A note on one-loop cluster adjacency in \( \mathcal{N} \) = 4 SYM

Abstract

We study cluster adjacency conjectures for amplitudes in maximally supersymmetric Yang-Mills theory. We show that the n-point one-loop NMHV ratio function satisfies Steinmann cluster adjacency. We also show that the one-loop BDS-like normalized NMHV amplitude satisfies cluster adjacency between Yangian invariants and final symbol entries up to 9-points. We present conjectures for cluster adjacency properties of Plücker coordinates, quadratic cluster variables, and NMHV Yangian invariants that generalize the notion of weak separation.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    S. Fomin and A. Zelevinsky, Cluster algebras I: foundations, J. Amer. Math. Soc. 15 (2002) 497 [math/0104151].

  2. [2]

    S. Fomin, L. Williams and A. Zelevinsky, Introduction to cluster algebras. Chapters 1–3, arXiv:1608.05735.

  3. [3]

    S. Fomin, L. Williams and A. Zelevinsky, Introduction to cluster algebras. Chapters 4–6, arXiv:1707.07190.

  4. [4]

    M. Gekhtman, M.Z. Shapiro and A.D. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003) 899 [math/0208033].

  5. [5]

    J.S. Scott, Grassmanniand and cluster algebras, Proc. London Math. Soc. 92 (2006) 345380.

    Article  Google Scholar 

  6. [6]

    J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].

    ADS  Article  Google Scholar 

  7. [7]

    J. Golden, M.F. Paulos, M. Spradlin and A. Volovich, Cluster polylogarithms for scattering amplitudes, J. Phys. A 47 (2014) 474005 [arXiv:1401.6446] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  8. [8]

    J. Golden and M. Spradlin, A cluster bootstrap for two-loop MHV amplitudes, JHEP 02 (2015) 002 [arXiv:1411.3289] [INSPIRE].

    ADS  Article  Google Scholar 

  9. [9]

    T. Harrington and M. Spradlin, Cluster functions and scattering amplitudes for six and seven points, JHEP 07 (2017) 016 [arXiv:1512.07910] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. [10]

    J. Golden and A.J. Mcleod, Cluster algebras and the subalgebra constructibility of the seven-particle remainder function, JHEP 01 (2019) 017 [arXiv:1810.12181] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  11. [11]

    J. Drummond, J. Foster and O. Gürdoğan, Cluster adjacency properties of scattering amplitudes in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 120 (2018) 161601 [arXiv:1710.10953] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    J. Golden, A.J. McLeod, M. Spradlin and A. Volovich, The Sklyanin bracket and cluster adjacency at all multiplicity, JHEP 03 (2019) 195 [arXiv:1902.11286] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  13. [13]

    J. Mago, A. Schreiber, M. Spradlin and A. Volovich, Yangian invariants and cluster adjacency in \( \mathcal{N} \) = 4 Yang-Mills, JHEP 10 (2019) 099 [arXiv:1906.10682] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  14. [14]

    T. Łukowski, M. Parisi, M. Spradlin and A. Volovich, Cluster adjacency for m = 2 Yangian invariants, JHEP 10 (2019) 158 [arXiv:1908.07618] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. [15]

    J. Drummond, J. Foster and O. Gürdoğan, Cluster adjacency beyond MHV, JHEP 03 (2019) 086 [arXiv:1810.08149] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  16. [16]

    J. Drummond, J. Foster, O. Gürdoğan and G. Papathanasiou, Cluster adjacency and the four-loop NMHV heptagon, JHEP 03 (2019) 087 [arXiv:1812.04640] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    Ö. Gürdoğan and M. Parisi, Cluster patterns in Landau and leading singularities via the Amplituhedron, to appear.

  18. [18]

    J. Drummond, J. Foster, O. Gürdoğan and C. Kalousios, Tropical fans, scattering equations and amplitudes, arXiv:2002.04624 [INSPIRE].

  19. [19]

    N. Henke and G. Papathanasiou, How tropical are seven- and eight-particle amplitudes?, JHEP 08 (2020) 005 [arXiv:1912.08254] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  20. [20]

    B. Leclerc and A. Zelevinsky, Quasicommuting families of quantum Plücker coordinates, Amer. Math. Soc. Transl. 181 (1998) 85.

    MATH  Google Scholar 

  21. [21]

    S. Oh, A. Postnikov and D.E. Speyer, Weak separation and plabic graphs, Proc. Lond. Math. Soc. 110 (2015) 721 [arXiv:1109.4434].

    MathSciNet  Article  Google Scholar 

  22. [22]

    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  23. [23]

    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. [24]

    E.K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl. 16 (1982) 263 [Funkt.Anal.Pril. 16N4 (1982) 27] [INSPIRE].

  25. [25]

    S. Caron-Huot et al., The cosmic Galois group and extended steinmann relations for planar \( \mathcal{N} \) = 4 SYM amplitudes, JHEP 09 (2019) 061 [arXiv:1906.07116] [INSPIRE].

  26. [26]

    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 (2005) 085001 [hep-th/0505205] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. [28]

    L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for Scattering Amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  29. [29]

    O. Steinmann, Über den Zusammenhang zwischen den Wightmanfunktionen und der retardierten Kommutatoren, Helv. Phys. Acta 33 (1960) 257.

    MathSciNet  MATH  Google Scholar 

  30. [30]

    O. Steinmann, Wightman-Funktionen und retardierten Kommutatoren II, Helv. Phys. Acta 33 (1960) 347.

    MathSciNet  MATH  Google Scholar 

  31. [31]

    K.E. Cahill and H.P. Stapp, Optical theorems and Steinmann relations, Annals Phys. 90 (1975) 438 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. [32]

    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Generalized unitarity for N = 4 super-amplitudes, Nucl. Phys. B 869 (2013) 452 [arXiv:0808.0491] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  33. [33]

    H. Elvang, D.Z. Freedman and M. Kiermaier, Dual conformal symmetry of 1-loop NMHV amplitudes in N = 4 SYM theory, JHEP 03 (2010) 075 [arXiv:0905.4379] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  34. [34]

    L.J. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. [35]

    J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].

  36. [36]

    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  37. [37]

    A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Marcus Spradlin.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2005.07177

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mago, J., Schreiber, A., Spradlin, M. et al. A note on one-loop cluster adjacency in \( \mathcal{N} \) = 4 SYM. J. High Energ. Phys. 2021, 84 (2021). https://doi.org/10.1007/JHEP01(2021)084

Download citation

Keywords

  • Scattering Amplitudes
  • Supersymmetric Gauge Theory
  • Differential and Algebraic Geometry