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  • Open Access

Elliptic Leading Singularities and Canonical Integrands

E. Chaubey1,* and V. Sotnikov2,†

  • *Contact author: eekta@uni-bonn.de
  • †Contact author: vasily.sotnikov@physik.uzh.ch
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Phys. Rev. Lett. 135, 101903 – Published 4 September, 2025

DOI: https://doi.org/10.1103/4fjc-lfnx

Abstract

In the well-studied genus zero case, bases of d log integrands with integer leading singularities define Feynman integrals that automatically satisfy differential equations in canonical form. Such integrand bases can be constructed without input from the differential equations and without explicit involvement of dimensional regularization parameter ε. We propose a generalization of this construction to genus one geometry arising from the appearance of elliptic curves. We argue that a particular choice of algebraic 1-forms of the second kind that avoids derivatives is crucial. We observe that the corresponding Feynman integrals satisfy a special form of differential equations that has not been previously reported, and that their solutions order by order in ε yield pure functions. We conjecture that our integrand-level construction universally leads to such differential equations.

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References (95)

  1. A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254, 158 (1991).
  2. A. V. Kotikov, Differential equation method: The calculation of N point Feynman diagrams, Phys. Lett. B 267, 123 (1991); 295, 409(E) (1992).
  3. Z. Bern, L. J. Dixon, and D. A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B412, 751 (1994).
  4. E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cimento A 110, 1435 (1997).
  5. T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B580, 485 (2000).
  6. F. V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. 100B, 65 (1981).
  7. K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate beta functions in 4 loops, Nucl. Phys. B192, 159 (1981).
  8. S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15, 5087 (2000).
  9. A. V. Smirnov and A. V. Petukhov, The number of master integrals is finite, Lett. Math. Phys. 97, 37 (2011).
  10. J. M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110, 251601 (2013).
  11. A. B. Goncharov, M. Spradlin, C. Vergu, and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105, 151605 (2010).
  12. C. Duhr, Hopf algebras, coproducts and symbols: An application to Higgs boson amplitudes, J. High Energy Phys. 08 (2012) 043.
  13. D. Chicherin and V. Sotnikov, Pentagon functions for scattering of five massless particles, J. High Energy Phys. 12 (2020) 167.
  14. R. N. Lee, Reducing differential equations for multiloop master integrals, J. High Energy Phys. 04 (2015) 108.
  15. M. Prausa, epsilon: A tool to find a canonical basis of master integrals, Comput. Phys. Commun. 219, 361 (2017).
  16. S. Abreu, B. Page, and M. Zeng, Differential equations from unitarity cuts: Nonplanar hexa-box integrals, J. High Energy Phys. 01 (2019) 006.
  17. C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun. 222, 295 (2018).
  18. R. N. Lee, Libra: A package for transformation of differential systems for multiloop integrals, Comput. Phys. Commun. 267, 108058 (2021).
  19. C. Dlapa, J. Henn, and K. Yan, Deriving canonical differential equations for Feynman integrals from a single uniform weight integral, J. High Energy Phys. 05 (2020) 025.
  20. L. Adams, C. Bogner, and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. (N.Y.) 55, 102301 (2014).
  21. M. Søgaard and Y. Zhang, Elliptic functions and maximal unitarity, Phys. Rev. D 91, 081701(R) (2015).
  22. M. F. Paulos, M. Spradlin, and A. Volovich, Mellin amplitudes for dual conformal integrals, J. High Energy Phys. 08 (2012) 072.
