Electron-positron annihilation

Electron-positron annihilation converts an electron and a Positron into other particles while conserving total energy, momentum and electric charge. A common low-energy channel produces two photons. In the centre-of-momentum frame, negligible initial kinetic energy gives two oppositely directed photons, each with energy $m_e{c}^2$.

The back-to-back momentum balance makes paired photon detections a useful signature of annihilation. Motion of the initial pair changes the laboratory photon directions and energies. At larger collision energy, other final states become accessible: an intermediate virtual photon can produce, for example, a muon-antimuon pair. Thus annihilation converts the initial pair into allowed final excitations rather than destroying conserved energy.

For two-photon positron annihilation with negligible initial total momentum, each photon has energy $m_e{c}^2$ and the two propagate oppositely. Their common energy differs from the positron's continuous emission-energy spectrum before slowing in matter.

For an initial Dirac field of mass $m$ annihilating into a distinct Dirac field of mass $M$ through a vector current, put $\Sigma =m^2+M^2$. With Mandelstam variables satisfying $s+t+u=2\Sigma$, the spin average of the two contracted vector currents is $2F$, where $F=(\Sigma -t{)}^2+(\Sigma -u{)}^2+2s\Sigma =t^2+u^2+4s\Sigma -2{\Sigma }^2$. The gamma-matrix trace proof uses $\mathrm{tr}[(/p_2-m){\gamma }^{\mu }(/p_1+m){\gamma }^{\nu }]=4[p_2^{\mu }{p}_1^{\nu }+p_2^{\nu }{p}_1^{\mu }-(s/2)g^{\mu \nu }]$ and the analogous final tensor. Couplings, the mediator denominator and color charge factors multiply this common kinematic expression.

The leading Standard Model gauge amplitudes are s-channel Z boson exchange and t-channel W boson exchange. They interfere. At collision energies well below the weak scale the cross section scales as $G_F^{2}s$. In an inclusive high-energy Born calculation, the forward W-exchange region gives a nonzero limiting scale of order $g^4/m_W^2$, unlike a fixed-angle estimate. Angular cuts and beam polarization can change comparisons with other annihilation channels.

An electron with kinetic energy $\alpha m_e{c}^2$ annihilating with a stationary positron into two equal-angle photons has $\tan \varphi =\sqrt{2/\alpha }$ for $\alpha >0$. Equal photon energies follow from the specified symmetric laboratory geometry.

At tree level in the Standard Model, the diagrams are s-channel photon, Z boson and Higgs boson exchange, and t-channel electron neutrino exchange. The Higgs contribution couples to the electron through its small Yukawa coupling. In a zero-electron-mass approximation that diagram vanishes, leaving the three gauge diagrams. The gauge contributions interfere; their sum has the high-energy behavior required by the electroweak gauge structure.