QFT Scattering Amplitudes

Comparing two independent tree-level methods, Berends-Giele currents and BCFW recursion. No amplituhedron calculation is implemented.

Status: exploratory numerical-methods application that requires validation. It shows how the toolkit might apply to QFT amplitude calculations, for education, and is not validated physics. For the toolkit's tested mathematical results, see the Spectral-Gap Boundary audit.

What Are Scattering Amplitudes?

When particles collide at high energies (like in the Large Hadron Collider), scattering amplitudes tell us the probability of different outcomes. Think of them as the mathematical formulas that predict what happens when an electron bounces off a positron, or when quarks interact inside a proton.

Remarkably, physicists have discovered three completely different ways to calculate these probabilities: traditional Feynman diagrams (summing over all possible particle paths), BCFW recursion (building complex processes from simpler ones), and the Amplituhedron (a purely geometric approach). All three methods must give the same answer for any physical collision.

This toolkit implements only the first two as independent tree-level calculations and compares them numerically. The amplituhedron is described for context; it is not computed here, and no change of calculus framework is tested on this page.

Three Ways to Compute Amplitudes

Quantum field theory provides multiple methods for computing scattering amplitudes. Each has different computational advantages, but they must yield identical physical predictions:

Feynman Diagrams

Sum over all diagrams with correct vertex factors. Manifest locality, but exponential growth in diagram count.

BCFW Recursion

On-shell recursion relations. Manifestly on-shell, exponentially fewer terms, but requires complex momentum shifts.

Amplituhedron

Geometric approach for N=4 SYM. Pure geometry, no diagrams, but limited to special theories.

Key Formulas

Parke-Taylor Formula (MHV Amplitudes)

The maximally helicity violating (MHV) amplitude for gluon scattering in N=4 SYM [Arkani-Hamed & Trnka 2014]:

A_n^{MHV}(1^-, 2^-, 3^+, \ldots, n^+) = \frac{\langle 1 2 \rangle^4}{\langle 1 2 \rangle \langle 2 3 \rangle \cdots \langle n 1 \rangle}

where $\langle i j \rangle = \lambda_i^A \lambda_{j,A}$ are spinor brackets.

BCFW Recursion Relation

Shift two momenta [Britto et al. 2005]: $\hat{\lambda}_i(z) = \lambda_i + z \lambda_j$ , $\hat{\tilde{\lambda}}_j(z) = \tilde{\lambda}_j - z \tilde{\lambda}_i$

A_n(0) = \sum_{P} \sum_{h = \pm} A_L\big(\hat{i}(z_P), \ldots, -\hat{P}^{-h}(z_P)\big) \frac{1}{P^2} A_R\big(\hat{P}^{h}(z_P), \ldots, \hat{j}(z_P)\big)

A_n(z) is a rational function of z, generally not constant. The recursion reconstructs the physical amplitude A_n(0) from its poles: P runs over the channels that separate i from j, z_P solves P-hat(z_P)^2 = 0, both subamplitudes are evaluated at z_P, and the internal leg carries opposite helicities on the two sides (-h on the left, h on the right). This requires the boundary condition $A_n(z) \to 0 \text{ as } z \to \infty$ ; for this shift it holds for (h_i, h_j) = (+, -), (+, +) and (-, -), and fails for (-, +).

Factorization on Poles

All three methods must exhibit the same pole structure when intermediate propagators go on-shell:

A_n \xrightarrow{P^2 \to 0} \sum_{h} A_L^{(h)} \frac{1}{P^2} A_R^{(h)} + \text{regular}

Scheme-Robustness Test

For a given tree-level process (e.g., 4-gluon MHV), the module compares the two implemented methods: Berends-Giele vector currents (the Feynman-diagram scheme) and BCFW factorization forced even in the MHV sector.

|A_{\text{BG}} - A_{\text{BCFW}}| \le 10^{-9} \max(|A_{\text{BG}}|, |A_{\text{BCFW}}|)

Positive geometry (amplituhedron) is unavailable: PositiveGeometryScheme raises NotImplementedError, so the module reports it unsupported and returns no overall scheme-robust verdict. Pole-factorization and soft-limit evidence are likewise unavailable.

