Past exam of the particle and nuclear physics course of the University of Cambridge / 2012 / Paper 1 / 3

A Feynman diagram specifies a term in the perturbative scattering amplitude, not a measured microscopic trajectory. Assign external spinors or polarization vectors, an interaction factor to each Feynman vertex and a propagator to each internal line; impose four-momentum conservation and integrate over independent loop momenta. Sum amplitudes for indistinguishable alternatives, then square, sum final spins and average unpolarized initial spins. A decay rate uses $d\Gamma =|\mathcal{M}{|}^{2}d{\Phi }_n/(2M)$ in natural units; a cross section also divides by incident flux. interference means probabilities of individual diagrams must not simply be added.

The electromagnetic fermion vertex is $f\bar{f}\gamma$, with coupling proportional to $e{Q}_f{\gamma }^{\mu }$; it preserves fermion flavour. There is no elementary three-photon or four-photon vertex in quantum electrodynamics. The weak charged current has $W\ell \nu$ and $W{u}_i{d}_j$ vertices, with a left-chiral projector and a Cabibbo-Kobayashi-Maskawa matrix element at the quark vertex. The weak neutral current has $Zf\bar{f}$ vertices with vector and axial couplings; it includes neutrinos and is flavour diagonal at tree level. Electroweak gauge interactions also include $WW\gamma$, $WWZ$ and quartic $WWWW$, $WWZZ$, $WWZ\gamma$ and $WW\gamma \gamma$ vertices. Higgs-fermion and Higgs-vector interactions complete the mass-related scalar sector.^[ref]

Quantum chromodynamics has $q\bar{q}g$ vertices proportional to $g_s{\gamma }^{\mu }{T}^a$, plus three-gluon and four-gluon Feynman vertices because gluons carry color charge. A quark-gluon vertex preserves quark flavour. Internal particles need not be on their mass shells, but electric charge and four-momentum are conserved at each vertex. The figures below show representative leading flavour topologies; binding and hadronization are implicit, and additional strong-interaction contributions can modify hadronic amplitudes.

The weak charged current spectator topology is $c\to q{W}^{+\ast }$, $W^{+\ast }\to u{\bar{q}}^{\prime }$, with the initial $\bar{u}$ a spectator. The pair $q\bar{u}$ forms the negative meson and $u{\bar{q}}^{\prime }$ the positive meson. The amplitude contains $V_{cq}^{\ast }{V}_{u{q}^{\prime }}$ in one common charged-current convention; complex conjugating all factors changes no rate classification. With $c_C=\cos {\theta }_C$ and $s_C=\sin {\theta }_C$, the CKM magnitudes areThus ${\pi }^{+}{K}^{-}$ is favoured, both equal-flavour pairs are singly Cabibbo suppressed, and $K^{+}{\pi }^{-}$ is doubly suppressed. The corresponding parametric rate hierarchy is $1:{\tan }^2{\theta }_C:{\tan }^4{\theta }_C$ after normalizing to the favoured mode. Cabibbo suppression counting alone does not rank the middle two: phase space, different matrix elements and hadronic final-state interactions matter. With identical constant matrix elements, the larger pion phase space would favour ${\pi }^{+}{\pi }^{-}$, whereas experimentally $K^{+}{K}^{-}$ is faster. The empirical decreasing order isAs a later cross-check, HFLAV's 2024 averages give approximately $4.00\%$, $0.411\%$, $0.149\%$ and $0.0138\%$ respectively. The exact middle ordering is an experimental input, not a consequence of the supplied masses and Cabibbo angle alone.^[ref]

The initial strangeness is $-2$. The $\Lambda {\pi }^0$ final state has strangeness $-1$, so its allowed nonleptonic amplitude is weak. One representative topology is $s\to u{W}^{-\ast }$ followed by $W^{-\ast }\to d\bar{u}$; spectator $u,s$ combine with the produced $d$ into the $\Lambda$, while $u\bar{u}$ contributes to the neutral pion. The available rest energy is $1314.8-1115.7-135.0=64.1\mathrm{MeV}$.

