1,720,971 research outputs found
Cutting Feynman Amplitudes: from Adaptive Integrand Decomposition to Differential Equations on Maximal Cut
Full text linkIn this thesis we discuss, within the framework of the Standard Model (SM) of particle physics, advanced methods for the computation of scattering amplitudes at higher-order in perturbation theory. We offer a new insight into the role played by the unitarity of scattering amplitudes in the theoretical understanding and in the computational simplification of multi-loop calculations, at both the algebraic and the analytical level.
On the algebraic side, generalized unitarity can be used, within the integrand reduction method, to express the integrand associated to a multi-loop amplitude as a sum of fundamental, irreducible contributions, yielding to a decomposition of the amplitude as a linear combination of master integrals. In this framework, we propose an adaptive formulation of the integrand decomposition algorithm, which systematically adjusts to the kinematics of the individual integrands the dimensionality of the momentum space, where unitarity cuts are performed. This new formulation makes the integrand decomposition method, which in the past played a key role in streamlining one-loop computations, an efficient tool also at multi-loop level. We provide evidence of the generality of the proposed method by determining a universal parametrization
of the integrand basis for two-loop amplitudes in arbitrary kinematics and we illustrate its technical feasibility in the first automated implementation of the analytic integrand decomposition at one- and two-loop level.
On the analytic side, we discuss the role of maximal-unitarity for the solution of differential equations for dimensionally regulated Feynman integrals. The determination of the analytic expression of the master integrals as a Laurent expansion in the dimensional regulating
parameter requires the knowledge of the solutions of the homogeneous part of their differential equations at d=4. In all cases where Feynman integrals fulfil genuine first-order differential equations with a linear dependence on d, the corresponding homogeneous solutions can be determined through the Magnus exponential expansion.
In this work we apply the latter to two-loop corrections to several SM processes such as the Higgs decay to weak vector bosons, H → WW, triple gauge couplings ZWW and γ∗WW and to the elastic scattering μe → μe in quantum electrodynamics.
In some cases, the inadequacy of the Magnus method hints at the presence of master integrals that obey higher-order differential equations, for which no general theory exists. In this thesis we show that maximal-cuts of Feynman integrals solve, by construction, such homogeneous equations regardless of their order and complexity. Hence, whenever a Feynman integral obeys an irreducible higher-order differential equation, the computation of its maximal-cut along independent contours provides a closed integral representation of the full set of independent homogeneous solutions. We apply this strategy to the two-loop elliptic integrals that appear in heavy-quark mediated corrections to gg → gg and gg → gH as well as to the three-loop massive banana graph, which constitute the first example of Feynman integral that obeys a third-order differential equation.
In the light of the results presented in this thesis, generalized unitarity emerges as a powerful tool not only for handling the algebraic complexity of perturbative calculations but also for investigating the nature of new classes of mathematical functions encountered in particle physics
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
Full text linkWe consider the calculation of the master integrals of the three-loop massive banana graph. In the case of equal internal masses, the graph is reduced to three master integrals which satisfy an irreducible system of three coupled linear differential equations. The solution of the system requires finding a 3×3 matrix of homogeneous solutions. We show how the maximal cut can be used to determine all entries of this matrix in terms of products of elliptic integrals of first and second kind of suitable arguments. All independent solutions are found by performing the integration which defines the maximal cut on different contours. Once the homogeneous solution is known, the inhomogeneous solution can be obtained by use of Euler's variation of constants
On the maximal cut of Feynman integrals and the solution of their differential equations
Full text linkThe standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in ε=(4−d)/2 , where d are the space–time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations
BCJ identities and d-dimensional generalized unitarity
Full text linkWe present a set of relations between one-loop integral coefficients for dimensionally regulated QCD amplitudes. Within dimensional regularization, the combined use of color-kinematics duality and integrand reduction yields the existence of relations between the integrand residues of partial amplitudes with different orderings of the external particles. These relations can be established for the cut-constructible contributions as well for the ones responsible for rational terms. Starting from the general parametrization of one-loop residues and applying Laurent expansion in order to extract the coefficients of the amplitude decomposition in terms of master integrals, we show that the full set of relations can be obtained by considering the BCJ identities between d-dimensional tree-level amplitudes. We provide explicit examples for multi-gluon scattering amplitudes at one loop
Master integrals for the NNLO virtual corrections to scattering in QED: the planar graphs
Full text linkWe evaluate the master integrals for the two-loop, planar box-diagrams
contributing to the elastic scattering of muons and electrons at
next-to-next-to leading-order in QED. We adopt the method of differential
equations and the Magnus exponential series to determine a canonical set of
integrals, finally expressed as a Taylor series around four space-time
dimensions, with coefficients written as combination of generalised
polylogarithms. The electron is treated as massless, while we retain full
dependence on the muon mass. The considered integrals are also relevant for
crossing-related processes, such as di-muon production at -colliders,
as well as for the QCD corrections to -pair production at hadron
colliders.Comment: article unchanged; updated ancillary file <arXiv-fam1.m
Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a pair and to the triple gauge couplings and
Full text linkWe compute the two-loop master integrals required for the leading QCD corrections to the interaction vertex of a massive neutral boson X, e.g. H, Z or , with a pair of bosons, mediated by a SU(2) quark doublet composed of one massive and one massless flavor. All the external legs are allowed to have arbitrary invariant masses. The Magnus exponential is employed to identify a set of master integrals that, around d = 4 space-time dimensions, obey a canonical system of differential equations. The canonical master integrals are given as a Taylor series in ϵ = (4 − d)/2, up to order four, with coefficients written as combination of Goncharov polylogarithms, respectively up to weight four. In the context of the Standard Model, our results are relevant for the mixed EW-QCD corrections to the Higgs decay to a W pair, as well as to the production channels obtained by crossing, and to the triple gauge boson vertices and
Adaptive Integrand Decomposition
Full text linkWe present a simplified variant of the integrand reduction algorithm for multiloop scattering amplitudes in dimensions, which exploits the decomposition of the integration momenta in parallel and orthogonal subspaces, , where 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, ar
Off-shell currents and color–kinematics duality
No full textAbstractWe elaborate on the color–kinematics duality for off-shell diagrams in gauge theories coupled to matter, by investigating the scattering process gg→ss,qq¯,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→qq¯g
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