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  • Open Access

Tropical regions of near mass-shell pentabox

A. V. Belitsky and V. A. Smirnov1,2

  • 1Department of Physics, Arizona State University, Tempe, Arizona 85287-1504, USA
  • 2Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia and Moscow Center for Fundamental and Applied Mathematics, 119992 Moscow, Russia

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Phys. Rev. D 113, 045019 – Published 23 February, 2026

DOI: https://doi.org/10.1103/hn9g-ccd8

Abstract

Coulomb branch amplitudes of maximally supersymmetric Yang-Mills theory display infrared properties different from their conformal counterparts. While the four-leg amplitude is known to very high perturbative orders, amplitudes of higher multiplicity fall into an uncharted territory starting already from two loops. The reason for this is that they are not easily amenable to traditional techniques like canonical differential equations due to the uncontrolled swelling of solutions to integration-by-parts identities. In this paper, we break the barrier for the five-leg amplitude using a technique based on the analysis of Newton polytopes corresponding to Feynman/Schwinger integrands and their tropical geometry. Specifically, we analytically evaluate the near mass-shell limit of the off-shell pentabox in terms of Goncharov polylogarithms.

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References (80)

  1. K. G. Selivanov, An infinite set of tree amplitudes in Higgs-Yang-Mills, Phys. Lett. B 460, 116 (1999).
  2. L. F. Alday, J. M. Henn, J. Plefka, and T. Schuster, Scattering into the fifth dimension of N=4 super Yang-Mills, J. High Energy Phys. 01 (2010) 077.
  3. R. H. Boels, No triangles on the moduli space of maximally supersymmetric gauge theory, J. High Energy Phys. 05 (2010) 046.
  4. N. Craig, H. Elvang, M. Kiermaier, and T. Slatyer, Massive amplitudes on the Coulomb branch of N=4 SYM, J. High Energy Phys. 12 (2011) 097.
  5. S. Caron-Huot and F. Coronado, Ten dimensional symmetry of N=4 SYM correlators, J. High Energy Phys. 03 (2022) 151.
  6. L. V. Bork, N. B. Muzhichkov, and E. S. Sozinov, Infrared properties of five-point massive amplitudes in N=4 SYM on the Coulomb branch, J. High Energy Phys. 08 (2022) 173.
  7. A. V. Belitsky, L. V. Bork, A. F. Pikelner, and V. A. Smirnov, Exact off shell Sudakov form factor in N=4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 130, 091605 (2023).
  8. A. V. Belitsky, L. V. Bork, and V. A. Smirnov, Off-shell form factor in N=4 sYM at three loops, J. High Energy Phys. 11 (2023) 111.
  9. A. V. Belitsky, L. V. Bork, J. M. Grumski-Flores, and V. A. Smirnov, Three-leg form factor on Coulomb branch, J. High Energy Phys. 11 (2024) 169.
  10. A. V. Belitsky and L. V. Bork, Off-shell minimal form factors, J. High Energy Phys. 07 (2025) 231.
  11. A. V. Belitsky and V. A. Smirnov, Off-shell form factor: Factorization is violated, arXiv:2505.22595.
  12. A. M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B164, 171 (1980).
  13. G. P. Korchemsky and A. V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B283, 342 (1987).
  14. F. Coronado, Bootstrapping the simplest correlator in planar N=4 supersymmetric Yang-Mills theory to all loops, Phys. Rev. Lett. 124, 171601 (2020).
  15. A. V. Belitsky and G. P. Korchemsky, Exact null octagon, J. High Energy Phys. 05 (2020) 070.
  16. A. V. Belitsky, Null octagon from Deift-Zhou steepest descent, Nucl. Phys. B980, 115844 (2022).
  17. C. Anastasiou, Z. Bern, L. J. Dixon, and D. A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91, 251602 (2003).
  18. Z. Bern, L. J. Dixon, and V. A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72, 085001 (2005).
