arXiv subjects

Tiziano Peraro

Publications and source records attributed to Tiziano Peraro.

At least 19 recordsLinked to original sources

arXiv2026

Two-loop QCD amplitudes for $t\bar{t}W$ production at the LHC in the leading-colour approximation

We present a numerical computation of the two-loop QCD scattering amplitudes for the production of a top-antitop quark pair in association with a $W$ boson ($t\bar{t}W$) at the LHC in the generalised leading-colour approximation, retaining the exact dependence on the top-quark and $W$-boson masses. Rather than pursuing a fully analytic calculation, we employ a hybrid framework that combines numerical evaluation with strong algebraic and analytic control, allowing ultraviolet and infrared singularities as well as large intermediate cancellations to be treated exactly. This is achieved by expressing the finite remainder in terms of a set of special functions with rational coefficients. The special functions are evaluated numerically by solving differential equations through power-series expansions, while the values of the rational coefficients are reconstructed, point by point, from finite-field evaluations. The calculation is performed in the 't Hooft-Veltman scheme and validated against an independent implementation in conventional dimensional regularisation employing a substantially different computational strategy. We finally provide the colour- and polarisation-summed hard functions evaluated on the phase-space grid used in a previous computation of the next-to-next-to-leading-order QCD corrections to the $t\bar{t}W$ cross section.

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arXiv2026

NNLO QCD predictions for $t\bar t W$ production at hadron colliders

The production of a top-antitop quark pair in association with a $W$ boson constitutes one of the heaviest final states currently studied at the Large Hadron Collider (LHC) at CERN. Measurements of its production rate have consistently exceeded Standard Model predictions. Owing to the complexity of the two-loop amplitudes entering the double-virtual correction, next-to-next-to-leading-order (NNLO) QCD calculations for this process have so far employed dynamical approximations for the two-loop contribution. We present NNLO QCD predictions based, for the first time, on a direct computation of the required two-loop amplitudes in the generalised leading-colour limit.

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arXiv2026

Two-loop Six-point Planar Massless Feynman Integrals to Higher $ε$ Orders

In this work, we calculate two-loop six-point planar massless Feynman integrals at higher orders in the dimensional regulator $ε$, corresponding to higher transcendental weights. In previous works, these integrals were calculated up to weight four for the purpose of two-loop gauge theory amplitudes. Using modern rational reconstruction methods, we identify the complete alphabet with $269$ letters relevant to all weights, derive the analytic canonical differential equation and obtain the symbols up to weight six. As a proof of concept, using a new method with Chebyshev pseudospectral transport, we show that the corresponding pure basis can be efficiently evaluated up to weight six, i.e., to $ \mathcal{O}(ε^2)$ in a physical scattering region. The results of this work can be applied to future three-loop amplitudes and provide new data for the formal study of symbols and cluster algebras.

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arXiv2025

CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators

We elaborate on the method of parametric annihilators for deriving integral relations. Parametric annihilators are differential operators that annihilate multivalued integration kernels appearing in suitable integral representations of special functions. We illustrate this approach in a way that applies to a broad variety of integral representations. We describe a method for computing parametric annihilators based on efficient linear solvers and use them to derive relations between a wide class of special functions related to important problems in high-energy physics. We also formulate a similar method for deriving differential equations satisfied by the independent integrals within an integral family. We show applications to several classes of special functions, including hypergeometric functions, loop integrals in various representations (including Baikov, loop-by-loop Baikov, Lee-Pomeransky and Schwinger representations) and duals of loop integrals. We finally present the public Mathematica package CALICO for computing parametric annihilators and its usage in several examples of high relevance in theoretical particle physics.

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arXiv2025

Two-loop Feynman integrals for leading colour $t\bar{t}W$ production at hadron colliders

We compute a complete set of the two-loop Feynman integrals that are required for the next-to-next-to-leading order QCD corrections to on-shell top-pair production in association with a $W$ boson at hadron colliders in the leading colour approximation. These Feynman integrals also contribute to Higgs or $Z$-boson production in association with a top pair. We employ the method of differential equations (DEs), facilitated by the use of finite field methods to handle the algebraic complexity stemming from the seven-scale kinematics. The presence of the top quark in the virtual propagators, in addition to the mass of the external $W$ boson, gives rise to nested square roots and three elliptic curves. We obtain DEs that depend at most quadratically on the dimensional regulator $ε$ for sectors where these analytic structures appear, and are $ε$-factorised otherwise. We express the DEs in terms of a minimal set of differential one-forms, separating the logarithmic ones. We solve the DEs numerically in the physical kinematic region, with the method of generalised power series expansions.

