}^{-1}\mathrm{eV}^{-3}, \quad b=0.2~\mathrm{s}^{-1}. \tag{181}\]

The values of \(\kappa\) and \(b\) are illustrative; they only set the scale.

1. Show that the relative change of the signal due to the mass is \(-3m_\beta^2/(2w^2)\). Compute it for \(m_\beta^2=0.2~\mathrm{eV}^2\) at \(w=2\), \(7\) and \(20~\mathrm{eV}\). Why is the point right at the endpoint, where the effect is largest, not necessarily the most favorable one?

2. At the point \(w\) one measures for a time \(t\) and obtains \(n\sim\operatorname{Poisson}(tR)\). Show that the error of the estimate of \(m_\beta^2\) from a single point at \(m_\beta^2\simeq0\) is

   \[ \sigma^2(m_\beta^2)=\frac{1}{t\,I(w)}, \qquad I(w)=\frac{1}{R}\left(\frac{\partial R}{\partial m_\beta^2}\right)^2 =\frac{9\kappa^2w^2}{4(\kappa w^3+b)}. \tag{182}\]

3. Find the value \(w_{\mathrm{opt}}\) at which \(I(w)\) is maximal, and show that at this point the signal is twice the background: \(\kappa w_{\mathrm{opt}}^3=2b\). Compute \(w_{\mathrm{opt}}\).

4. Let the whole time \(T=259\) days be spent at \(w_{\mathrm{opt}}\). Find \(\sigma(m_\beta^2)\) and compare with the statistical error \(0.108~\mathrm{eV}^2\) of the first five KATRIN campaigns. Name two reasons why the toy estimate is too optimistic.

5. Show that at the optimal point

   \[ \sigma(m_\beta^2)\propto\kappa^{-2/3}\,b^{1/6}\,T^{-1/2}. \tag{183}\]

   By what factor does the error decrease if the background is halved? If the signal is doubled?

6. The shifted analyzing region reduced the background roughly by half but widened the transmission edge from \(0.95\) to \(2.7~\mathrm{eV}\). Why does a known edge width, unlike the unaccounted broadening of Exercise 7.3, not shift \(m_\beta^2\), yet still reduce the sensitivity? Why is the choice between the two configurations made on the basis of the total error rather than of either of these numbers alone?

State the main conclusion: why does a real scan distribute the time over many points below and above the endpoint instead of spending it at a single optimal point?

Exercise 7.8. Calibration in the likelihood function

Suppose that near the minimum the spectral data are sensitive only to a combination of the squared mass and the optical depth: a shift of \(\tau\) by \(\delta\tau\) can be compensated by a shift of \(m^2\equiv m_\beta^2\) by \(-k\,\delta\tau\). Then

\[ D_\beta(m^2,\tau)= \frac{\bigl[m^2+k(\tau-\tau_0)-\mu\bigr]^2}{\sigma_s^2}, \tag{184}\]

while the electron gun gave \(\tau_{\mathrm{cal}}\pm\sigma_\tau\).

1. Show that \(D_\beta\) has no unique minimum in the \((m^2,\tau)\) plane. What does the region of small \(D_\beta\) look like, and what does it mean physically?

2. Add the calibration term \((\tau-\tau_{\mathrm{cal}})^2/\sigma_\tau^2\) and minimize over \(\tau\) at fixed \(m^2\). Show that

   \[ -2\ln L_{\mathrm{prof}}(m^2) =\frac{\bigl[m^2+k(\tau_{\mathrm{cal}}-\tau_0)-\mu\bigr]^2} {\sigma_s^2+k^2\sigma_\tau^2}. \tag{185}\]

   What is the error of \(m^2\)? Consider the limits \(\sigma_\tau\to0\) and \(\sigma_\tau\to\infty\).

3. What happens if one simply fixes \(\tau=\tau_{\mathrm{cal}}\)? Why does such an analysis underestimate the mass error?

4. The contributions to \(\sigma(m_\beta^2)\) in the first five KATRIN campaigns, in \(\mathrm{eV}^2\), are: statistics \(0.108\); optical depth \(0.052\); energy-loss distribution \(0.034\); background dependent on the step duration \(0.027\); changes of the source potential \(0.022\); non-Poissonian background fluctuations \(0.015\). Add the systematic contributions in quadrature, then add the statistics. Compare with the result \(m_\beta^2=-0.14^{+0.13}_{-0.15}~\mathrm{eV}^2\). What fraction of the total variance comes from statistics? Why is such a sum only an estimate?

5. Assume that for a zero true mass the \(90\%\) sensitivity corresponds to \(m_\beta^2\simeq1.64\,\sigma(m_\beta^2)\). Estimate the sensitivity for the total error from item 4. What total error is needed for a sensitivity of \(0.30~\mathrm{eV}\)? Compare it with the statistical error scaled from \(259\) to \(1000\) days by the \(1/\sqrt t\) law. What, besides the data-taking time, has to improve?

State the main conclusion: why is the calibration included in the joint fit as an additional term rather than simply substituting the measured value?

Exercise 7.9. A kink from a heavy neutrino

Suppose that, in addition to the light states, there is a mass state \(m_4\) of order keV with admixture \(|V_{e4}|^2\ll1\). With the notation \(x=E_0-E\) and the spectrum of Exercise 04.1,

\[ S(E)=\bigl(1-|V_{e4}|^2\bigr)S_0(E)+|V_{e4}|^2S_{m_4}(E), \qquad S_m(E)=C\,p_e(E+m_e)\,x\sqrt{x^2-m^2}\,\Theta(x-m). \tag{186}\]

1. Show that the relative distortion of the spectrum is

   \[ r(E)\equiv\frac{S}{S_0}-1= \begin{cases} -|V_{e4}|^2, & x<m_4,\\[1mm] |V_{e4}|^2\left(\sqrt{1-m_4^2/x^2}-1\right), & x>m_4. \end{cases} \tag{187}\]

   Why does the region \(E>E_0-m_4\) say nothing about the heavy state if the normalization of the spectrum is a free parameter?

2. Show that near the threshold \(S_{m_4}\propto\sqrt{x-m_4}\), and explain why a kink appears in the spectrum. Compute \(r/|V_{e4}|^2\) for \(x/m_4=1.01\), \(1.1\), \(1.5\) and \(2\).

3. Find numerically the fraction of all decays with \(E<E_0-m_4\) for \(m_4=2\), \(5\) and \(10~\mathrm{keV}\) (take the Fermi function equal to one). Where in the spectrum should one look for the kink at \(m_4=10~\mathrm{keV}\), and are there many electrons there?

4. The distortion is of order \(|V_{e4}|^2\). Estimate how many electrons \(N\) are needed to see it at the level of three standard deviations for \(|V_{e4}|^2=10^{-6}\). How long does it take to collect them at a count rate of \(10^8~\mathrm{s}^{-1}\)? Why is statistics not the main difficulty?

5. Why is the integral mode, in which the points are collected by sequential scanning of the voltage, inconvenient for this search, and what does a detector that measures the energy of each electron give?

Compare the two searches: for \(m_\beta\) the effect is noticeable but there are few events near the endpoint; for \(m_4\) there are many events but the effect is small. What limits each of them?

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Exercise 8.1. From phase modulation to the CRES spectrum

Consider the model of the received signal

\[ V(t)=V_0\cos\!\left[2\pi f_ct+h\sin(2\pi f_zt)\right]. \tag{188}\]

1. Compute the instantaneous frequency

   \[ f_{\mathrm{inst}}(t)=\frac{1}{2\pi}\frac{d\Phi}{dt} \tag{189}\]

   and show that the amplitude of its deviation is \(\Delta f=h f_z\).

2. Using the Jacobi–Anger identity

   \[ e^{ih\sin\theta}=\sum_{n=-\infty}^{\infty}J_n(h)e^{in\theta}, \tag{190}\]

   obtain the frequencies \(f_n=f_c+n f_z\) and the relative powers \(J_n^2(h)\).

3. In the limit \(h\ll1\) use

   \[ J_0(h)\simeq1-\frac{h^2}{4}, \qquad J_{\pm1}(h)\simeq\pm\frac h2 \tag{191}\]

   and find the fraction of power in the carrier and in the first sidebands.

4. Using scipy.special.jv construct the line weights for \(h=0.2\), \(1.2\) and \(3.0\). Check \(\sum_nJ_n^2(h)=1\) numerically.

Exercise 8.2. A real trap and a finite CRES track

Suppose that near the center of the magnetic trap

\[ B(z)=B_0\left(1+\frac{z^2}{L^2}\right), \qquad z(t)=z_{\max}\cos(2\pi f_zt), \tag{192}\]

and that the energy loss over one longitudinal period can be neglected.

1. Substitute \(z(t)\) into \(B(z)\) and show that the magnetic modulation of the frequency occurs at twice the longitudinal frequency \(2f_z\).

2. Find the mean cyclotron frequency and the amplitude of its modulation \(\Delta f_B\) in terms of \(f_{c0}\) and \(z_{\max}/L\).

3. Integrate the instantaneous frequency and show that the index of the magnetic phase modulation is equal to

   \[ h_B=\frac{\Delta f_B}{2f_z}. \tag{193}\]

4. Explain why the signal propagation phase, proportional to \(z(t)\), also creates sidebands with spacing \(f_z\).

5. Compute the Fourier transform of a harmonic observed only on the interval \(0<t<T_{\mathrm{obs}}\), and obtain the shape \(T_{\mathrm{obs}}^2\operatorname{sinc}^2[\pi(f-f_n)T_{\mathrm{obs}}]\).

6. For what \(T_{\mathrm{obs}}\) does the full width at half maximum of the Fourier peak correspond to \(\Delta E=0.10~\mathrm{eV}\), that is, \(\Delta f_c=5.1~\mathrm{kHz}\) near the endpoint of the tritium spectrum? Use a rectangular time window. Explain why the center of the line can be determined more precisely than its width, and how the signal-to-noise ratio enters this precision.

Exercise 8.3. The calorimetric spectrum of \({}^{163}\mathrm{Ho}\)

In a calorimetric experiment the entire atomic excitation energy \(E_c\) is measured, and the neutrino carries away

\[ E_\nu=Q-E_c. \tag{194}\]

Use the model

\[ \frac{d\Gamma}{dE_c} =C\,(Q-E_c)\sqrt{(Q-E_c)^2-m_\nu^2}\, \Theta(Q-E_c-m_\nu) \sum_j w_jL_j(E_c), \tag{195}\]

where the atomic line has the shape

\[ L_j(E_c)=\frac{\Gamma_j/(2\pi)} {(E_c-E_j)^2+\Gamma_j^2/4}. \tag{196}\]

1. Show that \(\int_{-\infty}^{\infty}L_j(E_c)\,dE_c=1\).

2. Find the maximum calorimetric energy and explain why the neutrino mass is again measured from the shape of the spectrum near the endpoint.

