On differential properties of feynman integrals

Integration-by-parts identities and differential equations for parametrised Feynman integrals

“…While general algorithms were not developed at the time, two of Regge's collaborators, Barucchi and Ponzano, were able to construct a concrete application of the general formalism for one-loop diagrams [30,31]. In those papers, they show that for one-loop diagrams it is always possible to organise the relevant Feynman integrals into sets (that we would now call 'families'), and find a system of linear homogeneous differential equation in the Mandelstam invariants that closes on these sets, with the maximum required size of the system being 2 n − 1 for graphs with n propagators.…”

“…In the present paper, we start from the ideas of refs. [16][17][18][19] and the concrete results of Barucchi and Ponzano [30,31] to propose a projective framework to derive IBP identities and systems of linear differential equations for Feynman integrals. In order to do so, we JHEP03(2024)096 need to generalise the Barucchi-Ponzano results in several directions.…”

“…We have emphasised the significance of the early mathematical results reported in [16][17][18][19], which resonate strikingly with contemporary research. These ideas from algebraic topology were turned into a concrete application to one-loop diagrams by Barucchi and Ponzano [30,31]. In order to apply these results in the modern context, we have shown how the analysis extends naturally to dimensional regularisation, we have generalised the results to the two-loop level (indeed we expect the technique to be applicable to all orders), and we have emphasised the role played by boundary terms in the IBP identities, noting that they do not vanish in general, and in fact they provide a useful tool to link complicated integrals to simple ones.…”

“…In a direct implementation of the algorithm proposed in [30,31], the chosen basis integrals are I(1, 1, 1, 1; 2ϵ), I(2, 1, 2, 1; 2ϵ), I(1, 2, 1, 2; 2ϵ) and I(2, 2, 2, 2; 2ϵ): all the other relevant integrals are determined by the linear system presented below. We find…”

IBPs and differential equations in parameter space

“…

We present a projective framework for the construction of Integration by Parts (IBP) identities and differential equations for Feynman integrals, working in Feynman-parameter space. This framework originates with very early results which emerged long before modern techniques for loop calculations were developed [17][18][19][20][21][22]. Adapting and generalising these results to the modern language, we use simple tools of projective geometry to generate sets of IBP identities and differential equations in parameter space, with a technique applicable to any loop order.

…”