Exercises: Neutrinos. First Contact

Lecture 01 · Introduction to Neutrino Physics

Author

Dmitry V. Naumov

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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?

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.