Exercises: Neutrino Oscillations in Vacuum. Two Flavors

Lecture 09 · Introduction to Neutrino Physics

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Exercises

Exercise 9.1. Global and relative phases

Consider a normalized superposition of two energy eigenstates:

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

1. Show that the replacement

   \[ |\psi(t)\rangle\longrightarrow e^{i\chi(t)}|\psi(t)\rangle \tag{2}\]

   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

   \[ |a\rangle =\cos\eta\,|E_1\rangle +e^{i\varphi}\sin\eta\,|E_2\rangle. \tag{3}\]

   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

\[ |\nu_\alpha\rangle =c_\theta|\nu_1\rangle+s_\theta|\nu_2\rangle, \qquad |\nu_\beta\rangle =-s_\theta|\nu_1\rangle+c_\theta|\nu_2\rangle, \tag{4}\]

where

\[ H_0|\nu_i\rangle=E_i|\nu_i\rangle, \qquad c_\theta=\cos\theta, \quad s_\theta=\sin\theta. \tag{5}\]

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

   \[ \operatorname{Var}(H_0) =\langle H_0^2\rangle-\langle H_0\rangle^2; \tag{6}\]

5. show that

   \[ \operatorname{Var}_\alpha(H_0) =\operatorname{Var}_\beta(H_0) =\frac14\sin^2 2\theta\,(E_2-E_1)^2. \tag{7}\]

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 \(t=0\):

\[ |\psi(0)\rangle=|\nu_\alpha\rangle. \tag{8}\]

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

   \[ \mathcal A_{\alpha\to\alpha}(t) =c_\theta^2e^{-iE_1t}+s_\theta^2e^{-iE_2t}, \tag{9}\]

   \[ \mathcal A_{\alpha\to\beta}(t) =s_\theta c_\theta \left(e^{-iE_2t}-e^{-iE_1t}\right). \tag{10}\]

4. Introduce

   \[ \overline E=\frac{E_1+E_2}{2}, \qquad \Delta E=E_2-E_1, \tag{11}\]

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

5. Check unitarity:

   \[ P_{\alpha\to\alpha}(t)+P_{\alpha\to\beta}(t)=1. \tag{12}\]

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

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

   \[ P_{\alpha\to\beta}(t) \simeq \sin^2 2\theta\,\frac{(\Delta E)^2t^2}{4}. \tag{13}\]

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

\[ e^{-i(E_it-p_iL)}. \tag{14}\]

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

   \[ \Delta\varphi(t,L)=\Delta E\,t-\Delta p\,L. \tag{15}\]

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

   \[ \Delta E =\frac{\Delta m^2}{E_2+E_1}, \qquad \Delta m^2=m_2^2-m_1^2. \tag{16}\]

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

   \[ \Delta\varphi\simeq\frac{\Delta m^2L}{2E}. \tag{17}\]

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

   \[ \hbar c=1.97327\cdot10^{-7}~\mathrm{eV\,m}, \tag{18}\]

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

   \[ P_{\alpha\to\beta} =\sin^2 2\theta\, \sin^2\!\left[ 1.267\, \frac{\Delta m^2[\mathrm{eV}^2]L[\mathrm{km}]} {E[\mathrm{GeV}]} \right]. \tag{19}\]

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

   \[ \Delta E \simeq \Delta m\left(1-\frac{p^2}{2m_1m_2}\right), \qquad \Delta m=m_2-m_1. \tag{20}\]

   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

\[ P_{\alpha\to\beta}^{\max}=0.64. \tag{21}\]

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

   \[ E=0.30~\mathrm{GeV}, \qquad E=0.60~\mathrm{GeV}, \qquad E=1.20~\mathrm{GeV}. \tag{22}\]

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

   \[ T=\frac{\ell^2}{2} \left(m_1\dot\varphi_1^2+m_2\dot\varphi_2^2\right), \tag{23}\]

   \[ V=\frac{\ell^2}{2} \left[ \omega_0^2(m_1\varphi_1^2+m_2\varphi_2^2) +k(\varphi_1-\varphi_2)^2 \right]. \tag{24}\]

2. Obtain the matrix equation

   \[ \mathsf M\ddot{\boldsymbol\varphi} +\mathsf K\boldsymbol\varphi=0 \tag{25}\]

   and write down \(\mathsf M\) and \(\mathsf K\).

