arXiv subjects

Pierpaolo Mastrolia

Publications and source records attributed to Pierpaolo Mastrolia.

At least 55 records · Page 3Linked to original sources

arXiv2016

Adaptive Integrand Decomposition

We present a simplified variant of the integrand reduction algorithm for multiloop scattering amplitudes in $d = 4 - 2ε$ dimensions, which exploits the decomposition of the integration momenta in parallel and orthogonal subspaces, $d=d_\parallel+d_\perp$, where $d_\parallel$ is the dimension of the space spanned by the legs of the diagrams. We discuss the advantages of a lighter polynomial division algorithm and how the orthogonality relations for Gegenbauer polynomilas can be suitably used for carrying out the integration of the irreducible monomials, which eliminates spurious integrals. Applications to one- and two-loop integrals, for arbitrary kinematics, are discussed.

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arXiv2016

Adaptive Integrand Decomposition in parallel and orthogonal space

We present the integrand decomposition of multiloop scattering amplitudes in parallel and orthogonal space-time dimensions, $d=d_\parallel+d_\perp$, being $d_\parallel$ the dimension of the parallel space spanned by the legs of the diagrams. When the number $n$ of external legs is $n\le 4$, the corresponding representation of the multiloop integrals exposes a subset of integration variables which can be easily integrated away by means of Gegenbauer polynomials orthogonality condition. By decomposing the integration momenta along parallel and orthogonal directions, the polynomial division algorithm is drastically simplified. Moreover, the orthogonality conditions of Gegenbauer polynomials can be suitably applied to integrate the decomposed integrand, yielding the systematic annihilation of spurious terms. Consequently, multiloop amplitudes are expressed in terms of integrals corresponding to irreducible scalar products of loop momenta and external momenta. We revisit the one-loop decomposition, which turns out to be controlled by the maximum-cut theorem in different dimensions, and we discuss the integrand reduction of two-loop planar and non-planar integrals up to $n=8$ legs, for arbitrary external and internal kinematics. The proposed algorithm extends to all orders in perturbation theory.

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arXiv2016

Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering

We present the calculation of the master integrals needed for the two-loop QCDxEW corrections to $ q + \bar{q} \to l^- + l^+$ and $ q + \bar{q}' \to l^- + \overlineν \, , $ for massless external particles. We treat W and Z bosons as degenerate in mass. We identify three types of diagrams, according to the presence of massive internal lines: the no-mass type, the one-mass type, and the two-mass type, where all massive propagators, when occurring, contain the same mass value. We find a basis of 49 master integrals and evaluate them with the method of the differential equations. The Magnus exponential is employed to choose a set of master integrals that obeys a canonical system of differential equations. Boundary conditions are found either by matching the solutions onto simpler integrals in special kinematic configurations, or by requiring the regularity of the solution at pseudo-thresholds. The canonical master integrals are finally given as Taylor series around d=4 space-time dimensions, up to order four, with coefficients given in terms of iterated integrals, respectively up to weight four.

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arXiv2015

Off-shell Currents and Color-Kinematics Duality

We elaborate on the color-kinematics duality for off-shell diagrams in gauge theories coupled to matter, by investigating the scattering process $gg\to ss, q\bar q, gg$, and show that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a color-kinematics violating term due to the contributions of sub-graphs only. Such anomaly vanishes when the four particles connected by the Jacobi relation are on their mass shell with vanishing squared momenta, being either external or cut particles, where the validity of the color-kinematics duality is recovered. We discuss the role of the off-shell decomposition in the direct construction of higher-multiplicity numerators satisfying color-kinematics identity in four as well as in $d$ dimensions, for the latter employing the Four Dimensional Formalism variant of the Four Dimensional Helicity scheme. We provide explicit examples for the QCD process $gg\to q\bar{q}g$.

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arXiv2015

QCD Amplitudes: new perspectives on Feynman integral calculus

I analyze the algebraic patterns underlying the structure of scattering amplitudes in quantum field theory. I focus on the decomposition of amplitudes in terms of independent functions and the systems of differential equations the latter obey. In particular, I discuss the key role played by unitarity for the decomposition in terms of master integrals, by means of generalized cuts and integrand reduction, as well as for solving the corresponding differential equations, by means of Magnus exponential series.

