TreeLevel in a course

What your students handle themselves: a theory, a diagram, and the whole computation through to the cross section — with the intermediate steps left readable.

The computation stays visible

A student who reads “1,977 pb” has learnt nothing unless they see where the number comes from. Every step is displayed in the window:

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

Vertices, derived in front of them

Each interaction term of the Lagrangian becomes a vertex, with its combinatorial factor, its momenta and its colour and Dirac structure. The Feynman rules and amplitude tab gives them in LaTeX.

The amplitude, assembled

$i\mathcal M$ is written out: spinor chains, propagators and vertex factors, where the diagram puts them. The student finds their own worksheet back, term by term.

The spin sum

$\langle|\mathcal M|^2\rangle$ is computed in $s$ , $t$ and $u$ through Dirac traces, polarisation sums and colour factors, then displayed as a formula rather than a number.

Two routes, one result

The symbolic computation and the numerical helicity computation are independent and agree to $10^{-8}$ . A student redoing the work by hand has a reference to correct themselves against.

From the amplitude to what the apparatus records

This is the step students take worst: between “theory gives 2.99 pb” and “the experiment counted so many events”, a course usually passes in one sentence. Two further workspaces make them cross it for real.

A machine, not a process

The Event source asks what a real experiment asks: I fire x at y at such an energy — what comes out? Everything that can happen happens, each with its probability. Only the channels those beams can take are offered, and the app says why the others go dark.

Measuring instead of computing

The cross section is counted on the event sample instead of being integrated, and appears with its error bar next to the one the tree computed. The two do not agree, and that is the whole exercise.

What a detector would see

Tracker, electromagnetic and hadronic calorimeters, muon chambers, solenoid field: radii, $\Delta\eta\times\Delta\phi$ cells, resolutions, thresholds and efficiencies are all adjustable. The student changes the field or the resolution and sees at once what the apparatus loses.

Tracks, deposits, missing energy

The $x\text{–}y$ and $z\text{–}\rho$ cross sections show the event: helices of the charged particles, hits layer by layer, deposits per cell, anti- $k_T$ jets, missing transverse energy. The zoom runs from the metre of the detector to the femtometre of the vertex.

The example worth a whole lecture

$e^+e^-\to\mu^+\mu^-$ at 200 GeV, measured on the sample and then set against the calculation:

What is counted	$\sigma$
raw count, any muon pair	$7.2\ \mathrm{pb}$
requiring $m(\mu\mu) > 0.7\sqrt s$	$2.86\pm0.14\ \mathrm{pb}$
$\sigma$ computed at tree level	$2.99\ \mathrm{pb}$

The raw count is two and a half times the calculation, and that is not an error: a photon emitted before the collision lowers the energy the mediator actually had, and above the Z it most often brings the collision back onto the resonance. Those events are the same process at another energy, and they dominate. The threshold is adjustable, and between $0.6$ and $0.75\sqrt s$ the answer is flat: that plateau shows the student that a cut is not an arbitrary setting.

Twenty course processes, already built

The app ships with twenty ready-to-open documents, chosen to cover an introductory course. Each one opens, changes and recomputes.

Course topic	Examples supplied
Elementary QED	Bhabha ( $s$ and $t$ channels, interference), Compton
Electroweak, the Z peak	$e^+e^-\to\mu^+\mu^-$ (photon and Z summed), $Z\to b\bar b$
Non-abelian gauge structure	$e^+e^-\to W^+W^-$ : the cancellation between the three diagrams, visible by unticking them one at a time
Weak decays	muon, neutron, $\tau\to\pi\nu$
Yukawa coupling	$h\to\tau^+\tau^-$
Colour	$u\bar u\to t\bar t$
Hadrons and resonances	$J/\psi\to\mu^+\mu^-$ , $e^+e^-\to\pi^+\pi^-$ ( $\rho$ ), $\pi^+p\to\pi^+p$ ( $\Delta$ ), $\Lambda\to p\pi^-$ , $\Omega^-\to\Lambda K^-$
Axial anomaly	$\pi^0\to\gamma\gamma$
Majorana neutrinos	$e^-e^-\to W^-W^-$
Beyond the Standard Model	SU(5): $X\to uu$ and $uu\to e^+\bar d$ ; MSSM imported from a UFO folder

The Generate button enumerates every tree-level diagram of a process: the student proposes their list, then compares it with the app's.

What you can check before handing it to them

The rules are not transcribed

They are derived from the Lagrangian every time it is opened. The 129 Standard Model vertices match those of MadGraph one for one — the correspondence table is published on the Theories page.

The theories are open

Full Standard Model in Feynman gauge with a complex CKM, QED, $\phi^4$ , an effective hadronic model, minimal SU(5). You can write your own in a small readable language, or import a UFO folder produced by FeynRules.

Orders of magnitude can be checked

62 pb at 40 GeV for $e^+e^-\to\mu^+\mu^-$ , that is $4\pi\alpha^2/3s$ ; 1,977 pb at the Z peak. A $\sigma(\sqrt s)$ scan over 641 energies takes one second — enough to plot the curve during the lecture.

Nothing leaves the machine

No account, no connection, no data sent. The app works offline, which settles the question in the lab room as well as in an exam.

In the room

macOS 14 or later (Apple silicon and Intel), iPadOS 17 or later, Windows 10 (2004) and 11 on x64 and ARM64. The interface exists in 11 languages, so it can be handed to an international group as it is.

Universal Mac + iPad purchase at CHF 40 on the App Store; CHF 40 on the Microsoft Store. Institutions equipping a group buy in volume through Apple School Manager. First computation in 10 minutes →

Tell me what your course is missing

TreeLevel is written by a physicist, not by a marketing department: if a process from your syllabus is not among the examples, if a step of the computation deserves to be shown differently, or if your department is considering equipping a group, write to me — this is what decides what gets built next.

Write to the author