Numerical Determination of Feynman Integrals Using Complete Monotonicity

A real function is said to be completely monotone (CM) in a region if the function and all its signed derivatives are positive at every point in that region. Complete monotonicity imposes infinitely many constraints on the function and its derivatives, and the space of CM functions in a region is convex.

In this talk, we will consider cases where this space is finite-dimensional - for instance, when the function’s Taylor coefficients satisfy a recursion relation or the function obeys a differential equation. Adopting a physics-inspired approach, we will discuss how recursion and positivity can be combined to formulate a convex optimisation problem that numerically constrains the values of the function throughout the CM region.

Our focus will be on scalar Feynman integrals, which are completely monotone in the Euclidean region under generic kinematics. These integrals are transcendental functions of the kinematic parameters and masses. However, when these parameters are rational, they evaluate to periods in the sense of Kontsevich and Zagier - that is, complex numbers given by integrals of algebraic differential forms over domains defined by polynomial inequalities with rational coefficients.

In our framework, the differential equations satisfied by the Feynman integrals provide the recursion structure, while complete monotonicity imposes positivity constraints. We will describe how to formulate the resulting numerical problem both as a linear programming problem (LPP) and as a semi-definite programming problem (SDPB), and present the bounds obtained.

Finally, time permitting, we will also discuss how this approach leads to rational approximations of certain transcendental functions.