# 5-Point 1PI Proper Vertex Function of Gluon and the S-Matrix Element

## TL;DR

This paper derives an all-orders expression for the S-matrix element of gluon scattering in QCD using 1PI vertex functions and propagators, providing a foundational framework for understanding UV divergences and scattering amplitudes.

## Contribution

It presents a novel formulation expressing the gluon scattering S-matrix in terms of 5-point, 4-point, and 3-point 1PI vertex functions at all orders in the coupling.

## Key findings

- Expressed the S-matrix element in terms of 1PI vertex functions and propagators.
- Provided a path integral formulation for all-order scattering amplitudes.
- Clarified the role of UV divergences in the vertex functions.

## Abstract

As far as the renormalization in perturbative QCD is concerned the n-point one particle irreducible (1PI) proper vertex function is the basic building block where the ultra-violet (UV) divergence occurs when the loop momentum integration limit goes to infinity. In this paper we express the S-matrix element for the $gg \rightarrow ggg$ scattering process at all orders in coupling constant in terms of 5-point, 4-point, 3-point 1PI proper vertex functions and the (full) propagator by using the path integral formulation of QCD.