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arXiv2025

Reduction to master integrals and transverse integration identities

The reduction of Feynman integrals to a basis of linearly independent master integrals is a pivotal step in loop calculations, but also one of the main bottlenecks. In this paper, we assess the impact of using transverse integration identities for the reduction to master integrals. Given an integral family, some of its sectors correspond to diagrams with fewer external legs or to diagrams that can be factorized as products of lower-loop integrals. Using transverse integration identities, i.e. a tensor decomposition in the subspace that is transverse to the external momenta of the diagrams, one can map integrals belonging to such sectors and their subsectors to (products of) integrals belonging to new and simpler integral families, characterized by either fewer generalized denominators, fewer external invariants, fewer loops or combinations thereof. Integral reduction is thus drastically simpler for these new families. We describe a proof-of-concept implementation of the application of transverse integration identities in the context of integral reduction. We include some applications to cutting-edge integral families, showing significant improvements over traditional algorithms.

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arXiv2025

Complete function space for planar two-loop six-particle scattering amplitudes

We derive the full system of canonical differential equations for all planar two-loop massless six-particle master integrals, and determine analytically the boundary conditions. This fully specifies the solutions, which may be written as Chen iterated integrals. We argue that this is sufficient information for evaluating any scattering amplitude in four dimensions up to the finite part. We support this claim by reducing, for the most complicated integral topologies, integrals with typical Yang-Mills numerators. We use the analytic solutions to the differential equations, together with dihedral symmetry, to provide the full solution space relevant for two-loop six-particle computations. This includes the relevant function alphabet, as well as the independent set of iterated integrals up to weight four. We also provide the answer for all master integrals in terms of iterated integrals that can be readily evaluated numerically. As a proof of concept, we provide a numerical implementation that evaluates the integrals in part of the Euclidean region, and validate this against numerical evaluation of the Feynman integrals. Our result removes the bottleneck of Feynman integral evaluation, paving the way to future analytic evaluations of six-particle scattering amplitudes.

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arXiv2024

A computation of two-loop six-point Feynman integrals in dimensional regularization

We compute three families of two-loop six-point massless Feynman integrals in dimensional regularization, namely the double-box, the pentagon-triangle, and the hegaxon-bubble family. This constitutes the first analytic computation of two-loop master integrals with eight scales. We use the method of canonical differential equations. We describe the corresponding integral basis with uniform transcendentality, the relevant function alphabet, and analytic boundary values at a particular point in the Euclidean region up to the fourth order in the regularization parameter $ε$. The results are expressed as one-fold integrals over classical polylogarithms suitable for fast and high-precision evaluation.

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arXiv2023

Reduction to master integrals via intersection numbers and polynomial expansions

Intersection numbers are rational scalar products among functions that admit suitable integral representations, such as Feynman integrals. Using these scalar products, the decomposition of Feynman integrals into a basis of linearly independent master integrals is reduced to a projection. We present a new method for computing intersection numbers that only uses rational operations and does not require any integral transformation or change of basis. We achieve this by systematically employing the polynomial series expansion, namely the expansion of functions in powers of a polynomial. We also introduce a new prescription for choosing dual integrals, de facto removing the explicit dependence on additional analytic regulators in the computation of intersection numbers. We describe a proof-of-concept implementation of the algorithm over finite fields and its application to the decomposition of Feynman integrals at one and two loops.

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arXiv2023

Two-loop QCD corrections to the $V\to q\bar{q}g$ helicity amplitudes with axial-vector couplings

We compute the two-loop corrections to the helicity amplitudes for the coupling of a massive vector boson to a massless quark-antiquark pair and a gluon, accounting for vector and axial-vector couplings of the vector boson and distinguishing isospin non-singlet and singlet contributions. A new four-dimensional basis for the decomposition of the amplitudes into 12 invariant tensor structures is introduced. The associated form factors are then computed up to two loops in QCD using dimensional regularization. After performing renormalization and infrared subtraction, the finite parts of the renormalized non-singlet vector and axial-vector form factors are shown agree with each other, and to reproduce the previously known two-loop amplitudes. The singlet axial-vector amplitude receives a contribution from the axial anomaly from two loops onwards. This amplitude is computed for massless and massive internal quarks. Our results provide the last missing two-loop amplitudes entering the NNLO QCD corrections of vector-boson-plus-jet production at hadron colliders.

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arXiv2023

Isolated photon production in association with a jet pair through next-to-next-to-leading order in QCD

In this work, we provide a comprehensive set of differential cross-section distributions for photon + di-jet production in proton-proton collisions with next-to-next-to-leading order precision in massless QCD. The event selection corresponds to recent measurements by the ATLAS collaboration. We observe an improved description of data in comparison to lower-order calculations in the case of observables that are expected to be well described by perturbation theory. The results also show better agreement with data than parton-shower-matched and multi-jet-merged predictions generated for the ATLAS analysis using the \textsc{Sherpa} Monte Carlo. A particular highlight of our study is the use of exact five-point two-loop virtual amplitudes. This is the first calculation of a complete two-to-three hadron-collider process at next-to-next-to-leading order in QCD that does not rely on the leading-colour approximation at two loops. We demonstrate, nevertheless, that the sub-leading-colour effects present in the infrared- and ultraviolet-finite double-virtual contributions are negligible in view of the remaining scale uncertainties.