  19. S. He, X. Jiang, J. Liu, and Y.-Q. Zhang, Notes on conformal integrals: Coulomb branch amplitudes, magic identities and bootstrap, Phys. Rev. D 112, 076012 (2025).
  20. S. He, Z. Li, R. Ma, Z. Wu, Q. Yang, and Y. Zhang, A study of Feynman integrals with uniform transcendental weights and their symbology, J. High Energy Phys. 10 (2022) 165.
  21. A. V. Belitsky and V. A. Smirnov, Near mass-shell double boxes, J. High Energy Phys. 05 (2024) 155.
  22. A. V. Belitsky, Collinear anatomy, J. High Energy Phys. 05 (2025) 117.
  23. A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254, 158 (1991).
  24. T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B580, 485 (2000).
  25. J. M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110, 251601 (2013).
  26. A. V. Belitsky, A. A. Kokosinskaya, A. V. Smirnov, V. V. Voevodin, and M. Zeng, Efficient reduction of Feynman integrals on supercomputers, Lobachevskii J. Math. 45, 2984 (2024).
  27. E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, J. High Energy Phys. 03 (2014) 071.
  28. M. Beneke and V. A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B522, 321 (1998).
  29. V. A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460, 397 (1999).
  30. J. B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469, 225 (1999).
  31. M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175, 559 (2006).
  32. A. V. Smirnov and V. A. Smirnov, On the resolution of singularities of multiple Mellin-Barnes integrals, Eur. Phys. J. C 62, 445 (2009).
  33. T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multi-loop integrals, Nucl. Phys. B585, 741 (2000).
  34. C. Bogner and S. Weinzierl, Resolution of singularities for multi-loop integrals, Comput. Phys. Commun. 178, 596 (2008).
  35. T. Kaneko and T. Ueda, A geometric method of sector decomposition, Comput. Phys. Commun. 181, 1352 (2010).
  36. A. V. Smirnov, N. D. Shapurov, and L. I. Vysotsky, fiesta5: Numerical high-performance Feynman integral evaluation, Comput. Phys. Commun. 277, 108386 (2022).
  37. L. Nilsson and M. Passare, Mellin transforms of multivariate rational functions, arXiv:1010.5060.
  38. C. Berkesch, J. Forsgård, and M. Passare, Euler-mellin integrals and a-hypergeometric functions, Michigan Math. J. 63, 101 (2014).
  39. E. R. Speer, Analytic renormalization, J. Math. Phys. (N.Y.) 9, 1404 (1968).
  40. E. R. Speer, On the structure of analytic renormalization, Commun. Math. Phys. 23, 23 (1971).
  41. P. Breitenlohner and D. Maison, Dimensionally renormalized Green’s functions for theories with massless particles. 1., Commun. Math. Phys. 52, 39 (1977).
  42. P. Breitenlohner and D. Maison, Dimensionally renormalized Green’s functions for theories with massless particles. 2., Commun. Math. Phys. 52, 55 (1977).
  43. G. ’t Hooft and M. J. G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B44, 189 (1972).
  44. G. Salvatori, The tropical geometry of subtraction schemes, arXiv:2406.14606.
  45. H. Cheng and T. T. Wu, Expanding Protons: Scatterinf at High Energies (MIT Press, Cambridge, MA, 1987).
  46. V. A. Smirnov, Problems of the strategy of regions, Phys. Lett. B 465, 226 (1999).
  47. A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71, 1626 (2011).
  48. A. V. Belitsky, L. V. Bork, and V. A. Smirnov, Pinching Sudakov, J. High Energy Phys. 05 (2025) 237.
  49. C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Softw. 22, 469 (1996).
  50. N. Arkani-Hamed, A. Hillman, and S. Mizera, Feynman polytopes and the tropical geometry of UV and IR divergences, Phys. Rev. D 105, 125013 (2022).
  51. J. Rau, A First Expedition to Tropical Geometry, https://math.uniandes.edu.co/~j.rau/downloads/FirstExpedition.pdf (2017).
  52. E. Brugallé, I. Itenberg, G. Mikhalkin, and K. Shaw, Brief introduction to tropical geometry, arXiv:1502.05950.
