17. Extending to 2→2 Scattering

The gap, stated honestly

Everything in feynlag.pheno before this chapter computes a decay: one particle in, two (or three, off-shell) out. It cannot compute a cross section at all — there is no notion of two incoming particles, no Mandelstam invariants, no flux factor, and no way to combine two vertices joined by a propagator into an amplitude rather than an already-squared number. The covariant-trace design was chosen partly with this in mind (see CLAUDE.md’s note that it gives “a cleaner path to a later 2→2”), and the repository already carries two MadGraph-validated cross-section benchmarks — \(\sigma(e^+e^-\to\mu^+\mu^-) = 2.7878\pm0.0027\) pb and \(\sigma(e^+e^-\to W^+W^-) = 19.498\pm0.058\) pb at \(\sqrt s = 200\) GeV (docs/benchmark.md) — so the acceptance oracles for a native 2→2 engine already exist and cost nothing new to obtain.

Two facts drive the whole design that follows.

The ε (γ₅) term does not vanish for 2→2. The 1→2 engine’s reduce_projectors() proves the γ₅ trace term is zero whenever a chiral fermion current meets at most 2 independent momenta and 2 free Lorentz indices — true for every decay, because a two-body final state supplies only \(P=p_1+p_2\), one combination, and the polarization sums it meets are symmetric. A 2→2 process has 3 independent momenta (\(k_4=k_1+k_2-k_3\) is not independent), and with two chiral fermion currents \(|\mathcal M|^2\) genuinely contains an \(\varepsilon\cdot\varepsilon\) term — the forward–backward asymmetry, a real, physically necessary piece of the answer, not a computational nuisance. The guard’s threshold was n_momenta > 3, which passes at exactly the 2→2 value and would have silently returned a wrong \(d\sigma/d\cos\theta\); Tier 1 tightens it to n_momenta > 2 (§17.5).

The 1→2 squared-amplitude functions are terminal and cannot be extended. ffs_squared()/ffv_squared return an already-squared scalar, with the 1→2 mass-interference term and the parent’s spin average (ffv_squared’s hardcoded /3) baked in as literals for that one topology. There is no partial expression to attach a propagator or a second diagram to. Tier 1 therefore builds the amplitude explicitly — open fermion chains, propagators, a genuine sum over diagrams — reusing the covariant engine (pheno/lorentz.py) unchanged underneath.

17.1 Tier 1 — kinematics, one diagram, no interference ✅ (delivered)

Channels: \(e^+e^-\to\mu^+\mu^-\) through a single photon (pure QED); \(f_1\bar f_1\to f_2\bar f_2\) through one s-channel scalar. Effort: small–medium.

What Tier 1 deliberately does not attempt: more than one diagram (γ/Z interference — Tier 3). This is a hard guard, not an omission: the engine raises NotImplementedError rather than returning a number that would be wrong by an \(O(1)\) fraction. (Chiral couplings on both vertices of the same diagram — the ε·ε term above — were Tier 1’s other hard guard; §17.2 below computes it instead.)

What was built

• TwoToTwoKinematics — Mandelstam invariants s (primary, positive=True) and t (primary, real=True, negative throughout the physical region), with u a derived property so momentum conservation holds by construction rather than assertion; the ten-entry on-shell dot table (including a dedicated momentum head for the dependent \(k_4\), so conservation is a testable invariant of the table, not expression-level algebra colliding with TensAdd expansion inside dirac_trace()); the flux factor \(1/(2\sqrt{\lambda(s,m_1^2,m_2^2)})\); t_bounds/cos_theta/t_of_cos; and the \(d\sigma/dt\), \(d\sigma/d\cos\theta\) conversion factors. s and t are kept as the primary symbols rather than (s,\cos\theta) because the dot table is then linear with no radicals — a \(\cos\theta\) parametrization would drag \(\sqrt{\lambda_i\lambda_f}\) into every table entry and hence into every Dirac trace, exactly the reason ThreeBodyKinematics confines its own radicals to s12_bounds.

• diagrams — the amplitude-level object layer: Leg, ChainVertex (with .bar(), matching feynlag.dirac.dirac_conjugate(), and .epsilon_coefficient()), SpinorChain (the open, γ₅-free trace of one fermion line — diagonal and chirality-mixing pieces derived the same way as ffs_squared/ffv_squared, but left open on external Lorentz indices instead of immediately reduced), BosonPropagator and Diagram/Amplitude. Amplitude.squared combines two chains through one propagator and reduces to on-shell dot products with a single call to contract_to_dots() — never per-chain — matching the library’s existing rule that that function runs once, on a fully-contracted expression.