The $\Lambda {\bar{K}}^0$ state would conserve strangeness and could have a strong vertex, but its threshold exceeds the parent mass by $1115.7+497.6-1314.8=298.5\mathrm{MeV}$. It is kinematically forbidden. Coupling strength cannot rescue a closed decay channel.

The radiative channel also changes strangeness. It is a weak radiative hyperon decay, not an ordinary electromagnetic decay between states of the same flavour. A representative two-quark contribution is W exchange $su\to ud$ with a photon attached to a charged line; the other allowed attachments, hadronic intermediate states and short-distance loop contributions must be included for a complete amplitude. Its extra electromagnetic coupling and hadronic dynamics suppress it relative to the nonleptonic channel. ConsequentlyNA48 Collaboration measured a radiative branching fraction of about $1.16\times 10^{-3}$, corroborating the suppression rather than a purely electromagnetic interpretation.^[ref]

For the neutral quark-antiquark mesons here, $C=(-1{)}^{L+S}$. Thus $C({\eta }^{\prime })=C({\pi }^0)=+1$, while $C({\rho }^0)=C(\gamma )=-1$. The radiative ${\rho }^0\gamma$ mode conserves charge conjugation and admits a quark-line photon-emission diagram. Both components of the supplied ${\rho }^0$ wavefunction contribute: the charge-weighted flavour overlap is proportional to $Q_u-Q_d=1$, so it does not cancel. Its photon energy is $(m_{{\eta }^{\prime }}^2-m_{\rho }^2)/(2m_{{\eta }^{\prime }})\simeq 165\mathrm{MeV}$.

The ${\pi }^0\gamma$ mode changes charge conjugation and, more decisively, a spin-zero particle cannot decay into another spin-zero particle and one real photon. In the parent rest frame the photon has helicity $\pm 1$, while the opposite-moving pion has helicity zero; the spin-zero two-body helicity selection rule cannot be satisfied. Its on-shell two-body amplitude vanishes even if a weak interaction violates C. A virtual photon subsequently making another pair is a different final state.

For ${\pi }^0{\rho }^0$, angular-momentum conservation requires relative orbital angular momentum $L=1$. Its parity is then negative, matching the parent, but its charge-conjugation eigenvalue is $C_{\pi }{C}_{\rho }=-1$: the two distinct self-conjugate particles are not interchanged by C, so there is no additional $(-1{)}^L$ factor in this C eigenvalue. The decay is forbidden by strong and electromagnetic charge-conjugation conservation despite being above threshold. A weak contribution would require CP violation as well. At the strong/electromagnetic accuracy of the diagram classification,If extremely small weak CP-violating amplitudes are retained, ${\pi }^0{\rho }^0$ need not be exactly zero; the one-real-photon spin prohibition remains exact for ${\pi }^0\gamma$.

For electron-positron annihilation into a muon pair the gauge diagrams are s-channel photon and Z exchange. For electron-positron annihilation into electron neutrinos there is s-channel Z exchange and t-channel W exchange, with interference. For electron-positron annihilation into W-boson pairs there are s-channel photon and Z diagrams and a t-channel electron-neutrino diagram. All three gauge amplitudes are required for the proper high-energy cancellation. Electron-Yukawa-suppressed s-channel Higgs diagrams can also produce the massive muon or W pair; they vanish in the usual zero-electron-mass approximation used for the leading gauge comparison.

There is no energy-independent ordering. For unpolarized beams and inclusive Born cross sections the useful regimes are:

The Z-pole ratio follows $[\frac{1}{2}{]}^2/[(-\frac{1}{2}+{\sin }^2{\theta }_W{)}^2+{\sin }^4{\theta }_W]\simeq 1.99$. Treating every weak reaction as uniformly slower misses both the resonance and forward exchange.