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arXiv2014

Multi-leg One-loop Massive Amplitudes from Integrand Reduction via Laurent Expansion

We present the application of a novel reduction technique for one-loop scattering amplitudes based on the combination of the integrand reduction and Laurent expansion. We describe the general features of its implementation in the computer code NINJA, and its interface to GoSam. We apply the new reduction to a series of selected processes involving massive particles, from six to eight legs.

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arXiv2014

NLO QCD corrections to Higgs boson production in association with a top quark pair and a jet

We present the calculation of the cross section for Higgs boson production in association with a top quark pair plus one jet, at next-to-leading-order (NLO) accuracy in QCD. All mass dependence is retained without recurring to any approximation. After including the complete NLO QCD corrections, we observe a strong reduction in the scale dependence of the result. We also show distributions for the invariant mass of the top quark pair, with and without the additional jet, and for the transverse momentum and the pseudorapidity of the Higgs boson. Results for the virtual contributions are obtained with a novel reduction approach based on integrand decomposition via Laurent expansion, as implemented in the library Ninja. Cross sections and differential distributions are obtained with an automated setup which combines the GoSam and Sherpa frameworks.

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arXiv2014

On the Subleading-Soft Behaviour of QCD Amplitudes

We elaborate on the radiative behaviour of tree-level scattering amplitudes in the soft regime. We show that the sub-leading soft term in single-gluon emission of quark-gluon amplitudes in QCD is controlled by differential operators, whose universal form can be derived from both on-shell recursion relation and gauge invariance, as it was shown to hold for graviton- and gluon-scattering.

hep-th↗

arXiv2014

Three-loop master integrals for ladder-box diagrams with one massive leg

The three-loop master integrals for ladder-box diagrams with one massive leg are computed from an eighty-five by eighty-five system of differential equations, solved by means of Magnus exponential. The results of the considered box-type integrals, as well as of the tower of vertex- and bubble-type master integrals associated to subtopologies, are given as a Taylor series expansion in the dimensional regulator parameter epsilon = (4-d)/2. The coefficients of the series are expressed in terms of uniform weight combinations of multiple polylogarithms and transcendental constants up to weight six. The considered integrals enter the next-to-next-to-next-to-leading order virtual corrections to scattering processes like the three-jet production mediated by vector boson decay, V* -> jjj, as well as the Higgs plus one-jet production in gluon fusion, pp -> Hj.

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arXiv2014

On the Four-Dimensional Formulation of Dimensionally Regulated Amplitudes

We propose a pure four-dimensional formulation (FDF) of the d-dimensional regularization of one-loop scattering amplitudes. In our formulation particles propagating inside the loop are represented by massive internal states regulating the divergences. The latter obey Feynman rules containing multiplicative selection rules which automatically account for the effects of the extra-dimensional regulating terms of the amplitude. The equivalence between the FDF and the Four Dimensional Helicity scheme is discussed. We present explicit representations of the polarization and helicity states of the four-dimensional particles propagating in the loop. They allow for a complete, four-dimensional, unitarity-based construction of d-dimensional amplitudes. Generalized unitarity within the FDF does not require any higher-dimensional extension of the Clifford and the spinor algebra. Finally we show how the FDF allows for the recursive construction of $d$-dimensional one-loop integrands, generalizing the four-dimensional open-loop approach.

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arXiv2014

Magnus and Dyson Series for Master Integrals

We elaborate on the method of differential equations for evaluating Feynman integrals. We focus on systems of equations for master integrals having a linear dependence on the dimensional parameter. For these systems we identify the criteria to bring them in a canonical form, recently identified by Henn, where the dependence of the dimensional parameter is disentangled from the kinematics. The determination of the transformation and the computation of the solution are obtained by using Magnus and Dyson series expansion. We apply the method to planar and non-planar two-loop QED vertex diagrams for massive fermions, and to non-planar two-loop integrals contributing to 2 -> 2 scattering of massless particles. The extension to systems which are polynomial in the dimensional parameter is discussed as well.

