Three-body interaction for particle action

Consider a scalar quantum field theory with the following interaction vertices:\[g_{F} |\Phi|^{2} A + \frac{h_{F}}{6}A^{3}\]where \(g_{F}\) and \(h_{F}\) are two coupling strengths for cubic interactions. In the particle theory we can consider a three-body system with a three-body interaction (i.e. not pair-wise) that has the form\[S_{3}\left[ q_{14}, \, q_{25}, \, q_{36} \right] \equiv g_{P}^{3} h_{F} \int\limits_{0}^{T_{14}} \mathrm{d}\tau \int\limits_{0}^{T_{25}} \mathrm{d}\sigma \int\limits_{0}^{T_{36}} \mathrm{d}\rho \int \mathrm{d}Y \, G_{A}\left[ q_{14}(\tau)| Y \right] G_{A}\left[ q_{25}(\sigma)| Y \right] G_{A}\left[ q_{36}(\rho)| Y \right] \]In the eikonal approximation we will get a three-body Gram invariant from this term in the action.