Feynman Integrals from Positivity Constraints

TL;DR

This work introduces a novel positivity-based framework for numerically bounding and determining Feynman integrals by recasting integrals as linear combinations of a small master set and enforcing an infinite family of positivity constraints via semidefinite programming. The authors develop and compare two complementary implementations: positivity constraints in loop momentum space and in Feynman parameter space, including systematic ε-expansion constraints that yield a Hankel-structure relation among expansion coefficients. They demonstrate the method on the one-loop bubble family and extend it to a nontrivial three-loop banana diagram with unequal masses, obtaining high-precision results for 11 master integrals and revealing consistency relations across ε-orders. The results indicate rapid convergence of the SDP-based bounds and highlight the potential of inequality-based bootstrap-like techniques as a complementary tool to differential-equation methods for Feynman integrals, with implications for precision perturbative calculations and beyond.

Abstract

We explore inequality constraints as a new tool for numerically evaluating Feynman integrals. A convergent Feynman integral is non-negative if the integrand is non-negative in either loop momentum space or Feynman parameter space. Applying various identities, all such integrals can be reduced to linear sums of a small set of master integrals, leading to infinitely many linear constraints on the values of the master integrals. The constraints can be solved as a semidefinite programming problem in mathematical optimization, producing rigorous two-sided bounds for the integrals which are observed to converge rapidly as more constraints are included, enabling high-precision determination of the integrals. Positivity constraints can also be formulated for the $ε$ expansion terms in dimensional regularization and reveal hidden consistency relations between terms at different orders in $ε$. We introduce the main methods using one-loop bubble integrals, then present a nontrivial example of three-loop banana integrals with unequal masses, where 11 top-level master integrals are evaluated to high precision.

Figures (14)

• Figure 1: The one-loop bubble integral with external legs of virtuality $p^2$ and two internal massive line with the same squared mass $m^2$.
• Figure 2: Comparison between ad hoc positivity bounds Eq. \ref{['eq:bubbleCrudeResult4D']} and the analytic result for the bubble integral $\hat{I}^{d=4}_{2,1}$ normalized according to Eq. \ref{['eq:bubbleNormalizedA1A2']}.
• Figure 3: Three eigenvalues of $\widetilde{\mathbb M}$ defined in Eq. \ref{['eq:tildeMpos']} as a function of $\hat{I}^d_{2,1}$, at the kinematic point Eq. \ref{['eq:bubbleEucNumPoint']}. The lowest eigenvalue corresponds to the bottom orange curve and the remaining two eigenvalues correspond to the upper black curves.
• Figure 4: Magnified version of the vicinity of a small region of Fig. \ref{['fig:momSpaceThreeEigs']} in which the lowest eigenvalue of $\widetilde{\mathbb M}$, shown in the curve, is non-negative.
• Figure 5: Eigenvalues as a function $\hat{I}^d_{2,1}$, for the $10\times 10$ symmetric matrix that represent the quadratic dependence of the RHS of Eq. \ref{['eq:posAnsatzSize10']} on the $\alpha_i$ parameters after factoring out $I^d_{3,0}$.