We compute the potential-graviton contributions to the conservative scattering angle of two non-spinning bodies in maximal supergravity at fifth order in Newton's constant, including second-order self-force effects. Our goal is to tackle the challenging integrals arising at this order in Einstein gravity, but within the technically simpler framework of supergravity. The calculation employs the scattering-amplitude framework, effective field theory, and multi-loop integration techniques based on integration by parts and differential equations. The final result is expressed as a series expansion around the static limit, thereby avoiding the explicit evaluation of intricate special functions. This series solution for the master integrals applies, as well, to the corresponding computation in general relativity. Remarkably, we observe nontrivial cancellations among contributions associated with Calabi-Yau integrals, alongside a distinct contribution governed by a Heun differential equation.

FIRE is a program which performs integration-by-parts (IBP) reduction of Feynman integrals. Originally, the C++ version of FIRE relies on the computer algebra system Fermat by Robert Lewis to simplify rational functions. We present an upgrade of FIRE which incorporates a new library FUEL initially described in a separate publication, which enables a flexible choice of third-party computer algebra systems as simplifiers, as well as efficient communications with some of the simplifiers as C++ libraries rather than through Unix pipes. We achieve significant speedups for IBP reduction of Feynman integrals involving many kinematic variables, when using an open source backend based on FLINT newly added in this work, or the Symbolica backend developed by Ben Ruijl as a potential successor of FORM.

We prove a recent conjecture of Dragovic et al arXiv2504.20515 stating that the magnetic geodesic flow on the standard sphere $S^n\subset \mathbb R^{n+1}$ whose magnetic 2-form is the restriction of a constant 2-form from $\mathbb{R}^{n+1}$ is Liouville integrable. The integrals are quadratic and linear in momenta.

We address the problem of evaluation of multiloop Feynman integrals by means of their Mellin-Barnes representation. After a brief overview of available capabilities though open source toolkits and their application in various circumstances, we introduce a new code MBcreate which allows one to automatically deduce a concise Mellin-Barnes representation for a given parametric integral. A thorough discussion of its implementation and use is provided.

In this paper we present a new release of the FIESTA program (Feynman Integral Evaluation by a Sector decomposiTion Approach). FIESTA5 is performance-oriented - we implemented improvements of various kinds in order to make Feynman integral evaluation faster. We plugged in two new integrators, the Quasi Monte Carlo and Tensor Train. At the same time the old code of FIESTA4 was upgraded to the C++17 standard and mostly rewritten without self-made structures such as hash tables. There are also several essential improvements which are most relevant for complex integrations - the new release is capable of producing results where previously impossible.

We compute the potential-graviton contributions to the conservative scattering angle of two non-spinning bodies in maximal supergravity at fifth order in Newton's constant, including second-order self-force effects. Our goal is to tackle the challenging integrals arising at this order in Einstein gravity, but within the technically simpler framework of supergravity. The calculation employs the scattering-amplitude framework, effective field theory, and multi-loop integration techniques based on integration by parts and differential equations. The final result is expressed as a series expansion around the static limit, thereby avoiding the explicit evaluation of intricate special functions. This series solution for the master integrals applies, as well, to the corresponding computation in general relativity. Remarkably, we observe nontrivial cancellations among contributions associated with Calabi-Yau integrals, alongside a distinct contribution governed by a Heun differential equation.

We present analytical results for all master integrals for massless three-point functions, with one off-shell leg, at four loops. Our solutions were obtained using differential equations and direct integration techniques. We review the methods and provide additional details.

FIRE7 is a major update to the FIRE program for integration-by-parts (IBP) reduction of Feynman integrals. A large part of improvements is related to the automatic reduction and reconstruction with the modular arithmetic approach, while the performance of the classical rational polynomial approach is also significantly increased. An improved presolve algorithm performs Gaussian elimination to simplify IBP identities before substituting numerical indices as in the Laporta algorithm. Various new command line tools are included to facilitate tasks such as applying an IBP reduction table to reduce a loop integrand as a linear combination of individual integrals.

In this work, we solve the fundamental problem of describing the coordinate transformations that preserve the upper triangular Toeplitz form of the given operator field. Surprisingly, this problem is closely related to the description of all Nijenhuis operators in the same form. This description, as well as the formulas for the aforementioned coordinate transformations, are given by the implicit formulas involving matrix-valued functions.

We compute the photon-quark and Higgs-gluon form factors to four-loop order within massless perturbative Quantum Chromodynamics. Our results constitute ready-to-use building blocks for N${}^4$LO cross sections for Drell-Yan processes and gluon-fusion Higgs boson production at the LHC. We present complete analytic expressions for both form factors and show several of the most complicated master integrals.

