Executive summary: this article explores how simplified models are used to assess the physics potential of a muon collider and describes in detail the calculation of rates for annihilation, neutral VBF, and charged VBF processes using standard tools such as FeynRules, Whizard, and MadGraph5. Simplified models allow new interactions to be compactly parameterized through a reduced number of parameters such as masses, couplings, and widths, facilitating direct comparisons between theoretical studies and experimental simulations.
Concept of simplified models: a simplified model defines the relevant particle content and the minimal interactions needed to describe a signal at a collider. In the context of muon colliders, scalar mediators, neutral vectors, or charged vectors with couplings to muons and electroweak bosons are typically considered. These constructions are ideal for generating model files compatible with FeynRules, which in turn exports UFO formats or interfaces for Whizard and MadGraph5.
Practical implementation with FeynRules: the first step is to encode the simplified model Lagrangian in FeynRules to obtain Feynman rules and export to interoperable formats. FeynRules generates UFO files for MadGraph5 and modules for Whizard, ensuring that vertices, color factors, and particle widths are correctly propagated to the event generators. It is advisable to validate vertices and decays analytically before moving to numerical simulation.
Calculation of cross sections and rates: MadGraph5 and Whizard compute amplitude matrices and perform numerical integration over phase space to obtain differential and total cross sections. The expected event rate is obtained by multiplying the cross section by the integrated luminosity and by the experimental selection efficiency, i.e., rate = cross section × integrated luminosity × efficiency. It is essential to include initial-state radiation and beamstrahlung effects specific to muon colliders when applicable, and to consider beam energy broadening for realistic estimates.
s-channel annihilation: in typical s-channel annihilation processes, two muons collide to produce a mediator that decays into the sought final states. MadGraph5 and Whizard calculate the s-channel amplitude including finite widths and possible interference with electroweak background. In practice, spectral lines and resonances are analyzed, mediator masses and couplings are scanned, and sensitivities are extracted as a function of collider luminosity and energy resolution.
Neutral VBF and charged VBF: vector boson fusion (VBF) includes t-channel configurations where each muon emits a vector boson that subsequently interacts. For neutral VBF, the intermediate mediators are typically Z or virtual photons, while charged VBF involves virtual W bosons that generate charged states. Whizard is particularly efficient for processes with multiple particles in the final state and for precise treatments of beam structures and phase-space selection. In both cases, it is crucial to impose acceptance cuts to regulate collinear regions and evaluate background contributions.
Comparison Whizard vs MadGraph5: MadGraph5 offers flexibility and a broad community of UFO models, ideal for rapid sensitivity studies and distribution generation. Whizard excels in leptonic colliders due to its advanced beam treatment and efficient integration for processes with many diagrams. In benchmarking studies, it is recommended to use both and compare results to estimate theoretical and numerical uncertainties.
Recommended steps for a benchmarking study: 1) Define simplified models and parameter ranges; 2) Implement the Lagrangian in FeynRules and export to UFO/Whizard; 3) Generate events and calculate cross sections with MadGraph5 and Whizard; 4) Apply experimental cuts and beam effects; 5) Evaluate rates with different luminosities and estimate statistical significance and limits on couplings and masses; 6) Repeat the analysis for annihilation, neutral VBF, and charged VBF and compare relative sensitivity.
Experimental and theoretical considerations: in addition to event simulation, studies should incorporate estimates of detector efficiency, background noise, and cybersecurity effects in data handling. It is also important to quantify theoretical uncertainties from scale choice and radiation lines, as well as to validate models against full detector simulations when possible.
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Conclusion: simplified models, implemented with FeynRules and studied with MadGraph5 and Whizard, constitute an efficient way to benchmark the physics potential of a muon collider. Evaluating annihilation, neutral VBF, and charged VBF with realistic rate estimates allows defining design goals and experimental strategies. If you seek support to integrate simulation, custom software development, artificial intelligence analysis, or secure deployments on AWS and Azure cloud services, the Q2BSTUDIO team is prepared to collaborate at every step of the process.
