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Samuel Abreu

Publications and source records attributed to Samuel Abreu.

At least 37 records · Page 2Linked to original sources
arXiv2022

The analytic two-loop soft function for leading-jet $p_T$

We present the calculation of the two-loop soft function for the transverse momentum distribution of the leading jet produced in association with any colour-singlet system (e.g.~a Higgs or a $Z$ boson). This constitutes a central ingredient for the resummation of the above distribution as well as the jet-vetoed cross section at the next-to-next-to-next-to-leading logarithmic order, both of which play an important role in the precision physics programme at the Large Hadron Collider. The calculation is performed in soft-collinear effective theory with an appropriate regularisation of the rapidity divergences that occur in the phase-space integrals. We obtain analytic results by employing an exponential regulator and by taking a Laurent expansion in the jet radius $R$. All expressions are presented as ancillary files with this article.

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arXiv2021

Two-Loop Hexa-Box Integrals for Non-Planar Five-Point One-Mass Processes

We present the calculation of the three distinct non-planar hexa-box topologies for five-point one-mass processes. These three topologies are required to obtain the two-loop virtual QCD corrections for two-jet-associated W, Z or Higgs-boson production. Each topology is solved by obtaining a pure basis of master integrals and efficiently constructing the associated differential equation with numerical sampling and unitarity-cut techniques. We present compact expressions for the alphabet of these non-planar integrals, and discuss some properties of their symbol. Notably, we observe that the extended Steinmann relations are in general not satisfied. Finally, we solve the differential equations in terms of generalized power series and provide high-precision values in different regions of phase space which can be used as boundary conditions for subsequent evaluations.

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arXiv2021

The diagrammatic coaction beyond one loop

The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the $ε$ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

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arXiv2019

Diagrammatic Coaction of Two-Loop Feynman Integrals

It is known that one-loop Feynman integrals possess an algebraic structure encoding some of their analytic properties called the coaction, which can be written in terms of Feynman integrals and their cuts. This diagrammatic coaction, and the coaction on other classes of integrals such as hypergeometric functions, may be expressed using suitable bases of differential forms and integration contours. This provides a useful framework for computing coactions of Feynman integrals expressed using the hypergeometric functions. We will discuss examples where this technique has been used in the calculation of two-loop diagrammatic coactions.

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arXiv2019

Generalized hypergeometric functions and intersection theory for Feynman integrals

Feynman integrals that have been evaluated in dimensional regularization can be written in terms of generalized hypergeometric functions. It is well known that properties of these functions are revealed in the framework of intersection theory. We propose a new application of intersection theory to construct a coaction on generalized hypergeometric functions. When applied to dimensionally regularized Feynman integrals, this coaction reproduces the coaction on multiple polylogarithms order by order in the parameter of dimensional regularization.

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arXiv2019

Three-loop contributions to the $ρ$ parameter and iterated integrals of modular forms

We compute fully analytic results for the three-loop diagrams involving two different massive quark flavours contributing to the $ρ$ parameter in the Standard Model. We find that the results involve exactly the same class of functions that appears in the well-known sunrise and banana graphs, namely elliptic polylogarithms and iterated integrals of modular forms. Using recent developments in the understanding of these functions, we analytically continue all the iterated integrals of modular forms to all regions of the parameter space, and in each region we obtain manifestly real and fast-converging series expansions for these functions.

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arXiv2019

From positive geometries to a coaction on hypergeometric functions

It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally-regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter $ε$. We show that the coaction defined on this class of integral is consistent, upon expansion in $ε$, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric ${}_{p+1}F_p$ and Appell functions.

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arXiv2019

The two-loop five-point amplitude in $\mathcal N=8$ supergravity

We compute the symbol of the two-loop five-point amplitude in $\mathcal N=8$ supergravity. We write an ansatz for the amplitude whose rational prefactors are based on not only 4-dimensional leading singularities, but also $d$-dimensional ones, as the former are insufficient. Our novel $d$-dimensional unitarity-based approach to the systematic construction of an amplitude's rational structures is likely to have broader applications, for example to analogous QCD calculations. We fix parameters in the ansatz by performing numerical integration-by-parts reduction of the known integrand. We find that the two-loop five-point $\mathcal N=8$ supergravity amplitude is uniformly transcendental. We then verify the soft and collinear limits of the amplitude. There is considerable similarity with the corresponding amplitude for $\mathcal N=4$ super-Yang-Mills theory: all the rational prefactors are double copies of the Yang-Mills ones and the transcendental functions overlap to a large degree. As a byproduct, we find new relations between color-ordered loop amplitudes in $\mathcal N=4$ super-Yang-Mills theory.

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arXiv2019

Differential equations from unitarity cuts: nonplanar hexa-box integrals

We compute $ε$-factorized differential equations for all dimensionally-regularized integrals of the nonplanar hexa-box topology, which contribute for instance to 2-loop 5-point QCD amplitudes. A full set of pure integrals is presented. For 5-point planar topologies, Gram determinants which vanish in $4$ dimensions are used to build compact expressions for pure integrals. Using unitarity cuts and computational algebraic geometry, we obtain a compact IBP system which can be solved in 8 hours on a single CPU core, overcoming a major bottleneck for deriving the differential equations. Alternatively, assuming prior knowledge of the alphabet of the nonplanar hexa-box, we reconstruct analytic differential equations from 30 numerical phase-space points, making the computation almost trivial with current techniques. We solve the differential equations to obtain the values of the master integrals at the symbol level. Full results for the differential equations and solutions are included as supplementary material.

