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A Geometrical Framework for Amplitude Recursions: Bridging Between Trees and Loops

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Anti-Differentiation and the Calculation of Feynman Amplitudes

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Abstract

Various methods for the recursive evaluation of scattering amplitudes in quantum field theory and string theory have been put forward during the last couple of years. In these proceedings we describe a geometrical framework, which is believed to be capable of treating many of these recursions in a unified way. Our recursive framework is based on manipulating iterated integrals on Riemann surfaces with boundaries. A geometric parameter appears as variable of a differential equation of KZ or KZB type. The parameter interpolates between two associated regularized boundary values, which contain iterated integrals closely related to scattering amplitudes defined on two different geometries.

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Notes

  1. 1.

    The polynomials \(\mathcal {N}\) and \(\mathcal {D}\) are very closely related to the Symanzik polynomials.

  2. 2.

    For simplicity, we consider real integration paths here exclusively. More general quantities a, for example complex functions of complex parameters, will lead to the hyperlogarithms discussed in Erik Panzer’s talk.

  3. 3.

    We use the notation \(\prod _{x_a\leq x_i<x_j\leq x_b}=\prod _{i,j\in \{1,2,\dots ,L\}:\,x_a\leq x_i<x_j\leq x_b}\).

  4. 4.

    The limit 𝜖 → 0 is assumed to be taken within the unit interval.

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Acknowledgements

Both authors would like to thank the Kolleg Mathematik und Physik Berlin for supporting the conference “Antidifferentiation and the Calculation of Feynman Amplitudes”. AK would like to thank the IMPRS for Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory, of which he is a member and which renders his studies possible.

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Correspondence to Johannes Broedel .

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Broedel, J., Kaderli, A. (2021). A Geometrical Framework for Amplitude Recursions: Bridging Between Trees and Loops. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_6

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  • DOI: https://doi.org/10.1007/978-3-030-80219-6_6

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