arXiv subjects

Giulio Crisanti

Publications and source records attributed to Giulio Crisanti.

12 recordsLinked to original sources

arXiv2026

Magic Relations and Critical Varieties of Feynman Integrals

Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break down, they are important to detect and understand. In this paper, we take a first step toward a systematic characterization of such identities. Specifically, we observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.

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arXiv2026

New Tools in the Landau Bootstrap

We describe recent advances in our understanding of the analytic structure of Feynman integrals. In particular, we describe two new classes of constraints on such integrals, that identify discontinuities that either cannot be repeated, or that always give rise to the same result (no matter which other discontinuities are computed first). These new constraints hold at all orders in dimensional regularization, and provide us with new input for the Landau bootstrap, where information about the singularities and discontinuities of individual Feynman integrals is used to construct their functional form.

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arXiv2026

Landau's Leviathans

We present a new method together with a proof-of-concept implementation for determining the Landau singularities of Feynman integrals, read off directly from where the Euler characteristic of the associated integral drops. Working over finite fields makes the requisite elimination tractable for multi-scale integrals at the multi-loop frontier. The algorithm returns the genuine and complete set of singularities, subject to a set of conditions which are practically testable. We apply these methods to classes of Feynman integrals beyond the reach of current methods, including non-planar six-point diagrams at two loops, as well as a fully massive three-loop envelope graph. Several of the newly found singularities, both in $d$- and 4-dimensional external kinematics, are of unexpected complexity when compared to previously known singularities for these examples.

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arXiv2026

Three-loop helicity amplitudes of four-lepton scattering in QED

We present the analytic expressions of the three-loop virtual corrections to the helicity amplitudes of 2 -> 2 four-fermion scattering processes in massless QED. The contributing Feynman diagrams are grouped into integrand families characterised by independent Symanzik polynomials and decomposed in terms of master integrals using an optimised integration-by-parts strategy. Upon the renormalisation of the ultraviolet divergences and the extraction of the universal infrared pole structure, the finite results are expressed in terms of generalised polylogarithms up to transcendental weight six. Amplitudes for dimuon production in electron-positron annihilations, electron-muon scattering, and Bhabha scattering are explicitly derived.

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arXiv2026

High-energy evolution in planar QCD to three loops: the non-conformal contribution

The Balitsky-Kovchegov (BK) equation offers a tractable description of the high-energy growth of gauge-theory scattering amplitudes and the nonlinear saturation effects that eventually tame it. Motivated by the upcoming Electron-Ion Collider (EIC), whose extended kinematic reach promises more decisive tests of saturation at high energies, we present a framework based on the spacelike-timelike correspondence that streamlines the computation of multi-loop corrections to the BK equation. We explicitly verify the correspondence at three loops in the large-flavor limit and predict the full nonconformal component of the three-loop BK Hamiltonian in the planar limit of a generic gauge theory, treating the numbers of fermions and scalars as free parameters.

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arXiv2025

Sampling Polynomial Rational Remainders with SP$\mathbb{Q}$R: A new Package for Polynomial Division and Elimination

We introduce SP$\mathbb{Q}$R, a new Mathematica package for the division and elimination of variables from polynomial systems. SP$\mathbb{Q}$R works by sampling and reconstructing results over finite fields, in an analogous manner to many state of the art Integration by Parts algorithms for Feynman integrals. This allows SP$\mathbb{Q}$R to effectively overcome expression swell during the construction of Gröbner bases, which in many cases is the major bottleneck in such computations. Benchmarks on state of the art Macaulay resultants show that SP$\mathbb{Q}$R can deliver substantial gains over symbolic computer algebra workflows -- reducing both runtime and memory footprint by multiple orders of magnitude. Likewise when applied to study Feynman integrals, we show how SP$\mathbb{Q}$R can be used to find previously unknown Landau singularities.

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arXiv2025

Gravitational waveforms from restriction theory and rapid-decay homology

We present a systematic framework for computing frequency-domain gravitational waveforms from relativistic binary scattering in different asymptotic regimes. The method yields a controlled series expansion that can in principle be extended to arbitrary order in the relevant kinematic parameter. By combining differential-equation techniques with restriction theory and algebraic-geometry methods for impact-parameter-space Fourier integrals, we derive recursion relations that generate the leading-order (tree-level) waveform in both the soft-emission and post-Newtonian regimes, establishing a proof of principle for extending the approach to higher-loop computations. Finally, following constraints from rapid-decay homology, we show that the Fourier integrals underlying the waveform satisfy epsilon-form differential equations mixing Bessel- and exponential-type kernels, marking a first step toward uncovering the analytic structure of the exact solution.

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arXiv2024

Gluing via Intersection Theory

Higher-point functions in N = 4 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 re-gluing 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.

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arXiv2024

Feynman Integral Reductions by Intersection Theory with Orthogonal Bases and Closed Formulae

We present a prescription for choosing orthogonal bases of differential $n$-forms belonging to quadratic twisted period integrals, with respect to the intersection number inner product. To evaluate these inner products, we additionally propose a new closed formula for intersection numbers beyond $\mathrm{d} \log$ forms. These findings allow us to systematically construct orthonormal bases between twisted period integrals of this type. In the context of Feynman integrals, this represents all diagrams at one-loop.

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arXiv2024

Fourier Calculus from Intersection Theory

Building on recent advances in studying the co-homological 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.

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arXiv2024

Two-loop Vertices with Vacuum Polarization Insertion

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.

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arXiv2023

Intersection Numbers, Polynomial Division and Relative Cohomology

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.

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