We address the problem of unambiguous reconstruction of rational functions of many variables. This is particularly relevant for recovery of exact expansion coefficients in integration-by-parts identites (IBPs) based on modular arithmetic. These IBPs are indispensable in modern approaches to evaluation of multiloop Feynman integrals by means of differential equations. Modular arithmetic is far more superior to algebraic implementations when one deals with high-multiplicity situations involving a large number of Lorentz invariants. We introduce a new method based on balanced relations which allows one to achieve the goal of a robust functional restoration with minimal data input. The technique is implemented as a Mathematica package Reconstruction.m in the FIRE6 environment and thus successfully demonstrates a proof of concept.

We consider a linearization problem for Nijenhuis operators in dimension two around a point of scalar type in analytic category. The problem was almost completely solved in arXiv:1903.06411. One case, however, namely the case of left-symmetric algebra $\mathfrak b_{1, \alpha}$, proved to be difficult. Here we solve it and, thus, complete the solution of the linearization problem for Nijenhuis operators in dimension two. The problem turns out to be related to classical results on the linearization of vector fields and their monodromy mappings.

Tamanoi equation is a Macdonald type equation for the orbifold Euler characteristic and for its analogues of higher orders. It claims that the generating series of the orbifold Euler characteristics of a fixed order of analogues of the symmetric powers for a space with a finite group action can be represented as a certain unified (explicitly written) power series in the exponent equal to the orbifold Euler characteristic of the same order of the space itself. In the paper, in particular, we explain how the Tamanoi equation follows from its verification for actions of (finite) groups on the one-point space. Statements used for that are generalized to analogues of the orbifold Euler characteristic corresponding to finitely generated groups. It is shown that, for these generalizatins, the analogue of the Tamanoi equation does not hold in general.

We study multi-propagator angular integrals, a class of phase-space integrals relevant to processes with multiple observed final states and a test-bed for transferring loop-integral technology to phase space integrals without reversed unitarity. We present an Euler integral representation similar to Lee-Pomeransky representation and explicitly describe a recursive IBP reduction and dimensional shift relations for the general case of $n$ denominators. On the level of master integrals, applying a differential equation approach, we explicitly calculate the previously unknown angular integrals with four denominators for any number of masses to finite order in $\varepsilon$. Extending the idea of dimensional recurrence, we explore the decomposition of angular integrals into branch integrals reducing the number of scales in the master integrals from $(n+1)n/2$ to $n+1$. To showcase the potential of this method, we calculate the massless three denominator integral to establish all-order results in $\varepsilon$ including a resummation of soft logarithms.

Two-loop multi-leg form factors in off-shell kinematics require knowledge of planar and nonplanar double box Feynman diagrams with massless internal propagators. These are complicated functions of Mandelstam variables and external particle virtualities. The latter serve as regulators of infrared divergences, thus making these observables finite in four space-time dimensions. In this paper, we use the method of canonical differential equations for calculation of (non)planar double box integrals in the near mass-shell kinematical regime, i.e., where virtualities of external particles are much smaller than the Mandelstam variables involved. We deduce a basis of master integrals with uniform transcendental weight based on the analysis of leading singularities by means of the Baikov representation as well as an array of complementary techniques. We dub the former asymptotically canonical since it is valid in the near mass-shell limit of interest. We iteratively solve resulting differential equations up to weight four in terms of multiple polylogarithms.

In the article a technique of the usage of $f$-continuous functions (on mappings) and their families is developed. A proof of the Urysohn's Lemma for mappings is presented and a variant of the Brouwer-Tietze-Urysohn Extension Theorem for mappings is proven. Characterizations of the normality properties of mappings are given and the notion of a perfect normality of a mapping is introduced. It seems to be the most optimal in this approach.

Coulomb branch amplitudes of maximally supersymmetric Yang-Mills theory display infrared properties different from their conformal counterparts. While the four-leg amplitude is known to very high perturbative orders, amplitudes of higher multiplicity fall into an uncharted territory starting already from two loops. The reason for this is that they are not easily amenable to traditional techniques like canonical differential equations due to the uncontrolled swelling of solutions to integration-by-parts identities. In this paper, we break the barrier for the five-leg amplitude using a technique based on the analysis of Newton polytopes corresponding to Feynman/Schwinger integrands and their tropical geometry. Specifically, we analytically evaluate the near mass-shell limit of the off-shell pentabox in terms of Goncharov polylogarithms.

We calculate the total Born cross section of the $e^+e^-$-pair production by an electron in the field of a nucleus (trident process) using the modern multiloop methods. For general energies we obtain the cross section in terms of converging power series. The threshold asymptotics and the high-energy asymptotics are obtained analytically. In particular, we obtain additional contribution to the Racah formula due to the identity of the final electrons. Besides, our result for the leading term of the high-energy asymptotics reveals a typo in an old Racah paper [Racah1937].

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional nilpotent Leibniz algebras. In particular, the variety of complex $5$-dimensional nilpotent Leibniz algebras has dimension $24$ it has $10$ irreducible components (there is only one rigid algebra in this variety).