3. Construct a toy spectrum for two lines with \((E_j,\Gamma_j,w_j)\) of your choice at \(m_\nu=0\) and \(m_\nu=1~\mathrm{eV}\).

4. Let the activity of a single pixel be \(a\), and let two pulses be unresolved if the interval between them is smaller than \(\tau_r\). For \(a\tau_r\ll1\) show that the fraction of unresolved pile-up events is of order

   \[ f_{\mathrm{pu}}\simeq a\tau_r, \tag{197}\]

   and that their energy spectrum is proportional to the convolution \((d\Gamma/dE_c)*(d\Gamma/dE_c)\).

5. Estimate \(f_{\mathrm{pu}}\) for \(a=10~\mathrm{s}^{-1}\) and \(\tau_r=1~\mathrm{\mu s}\). Why does increasing the activity improve the statistics but at the same time strengthen this background?

6. In one paragraph, compare the systematics of the KATRIN spectrometer and of the calorimeter: where the source is located, which energy is measured, and why the calorimeter eliminates the losses on the electron’s exit from the source but acquires the problem of unresolved pile-up.

Exercise 8.4. The radio chain: cutoff, cavity, cascade

Femtowatt-level power must be brought to the ADC without losing anything in frequency. The formulas of the chain are given in the lecture without derivation; derive them yourself.

1. The field in a waveguide vanishes at the metal walls. From the condition that at least a half-wavelength must fit across a tube of diameter \(d\), obtain an estimate of the cutoff frequency

   \[ f_{\mathrm{cr}}\simeq\frac{c}{2d} \tag{198}\]

   and compute it for \(d=10\) mm. The exact solution for a circular tube of radius \(a\) gives \(f_{\mathrm{cr}}=1.841\,c/2\pi a\). Compare with the estimate and explain the sign of the discrepancy.

2. The quality factor \(Q=f/\Delta f\) is equal to \(2\pi\) times the ratio of the energy stored in the cavity to the losses per period. Show that the field decays over a time \(\tau_Q=Q/2\pi f\), and compute \(\tau_Q\) and the bandwidth \(\Delta f\) for \(Q=10^4\) at \(26\) GHz. Compare \(\tau_Q\) with the track duration \(100\) μs.

3. An amplifier with gain \(G\) and noise temperature \(T\) adds noise power \(kT\Delta f\). Refer the noise of two successive stages to the input of the first and obtain

   \[ T_{\mathrm{sys}}=T_1+\frac{T_2}{G_1}, \tag{199}\]

   then generalize to \(n\) stages.

4. Let the first stage be cooled (\(T_1=20\) K, \(G_1=20\) dB), and the second operate at room temperature (\(T_2=300\) K, \(G_2=20\) dB). Find \(T_{\mathrm{sys}}\). Swap the stages and find it again. By what factor does the noise power in the band change?

5. A mixer multiplies \(\cos(2\pi f_ct)\) and \(\cos(2\pi f_{\mathrm{ref}}t)\). Write down the result and state which of the two frequencies the filter keeps. For what \(f_{\mathrm{ref}}\) do the signals \(26.0\) and \(26.1\) GHz end up at the same intermediate frequency?

6. The ADC records a band \(B\) around the intermediate frequency. Formulate the Nyquist condition and estimate the sampling frequency for \(B=100\) MHz. Where will the line \(1.52\) GHz end up if it is digitized at a frequency of \(1\) GHz?

Exercise 8.5. A femtowatt: power, noise, and observation time

1. From the Larmor formula for an accelerated charge obtain the power of the cyclotron radiation

   \[ P_{\mathrm{rad}} =\frac{e^4B^2}{6\pi\varepsilon_0m_e^2c}\,\gamma^2\beta^2\sin^2\alpha \tag{200}\]

   and compute it for \(E=18.6\) keV, \(B=1\) T, \(\alpha=90^\circ\).

2. How much energy does the electron lose in \(T_{\mathrm{obs}}=100\) μs? Express the answer in eV and compare with the target precision \(0.1\) eV.

3. The receiver noise gives a power \(kT_{\mathrm{n}}\) per hertz of bandwidth, and the bandwidth of a single Fourier line is \(\Delta f=0.886/T_{\mathrm{obs}}\). Find the noise power in the line for \(T_{\mathrm{n}}=132\) K and \(T_{\mathrm{obs}}=100\) μs and the signal-to-noise ratio in the ideal case where all the radiated power reaches the amplifier.

4. Show that for a fixed signal power the signal-to-noise ratio in a single line grows in proportion to \(T_{\mathrm{obs}}\). What prevents lengthening the recording indefinitely? Take into account item 2 and the inelastic collisions with gas molecules from Lecture 07.

5. A smartphone antenna receives about \(10^{-10}\) W from a base station, and about \(1.4\cdot10^{-16}\) W from a GPS satellite. Place on one logarithmic scale these quantities, the power of the phone’s transmitter (\(0.2\) W), the CRES signal, and the noise from item 3. Which of the comparisons determines the detectability of the signal?

Exercise 8.6. Missing momentum: the pion and the tau

1. For the decay of a pion at rest \(\pi^+\to\mu^++\nu_\mu\) obtain

   \[ m_\nu^2=m_\pi^2+m_\mu^2-2m_\pi\sqrt{p_\mu^2+m_\mu^2}. \tag{201}\]

2. Set \(m_\nu=0\) and find \(p_\mu=(m_\pi^2-m_\mu^2)/2m_\pi\) numerically for \(m_\pi=139.57\) MeV and \(m_\mu=105.66\) MeV.

3. Differentiate Equation 201 and obtain

   \[ \delta m_\nu^2=-2m_\pi\frac{p_\mu}{E_\mu}\,\delta p_\mu. \tag{202}\]

   What relative precision of the measurement of \(p_\mu\) is needed to be sensitive to \(m_\nu=1\) eV? Compare with the limit achieved, \(170\) keV.

4. For the decay \(\tau^-\to X^-+\nu_\tau\) in the τ rest frame obtain

   \[ E_X^*=\frac{m_\tau^2+m_X^2-m_\nu^2}{2m_\tau}, \qquad m_X\le m_\tau-m_\nu. \tag{203}\]

5. In the lab the τ moves with \(\gamma_\tau\), \(\beta_\tau\). Show that for a given \(m_X\) the energy of the visible system fills the band \(E_X=\gamma_\tau\left(E_X^*\pm\beta_\tau p_X^*\right)\), where \(p_X^*=\sqrt{(E_X^*)^2-m_X^2}\). Why is the mass sought from the two-dimensional boundary in the \((m_X,E_X)\) plane?

6. By what factor are the limits \(m_{\nu_\mu}<170\) keV and \(m_{\nu_\tau}<18.2\) MeV weaker than the tritium limit \(m_\beta<0.45\) eV? Explain the loss in one sentence about the scale of the energy release.

Exercise 8.7. Field, frequency and track jumps

A CRES cell sits in a field \(B=0.959~\mathrm{T}\); \(m_ec^2=511.0~\mathrm{keV}\), the electron energy near the endpoint of the tritium spectrum is \(E=18.6~\mathrm{keV}\), and the radiated power is \(P_{\mathrm{rad}}=1.2~\mathrm{fW}\). The cross section for inelastic scattering of an electron on a \(\mathrm T_2\) molecule at this energy is \(\sigma_{\mathrm{inel}}\simeq3.4\cdot10^{-18}~\mathrm{cm}^2\), and the typical energy loss in a single collision is \(12.8~\mathrm{eV}\).

1. Find \(f_0=eB/(2\pi m_e)\) and the cyclotron frequency

   \[ f_c=\frac{f_0}{1+E/m_ec^2} \tag{204}\]

   near the endpoint of the spectrum and for the krypton line \(E=17.83~\mathrm{keV}\). How many kilohertz correspond to \(1~\mathrm{eV}\)?

2. Show that for simultaneous errors in the field and in the frequency

   \[ \delta E=(m_ec^2+E)\left(\frac{\delta B}{B}-\frac{\delta f_c}{f_c}\right). \tag{205}\]

   What \(\delta B/B\) gives \(\delta E=0.10~\mathrm{eV}\)? Express \(\delta B\) in microtesla and compare with the Earth’s magnetic field (\(\sim50~\mathrm{\mu T}\)). Why is the field calibrated with the electrons of known energy themselves, and not only with a magnetometer?

3. Find the frequency drift rate \(\dot f_c=f_cP_{\mathrm{rad}}/(m_ec^2+E)\) and the frequency shift over \(100~\mathrm{\mu s}\). In which direction does the frequency drift, and why?

4. Find the frequency jump after a collision with a loss of \(12.8~\mathrm{eV}\). Compare it with the drift from item 3 and with the linewidth \(0.886/T_{\mathrm{obs}}\) for \(T_{\mathrm{obs}}=100~\mathrm{\mu s}\). Why do collisions show up on the track as separate steps?

5. At what number density of molecules \(n\) is the mean time between collisions \(1/(n\sigma_{\mathrm{inel}}v)\) equal to \(100~\mathrm{\mu s}\)? Compare with the mean number density in the KATRIN source, where the column density \(4\cdot10^{17}~\mathrm{cm}^{-2}\) is accumulated over a length of \(10~\mathrm{m}\).

6. Estimate the activity of \(1~\mathrm{cm}^3\) of such a gas (\(T_{1/2}=12.32\) years, two tritium atoms per molecule) and the number of decays per year with the electron in the last \(10~\mathrm{eV}\), if the fraction of decays in the last interval \(\Delta E\) is approximately \(2\cdot10^{-13}\,(\Delta E/1~\mathrm{eV})^3\).

State the main conclusion: why is CRES limited not by its resolution but by the volume of the source, and why do the next stages need a volume of the order of a cubic meter?

Exercise 8.8. The microcalorimeter: heating, noise and the choice of isotope

An absorber with heat capacity \(C=10^{-12}~\mathrm{J/K}\) operates at \(T=20~\mathrm{mK}\); the energy release of the \(^{163}\mathrm{Ho}\) decay is \(Q\simeq2.86~\mathrm{keV}\). Take the heat capacity to be independent of temperature over a small interval. The half-lives are \(4570\) years for \(^{163}\mathrm{Ho}\) and \(4.3\cdot10^{10}\) years for \(^{187}\mathrm{Re}\).