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arXiv2018

Coaction for Feynman integrals and diagrams

We propose a general coaction for families of integrals appearing in the evaluation of Feynman diagrams, such as multiple polylogarithms and generalized hypergeometric functions. We further conjecture a link between this coaction and graphical operations on Feynman diagrams. At one-loop order, there is a basis of integrals for which this correspondence is fully explicit. We discuss features and present examples of the diagrammatic coaction on two-loop integrals. We also present the coaction for the functions ${}_{p+1}F_p$ and Appell $F_1$.

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arXiv2018

Five-Point Two-Loop Amplitudes from Numerical Unitarity

We present advances in the development of the numerical unitarity method for the computation of multi-loop amplitudes in QCD. As an application, we show results for all the leading-color two-loop five-gluon helicity amplitudes. The amplitudes are reduced to a linear combination of master integrals by employing unitarity-compatible integration-by-parts identities, and the corresponding integral coefficients are computed in an exact manner on rational phase-space points through finite fields arithmetics.

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arXiv2018

The diagrammatic coaction and the algebraic structure of cut Feynman integrals

We present a new formula for the coaction of a large class of integrals. When applied to one-loop (cut) Feynman integrals, it can be given a diagrammatic representation purely in terms of pinches and cuts of the edges of the graph. The coaction encodes the algebraic structure of these integrals, and offers ways to extract important properties of complicated integrals from simpler functions. In particular, it gives direct access to discontinuities of Feynman integrals and facilitates a straightforward derivation of the differential equations they satisfy, which we illustrate in the case of the pentagon.

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arXiv2018

Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case

We construct a diagrammatic coaction acting on one-loop Feynman graphs and their cuts. The graphs are naturally identified with the corresponding (cut) Feynman integrals in dimensional regularization, whose coefficients of the Laurent expansion in the dimensional regulator are multiple polylogarithms (MPLs). Our main result is the conjecture that this diagrammatic coaction reproduces the combinatorics of the coaction on MPLs order by order in the Laurent expansion. We show that our conjecture holds in a broad range of nontrivial one-loop integrals. We then explore its consequences for the study of discontinuities of Feynman integrals, and the differential equations that they satisfy. In particular, using the diagrammatic coaction along with information from cuts, we explicitly derive differential equations for any one-loop Feynman integral. We also explain how to construct the symbol of any one-loop Feynman integral recursively. Finally, we show that our diagrammatic coaction follows, in the special case of one-loop integrals, from a more general coaction proposed recently, which is constructed by pairing master integrands with corresponding master contours.

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arXiv2017

Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity

We present a calculation of the planar two-loop five-gluon amplitudes. The amplitudes are obtained in a variant of the generalized unitarity approach suitable for numerical computations, which we extend for use with finite field arithmetics. Employing a new method for the generation of unitarity-compatible integration-by-parts identities, all helicity amplitudes are reduced to a linear combination of master integrals for the first time. The approach allows us to compute exact values for the integral coefficients at rational phase-space points. All required master integrals are known analytically, and we obtain arbitrary-precision values for the amplitudes.

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arXiv2017

Cuts from residues: the one-loop case

Using the multivariate residue calculus of Leray, we give a precise definition of the notion of a cut Feynman integral in dimensional regularization, as a residue evaluated on the variety where some of the propagators are put on shell. These are naturally associated to Landau singularities of the first type. Focusing on the one-loop case, we give an explicit parametrization to compute such cut integrals, with which we study some of their properties and list explicit results for maximal and next-to-maximal cuts. By analyzing homology groups, we show that cut integrals associated to Landau singularities of the second type are specific combinations of the usual cut integrals, and we obtain linear relations among different cuts of the same integral. We also show that all one-loop Feynman integrals and their cuts belong to the same class of functions, which can be written as parametric integrals.

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arXiv2017

The algebraic structure of cut Feynman integrals and the diagrammatic coaction

We study the algebraic and analytic structure of Feynman integrals by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour. This operation is a coaction. It reduces to the known coaction on multiple polylogarithms, but applies more generally, e.g. to hypergeometric functions. The coaction also applies to generic one-loop Feynman integrals with any configuration of internal and external masses, and in dimensional regularization. In this case, we demonstrate that it can be given a diagrammatic representation purely in terms of operations on graphs, namely contractions and cuts of edges. The coaction gives direct access to (iterated) discontinuities of Feynman integrals and facilitates a straightforward derivation of the differential equations they admit. In particular, the differential equations for any one-loop integral are determined by the diagrammatic coaction using limited information about their maximal, next-to-maximal, and next-to-next-to-maximal cuts.

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arXiv2015

Cuts and coproducts of massive triangle diagrams

Relations between multiple unitarity cuts and coproducts of Feynman integrals are extended to allow for internal masses. These masses introduce new branch cuts, whose discontinuities can be derived by placing single propagators on shell and identified as particular entries of the coproduct. First entries of the coproduct are then seen to include mass invariants alone, as well as threshold corrections for external momentum channels. As in the massless case, the original integral can possibly be recovered from its cuts by starting with the known part of the coproduct and imposing integrability contraints. We formulate precise rules for cuts of diagrams, and we gather evidence for the relations to coproducts through a detailed study of one-loop triangle integrals with various combinations of external and internal masses.

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