Exercises: Three Flavors and the PMNS Matrix

Lecture 10 · Introduction to Neutrino Physics

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Exercises

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

   \[ |U_0|^2= \begin{pmatrix} 2/3&1/3&0\\ 1/6&1/3&1/2\\ 1/6&1/3&1/2 \end{pmatrix}. \tag{1}\]

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

\[ U=R_{23}U_{13}(\delta)R_{12}. \tag{2}\]

1. Multiply the matrices and obtain the first row and the third column of \(U\).

2. Prove the relations

   \[ |U_{e3}|=\sin\theta_{13}, \qquad \frac{|U_{e2}|}{|U_{e1}|}=\tan\theta_{12}, \qquad \frac{|U_{\mu3}|}{|U_{\tau3}|}=\tan\theta_{23}. \tag{3}\]

3. For the NuFIT 6.1 best-fit point with normal ordering use

   \[ \sin^2\theta_{12}=0.3088, \quad \sin^2\theta_{23}=0.470, \quad \sin^2\theta_{13}=0.02248 \tag{4}\]

   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

   \[ \widetilde U =U\operatorname{diag} \left(1,e^{i\alpha_{21}/2},e^{i\alpha_{31}/2}\right). \tag{5}\]

   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

\[ |\nu_\alpha\rangle =\sum_{i=1}^3U_{\alpha i}^*|\nu_i\rangle. \tag{6}\]

1. Carry out the evolution of the mass components and obtain the ultrarelativistic amplitude

   \[ \mathcal A_{\alpha\to\beta}(L,E) =\sum_iU_{\beta i}U_{\alpha i}^* \exp\!\left(-i\frac{m_i^2L}{2E}\right). \tag{7}\]

2. Square the amplitude and obtain a double sum over \(i,j\).

3. Combine the terms \((i,j)\) and \((j,i)\), introduce

   \[ X_{\alpha\beta}^{ij} =U_{\alpha i}^*U_{\beta i} U_{\alpha j}U_{\beta j}^*, \qquad \Delta_{ij}=\frac{\Delta m_{ij}^2L}{4E}, \tag{8}\]

   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

   \[ \sum_\beta P_{\alpha\to\beta}=1 \tag{9}\]

   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

   \[ \Delta m_{31}^2=\Delta m_{32}^2+\Delta m_{21}^2 \tag{10}\]

   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

   \[ A_{\alpha\beta}^{\mathrm{CP}} =P(\nu_\alpha\to\nu_\beta) -P(\bar\nu_\alpha\to\bar\nu_\beta) =4\sum_{i>j}\operatorname{Im}X_{\alpha\beta}^{ij} \sin2\Delta_{ij}. \tag{11}\]

3. Prove that \(A_{\alpha\alpha}^{\mathrm{CP}}=0\) in vacuum.

4. For the quartet

   \[ J_{\mathrm{CP}} =\operatorname{Im} \left(U_{e2}^*U_{\mu2}U_{e1}U_{\mu1}^*\right) \tag{12}\]

   check the invariance under rephasing of the rows and columns of the mixing matrix.

5. From the standard PMNS parametrization obtain

   \[ J_{\mathrm{CP}} =\frac18 \sin2\theta_{12}\sin2\theta_{23}\sin2\theta_{13} \cos\theta_{13}\sin\delta_{\mathrm{CP}}. \tag{13}\]

6. Using \(\Delta_{31}=\Delta_{32}+\Delta_{21}\) and trigonometric identities, derive

   \[ A_{e\mu}^{\mathrm{CP}} =16J_{\mathrm{CP}} \sin\Delta_{21}\sin\Delta_{32}\sin\Delta_{31}. \tag{14}\]

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

   \[ \theta_{12}=33.76^\circ, \quad \theta_{23}=43.29^\circ, \quad \theta_{13}=8.62^\circ, \quad \delta_{\mathrm{CP}}=212^\circ \tag{15}\]

   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

   \[ \Delta m_{21}^2=7.537\cdot10^{-5}~\mathrm{eV}^2, \qquad \Delta m_{31}^2=2.511\cdot10^{-3}~\mathrm{eV}^2. \tag{16}\]

   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:

\[ \begin{aligned} \sin^2\theta_{12}&=0.3088,& \sin^2\theta_{23}&=0.470,& \sin^2\theta_{13}&=0.02248,\\ \Delta m_{21}^2&=7.537\cdot10^{-5}~\mathrm{eV}^2,& \Delta m_{31}^2&=2.511\cdot10^{-3}~\mathrm{eV}^2,& \delta_{\mathrm{CP}}&=212^\circ. \end{aligned} \tag{17}\]