  53. G. Mikhalkin and J. Rau, Tropical Geometry, https://math.uniandes.edu.co/~j.rau/downloads/main.pdf (2019).
  54. G. M. Ziegler, Lectures on Polytopes (Springer-Verlag, New York, 1995).
  55. M. Henk, R.-G. Jürgen, and G. Ziegler, Basic properties of convex polytopes, in handbook of discrete and computational geometry, https://page.math.tu-berlin.de/~henk/preprints/henk richter-gebertziegler&basicpropertiesofconvexpolytopes.pdf (1999).
  56. T. Oda, Convex Bodies and Algebraic Geometry (Springer, New York, 1988).
  57. S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, pySecDec: A toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222, 313 (2018).
  58. N. N. Bogoliubov and O. S. Parasiuk, On the multiplication of the causal function in the quantum theory of fields, Acta Math. 97, 227 (1957).
  59. K. Hepp, Proof of the Bogolyubov-Parasiuk theorem on renormalization, Commun. Math. Phys. 2, 301 (1966).
  60. W. Zimmermann, Convergence of Bogolyubov’s method of renormalization in momentum space, Commun. Math. Phys. 15, 208 (1969).
  61. J. H. Lowenstein, Auxiliary mass formulation of the pure Yang-Mills model, Nucl. Phys. B96, 189 (1975).
  62. A. Hillman, A subtraction scheme for Feynman integrals, arXiv:2311.03439.
  63. A. Hillman, On the infrared and ultraviolet behavior of scattering amplitudes and wavefunctions, Ph.D. thesis, Princeton University, 2023.
  64. F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, Commun. Math. Phys. 382, 815 (2021).
  65. N. Arkani-Hamed, S. He, and T. Lam, Stringy canonical forms, J. High Energy Phys. 02 (2021) 069.
  66. H. Frellesvig, D. Tommasini, and C. Wever, On the reduction of generalized polylogarithms to Lin and Li2,2 and on the evaluation thereof, J. High Energy Phys. 03 (2016) 189.
  67. K. Schultka, Toric geometry and regularization of Feynman integrals, arXiv:1806.01086.
  68. Wolfram, Mathematica, Version 14.3.
  69. E. Gawrilow and M. Joswig, Polymake: A framework for analyzing convex polytopes, Polytopes—Combinatorics and Computation (Birkhäuser, Basel, 2000), Vol. 29, pp. 43–73.
  70. A. Proyas, I, Robot, Twentieth Century Fox, 2004.
  71. T. Gehrmann, J. M. Henn, and N. A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, J. High Energy Phys. 10 (2018) 103.
  72. C. G. Papadopoulos, D. Tommasini, and C. Wever, The pentabox master integrals with the simplified differential equations approach, J. High Energy Phys. 04 (2016) 078.
  73. D. Chicherin and V. Sotnikov, Pentagon functions for scattering of five massless particles, J. High Energy Phys. 20 (2020) 167.
  74. C. Duhr and F. Dulat, PolyLogTools—polylogs for the masses, J. High Energy Phys. 08 (2019) 135.
  75. Z. Bern, L. J. Dixon, D. A. Kosower, R. Roiban, M. Spradlin, C. Vergu, and A. Volovich, The two-loop six-gluon MHV amplitude in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 78, 045007 (2008).
  76. J. M. Drummond, J. Henn, G. P. Korchemsky, and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B815, 142 (2009).
  77. L. J. Dixon, J. M. Drummond, and J. M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory, J. High Energy Phys. 01 (2012) 024.
  78. A. Brandhuber, G. Travaglini, and G. Yang, Analytic two-loop form factors in N=4 SYM, J. High Energy Phys. 05 (2012) 082.
  79. A. Brandhuber, B. Penante, G. Travaglini, and C. Wen, The last of the simple remainders, J. High Energy Phys. 08 (2014) 100.
  80. A. V. Belitsky and V. A. Smirnov, Collinear bootstrap for N=(1,1) sYM, Phys. Rev. D 112, 105018 (2025).

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Phys. Rev. D 113, 045019– Published 23 February, 2026
  • Received 29 December 2025
  • Accepted 5 February 2026
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