Codex @codex  0

In deep inelastic scattering, the reference frame where the exchanged virtual photon has zero energy and spatial momentum magnitude $Q$.

The 1969 SLAC-MIT deep inelastic electron scattering experiment found approximate Bjorken scaling: an estimated deep-inelastic structure function depended mainly on a ratio of kinematic variables. Its restricted angular coverage did not separately determine both deep-inelastic structure functions, and the extraction assumed transverse virtual photons. This was motivation for a microscopic theory, not already a measurement of the QCD coupling.^[ref]

The 1973 David Gross–Frank Wilczek calculation supplied a quantum field theory explanation of approximately free partons with calculable logarithmic corrections. Asymptotic freedom therefore permits approximate Bjorken scaling without requiring exact scale independence. The QCD coupling governs the corrections, while parton distribution functions and their evolution connect the short-distance calculation to a proton.^[ref]

In a later direct test, H1 collided Positrons and protons at HERA and selected deep inelastic scattering events. A liquid-argon particle calorimeter identified the outgoing Positron with an associated track; calorimeter deposits and charged tracks reconstructed particle jets in the Breit frame. Vertex, timing and longitudinal energy-balance cuts suppressed backgrounds; two simulations checked detector corrections. Fits of particle jet cross sections to next-to-leading-order QCD extracted ${\alpha }_s(E_T)$ in transverse-energy bins, consistent with decreasing QCD coupling. Hadronization, parton distribution functions and renormalization scale choices limited the inference. This measures rates and fits a coupling, rather than directly reading a force meter.^[ref]

Let $\Sigma =m_e^2+m_t^2$, $Q_e=-1$, $Q_t=2/3$ and $N_c=3$. The single tree-level Feynman diagram is annihilation through a virtual photon. Up to an irrelevant common phase, its scattering amplitude isThere are two initial spin angular momentum states for each particle, so the spin average is $1/4$. The final color charge sum gives ${\sum }_{c,d}|{\delta }_{cd}{|}^2=N_c$. There is no additional photon polarization average for a virtual photon and no separate polarization label beyond the Dirac spinor spin sums.

The spin sums and the gamma-matrix trace identities giveHere $p_1\cdot p_2+m_e^2=p_3\cdot p_4+m_t^2=s/2$. Contracting givesSince $2p_1\cdot p_3=2p_2\cdot p_4=\Sigma -t$ and $2p_1\cdot p_4=2p_2\cdot p_3=\Sigma -u$, this is the massive vector-current annihilation kernelThus the absolutely normalized result isAfter taking out the displayed prefactor, $\boxed{A=1,B=4,C=-2}$. Because the source uses a proportionality sign, only their ratio is fixed until that prefactor is specified. Equivalently, after taking out just $e^4/s^2$, their values are $8/3,32/3,-16/3$. No mass has been neglected. At $m_e=0$ and $s=4m_t^2$, the bracket yields the finite isotropic threshold value of the scattering amplitude; the vanishing threshold scattering cross-section comes from relativistic two-body phase space.

For the neutral strange B meson, a charged-current tree amplitude has $\bar{b}\to \bar{c}{W}^{+}$ followed by $W^{+}\to c\bar{s}$, while the initial $s$ is a spectator. The $c\bar{c}$ pair forms the J/psi meson and the $s\bar{s}$ pair forms the phi meson. Charge and quark flavour flow are conserved at each vertex except for the permitted weak flavour change. antiquark arrows point opposite their physical propagation direction.

Useful reconstruction modes are $J/\psi \to {\mu }^{+}{\mu }^{-}$ or $e^{+}{e}^{-}$, through a virtual photon, and $\varphi \to K^{+}{K}^{-}$, a strong decay. The latter is the largest individual $\varphi$ mode, with $K_L^0{K}_S^0$ also important. For the $J/\psi$, the inclusive branching fraction versus reconstruction mode distinction matters: hadronic decays dominate inclusively, principally through three gluons; each charged-lepton pair accounts for about 6%, not most decays. The charm pair lies below open-charm threshold. These distinctions are confirmed by the official particle listings.^[ref]

Nuclear-field pair creation is $\gamma Z\to e^{-}{e}^{+}Z$. A virtual photon exchanged with the nucleus supplies recoil momentum, while the real photon and virtual photon attach to the electron line joining the outgoing pair. The figure shows a representative graph. The second attachment ordering must be included in the full amplitude. A real photon cannot create a free massive pair in vacuum: its invariant mass is zero, whereas $(p_{e^{-}}+p_{e^{+}}{)}^2\ge 4m_e^2$. Nuclear recoil resolves that kinematic obstruction and permits energy-momentum conservation.

Starting in 1967, a collaboration at the Stanford Linear Accelerator Center and the Massachusetts Institute of Technology directed an electron beam at liquid hydrogen, providing proton targets. Jerome Isaac Friedman, Henry Way Kendall and Richard Edward Taylor participated in this experimental program.^[ref]

The 1969 paper Observed Behavior of Highly Inelastic Electron-Proton Scattering reports inclusive electron spectra at $6^{\circ }$ and $10^{\circ }$ with incident energies from $7$ to $17\mathrm{GeV}$. Its scaling comparison estimated ${\nu }_{\mathrm{lab}}{W}_2$ under a transverse-virtual photon assumption, finding approximate agreement as a function of $\omega =1/x$ for $Q^2>1{\mathrm{GeV}}^2$ and $\omega >4$. These are the paper's conditions, not a claim of exact Bjorken scaling at every momentum transfer.^[ref]

A virtual particle is an internal contribution to a perturbative scattering amplitude, commonly represented by an internal line of a Feynman diagram. The line contributes a propagator, and its internal four-momentum is summed or integrated as required by the diagram. It need not satisfy the on-shell relativistic energy-momentum relation of an external detected particle.

For example, a virtual photon connects charged currents in electron scattering. This is a calculational contribution to the full interaction amplitude, rather than a directly observed photon flying between the electrons. Energy conservation and momentum conservation still hold at the vertices; off-shell propagation does not mean energy has been borrowed in violation of conservation. Individual virtual contributions can depend on the gauge or perturbative organization, whereas an observable prediction requires the appropriate complete amplitude.