Tutorials

Four walks, from the first diagram to physics beyond the Standard Model. Each takes about ten minutes; the numbers quoted are those TreeLevel displays. The interface follows the system language; the labels used below are the English ones.

1. e⁺e⁻ → μ⁺μ⁻: a first computation, then the Z peak

1. New document (⌘N). In the palette the StandardModel is selected; leave the direction of time on time →.
2. Vertex tool: click twice on the canvas, left and right, to place two vertices.
3. Line tool, particle γ: drag from the left vertex to the right one. That is the photon propagator.
4. Particle e⁻ / e⁺: drag from empty space, bottom left, to the left vertex — the palette announces that an e⁻ enters. Repeat from the top left: the fermion-flow arrow will run backwards in time, the leg is read as an incoming e⁺. (The order in which you drag the two legs does not matter: what counts is the direction of the arrow.)
5. Particle μ⁻ / μ⁺: drag from the right vertex into empty space, top right (outgoing μ⁻), then from empty space at the bottom right to the vertex (the arrow comes back to the vertex: outgoing μ⁺).
6. Look at the Analysis tab: Process: e⁻ e⁺ → μ⁻ μ⁺, consistent diagram. If the analysis reports a non-existent vertex, one of the legs is read the wrong way: select it and tick Antiparticle, or move its end to the other side of the vertex.
7. Expand Feynman rules and amplitude: the vertex −ieγ^μ twice, the propagator −ig_μν/(p₁+p₂)², the spinors v̄(p₂)…u(p₁) and ū(p₃)…v(p₄).
8. Compute tab: √s = 40 GeV, Compute. σ = 62.3 pb. Check by hand: σ = 4πα²/3s with α = 1/127.9 (the model's value at M_Z) and 1 GeV⁻² = 3.894 × 10⁸ pb gives 62.3 pb.
9. Back in the palette, click Generate › Add the missing diagrams: the Z diagram appears — together with h and G⁰ exchange, proportional to m_e and negligible (the No scalar exchange option of the Generate menu avoids them). The computation now bears on the sum.
10. Plot against energy: from 40 to 200 GeV, 300 points, log scale, Plot. The Z peak reaches 1,979 pb at √s = M_Z = 91.19 GeV, with the width Γ_Z = 2.495 GeV declared in the model. The curve goes through a minimum near 70 GeV (25 pb) before rising towards the resonance; the γ–Z interference, proportional to the small vector couplings g_V, stays discreet in the total σ — it shows up in the forward–backward asymmetry.

2. The muon lifetime

1. New document, model StandardModel. Place two vertices.
2. Line W⁺ / W⁻ between the two vertices: a virtual W.
3. Particle μ⁻ / μ⁺: drag from empty space, on the left, to the first vertex (incoming μ⁻). Particle ν_μ: drag from the first vertex into empty space (outgoing ν_μ).
4. At the second vertex: outgoing e⁻ (from the vertex into empty space) and outgoing ν̄_e — drag the leg from empty space to the vertex so that the arrow runs backwards in time.
5. Check in Analysis: μ⁻ → ν_μ e⁻ ν̄_e, consistent. A classic mistake is to draw the ν̄_e as a particle: the requested vertex ve e W~ does not exist, and the inspector lists the neighbouring vertices that do (ve~ e W, e~ ve W~) — in other words, one leg must be reversed.
6. Compute tab: the decay is three-body, phase space is integrated exactly (Dalitz). Γ = 3.007 × 10⁻¹⁹ GeV, τ = 2.19 µs, T½ = 1.52 µs. The measured value is 2.197 µs; the difference is due to radiative corrections, absent at tree level.

Variant: in the Hadrons model, draw n → p e⁻ ν̄_e by W exchange (τ = 946 s at tree level with g_A = 1.27, against 879 s measured) or π⁺ → μ⁺ν_μ, which has a single contact vertex.

3. e⁺e⁻ → W⁺W⁻: generate, sum, and see the gauge cancellation

1. New document. Place two vertices, drag the external legs: incoming e⁻ and e⁺ on the left, outgoing W⁻ and W⁺ on the right (particle W⁺ / W⁻: dragged from the vertex into empty space the line is an outgoing W⁺; from empty space to the vertex, i.e. against time, an outgoing W⁻). Do not draw an internal line yet.
2. Generate › Replace by all tree diagrams: four diagrams — γ, Z and h exchange (negligible, ∝ m_e) and t-channel ν_e exchange.
3. Compute tab, √s = 200 GeV: σ = 19.2 pb, as at LEP (17 pb measured, radiative corrections included).
4. Delete the neutrino diagram (select it in the list, −) and recompute: 18.1 pb. Almost nothing changes… at 200 GeV.
5. Plot against energy, from 170 to 1,000 GeV, log scale: without the neutrino σ grows (118 pb at 500 GeV, 385 pb at 1 TeV); with the four diagrams it falls as it should (7.1 pb at 500 GeV, 2.4 pb at 1 TeV). That is the cancellation between the gauge diagrams, and the reason for the triple-boson couplings of the Standard Model.

4. Beyond the Standard Model: SU(5) and the MSSM

SU(5): a boson that turns quarks into leptons

1. Choose the SU5 model. The palette gains two bosons, X (charge 4/3) and Y (1/3), colour triplets of mass 10¹⁵ GeV.
2. Draw X → u u: one vertex, incoming X, two outgoing u. Analysis: consistent, with the colour factor ε_abc in the rule — a vertex that violates baryon number.
3. Compute: Γ(X → uu) = 4.17 × 10¹² GeV. Draw X → e⁺ d̄: the same width. The branching ratios ½ / ½ are those of Georgi–Glashow; for Y you will find ½ (u d), ¼ (e⁺ ū), ¼ (ν̄ d̄), and Γ_Y = Γ_X, as befits two members of the same doublet.
4. Draw u u → e⁺ d̄ and generate: three diagrams (s-channel X, t- and u-channel Y). At √s = 2 × 10¹⁵ GeV, σ = 7 × 10⁻²⁶ pb; the Analysis tab gives the full symbolic expression in s, t, M_X, M_Y.

The model file (Theory tab) shows how these couplings are written: bilinears ψᵀCγ^μP_Lχ with the charge-conjugation matrix C. See the Theories page.

MSSM: importing a UFO

1. Download the MSSM_SLHA2 UFO folder from the FeynRules model database (or any UFO model shipped with MadGraph). In the palette, button 📁 Import a UFO folder, choose the folder.
2. A second later: “UFO imported: 50 fields, 822 vertices”. The model appears in My models; the palette lists neutralinos, charginos, squarks, sleptons and the gluino with their PDG codes (1000022 for χ̃⁰₁…).
3. Draw e⁻ e⁺ → χ̃⁰₁ χ̃⁰₁ (two outgoing n1 legs; the neutralino is Majorana, there is no antiparticle), then Generate: s-channel Z exchange, t- and u-channel ẽ_L and ẽ_R exchange.
4. Compute at √s = 500 GeV: σ = 269 fb at the SPS1a point, within the range of the linear-collider studies. The row/column roles of the external spinors change from one diagram to the next: TreeLevel applies Denner's fermion-flow rules, and the result does not depend on the chosen orientation.