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

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