1. Find the heating \(\Delta T\) from a decay with \(E_c=Q\) and the relative change of the magnetization of a paramagnetic sensor obeying the Curie law \(M\propto1/T\).

2. In the canonical ensemble the energy of a body fluctuates:

   \[ \langle\delta E^2\rangle=k_BT^2C. \tag{206}\]

   Derive this relation from \(\langle E\rangle=-\partial\ln Z/\partial\beta\), \(\beta=1/k_BT\). Compute \(\sqrt{\langle\delta E^2\rangle}\) in electronvolts and the corresponding full width at half maximum \(2.355\sqrt{\langle\delta E^2\rangle}\). Compare with the resolution of about \(6~\mathrm{eV}\) achieved by HOLMES.

3. A superconducting thermometer with resistance \(R(T)\) is held at constant voltage, so the Joule power is \(P_J=V^2/R\). Introduce \(\alpha=d\ln R/d\ln T\) and linearize the heat-balance equation \(C\,d\Delta T/dt=-G\Delta T+\Delta P_J\). Show that

   \[ \tau_{\mathrm{eff}}=\frac{C/G}{1+\mathcal L}, \qquad \mathcal L=\frac{\alpha P_J}{GT}. \tag{207}\]

   Why is the pulse shorter than \(C/G\) for \(\alpha>0\), and how does this help at high activity?

4. Find the number of atoms and the mass of the isotope needed for an activity \(a=10~\mathrm{s}^{-1}\) in a single pixel, for \(^{163}\mathrm{Ho}\) and for \(^{187}\mathrm{Re}\). By what factor do the numbers of atoms differ?

5. Explain why holmium can be implanted into a small absorber, whereas rhenium would have to be the absorber itself, and how this affects \(C\) and the heating from item 1.

State the main conclusion: why is a small \(Q\) useful only together with a short half-life and a small heat capacity?

Exercise 8.9. What did supernova 1987A tell us about the mass?

A neutrino of mass \(m\) and energy \(E_\nu\) is delayed by

\[ \Delta t\simeq5.15~\mathrm{ms} \left(\frac{L}{10~\mathrm{kpc}}\right) \left(\frac{m}{1~\mathrm{eV}}\right)^2 \left(\frac{10~\mathrm{MeV}}{E_\nu}\right)^2. \tag{208}\]

Supernova 1987A exploded in the Large Magellanic Cloud, \(L\simeq50~\mathrm{kpc}\). The Kamiokande-II detector recorded about a dozen events within \(\simeq12.4~\mathrm{s}\) with energies from roughly \(7.5\) to \(35~\mathrm{MeV}\); take the neutrino energies equal to these values.

1. Find the delay for \(E_\nu=10\) and \(30~\mathrm{MeV}\) at \(m=1\) and \(10~\mathrm{eV}\).
2. Require that the difference between the delays at \(7.5\) and \(35~\mathrm{MeV}\) not exceed the duration of the signal. What upper bound on \(m\) does this give? What assumption about the emission time of neutrinos of different energies is made here? Compare the bound with \(15~\mathrm{eV}\) from ECHo and \(0.45~\mathrm{eV}\) from KATRIN.
3. Can neutrinos outrun light? The neutrinos arrived about three hours before the light. What relative velocity difference \((v-c)/c\) would this correspond to? Explain why in reality the light simply left the star later. In 2011 a measurement of \((v-c)/c\simeq2.5\cdot10^{-5}\) was reported for accelerator neutrinos; the cause later turned out to be an equipment fault. How many years before the light would the neutrinos from SN 1987A have arrived at such a velocity?
4. Suppose that for a future supernova at a distance of \(10~\mathrm{kpc}\) the signal is cut off abruptly, and the moment of the cutoff for \(E_\nu=10\) and \(30~\mathrm{MeV}\) can be compared with a precision of \(1~\mathrm{ms}\). What mass could one be sensitive to?
5. Each mass state is delayed in its own way. Estimate the delay for \(m=0.45~\mathrm{eV}\), \(E_\nu=10~\mathrm{MeV}\), \(L=10~\mathrm{kpc}\) and the difference between the delays of two states if \(|m_3^2-m_1^2|=2.5\cdot10^{-3}~\mathrm{eV}^2\). Can the mass states be separated by their arrival times?

State the main conclusion: why does the time-of-flight method not need a model of the β spectrum, but does need a model of the source?

Exercise 8.10. Three ways to weigh: \(m_\beta\), \(m_{\beta\beta}\) and \(\Sigma\)

Use rounded parameters: \(|V_{e2}|^2=0.30\), \(|V_{e3}|^2=0.022\), \(|V_{e1}|^2=1-|V_{e2}|^2-|V_{e3}|^2\), \(\Delta m_{21}^2=7.5\cdot10^{-5}~\mathrm{eV}^2\), \(|\Delta m_{31}^2|=2.5\cdot10^{-3}~\mathrm{eV}^2\). The three observables are

\[ m_\beta^2=\sum_i|V_{ei}|^2m_i^2, \qquad m_{\beta\beta}=\Bigl|\sum_i|V_{ei}|^2e^{i\phi_i}m_i\Bigr|, \qquad \Sigma=m_1+m_2+m_3, \tag{209}\]

where the phases \(\phi_i\) are unknown.

1. Find the three masses for the normal ordering with \(m_1=0\) and for the inverted ordering with \(m_3=0\) (for the inverted ordering \(m_1^2=|\Delta m_{31}^2|\), \(m_2^2=m_1^2+\Delta m_{21}^2\)).
2. Compute \(m_\beta\) and \(\Sigma\) for both orderings.
3. Find the smallest and the largest \(m_{\beta\beta}\) for arbitrary phases for both orderings. Hint: the modulus of a sum of three vectors of lengths \(a_i\) lies between \(\max\bigl(0,\,2\max a_i-\sum a_i\bigr)\) and \(\sum a_i\).
4. Plot the three quantities as functions of the lightest mass from \(10^{-4}\) to \(1~\mathrm{eV}\) for both orderings. Find the interval of \(m_1\) in which \(m_{\beta\beta}\) can vanish for the normal ordering. Why does \(m_\beta\) never vanish?
5. The KATRIN limit \(m_\beta<0.45~\mathrm{eV}\) lies in the region of nearly degenerate masses. What constraint on \(\Sigma\) follows from it? Which mass ordering can an experiment with a sensitivity of \(m_\beta\simeq40~\mathrm{meV}\) test, and why is such a goal insufficient for the normal ordering?

State the main conclusion: why can the three mass measurements not be compared as three measurements of the same number?

Seminar

S8.1. How long to listen to the electron

A long recording narrows the line, but during that time the electron manages to lose energy to radiation. Let us find the compromise.

1. Convert the linewidth into an energy error, \(\delta E_{\mathrm{lin}}=(m_ec^2+E)\,\Delta f/f_c\) at \(\Delta f=0.886/T_{\mathrm{obs}}\), and fill in the table for \(T_{\mathrm{obs}}=30\), \(100\), \(300\) and \(1000\) μs.
2. For the same times find the total energy loss to radiation \(\Delta E_{\mathrm{rad}}=P_{\mathrm{rad}}T_{\mathrm{obs}}\).
3. Suppose the track model reconstructs this loss with a precision of \(10\%\), so that the residual systematic is \(0.1\,\Delta E_{\mathrm{rad}}\). Add it to \(\delta E_{\mathrm{lin}}\) in quadrature and find the optimal \(T_{\mathrm{obs}}\).
4. Discuss what else terminates the track: inelastic scattering on the gas and the escape of the electron from the trap. How do these processes change the optimum you found?

S8.2. The trap: statistics versus resolution

Let the field at the edges of the trap be \(B_{\max}=B_0(1+\eta)\), \(\eta\ll1\).

1. From the reflection condition obtain \(|\cos\alpha_0|<\sqrt{\eta/(1+\eta)}\simeq\sqrt\eta\).
2. For an isotropic source \(\cos\alpha_0\) is uniformly distributed on \([-1,1]\). Show that the fraction of trapped electrons is \(\sqrt\eta\), and compute it for \(\eta=10^{-1}\), \(10^{-2}\), \(10^{-3}\).
3. A trapped electron passes through fields from \(B_0\) to \(B_{\max}\). Show that without a trajectory correction this smears the reconstructed energy by \(\delta E\simeq(m_ec^2+E)\,\eta\), and compute \(\delta E\) for the same \(\eta\).
4. Choose \(\eta\) such that the fraction of trapped electrons is at least \(1\%\), and the smearing is at most \(0.1\) eV. Does it work out? What conclusion follows about the need to reconstruct the trajectory from the sidebands?

S8.3. Neutrino mass from arrival times

A continuation of the applet “Mass from arrival times” and a bridge to the course on statistical analysis.

1. From \(\Delta t=L\,m^2c^4/2cE_\nu^2\) obtain the working form

   \[ \Delta t\simeq5.15~\text{ms} \left(\frac{L}{10~\text{kpc}}\right) \left(\frac{m}{1~\text{eV}}\right)^2 \left(\frac{10~\text{MeV}}{E_\nu}\right)^2. \tag{210}\]

2. Let \(N\) neutrinos be produced at the moment \(t_0\) with a Gaussian spread \(\sigma_t\), and let pairs \((E_i,t_i)\) be measured. Write down \(-2\ln L(m^2,t_0)\), eliminate \(t_0\) analytically, and show that what remains is the sum of squared deviations from a straight line in the variable \(x_i=(10~\text{MeV}/E_i)^2\).

3. Obtain the estimate of the slope and its variance:

   \[ \hat m^2=\frac{S_{xt}}{5.15\,S_{xx}}, \qquad \sigma(\hat m^2)=\frac{\sigma_t}{5.15\sqrt{S_{xx}}}, \tag{211}\]

   where \(S_{xx}=\sum_i(x_i-\bar x)^2\), \(S_{xt}=\sum_i(x_i-\bar x)(t_i-\bar t)\). Explain why \(\hat m^2\) can turn out to be negative.

4. Take \(N=15\) energies uniformly covering \(6\)–\(35\) MeV and \(\sigma_t=2\) ms. Compute \(\sigma(\hat m^2)\). How many events are needed for \(\sigma(\hat m^2)=0.1\) eV²? Compare with the two dozen events from SN 1987A.

