Gamma-UPC+MadGraph5_aMC@NLO Framework

• Gamma-UPC+MadGraph5_aMC@NLO Framework is an automated computational platform that integrates precise photon flux modeling with fully automated NLO QCD/EW corrections.
• It employs both EDFF and ChFF parameterizations to model realistic impact-parameter dependent photon fluxes and survival probabilities in ultraperipheral collisions.
• The framework seamlessly couples with MadGraph5_aMC@NLO, enabling detailed simulations of dilepton, multiboson, and heavy-quark final states with mixed QCD/EW corrections.

The Gamma-UPC+MadGraph5_aMC@NLO framework is an automated computational platform that enables next-to-leading order (NLO) QCD and electroweak (EW) calculations for photon–photon (γγ) initiated processes in ultraperipheral collisions (UPCs) of protons and ions at high-energy colliders. By integrating the gamma-UPC module (which supplies precise, impact-parameter–dependent photon fluxes based on realistic electromagnetic structure of hadrons and nuclei) with the MadGraph5_aMC@NLO infrastructure (for automated perturbative calculations), the framework addresses the production of exclusive and inclusive final states via γγ fusion and delivers NLO (QCD and/or EW) corrections in a fully automated manner. This unified machinery represents the first deployment of general-purpose NLO predictions for photon-induced UPC processes spanning a wide range of Standard Model and beyond-the-Standard-Model (BSM) scenarios.

1. Photon Flux Modeling in Ultraperipheral Collisions

In UPCs, highly relativistic protons or ions act as sources of quasi-real photons through their Lorentz-boosted electromagnetic fields. The gamma-UPC module provides two primary parameterizations for the photon flux:

• Electric Dipole Form Factor (EDFF): Implements the standard analytical formula for the transverse photon number density, which depends on the impact parameter $b$, the Lorentz boost factor $\gamma_L$, and the projectile charge $Z$. The photon flux is given by

$$N^{\text{EDFF}}_{\gamma/Z}(E_\gamma, b) = \frac{Z^2\alpha}{\pi^2} \frac{\xi^2}{b^2} \left[ K_1^2(\xi) + \frac{1}{\gamma_L^2} K_0^2(\xi) \right],\qquad \xi = \frac{E_\gamma b}{\gamma_L},$$

where $K_n$ are modified Bessel functions.

• Charge Form Factor (ChFF): Utilizes the Fourier transform of the hadron/nuclear charge distribution to yield a photon flux accounting for nuclear or proton spatial substructure, ensuring physically realistic behavior at small $b$. For heavy ions, this uses the Woods–Saxon charge density.

To ensure collision exclusivity, the framework folds in a no–hadronic–interaction (survival) probability $P_{\text{no\,inel}}(b_1, b_2)$, typically modeled by an eikonal function or the optical-Glauber model:

$$S^2 = \frac{\int d^2b_1\, d^2b_2\, P_{\text{no\,inel}}(b_1, b_2)\, N_{\gamma/Z_1}(E_{\gamma_1}, b_1)\, N_{\gamma/Z_2}(E_{\gamma_2}, b_2)}{\int d^2b_1\, d^2b_2\, N_{\gamma/Z_1}(E_{\gamma_1}, b_1)\, N_{\gamma/Z_2}(E_{\gamma_2}, b_2)}.$$

The exclusive cross section for producing a final state $X$ in $A_1A_2$ UPCs is then computed as

$$\sigma(A_1A_2 \to A_1 X A_2) = \int \frac{dE_{\gamma_1}}{E_{\gamma_1}} \frac{dE_{\gamma_2}}{E_{\gamma_2}}\, \left[ \frac{d^2N^{(A_1A_2)}_{\gamma_1/Z_1, \gamma_2/Z_2}}{dE_{\gamma_1} dE_{\gamma_2}} \right]\, \sigma_{\gamma\gamma\to X}(W).$$

This structure, incorporating the full collision geometry, is essential for simulating processes where the spatial extent of the nuclei and their electromagnetic fields play a key phenomenological role (Shao et al., 2022, Eskola et al., 2023, Eskola et al., 6 May 2024, Shao et al., 28 Aug 2025).

