feynman-engine 0.2.1

Feynman diagram generator + amplitude/cross-section calculator wrapping QGRAF, FORM, LoopTools, OpenLoops, and LHAPDF

FeynmanEngine

A Feynman diagram generator and amplitude calculator for particle physics. Type a process like e+ e- -> mu+ mu- and get back enumerated diagrams (SVG/TikZ), the symbolic spin-averaged $|\overline{\mathcal{M}}|^2$, integrated cross-sections at LO and NLO, decay widths, and 1-loop scalar integrals.

The browser UI shipped with the package; 128+ pre-loaded examples in the sidebar

Built on proven HEP tooling:

• QGRAF for diagram enumeration
• FORM for symbolic algebra
• LoopTools for 1-loop scalar/tensor integrals
• OpenLoops 2 for tree + 1-loop SM amplitudes
• LHAPDF for PDFs (CT18LO default)
• SymPy for Dirac trace computation
• TikZ-Feynman for diagram rendering
• FastAPI and SciPy

Contents

• Installation
• Capabilities
  • Theory coverage
  • LHC validation at √s = 13 TeV
  • What's intentionally out of scope
• Quick start
  • Examples
• Architecture
• Citations

Installation

pip install feynman-engine
feynman setup     # builds QGRAF, FORM, LoopTools, LHAPDF, OpenLoops 2 (10-20 min, one-time)
feynman serve     # http://localhost:8000

For platform-specific prerequisites, the lightweight install path, Docker, and troubleshooting, see INSTALLATION.md.

Capabilities

• Feynman diagrams by topology, rendered to SVG/TikZ
• Tree amplitudes via 140 curated formulas (15 QED, 56 QCD, 68 EW, 1 BSM template, including per-quark-flavour Drell-Yan and charged-current variants), FORM color algebra, or SymPy γ-matrix traces
• 1-loop amplitudes via 35 curated formulas, Passarino-Veltman reduction, and analytic A0/B0/C0/D0 (pure Python, no Fortran needed for the closed-form integrals; LoopTools fallback for general kinematics)
• Cross-sections via scipy.quad (2→2) and RAMBO/Vegas Monte Carlo (2→N) with full massive Källén kinematics
• Decay widths for nine Higgs channels plus all Z/W/top channels, agreeing with PDG 2024 within 3% per channel and within 0.1% on the summed Higgs width; off-shell H → VV → 4f handled via 2-D Breit-Wigner integration (Pocsik-Zsigmond / Cahn)
• Differential observables for cosθ, pT, η, y, M_inv, M_ll, ΔR
• Hadronic σ with built-in LO PDF (factor-of-2-3 accuracy) or LHAPDF + CT18LO (percent-level)
• NLO in six regimes: 34 tabulated LHC K-factors, universal QED via charge-correlator (K = 1 + 3α/(4π) exact for textbook cases), EW Sudakov LL+NLL, first-principles Catani-Seymour Drell-Yan (K(pp→DY @ 13 TeV) = 1.19 vs YR4 1.21), tabulated NLO QCD K-factors for partial decay widths (H → gg K=1.66, H → bb̄ K=1.13, H → cc̄ K=1.24), and OpenLoops virtuals for arbitrary QCD processes
• Trust labels on every numerical result (validated / approximate / rough / blocked); the API refuses (HTTP 422) processes that would otherwise return wrong numbers
• REST API + Python API + browser UI all from one pip install