5. Check your formulas with the applet: for zero true mass estimate how often the estimate comes out negative.

S8.4. Holmium: pixels versus pile-up

1. For a Poisson stream with activity \(a\), show that the fraction of events with an unresolved neighbor within the resolving time \(\tau_r\) is equal to \(1-e^{-a\tau_r}\simeq a\tau_r\).

2. Let the total activity \(A\) be distributed over \(N\) identical pixels. Obtain the rate of unresolved pairs

   \[ R_{\mathrm{par}}\simeq\frac{A^2\tau_r}{N} \tag{212}\]

   and explain why, at fixed \(A\), splitting into pixels helps.

3. Calculate \(R_{\mathrm{par}}\) for \(A=10^6~\text{s}^{-1}\), \(\tau_r=1\) μs at \(N=10^3\) and \(N=10^5\).

4. Estimate the fraction of events in the last \(10\) eV of the holmium spectrum, assuming the phase space \(\propto(Q-E_c)^2\) at \(Q=2.86\) keV. Compare the useful count rate with \(R_{\mathrm{par}}\) from item 3 and say whether splitting into pixels alone is sufficient.

5. From \(\Delta T=E_c/C\) find the heating at \(E_c=2.86\) keV and \(C=10^{-12}\) J/K. How does the choice of \(C\) and the thermal coupling \(G\) conflict with the requirement of small \(\tau_r\)?

---

Exercise 9.1. Global and relative phases

Consider a normalized superposition of two energy eigenstates:

\[ |\psi(t)\rangle =c_1e^{-iE_1t}|E_1\rangle+c_2e^{-iE_2t}|E_2\rangle, \qquad |c_1|^2+|c_2|^2=1. \tag{213}\]

1. Show that the replacement

   \[ |\psi(t)\rangle\longrightarrow e^{i\chi(t)}|\psi(t)\rangle \tag{214}\]

   does not change the expectation value of an arbitrary operator \(\widehat A\).

2. Find the density matrix \(\rho(t)=|\psi(t)\rangle\langle\psi(t)|\) in the energy basis. Which of its elements depend on time?

3. Explain why the probabilities of measuring \(E_1\) and \(E_2\) are constant.

4. Introduce the state

   \[ |a\rangle =\cos\eta\,|E_1\rangle +e^{i\varphi}\sin\eta\,|E_2\rangle. \tag{215}\]

   Compute \(P(a,t)=|\langle a|\psi(t)\rangle|^2\) and explicitly isolate the interference term that depends on \((E_2-E_1)t\).

5. Under what conditions does this interference term vanish? Distinguish between the absence of one of the components, degeneracy of the energies, and a special choice of the measurement basis.

State the main physical conclusion: why is the global phase unobservable, while the relative phase can change probabilities?

Exercise 9.2. Energy of a flavor state

Two flavor states are related to the mass states by a rotation

\[ |\nu_\alpha\rangle =c_\theta|\nu_1\rangle+s_\theta|\nu_2\rangle, \qquad |\nu_\beta\rangle =-s_\theta|\nu_1\rangle+c_\theta|\nu_2\rangle, \tag{216}\]

where

\[ H_0|\nu_i\rangle=E_i|\nu_i\rangle, \qquad c_\theta=\cos\theta, \quad s_\theta=\sin\theta. \tag{217}\]

For each of the two flavor states:

1. find the probabilities of obtaining the results \(E_1\) and \(E_2\) in a measurement of the energy;

2. compute \(\langle H_0\rangle\);

3. compute \(\langle H_0^2\rangle\);

4. obtain the variance

   \[ \operatorname{Var}(H_0) =\langle H_0^2\rangle-\langle H_0\rangle^2; \tag{218}\]

5. show that

   \[ \operatorname{Var}_\alpha(H_0) =\operatorname{Var}_\beta(H_0) =\frac14\sin^2 2\theta\,(E_2-E_1)^2. \tag{219}\]

Explain why a nonzero variance indicates the presence of several energy components, but does not mean that the energy itself oscillates in time. For what values of the parameters does the variance vanish?

Exercise 9.3. Complete derivation of the two-flavor probability

Let a neutrino of flavor \(\alpha\) be produced at \(t=0\):

\[ |\psi(0)\rangle=|\nu_\alpha\rangle. \tag{220}\]

1. Expand the initial state in terms of \(|\nu_1\rangle\) and \(|\nu_2\rangle\) and carry out its time evolution.

2. Write down the inverse rotation from the mass basis to the flavor basis.

3. Obtain the amplitudes

   \[ \mathcal A_{\alpha\to\alpha}(t) =c_\theta^2e^{-iE_1t}+s_\theta^2e^{-iE_2t}, \tag{221}\]

   \[ \mathcal A_{\alpha\to\beta}(t) =s_\theta c_\theta \left(e^{-iE_2t}-e^{-iE_1t}\right). \tag{222}\]

4. Introduce

   \[ \overline E=\frac{E_1+E_2}{2}, \qquad \Delta E=E_2-E_1, \tag{223}\]

   isolate the global phase and obtain the appearance and survival probabilities.

5. Check unitarity:

   \[ P_{\alpha\to\alpha}(t)+P_{\alpha\to\beta}(t)=1. \tag{224}\]

6. Find the maximum transition probability and the oscillation period.

7. Expand the transition probability at \(|\Delta E|t\ll1\) and show that

   \[ P_{\alpha\to\beta}(t) \simeq \sin^2 2\theta\,\frac{(\Delta E)^2t^2}{4}. \tag{225}\]

Why does the probability near \(t=0\) grow quadratically, although the transition amplitude grows linearly?

Exercise 9.4. From the plane-wave phase to the dependence on L/E

For a mass component of a plane wave, use the phase

\[ e^{-i(E_it-p_iL)}. \tag{226}\]

Denote \(\varphi_i=E_it-p_iL\) and \(\Delta\varphi=\varphi_2-\varphi_1\).

1. Show that the relative phase of the two components is equal to

   \[ \Delta\varphi(t,L)=\Delta E\,t-\Delta p\,L. \tag{227}\]

2. First set \(p_1=p_2=p\). Multiplying the difference of the square roots by the conjugate sum, prove the exact equality

   \[ \Delta E =\frac{\Delta m^2}{E_2+E_1}, \qquad \Delta m^2=m_2^2-m_1^2. \tag{228}\]

3. In the ultrarelativistic limit \(E_i\simeq p\simeq E\) and for \(t\simeq L\) obtain

   \[ \Delta\varphi\simeq\frac{\Delta m^2L}{2E}. \tag{229}\]

4. Repeat the derivation by setting \(E_1=E_2=E\). Find \(\Delta p\) and show that the same result now arises from the term \(-\Delta pL\).

5. For equal energies, factor the global time phase out of the state. Show that the probability at a fixed point does not depend on \(t\), but does depend on \(L\) when different detection points are compared. Which term in \(\Delta\varphi\) is responsible for this dependence?

6. Explain why the two plane-wave idealizations lead to the same phase only at leading ultrarelativistic order.

7. Using

   \[ \hbar c=1.97327\cdot10^{-7}~\mathrm{eV\,m}, \tag{230}\]

   independently obtain the coefficient \(1.267\) in the formula

   \[ P_{\alpha\to\beta} =\sin^2 2\theta\, \sin^2\!\left[ 1.267\, \frac{\Delta m^2[\mathrm{eV}^2]L[\mathrm{km}]} {E[\mathrm{GeV}]} \right]. \tag{231}\]

8. Finally, for a general momentum \(p\ll m_i\) obtain

   \[ \Delta E \simeq \Delta m\left(1-\frac{p^2}{2m_1m_2}\right), \qquad \Delta m=m_2-m_1. \tag{232}\]

   Write down the leading nonrelativistic probability as a function of time. Why, when passing to distance, does the velocity enter explicitly and the universal dependence on \(L/E\) alone disappear?

Exercise 9.5. A mock-up of an oscillation experiment

In a certain two-flavor experiment the maximum transition probability is equal to

\[ P_{\alpha\to\beta}^{\max}=0.64. \tag{233}\]

The first maximum at energy \(E=0.60~\mathrm{GeV}\) is observed at a distance \(L=295~\mathrm{km}\).

1. In the convention \(0\leq\theta\leq45^\circ\) find the mixing angle.

2. Using the practical probability formula, find \(|\Delta m^2|\).

3. Find the oscillation length \(L_{\mathrm{osc}}\) at the given energy.

4. At the same distance compute the appearance and survival probabilities for

   \[ E=0.30~\mathrm{GeV}, \qquad E=0.60~\mathrm{GeV}, \qquad E=1.20~\mathrm{GeV}. \tag{234}\]

5. Plot \(P_{\alpha\to\beta}(E)\) and \(P_{\alpha\to\alpha}(E)\) in the range \(0.2\leq E\leq1.5~\mathrm{GeV}\).

6. Can the sign of \(\Delta m^2\) be determined from these measurements?

7. Compare three reasons for a vanishing transition probability:

   • \(\sin2\theta=0\);
   • \(\Delta m^2=0\);
   • \(\Delta m^2L/(4E)=n\pi\).

   Which of them forbid oscillations altogether, and which one creates only isolated nodes of the probability?

Exercise 9.6. Coupled pendulums of different masses

Two pendulums of the same length \(\ell\) and different masses \(m_1\) and \(m_2\) are connected by a weak spring of stiffness \(k\). The deflection angles \(\varphi_i\) are small, and \(\omega_0^2=g/\ell\).

1. Show that in the quadratic approximation

   \[ T=\frac{\ell^2}{2} \left(m_1\dot\varphi_1^2+m_2\dot\varphi_2^2\right), \tag{235}\]

   \[ V=\frac{\ell^2}{2} \left[ \omega_0^2(m_1\varphi_1^2+m_2\varphi_2^2) +k(\varphi_1-\varphi_2)^2 \right]. \tag{236}\]

2. Obtain the matrix equation

   \[ \mathsf M\ddot{\boldsymbol\varphi} +\mathsf K\boldsymbol\varphi=0 \tag{237}\]

   and write down \(\mathsf M\) and \(\mathsf K\).

3. Introduce mass-normalized coordinates

   \[ \boldsymbol x=\mathsf M^{1/2}\boldsymbol\varphi \tag{238}\]

   and obtain the real symmetric matrix \(\Omega^2\) in the equation \(\ddot{\boldsymbol x}+\Omega^2\boldsymbol x=0\).

