Past exam of the particle physics course of the University of Oxford / 2018 / 4

Use dimensionless physical helicity $h=\mathbf{S}\cdot \hat{\mathbf{p}}$, with spin eigenvalues $\pm 1/2$ in natural units. In the Dirac representation,The entries $1$ in the supplied transformation denote two-by-two identities. Since its matrix acts only on the upper/lower spinor blocks, it commutes with the repeated Pauli block. Therefore $u^{-1}=u$ andThe eigenvalues remain the physical spin projection along the direction of motion, $+1/2$ or $-1/2$. A convention omitting the factor one half instead labels them $\pm 1$; this changes no measurement probabilities. A change of gamma-matrix representation cannot change the spin observable.

The two Weyl spinors transform in the inequivalent left- and right-chiral representations of the proper Lorentz group. They are selected by $P_{L,R}=(1\mp {\gamma }^5)/2$ and have different boost transformation laws; parity exchanges the two. A massive Dirac equation couples them, whereas in the massless limit they can propagate independently.

For the particular $u$ printed in this paper, transforming its supplied Dirac ${\gamma }^5$ givesThus the upper block is right chiral and the lower block left chiral in this convention. Many references use the opposite block order; neither convention changes physics, but they must not be mixed. Chirality is not generally the same as helicity for a massive electron. The weak charged current selects the left-chiral electron component, reducing to negative helicity at high energy.

Put $r=p/(E+m)$ and assume $p>0$. The common plane wave has unit modulus. The component norm isFor unit spinor probability density choosewhere $E^2=p^2+m^2$ was used. This is spinor normalization, not square normalization of a plane wave over infinite space; a box or wavepacket gives the corresponding spatial normalization. If the convention is instead ${\psi }^{†}\psi =2E$, use $N=\sqrt{(E+m)/10}$.

For momentum along $+z$, the positive-helicity components are the first and third, while the second and fourth have negative helicity. HenceBoth are independent of speed. In the source's chiral block order the exact left-chiral weight is $1/2+3p/(10E)$, tending to $4/5$ at high energy; helicity and chirality probabilities agree only in that limit.

The charged-current process iswith a $t$-channel $W^{-}$ emitted at the electron-to-neutrino vertex and absorbed at the antimuon-to-antineutrino vertex. It is not an $s$-channel Annihilation of the unlike-flavour charged leptons into one $W$.

At high momentum transfer the incident electron's left-chiral fraction tends to $4/5$. In the collision centre-of-mass frame, the incoming antimuon moves along $-z$ and its charged-current allowed right helicity has spin along $-z$. A rest-frame spin at Angle $\theta$ to $+z$ projects onto that spin direction with probability ${\sin }^2(\theta /2)$. Therefore, if ${\sigma }_0$ denotes the cross section for a pure left-helicity electron and fully selected antimuon spin,It vanishes at $\theta =0$ in the ultrarelativistic selection limit and is maximal at $\theta =\pi$. Finite-mass corrections can fill the suppressed orientation. The absolute normalization depends on beam energy and the exchanged-boson propagator; the spin-angle dependence is what is fixed here.

For an incoming ${\mu }^{-}$, a putative $e^{-}{\mu }^{-}\to {\nu }_e{\nu }_{\mu }$ process would change total electric charge from $-2$ to zero. It is forbidden: both negative charged leptons would have to emit a $W^{-}$, and a single exchanged charged boson cannot connect those two emission vertices. Ordinary elastic $e^{-}{\mu }^{-}\to e^{-}{\mu }^{-}$ scattering remains possible through $t$-channel photon or $Z$ exchange, shown in the figure, but this is not the charged-current reaction of part d. The neutral-current weak contribution is distinct from the electromagnetic one.

A proton contains up quarks on which the charged current can act:through $W^{-}$ exchange, weighted by $V_{ud}$. Spectator quarks complete the hadronic final state. At the exclusive nucleon level the charge assignment is $e^{-}p\to {\nu }_{e}n$; at the high momentum transfers specified, the generally dominant description is inclusive $e^{-}p\to {\nu }_{e}X$ with proton structure and hadronic breakup, not an unsuppressed pointlike proton-to-neutron transition. The quark-level diagram in the figure displays this allowed charge flow. The electron's left-chiral selection is the same as in part d.