Approximating Feynman integrals using complete monotonicity and Stieltjes properties

Technical Summary: Approximating Feynman Integrals using Complete Monotonicity and Stieltjes Properties

Problem Statement
The numerical evaluation of multi-loop Feynman integrals remains a central challenge in perturbative quantum field theory. While various methods exist—including sector decomposition, differential equation techniques, auxiliary mass flow, and Monte Carlo integration—efficient and robust evaluation across different kinematic regions is still difficult. Existing approaches often struggle with the complexity of multi-loop structures and the need for precise boundary data or extensive computational resources.

Methodology
The authors propose two novel approaches that leverage universal analytic properties related to positivity: Complete Monotonicity (CM) and Stieltjes properties.

1. The CM Bootstrap:

   • Foundation: Scalar Feynman integrals in the Euclidean region are proven to be completely monotonic functions. A function $f(x)$ is completely monotonic if $(-1)^n \frac{d^n}{dx^n}f(x) \geq 0$ for all $n \in \mathbb{N}_0$.
   • Mechanism: This method combines the differential equations satisfied by master integrals ($\frac{d}{dx}\mathbf{f}(x) = A(x)\mathbf{f}(x)$) with the CM positivity constraints. By recursively defining matrices $Q_n(x)$ derived from the differential equation matrix $A(x)$, the CM condition is transformed into a set of linear inequalities: $Q_n(x)\mathbf{f}(x) \geq 0$.
   • Advantage: This allows for the reconstruction of integrals and the derivation of rigorous bounds without requiring explicit boundary conditions. In favorable kinematic regions, these bounds are two-sided and converge rapidly.

2. Stieltjes Function Extension and Padé Approximants:

   • Foundation: The authors prove that under specific conditions, Feynman integrals belong to the class of Stieltjes functions. A Stieltjes function admits an integral representation $f(z) = \int_0^{1/R} \frac{\rho(u)}{1+uz} du$ with $\rho(u) \geq 0$.
   • Conditions: For a Feynman integral in Feynman parametrization to be Stieltjes, the exponent of the second Symanzik polynomial must satisfy $0 < \sum_i \nu_i - LD/2 \leq 1$ (where $L$ is loop order, $D$ is dimension, and $\nu_i$ are propagator powers). Additionally, the Euclidean region must exist such that the second Symanzik polynomial can be written as $F = Ax + B$ with $A, B \geq 0$. This holds for general planar integrals and is evidenced for non-planar cases.
   • Mechanism: Stieltjes functions possess the property that their Padé approximants converge rapidly in the cut complex plane with rigorous error bounds. Specifically, diagonal and near-diagonal Padé approximants ($P_{N-1}^N$ and $P_N^N$) provide upper and lower bounds that converge to the true function value.
   • Algorithm:
     1. Identify a kinematic point in the Euclidean region where the CM bootstrap yields fast-converging two-sided bounds.
     2. Compute the integral value and its derivatives at this point.
     3. Construct a Taylor expansion and build Padé approximants.
     4. Evaluate these rational functions across the complex plane for fast numerical approximation and analytic continuation.

Key Contributions and Results

• Rigorous Bounds without Boundary Data: The CM bootstrap successfully reconstructs integrals and provides rigorous bounds using only differential equations and positivity, eliminating the need for traditional boundary data.
• Proof of Stieltjes Nature: The paper establishes that a broad class of scalar Feynman integrals (including massive bubbles, banana integrals, zig-zag, and ladder integrals in specific dimensions) are Stieltjes functions, provided propagator powers are chosen appropriately.
• Efficient Numerical Evaluation:
  • Massive Bubble Integral: The method was illustrated on the massive one-loop bubble. The CM bootstrap provided tight two-sided bounds in the region $x \in (-2, 0)$ and a very close lower bound for $x > 0$.
  • Multi-loop Banana Integrals: The approach was applied to banana integrals up to four loops. The evaluation time scales with the number of master integrals rather than the analytic complexity of the function space (e.g., Calabi-Yau manifolds), performing competitively against AMFlow.
  • 20-Loop Banana Integral: Demonstrating independence from differential equations, the authors used the Bessel integral representation to obtain a Taylor expansion for a 20-loop banana integral. Padé approximants were computed within seconds, achieving high precision (agreement of at least 18 digits in specific regions) using only a few derivatives.
• Analytic Continuation: The Padé approximant method enables fast analytic continuation into the cut complex plane, overcoming the limitations of standard Taylor expansions which converge slowly near branch cuts.

Significance and Claims
The paper claims that these methods offer a new set of tools for precision calculations in quantum field theory. The significance lies in:

1. Robustness: Providing rigorous error bounds and convergence guarantees based on fundamental analytic properties rather than heuristic numerical methods.
2. Efficiency: Offering fast numerical evaluation for high-loop integrals (demonstrated up to 20 loops) and multi-scale problems where traditional Integration-by-Parts (IBP) approaches reach their limits.
3. Generality: The structural property of complete monotonicity is shared by a wide class of quantities in quantum field theory. The authors suggest this approach may extend beyond Feynman integrals to areas such as gravitational physics and cosmology.

Limitations and Future Work
The authors note that while the method works well in the Euclidean region, the precision of the expansion decreases rapidly near the branch cut, making continuation to the physical region numerically challenging. Future work aims to improve convergence in this regime. Additionally, while the current method handles multi-scale integrals by fixing all but one variable, a genuine multivariate extension of the Stieltjes property is identified as a necessary future development. The extension to dimensional regularization (reconstructing Laurent expansions via numerical sampling) is also proposed as a natural next step.