TreeLevel Feynman diagrams that compute themselves.

A theory written as a Lagrangian, diagrams drawn by hand or generated, and in return: the Feynman rules, the amplitude, $\langle|\mathcal M|^2\rangle$ , cross sections, decay widths and lifetimes. Native, fast, on Mac and Windows.

Read the guide

macOS 14 or later, Apple silicon and Intel; iPadOS 17 or later. Universal purchase Mac + iPad, 40 CHF. On PC: Windows 10 (2004) and 11, x64 and ARM64, 40 CHF on the Microsoft Store. Interface in 11 languages. First computation in 10 minutes →

The TreeLevel window: particle palette on the left, canvas in the middle, inspector on the right

From the theory to the observables

In a single window, TreeLevel chains together what usually takes a table of rules, a symbolic algebra program and an integration code.

Lagrangian → Feynman rules → diagrams → amplitude → spin sum → observables

The theory comes first

The full Standard Model (Feynman gauge, complex CKM), QED, $\phi^4$ , an effective hadron model, minimal SU(5). Write your own in a small readable language, or import a UFO folder (MadGraph/FeynRules): the MSSM imports completely, 822 vertices.

Rules, derived

Every interaction term becomes a vertex: combinatorial factors, momenta, colour and Dirac structure. Nothing is copied from a table — the 129 Standard Model vertices agree one by one with MadGraph's.

Draw, check

Three tools are enough. The canvas reads the diagram along the time axis: a line running backwards in time is an antiparticle. The inspector tells you at once whether the diagram is consistent, and if not, why.

Sum the diagrams

The Generate button enumerates every tree diagram of a process. The valid diagrams of one process are summed with their relative signs, Majorana fermions included.

Compute fast

Numerical helicity amplitudes on all cores: a $\sigma(\sqrt s)$ scan over 641 energies takes a second. Quadrature for $2\to2$ , closed formulas for $1\to2$ , Dalitz for $1\to3$ , RAMBO Monte Carlo beyond.

And see the formula

The symbolic engine computes $\langle|\mathcal M|^2\rangle$ in $s$ , $t$ , $u$ with Dirac traces, polarisation sums and colour factors; rules and amplitude are displayed in LaTeX, and agree with the numerical result to $10^{-8}$ .

What it looks like

e+e- → μ+μ- by photon and Z exchange — $e^+e^-\to\mu^+\mu^-$ : the two tree diagrams, generated and summed.

e+e- → μ+μ- cross section against √s, with the Z peak — The cross section plotted from 40 to 200 GeV: 62 pb at 40 GeV ( $4\pi\alpha^2/3s$ ), 1,977 pb at the Z peak.

Quantity	TreeLevel	Reference
Muon lifetime	2.19 µs	2.197 µs (measured)
Z, W and Higgs widths (tree-level channels)	partial widths and branching ratios	Particle Data Group
$\Gamma(\pi^0\to\gamma\gamma)$ (effective model, anomaly)	7.79 eV	7.7 eV
Neutron lifetime (tree level, $g_A = 1.27$ )	946 s	879 s (no radiative corrections)
$e^+e^-\to W^+W^-$	gauge cancellation between $\gamma$ , $Z$ and $\nu$	correct high-energy behaviour
$\sigma(e^+e^-\to\tilde\chi^0_1\tilde\chi^0_1)$ at 500 GeV, MSSM SPS1a	269 fb	250–300 fb (ILC studies)
SU(5): branching ratios of X and Y	$\tfrac12\ \tfrac12$ ; $\tfrac12\ \tfrac14\ \tfrac14$	Georgi–Glashow

Generate events, simulate a detector

Three workspaces in one window. Compute: the diagrams, the amplitude, $\sigma$ and $\Gamma$ . Generation: thousands of unweighted events of the process, drawn in phase space with probability $|\mathcal M|^2$ , each event shown in projection with its particle table, statistics as histograms ( $p_T$ , $\eta$ , $\cos\theta$ , invariant masses, missing transverse energy, thrust…), a selection with its efficiency and the number of events expected at a given luminosity; export to Les Houches (.lhe) and HepMC3 for Pythia or Herwig, import of their files. Detector: a parametrised cylindrical detector (tracker, calorimeters, muon chambers, magnetic field, resolutions), its cross-section with the helices, hits and deposits of each event, the reconstructed objects, anti- $k_T$ jets, resolution and efficiency per object type.

TreeLevel's Generation workspace on the Mac: an e+e- → b b̄ event from Pythia 8 in projection, its particle table and the angular distribution of the sample — The Generation workspace: $e^+e^-\to b\bar b$ at 200 GeV, showered and hadronised by Pythia 8 — one event in projection with its particle table; on the right, the angular distribution of the 2,000 events.

TreeLevel's Detector workspace on the Mac: a proton-proton collision at 13 TeV in the x–y and z–ρ cuts, with its tracks, deposits and reconstructed jets — The Detector on a 13 TeV proton-proton collision (Machine source, Pythia 8): 122 tracks curling in the field, more than 2,000 cells hit, two jets and a photon reconstructed.

On Windows too

The same application on PC, Windows 10 and 11, x64 and ARM64: the same engine, the same .feyndiag documents interchangeable with the Mac, the same interface in 11 languages.

TreeLevel on Windows: e+e- → μ+μ- and the cross-section plot around the Z peak — On Windows: $e^+e^-\to\mu^+\mu^-$ and the cross section from 40 to 200 GeV.

TreeLevel on Windows: e+e- → neutralinos in the imported MSSM, angular distribution — The imported MSSM (UFO): $e^+e^-\to\tilde\chi^0_1\tilde\chi^0_1$ at 500 GeV, angular distribution.

Who is it for?

For the student learning Feynman rules who wants to see, on their own diagram, what each vertex becomes; for the teacher preparing an exercise and its solution; for the physicist who needs a tree-level order of magnitude without launching a MadGraph chain, or who wants to check the rules of a model they have just written.

What TreeLevel does not do (yet). No loops, no parton distribution functions: cross sections are partonic or leptonic. Resonance widths enter the propagators, but radiative corrections do not. See the Physics page.