Abstract
The Bern-Kosower formalism, originally developed around 1990 as a novel way of obtaining on-shell amplitudes in field theory as limits of string amplitudes, has recently been shown to be extremely effcient as a tool for obtaining form factor decompositions of the N - gluon vertices. Its main advantages are that gauge invariant structures can be generated by certain systematic integration-by-parts procedures, making unnecessary the usual tedious analysis of the non-abelian off-shell Ward identities, and that the scalar, spinor and gluon loop cases can be treated in a unified way. After discussing the method in general for the N - gluon case, I will show in detail how to rederive the Ball- Chiu decomposition of the three - gluon vertex, and finally present two slightly different decompositions of the four - gluon vertex, one generalizing the Ball Chiu one, the other one closely linked to the QCD effective action.
Highlights
Multi-gluon amplitudes in QCD pose two very different computational challenges, depending on whether one can impose on-shell conditions or not
A very different challenge is posed by the off-shell one-particle irreducible amplitudes (“vertices”). Their calculation is much more difficult, and compared to the on-shell case, relatively little progress has been made in recent years in advancing the available tools for their calculation
Let us start our discussion with the N-gluon vertex at the one-loop level
Summary
TA = gμ1μ2 (k1 − k2)μ3 , TB = gμ1μ2 (k1 + k2)μ3 , TC = −[(k1k2)gμ1μ2 − k1μ2 k2μ1 ](k1 − k2)μ3 , TF = [(k1k2)gμ1μ2 − k1μ2 k2μ1 ][k1μ3 (k2k3) − k2μ3 (k1k3)] , TH. This decomposition is valid for any spin in the loop, and for higher loop corrections. It involves six universal tensor structures TA, TB, TC, TF, TH, TS with scalar coefficient functions A, B, C, F, H, S. TA is just the tree-level vertex, at tree-level one has A = 1 with the other coefficient functions vanishing. F and H possess true three-point kinematics, and are transversal, while A, B, C have (pinched) two-point kinematics, and have a longitudinal part
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