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Scattering equations: from projective spaces to tropical grassmannians

  • Freddy Cachazo
  • Nick Early
  • Alfredo GuevaraEmail author
  • Sebastian Mizera
Open Access
Regular Article - Theoretical Physics

Abstract

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ1, to higher-dimensional projective spaces ℂℙk − 1. The standard, k = 2 Mandelstam invariants, sab, are generalized to completely symmetric tensors \( {\mathrm{s}}_{a_1{a}_2\dots {a}_k} \) subject to a ‘massless’ condition \( {\mathrm{s}}_{a_1{a}_2\dots {a}_{k-2}bb}=0 \) and to ‘momentum conservation’. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all ‘biadjoint amplitudes’ for (k, n) = (3, 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.

Keywords

Scattering Amplitudes Differential and Algebraic Geometry 

Notes

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.

References

  1. [1]
    F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].Google Scholar
  2. [2]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].CrossRefGoogle Scholar
  3. [3]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].CrossRefzbMATHGoogle Scholar
  4. [4]
    H. Kawai, D.C. Lewellen and S.-H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  5. [5]
    Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].MathSciNetGoogle Scholar
  6. [6]
    S. Mizera, Scattering amplitudes from intersection theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  7. [7]
    P. Mastrolia and S. Mizera, Feynman integrals and intersection theory, JHEP 02 (2019) 139 [arXiv:1810.03818] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    H. Frellesvig et al., Decomposition of Feynman integrals on the maximal cut by intersection numbers, JHEP 05 (2019) 153 [arXiv:1901.11510] [INSPIRE].CrossRefGoogle Scholar
  9. [9]
    L. Dolan and P. Goddard, The polynomial form of the scattering equations, JHEP 07 (2014) 029 [arXiv:1402.7374] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].CrossRefGoogle Scholar
  11. [11]
    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, U.K. (2016) [arXiv:1212.5605] [INSPIRE].CrossRefzbMATHGoogle Scholar
  12. [12]
    F. Cachazo, N. Early, A. Guevara and S. Mizera, Δ-algebra and scattering amplitudes, JHEP 02 (2019) 005 [arXiv:1812.01168] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    S. Franco, D. Galloni, B. Penante and C. Wen, Non-planar on-shell diagrams, JHEP 06 (2015) 199 [arXiv:1502.02034] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J.L. Bourjaily, S. Franco, D. Galloni and C. Wen, Stratifying on-shell cluster varieties: the geometry of non-planar on-shell diagrams, JHEP 10 (2016) 003 [arXiv:1607.01781] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M.M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math. 16 (1993) 29 [alg-geom/9210002].MathSciNetzbMATHGoogle Scholar
  16. [16]
    J. Sekiguchi, The versal deformation of the E 6 -singularity and a family of cubic surfaces, J. Math. Soc. Jpn. 46 (1994) 355.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J. Sekiguchi, Cross ratio varieties for root systems, Kyushu J. Math. 48 (1994) 123.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J. Sekiguchi and M. Yoshida, W(E 6)-action on the configuration space of six lines on the real projective plane, Kyushu J. Math. 51 (1997) 297.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J. Sekiguchi, Configurations of seven lines on the real projective plane and the root system of type E 7, J. Math. Soc. Jpn. 51 (1999) 987.CrossRefzbMATHGoogle Scholar
  20. [20]
    J. Sekiguchi, Cross ratio varieties for root systems II: the case of the root system of type E 7, Kyushu J. Math. 54 (2000) 7.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Keel and J. Tevelev, Chow quotients of Grassmannians II, math.AG/0401159.
  22. [22]
    S. Keel and J. Tevelev, Geometry of Chow quotients of Grassmannians, Duke Math. J. 134 (2006) 259.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004) 389 [math.AG/0304218].
  24. [24]
