Multiloop integrals in dimensional regularization made simple.

By J. Henn. Physical Review Letters, 2013

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Scattering amplitudes at loop level can be expressed in terms of Feynman integrals. The latter satisfy partial differential equations in the kinematical variables. We argue that a good choice of basis for (multi)loop integrals can lead to significant simplifications of the differential equations, and propose criteria for finding an optimal basis. This builds on experience obtained in supersymmetric field theories that can be applied successfully to generic quantum field theory integrals. It involves studying leading singularities and explicit integral representations. When the differential equations are cast into canonical form, their solution becomes elementary. The class of functions involved is easily identified, and the solution can be written down to any desired order in ϵ within dimensional regularization. Results obtained in this way are particularly simple and compact. In this Letter, we outline the general ideas of the method and apply them to a two-loop example.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/538f3b098023eac68f040ffb99c776cfd86989f4

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Citations in the corpus

High precision determination of the gluon fusion Higgs boson cross-section at the LHC (2016, background): “[74] J.”
Integral reduction with Kira 2.0 and finite field methods (2020, background): “# A master equation with 3 elements box[1,1,1,1]*(1) box[1,1,1,2]*(s) box[1,2,1,1]*(s)”
Cuts of Feynman Integrals in Baikov representation (2017, background): “We gratefully acknowledge fruitful discussions with P. A. Baikov, V. A. Smirnov, M. Czakon, R. N. Lee, E. Panzer, C. Anastasiou, K. J. Larsen, J. Henn and C. Duhr during various stages of this project.”
Feynman integrals and intersection theory (2018, methodology): “The resolvent matrix is employed to change the basis of MIs, and to define a special set of basic integrals that obey canonical systems of differential equations [8, 55], for which the differential equation matrix has a simple D-dependent term, factored out of the kinematics.”
On the maximal cut of Feynman integrals and the solution of their differential equations (2016, background): “The choice of master integrals is, of course, arbitrary and more recently it was shown that by properly selecting the basis of integrals one can simplify the form of the differential equations substantially, bringing them to a so-called canonical form [12].”
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph (2017, methodology): “With this, we can therefore imagine a way to generalize the idea originally presented in [7], where it was claimed that master integrals with unit leading singularity are expected to fulfil canonical differential equations.”
Higgs boson production at hadron colliders at N3LO in QCD (2018, background): “The nominally smallest corrections for each channel can be found in an interval of μ ∈ [40, 90] GeV.”
Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report (2017, methodology): “1 Loop integrals Most of the new analytic results for two-loop integrals and beyond have been calculated employing the differential equations technique [5, 6], which got a significant boost through Henn’s canonical form [7].”
pySecDec: A toolbox for the numerical evaluation of multi-scale integrals (2017, background): “loop integral import * >>> internal lines = [[’0’,[1,2]], [’m’,[2,3]], [’m’,[3,1]]] >>> external lines = [[’p1’,1],[’p2’,2],[’-p12’,3]] >>> li = LoopIntegralFromGraph(internal lines, external lines)”
Lectures on differential equations for Feynman integrals (2014, methodology): “As an example, we fully explain the basis choice made in [9] using these concepts.”
Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series (2018, background): “re) assigns a matrix of symbols not just to pure objects, but more generally to unipotent objects. This may have implications when thinking about canonical diﬀerential equations for Feynman integrals [83], because unipotency seems to be a more general concept than purity. 5 Elliptic symbol calculus ”
Decomposition of Feynman integrals on the maximal cut by intersection numbers (2019, methodology): “Together with novel algorithms for simplifying the solution of systems of differential equations [9], which triggered further studies [33–38], the calculations of several multi-scale, multi-loop, multi-particle amplitudes became feasible [39–57].”
Reducing differential equations for multiloop master integrals (2014, background): “[9] two guiding principles have been suggested.”