• propagator gained amplitude-level (not squared-modulus) propagators — propagator_denominator, scalar_propagator, vector_propagator — alongside the existing breit_wigner()/vector_propagator_numerator, which Tier 1 reuses verbatim rather than duplicating.

• ExternalState — spin/colour degrees of freedom for an external scattering state, and the home of the averaging factor. This is the direct fix for the anti-pattern in ffv_squared’s hardcoded /3: an initial-state spin average masquerading as part of the Lorentz algebra, which made that function unusable the moment the vector is internal rather than the parent. average_factor() computes \(1/\prod_i \mathrm{dof}_i\) from declared data, and cross_section()/differential_cross_section apply it exactly once, never inside a squared amplitude.

• scattering — average_factor, differential_cross_section, cross_section (symbolic t-integral), and the two worked Tier-1 assemblers, ffv_s_channel_squared/ ffs_s_channel_squared, in the same coupling-in/number-out style as scalar_offshell_vv_width().

reduce_projectors: tightened now, retired for chains in Tier 2

n_momenta > 3 became n_momenta > 2 in reduce_projectors() — the only two callers, ffs_squared/ffv_squared, always pass n_momenta=2 (a chain’s own two legs), so the change is invisible to the 1→2 suite; a regression test now pins that n_momenta=3 raises (tests/test_pheno.py::test_gamma5_guard_raises_outside_two_body). This function still correctly proves each individual chain’s own ε term is zero — that reasoning is unaffected by the overall process being 2→2, since a chain’s trace only ever involves its own two legs. What it cannot see is the cross-chain ε·ε term that appears when two chiral chains meet through a propagator; Amplitude computes that term directly now (§17.2, delivered) instead of refusing it, and SpinorChain.trace no longer calls reduce_projectors at all — it stays frozen as the 1→2-only helper ffs_squared/ffv_squared use (tests/test_scattering.py::test_reduce_projectors_is_no_longer_on_the_chain_path).

Verified

Against the textbook QED closed form (Peskin & Schroeder §5.1 [PS95]) \(\sigma(e^+e^-\to\mu^+\mu^-) = \frac{4\pi\alpha^2}{3s}\cdot\frac{\beta(3-\beta^2)}{2}\), \(\beta=\sqrt{1-4m_\mu^2/s}\) — symbolically exact, plus the massless limit \(4\pi\alpha^2/3s\) — and against an independent explicit-4×4-Dirac-matrix oracle built in the CM frame (sharing no code with the covariant engine, in the same spirit as tests/test_pheno.py’s _oracle_*). At the exact docs/benchmark.md parameter point (\(\alpha^{-1}=132.50698\), \(\sqrt s=200\) GeV), the QED-only cross section comes out 2.322 pb — the photon-only fraction of MadGraph’s full 2.7878 pb, with the \(\approx20\%\) gap being exactly the \(\gamma\)/\(Z\) interference Tier 3 supplies, so this number cannot be accidentally “passed” here. tests/test_scattering.py also pins that a chiral coupling on both vertices genuinely differs from the naive “half the vector result” guess that does hold for a 1→2 decay (test_ffv_chiral_is_half_the_vector_result in test_pheno.py) — by up to \(O(1)\), at several angles — which is what justified Tier 1’s refusal to compute that case rather than silently guessing, before §17.2 computed it for real.

17.2 Tier 2 — the ε (γ₅) algebra ✅ (delivered)

Channels: \(e^+e^-\to\mu^+\mu^-\) through the \(Z\) alone; any process with a single diagram carrying two chiral fermion currents. Effort: large.

This landed before Tier 3, not folded into it. It was needed the moment one diagram has two chiral currents — no interference required — so it could be validated in isolation, against one closed form (\(d\sigma/d\cos\theta \propto (1+\cos^2\theta) + A_{FB}\cos\theta\), with the sign and magnitude of the forward–backward asymmetry pinned against [LEPEWWG06]), with exactly one new failure mode. Folded into Tier 3 alongside multi-diagram interference, a wrong \(A_{FB}\) could equally have been the ε reduction, a relative diagram sign, or fermion-flow bookkeeping — three candidate bugs instead of one.