2. Automation of NLO QCD and Electroweak Corrections

The framework integrates the modular, fully-automated NLO generator MadGraph5_aMC@NLO, which is extended for coherent γγ physics as follows:

• Perturbative Expansion: For processes with a given LO coupling structure $\sim \alpha^k$, the NLO level adds all $\mathcal{O}(\alpha^{k+1})$ (EW) and/or $\mathcal{O}(\alpha^{k}\alpha_s)$ (QCD) corrections as appropriate. For dilepton production ($\gamma\gamma\to\ell^+\ell^-$), for instance, only EW corrections emerge at NLO.
• Hybrid Renormalization Schemes: To maintain consistency with the quasi-real nature of the incoming photons, couplings at the external photon vertices are renormalized in the $\alpha(0)$ scheme (on-shell), while high-scale virtual corrections and real emission vertices use Gμ or $\alpha(M_Z)$. This hybrid approach avoids artificially large NLO corrections that otherwise result from scheme mismatches (Shao et al., 14 Apr 2025, Pagani et al., 2021, Jiang et al., 29 Oct 2024, Shao et al., 28 Aug 2025).
• FKS Subtraction Method with Coherent Photons: The infrared subtraction procedure is adapted so that initial-state photon splitting (which corresponds to resolved-photon evolution in pp collisions) is forbidden—initial photons must remain coherent. In the FKS algorithm, PDF counterterms for initial photons are set to zero or tailored appropriately; for EW corrections, the counterterm has the form

$$K_\gamma^{\text{(EW)}}(z) = -\log(\mu^2\, \xi_A^2\, R_A^2)\, P_{\gamma}^{\text{(EW)}}(z,0),$$

where $R_A$ is the emitting nucleus radius and $\xi_A$ an uncertainty parameter of order unity (Shao et al., 14 Apr 2025, Shao et al., 28 Aug 2025).

• Automated Workflow: Users generate processes with a photon–photon initial state by specifying coherent photons (not partons) in the MadGraph syntax, e.g.,

  import model myNLOmodel_w_qcd_qed-restrict_card_w_a0 generate !a! !a! > X [QCD QED]

  where !a! denotes an initial coherent photon (Shao et al., 28 Aug 2025).

3. Modeling Photon-Induced Final States and Precision Observables

The automated NLO workflow supports a wide array of γγ-initiated final states, including:

• Dilepton Production: Used for flux calibration and new physics searches (anomalous magnetic moments, quartic couplings). NLO EW corrections to $\gamma\gamma\to\tau^+\tau^-$ reach –3%, with the weak correction (–4%) dominant over the positive QED part (+1%). The cross section including the anomalous magnetic moment $a_\tau$ is

$$\sigma_{a_\tau} = \sigma_{\rm LO} + \delta \sigma_{a_\tau}$$

with the $a_\tau$-dependent piece parameterized via the $\gamma\tau\tau$ form factors (Jiang et al., 29 Oct 2024). Differential K-factors for NLO EW corrections display different shapes from the $a_\tau$-sensitive terms, emphasizing the need to disentangle them in precision analyses.

• Light-by-Light Scattering and W-Pair Production: For multiboson final states, both QCD and EW NLO corrections can be present, and the impact of the renormalization scheme choice and IR subtraction is crucial for perturbative reliability (Shao et al., 14 Apr 2025).
• Heavy Quark and Top-Quark Final States: The framework handles production of $t\bar{t}$, $t\bar{t}\gamma$ etc., and supports studies of high-mass BSM resonances and anomalous quartic gauge couplings.
• Azimuthal Modulations and Transverse-Momentum Dependence: New features include kinematic “smearing” to restore the transverse-momentum and azimuthal angle dependence of the outgoing leptons, revealing the polarization structure of the photon flux and leading to $\cos 2\Delta\phi$ and $\cos 4\Delta\phi$ modulations in the dilepton spectrum:

$$\frac{d\sigma}{d(\Delta\phi)} \propto A + B\cos(2\Delta\phi) + C\cos(4\Delta\phi)$$

with coefficients read off from transverse-momentum–dependent photon densities (Crépet et al., 27 Sep 2024).