4. Find the eigenfrequencies and the normalized eigenvectors. Show that

   \[ \omega_+^2=\omega_0^2, \qquad \omega_-^2 =\omega_0^2+k\left(\frac1{m_1}+\frac1{m_2}\right). \tag{239}\]

5. Introducing the mixing angle through the eigenvectors, prove

   \[ \tan^2\theta=\frac{m_2}{m_1}, \qquad \sin^2 2\theta =\frac{4m_1m_2}{(m_1+m_2)^2}. \tag{240}\]

6. Let \(\Omega\) be the positive square root of \(\Omega^2\). Introduce

   \[ \boldsymbol a =\boldsymbol x+i\Omega^{-1}\dot{\boldsymbol x} \tag{241}\]

   and prove the exact equation \(i\dot{\boldsymbol a}=\Omega\boldsymbol a\), as well as the conservation of the norm \(\boldsymbol a^\dagger\boldsymbol a\).

7. If at the initial moment only the first pendulum is excited, obtain

   \[ P_{1\to2}(t) =\sin^2 2\theta\, \sin^2\frac{(\omega_--\omega_+)t}{2}. \tag{242}\]

8. For

   \[ \frac{m_2}{m_1}=\frac14, \qquad \frac{k}{m_1\omega_0^2}=0.10 \tag{243}\]

   find the maximum value of \(P_{1\to2}\), \(\omega_-/\omega_0\) and the time of the first maximum.

9. At what mass ratio is complete transfer possible?

10. Explain why \(P_i=|a_i|^2/\|\boldsymbol a\|^2\) is exactly the fraction of the norm of the complex amplitude, but coincides with the fraction of the averaged local energy only in the weak-coupling limit.

Exercise 9.7. Oscillation scales: from a reactor to the Sun

In practical units the two-flavor transition probability reads

\[ P_{\alpha\to\beta} =\sin^2 2\theta\, \sin^2\!\left[ 1.267\, \frac{\Delta m^2[\mathrm{eV}^2]L[\mathrm{km}]} {E[\mathrm{GeV}]} \right]. \tag{244}\]

Use two scales of the mass splitting, the atmospheric and the solar one:

\[ \Delta m^2_{\mathrm{atm}}=2.5\cdot10^{-3}~\mathrm{eV}^2, \qquad \Delta m^2_{\mathrm{sol}}=7.5\cdot10^{-5}~\mathrm{eV}^2. \tag{245}\]

1. Show that the oscillation length in practical units is

   \[ L_{\mathrm{osc}}[\mathrm{km}] \simeq2.48\, \frac{E[\mathrm{GeV}]}{|\Delta m^2|[\mathrm{eV}^2]}, \tag{246}\]

   and that the same coefficient is obtained for \(L\) in meters and \(E\) in MeV.

2. Find the distance to the first transition maximum, \(L_{\mathrm{osc}}/2\), for both scales and three sources: reactor antineutrinos with \(E=4~\mathrm{MeV}\), an accelerator beam with \(E=0.6~\mathrm{GeV}\) and atmospheric neutrinos with \(E=1~\mathrm{GeV}\). Collect the results in a table.

3. Atmospheric neutrinos with \(E\simeq1~\mathrm{GeV}\) reach the detector from above, after traveling about \(15~\mathrm{km}\), and from below, after crossing the Earth over up to \(12\,700~\mathrm{km}\). Estimate the phase \(1.267\,\Delta m^2_{\mathrm{atm}}L/E\) for both directions. Why does a comparison of the fluxes from above and from below test oscillations while hardly relying on a calculation of the absolute flux?

4. The near detector of a reactor experiment is located at \(L=0.3~\mathrm{km}\). Estimate the disappearance probability for it at \(\sin^2 2\theta=0.088\) and \(\Delta m^2_{\mathrm{atm}}\). Compare the exact answer with the small-phase approximation. What is such a detector for?

5. For solar neutrinos with \(E=1~\mathrm{MeV}\) and \(\Delta m^2_{\mathrm{sol}}\), find \(L_{\mathrm{osc}}\). How many periods fit into the Sun–Earth distance \(L=1.496\cdot10^{8}~\mathrm{km}\)? Over a year this distance changes by about \(5\cdot10^{6}~\mathrm{km}\) because of the eccentricity of the orbit. Can the annual variation of the oscillation phase be observed?

State the main conclusion: why do experiments with energies from MeV to GeV measure the same splittings, and what condition on \(L/E\) makes an experiment sensitive to a given \(\Delta m^2\)?

Exercise 9.8. Averaging of oscillations and the 1/2 bound

A detector registers neutrinos of different energies, so it measures not \(P_{\alpha\to\beta}(L,E)\) at a single point but its average over the distribution of the phase \(\phi=\Delta m^2L/(2E)\).

1. Show that the two-flavor probability can be written as

   \[ P_{\alpha\to\beta} =\frac12\sin^2 2\theta\,(1-\cos\phi). \tag{247}\]

2. Let the phase be uniformly distributed over an interval much larger than \(2\pi\). Find \(\langle P_{\alpha\to\beta}\rangle\) and \(\langle P_{\alpha\to\alpha}\rangle\). Show that \(\langle P_{\alpha\to\alpha}\rangle\geq1/2\) for any \(\theta\).

3. Let the phase be Gaussian distributed with mean \(\phi_0\) and width \(\sigma_\phi\). Prove that

   \[ \langle\cos\phi\rangle=e^{-\sigma_\phi^2/2}\cos\phi_0 \tag{248}\]

   and write down \(\langle P_{\alpha\to\beta}\rangle\). The factor \(e^{-\sigma_\phi^2/2}\) is called the oscillation contrast.

4. At fixed \(L\) the spread of the phase comes from the spread of the energy. Show that for \(\sigma_E/E\ll1\)

   \[ \sigma_\phi\simeq\phi_0\,\frac{\sigma_E}{E}. \tag{249}\]

   For \(\sigma_E/E=0.05\) find the contrast at the first maximum \(\phi_0=\pi\). At what \(\phi_0\) does the contrast drop to \(1/2\)? How many oscillation periods is this?

5. Compare with an incoherent mixture: the source emits \(|\nu_1\rangle\) with probability \(\cos^2\theta\) and \(|\nu_2\rangle\) with probability \(\sin^2\theta\). Find the probability of detecting the flavor \(\alpha\) and show that it coincides with the fully averaged \(\langle P_{\alpha\to\alpha}\rangle\).

6. For solar electron neutrinos the measured values are \(P_{ee}\approx0.55\) at energies below \(1~\mathrm{MeV}\) and \(P_{ee}\approx1/3\) at energies of a few MeV. The solar mixing angle is \(\sin^2\theta=0.31\). Compute the fully averaged two-flavor vacuum survival probability. Which of the two measurements does it explain? Is there an angle \(\theta\) for which it equals \(1/3\)?

State the main physical conclusion: what information does a detector lose in the fully averaged regime, and why does \(P_{ee}\approx1/3\) require a mechanism beyond vacuum oscillations?

Exercise 9.9. When the mass components stop interfering

The formula for \(P_{\alpha\to\beta}(L,E)\) was derived for plane waves. A real neutrino is produced localized: it is a wave packet of size \(\sigma_x\). Mass components of the same energy \(E\) move with different group velocities. Use \(\hbar=c=1\) and

\[ \hbar c=1.97327\cdot10^{-7}~\mathrm{eV\,m}. \tag{250}\]

1. Show that in the ultrarelativistic limit

   \[ v_i\simeq1-\frac{m_i^2}{2E^2}, \qquad |v_1-v_2|\simeq\frac{|\Delta m^2|}{2E^2}. \tag{251}\]

2. The packets of the mass components stop overlapping when their relative displacement becomes comparable to \(\sigma_x\). Obtain the coherence length

   \[ L_{\mathrm{coh}}\simeq\frac{2E^2\sigma_x}{|\Delta m^2|} \tag{252}\]

   and show that

   \[ \frac{L_{\mathrm{coh}}}{L_{\mathrm{osc}}} \simeq\frac{\sigma_x}{\lambda}, \qquad \lambda=\frac{2\pi}{E}. \tag{253}\]

   What is the meaning of this ratio?

3. A reactor experiment observes oscillations with full contrast at \(E=4~\mathrm{MeV}\), \(|\Delta m^2|=2.5\cdot10^{-3}~\mathrm{eV}^2\) and \(L=1.6~\mathrm{km}\). What lower bound on \(\sigma_x\) follows from this? Compare it with the wavelength \(\lambda\) in meters.

4. For solar neutrinos with \(E=1~\mathrm{MeV}\), \(|\Delta m^2|=7.5\cdot10^{-5}~\mathrm{eV}^2\) and \(L=1.496\cdot10^{11}~\mathrm{m}\), find how large \(\sigma_x\) must be for the packets still to overlap at the Earth. Does the observed survival probability change if they have separated? Compare the probability for an incoherent mixture of mass states with the fully averaged oscillation probability.

5. Coherence is also needed at production. A source localized within a region \(\sigma_x\) has a momentum spread \(\sigma_p\sim1/(2\sigma_x)\). At equal energy the momenta of the mass components differ by \(|\Delta p|\simeq|\Delta m^2|/(2E)\). Show that the condition \(\sigma_p\gg|\Delta p|\) (the source cannot determine which mass state was emitted) is equivalent to

   \[ \sigma_x\ll\frac{E}{|\Delta m^2|}=\frac{L_{\mathrm{osc}}}{4\pi}. \tag{254}\]

6. Combine the two conditions into a window for \(\sigma_x\) and estimate its bounds in meters for the reactor example of item 3.

State the main conclusion: what implicit assumptions about the size of the wave packet does the plane-wave formula contain, and why does it work so well in terrestrial experiments?

---

Seminar

Exercise 10.1. Build the matrix from three hints

Work in the first approximation: \(U_{e3}=0\), the atmospheric mixing angle is close to \(45^\circ\), and the matrix may be chosen real.

1. For high-energy solar neutrinos take \(P_{ee}\simeq1/3\). For an adiabatic exit from the Sun this gives \(|U_{e2}|^2\simeq1/3\). Using the normalization of the electron row, fill in \(|U_{e1}|^2\) and \(|U_{e3}|^2\).

2. Atmospheric disappearance is described in the leading approximation by the depth \(4|U_{\mu3}|^2(1-|U_{\mu3}|^2)\simeq1\). Find \(|U_{\mu3}|^2\). Fill in \(|U_{\tau3}|^2\) from the normalization of the third column.

