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A theoretical particle physicist who coauthors the Particle Data Group review of the electroweak interaction with J. Erler. It connects measured observables to renormalization, radiative corrections and fits of Standard Model parameters. Comparing many precision observables with one consistent theory makes small departures from its predictions a quantitative constraint on proposed new interactions.^[ref]

A theoretical particle physicist who coauthors a Particle Data Group review of precision electroweak interactions with A. Freitas. Its combined treatment of neutrino scattering, parity-sensitive measurements and massive-boson observables helps test whether the same Standard Model parameters describe very different experiments. Global fits can then constrain extensions without relying on a single anomalous measurement.^[ref]

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.

Assume first an ultrarelativistic charged-current interaction with negligible electron mass in the centre-of-mass spin analysis. Choose the incident antineutrino along $+z$. Its right helicity gives spin $+z$; the interacting left-helicity electron moves along $-z$ and also has spin $+z$. The total initial projection is $+1$. The final antineutrino and selected electron likewise have total spin $+1$ along the outgoing antineutrino direction. The spin-one rotation therefore givesAn unpolarized electron target changes the normalization through the spin average, not this selected-current shape. For electron antineutrinos the charged-current diagram is the $s$-channel $W^{-}$ graph. A contact approximation further requires $m_e^2\ll s\ll m_W^2,m_Z^2$.

The full Standard Model also has neutral-current $Z$ exchange. In the contact limit it supplies a right-electron contribution as well as interference in the left-electron channel. With $g_L=1/2+{\sin }^2{\theta }_W$, $g_R={\sin }^2{\theta }_W$, the full high-energy shape is proportional to $g_L^2(1+\cos {\theta }^{\star }{)}^2/4+g_R^2$. Thus the pure squared angular form is an explicitly stated charged-current approximation, not an exact prediction for all Standard Model contributions. At momentum transfers comparable to the weak-boson masses, propagators must also be retained.^[ref]

Let $z=\cos {\theta }^{\star }$. Rotating the spin-one intermediate state between initial and final axes gives probability amplitudes proportional to $d_{\pm 1,\pm 1}^1$ for equal fermion helicities and $d_{\mp 1,\pm 1}^1$ for opposite ones. Thus, apart from a common positive normalization,

The common factor $1/4$ from the rotation elements is absorbed into the normalization. Distinct external helicities are orthogonal, so an unpolarized rate adds their squared probability amplitudes incoherently.

The source interchanges the conventional left and right electron coupling labels. With $g_L=T_3-Q{\sin }^2{\theta }_W$ and $g_R=-Q{\sin }^2{\theta }_W$, the physical electron values areAt ${\sin }^2{\theta }_W=0$, for example, the left electron still carries weak isospin $-1/2$, while the right electron is a weak singlet; the printed assignment gives the opposite result. This fixes how the helicity-channel table should be interpreted.^[ref]

For a muon final state, lepton universality gives the same pair of couplings. Crucially, swapping both labels does not change the summed distribution. Put $L=g_L$, $R=g_R$; thenHere $\mathcal{N}$ includes the resonance propagator, initial spin average and common coupling normalization. Equivalently,The source's simultaneous label interchange reverses ${\mathcal{A}}_{\ell }$ but leaves its square unchanged.