Exercises: Neutrino Spin. Helicity and Polarization
Lecture 03 · Introduction to Neutrino Physics
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Exercises
Exercise 3.1. Nuclear recoil in the Goldhaber experiment
In the Goldhaber experiment, electron capture leads to the reaction
\[ {}^{152m}\mathrm{Eu}+e^- \longrightarrow {}^{152}\mathrm{Sm}^*+\nu_e. \tag{1}\]
Take
\[ E_\nu\simeq1~\mathrm{MeV}, \qquad M_{B^*}\simeq152~u, \qquad uc^2=931.5~\mathrm{MeV}. \tag{2}\]
Neglecting the momentum of the captured electron, show that \(p_{B^*}\simeq p_\nu\simeq E_\nu/c\).
Calculate the recoil energy
\[ T_{B^*}\simeq\frac{E_\nu^2}{2M_{B^*}c^2} \tag{3}\]
and the nuclear velocity \(v_{B^*}=p_{B^*}/M_{B^*}\).
If the nucleus loses the direction of recoil over a length \(\ell=1\)–\(5~\text{Å}\), estimate \(\tau_{\mathrm{stop}}\sim\ell/v_{B^*}\).
Compare the result with the lifetime of the excited state \(\tau(B^*)\simeq0.03~\mathrm{ps}\).
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:
\[ P_{\mathrm{opp}}=\frac{1-\beta_\nu}{2}, \qquad \beta_\nu=\frac{p_\nu}{E_\nu}. \tag{4}\]
Using \(E_\nu^2=p_\nu^2+m_\nu^2\), show that for \(E_\nu\gg m_\nu\)
\[ P_{\mathrm{opp}}\simeq \left(\frac{m_\nu}{2E_\nu}\right)^2. \tag{5}\]
For \(E_\nu=1~\mathrm{MeV}\) and \(m_\nu<0.45~\mathrm{eV}\) obtain the upper bound \(P_{\mathrm{opp}}\).
Why do helicity and chirality practically coincide for an ultrarelativistic neutrino, even though they are different concepts?
Which of these characteristics can change sign upon a transition to another inertial frame?
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
\[ A(J=0)+e^- \longrightarrow B^*(J=1)+\nu_e. \tag{6}\]
Direct the axis \(z\) along the neutrino momentum; neglect the orbital angular momentum of the final state.
Write down the conservation law for the projection of the total angular momentum:
\[ m_e=m_{B^*}+m_\nu. \tag{7}\]
List the allowed values of \(m_e=\pm\tfrac12\), \(m_\nu=\pm\tfrac12\) and \(m_{B^*}=-1,0,+1\).
Construct a table of the allowed triples \((m_e,m_{B^*},m_\nu)\).
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?
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.
From the transformation
\[ E_\gamma^{\mathrm{lab}} =\gamma E_\gamma(1+\beta\cos\theta) \tag{8}\]
obtain, to first order in \(v/c\),
\[ E_\gamma^{\mathrm{lab}} \simeq E_\gamma \left(1+\frac vc\cos\theta\right). \tag{9}\]
Find \(E_{\max}\), \(E_{\min}\) and the width \(E_{\max}-E_{\min}\).
Using the velocity from Exercise 03.1, estimate the relative Doppler shift.
In both emission and absorption of a photon a free nucleus of mass \(M\) receives the recoil energy
\[ E_R\simeq\frac{E_\gamma^2}{2Mc^2}, \tag{10}\]
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.
Why does resonant absorption select a narrow range of angles?
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
\[ p_\nu'=\gamma_u \left(p_\nu-\frac{uE_\nu}{c^2}\right). \tag{11}\]
Obtain the condition under which \(p_\nu'\) changes sign. Check the same result with the velocity-addition law \(v_\nu'=(v_\nu-u)/(1-uv_\nu/c^2)\).
Show that for \(E_\nu\gg m_\nu c^2\)
\[ 1-\beta_\nu\simeq \frac{m_\nu^2c^4}{2E_\nu^2}. \tag{12}\]
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 \(c\).
Estimate the minimum Lorentz factor of such an observer and compare it with \(\gamma_\nu=E_\nu/(m_\nu c^2)\).
Why does the helicity change sign after the overtaking? Why is the same thought experiment impossible for a strictly massless particle?
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
\[ N_+=0.988\times10^6, \qquad N_-=1.012\times10^6. \tag{13}\]
The count asymmetry
\[ \mathcal A=\frac{N_+-N_-}{N_++N_-} \tag{14}\]
is related to the circular polarization of the photons by
\[ \mathcal A\simeq k\,h_\gamma, \qquad k=f_{\mathrm{res}}P_ea_\gamma, \tag{15}\]
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 \(k=0.012\pm0.003\).
Compute \(\mathcal A\).
Treat each detected photon as an independent trial that falls into the count \(N_+\) with probability \(p=N_+/(N_++N_-)\). Show that
\[ \sigma_{\mathcal A}\simeq \sqrt{\frac{1-\mathcal A^2}{N_++N_-}}, \tag{16}\]
and compute \(\sigma_{\mathcal A}\).
Find \(h_\gamma\) and its statistical uncertainty. Why, in the geometry selected by the resonance, is this the neutrino helicity?
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 \(h(\nu_e)=-1.0\pm0.3\).
