Validated against MadGraph¶

The strongest correctness statement a Lagrangian→Feynman-rule tool can make is that its output computes the right numbers in a downstream generator. feynlag’s Standard-Model UFO is round-tripped through MadGraph5_aMC@NLO: the exported model is imported, real cross sections are computed, and they are compared to the standard sm model shipped with MadGraph at the same parameter point.

Reproduce it with:

python scripts/madgraph_roundtrip.py

which downloads MadGraph if needed, exports the SM UFO (scripts/export_sm_ufo.py), and runs both models on both processes.

Parameter point¶

The export uses MadGraph’s stock electroweak input scheme \((\alpha_{\rm EW}^{-1}, G_F, M_Z) = (132.50698,\ 1.16639\times10^{-5},\ 91.1876)\), from which the same tree-level relations the stock sm uses give

\[M_W = 80.4185\ \text{GeV},\quad M_Z = 91.1876\ \text{GeV},\quad v = 246.22\ \text{GeV},\]

so the two models have numerically identical couplings by construction. Lepton-collider beams: \(\sqrt s = 200\) GeV, no PDF (lpp=0).

Cross sections¶

Process	feynlag UFO	stock `sm`	agreement
\(e^+e^-\to\mu^+\mu^-\)	\(2.7878 \pm 0.0027\) pb	\(2.7878 \pm 0.0027\) pb	0.00% (0.0σ)
\(e^+e^-\to W^+W^-\)	\(19.498 \pm 0.058\) pb	\(19.497 \pm 0.058\) pb	0.01% (0.0σ)

Both agree with the stock model to within Monte-Carlo error.

What each process validates¶

\(e^+e^-\to\mu^+\mu^-\) exercises the photon and \(Z\) FFV couplings and the \(Z\) propagator (γ and \(Z\) \(s\)-channel interference).
\(e^+e^-\to W^+W^-\) is the gauge-cancellation acid test. The \(\nu\) \(t\)-channel diagram grows with energy and is tamed only by a precise cancellation against the \(\gamma/Z\) \(s\)-channel diagrams — which requires the relative signs across the \(W\bar\nu e\) FFV vertex and the \(WW\gamma/WWZ\) triple-gauge vertices to be exactly right. A wrong relative sign gives ≈98 pb instead of ≈19.5 pb; getting 19.5 pb confirms the cancellation holds.

Bugs the round-trip caught¶

Building this benchmark surfaced two real UFO-export bugs (both fixed, now regression-tested) and one convention subtlety — exactly the kind of thing that is invisible to unit tests but breaks a real generator run:

Relative imports in the generated UFO (from .object_library …) broke MadGraph’s param-card generation, which runs the modules as standalone scripts. Real UFO models use absolute imports; feynlag now does too (test_ufo_export.py::test_ufo_uses_absolute_imports).
The FFV export dropped the Feynman-rule \(i\) that the UFO coupling convention carries (the bosonic and VVV couplings already had it). This is invisible in \(e^+e^-\to\mu^+\mu^-\) (an overall phase cancels in \(|\mathcal M|^2\)) but breaks the FFV↔VVV interference in \(e^+e^-\to W^+W^-\). Fixed in add_fermion_vertex.
The triple-gauge coupling needs a sign flip for the electroweak vertices but not for the gluon. This looked for a long time like an unresolved MadGraph convention mismatch; it is neither a mismatch nor a bug. feynlag’s symbols label fields, a UFO leg labels a particle, and the field \(W^+\) annihilates a \(W^+\) but creates a \(W^-\) — so the leg carrying the symbol Wp is UFO’s W- leg. Emitting legs under naive names transposes each conjugate pair, and VVV1 is totally antisymmetric: one conjugate pair \(\Rightarrow -1\) (the electroweak cubics), none \(\Rightarrow +1\) (\(ggg\)). gauge_basis.ufo_leg_sign computes it, and the export no longer applies anything by hand.