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arXiv2013

Multi-loop Integrand Reduction via Multivariate Polynomial Division

We present recent developments on the topic of the integrand reduction of scattering amplitudes. Integrand-level methods allow to express an amplitude as a linear combination of Master Integrals, by performing operations on the corresponding integrands. This approach has already been successfully applied and automated at one loop, and recently extended to higher loops. We describe a coherent framework based on simple concepts of algebraic geometry, such as multivariate polynomial division, which can be used in order to obtain the integrand decomposition of any amplitude at any loop order. In the one-loop case, we discuss an improved reduction algorithm, based on the application of the Laurent series expansion to the integrands, which has been implemented in the semi-numerical library Ninja. At two loops, we present the reduction of five-point amplitudes in N=4 SYM, with a unitarity-based construction of the integrand. We also describe the multi-loop divide-and-conquer approach, which can always be used to find the integrand decomposition of any Feynman graph, regardless of the form and the complexity of the integrand, with purely algebraic operations.

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arXiv2013

Integrand Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division

We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multiparticle cut, whose knowledge is a mandatory prerequisite for applying the integrand-reduction procedure. By using the division modulo Groebner basis, we can derive a simple integrand recurrence relation that generates the multiparticle pole decomposition for integrands of arbitrary multiloop amplitudes. We apply the new reduction algorithm to the two-loop planar and nonplanar diagrams contributing to the five-point scattering amplitudes in N=4 super Yang-Mills and N=8 supergravity in four dimensions, whose numerator functions contain up to rank-two terms in the integration momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and subtopologies. We obtain the integral basis for the decomposition of each diagram from the polynomial form of the residues. Our approach is well suited for a seminumerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of the integrand-reduction method to all orders in perturbation theory.

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arXiv2012

Integrand reduction of one-loop scattering amplitudes through Laurent series expansion

We present a semi-analytic method for the integrand reduction of one-loop amplitudes, based on the systematic application of the Laurent expansions to the integrand-decomposition. In the asymptotic limit, the coefficients of the master integrals are the solutions of a diagonal system of equations, properly corrected by counterterms whose parametric form is konwn a priori. The Laurent expansion of the integrand is implemented through polynomial division. The extension of the integrand-reduction to the case of numerators with rank larger than the number of propagators is discussed as well.

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arXiv2012

Scattering Amplitudes from Multivariate Polynomial Division

We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the Weak Nullstellensatz Theorem and on the division modulo the Groebner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrised by a number of coefficients equal to the number of solutions of the cut itself.

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arXiv2012

Automated One-Loop Calculations with GoSam

We present the program package GoSam which is designed for the automated calculation of one-loop amplitudes for multi-particle processes in renormalisable quantum field theories. The amplitudes, which are generated in terms of Feynman diagrams, can be reduced using either D-dimensional integrand-level decomposition or tensor reduction. GoSam can be used to calculate one-loop QCD and/or electroweak corrections to Standard Model processes and offers the flexibility to link model files for theories Beyond the Standard Model. A standard interface to programs calculating real radiation is also implemented. We demonstrate the flexibility of the program by presenting examples of processes with up to six external legs attached to the loop.

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arXiv2011

Automation of One-Loop Calculations with GoSam: Present Status and Future Outlook

In this presentation, we describe the GoSam (Golem/Samurai) framework for the automated computation of multi-particle scattering amplitudes at the one-loop level. The amplitudes are generated analytically in terms of Feynman diagrams, and can be evaluated using either D-dimensional integrand reduction or tensor decomposition. GoSam can be used to compute one-loop corrections to Standard Model (QCD and EW) processes, and it is ready to link generic model files for theories Beyond SM. We show the main features of GoSam through its application to several examples of different complexity.

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arXiv2011

On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes

We propose a first implementation of the integrand-reduction method for two-loop scattering amplitudes. We show that the residues of the amplitudes on multi-particle cuts are polynomials in the irreducible scalar products involving the loop momenta, and that the reduction of the amplitudes in terms of master integrals can be realized through polynomial fitting of the integrand, without any apriori knowledge of the integral basis. We discuss how the polynomial shapes of the residues determine the basis of master integrals appearing in the final result. We present a four-dimensional constructive algorithm that we apply to planar and non-planar contributions to the 4- and 5-point MHV amplitudes in N=4 SYM. The technique hereby discussed extends the well-established analogous method holding for one-loop amplitudes, and can be considered a preliminary study towards the systematic reduction at the integrand-level of two-loop amplitudes in any gauge theory, suitable for their automated semianalytic evaluation.

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