    F. Cachazo, S. Mizera and G. Zhang, Scattering equations: real solutions and particles on a line, JHEP 03 (2017) 151 [arXiv:1609.00008] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    P. Tourkine, Tropical amplitudes, Annales Henri Poincaré 18 (2017) 2199 [arXiv:1309.3551] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    S. Herrmann, A. Jensen, M. Joswig and B. Sturmfels, How to draw tropical planes, Electron. J. Combinat. 16 (2009) 6 [arXiv:0808.2383].MathSciNetzbMATHGoogle Scholar
  27. [27]
    D.E. Roberts, Mathematical structure of dual amplitudes, Ph.D. thesis, Durham University, Durham, U.K. (1972) [INSPIRE].
  28. [28]
    D.B. Fairlie and D.E. Roberts, Dual models without tachyonsa new approach, (1972) [INSPIRE].
  29. [29]
    D.B. Fairlie, A coding of real null four-momenta into world-sheet coordinates, Adv. Math. Phys. 2009 (2009) 284689 [arXiv:0805.2263] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Spradlin and A. Volovich, From twistor string theory to recursion relations, Phys. Rev. D 80 (2009) 085022 [arXiv:0909.0229] [INSPIRE].MathSciNetGoogle Scholar
  31. [31]
    F. Cachazo, S. He and E.Y. Yuan, Scattering in three dimensions from rational maps, JHEP 10 (2013) 141 [arXiv:1306.2962] [INSPIRE].CrossRefGoogle Scholar
  32. [32]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of residues and Grassmannian dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    C. Kalousios, Massless scattering at special kinematics as Jacobi polynomials, J. Phys. A 47 (2014) 215402 [arXiv:1312.7743] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  34. [34]
    K. Aomoto, On vanishing of cohomology attached to certain many valued meromorphic functions, J. Math. Soc. Jpn. 27 (1975) 248.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Integration rulesfor scattering equations, JHEP 09 (2015) 129 [arXiv:1506.06137] [INSPIRE].CrossRefzbMATHGoogle Scholar
  36. [36]
    H. Gomez, Λ scattering equations, JHEP 06 (2016) 101 [arXiv:1604.05373] [INSPIRE].CrossRefGoogle Scholar
  37. [37]
    X. Gao, S. He and Y. Zhang, Labelled tree graphs, Feynman diagrams and disk integrals, JHEP 11 (2017) 144 [arXiv:1708.08701] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    H. Gomez and E.Y. Yuan, N-point tree-level scattering amplitude in the new Berkovitsstring, JHEP 04 (2014) 046 [arXiv:1312.5485] [INSPIRE].CrossRefGoogle Scholar
  39. [39]
    W. Siegel, Amplitudes for left-handed strings, arXiv:1512.02569 [INSPIRE].
  40. [40]
    E. Casali and P. Tourkine, On the null origin of the ambitwistor string, JHEP 11 (2016) 036 [arXiv:1606.05636] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    S. Mizera and G. Zhang, A string deformation of the Parke-Taylor factor, Phys. Rev. D 96 (2017) 066016 [arXiv:1705.10323] [INSPIRE].MathSciNetGoogle Scholar
  42. [42]
    E. Casali and P. Tourkine, Windings of twisted strings, Phys. Rev. D 97 (2018) 061902 [arXiv:1710.01241] [INSPIRE].Google Scholar
  43. [43]
    J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Abelian Z-theory: NLSM amplitudes and α-corrections from the open string, JHEP 06 (2017) 093 [arXiv:1608.02569] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    C.R. Mafra and O. Schlotterer, Non-Abelian Z-theory: Berends-Giele recursion for the α-expansion of disk integrals, JHEP 01 (2017) 031 [arXiv:1609.07078] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Semi-Abelian Z-theory: NLSM+ϕ 3 from the open string, JHEP 08 (2017) 135 [arXiv:1612.06446] [INSPIRE].
  46. [46]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].CrossRefGoogle Scholar
  47. [47]
    Y. Geyer, Ambitwistor strings: worldsheet approaches to perturbative quantum field theories, Ph.D. thesis, Oxford U., Inst. Math., Oxford, U.K. (2016) [arXiv:1610.04525] [INSPIRE].
  48. [48]
    D.R. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
  49. [49]
    C. Jost, Computing characteristic classes and the topological Euler characteristic of complex projective schemes, J. Softw. Alg. Geom. 7 (2015) 31.MathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Helmer, Computing characteristic classes of subschemes of smooth toric varieties, J. Alg. 476 (2017) 548.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Massachusetts Institute of TechnologyCambridgeU.S.A.
  3. 3.Department of Physics & AstronomyUniversity of WaterlooWaterlooCanada
  4. 4.CECs Valdivia & Departamento de FísicaUniversidad de ConcepciónConcepciónChile

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