The same rule explains why VVS/VVSS/VVVV never needed a flip — their structures are invariant under that transposition — and it reproduces MadGraph’s GC_4, GC_53, GC_10, GC_3 and GC_61 with no other adjustment. It is also behaviour-preserving here: the old export emitted \((\gamma,W^+,W^-)\) with \(-ie\), the same vertex as the \((\gamma,W^-,W^+)\) with \(+ie\) emitted now, so the 19.50 pb below is unchanged.

This is the payoff of feynlag’s verification-first design: the round-trip is a harness that turns “the model looks right” into “the model computes the right cross section.”

Quartic gauge couplings: the \(\gamma\gamma\to W^+W^-\) Ward identity¶

Cross sections are not the only oracle MadGraph offers. Its check command evaluates a process’s matrix element at random phase-space points and runs three model-internal consistency tests — Lorentz invariance, the gauge/BRS (Ward) identity, and leg-permutation symmetry — with no beams and no Fortran cross-section run, so it is fast enough to use as a routine acceptance test.

\(\gamma\gamma\to W^+W^-\) is the ideal probe for the quartic: its only diagrams are \(t\)- and \(u\)-channel \(W\) exchange plus the \(\gamma\gamma W^+W^-\) contact term, and gauge invariance holds only if the quartic’s normalization relative to the exchange diagrams is exactly right. A wrong quartic cannot hide.

mg5_aMC <<< 'import model /path/to/FEYNLAG_SM
check a a > w+ w-'

Model	Lorentz invariance	Gauge (BRS) ratio	Result
stock `sm`	\(3.1\times10^{-15}\)	\(5.0\times10^{-28}\)	Passed
feynlag SM UFO	\(2.6\times10^{-15}\)	\(1.0\times10^{-27}\)	Passed
feynlag, quartic \(\times 3\)	\(3.7\times10^{-1}\)	\(2.6\times10^{-2}\)	Failed

The third row is the point. Before this was validated, assemble_vvvv reconstructed 3× the true vertex (the three UFO VVVV1/2/3 structures are linearly dependent, so the decomposition is a 1-parameter family and the symmetric representative carries a \(1/3\) that the hard-coded normalization omitted). Re-exporting with the old value makes both checks fail catastrophically and inflates the matrix element by ~500×. The unit tests could not see it, because the “independent” ground truth had been built in the same over-complete convention and compared coefficient lists, which cannot detect a common factor.

The exported couplings also match stock sm entry by entry at the shared parameter point, compared in the convention-free metric-pair basis since MadGraph’s VVVV basis differs from feynlag’s (tests/test_ufo_sm_bosonic.py, in CI):

Vertex	feynlag	stock `sm`
\(\gamma\gamma W^+W^-\)	\(ie^2\)	`GC_5`
\(W^+W^-W^+W^-\)	\(-ig^2\)	`GC_35`
\(W^+W^-ZZ\)	\(ig^2c_w^2\)	`GC_36`
\(\gamma W^+W^-Z\)	matches on MG’s two-structure `VVVV5` shape	`GC_57`
\(hW^+W^-\), \(hZZ\), \(hhW^+W^-\), \(hhZZ\)	extractor output, unrescaled	`GC_72`, `GC_81`, `GC_34`, `GC_65`

QCD: the \(u\bar u\to gg\) gauge check¶

Until this, feynlag’s QCD sector had never been exercised in any process. The cross-section round-trip covers \(e^+e^-\to\mu^+\mu^-\) and \(e^+e^-\to W^+W^-\), both QCD-free, and the \(\gamma\gamma\to W^+W^-\) check above is electroweak. Every QCD result in the repo was a symbolic pin of feynlag’s own output — which is exactly how three separate colour defects survived in it.