  23. S. Caron-Huot and K. J. Larsen, Uniqueness of two-loop master contours, J. High Energy Phys. 10 (2012) 026.
  24. L. Adams, C. Bogner, and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys. (N.Y.) 54, 052303 (2013).
  25. S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theory 148, 328 (2015).
  26. J. Broedel, C. R. Mafra, N. Matthes, and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, J. High Energy Phys. 07 (2015) 112.
  27. C. Bogner, A. Schweitzer, and S. Weinzierl, Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral, Nucl. Phys. B922, 528 (2017).
  28. J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C. G. Raab, C. S. Radu, and C. Schneider, Iterated elliptic and hypergeometric integrals for Feynman diagrams, J. Math. Phys. (N.Y.) 59, 062305 (2018).
  29. J. Broedel, C. Duhr, F. Dulat, and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: An application to the sunrise integral, Phys. Rev. D 97, 116009 (2018).
  30. C. Vergu and M. Volk, Traintrack Calabi-Yaus from twistor geometry, J. High Energy Phys. 07 (2020) 160.
  31. J. L. Bourjaily, N. Kalyanapuram, C. Langer, and K. Patatoukos, Prescriptive unitarity with elliptic leading singularities, Phys. Rev. D 104, 125009 (2021).
  32. C. Duhr, A. Klemm, F. Loebbert, C. Nega, and F. Porkert, Geometry from integrability: Multi-leg fishnet integrals in two dimensions, J. High Energy Phys. 07 (2024) 008.
  33. C. Duhr, F. Gasparotto, C. Nega, L. Tancredi, and S. Weinzierl, On the electron self-energy to three loops in QED, J. High Energy Phys. 11 (2024) 020.
  34. S. Badger, M. Becchetti, N. Giraudo, and S. Zoia, Two-loop integrals for tt¯+jet production at hadron colliders in the leading colour approximation, J. High Energy Phys. 07 (2024) 073.
  35. M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, Emergence of Calabi-Yau manifolds in high-precision black hole scattering, Nature (London) 641, 603 (2025).
  36. C. Duhr, F. Porkert, and S. F. Stawinski, Canonical differential equations beyond genus one, J. High Energy Phys. 02 (2025) 014.
  37. C. Duhr and S. Maggio, Feynman integrals, elliptic integrals and two-parameter K3 surfaces, J. High Energy Phys. 06 (2025) 250.
  38. J. Chen, L. L. Yang, and Y. Zhang, On an approach to canonicalizing elliptic Feynman integrals, arXiv:2503.23720.
  39. C. Duhr, A. Klemm, C. Nega, and L. Tancredi, The ice cone family and iterated integrals for Calabi-Yau varieties, J. High Energy Phys. 02 (2023) 228.
  40. M. Becchetti, C. Dlapa, and S. Zoia, Canonical differential equations for the elliptic two-loop five-point integral family relevant to tt¯+jet production at leading colour, arXiv:2503.03603.
  41. J. Broedel, C. Duhr, F. Dulat, B. Penante, and L. Tancredi, Elliptic symbol calculus: From elliptic polylogarithms to iterated integrals of Eisenstein series, J. High Energy Phys. 08 (2018) 014.
  42. F. Brown, Notes on motivic periods, arXiv:1512.06410.
  43. L. Adams, E. Chaubey, and S. Weinzierl, Simplifying differential equations for multiscale Feynman integrals beyond multiple polylogarithms, Phys. Rev. Lett. 118, 141602 (2017).
  44. L. Görges, C. Nega, L. Tancredi, and F. J. Wagner, On a procedure to derive ε-factorised differential equations beyond polylogarithms, J. High Energy Phys. 07 (2023) 206.
  45. P. Mastrolia and S. Mizera, Feynman integrals and intersection theory, J. High Energy Phys. 02 (2019) 139.
  46. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot, and J. Trnka, The all-loop integrand for scattering amplitudes in planar N=4 SYM, J. High Energy Phys. 01 (2011) 041.
  47. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, and J. Trnka, Local integrals for planar scattering amplitudes, J. High Energy Phys. 06 (2012) 125.
