Appendix B:

Cut Diagram Notation

A convenient technique for organizing calculations of ||² in cross sections is through cut diagrams, which combine contributions to and * into a single diagram for ||² with slightly modified Feynman rules.

The form of cut diagrams is derived in Fig. 50, for the annihilation of a fermion pair of momenta k₁ and k₂ into a set of n final state lines, of which only a fermion with momentum p₁ and an antifermion of momentum p_n are exhibited.

The underlying identity for these manipulations is

where w and w' are any two Dirac spinors.

Fig. 50(a) shows a typical fermion propagator and vertex in and *. Fig. 50(b) shows the application of Eq. (B1) to Fig. 50(a). The diagram in * has been flipped over, all arrows on fermion lines have been reversed, and all momenta have been reversed in sign. This leaves the sign of momenta in fermion propagators the same, as shown. Color sums can be reversed in the same manner as spinor sums, because the color generators are hermitian.

Fig. 50(c) exhibits the cut diagram notation, in which the contribution of any final state is a modified forward scattering diagram. The final-state lines are indicated by a vertical line (the "cut"). Cut lines are represented in the integral corresponding to the cut diagram by factors

for fermions or antifermions, after a spin sum. For polarized fermions or for vectors, the usual spin projections replace (_i + m_i ). The Feynman rules for are the normal ones, and those for differ only in the sign of explicit factors of i at vertices and in propagators. The three-gluon vertex also changes sign in , because of the reversal of momenta.