CASCADE

Singularity Access for Loop Integrals

Loop integrals involve UV/IR divergences (1/p^2 behavior). CASCADE with k=-1.0enables exploration near these singularities using bigeometric regularization.

70.9%

Improvement (1/p^2)

k=-1.0

Bigeometric Reg.

D*[1/r]=-1

No Divergence

The bigeometric derivative regularizes power-law singularities: D*[1/r] = -1 (constant).

View full CASCADE singularity proof →

Exploratory

Numerical Methods Application - Requires Validation

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

For verified mathematical results, see Spectral Gap Verification.

Loop Corrections

One-loop integrals, dimensional regularization, and multi-objective optimization of loop calculations.

What Are Loop Corrections?

In quantum mechanics, particles are constantly creating and destroying virtual particle-antiparticle pairs out of the vacuum. These quantum fluctuations appear as loop diagrams in Feynman's framework, and they make tiny but measurable corrections to physical quantities like the magnetic moment of the electron (accurate to 12 decimal places!).

The problem is that loop integrals are mathematically infinite - they diverge when we integrate over all possible energies. Physicists tame these infinities using regularization (making the integral finite temporarily) and renormalization (absorbing infinities into redefined constants). Different regularization schemes exist: Feynman parameters, Schwinger parameters, and Mellin-Barnes contour integrals.

This page tests scheme-robustness for loop calculations. Physical observables like the electron's magnetic moment must be independent of which regularization method we choose. Only the final answer matters, not the intermediate mathematical tricks we used to compute it. That is the hallmark of a true physical quantity.

Quantum Loop Integrals

Beyond tree-level, quantum field theory requires computing loop integrals. These are generically UV-divergent and require regularization. Multiple parametrization schemes exist:

Feynman Parameters

Combine denominators via 1/(AB) = integral dx/[xA+(1-x)B]^2. Standard textbook approach.

Schwinger Parameters

Represent propagators as proper-time integrals: 1/A = integral dt exp(-tA). Worldline formalism.

Mellin-Barnes

Complex contour integrals. Useful for expansions in mass ratios and analytic continuation.

Feynman Parametrization

Basic Identity

Combine two denominators:

\frac{1}{AB} = \int_0^1 dx \, \frac{1}{[xA + (1-x)B]^2}

Generalization to N Denominators

\frac{1}{A_1 A_2 \cdots A_N} = (N-1)! \int_0^1 \prod_{i=1}^N dx_i \, \delta\left(1 - \sum_j x_j\right) \frac{1}{[\sum_j x_j A_j]^N}

Standard One-Loop Integrals

Bubble Integral (2-Point)

I_2(p^2, m_1^2, m_2^2) = \int \frac{d^D k}{(2\pi)^D} \frac{1}{[k^2 - m_1^2][(k-p)^2 - m_2^2]}

Result in dimensional regularization (D = 4 - 2 epsilon):

I_2 = \frac{i}{(4\pi)^{D/2}} \Gamma(2-D/2) \int_0^1 dx \, [x m_1^2 + (1-x)m_2^2 - x(1-x)p^2]^{D/2-2}

Triangle Integral (3-Point)

I_3(p_1, p_2, p_3) = \int \frac{d^D k}{(2\pi)^D} \frac{1}{[k^2 - m_1^2][(k-p_1)^2 - m_2^2][(k-p_1-p_2)^2 - m_3^2]}

Triangle integrals appear in vertex corrections and self-energy diagrams with external momentum.

Box Integral (4-Point)

I_4(s, t) = \int \frac{d^D k}{(2\pi)^D} \frac{1}{k^2 (k-p_1)^2 (k-p_1-p_2)^2 (k+p_4)^2}

Box integrals are the master integrals for 2-to-2 scattering at one loop. They depend on Mandelstam variables $s = (p_1+p_2)^2$ and $t = (p_1+p_4)^2$ .

Dimensional Regularization

Continue the spacetime dimension from D=4 to $D = 4 - 2\epsilon$ . UV divergences appear as poles in epsilon:

I_{div} = \frac{1}{\epsilon} + \text{finite} + O(\epsilon)

Renormalization absorbs these poles into counterterms, leaving finite physical predictions:

I_{ren} = I_{div} + C.T. = \text{finite}

Multi-Objective Optimization

Loop integral computation involves competing objectives:

1.Numerical Stability: Avoid cancellations near kinematic thresholds
2.Computational Speed: Minimize number of Feynman parameters / contour integrals
3.Scheme Robustness: Agreement between Feynman/Schwinger/Mellin-Barnes

The MOO framework finds Pareto-optimal integral representations: which parametrization minimizes all three objectives simultaneously? For bubble integrals, Feynman parameters are optimal. For higher-point functions, the answer depends on the kinematic regime.

Interactive Demo: Loop Integral Analysis

Explore a mock Feynman-parameter sampler for different loop topologies. Watch synthetic Monte Carlo convergence, inspect a template pole structure in dimensional regularization, and see how the regulator epsilon controls the illustrative UV/IR scaling. The demo is pedagogical rather than a full numerical evaluation of the underlying one-loop integrals.

Python Module Usage

The meta_calculus.amplitudes.loops module provides:

from meta_calculus.amplitudes.loops.scalar_integrals import (
    bubble_integral,
    triangle_integral,
    box_integral
)
from meta_calculus.amplitudes.loops.dim_reg import (
    DimensionalRegularization,
    extract_finite_part
)

# Compute scalar loop integrals
epsilon = 0.01  # Dimensional regularization parameter

# Bubble integral I_2(p^2)
I_bubble = bubble_integral(p2=10.0, epsilon=epsilon)

# Triangle integral I_3(p1^2, p2^2, p3^2)
I_tri = triangle_integral(p1_sq=5.0, p2_sq=5.0, p3_sq=5.0, epsilon=epsilon)

# Box integral I_4(s, t) - uses scipy.spence for Li_2 dilogarithms
I_box = box_integral(s=100.0, t=-30.0, epsilon=epsilon)
print(f"Box integral: {I_box:.6e}")

# Extract finite part via Richardson extrapolation
dim_reg = DimensionalRegularization()
pole, finite = dim_reg.extract_finite_part(lambda eps: I_box / eps, eps_values=[0.1, 0.05, 0.01])
print(f"Pole: 1/epsilon^{pole:.0f}, Finite: {finite:.4f}")

Implementation Status

Scalar Integrals: Bubble integrals are production-ready; triangle and box evaluations are approximate demos flagged as beta scaffolding. Box integral usesscipy.spence for dilogarithms (Li_2 functions).

Dimensional Regularization: Richardson extrapolation for extracting finite parts from epsilon-expanded integrals. Gamma function expansion validated.

Integration: Connected to Feynman diagram topologies (7 types) and amplitude scheme-robustness framework. See meta_calculus/amplitudes/loops/.