Analytic treatment of the two loop equal mass sunrise graph

By S. Laporta, Ettore Remiddi. 2004

238 citations recorded by the source corpus.

Abstract The two loop equal mass sunrise graph is considered in the continuous d -dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d = 2 and d = 4 , the second order differential equation for the scalar master integral is expanded in ( d − 2 ) and solved by the variation of the constants method of Euler up to first order in ( d − 2 ) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/daba66012d8c74265007ff2b76dab3cac54cfb5a

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In Astronomy and Astrophysics

In Nuclear and High Energy Physics

Citations in the corpus

On the maximal cut of Feynman integrals and the solution of their differential equations (2016, methodology): “In [21], for example, the second order differential equation satisfied by the two-loop massive sunrise graph was solved by inferring the homogeneous solution from the calculation of the imaginary part of the graph.”
Higgs boson production at hadron colliders at N3LO in QCD (2018, methodology): “[57] we define two integrals I1 = ∫ r3 r2 dx 1”
The elliptic dilogarithm for the sunset graph (2013, methodology): “This differential equation has been derived in [16] using the properties of the Feynman integrals or in [17] using cohomological methods.”
Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral (2016, methodology): “where the last integral can be performed by standard techniques using the integral representation for J (0,1) 1 (u), see for example [13].”
The two-loop sunrise graph with arbitrary masses (2013, background): “Setting m1 = m2 = m3 = m it has been shown by Laporta and Remiddi [9] that the system of four coupled first-order differential equations reduces to a single secondorder differential equation.”
Decomposition of Feynman integrals on the maximal cut by intersection numbers (2019, background): “The number of MIs for a given integral family, the order of the differential equations they obey, and the classification of the homogeneous solutions according to the independent components of the integration domain, revealed a natural correspondence between the MIs and the geometric properties of t”