Theory coverage

Theory	Particles	Examples
QED	leptons, γ	e⁺e⁻→μ⁺μ⁻, Bhabha, Compton, e⁺e⁻→γγ, μμγ brem, 1-loop VP/vertex/box
QCD	quarks, gluons, ghosts	qq̄→gg, gg→gg, qg→qg, qq̄→tt̄ (massive top), 2→3 multi-jet, 1-loop
QCDQED	QCD + photon	qq̄→γγ, qq̄→γg, qg→qγ, γg→qq̄ (per quark flavour)
EW	full SM (γ, Z, W±, H, leptons, quarks)	qq̄→ZH/ZZ/W⁺W⁻ (5 quark flavours each), e⁺e⁻→W⁺W⁻, all Z/W/H decays, t→bW, loop-induced H→γγ/Zγ/gg, gg→H
BSM	Z′, scalar dark matter	e⁺e⁻→χχ̄, Z′→χχ̄, χχ̄→ll (DM annihilation), Z′ decays

LHC validation at √s = 13 TeV

Process	Engine σ	Reference	Status
pp → tt̄ (LO)	518 pb	MG5 LO 504 pb	within 3% of MG5
pp → tt̄ (NLO, K=1.6)	828 pb	LHC NLO 700-830 pb	in band
pp → DY (60 < M_ll < 120)	1530 pb	~2000 pb	within 25%
pp → ZZ	8.8 pb	~10 pb	within 12%
pp → H (ggF, NLO K=1.7)	38.6 pb	YR4 NNLO ~44 pb	within 12%
pp → ZH	0.58 pb	~0.5 pb	within 16%
pp → H + jj (VBF)	3.78 pb	3.78 pb	exact (calibrated)
pp → γγ (pT_γ > 30 GeV)	30 pb	30-50 pb	in range

Higgs decay validation (m_H = 125.20 GeV)

All nine SM Higgs partial widths agree with PDG 2024 within 3%; sum of partial widths reproduces the PDG total Γ_H to 0.1%.

Channel	Engine	PDG 2024	Δ
H → bb̄ (LO + NLO QCD K=1.13)	2.413 MeV	2.41 MeV	+0.12%
H → cc̄ (LO + NLO QCD K=1.24)	0.117 MeV	0.117 MeV	+0.24%
H → τ⁺τ⁻	0.259 MeV	0.257 MeV	+0.83%
H → gg (LO + NLO QCD K=1.66)	0.335 MeV	0.336 MeV	-0.25%
H → γγ (exact loop FF)	9.16 keV	9.31 keV	-1.6%
H → Zγ (loop FF)	6.41 keV	6.31 keV	+1.5%
H → WW → 4f (off-shell BW)	0.855 MeV	0.881 MeV	-3.0%
H → ZZ → 4f (off-shell BW)	0.108 MeV	0.108 MeV	-0.02%
Σ Γ_H (sum)	4.10 MeV	4.10 MeV	+0.1%

What's intentionally out of scope

• 2-loop and higher diagram generation. The 35 curated 1-loop entries cover textbook self-energies, vertex form factors, DGLAP NLO splitting kernels, α_s 2-loop running, and loop-induced Higgs decays, but not 2-loop diagrams from scratch.
• Full SUSY / SMEFT / 2HDM models. The bundled BSM model is intentionally minimal (Z′ + scalar dark matter). Larger frameworks should use the register_curated_amplitude extension API; see examples/for_bsm_theorist.ipynb.
• NNLO precision physics. For sub-percent NNLO predictions, use NNLOJET or MCFM. This package targets fast LO and NLO with diagrams and amplitudes returned in one call.

Quick start

The Python API:

from feynman_engine.amplitudes.cross_section import total_cross_section
from feynman_engine.amplitudes.hadronic import hadronic_cross_section

r = total_cross_section("e+ e- -> mu+ mu-", "QED", sqrt_s=91.0)
print(r["sigma_pb"], r["trust_level"])    # 10.47 pb, "validated"

r = hadronic_cross_section("p p -> t t~", sqrt_s=13000.0, theory="QCD", order="NLO")
print(r["sigma_pb"], r["k_factor"])       # 828 pb, K=1.6 (tabulated NLO)

The decay-width route accepts an order parameter:

from feynman_engine.api.routes import get_decay_width

r = get_decay_width(process="H -> g g", theory="QCD", order="NLO")
print(r["width_mev"], r["k_factor_nlo"]) # 0.335 MeV, K=1.66 (Spira 1995)

The full REST API surface is documented at http://localhost:8000/docs (Swagger) once you run feynman serve.

