A bstract We present a simplification of the recursive algorithm for evaluation intersection numbers differential n -forms, by combining advantages emerging from choice delta-forms as generators relative twisted cohomology groups and polynomial division technique, recently proposed in literature. show that capture leading behaviour presence evanescent analytic regulators, whose use is, therefore, bypassed. This simplified is applied to derive complete decomposition two-loop planar non-planar...

A bstract The reduction of Feynman integrals to a basis linearly independent master is pivotal step in loop calculations, but also one the main bottlenecks. In this paper, we assess impact using transverse integration identities for integrals. Given an integral family, some its sectors correspond diagrams with fewer external legs or that can be factorized as products lower-loop Using identities, i.e. tensor decomposition subspace momenta diagrams, map belonging such and their subsectors...

A bstract We elaborate on the connection between Gel’fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations Feynman Integrals. propose a novel, more efficient algorithm to compute Macaulay matrices, which are used derive systems of differential equations. The matrices then employed obtain linear relations $$ \mathcal{A} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> -hypergeometric (Euler) integrals...

A bstract We propose a new method for the evaluation of intersection numbers twisted meromorphic n -forms, through Stokes’ theorem in dimensions. It is based on solution an -th order partial differential equation and multivariate residues. also present algebraic expression contribution from each residue. illustrate our approach with number simple examples mathematics physics.

A bstract This work studies limits of Pfaffian systems, a class first-order PDEs appearing in the Feynman integral calculus. Such appear naturally context scattering amplitudes when there is separation scale given set kinematic variables. We model these limits, which are often singular, via restrictions $$ \mathcal{D} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> -modules. thereby develop two different restriction algorithms: one based on gauge...

High precision calculations in perturbative QFT often require evaluation of big collection Feynman integrals. Complexity this task can be greatly reduced via the usage linear identities among Based on mathematical theory intersection numbers, recently a new method for derivation such and decomposition integrals was introduced applied to many non-trivial examples. In note we discuss latest developments algorithms their application reduction

A bstract We study the six-particle amplitude in planar $$ \mathcal{N} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 super Yang-Mills theory double scaling (DS) limit, only nontrivial codimension-one boundary of its positive kinematic region. construct relevant function space, which is significantly constrained due to extended Steinmann relations, up weight 13 coproduct form, and 12 as an explicit polylogarithmic representation. Expanding latter...

We present a simplification of the recursive algorithm for evaluation intersection numbers differential $n$-forms, by combining advantages emerging from choice delta-forms as generators relative twisted cohomology groups and polynomial division technique, recently proposed in literature. show that capture leading behaviour presence evanescent analytic regulators, whose use is, therefore, bypassed. This simplified is applied to derive complete decomposition two-loop planar non-planar Feynman...

The reduction of Feynman integrals to a basis linearly independent master is pivotal step in loop calculations, but also one the main bottlenecks. In this paper, we assess impact using transverse integration identities for integrals. Given an integral family, some its sectors correspond diagrams with fewer external legs or that can be factorized as products lower-loop Using identities, i.e. tensor decomposition subspace momenta diagrams, map belonging such and their subsectors (products of)...

The finite remainder function for planar, color-ordered, maximally helicity violating scattering processes in $$ \mathcal{N} = 4 super Yang-Mills theory possesses a non-vanishing multi-Regge limit that depends on the choice of Mandelstam region. We analyze combined collinear all regions through an analytic continuation Wilson loop OPE. At leading order, former is determined by gluon excitation Gubser-Klebanov-Polyakov string. illustrate general procedure at example heptagon two loops. In...

This work studies limits of Pfaffian systems, a class first-order PDEs appearing in the Feynman integral calculus. Such appear naturally context scattering amplitudes when there is separation scale given set kinematic variables. We model these limits, which are often singular, via restrictions D-modules. thereby develop two different restriction algorithms: one based on gauge transformations, and another relying Macaulay matrix. These algorithms output systems containing fewer variables...

We study the six-particle amplitude in planar $\mathcal{N} = 4$ super Yang-Mills theory double scaling (DS) limit, only nontrivial codimension-one boundary of its positive kinematic region. construct relevant function space, which is significantly constrained due to extended Steinmann relations, up weight 13 coproduct form, and 12 as an explicit polylogarithmic representation. Expanding latter collinear DS using Pentagon Operator Product Expansion, we compute non-divergent coefficient a...

We elaborate on the connection between Gel'fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations Feynman integrals. propose a novel, more efficient algorithm to compute Macaulay matrices, which are used derive systems of differential equations. The matrices then employed obtain linear relations ${\cal A}$-hypergeometric (Euler) integrals integrals, through recurrence projections by intersection numbers.

High precision calculations in perturbative QFT often require evaluation of big collection Feynman integrals. Complexity this task can be greatly reduced via the usage linear identities among Based on mathematical theory intersection numbers, recently a new method for derivation such and decomposition integrals was introduced applied to many non-trivial examples. In note we discuss latest developments algorithms their application reduction

We propose a new method for the evaluation of intersection numbers twisted meromorphic $n$-forms, through Stokes' theorem in $n$ dimensions. It is based on solution an $n$-th order partial differential equation and multivariate residues. also present algebraic expression contribution from each residue. illustrate our approach with number simple examples mathematics physics.