  48. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov, and J. Trnka, Grassmannian Geometry of Scattering Amplitudes (Cambridge University Press, Cambridge, England, 2016), 10.1017/CBO9781316091548.
  49. F. Cachazo, Sharpening the leading singularity, arXiv:0803.1988.
  50. J. L. Bourjaily, N. Kalyanapuram, C. Langer, K. Patatoukos, and M. Spradlin, Elliptic, Yangian-invariant “leading singularity,” Phys. Rev. Lett. 126, 201601 (2021).
  51. H. Frellesvig, On epsilon factorized differential equations for elliptic Feynman integrals, J. High Energy Phys. 03 (2022) 079.
  52. C. Duhr, S. Maggio, C. Nega, B. Sauer, L. Tancredi, and F. J. Wagner, Aspects of canonical differential equations for Calabi-Yau geometries and beyond, J. High Energy Phys. 06 (2025) 128.
  53. M. Becchetti, D. Canko, V. Chestnov, T. Peraro, M. Pozzoli, and S. Zoia, Two-loop Feynman integrals for leading colour tt¯W production at hadron colliders, J. High Energy Phys. 07 (2025) 001.
  54. J. Broedel, C. Duhr, F. Dulat, B. Penante, and L. Tancredi, Elliptic Feynman integrals and pure functions, J. High Energy Phys. 01 (2019) 023.
  55. J. Henn, B. Mistlberger, V. A. Smirnov, and P. Wasser, Constructing d-log integrands and computing master integrals for three-loop four-particle scattering, J. High Energy Phys. 04 (2020) 167.
  56. H. Frellesvig and S. Weinzierl, On ϵ-factorised bases and pure Feynman integrals, SciPost Phys. 16, 150 (2024).
  57. P. A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Methods Phys. Res., Sect. A 389, 347 (1997).
  58. H. Frellesvig and C. G. Papadopoulos, Cuts of Feynman Integrals in Baikov representation, J. High Energy Phys. 04 (2017) 083.
  59. J. Bosma, M. Sogaard, and Y. Zhang, Maximal cuts in arbitrary dimension, J. High Energy Phys. 08 (2017) 051.
  60. E. Herrmann and J. Parra-Martinez, Logarithmic forms and differential equations for Feynman integrals, J. High Energy Phys. 02 (2020) 099.
  61. J. L. Bourjaily, A. J. McLeod, M. Spradlin, M. von Hippel, and M. Wilhelm, Elliptic double-box integrals: Massless scattering amplitudes beyond polylogarithms, Phys. Rev. Lett. 120, 121603 (2018).
  62. S. Weinzierl, Feynman Integrals. A Comprehensive Treatment for Students and Researchers, UNITEXT for Physics (Springer, New York, 2022), 10.1007/978-3-030-99558-4.
  63. R. N. Lee, Presenting LiteRed: A tool for the Loop InTEgrals REDuction, arXiv:1212.2685.
  64. T. Peraro, FiniteFlow: Multivariate functional reconstruction using finite fields and dataflow graphs, J. High Energy Phys. 07 (2019) 031.
  65. K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83, 831 (1977).
  66. L. Adams, E. Chaubey, and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, J. High Energy Phys. 10 (2018) 206.
  67. O. V. Tarasov, Hypergeometric representation of the two-loop equal mass sunrise diagram, Phys. Lett. B 638, 195 (2006).
  68. M. Caffo, H. Czyz, S. Laporta, and E. Remiddi, The master differential equations for the two loop sunrise selfmass amplitudes, Nuovo Cimento A 111, 365 (1998), https://en.sif.it/journals/sif/nca/econtents/1998/111/04/article/3.
  69. L. Adams, C. Bogner, and S. Weinzierl, The sunrise integral around two and four space-time dimensions in terms of elliptic polylogarithms, Acta Phys. Pol. B 46, 2131 (2015).
  70. S. Müller-Stach, S. Weinzierl, and R. Zayadeh, A second-order differential equation for the two-loop sunrise graph with arbitrary masses, Commun. Num. Theor. Phys. 6, 203 (2012).
  71. C. Bogner, S. Müller-Stach, and S. Weinzierl, The unequal mass sunrise integral expressed through iterated integrals on M¯1,3, Nucl. Phys. B954, 114991 (2020).
  72. A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, J. High Energy Phys. 06 (2017) 127.
  73. M. Becchetti, R. Bonciani, L. Cieri, F. Coro, and F. Ripani, Two-loop form factors for diphoton production in quark annihilation channel with heavy quark mass dependence, J. High Energy Phys. 12 (2023) 105.