What was built

• epsilon (new) — the only module with ε knowledge. SymPy’s own sympy.physics.hep.gamma_matrices ships neither a γ₅ object nor an ε-aware gamma_trace (checked directly against the installed version, not assumed), so both identities below are hand-rolled rather than delegated to a SymPy primitive:

  • \(\mathrm{Tr}[\gamma^a\gamma^b\gamma^c\gamma^d\gamma_5] = \kappa\,\varepsilon^{abcd}\) — gamma5_trace_coefficient() reads \(\kappa\) off the literal \(\gamma^0\gamma^1\gamma^2\gamma^3\gamma_5\) matrix from feynlag.dirac._dirac_rep, against a stated \(\varepsilon^{0123}=+1\) normalization (levi_civita_array()) — \(\kappa=-4i\), derived, not quoted, the same discipline dirac.majorana_symmetry_sign uses for its own sign.

  • \(\varepsilon^{a\ldots}\varepsilon^{b\ldots} = s_{\det}\!\cdot\!\det[g^{a_ib_j}]\) — epsilon_product_sign() derives \(s_{\det}=-1\) the same way, against the \((+,-,-,-)\) metric. epsilon_pair_tensor() expands this as 24 signed products of four metrics with all free Lorentz indices left open, each slot resolved by content (two momenta contract through a fresh dummy per pairing; a momentum meeting a free index just evaluates the head there; two free indices meet the ordinary LorentzIndex.metric) — no Eps tensor object needed, and the all-momentum case reduces on its own to the roadmap’s \(-\det[p_i\cdot p'_j]\) Gram determinant.

  • assert_epsilon_single_vanishes() — the single-ε cross terms (one chain’s ε piece times the other chain’s ordinary trace) reduce to a scalar times \(\varepsilon\)(four momenta drawn from the diagram’s own external legs); since \(\varepsilon(a,b,c,d)^2=-\det[\cdot]\) (identity 2 again), a vanishing Gram determinant over every 4-subset of the diagram’s momenta proves every such term is zero — a computed prove-or-refuse guard reading the diagram’s actual kin.dot table, not a momentum-count proxy that would need re-tuning for a future topology.

• epsilon_structure() — a chain’s ε piece as (prefactor, slots) or None (a scalar chain, or a pure-vector g_L=g_R chain, has none — exactly the cases the pre-Tier-2 engine already handled correctly, so those results are unchanged bit-for-bit). SpinorChain.trace no longer calls reduce_projectors — see the retirement note above.

• squared() assembles the ε contribution alongside the existing non-ε piece — both are separately fully-contracted scalars before being summed, so the house rule that contract_to_dots() runs exactly once, on a fully-contracted expression, survives literally. Contracting the two propagator numerators against the ε tensor stepwise (one numerator at a time, with .expand() between them) is ~3.5× faster than contracting both at once for a massive mediator.

• forward_backward_asymmetry() — the Tier-2 acceptance observable, \(A_{FB}=(\sigma_F-\sigma_B)/(\sigma_F+\sigma_B)\) over the \(\cos\theta\gtrless0\) hemispheres. Needs no ExternalState\ s, unlike differential_cross_section/cross_section: the spin average, any identical-particle symmetry factor, and the \(d\sigma/d\cos\theta\) conversion factor are all \(\cos\theta\)-independent and cancel exactly in the ratio.

Verified

The headline test, test_chiral_squared_amplitude_matches_explicit_matrix_oracle: with fully symbolic, independent chiral couplings \(g_L,g_R,h_L,h_R\) on both vertices (not the pure-vector case, where the ε term is trivially zero) through a massless mediator, the covariant engine matches the independent explicit-4×4-matrix oracle exactly (tests/test_scattering.py’s _oracle_qed_general) — this one identity simultaneously fixes \(\kappa\), the \(\mp\) sign in the \(P_{L,R}\) split, and the \(\varepsilon_{0123}\) convention, and its full generality is what shows the single-ε cross terms genuinely cancel rather than happening to vanish at one lucky point. test_z_massive_mediator_matches_explicit_matrix_oracle extends the oracle to a massive mediator (the only test needing that extension) and checks vector_propagator_numerator’s sign convention composes correctly with the ε piece; test_massive_mediator_epsilon_piece_drops_the_qq_term separately confirms the propagator’s \(q_\mu q_\nu/M^2\) term drops out of the ε contribution automatically (a repeated momentum in the same ε ⟹ two equal Gram-determinant rows ⟹ 0), for four symbolic external masses.