To register your own $|\overline{\mathcal{M}}|^2$ for a custom or BSM process:

import sympy as sp
from feynman_engine.physics.amplitude import register_curated_amplitude

s, t, u = sp.symbols("s t u", positive=True)
register_curated_amplitude(
    "my+ my- -> custom_X", "BSM",
    msq=2 * sp.Symbol("g_X")**4 * (t**2 + u**2) / s**2,
    description="Custom BSM 2->2 via single mediator",
)
# total_cross_section(), differential_distribution(), and hadronic
# enumeration all use the registered formula automatically.

Examples

Five Jupyter notebooks in examples/ cover different audiences:

• getting_started.ipynb: install, first diagram, browser UI
• for_undergrad_qft.ipynb: teaching particle physics with diagrams + amplitudes side by side
• for_lhc_experimentalist.ipynb: LHC observables, K-factors, dσ/dM_ll across the Z peak
• for_bsm_theorist.ipynb: registering custom BSM amplitudes (heavy scalar mediator, SMEFT operator, leptophobic Z′)
• nlo_quickstart.ipynb: five-step tour of the NLO machinery (universal QED, EW Sudakov, hadronic DY)

Architecture

feynman_engine/
├── __main__.py            CLI: serve, generate, install-*, doctor
├── core/                  Diagram model, QGRAF interface, generator
├── render/                TikZ → PDF → SVG pipeline
├── amplitudes/
│   ├── symbolic.py        SymPy γ-matrix tree amplitudes
│   ├── form_trace.py      FORM trace + SU(3) color algebra
│   ├── loop.py            Passarino-Veltman decomposition
│   ├── analytic_integrals.py   Closed-form A0/B0/C0/D0
│   ├── looptools_bridge.py     LoopTools numerical evaluation
│   ├── cross_section.py        scipy.quad / RAMBO / Vegas integrators
│   ├── nlo_cross_section.py    Analytic K, universal QED/EW NLO routes
│   ├── nlo_qed_general.py      Universal QED NLO (charge correlator)
│   ├── nlo_ew_general.py       EW Sudakov LL+NLL
│   ├── nlo_general.py          Catani-Seymour generic NLO
│   ├── loop_curated.py         35 textbook 1-loop formulas
│   ├── differential.py         Histogrammed observables
│   ├── pdf.py                  Built-in PDF + LHAPDF auto-discovery
│   ├── hadronic.py             pp σ via PDF convolution
│   ├── dipole_subtraction.py   Catani-Seymour dipoles
│   └── openloops_bridge.py     OpenLoops 2 wrapper
├── physics/
│   ├── amplitude.py            Amplitude registry + backend chain
│   ├── trust.py                Trust-level enforcement (BLOCKED → 422)
│   ├── nlo_k_factors.py        Tabulated LHC NLO/LO ratios
│   ├── theories/               Particle + vertex registries per theory
│   └── translator.py           Process-string parser
├── api/                   FastAPI routes + Pydantic schemas
├── frontend/              Browser UI (vanilla JS + SVG rendering)
└── resources/             Bundled HEP source archives + QGRAF model files

The trust labelling in physics/trust.py is the safety boundary. Every endpoint that returns a number classifies the request first and refuses (HTTP 422 with a structured block_reason and workaround) for processes known to produce wrong values.

Citations

If you use FeynmanEngine in research, cite the software (Zenodo DOI in the badge above) and the wrapped tools that contributed to your specific workflow.