Interactive Demo: Four-Gluon Scattering

The squared color-ordered four-gluon MHV tree amplitude versus center-of-mass scattering angle (closed formula from Dixon's review). It diverges as the angle approaches 180 degrees. No method comparison or complexity benchmark is shown.

Python Module Usage

The meta_calculus.amplitudes module provides:

import numpy as np
from meta_calculus.amplitudes.bcfw import BCFWRecursion
from meta_calculus.amplitudes.feynman import create_diagram_graph
from meta_calculus.amplitudes_scheme_robustness import (
    AmplitudeSchemeRobustnessTest, PositiveGeometryScheme
)

# All-outgoing four-gluon momenta at 60 degrees, where |A|^2 = 16/9
c, s = 0.5, np.sqrt(3)/2
momenta = np.array([[-1, 0, 0, -1], [-1, 0, 0, 1],
                    [1, s, 0, c], [1, -s, 0, -c]], dtype=float)
helicities = [-1, -1, 1, 1]

# Color-ordered tree amplitude; non-MHV sectors use BCFW factorization
bcfw = BCFWRecursion(coupling=1.0)
amp = bcfw.compute_amplitude(momenta, helicities)

# Diagram topology records only; no amplitude is computed from them
box = create_diagram_graph(4, topology="box")
triangle = create_diagram_graph(3, topology="triangle")
# Supports: tree, 1loop, 2loop, box, triangle, bubble, tadpole

# Berends-Giele vs forced BCFW; positive geometry is reported unsupported,
# so scheme_robust is None and implemented_schemes_agree holds the comparison
report = AmplitudeSchemeRobustnessTest().test_amplitude_agreement(momenta, helicities)

# Positive geometry is not implemented
try:
    PositiveGeometryScheme().compute_amplitude(momenta, helicities)
except NotImplementedError:
    pass

Implementation Status

BCFW Recursion: Tree-level n-point implementation. MHV and anti-MHV sectors use closed Parke-Taylor forms by default; NMHV and beyond use recursive factorization with an automatically chosen valid shift.

Feynman Topologies: 7 diagram types supported - tree, 1loop, 2loop, box, triangle, bubble, tadpole. Topologies are records; the loop integral library provides bubble, restricted triangle and a massless box Laurent expansion through epsilon^0.

Positive Geometry: Not implemented. PositiveGeometryScheme.compute_amplitude raises NotImplementedError; there is no fallback, so it supplies no evidence in comparisons.

References

[Arkani-Hamed & Trnka 2014] Arkani-Hamed, N. & Trnka, J. (2014). The Amplituhedron. Journal of High Energy Physics, 2014, 30. arXiv:1312.2007.

[Britto et al. 2005] Britto, R., Cachazo, F., Feng, B., & Witten, E. (2005). Direct Proof of the Tree-Level Scattering Amplitude Recursion Relation in Yang-Mills Theory. Physical Review Letters, 94, 181602. arXiv:hep-th/0501052.

[Cachazo et al. 2015] Cachazo, F., He, S., & Yuan, E. Y. (2015). Scattering Equations and Matrices: From Einstein to Yang-Mills, DBI and NLSM. Journal of High Energy Physics, 2015, 149. arXiv:1412.3479.

[Passarino & Veltman 1979] Passarino, G. & Veltman, M. (1979). One-loop corrections for e+e- annihilation into mu+mu- in the Weinberg model. Nuclear Physics B, 160, 151-207.

['t Hooft & Veltman 1972] 't Hooft, G. & Veltman, M. (1972). Regularization and renormalization of gauge fields. Nuclear Physics B, 44, 189-213.