Automated NRQCD and NRQED simulations of quarkonium and leptonium production with P-wave states and physical-mass effects
We present the $\texttt{MadSONS}$ module, which extends the $\texttt{MadGraph5_aMC@NLO}$ framework by providing a fully automated solution for the simulation of arbitrary tree-level production processes involving non-relativistic bound states. More specifically, we focus on the production of quarkonia and leptonia in non-relativistic QCD and QED, respectively. This work constitutes the next step in the programme initiated in arXiv:2510.26773, where the colour and spin projectors required for S-wave states were first incorporated into $\texttt{MadGraph5_aMC@NLO}$. In the present development, we implement the remaining projectors associated with orbital and total angular momentum, thereby enabling the event generation of processes involving P-wave states. The derivatives required for the orbital-angular-momentum projection are evaluated using dual numbers. In addition, we implement a momentum-reshuffling procedure that accounts for physical bound-state mass effects in the phase space. The resulting framework applies to a wide range of collider environments while remaining compatible with standard multi-purpose Monte Carlo event generators for parton showering and hadronisation. As a first phenomenological application, we revisit $J/\psi + \psi(2{\rm S})$ production at the LHC and obtain an improved description of the LHCb data.