3. Choose the same signs for \(U_{\mu3}\) and \(U_{\tau3}\). What does the orthogonality of the second and third columns say about the relation between \(U_{\mu2}\) and \(U_{\tau2}\)? Normalize the second column and fill in \(|U_{\mu2}|^2\), \(|U_{\tau2}|^2\).

4. Use the normalizations of the two remaining rows. Show that

   \[ |U_0|^2= \begin{pmatrix} 2/3&1/3&0\\ 1/6&1/3&1/2\\ 1/6&1/3&1/2 \end{pmatrix}. \tag{255}\]

5. Choose the signs of the elements so that the resulting real matrix \(U_0\) satisfies \(U_0^{\mathsf T}U_0=I\). Write out all three ones and all six zeros in the product.

6. Which entry does the reactor result \(|U_{e3}|^2\simeq0.0225\) change first? Explain which conclusions of the approximate matrix remain qualitatively correct. Can this construction be used to independently check unitarity, if it was used at every step?

Exercise 10.2. How to read the PMNS matrix

Use the standard factorization

\[ U=R_{23}U_{13}(\delta)R_{12}. \tag{256}\]

1. Multiply the matrices and obtain the first row and the third column of \(U\).

2. Prove the relations

   \[ |U_{e3}|=\sin\theta_{13}, \qquad \frac{|U_{e2}|}{|U_{e1}|}=\tan\theta_{12}, \qquad \frac{|U_{\mu3}|}{|U_{\tau3}|}=\tan\theta_{23}. \tag{257}\]

3. For the NuFIT 6.1 best-fit point with normal ordering use

   \[ \sin^2\theta_{12}=0.3088, \quad \sin^2\theta_{23}=0.470, \quad \sin^2\theta_{13}=0.02248 \tag{258}\]

   and compute \(|U_{e1}|\), \(|U_{e2}|\), \(|U_{e3}|\), \(|U_{\mu3}|\) and \(|U_{\tau3}|\).

4. Explain why the phase \(\delta\) becomes unobservable at \(s_{13}=0\).

5. An arbitrary unitary matrix \(3\times3\) contains three angles and six phases. Count how many phases can be removed by rephasing the fields in the Dirac case.

6. For the Majorana case set

   \[ \widetilde U =U\operatorname{diag} \left(1,e^{i\alpha_{21}/2},e^{i\alpha_{31}/2}\right). \tag{259}\]

   Show that the additional phases cancel in the product \(\widetilde U_{\beta i}\widetilde U_{\alpha i}^*\) and therefore do not enter the oscillation probabilities.

Homework

Exercise 10.3. General three-flavor probability

For three active neutrinos

\[ |\nu_\alpha\rangle =\sum_{i=1}^3U_{\alpha i}^*|\nu_i\rangle. \tag{260}\]

1. Carry out the evolution of the mass components and obtain the ultrarelativistic amplitude

   \[ \mathcal A_{\alpha\to\beta}(L,E) =\sum_iU_{\beta i}U_{\alpha i}^* \exp\!\left(-i\frac{m_i^2L}{2E}\right). \tag{261}\]

2. Square the amplitude and obtain a double sum over \(i,j\).

3. Combine the terms \((i,j)\) and \((j,i)\), introduce

   \[ X_{\alpha\beta}^{ij} =U_{\alpha i}^*U_{\beta i} U_{\alpha j}U_{\beta j}^*, \qquad \Delta_{ij}=\frac{\Delta m_{ij}^2L}{4E}, \tag{262}\]

   and derive the general probability formula in terms of \(\operatorname{Re}X^{ij}\) and \(\operatorname{Im}X^{ij}\).

4. Obtain the formula for the survival probability \(P_{\alpha\to\alpha}\).

5. Using the unitarity \(U\), prove

   \[ \sum_\beta P_{\alpha\to\beta}=1 \tag{263}\]

   and show that for \(L=0\) \(P_{\alpha\to\beta}=\delta_{\alpha\beta}\).

6. Show how the general formula reduces to the two-flavor one if one mass component does not participate.

7. Prove

   \[ \Delta m_{31}^2=\Delta m_{32}^2+\Delta m_{21}^2 \tag{264}\]

   and explain why only two oscillation frequencies are independent.

8. Perform the replacement \(m_i^2\to m_i^2+C\). Show that the amplitude acquires only an overall phase factor, while the probability does not change.

9. Which experiments, unlike oscillations, are sensitive to the absolute scale of neutrino masses?

Exercise 10.4. CP violation and the Jarlskog invariant

1. Show that for antineutrinos the oscillation amplitude is obtained by the replacement \(U\to U^*\).

2. Using the general three-flavor probability, obtain

   \[ A_{\alpha\beta}^{\mathrm{CP}} =P(\nu_\alpha\to\nu_\beta) -P(\bar\nu_\alpha\to\bar\nu_\beta) =4\sum_{i>j}\operatorname{Im}X_{\alpha\beta}^{ij} \sin2\Delta_{ij}. \tag{265}\]

3. Prove that \(A_{\alpha\alpha}^{\mathrm{CP}}=0\) in vacuum.

4. For the quartet

   \[ J_{\mathrm{CP}} =\operatorname{Im} \left(U_{e2}^*U_{\mu2}U_{e1}U_{\mu1}^*\right) \tag{266}\]

   check the invariance under rephasing of the rows and columns of the mixing matrix.

5. From the standard PMNS parametrization obtain

   \[ J_{\mathrm{CP}} =\frac18 \sin2\theta_{12}\sin2\theta_{23}\sin2\theta_{13} \cos\theta_{13}\sin\delta_{\mathrm{CP}}. \tag{267}\]

6. Using \(\Delta_{31}=\Delta_{32}+\Delta_{21}\) and trigonometric identities, derive

   \[ A_{e\mu}^{\mathrm{CP}} =16J_{\mathrm{CP}} \sin\Delta_{21}\sin\Delta_{32}\sin\Delta_{31}. \tag{268}\]

7. List all the conditions under which the vacuum CP asymmetry vanishes. Why is there no fundamental CP violation in the two-flavor vacuum model?

8. For the NuFIT 6.1 best-fit point with normal ordering use

   \[ \theta_{12}=33.76^\circ, \quad \theta_{23}=43.29^\circ, \quad \theta_{13}=8.62^\circ, \quad \delta_{\mathrm{CP}}=212^\circ \tag{269}\]

   and compute \(J_{\mathrm{CP}}\).

9. Estimate \(A_{e\mu}^{\mathrm{CP}}\) at the first atmospheric maximum, defined by the condition \(\Delta_{31}=\pi/2\). Use

   \[ \Delta m_{21}^2=7.537\cdot10^{-5}~\mathrm{eV}^2, \qquad \Delta m_{31}^2=2.511\cdot10^{-3}~\mathrm{eV}^2. \tag{270}\]

   Why is the asymmetry additionally suppressed by the smallness of the solar phase?

Exercise 10.5. Numerical lab of three flavors

Write a program in Python, Julia, or Mathematica that computes vacuum oscillations of three active neutrinos. For Python you can use the template.

Use the NuFIT 6.1 best-fit point with normal ordering:

\[ \begin{aligned} \sin^2\theta_{12}&=0.3088,& \sin^2\theta_{23}&=0.470,& \sin^2\theta_{13}&=0.02248,\\ \Delta m_{21}^2&=7.537\cdot10^{-5}~\mathrm{eV}^2,& \Delta m_{31}^2&=2.511\cdot10^{-3}~\mathrm{eV}^2,& \delta_{\mathrm{CP}}&=212^\circ. \end{aligned} \tag{271}\]

1. Construct the PMNS matrix in the standard parametrization and numerically check \(U^\dagger U=I\).

2. Implement the amplitude \(\mathcal A_{\alpha\to\beta}\). If the rows denote the final flavor \(\beta\) and the columns the initial flavor \(\alpha\), then in matrix form

   \[ \mathcal A =U\operatorname{diag}(e^{-i\phi_i})U^\dagger, \qquad \phi_i=2(1.267)\frac{m_i^2[\mathrm{eV}^2]L[\mathrm{km}]} {E[\mathrm{GeV}]}. \tag{272}\]

   One can set \(m_1^2=0\), since the overall mass is not observable.

3. Compute the probability matrix \(P_{\beta\alpha}=|\mathcal A_{\beta\alpha}|^2\).

4. Automatically check

   \[ P_{\beta\alpha}(0)=\delta_{\beta\alpha}, \qquad \sum_\beta P_{\beta\alpha}=1 \tag{273}\]

   with a numerical accuracy no worse than \(10^{-12}\).

5. For the initial \(\nu_\mu\) plot \(P_{\mu\to e}\), \(P_{\mu\to\mu}\) and \(P_{\mu\to\tau}\) as functions of \(L/E\) in the range \(0\leq L/E\leq2000~\mathrm{km/GeV}\).

6. Repeat the calculation for antineutrinos, replacing \(U\to U^*\), and plot \(P_{\mu\to e}-\overline P_{\mu\to e}\).

7. Repeat the comparison at \(\delta_{\mathrm{CP}}=0\) and explain the result.

8. Add the same number \(C\) to all \(m_i^2\) and numerically check that the probabilities do not change.

9. By scanning over \(L/E\), find the point of the maximal vacuum difference \(P_{\mu\to e}-\overline P_{\mu\to e}\) in absolute value in the indicated range.

10. Represent the triple of probabilities

    \[ (P_{\alpha\to e},P_{\alpha\to\mu},P_{\alpha\to\tau}) \tag{274}\]

    as a trajectory inside the probability triangle. Why can the trajectory not leave the triangle?

11. In item 1 the matrix is unitary by construction. What independent experimental normalizations and channels would be needed to check unitarity without fixing it in the parametrization in advance?

In a real long-baseline experiment the comparison of neutrinos and antineutrinos is also modified by matter. In this exercise only the vacuum part needs to be computed.

Exercise 10.6. What the moduli of the matrix know about the phase \(\delta_{\mathrm{CP}}\)

In the standard parametrization the elements of the first column are

\[ U_{e1}=c_{12}c_{13}, \qquad U_{\mu1}=-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}, \qquad U_{\tau1}=s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}. \tag{275}\]

1. Show that

   \[ |U_{\mu1}|^2 =s_{12}^2c_{23}^2+c_{12}^2s_{23}^2s_{13}^2 +2s_{12}c_{12}s_{23}c_{23}s_{13}\cos\delta, \tag{276}\]

   and obtain the analogous expression for \(|U_{\tau1}|^2\).

