Automated event generation for S-wave quarkonium and leptonium production in NRQCD and NRQED

TL;DR

This work delivers a first-principles, automated LO framework for bound-state production in high-energy collisions by extending MG5_aMC@NLO to NRQCD and NRQED formalisms for $S$-wave states. It leverages a new UFO bound-state model, enables generation across diverse collider environments (pp, ep, ee, UPCs), and validates the approach against established tools while exposing rich phenomenology, including multi-bound-state production and Higgs-assisted processes. The results highlight that subleading NRQCD/NRQED contributions can defy naive scaling expectations, underscoring the need for comprehensive, automated tools. The work lays the groundwork for future NLO accuracy, P-wave inclusion, and global NRQCD analyses, with public availability through MG5_aMC distributions and NLOAccess.

Abstract

We present an extension of the MadGraph5_aMC@NLO framework that enables the automated calculation of leading-order cross sections for S-wave quarkonium and leptonium production within the non-relativistic QCD (NRQCD) and non-relativistic QED (NRQED) factorisation formalisms. The framework has been validated against a variety of benchmark processes, demonstrating robustness and flexibility for phenomenological studies. A key advantage of this implementation is its seamless integration with existing MadGraph5_aMC@NLO features, allowing computations not only within the Standard Model but also in a wide range of Beyond the Standard Model or Effective Field Theory scenarios via a modified Universal Feynman Output (UFO) interface. Furthermore, the framework maintains compatibility with standard Monte Carlo event generators for parton showering and hadronisation. Through numerous examples, we highlight that theoretical studies of quarkonium processes require careful consideration: the impact of subleading contributions is often difficult to predict using simple counting arguments based solely on the hierarchy of couplings and velocity-scaling rules.

Figures (7)

• Figure 1: Transverse momentum ($p_{T,\mathcal{Q}}$) distributions for $J/\psi$ production in association with $c\bar{c}$ in $pp$ collisions at $\sqrt{s}=13$ TeV. Panels (b-d) correspond to different $J/\psi$ Fock states, while panel (a) shows the total contribution obtained by combining them. The blue curve represents the fixed-order LO (fLO) prediction from MG5_aMC, and the orange curve shows the MG5_aMC LO events combined with PS from Pythia v8.315 (LO+PS). Both include their respective 7-point scale-variation uncertainty bands. The lower panels display the ratio to the fLO contribution in each case.
• Figure 2: Example Feynman diagram contributing to triple colour-singlet production. All Feynman diagrams in this article were created with FeynGamefeyngamefeyngame2feyngame3.
• Figure 3: Example Feynman diagrams: (a) the final-state Higgs radiates off a heavy-quark line, and (b) it radiates off a gluon via the effective operator in HEFT. The crossed vertex represents the effective coupling between two gluons and a Higgs boson, as given in Eq. \ref{['eq:lagrangian_heft']}.
• Figure 4: Transverse momentum ($p_{T,\mathcal{Q}}$) and rapidity ($y_{\mathcal{Q}}$) differential cross-section distributions for $J/\psi$ (top) and $\Upsilon$ (bottom) production in association with a Higgs boson $H$ in $pp$ collisions at $\sqrt{s}=13\,\mathrm{TeV}$. The contributions from the $^1S_0^{[8]}$ and $^3S_1^{[8]}$ Fock states are shown in orange and blue, respectively, with the error bands representing the standard 7-point scale variation.
• Figure 5: Differential cross sections with respect to the leptonium transverse momentum $p_{T,\mathcal{L}}$ at $\sqrt{s}=13$ TeV, shown for (a) $\mathcal{L}=\mathit{Ps}_1$, (b) $\mathcal{L}=\mathit{TM}_1$, (c) $\mathcal{L}=\mathcal{T}_1$, and (d) all three states combined. Fiducial jet cuts of $p_{T,j}>2\,\mathrm{GeV}$ and $|\eta_j|<5$ are applied. The error bands represent the standard 7-point renormalisation and factorisation scale variations.