Past exam of the particle physics course of the University of Oxford / 2021 / 4

The chiral projectors $P_{L,R}=(1\mp {\gamma }^5)/2$ obey ${\gamma }^{\mu }{P}_L=P_R{\gamma }^{\mu }$, while the Dirac adjoints are ${\bar{\psi }}_L=\bar{\psi }{P}_R$ and ${\bar{\psi }}_R=\bar{\psi }{P}_L$. Consequently the cross terms vanish and the vector current isThe vector interaction preserves fermion chirality at a vertex and couples equally to left and right components. This statement about the vertex does not assert that a fermion mass term preserves chirality. Under parity, left and right components are exchanged; their equal coefficients allow a parity-invariant contraction with a true vector field.

An axial vector must couple to the axial current to obtain the analogous parity-invariant interaction. Since ${\gamma }^5{\psi }_L=-{\psi }_L$ and ${\gamma }^5{\psi }_R={\psi }_R$,The relative sign changes; the overall sign is conventional. A pure axial interaction can itself conserve parity when $A_{\mu }$ transforms as an axial vector. Simply contracting the original vector current with an axial-vector field would instead produce a parity-odd interaction.

For a massless fermion, chirality is fixed by helicity. For a massless antifermion, the field-chirality label is opposite to its physical helicity: a left-chiral field creates negative-helicity particles and positive-helicity antiparticles. At $E\gg m$, wrong-chirality amplitudes are suppressed by $m/E$, so chirality-preserving vector vertices conserve helicity along a physical fermion line up to these corrections.

For electron-positron annihilation into a muon pair through a vector mediator, each annihilating or created pair therefore has opposite physical helicities. Writing each pair in the order (negative particle, positive particle), the nonzero possibilities areHere $L,R$ refer to physical helicity along each particle's own momentum, not to the sign of its spin projected on a single common axis. The incoming pair has total spin projection $m=\pm 1$ along the electron beam; the outgoing pair has $m^{\prime }=\pm 1$ along the muon direction. Same-helicity pairs vanish in the strictly massless vector-current limit.

Assume unpolarized beams, negligible electron and muon masses, a single scattering s-channel spin-one mediator with equal left and right vector couplings, and leading-order QED. Z boson exchange and radiative corrections are omitted. The angular factors are the Wigner D-matrix elements $d_{m{m}^{\prime }}^1({\theta }^{\star })$, with $m,m^{\prime }=\pm 1$. Up to common phases and normalization they are $(1+\cos {\theta }^{\star })$ for equal pair-spin projections and $(1-\cos {\theta }^{\star })$ for opposite projections. The four orthogonal helicity channels add incoherently, givingWith $s$ the squared center-of-mass energy, the normalized differential scattering cross-sections areThe second expression includes the solid angle Jacobian: a distribution per angle is different from one per $\cos {\theta }^{\star }$. Integration gives $\sigma =4\pi {\alpha }^2/(3s)$.

Parity reverses all momenta but leaves the spin axial vector unchanged, so it interchanges $L\leftrightarrow R$. The $LR\to LR$ and $RL\to RL$ amplitudes have equal magnitudes, as do $LR\to RL$ and $RL\to LR$. Thus every configuration has the same rate as its parity transform, proving parity conservation. The summed distribution is also forward-backward symmetric. Strictly, parity reverses both the beam and outgoing momenta, leaving their relative angle unchanged; forward-backward reflection by itself is not the definition of parity.

Assume the same left and right couplings $g_l,g_r$ at the electron and muon vertices. This lepton universality assumption is necessary to describe both vertices using just two numbers. Define $a=|g_l{|}^2$, $b=|g_r{|}^2$ and $z=\cos {\theta }^{\star }$. The two same-chirality channels have squared coupling weights $a^2,b^2$ and angular factor $(1+z{)}^2$; the other two each have weight $ab$ and factor $(1-z{)}^2$. Hence, for a common positive normalization $C$,If $|g_l/g_r|\ne 1$, then $a\ne b$, so a forward-backward term survives. At the amplitude level, parity interchanges the $g_l^2$ and $g_r^2$ channels; their unequal squared magnitudes show directly that the interaction violates parity. Equivalently the chiral current contains both vector current and axial current terms, with coefficients $(g_r+g_l)/2$ and $(g_r-g_l)/2$; unequal magnitudes preclude a pure parity eigen-current.

The odd-in-$z$ term is a useful signature within these assumptions, but it should not itself be called a parity-odd scalar: the unpolarized relative angle is unchanged when all momenta are reversed. If the two species have independent couplings, the coefficient is instead proportional to $(a_e-b_e)(a_{\mu }-b_{\mu })$, and a parity-violating coupling at only one vertex need not produce an unpolarized forward-backward asymmetry.

Integrating the distribution in d, use ${\int }_{-1}^1(1+z^2)dz=8/3$ and ${\int }_0^{1}zdz-{\int }_{-1}^{0}zdz=1$. ThereforeSubtract backgrounds and correct the measured forward and backward yields for detector acceptance, charge misidentification and selection efficiency; radiative effects and other mediators must be removed or included in a fit. With $t=\sqrt{4A_{\mathrm{fb}}/3}$ and $r=|g_l/g_r|$, inversion givesA single unpolarized measurement determines the ratio up to $r\leftrightarrow 1/r$. A polarized-beam observable or another signed chiral asymmetry is needed to choose which coupling is larger. Values outside $0\le A_{\mathrm{fb}}\le 3/4$ would contradict this ideal universal single-mediator model, apart from experimental uncertainties.