  74. T. Ahmed, E. Chaubey, M. Kaur, and S. Maggio, Two-loop non-planar four-point topology with massive internal loop, J. High Energy Phys. 05 (2024) 064.
  75. M. Becchetti, F. Coro, C. Nega, L. Tancredi, and F. J. Wagner, Analytic two-loop amplitudes for qq¯→γγ and gg→γγ mediated by a heavy-quark loop, J. High Energy Phys. 06 (2025) 033.
  76. T. Ahmed, A. Chakraborty, E. Chaubey, and M. Kaur, Two-loop helicity amplitudes for diphoton production with massive quark loop, arXiv:2502.03282.
  77. L. Adams, E. Chaubey, and S. Weinzierl, Planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularization parameter, Phys. Rev. Lett. 121, 142001 (2018).
  78. S. Badger, M. Becchetti, C. Brancaccio, H. B. Hartanto, and S. Zoia, Numerical evaluation of two-loop QCD helicity amplitudes for gg→tt¯g at leading colour, J. High Energy Phys. 03 (2025) 070.
  79. See Supplemental Material at http://link.aps.org/supplemental/10.1103/4fjc-lfnx for algebraic bases for all considered examples.
  80. L. Adams and S. Weinzierl, Feynman integrals and iterated integrals of modular forms, Commun. Num. Theor. Phys. 12, 193 (2018).
  81. M. Deneufchâtel, G. H. E. Duchamp, V. H. N. Minh, and A. I. Solomon, Independence of hyperlogarithms over function fields via algebraic combinatorics, arXiv:1101.4497.
  82. C. Duhr, F. Porkert, C. Semper, and S. F. Stawinski, Self-duality from twisted cohomology, J. High Energy Phys. 03 (2025) 053.
  83. S. Abreu, D. Chicherin, H. Ita, B. Page, V. Sotnikov, W. Tschernow, and S. Zoia, All two-loop Feynman integrals for five-point one-mass scattering, Phys. Rev. Lett. 132, 141601 (2024).
  84. T. Gehrmann, J. Henn, P. Jakubčík, J. Lim, C. C. Mella, N. Syrrakos, L. Tancredi, and W. J. Torres Bobadilla, Graded transcendental functions: An application to four-point amplitudes with one off-shell leg, J. High Energy Phys. 12 (2024) 215.
  85. G. De Laurentis, H. Ita, B. Page, and V. Sotnikov, Compact two-loop QCD corrections for Vjj production in proton collisions, J. High Energy Phys. 06 (2025) 093.
  86. J. Chen, X. Jiang, C. Ma, X. Xu, and L. L. Yang, Baikov representations, intersection theory, and canonical Feynman integrals, J. High Energy Phys. 07 (2022) 066.
  87. F. Moriello, Generalised power series expansions for the elliptic planar families of Higgs+jet production at two loops, J. High Energy Phys. 01 (2020) 150.
  88. M. Hidding, DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, Comput. Phys. Commun. 269, 108125 (2021).
  89. T. Armadillo, R. Bonciani, S. Devoto, N. Rana, and A. Vicini, Evaluation of Feynman integrals with arbitrary complex masses via series expansions, Comput. Phys. Commun. 282, 108545 (2023).
  90. R. M. Prisco, J. Ronca, and F. Tramontano, LINE: Loop integrals numerical evaluation, J. High Energy Phys. 07 (2025) 219.
  91. S. Caron-Huot and J. M. Henn, Iterative structure of finite loop integrals, J. High Energy Phys. 06 (2014) 114.
  92. D. Chicherin, V. Sotnikov, and S. Zoia, Pentagon functions for one-mass planar scattering amplitudes, J. High Energy Phys. 01 (2022) 096.
  93. S. Pögel, X. Wang, and S. Weinzierl, The three-loop equal-mass banana integral in ϵ-factorised form with meromorphic modular forms, J. High Energy Phys. 09 (2022) 062.
  94. J. Broedel, C. Duhr, F. Dulat, B. Penante, and L. Tancredi, Elliptic polylogarithms and Feynman parameter integrals, J. High Energy Phys. 05 (2019) 120.
  95. J. Broedel, C. Duhr, and N. Matthes, Meromorphic modular forms and the three-loop equal-mass banana integral, J. High Energy Phys. 02 (2022) 184.

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Phys. Rev. Lett. 135, 101903– Published 4 September, 2025
  • Received 20 May 2025
  • Revised 7 July 2025
  • Accepted 11 August 2025
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