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
\[ Mc^2\simeq1.42\times10^5~\mathrm{MeV}, \qquad \tau\simeq0.03~\mathrm{ps}, \qquad \hbar=6.58\times10^{-16}~\mathrm{eV\cdot s}, \qquad k_B=8.62\times10^{-5}~\mathrm{eV/K}. \tag{17}\]
From energy and momentum conservation, show that for a nucleus at rest
\[ E_\gamma^{\mathrm{em}}\simeq\Delta E-\frac{(\Delta E)^2}{2Mc^2}, \qquad E_\gamma^{\mathrm{abs}}\simeq\Delta E+\frac{(\Delta E)^2}{2Mc^2}. \tag{18}\]
Find the gap \(\delta=E_\gamma^{\mathrm{abs}}-E_\gamma^{\mathrm{em}}\) in electronvolts.
Find the natural width of the level \(\Gamma=\hbar/\tau\). By what factor is the gap larger than the width?
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
\[ \sigma_D\simeq\Delta E\sqrt{\frac{k_BT}{Mc^2}}, \tag{19}\]
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?
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?
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
\[ \gamma^0= \begin{pmatrix}0&I\\I&0\end{pmatrix}, \qquad \gamma^i= \begin{pmatrix}0&\sigma_i\\-\sigma_i&0\end{pmatrix}, \qquad \gamma^5= \begin{pmatrix}-I&0\\0&I\end{pmatrix}. \tag{20}\]
Consider a spinor with momentum \(\mathbf p\), \(p=|\mathbf p|\), and helicity \(h=\pm1\):
\[ u_h(\mathbf p)= \begin{pmatrix} \sqrt{E-hp}\;\xi_h\\[3pt] \sqrt{E+hp}\;\xi_h \end{pmatrix}, \qquad (\boldsymbol\sigma\cdot\widehat{\mathbf p})\,\xi_h=h\,\xi_h, \qquad \xi_h^\dagger\xi_h=1. \tag{21}\]
Show that
\[ \gamma^\mu p_\mu= \begin{pmatrix} 0&E-\boldsymbol\sigma\cdot\mathbf p\\ E+\boldsymbol\sigma\cdot\mathbf p&0 \end{pmatrix}, \tag{22}\]
and, using \(\sqrt{E-p}\,\sqrt{E+p}=m\), check that \(u_h\) satisfies the equation \((\gamma^\mu p_\mu-m)u_h=0\).
Write down the projectors \(P_{L,R}=(1\mp\gamma^5)/2\). Which components of the spinor do they select? Check that \(u_h^\dagger u_h=2E\).
Obtain the fractions of the chiral components
\[ f_{L,R}(h)=\frac{\|P_{L,R}u_h\|^2}{\|u_h\|^2} =\frac{1\mp h\beta}{2}, \qquad \beta=\frac pE. \tag{23}\]
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 \(m=0.05~\mathrm{eV}\).
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
\[ T_\nu\simeq1.95~\mathrm K, \qquad \langle p\rangle\simeq3.15\,k_BT_\nu, \qquad k_B=8.62\times10^{-5}~\mathrm{eV/K}. \tag{24}\]
Take the mass \(m=0.05~\mathrm{eV}\) and set \(\hbar=c=1\).
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
\[ [H,\gamma^5]=2m\,\beta\gamma^5, \tag{25}\]
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?
Estimate the probability of the “wrong” helicity \(h=+1\) at decoupling, \(P\simeq(m/2E)^2\) for \(E\simeq3~\mathrm{MeV}\).
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.
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.
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
\[ \pi^+\to\mu^++\nu_\mu, \qquad \pi^+\to e^++\nu_e. \tag{26}\]
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 \(c=1\).
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.
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?
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
\[ 1-\beta_\ell=\frac{m_\ell^2}{m_\pi E_\ell}, \tag{27}\]
and compute this fraction for the muon and for the positron.
A full calculation with the left-handed current gives
\[ \Gamma(\pi^+\to\ell^+\nu_\ell)\propto m_\ell^2\left(m_\pi^2-m_\ell^2\right)^2. \tag{28}\]
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?
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?
Exercises for discussion in the seminar
Seminar 03.1. How does helicity transform?
The helicity of a particle is defined as
\[ h=\frac{\mathbf S\cdot\mathbf p}{|\mathbf p|}. \tag{29}\]
Trace the transformation of \(\mathbf S\), \(\mathbf p\) and \(h\) under
- an ordinary spatial rotation;
- spatial inversion \(P\);
- 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.
- Draw the directions of the momenta of the neutrino, the recoil nucleus and the selected \(\gamma\)-quantum.
- Explain the chain “neutrino helicity \(\to\) recoil-nucleus polarization \(\to\) circular polarization of the photon” using conservation of angular momentum.
- 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
\[ E_{\mathrm{lab}}\simeq E_\gamma(1+\beta_R\cos\theta). \tag{30}\]
The absorber has a resonance energy \(E_{\mathrm{res}}\) and a width \(\Gamma\).
- Find the direction \(\cos\theta_{\mathrm{res}}\) that falls into resonance.
- Show that the width of the selected angular interval in \(\cos\theta\) is of order \(\Gamma/(\beta_RE_\gamma)\).
- 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 \(z\).
- Find its velocity \(\beta\).
- What minimum Lorentz factor does an observer need in order to move faster than the particle in the same direction?
- What happens to the direction of the momentum and to the helicity of the particle in the frame of such an observer?
- At which step does the argument stop working for \(m=0\)?