[RADCOR 2025] Beyond Contour Deformation: Accelerating Feynman Integrals via Domain Decomposition

TL;DR

Evaluating multi-loop Feynman integrals in the physical (Minkowski) regime has long been hindered by the complexities of contour deformation. A new method presented at RADCOR 2025 bypasses this entirely by splitting integrals into strictly positive real components. By remapping singularities to the integration boundaries using Generic Cylindrical Algebraic Decomposition (GCAD), the authors achieve performance gains of several orders of magnitude while solving the long-standing problem of numerical instability in extreme kinematic limits.

The Problem: The High Cost of "Dodging" Singularities

In the world of high-energy physics, precision is king. However, calculating the multi-loop amplitudes required for LHC phenomenology involves integrating over regions where the denominator polynomial, $\mathcal{F}(\mathbf{x};\mathbf{s})$, vanishes.

The standard "fix" for decades has been contour deformation: shifting variables into the complex plane ($x_{k}o{x}_k-ia{u}_k$). While automatable, this approach has three fatal flaws:

1. Numerical Cancellations: Complex integrands produce oscillating positive and negative contributions, leading to a massive loss of precision (often 5+ digits).
2. Computational Weight: Evaluating complex-deformed integrands is significantly slower (up to 2.67x per evaluation).
3. Pathological Failures: In high-energy or small-mass limits, Landau singularities pinch the contour, making it impossible to find a valid deformation.

Figure 1: Visualizing the precision loss and speed degradation when switching from Euclidean (orange) to Minkowski (blue) regimes using traditional contour deformation.

Methodology: The Power of Positive Thinking (and Integrands)

The core insight of Jones, Olsson, and Stone is that we don't have to go into the complex plane to resolve a singularity if we can isolate it at the boundary.

1. Sign-Based Domain Splitting

The authors decompose the integration domain into regions $S^{+}$ where $\mathcal{F}>0$ and $S^{-}$ where $\mathcal{F}<0$. This ensures that within each region, the integrand has a uniform sign.

2. Remapping to Boundary

Using the GCAD (Generic Cylindrical Algebraic Decomposition) algorithm, the boundaries where $\mathcal{F}=0$ are identified. By applying clever variable transformations—such as $x_i^{\prime }=\frac{x_i{x}_j}{f({\mathbf{x}}_{eqi})-x_i}$—the internal singularity is "pushed" to infinity or zero.

3. Factorizing the Complex Phase

Because the sign is now uniform, the complex part of the causal $i\delta$ prescription factors out as a simple global prefactor: $(-1-i\delta {)}^{-(u-LD/2)}$. The remaining integrals are purely over real, non-negative integrands, which are far more stable for Quasi-Monte Carlo (QMC) integration.

Figure 2: The Two-loop non-planar box (BNP6) and all-massive triangle used for benchmarking.

Experimental Results: Orders of Magnitude Faster

The authors benchmarked their method using the pySecDec ecosystem. Two primary cases stand out:

Case A: Two-Loop Non-Planar Box (BNP6)

For moderate kinematics, the non-deformed method is roughly 10x faster. In high-energy limits, the advantage is even more stark: traditional contour deformation fails to converge even at low precision, while the new method remains efficient.

Case B: All-Massive Triangle

As the mass $m^{2}o0$, contour deformation timings skyrocket due to endpoint singularities. The new method, having mapped these singularities to the boundary, remains almost immune to the small-mass limit.

Figure 3: Integration time vs. requested precision. Dashed lines (new method) consistently outperform solid lines (contour deformation).

Critical Insight & Future Outlook

The brilliance of this work lies in its use of Algebraic Geometry (GCAD) to solve a Numerical Integration problem. By treating the singular locus as a geometric boundary to be remapped, it circumvents the "art" of choosing deformation parameters.

Limitations:

• Scalability: GCAD scales poorly with the number of variables. For integrals with many propagators, finding the decomposition becomes a symbolic bottleneck.
• Algebraic Integrands: The current implementation often results in square roots in the integrand. Standard sector decomposition tools (like pySecDec) struggle with these, though the authors suggest this is a solvable software engineering challenge.

The Takeaway: This is a major step toward the "automation of everything" in higher-order corrections. By removing the instability of contour deformation, we pave the way for more robust and faster precision calculations for the next generation of collider experiments.