Against the literature: the massless-final-state angular distribution matches the LEP Born-level form, \(\Sigma|\mathcal M|^2=(g_L^2+g_R^2)(h_L^2+h_R^2)(1+\cos^2\theta)+2(g_L^2-g_R^2)(h_L^2-h_R^2)\cos\theta\) (Eq. (1.55) of [LEPEWWG06] at zero beam polarization), and forward_backward_asymmetry reproduces \(A_{FB}^{0,f}=\tfrac34A_eA_f\) (Eq. (1.66)) symbolically, plus a numeric pin at the LEP effective weak mixing angle \(\sin^2\theta_{\mathrm{eff}}^{\mathrm{lept}}=0.23153\) giving the standard leptonic \(A_\ell\approx0.147\), \(A_{FB}^{0,\ell}\approx0.016\). test_z_only_total_cross_section_is_epsilon_independent confirms the ε term is odd in \(\cos\theta\) and integrates away over the full range — a chiral mediator’s total cross section equals a pure-vector coupling’s with the same \(g_L^2+g_R^2\), so Tier 1’s 2.322 pb QED benchmark cannot shift from Tier 2 landing; \(A_{FB}\) is an angular observable only.

17.3 Tier 3 — interference and the first cross-section benchmark

Channels: the full \(e^+e^-\to f\bar f\) (\(\gamma\)/\(Z\) interference); s-, t-, u-channel topology enumeration; a ScatteringCalculator mirroring DecayCalculator. Effort: medium–large.

Amplitude drops its len(diagrams) == 1 guard and performs the full \(\sum_{d,d'} c_d\bar c_{d'}\) double sum, which needs relative fermion-flow signs and identical-particle exchange signs — the 2→2 analogue of DecayCalculator.channels, currently a one-line “vertex contains parent” search, becomes a genuine s/t/u topology enumeration over pairs of DecayVertex.

Oracle: \(\sigma(e^+e^-\to\mu^+\mu^-) = 2.7878\pm0.0027\) pb at the exact docs/benchmark.md parameter point — 20% above Tier 1’s QED-only 2.322 pb, so this genuinely tests the interference term and not just a repeat of the QED piece.

17.4 Tier 4 — derivative couplings and the gauge-cancellation acid test

Channels: \(e^+e^-\to W^+W^-\); any process needing a VVV/VSS vertex. Effort: large.

The blocker: VSS/VVV couplings carry momentum tags p(φ) from to_momentum_space(), which amplitude_squared() explicitly refuses to square (amplitudes.py’s SUPPORTED_VERTEX_TYPES excludes both). Tier 4 must resolve each tag to the kinematic TensorHead of the diagram leg φ is assigned to, incoming/outgoing sign included. The relative sign between the \(W\bar\nu e\) current and the triple-gauge coupling is exactly what separates a correct \(\approx19.5\) pb from the \(\approx98\) pb a flipped sign gives (docs/benchmark.md’s account of the MadGraph round-trip) — Tier 4 must derive that sign from the diagram construction, not import the export script’s empirical fix.

Oracle: \(\sigma(e^+e^-\to W^+W^-) = 19.498\pm0.058\) pb at \(\sqrt s=200\) GeV, plus a direct pin that \(|\mathcal M|^2\) does not grow like \(s^2\) at high energy — the gauge-cancellation signature itself, checked directly rather than only through one integrated number.

17.5 Tier 5 — coloured / hadronic 2→2 (parton level)

Channels: \(q\bar q\to q'\bar q'\), \(qg\to qg\), \(gg\to gg\). Effort: medium–large.

Needs a colour-flow layer producing the colour factor \(C_{dd'}=\sum_{\text{colour}} T_d\bar T_{d'}\) separated from the Lorentz part, colour averaging over the initial state (already slotted via color, unused until now), and lifting collect_decay_vertices/fermion_decay_vertices’s twin 3-leg hard filters so the VVVV (\(gggg\)) contact vertex — already assembled by export/ufo/vvvv.py’s assemble_vvvv — becomes reachable.