2. Show that \(|U_{\mu1}|^2+|U_{\tau1}|^2\) does not depend on \(\delta\). Which unitarity condition does this express?

3. For the NuFIT 6.1 best-fit point with normal ordering

   \[ \sin^2\theta_{12}=0.3088, \quad \sin^2\theta_{23}=0.470, \quad \sin^2\theta_{13}=0.02248, \quad \delta=212^\circ \tag{277}\]

   compute \(|U_{\mu1}|^2\) and \(|U_{\tau1}|^2\).

4. With the same angles, find the range of \(|U_{\mu1}|\) as \(\delta\) runs from \(0\) to \(360^\circ\). The global analysis gives the \(3\sigma\) interval from \(0.259\) to \(0.498\) for \(|U_{\mu1}|\) and from \(0.810\) to \(0.834\) for \(|U_{e1}|\). Why are the entries of the \(\mu\) and \(\tau\) rows known much worse than those of the electron row?

5. Prove that the phases \(\delta\) and \(360^\circ-\delta\) give the same matrix of moduli \(|U_{\alpha i}|\). What is \(J_{\mathrm{CP}}\) at \(\delta=212^\circ\) and at \(\delta=148^\circ\)?

6. Show that the nine moduli determine \(|J_{\mathrm{CP}}|\) but not its sign.

State the main conclusion: what part of the information on \(\delta_{\mathrm{CP}}\) can measurements sensitive only to \(|U_{\alpha i}|\), such as survival channels, provide, and which measurements are needed for the sign of \(\sin\delta_{\mathrm{CP}}\)?

Exercise 10.7. Two scales: when three flavors reduce to two

Use the NuFIT 6.1 best-fit point with normal ordering:

\[ \begin{aligned} \sin^2\theta_{12}&=0.3088,& \sin^2\theta_{23}&=0.470,& \sin^2\theta_{13}&=0.02248,\\ \Delta m_{21}^2&=7.537\cdot10^{-5}~\mathrm{eV}^2,& \Delta m_{31}^2&=2.511\cdot10^{-3}~\mathrm{eV}^2, \end{aligned} \tag{278}\]

together with

\[ P_{\alpha\to\alpha} =1-4\sum_{i>j}|U_{\alpha i}|^2|U_{\alpha j}|^2\sin^2\Delta_{ij}, \qquad \Delta_{ij}=1.267\, \frac{\Delta m^2_{ij}[\mathrm{eV}^2]L[\mathrm{km}]}{E[\mathrm{GeV}]}. \tag{279}\]

1. Find the ratio \(\Delta m_{21}^2/\Delta m_{31}^2\). Setting \(\Delta_{21}=0\), show that

   \[ P_{\alpha\to\alpha} \simeq1-4|U_{\alpha3}|^2\left(1-|U_{\alpha3}|^2\right) \sin^2\Delta_{31}. \tag{280}\]

2. For the electron flavor obtain \(P_{ee}\simeq1-\sin^2 2\theta_{13}\sin^2\Delta_{31}\). For a reactor detector with \(L=1.6~\mathrm{km}\) and \(E=4~\mathrm{MeV}\) compute \(\Delta_{21}\), \(\Delta_{31}\), and the approximate and exact probabilities. Where is the first atmospheric disappearance maximum located?

3. In the same limit obtain the appearance probability

   \[ P_{\alpha\to\beta} \simeq4|U_{\alpha3}|^2|U_{\beta3}|^2\sin^2\Delta_{31}, \qquad \alpha\ne\beta, \tag{281}\]

   using the orthogonality of the rows \(\alpha\) and \(\beta\) for the sum of the two terms with the same phase.

4. For a \(\nu_\mu\) beam with \(L=295~\mathrm{km}\) and \(E=0.6~\mathrm{GeV}\) find \(|U_{\mu3}|^2=s_{23}^2c_{13}^2\), the disappearance depth \(4|U_{\mu3}|^2(1-|U_{\mu3}|^2)\), \(P_{\mu\mu}\) and \(P_{\mu e}\simeq\sin^2 2\theta_{13}\,s_{23}^2\sin^2\Delta_{31}\). Which of the two probabilities is sensitive to the octant of \(\theta_{23}\)? Compare \(P_{\mu e}\) with the exact vacuum value at \(\delta_{\mathrm{CP}}=212^\circ\) (for example, from the program of Exercise 10.5) and explain the discrepancy.

5. For a reactor detector at \(L=180~\mathrm{km}\) with \(E=4~\mathrm{MeV}\) find \(\Delta_{31}\). Average the fast terms and show that

   \[ P_{ee}\simeq c_{13}^4 \left(1-\sin^2 2\theta_{12}\sin^2\Delta_{21}\right)+s_{13}^4. \tag{282}\]

   Compute this probability.

6. Which two small parameters control the accuracy of the two-flavor approximations? Which parameters of the matrix does each of the three experiments considered here measure?

State the main conclusion: why did two-flavor formulas describe individual experiments so well, even though there are three flavors?

Exercise 10.8. Mass ordering in the reactor antineutrino spectrum

The sign \(\Delta m_{21}^2>0\) is known; the position of the third mass state is not. Compare two options with the same angles:

\[ \begin{aligned} \text{NO:}&\quad\Delta m_{31}^2=+2.511\cdot10^{-3}~\mathrm{eV}^2, &\Delta m_{32}^2&=\Delta m_{31}^2-\Delta m_{21}^2,\\ \text{IO:}&\quad\Delta m_{32}^2=-2.511\cdot10^{-3}~\mathrm{eV}^2, &\Delta m_{31}^2&=\Delta m_{32}^2+\Delta m_{21}^2, \end{aligned} \tag{283}\]

where \(\Delta m_{21}^2=7.537\cdot10^{-5}~\mathrm{eV}^2\), \(\sin^2\theta_{12}=0.3088\), \(\sin^2\theta_{13}=0.02248\). In this model comparison the magnitudes of the two large splittings simply swap places.

1. Using \(|U_{e1}|=c_{12}c_{13}\), \(|U_{e2}|=s_{12}c_{13}\), \(|U_{e3}|=s_{13}\), obtain

   \[ P_{ee} =1-c_{13}^4\sin^2 2\theta_{12}\sin^2\Delta_{21} -\sin^2 2\theta_{13} \left(c_{12}^2\sin^2\Delta_{31}+s_{12}^2\sin^2\Delta_{32}\right). \tag{284}\]

2. Show that \(P_{ee}\) in vacuum does not change when the signs of all \(\Delta m_{ij}^2\) are flipped simultaneously. How, then, can the mass ordering enter the probability?

3. Define \(\Delta_a=1.267\cdot2.511\cdot10^{-3}\,L/E\) and \(\Delta_b=\Delta_a-\Delta_{21}\) (in practical units). Prove that

   \[ P_{ee}^{\mathrm{NO}}-P_{ee}^{\mathrm{IO}} =-\sin^2 2\theta_{13}\cos2\theta_{12}\, \sin(\Delta_a+\Delta_b)\sin\Delta_{21}. \tag{285}\]

   Find the largest possible value of \(|P_{ee}^{\mathrm{NO}}-P_{ee}^{\mathrm{IO}}|\).

4. For \(L=52.5~\mathrm{km}\) (the JUNO baseline) plot \(P_{ee}(E)\) for NO and IO in the range \(2\leq E\leq8~\mathrm{MeV}\), together with their difference. At what energy is the difference largest? Relate the answer to the first solar maximum \(\Delta_{21}=\pi/2\).

5. What would happen for \(\theta_{12}=45^\circ\)? For \(\theta_{13}=0\)?

6. Estimate the period of the fast oscillations in energy near \(E=4~\mathrm{MeV}\), \(\delta E\simeq\pi E/\Delta_{31}\). What requirement on the energy resolution of the detector follows from it? What remains of the mass-ordering effect if the fast terms are averaged out?

State the main physical conclusion: how does the mass ordering show up in vacuum if the probability is sensitive only to the magnitudes of the splittings?

Exercise 10.9. The unitarity triangle and the number of phases

The orthogonality of the \(e\) and \(\mu\) rows is written as

\[ z_1+z_2+z_3=0, \qquad z_i=U_{ei}U_{\mu i}^*. \tag{286}\]

1. For the NuFIT 6.1 best-fit point with normal ordering

   \[ \sin^2\theta_{12}=0.3088, \quad \sin^2\theta_{23}=0.470, \quad \sin^2\theta_{13}=0.02248, \quad \delta_{\mathrm{CP}}=212^\circ \tag{287}\]

   construct the matrix \(U=R_{23}U_{13}(\delta)R_{12}\) and compute the three complex numbers \(z_i\). Check the closure and draw the triangle in the complex plane.

2. Find the side lengths \(|z_i|=|U_{ei}||U_{\mu i}|\). Which side is the shortest, and why?

3. Compute the area \(S=\frac12|\operatorname{Im}(z_1^*z_2)|\) and compare it with \(|J_{\mathrm{CP}}|/2\), where

   \[ J_{\mathrm{CP}} =\frac18\sin2\theta_{12}\sin2\theta_{23}\sin2\theta_{13} \cos\theta_{13}\sin\delta_{\mathrm{CP}}. \tag{288}\]

4. Under the rephasing \(U_{\alpha i}\to e^{i(\phi_\alpha-\psi_i)}U_{\alpha i}\) find how the \(z_i\) change. Which properties of the triangle are physical, and which depend on the convention?

5. Repeat the calculation for \(\delta_{\mathrm{CP}}=0\), \(180^\circ\) and \(270^\circ\). When does the triangle degenerate into a segment? What is the largest area for the given angles?

6. A unitary \(N\times N\) matrix has \(N^2\) real parameters. Show that \(N(N-1)/2\) of them are angles and \(N(N+1)/2\) are phases. Rephasing the fields removes \(2N-1\) phases for Dirac particles and \(N\) phases for Majorana particles. Fill in a table of the number of physical phases for \(N=2,3,4\).

7. Suppose the sides \(|z_i|\) have been measured independently, without assuming unitarity. What result would be a direct sign of a violation of unitarity of the three-neutrino matrix?

State the main conclusion: what does the area of the unitarity triangle mean, and why does a triangle built from the parametrization not test unitarity itself?

---

Exercise 11.1. The first Fresnel zone

The rings of the layer with boundaries \(r_m=z+m\lambda/2\) are called Fresnel zones. Find the radius \(b_1\) of the first zone for \(z\gg\lambda\) and the number of scatterers in it for a layer of thickness \(dz\) and concentration \(\mathcal N\). Explain why the growth of the ring area compensates the factor \(1/r\) of the spherical wave, and check that for \(z\gg ka^2\) the scattering amplitude hardly changes over the first zone.

