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Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles

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Abstract

We observe a property of orthogonality of the Mellin–Barnes transformation of triangle one-loop diagrams, which follows from our previous papers (Kondrashuk and Kotikov in JHEP 0808:106, 2008; Kondrashuk and Vergara in JHEP 1003:051, 2010; Allendes et al. in J Math Phys 51:052304, 2010). In those papers it has been established that Usyukina–Davydychev functions are invariant with respect to the Fourier transformation. This has been proved at the level of graphs and also via the Mellin–Barnes transformation. We partially apply to the one-loop massless scalar diagram the same trick in which the Mellin–Barnes transformation was involved and obtain the property of orthogonality of the corresponding MB transforms under integration over contours in two complex planes with certain weight. This property is valid in an arbitrary number of dimensions.

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Notes

  1. We omit the factor \(1/2\pi i\) in front of each contour integral in the complex plane.

  2. We do not know any explicit reference to this proof of the star-triangle relation via the MB transformation. I.K. explained this proof via MB transformation in his lectures on QFT at UdeC, Chile in May of 2009.

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Acknowledgements

The work of I.K. was supported in part by Fondecyt (Chile) Grants Nos. 1040368, 1050512 and 1121030, by DIUBB (Chile) Grant Nos. 125009, GI 153209/C and GI 152606/VC. Also, the work of I.K. is supported by Universidad del Bío-Bío and Ministerio de Educacion (Chile) within Project No. MECESUP UBB0704-PD018. He is grateful to the Physics Faculty of Bielefeld University for accepting him as a visiting scientist and for the kind hospitality and the excellent working conditions during his stay in Bielefeld. E.A.N.C. work was partially supported by project DIULS PR15151, Universidad de La Serena. The work of I.P.F. was supported in part by Fondecyt (Chile) Grant No. 1121030 and by Beca Conicyt (Chile) via Doctoral fellowship CONICYT-DAAD/BECAS Chile, 2016/91609937. I.P.F. is also supported by German Science Foundation DFG Grant CRC1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications”. This paper is based on the talk of I. K. at ICAMI 2017, San Andres, Colombia, November 27–December 1, 2017, and he is grateful to ICAMI organizers for inviting him. The financial support of I.K. participation in ICAMI 2017 has been provided by DIUBB via Fapei funding.

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Authors and Affiliations

  1. Instituto de Física y Astronomía, Universidad de Valparaíso, Av. Gran Bretaña 1111, Valparaiso, Chile

    Ivan Gonzalez

  2. Grupo de Matemática Aplicada and Grupo de Física de Altas Energías, Departamento de Ciencias Básicas, Universidad del Bío-Bío, Campus Fernando May, Av. Andres Bello 720, Casilla 447, Chillán, Chile

    Igor Kondrashuk

  3. Departamento de Matemáticas, Facultad de Ciencias, Universidad de La Serena, Av. Cisternas 1200, La Serena, Chile

    Eduardo A. Notte-Cuello

  4. Fakultät für Physik, Universität Bielefeld, Universitätsstraße 25, 33615, Bielefeld, Germany

    Ivan Parra-Ferrada

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  1. Ivan Gonzalez
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  4. Ivan Parra-Ferrada
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Correspondence to Igor Kondrashuk.

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Dedicated to the memory of Sasha Vasil’ev.

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Gonzalez, I., Kondrashuk, I., Notte-Cuello, E.A. et al. Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles. Anal.Math.Phys. 8, 589–602 (2018). https://doi.org/10.1007/s13324-018-0252-6

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