Twisted period integrals are ubiquitous in theoretical physics and mathematics, where they inhabit a finite-dimensional vector space governed by an inner product known as the intersection number. In this work, we uncover the associated tensor structures of intersection numbers and integrate them with the fibration method to develop a novel evaluation scheme. Companion matrices allow us to cast the computation of the intersection numbers in terms of a matrix operator calculus within the ambient tensor space. For illustrative purposes, our algorithm has been successfully applied to the numerical decomposition of a sample of two-loop integrals, coming from planar five-point massless functions, representing a significant advancement for the direct projection of Feynman integrals to master integrals via intersection numbers.

France

Italy

Giacomo Brunello

Vsevolod Chestnov

Pierpaolo Mastrolia

Intersection numbers from companion tensor algebra

Higher-point functions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> super Yang-Mills theory can be constructed using integrability by triangulating the surfaces on which Feynman graphs would be drawn. It remains hard to analytically compute the necessary regluing of the tiles by virtual particles. We propose a new approach to study a series of residues encountered in the two-particle gluing of the planar one-loop five-point function of stress tensor multiplets. After exposing the twisted period nature of the integral functions, we employ intersection theory to derive canonical differential equations and present a solution.

United Kingdom

Giulio Crisanti

Germany

Burkhard Eden

Maximilian Gottwald

Tobias Scherdin

Gluing via Intersection Theory

We present the analytical evaluation of the gravitational energy and angular momentum flux with tidal effects for inspiraling compact binaries, at the next-to-next-to-leading post-Newtonian (2PN) order, within the diagrammatic Effective Field Theory approach. We first compute the stress-energy tensor for a binary system, which requires the evaluation of two-point Feynman integrals, up to two loops. Then we extract the multipole moments of the system, which we present for generic orbits in center-of-mass coordinates, and which are needed to evaluate the total gravitational energy and the angular momentum flux for generic orbits. Finally, we provide the expressions for gauge invariant quantities for circular orbits, such as the fluxes, mode amplitudes, and phase of the emitted gravitational wave. Our results are useful for updating previous theoretical studies, as well as related phenomenological analyses and waveform models.

Manoj Mandal

Raj Patil

Jan Steinhoff

Radiating Love: adiabatic tidal fluxes and modes up to next-to-next-to-leading post-Newtonian order

Understanding the loop corrections to cosmological observables is of paramount importance for having control on the quantum consistency of a theory in an expanding universe as well as for phenomenological reasons. In the present work, we begin with a systematic study of such corrections in the context scalar toy models whose perturbative Bunch-Davies wave function enjoys an intrinsic definition in terms of cosmological polytopes, focusing on one-loop graphs. Owing to the underlying twisted period integral representation they admit, their combinatorial structure along with their vector space structure, emerging from polynomial ideals algebra and intersection theory, are exploited to set-up and analyze the differential equations that the two- and three-site one-loop corrections have to satisfy upon variation of the external kinematic variables. We find that, while the two-site contribution can be written in terms of multiple polylogarithms, this is no longer true for the three-site case, for which elliptic structures appear. As a nontrivial check, we consider the scattering amplitude limit, recovering the known result in terms of polylogarithms only.

Spain

Paolo Benincasa

Manoj K. Mandal

Francisco Vazão

One-loop corrections to the Bunch-Davies wave function of the universe

We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential n-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology groups and the polynomial division technique, recently proposed in the literature. We show that delta-forms capture the leading behaviour of the intersection numbers in presence of evanescent analytic regulators, whose use is, therefore, bypassed. This simplified algorithm is applied to derive the complete decomposition of two-loop planar and non-planar Feynman integrals in terms of a master integral basis. More generally, it can be applied to derive relations among twisted period integrals, relevant for physics and mathematical studies.

Denmark

Hjalte Frellesvig

Intersection numbers, polynomial division and relative cohomology

Building on recent advances in studying the cohomological properties of Feynman integrals, we apply intersection theory to the computation of Fourier integrals. We discuss applications pertinent to gravitational bremsstrahlung and deep inelastic scattering in the saturation regime. After identifying the bases of master integrals, the latter are evaluated by means of the differential equation method. Finally, new results with exact dependence on the spacetime dimension D are presented.

Canada

Mathieu Giroux

Sid Smith

Fourier calculus from intersection theory

We present the conservative effective two-body Hamiltonian at the third order in the post-Newtonian expansion with gravitoelectric quadrupolar dynamical tidal-interactions. Our derivation of the effective two-body Lagrangian is based on the diagrammatic effective field theory approach and it involves Feynman integrals up to three loops, which are evaluated within the dimensional regularization scheme. The elimination of the divergent terms occurring in the effective Lagrangian requires the addition of counterterms to ensure finite observables, thereby introducing a renormalization group flow to the post-adiabatic Love number. As a limiting case of the renormalized dynamical effective Hamiltonian, we also derive the effective Hamiltonian for adiabatic tides, and, in this regime, calculate the binding energy for a circular orbit, and the scattering angle in a hyperbolic scattering.

United States

Hector Silva

Renormalizing Love: tidal effects at the third post-Newtonian order

We present the analytic evaluation of the second-order corrections to the massive form factors, due to two-loop vertex diagrams with a vacuum polarization insertion, with exact dependence on the external and internal fermion masses, and on the squared momentum transfer. We consider vector, axial-vector, scalar and pseudoscalar interactions between the external fermion and the external field. After renormalization, the finite expressions of the form factors are expressed in terms of polylogarithms up to weight three.

Taushif Ahmed

Federico Gasparotto

Syed Hasan

Two-loop vertices with vacuum polarization insertion

We present a gravitoelectric quadrupolar dynamical tidal-interaction Hamiltonian for a compact binary system, that is valid to second order in the post-Newtonian expansion. Our derivation uses the diagrammatic effective field theory approach, and involves Feynman integrals up to two loops, evaluated with the dimensional regularization scheme. We also derive the effective Hamiltonian for adiabatic tides, obtained by taking the appropriate limit of the dynamical effective Hamiltonian, and we check its validity by verifying the complete Poincaré algebra. In the adiabatic limit, we also calculate two gauge-invariant observables, namely, the binding energy for a circular orbit and the scattering angle in a hyperbolic scattering. Our results are important for developing accurate gravitational waveform models for neutron-star binaries for present and future gravitational-wave observatories.

Gravitoelectric dynamical tides at second post-Newtonian order

We present the result of the quadratic-in-spin interaction Hamiltonian for binary systems of rotating compact objects with generic spins, up to N3LO corrections within the post-Newtonian expansion. The calculation is performed by employing the effective field theory diagrammatic approach, and it involves Feynman integrals up to three loops, evaluated within the dimensional regularization scheme. The gauge-invariant binding energy and the scattering angle, in special kinematic regimes and spin configurations, are explicitly derived. The results extend our earlier study on the spin-orbit interaction effects.