Exercise 11.2. Phase and attenuation

Let \(f(0)=a+ib\), where \(a,b>0\), and the medium is homogeneous over the length \(D\).

1. Find the additional phase of the transmitted wave.
2. Find the intensity ratio \(I(D)/I(0)\).
3. Using the optical theorem, express the result in terms of the total cross section.

Exercise 11.3. Two refractive indices

The matrix refractive index has the form

\[ \mathbf n=n_0\mathbb I+ \begin{pmatrix}\delta&\kappa\\\kappa&-\delta\end{pmatrix}. \tag{289}\]

1. Find the two eigen refractive indices.
2. Find the rotation angle from the original basis to the eigenstates.
3. Show that for \(\delta=0\) the original first state passes completely into the second after a suitable path length, if \(\kappa\ne0\).

Exercise 11.4. When coherence is lost

For each process, indicate whether it can be included in the real refractive index of the transmitted wave, or whether it requires a description of the loss of coherence:

• elastic scattering strictly forward without a change of the atom’s state;
• excitation of the atom;
• scattering through a finite angle;
• elastic scattering on two indistinguishable centers;
• absorption of a neutrino.

For each answer, formulate the physical criterion of distinguishability of the alternatives.

Exercise 11.5. A plate and the accumulated phase

Consider a transparent layer with a real refractive index. Neglect reflection at the boundaries.

1. At fixed frequency, obtain the ratios \(\lambda_{\mathrm{med}}/\lambda_0\) and \(v_{\mathrm{ph}}/c\).
2. Show that after exiting a layer of thickness \(D\) the additional phase is equal to \(\Delta\varphi=2\pi(n-1)D/\lambda_0\), even though the wavelength is again equal to \(\lambda_0\).
3. Calculate the phase for \(n=1.08\) and \(D=1.5\lambda_0\). What must the thickness be, for the same \(n\), to obtain a shift \(\pi\)?
4. Divide the layer into \(M\) parts and find the limit \([1+i\Delta\varphi/M]^M\) as \(M\to\infty\). Why does the small contribution of each layer allow a large total phase?
5. Check the calculation with the applet (in Russian). Find two pairs \(n,D\) with the same phase at the exit. Are the waves inside the plate the same?

Exercise 11.6. Two eigenstates

Let, at constant density,

\[ \begin{aligned} |\nu_e\rangle&=c_m|\nu_{\mathrm{mat},1}\rangle+s_m|\nu_{\mathrm{mat},2}\rangle,\\ |\nu_x\rangle&=-s_m|\nu_{\mathrm{mat},1}\rangle+c_m|\nu_{\mathrm{mat},2}\rangle, \end{aligned} \qquad c_m=\cos\theta_m,\quad s_m=\sin\theta_m. \tag{290}\]

Denote \(\Phi=E(n_{\mathrm{mat},1}-n_{\mathrm{mat},2})z\).

1. Write down the state at distance \(z\) for the initial \(\nu_e\). By projecting onto \(\nu_x\), obtain the amplitude and the probability \(P_{e\to x}=\sin^22\theta_m\sin^2(\Phi/2)\).
2. Show that the common shift \(n_{\mathrm{mat},i}\to n_{\mathrm{mat},i}+C\) changes only the overall phase. Decompose the increment \(\operatorname{diag}(\delta n_e,\delta n_x)\) into a common part and a part that distinguishes the flavors.
3. Calculate \(P_{e\to x}\) for \(\theta_m=30^\circ\) and \(\Phi=\pi\). How does the answer change for \(\theta_m=0\), \(\theta_m=45^\circ\), or equal refractive indices?
4. Find the first maximum and the spatial period of the probability. What does the angle change, and what does the difference of the refractive indices change?
5. Check your answers with the applet (in Russian). Which parameters are specified independently? How can they be found for a layer of given density from the weak interaction?

Exercise 11.7. A half-wave plate and neutrinos

The input is a pure state \(\nu_e\), the mass axes \(\nu_1,\nu_2\) are rotated by \(-\theta\) relative to the flavor axes. The phase difference of the waves \(\nu_1\) and \(\nu_2\) is \(\pi\).

1. Find the output state in the basis \((\nu_e,\nu_x)\) and verify that it is rotated by \(2\theta\) relative to the input.
2. Calculate \(P_{e\to x}\) for \(\theta=33.7^\circ\) and compare with the fractions of light intensity \(I_V\) and \(I_H\) for the same plate.
3. Does the output coincide with one of the crystal axes? For which \(\theta\) will it coincide with \(\nu_x\)?

Exercise 11.8. The refractive index operator

1. Starting from \(\hat n_{\mathrm{vac}}=\sum_i n_i|\nu_i\rangle\langle\nu_i|\) and the matrix \(U\), obtain the matrix \([\hat n_{\mathrm{vac}}]_f\) in the flavor basis.
2. Show that its trace and determinant do not depend on \(\theta\).
3. Add \(\delta\hat n=\mathrm{diag}(\delta n_e,\delta n_x)\) and find the angle \(\theta_m\) as a function of \(\theta\), \(n_2-n_1\), and \(\delta n_e-\delta n_x\). For what relation among these quantities is \(\theta_m=45^\circ\)?

Exercise 11.9. Vacuum and the Sun: two crystals of the same scale

A neutrino with energy \(E=10\) MeV, two flavors, \(\Delta m^2=m_2^2-m_1^2= 7.5\times10^{-5}\) eV\(^2\), \(\theta=33.7^\circ\). Constants: \(\hbar c=1.973\times10^{-7}\) eV\(\cdot\)m, \(G_F=1.166\times10^{-5}\) GeV\(^{-2}\), solar radius \(R_\odot=6.96\times10^{8}\) m. At the center of the Sun the electron density is \(\mathcal N_e=6.08\times10^{25}\) cm\(^{-3}\), and the charged current gives the difference of the additions (with \(\hbar=c=1\); take the formula as given)

\[ \delta n_e-\delta n_x=-\frac{\sqrt2\,G_F\mathcal N_e}{E}. \tag{291}\]

1. Find the neutrino wavelength \(\lambda=2\pi\hbar c/E\). How many wavelengths fit between the center of the Sun and its surface?
2. The vacuum crystal. From \(n_i\simeq1-m_i^2/(2E^2)\) find \(n_1-n_2\). Which of the two indices is larger? Over what length \(L_{\mathrm{vac}}\) does the phase difference \(E(n_1-n_2)z\) reach \(2\pi\), and how many wavelengths does it contain?
3. Compare with optics: for calcite \(n_o-n_e=0.172\) at \(\lambda_0=589\) nm. Find the thickness of a calcite half-wave plate and the length of a neutrino “half-wave plate” in vacuum. Express both in wavelengths.
4. Matter. Compute \(V=\sqrt2\,G_F\mathcal N_e\) in eV (convert the density to natural units with the factor \((\hbar c)^3\)) and \(\delta n_e-\delta n_x\). Find the ratio \[ A=\frac{\delta n_x-\delta n_e}{n_1-n_2}. \tag{292}\]
5. If the central density persisted up to the surface, what phase difference \(E|\delta n_e-\delta n_x|R_\odot\) would the neutrino accumulate? In the real Sun the density falls toward the edge, and about \(4\times10^{3}\) rad accumulates. What path length at the central density does this correspond to?
6. At constant central density the eigenstates of the sum are determined by \[ \tan2\theta_m=\frac{\sin2\theta}{\cos2\theta-A}, \qquad \frac{n_{\mathrm{mat},1}-n_{\mathrm{mat},2}}{n_1-n_2} =\sqrt{(\cos2\theta-A)^2+\sin^22\theta}. \tag{293}\] Find \(\theta_m\) (in which quadrant does \(2\theta_m\) lie?), the amplitude \(\sin^22\theta_m\) and the period \(L_{\mathrm{mat}}\). Compare with vacuum. What fraction of \(\nu_e\) belongs to \(\nu_{\mathrm{mat},2}\)?

State the main physical conclusion: why does a difference of indices of order \(10^{-19}\) not mean a small effect, and why can neither vacuum nor matter be neglected at the center of the Sun?

Exercise 11.10. A refractive index below one

An electron in an atom is an oscillator with natural frequency \(\omega_0\) and damping \(\gamma\). In Gaussian units, \(r_e=e^2/(mc^2)=2.818\times10^{-13}\) cm,

\[ f(0)=r_e\,\frac{\omega^2}{\omega_0^2-\omega^2-i\gamma\omega}, \qquad n-1=\frac{2\pi\mathcal N f(0)}{k^2}, \tag{294}\]

where \(\mathcal N\) is the electron density. Consider silicon: \(\rho=2.33\) g/cm\(^3\), molar mass \(M=28.09\) g/mol, \(Z=14\), the X-ray line \(\lambda=0.154\) nm (\(\hbar\omega\approx8.05\) keV, all \(\omega_0\) of the atom are much smaller than \(\omega\)).

1. Neglecting \(\gamma\), find the sign of \(n-1\) for \(\omega<\omega_0\) and for \(\omega>\omega_0\). Show that for \(\omega<\omega_0\) the index grows with frequency. Which light does a prism deflect more strongly, red or blue?
2. Show that for \(\omega\gg\omega_0\) \[ n=1-\delta,\qquad \delta=\frac{\mathcal N r_e\lambda^2}{2\pi}, \tag{295}\] and \(\omega_0\) drops out of the answer. Find \(\mathcal N\) and \(\delta\) for silicon.
3. A “paradox”: for \(n<1\) the phase velocity \(c/n\) exceeds \(c\). Write \(\delta=\omega_p^2/(2\omega^2)\), where \(\omega_p^2=4\pi\mathcal Ne^2/m\), and from \(k=n\omega/c\) find the group velocity \(v_g=d\omega/dk\). Show that \(v_g<c\) and \(v_{\mathrm{ph}}v_g\simeq c^2\). Compute \(\hbar\omega_p\).
4. An X-ray beam falls from vacuum onto a smooth silicon surface at a grazing angle \(\alpha\) to it. Using Snell’s law, show that for \(\alpha<\alpha_c\), \(\cos\alpha_c=n\), there is no refracted beam and total external reflection occurs. Find \(\alpha_c\approx\sqrt{2\delta}\) in mrad and in degrees. Why are the mirrors of X-ray telescopes set almost parallel to the beam?

State the main conclusion: what sets the sign of \(n-1\), and why does a neutrino refractive index below one not contradict causality?