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arXiv2022

A first look at the function space for planar two-loop six-particle Feynman integrals

Two-loop corrections to scattering amplitudes are crucial theoretical input for collider physics. Recent years have seen tremendous advances in computing Feynman integrals, scattering amplitudes, and cross sections for five-particle processes. In this paper, we initiate the study of the function space for planar two-loop six-particle processes. We study all genuine six-particle Feynman integrals, and derive the differential equations they satisfy on maximal cuts. Performing a leading singularity analysis in momentum space, and in Baikov representation, we find an integral basis that puts the differential equations into canonical form. The corresponding differential equation in the eight independent kinematic variables is derived with the finite-field reconstruction method and the symbol letters are identified. We identify the dual conformally invariant hexagon alphabet known from maximally supersymmetric Yang-Mills theory as a subset of our alphabet. This paper constitutes an important step in the analytic calculation of planar two-loop six-particle Feynman integrals.

hep-th↗

arXiv2021

Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders

We present an analytic computation of the gluon-initiated contribution to diphoton plus jet production at hadron colliders up to two loops in QCD. We reconstruct the analytic form of the finite remainders from numerical evaluations over finite fields including all colour contributions. Compact expressions are found using the pentagon function basis. We provide a fast and stable implementation for the colour- and helicity-summed interference between the one-loop and two-loop finite remainders in C++ as part of the NJet library.

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arXiv2021

Tensor decomposition for bosonic and fermionic scattering amplitudes

In this paper, we elaborate on a method to decompose multiloop multileg scattering amplitudes into Lorentz-invariant form factors, which exploits the simplifications that arise from considering four-dimensional external states. We propose a simple and general approach that applies to both fermionic and bosonic amplitudes and allows us to identify a minimal number of physically relevant form factors, which can be related one-to-one to the independent helicity amplitudes. We discuss explicitly its applicability to various four- and five-point scattering amplitudes relevant for LHC physics.

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arXiv2019

Analytic multi-loop results using finite fields and dataflow graphs with FiniteFlow

FiniteFlow is a public framework for defining and executing numerical algorithms over finite fields and reconstructing multivariate rational functions. The framework allows to build complex algorithms by combining basic building blocks into graphical representations of the calculation, known as dataflow graphs. It offers an easy-to-use Mathematica interface for implementing efficient custom algorithms without any low-level coding. We report on some new features of FiniteFlow which have been published after its initial release, give some simple example of usage for common tasks and review recent cutting-edge applications to two-loop five-point scattering and the four-loop cusp anomalous dimension.

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arXiv2019

FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs

Complex algebraic calculations can be performed by reconstructing analytic results from numerical evaluations over finite fields. We describe FiniteFlow, a framework for defining and executing numerical algorithms over finite fields and reconstructing multivariate rational functions. The framework employs computational graphs, known as dataflow graphs, to combine basic building blocks into complex algorithms. This allows to easily implement a wide range of methods over finite fields in high-level languages and computer algebra systems, without being concerned with the low-level details of the numerical implementation. This approach sidesteps the appearance of large intermediate expressions and can be massively parallelized. We present applications to the calculation of multi-loop scattering amplitudes, including the reduction via integration-by-parts identities to master integrals or special functions, the computation of differential equations for Feynman integrals, multi-loop integrand reduction, the decomposition of amplitudes into form factors, and the derivation of integrable symbols from a known alphabet. We also release a proof-of-concept C++ implementation of this framework, with a high-level interface in Mathematica.

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arXiv2019

A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons

We present the first numerical results for the two-loop helicity amplitudes for the scattering of four partons and a W-boson in QCD. We use a finite field sampling method to reduce directly from Feynman diagrams to the coefficients of a set of master integrals after applying integration-by-parts identities. Since the basis of master integrals is not yet fully known analytically, we identify a set of master integrals with a simple divergence structure using local numerator insertions. This allows for accurate numerical evaluation of the amplitude using sector decomposition methods.

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arXiv2019

Matter dependence of the four-loop cusp anomalous dimension

We compute analytically the matter-dependent contributions to the quartic Casimir term of the four-loop light-like cusp anomalous dimension in QCD, with $n_f$ fermion and $n_s$ scalar flavours. The result is extracted from the double pole of a scalar form factor. We adopt a new strategy for the choice of master integrals with simple analytic and infrared properties, which significantly simplifies our calculation. To this end we first identify a set of integrals whose integrands have a dlog form, and are hence expected to have uniform transcendental weight. We then perform a systematic analysis of the soft and collinear regions of loop integration and build linear combinations of integrals with a simpler infrared pole structure. In this way, only integrals with ten or fewer propagators are needed for obtaining the cusp anomalous dimension. These integrals are then computed via the method of differential equations through the addition of an auxiliary scale. Combining our result with that of a parallel paper, we obtain the complete $n_{f}$ dependence of the four-loop cusp anomalous dimension in QCD. Finally, using known numerical results for the gluonic contributions, we obtain an improved numerical prediction for the cusp anomalous dimension in $\mathcal{N}=4$ super Yang-Mills theory.

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