Foundational physics

• R. P. Feynman, "Space-Time Approach to Quantum Electrodynamics," Physical Review 76(6), 769-789 (1949), doi:10.1103/PhysRev.76.769
• G. Passarino and M. J. G. Veltman, "One-loop corrections for e+e- annihilation into mu+mu- in the Weinberg model," Nuclear Physics B 160(1), 151-207 (1979), doi:10.1016/0550-3213(79)90234-7
• A. Denner, "Techniques for the calculation of electroweak radiative corrections at the one-loop level and results for W-physics at LEP200," Fortschritte der Physik 41(4), 307-420 (1993), doi:10.1002/prop.2190410402
• S. Catani and M. H. Seymour, "A general algorithm for calculating jet cross sections in NLO QCD," Nuclear Physics B 485(1-2), 291-419 (1997), doi:10.1016/S0550-3213(96)00589-5

Wrapped tools

• QGRAF: P. Nogueira, "Automatic Feynman graph generation," Journal of Computational Physics 105(2), 279-289 (1993), doi:10.1006/jcph.1993.1074
• FORM: J. A. M. Vermaseren, "New features of FORM," arXiv:math-ph/0010025 (2000)
• LoopTools: T. Hahn and M. Perez-Victoria, "Automatized one-loop calculations in four and D dimensions," Computer Physics Communications 118(2-5), 153-165 (1999), doi:10.1016/S0010-4655(98)00173-8
• OpenLoops 2: F. Buccioni, J.-N. Lang, J. M. Lindert, P. Maierhöfer, S. Pozzorini, H. Zhang, M. F. Zoller, "OpenLoops 2," European Physical Journal C 79, 866 (2019), arXiv:1907.13071, doi:10.1140/epjc/s10052-019-7306-2
• COLLIER (used internally by OpenLoops): A. Denner, S. Dittmaier, L. Hofer, "COLLIER: a fortran-based Complex One-Loop LIbrary in Extended Regularizations," Computer Physics Communications 212, 220-238 (2017), doi:10.1016/j.cpc.2016.10.013
• OneLOop (used internally by OpenLoops): A. van Hameren, "OneLOop: For the evaluation of one-loop scalar functions," Computer Physics Communications 182(11), 2427-2438 (2011), arXiv:1007.4716
• CutTools (used internally by OpenLoops): G. Ossola, C. G. Papadopoulos, R. Pittau, "CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes," JHEP 0803, 042 (2008), arXiv:0711.3596
• LHAPDF6: A. Buckley et al., "LHAPDF6: parton density access in the LHC precision era," European Physical Journal C 75, 132 (2015), doi:10.1140/epjc/s10052-015-3318-8
• RAMBO: R. Kleiss, W. J. Stirling, S. D. Ellis, "A new Monte Carlo treatment of multiparticle phase space at high energies," Computer Physics Communications 40(2-3), 359-373 (1986), doi:10.1016/0010-4655(86)90119-0
• Vegas (when the vegas extra is installed): G. P. Lepage, "A new algorithm for adaptive multidimensional integration," Journal of Computational Physics 27, 192 (1978)

Specific physics formulas

• ggH cross-section in heavy-top limit: M. Spira, A. Djouadi, D. Graudenz, P. M. Zerwas, "Higgs boson production at the LHC," Nuclear Physics B 453(1-2), 17-82 (1995), doi:10.1016/0550-3213(95)00379-7
• pp→DY NLO K-factor benchmark: C. Anastasiou, L. J. Dixon, K. Melnikov, F. Petriello, "High precision QCD at hadron colliders: electroweak gauge boson rapidity distributions at NNLO," Physical Review D 69, 094008 (2004), arXiv:hep-ph/0312266
• EW Sudakov LL+NLL framework: S. Pozzorini, "Electroweak radiative corrections at high energies," Physical Review D 71, 053002 (2005); S. Catani and L. Comelli, Phys. Lett. B 446, 278 (1999)

Background textbooks

M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (1995). R. K. Ellis, W. J. Stirling, B. R. Webber, QCD and Collider Physics (1996). M. D. Schwartz, Quantum Field Theory and the Standard Model (2014).