Past exam of the particle and nuclear physics course of the University of Cambridge / 2017 / 1

Use the inverse beta decay threshold, with the charged-lepton mass generalized to $m_{\ell }$. For a stationary proton and a massless incident antineutrino, the Mandelstam invariant is $s=m_p^2+2m_p{E}_{\nu }$. At threshold the final particles have no relative momentum in their centre-of-momentum frame, so $s_{\mathrm{thr}}=(m_n+m_{\ell }{)}^2$. ThereforeUsing the supplied, rounded masses givesThe second term is the recoil correction; setting the threshold equal just to the final rest-mass excess would miss it.

Reactor electron antineutrinos have energies of a few million electronvolts. They can produce a Positron through inverse beta decay above the first threshold, but cannot produce a muon. Thus reactor neutrino oscillations are measured primarily as electron-antineutrino disappearance, through the event rate and energy spectrum relative to the unoscillated prediction. An oscillated muon antineutrino cannot be identified by the proposed muon-production channel at reactor energies; disappearance alone does not distinguish all possible final flavours.

The separate proton and neutron shell capacities are $2j+1=2,4,2,6,\dots$. Filled shells and paired particles have zero net spin and magnetic dipole moment. The single-particle nuclear magnetic moment is $\mu =g_{j}j{\mu }_N$, with orbital factors $g_{\ell }^p=1$, $g_{\ell }^n=0$ and $s=1/2$.

For nitrogen, the eight neutrons form a closed shell. The seventh proton occupies $1p_{1/2}$, so $\ell =1$, $j=1/2$ and the Landé g-factor givesHence the nuclear shell model predictsIts single-particle spin and parity is $1/2^{-}$. The negative moment results from the proton spin being predominantly opposed to the total angular momentum in this $j=\ell -1/2$ level.

For oxygen, eight protons and the first eight neutrons are paired. The ninth neutron occupies $1d_{5/2}$, with $\ell =2$, $j=5/2=\ell +1/2$. Since its orbital factor is zero, $g_j=g_n/(2\ell +1)=-3.826/5$. ThereforeThese are ideal single-particle predictions in units of the nuclear magneton, not fits incorporating nuclear configuration mixing.

For negligible final fermion masses, the chiral Z-boson partial width is proportional to $g_L^2+g_R^2$. Both channels have one color state. An Electron has $Q=-1$, $T_{3L}=-1/2$ and $T_{3R}=0$, givingFor one active neutrino species, $Q=0$, $T_{3L}=1/2$ and there is no coupled right-handed neutrino field, so $g_L^{\nu }=1/2$, $g_R^{\nu }=0$. The common coupling normalization and phase space cancel in the ratio:The same total decay width appears in the two branching fractions, soThere is no factor of three for a single specified flavour. Summing three active neutrino flavours would give an invisible branching fraction of approximately $20.0\%$ in this approximation.