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  • Letter
  • Open Access

Algorithm for symbol integrations for loop integrals

Song He (何颂)1,2,3,* and Yichao Tang (唐一朝)1,4,†

  • 1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 2School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, Hangzhou 310024, China and International Centre for Theoretical Physics Asia-Pacific, Beijing 100190, China
  • 3Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, People’s Republic of China
  • 4School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China

  • *songhe@itp.ac.cn
  • †tangyichao@itp.ac.cn
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Phys. Rev. D 108, L041702 – Published 14 August, 2023

DOI: https://doi.org/10.1103/PhysRevD.108.L041702

Abstract

We derive an algorithm for computing the total differentials of multiloop integrals expressed as onefold integrals of multiple polylogarithms, which can involve square roots of polynomials up to degree 4 and may evaluate to (elliptic) multiple polylogarithms [(e)MPLs]. This gives simple algebraic rules for computing the (W−1, 1) coproduct of the resulting weight-W functions up to period terms, and iterating it gives the symbol without actually performing any integration. In particular, our algorithm generalizes existing MPL integration rules and sidesteps the complicated rationalization procedure in the presence of square roots. We apply our algorithm to conformal double-D-gon integrals in D dimensions with generic kinematics and possibly massive circumferential propagators. We directly compute, for the first time, the total differential and symbol (up to period terms) of the D=3 double triangle and the D=4 double box, which in the special case with massless propagators represent the first appearance of eMPL functions in (two-loop) scattering amplitudes of Aharony-Bergman-Jafferis-Maldacena theory and N=4 super-Yang-Mills theory, respectively.

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Corrections

26 January, 2024

Correction: The last sentence of the Acknowledgments section contained errors and has been fixed (the grant numbers have been reordered, two were deleted, and a new one was added). In addition, a new funding statement has been added.

Article Text

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Outline

Information

Phys. Rev. D 108, L041702– Published 14 August, 2023
  • Received 19 June 2023
  • Accepted 4 July 2023
  • Corrected 26 January 2024
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