# Quantum Electrodynamics of Electron-Positron Annihilation to Muons

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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]
Contributed by: A. Ratti,D. Meliga,L. Lavagnino and S. Z. Lavagnino (June 2021)

Additional contribution by: A. Tabasso

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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 [1], 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).

References

[1] M. Böhm, *Gauge Theories of Strong and Electroweak Interactions*, 3rd ed. (A. Denner and H. Joos, trans.), Stuttgart: B.G. Teubner, 2001.

[2] T. D. Lee and M. Nauenberg, "Degenerate Systems and Mass Singularities," *Physical Review*, 133(6B), 1964. doi:10.1103/PhysRev.133.B1549.

[3] T. Kinoshita, "Mass Singularities of Feynman Amplitudes," *Journal of Mathematical Physics*, 3(4), 1962 pp. 650–677. doi:10.1063/1.1724268.

## Permanent Citation