4. Geometric and Nuclear Effects in Inclusive and Exclusive Processes

The convolution structure for exclusivity and for incorporating the finite size of the colliding ions demands:

• Realistic Nuclear Geometry: Photon fluxes and nuclear PDFs are both modeled with explicit impact-parameter dependence; the Woods–Saxon density profile is standard for heavy ions. The effective photon flux can be written as

$$f_{\gamma/A}^{\rm eff}(y) = \frac{1}{A} \int d^2r\, d^2s\, f_{\gamma/A}(y, r)\, T_B(s)\, \Gamma_{AB}(r-s),$$

where $T_B(s)$ is the nuclear thickness function and $\Gamma_{AB}$ the survival factor. This is critical for describing dijet photoproduction at high $z_\gamma$, where small impact parameter (“near-encounter”) events dominate and flux suppression from geometric overlap is maximal (Eskola et al., 2023, Eskola et al., 6 May 2024).

• Electromagnetic Dissociation and Neutron Emission: For Pb–Pb UPCs, nuclear breakup (characterized by forward neutron emission) is included by incorporating a further survival probability, modeled by a Poisson factor involving the photonuclear cross section, e.g.,

$$\Gamma_{AB}^{\rm e.m.}(b) = \exp\left[ - \int_0^1 dy\, f_{\gamma/B}(y, b) \sigma_{\gamma A \to A^*}(\sqrt{ys_{NN}}) \right].$$

(Crépet et al., 27 Sep 2024).

5. Computational Architecture and Technical Features

The principal computational and structural advancements comprise:

• Automated Matrix Element and Loop Reduction: MadGraph5_aMC@NLO’s MadLoop module generates one-loop amplitudes, employing several reduction algorithms, and its MadFKS module manages the modified subtraction for IR singularities.
• Process Flexibility and Output Formats: The framework accepts BSM models via UFO/FeynRules/ALOHA infrastructure, and can generate code output (Fortran, C++, or Python) for interfacing with shower Monte Carlos and detector simulation packages, such as Pythia 8 (Alwall et al., 2011).
• Precision and Validation: Comparisons with LHC data on exclusive WW, ZZ, $\ell^+\ell^-$, and $t\bar{t}$ final states show agreement at the percent level when flux and survival factor modeling is realistic. The accuracy of the NLO calculation is systematically improved by controlling the inclusion of QCD/EW corrections, precise coupling/renormalization scheme choices, and the underlying nuclear geometry.
• User Interface and Simulation Workflow: The seamless transition between hadronic, leptonic, and UPC environments is supported by dedicated process syntax, as well as options for selecting the photon flux model (pdlabel = edff or pdlabel = chff) (Shao et al., 28 Aug 2025).

6. Phenomenological Applications and Current Limitations

Key applications and boundaries of the Gamma-UPC+MadGraph5_aMC@NLO framework include:

• Process Scope: The automated approach currently covers exclusive and inclusive photon-photon final states with elementary particles. Support for bound states (e.g., quarkonia production including NRQCD matching) is under development.
• NLO+Parton Shower (PS): Fully automated NLO+PS matching is presently limited to processes free from additional jet complications and requires careful attention to IR-safe event selection and photon isolation (Shao et al., 14 Apr 2025).
• Scheme Dependence and Differential Observables: For processes such as $\gamma\gamma\to\tau^+\tau^-$, the stability of NLO corrections critically depends on hybrid scheme usage. Differential cross-section shapes for NLO and BSM-sensitive contributions can differ significantly (e.g., for anomalous moments or NRQCD matrix elements), requiring careful interpretation of measured data for precision BSM constraints (Jiang et al., 29 Oct 2024, Yedelkina et al., 2023).

7. Impact on Precision Collider Physics and Future Development

The Gamma-UPC+MadGraph5_aMC@NLO framework provides the HEP community with the first general-purpose, automated, NLO-accurate platform for simulating photon-fusion processes in ultraperipheral collisions. Its modular structure, ability to accommodate mixed QCD/EW corrections in a coherent photon environment, and flexibility for modeling the full geometric complexity of nuclear collisions, position it as a critical computational tool for Standard Model tests, BSM searches, and detailed benchmarking of photon flux models at the LHC, FCC-hh, EIC, and future high-energy colliders. Active development is ongoing to expand the elementary particle scope, improve integration with parton showers and final-state modeling, and extend to complex final states such as quarkonium pairs and tetraquark resonances (Yang et al., 21 Apr 2025, Shao et al., 2022, Shao et al., 14 Apr 2025, Shao et al., 28 Aug 2025).