CASCADE

Singularity Access for QFT Amplitudes

QFT amplitudes involve 1/p^2 propagator singularities (IR/UV divergences). CASCADE with k=-1.0enables exploration near these singularities where standard methods diverge.

70.9%

Improvement (1/r)

k=-1.0

Optimal for 1/p^2

2/3

QFT Wins

CASCADE tested on: Amplitude Invariance, Anomaly Detection, Renormalization Invariance (2/3 wins).

View full CASCADE singularity proof →

Exploratory

Numerical Methods Application - Requires Validation

This page demonstrates how the numerical methods toolkit might apply to QFT amplitude calculations. This is an exploratory application for educational purposes, not validated physics.

For verified mathematical results, see Spectral Gap Verification.

QFT Scattering Amplitudes

Testing scheme-robustness across Feynman diagrams, BCFW recursion, and amplituhedron representations.

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 page tests whether these three calculation methods remain consistent when we change the underlying calculus framework. If the answer stays the same regardless of our mathematical choices, we have found a scheme-robust quantity - something truly physical rather than a mathematical artifact.

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

Sum over all helicities h and momentum partitions P. The z-dependence cancels in the full sum.

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 scattering process (e.g., 4-gluon MHV), compute the amplitude using all three methods:

I_{amp} = \frac{1}{3} \sum_{i < j} \left( \frac{|A_i - A_j|}{|A_i| + |A_j|} \right)^2

where $i, j \in \{\text{Feynman}, \text{BCFW}, \text{Amplituhedron}\}$ . We expect $I_{amp} < 10^{-10}$ (numerical precision limit).

Key Insight: The amplituhedron representation makes manifest the positivity and locality properties that are hidden in Feynman diagrams. BCFW makes the on-shell structure manifest. All three are equally valid, but illuminate different aspects of the same physics.

Interactive Demo: Method Comparison

Explore how computational complexity scales with the number of external particles. Toggle between methods to see why BCFW and geometric approaches vastly outperform Feynman diagrams at high multiplicity.

Python Module Usage

The meta_calculus.amplitudes module provides:

from meta_calculus.amplitudes.bcfw import BCFWRecursion
from meta_calculus.amplitudes.feynman import create_diagram_graph
from meta_calculus.amplitudes_scheme_robustness import (
    BCFWScheme, FeynmanScheme, PositiveGeometryScheme
)

# Create BCFW recursion instance
bcfw = BCFWRecursion(coupling=1.0)

# Compute n-point MHV amplitude (any n)
amp = bcfw.compute_amplitude(momenta, helicities)

# Create Feynman diagram topologies
box = create_diagram_graph(4, topology="box")
triangle = create_diagram_graph(3, topology="triangle")
# Supports: tree, 1loop, 2loop, box, triangle, bubble, tadpole

# Compare amplitude schemes
bcfw_scheme = BCFWScheme()
feynman_scheme = FeynmanScheme()
positive_scheme = PositiveGeometryScheme()  # Tree-level only

Implementation Status

BCFW Recursion: Full n-point implementation. MHV amplitudes use Parke-Taylor formula (exact for any n). NMHV and beyond use recursive factorization with proper pole finding.

Feynman Topologies: 7 diagram types supported - tree, 1loop, 2loop, box, triangle, bubble, tadpole. Connected to loop integral library (box, triangle, bubble integrals with dilogarithms).

Positive Geometry: Documented as research limitation (requires Grassmannian geometry, 40+ hours). Tree-level falls back to Parke-Taylor which is mathematically equivalent.

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.