3. Introduce mass-normalized coordinates

   \[ \boldsymbol x=\mathsf M^{1/2}\boldsymbol\varphi \tag{26}\]

   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

   \[ \omega_+^2=\omega_0^2, \qquad \omega_-^2 =\omega_0^2+k\left(\frac1{m_1}+\frac1{m_2}\right). \tag{27}\]

5. Introducing the mixing angle through the eigenvectors, prove

   \[ \tan^2\theta=\frac{m_2}{m_1}, \qquad \sin^2 2\theta =\frac{4m_1m_2}{(m_1+m_2)^2}. \tag{28}\]

6. Let \(\Omega\) be the positive square root of \(\Omega^2\). Introduce

   \[ \boldsymbol a =\boldsymbol x+i\Omega^{-1}\dot{\boldsymbol x} \tag{29}\]

   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

   \[ P_{1\to2}(t) =\sin^2 2\theta\, \sin^2\frac{(\omega_--\omega_+)t}{2}. \tag{30}\]

8. For

   \[ \frac{m_2}{m_1}=\frac14, \qquad \frac{k}{m_1\omega_0^2}=0.10 \tag{31}\]

   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

\[ P_{\alpha\to\beta} =\sin^2 2\theta\, \sin^2\!\left[ 1.267\, \frac{\Delta m^2[\mathrm{eV}^2]L[\mathrm{km}]} {E[\mathrm{GeV}]} \right]. \tag{32}\]

Use two scales of the mass splitting, the atmospheric and the solar one:

\[ \Delta m^2_{\mathrm{atm}}=2.5\cdot10^{-3}~\mathrm{eV}^2, \qquad \Delta m^2_{\mathrm{sol}}=7.5\cdot10^{-5}~\mathrm{eV}^2. \tag{33}\]

1. Show that the oscillation length in practical units is

   \[ L_{\mathrm{osc}}[\mathrm{km}] \simeq2.48\, \frac{E[\mathrm{GeV}]}{|\Delta m^2|[\mathrm{eV}^2]}, \tag{34}\]

   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

   \[ P_{\alpha\to\beta} =\frac12\sin^2 2\theta\,(1-\cos\phi). \tag{35}\]

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

   \[ \langle\cos\phi\rangle=e^{-\sigma_\phi^2/2}\cos\phi_0 \tag{36}\]

   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 \(\sigma_E/E\ll1\)

   \[ \sigma_\phi\simeq\phi_0\,\frac{\sigma_E}{E}. \tag{37}\]

   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

\[ \hbar c=1.97327\cdot10^{-7}~\mathrm{eV\,m}. \tag{38}\]

1. Show that in the ultrarelativistic limit

   \[ v_i\simeq1-\frac{m_i^2}{2E^2}, \qquad |v_1-v_2|\simeq\frac{|\Delta m^2|}{2E^2}. \tag{39}\]

2. The packets of the mass components stop overlapping when their relative displacement becomes comparable to \(\sigma_x\). Obtain the coherence length

   \[ L_{\mathrm{coh}}\simeq\frac{2E^2\sigma_x}{|\Delta m^2|} \tag{40}\]

   and show that

   \[ \frac{L_{\mathrm{coh}}}{L_{\mathrm{osc}}} \simeq\frac{\sigma_x}{\lambda}, \qquad \lambda=\frac{2\pi}{E}. \tag{41}\]

   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

   \[ \sigma_x\ll\frac{E}{|\Delta m^2|}=\frac{L_{\mathrm{osc}}}{4\pi}. \tag{42}\]

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?