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

   \[ \mathcal A =U\operatorname{diag}(e^{-i\phi_i})U^\dagger, \qquad \phi_i=2(1.267)\frac{m_i^2[\mathrm{eV}^2]L[\mathrm{km}]} {E[\mathrm{GeV}]}. \tag{18}\]

   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

   \[ P_{\beta\alpha}(0)=\delta_{\beta\alpha}, \qquad \sum_\beta P_{\beta\alpha}=1 \tag{19}\]

   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

    \[ (P_{\alpha\to e},P_{\alpha\to\mu},P_{\alpha\to\tau}) \tag{20}\]

    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

\[ U_{e1}=c_{12}c_{13}, \qquad U_{\mu1}=-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}, \qquad U_{\tau1}=s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}. \tag{21}\]

1. Show that

   \[ |U_{\mu1}|^2 =s_{12}^2c_{23}^2+c_{12}^2s_{23}^2s_{13}^2 +2s_{12}c_{12}s_{23}c_{23}s_{13}\cos\delta, \tag{22}\]

   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

   \[ \sin^2\theta_{12}=0.3088, \quad \sin^2\theta_{23}=0.470, \quad \sin^2\theta_{13}=0.02248, \quad \delta=212^\circ \tag{23}\]

   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:

\[ \begin{aligned} \sin^2\theta_{12}&=0.3088,& \sin^2\theta_{23}&=0.470,& \sin^2\theta_{13}&=0.02248,\\ \Delta m_{21}^2&=7.537\cdot10^{-5}~\mathrm{eV}^2,& \Delta m_{31}^2&=2.511\cdot10^{-3}~\mathrm{eV}^2, \end{aligned} \tag{24}\]

together with

\[ P_{\alpha\to\alpha} =1-4\sum_{i>j}|U_{\alpha i}|^2|U_{\alpha j}|^2\sin^2\Delta_{ij}, \qquad \Delta_{ij}=1.267\, \frac{\Delta m^2_{ij}[\mathrm{eV}^2]L[\mathrm{km}]}{E[\mathrm{GeV}]}. \tag{25}\]

1. Find the ratio \(\Delta m_{21}^2/\Delta m_{31}^2\). Setting \(\Delta_{21}=0\), show that

   \[ P_{\alpha\to\alpha} \simeq1-4|U_{\alpha3}|^2\left(1-|U_{\alpha3}|^2\right) \sin^2\Delta_{31}. \tag{26}\]

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

   \[ P_{\alpha\to\beta} \simeq4|U_{\alpha3}|^2|U_{\beta3}|^2\sin^2\Delta_{31}, \qquad \alpha\ne\beta, \tag{27}\]

   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

   \[ P_{ee}\simeq c_{13}^4 \left(1-\sin^2 2\theta_{12}\sin^2\Delta_{21}\right)+s_{13}^4. \tag{28}\]

   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:

\[ \begin{aligned} \text{NO:}&\quad\Delta m_{31}^2=+2.511\cdot10^{-3}~\mathrm{eV}^2, &\Delta m_{32}^2&=\Delta m_{31}^2-\Delta m_{21}^2,\\ \text{IO:}&\quad\Delta m_{32}^2=-2.511\cdot10^{-3}~\mathrm{eV}^2, &\Delta m_{31}^2&=\Delta m_{32}^2+\Delta m_{21}^2, \end{aligned} \tag{29}\]

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

   \[ P_{ee} =1-c_{13}^4\sin^2 2\theta_{12}\sin^2\Delta_{21} -\sin^2 2\theta_{13} \left(c_{12}^2\sin^2\Delta_{31}+s_{12}^2\sin^2\Delta_{32}\right). \tag{30}\]

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

   \[ P_{ee}^{\mathrm{NO}}-P_{ee}^{\mathrm{IO}} =-\sin^2 2\theta_{13}\cos2\theta_{12}\, \sin(\Delta_a+\Delta_b)\sin\Delta_{21}. \tag{31}\]

   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

\[ z_1+z_2+z_3=0, \qquad z_i=U_{ei}U_{\mu i}^*. \tag{32}\]

1. For the NuFIT 6.1 best-fit point with normal ordering

   \[ \sin^2\theta_{12}=0.3088, \quad \sin^2\theta_{23}=0.470, \quad \sin^2\theta_{13}=0.02248, \quad \delta_{\mathrm{CP}}=212^\circ \tag{33}\]

   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

   \[ J_{\mathrm{CP}} =\frac18\sin2\theta_{12}\sin2\theta_{23}\sin2\theta_{13} \cos\theta_{13}\sin\delta_{\mathrm{CP}}. \tag{34}\]

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?