Oracle: the standard parton-level \(|\mathcal M|^2\) table (Ellis, Stirling & Webber [ESW96], e.g. \(q\bar q\to q'\bar q' = \tfrac49\cdot\tfrac{t^2+u^2}{s^2}\), \(gg\to gg = \tfrac92(3-\tfrac{tu}{s^2}-\tfrac{su}{t^2}-\tfrac{st}{u^2})\)), with colour factors \(C_F=4/3\), \(C_A=3\), \(T_R=1/2\). Explicitly excludes PDFs and hadronic luminosity — this tier stops at the parton-level \(2\to2\) cross section, matching the “no PDFs” scope already implicit in the rest of the library (no proton structure anywhere).

17.6 Summary

tier	channels	new machinery	effort	oracle
1 ✅	\(e^+e^-\to\mu^+\mu^-\) (γ only), \(f\bar f\to f\bar f\) via a scalar	TwoToTwoKinematics; SpinorChain/Diagram/Amplitude (done); amplitude-level propagators; ExternalState averaging	small–medium	Peskin QED closed form + explicit-matrix oracle
2 ✅	\(e^+e^-\to\mu^+\mu^-\) (Z only); any single chiral-current diagram	ε (γ₅) algebra (pheno.epsilon): \(\mathrm{Tr}[\gamma\gamma\gamma\gamma\gamma_5]\), ε·ε → Gram determinant, forward_backward_asymmetry	large	\(d\sigma/d\cos\theta\) shape + \(A_{FB}=\tfrac34A_eA_f\) vs [LEPEWWG06]; γ₅-carrying matrix oracle (massless + massive mediator)
3	full \(e^+e^-\to f\bar f\) (γ/Z interference)	multi-diagram double sum; s/t/u topology search; relative signs	medium–large	MadGraph 2.7878 pb
4	\(e^+e^-\to W^+W^-\); derivative couplings	VVV/VSS momentum-tag resolution; external vector polarization sums	large	MadGraph 19.498 pb; no \(s^2\) growth
5	\(q\bar q\to q\bar q\), \(qg\to qg\), \(gg\to gg\)	colour-flow algebra; colour averaging; 4-point contact vertices	medium–large	ESW parton-level table (no PDFs)

The recommended order is the table order: Tier 1 gave the first native cross section for the cost of an object layer that reuses the existing covariant engine unchanged; Tier 2 was the genuinely new algebra 2→2 requires that 1→2 never did; Tier 3 is where the two MadGraph benchmarks stop being aspirational and start being reproduced; Tier 4 is the hardest single piece (derivative couplings); Tier 5 generalizes to QCD but adds no new kind of physics beyond colour bookkeeping.

See 16. Extending the Decay Calculator for the parallel decay-side roadmap and its n_momenta/ε-drop precedent, and tests/test_scattering.py for what Tier 1 pins.

17.7 References

• [PS95] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley (1995), ISBN 0-201-50397-2 — Chapter 5, “Elementary Processes of Quantum Electrodynamics,” §5.1, the \(e^+e^-\to\mu^+\mu^-\) closed form Tier 1’s cross section reproduces.

• [ESW96] R. K. Ellis, W. J. Stirling and B. R. Webber, QCD and Collider Physics, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology 8, Cambridge University Press (1996), ISBN 978-0-521-54589-1 — the parton-level \(2\to2\) \(|\mathcal M|^2\) table Tier 5 targets.

• [LEPEWWG06] ALEPH, DELPHI, L3, OPAL, SLD Collaborations, LEP Electroweak Working Group, SLD Electroweak and Heavy Flavour Groups, “Precision Electroweak Measurements on the Z Resonance,” Phys. Rept. 427 (2006) 257–454, arXiv:hep-ex/0509008, doi:10.1016/j.physrep.2005.12.006 — Eq. (1.55) the Born-level \(e^+e^-\to f\bar f\) angular distribution, Eq. (1.56) the coupling asymmetry \(A_f\), Eq. (1.66) the forward–backward asymmetry \(A_{FB}^{0,f}=\tfrac34A_eA_f\) Tier 2’s forward_backward_asymmetry reproduces.

• [PDG] Particle Data Group, Review of Particle Physics — “Kinematics” review (Mandelstam invariants, the two-body \(t\)-range, \(d\sigma/dt\)); see the current edition at pdg.lbl.gov, cited without a year pin since PDG republishes annually and section numbers shift (same convention as 16. Extending the Decay Calculator §16.2’s use of the same review).