Quantum Electrodynamics of Electron-Positron Annihilation to Muons
This Demonstration considers electron-positron annihilation to muons as an application of quantum electrodynamics (QED). First-order radiative and virtual corrections are considered. As is well known, virtual and radiative corrections in QED are affected by soft and collinear divergences (when the masses of fermions are neglected), collectively designated as infrared (IR) divergences.[more]
In the radiative process, the divergences arise when integrating over the phase space of final state particles to get the cross section. The soft singularities are due to the emitted photon momentum going to zero, while the collinear ones arise when the photon momentum is collinear to the momentum of the emitting fermion .
The same kind of IR divergences occur in the virtual process when integrating over the loop momentum of the virtual photon that connects the two final state fermions. According to the Kinoshita–Lee–Nauenberg theorem, every QED (also QCD) amplitude is IR safe for final state corrections since the divergences arising from radiative and virtual corrections cancel one another [2, 3].
Use the "Feynman diagrams" button to open a panel showing the diagrams that contribute to the process, both those related to the amplitude and those belonging to the amplitude up to order . Selecting "" or " " shows the cut diagrams.
The "amplitude" button shows an approximation of the differential cross section plot when the components of the momenta are selected.[less]
Snapshot 1: The Feynman diagrams contributing to the amplitude of up to the next-to-leading order and their complex conjugates.
Snapshot 2: The diagrams of and are joined to show the cut diagrams leading to . The dashed lines of the cut diagrams represent the integration over the physical momenta crossing the cut lines so that the sum of the four cut diagrams gives the total cross section for .
Snapshot 3: The two possible diagrams associated to the radiative correction amplitudes and to its complex conjugates to order .
Snapshot 4: The four cut diagrams obtained by joining the and diagrams in pairs.
Snapshot 5: A sketch of the IR behavior of for . Either final state fermion momentum is fixed (lying on the - plane with inclination ), and we see how the differential cross varies with the momenta of the other particles. Since the final state momenta have mass-shell and momentum conservation constraints, we parametrize their left degrees of freedom using the Sudakov parametrization , which introduces a parameter related to the fraction of collinear momentum shared by each 4-vector and a parameter, representing the transverse component of momentum from the collinear direction:
Snapshot 6: Shows the Sudakov parametrization, which preserves the above constraints for small values.
Snapshot 7: The projection of the three final state particles momenta on the - plane. A collinear divergence appears for approaching 0, as well as a less obvious soft singularity when both and go to 0 or to 1 (when goes to 0).
 M. Böhm, Gauge Theories of Strong and Electroweak Interactions, 3rd ed. (A. Denner and H. Joos, trans.), Stuttgart: B.G. Teubner, 2001.
 T. D. Lee and M. Nauenberg, "Degenerate Systems and Mass Singularities," Physical Review, 133(6B), 1964. doi:10.1103/PhysRev.133.B1549.
 T. Kinoshita, "Mass Singularities of Feynman Amplitudes," Journal of Mathematical Physics, 3(4), 1962 pp. 650–677. doi:10.1063/1.1724268.