\(u\bar u\to gg\) is the right probe: \(t\)- and \(u\)-channel quark exchange (no \(ggg\)) interferes with the \(s\)-channel gluon (one \(ggg\)), so the amplitude is linear in the triple-gluon coupling and its sign is observable. \(gg\to gg\) will not do — it goes as \(ggg^2\) and is sign-blind.

python scripts/madgraph_qcd.py     # exports the UFO and runs all three models

Model	Lorentz invariance	Gauge (BRS) ratio	Result
stock `sm`	\(1.6\times10^{-16}\)	\(2.3\times10^{-30}\)	Passed
feynlag QCD UFO	\(3.0\times10^{-15}\)	\(2.0\times10^{-30}\)	Passed
feynlag, \(ggg\) sign flipped	\(3.6\times10^{-1}\)	\(4.0\)	Failed

The third row is the point, as before: a pass alone would only mean “nothing crashed”. scripts/madgraph_qcd.py exports the broken variant itself and exits non-zero unless it fails.

Two defects this caught, both invisible to everything symbolic¶

1. Every coupling was tagged QED. writer.py assigned order = {'QED': max(n_legs-2, 1)} unconditionally, so the gluon vertices carried a QED order. MadGraph rejects that outright — CRITICAL: Model with non QCD emission of gluon (found 6 of those) — and then builds the wrong diagram set, which made the first run of this check inconclusive rather than failing: the broken control passed too, because the \(ggg\) diagram was not in the amplitude at all. The order is now QCD whenever the colour tensor is not the singlet '1'; the power was already right (\(ggg\to1\), \(gggg\to2\), \(qqg\to1\)).

2. The \(qqg\) colour tensor was transposed. The writer emitted T(3,1,2) with its [bar, field, boson] leg order, reasoning from fermion_gauge_current’s T[r,c] Lagrangian index order. But UFO’s T(a,i,j) takes i as the fundamental index, so it must be T(3,2,1) — what MadGraph’s stock sm emits for [u~, u, g]. Transposing a hermitian \(T^a\) conjugates it, which flips the sign of the \(ggg\) interference: with T(3,1,2) the model failed both the Lorentz and gauge checks (rel. diff \(0.50\), BRS ratio \(3.4\)).

That second one is worth dwelling on. The failure was initially masked twice over — once by the coupling-order bug, and once by a regex in the check parser that dropped failing sections (MadGraph indents JAMP rows under a failure), so a hard failure first appeared as a missing section. And the naive reading of the surviving numbers was that flipping \(ggg\) “fixed” it — which would have been precisely the wrong conclusion, since \(ggg=-g_s\) matches stock GC_10. The colour index order was the real defect, and only the exported couplings matching stock (\(ggg=-g_s\), \(qqg=ig_s\), \(gggg=ig_s^2\)) made that legible.

Four-fermion operators: the muon-decay width¶

The four-fermion (FFFF) export is validated the same way, against an analytic result. scripts/madgraph_fermi.py exports a Fermi-theory UFO for

\[\mathcal L \supset -\tfrac{4G_F}{\sqrt2}(\bar\nu_\mu\gamma^\mu P_L\mu)(\bar e\,\gamma_\mu P_L\nu_e) + \text{h.c.}\]

and asks MadGraph for the muon partial width, comparing to the textbook

\[\Gamma(\mu^-\to e^-\bar\nu_e\nu_\mu) = \frac{G_F^2 m_\mu^5}{192\pi^3} = 3.009\times10^{-19}\ \text{GeV}\quad(\tau_\mu = 2.19\ \mu\text{s}).\]

Quantity	feynlag UFO (MadGraph)	analytic	agreement
\(\Gamma(\mu\to e\nu\nu)\)	\(3.007\times10^{-19}\) GeV	\(3.009\times10^{-19}\) GeV	0.07%

This confirms the exported FFFF* Lorentz structure, the fermion-flow leg assignment, and the coupling normalization are all physically correct — the part unit tests (symbolic extraction + coupling round-trip) cannot see. It also surfaced a UFO-export requirement specific to contact interactions: both the operator vertex and its Hermitian conjugate must be exported, since MadGraph needs the conjugate pair to establish a consistent fermion-number flow through the four-fermion vertex (a UFO carrying only one fails diagram generation). Not in CI (each launch compiles Fortran).