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Scattering of massless particles: scalars, gluons and gravitons

  • Freddy Cachazo
  • Song He
  • Ellis Ye Yuan
Open Access
Article

Abstract

In a recent note we presented a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. In this paper we show that a natural formulation also exists for a massless colored cubic scalar theory. In Yang-Mills, the formula is an integral over the space of n marked points on a sphere and has as integrand two factors. The first factor is a combination of Parke-Taylor-like terms dressed with U(N ) color structures while the second is a Pfaffian. The S-matrix of a U(N ) × U(Ñ ) cubic scalar theory is obtained by simply replacing the Pfaffian with a U(Ñ ) version of the previous U(N ) factor. Given that gravity amplitudes are obtained by replacing the U(N ) factor in Yang-Mills by a second Pfaffian, we are led to a natural color-kinematics correspondence. An expansion of the integrand of the scalar theory leads to sums over trivalent graphs and are directly related to the KLT matrix. Combining this and the Yang-Mills formula we find a connection to the BCJ color-kinematics duality as well as a new proof of the BCJ doubling property that gives rise to gravity amplitudes. We end by considering a special kinematic point where the partial amplitude simply counts the number of color-ordered planar trivalent trees, which equals a Catalan number. The scattering equations simplify dramatically and are equivalent to a special Y-system with solutions related to roots of Chebyshev polynomials. The sum of the integrand over the solutions gives rise to a representation of Catalan numbers in terms of eigenvectors and eigenvalues of the adjacency matrix of an A-type Dynkin diagram.

Keywords

Scattering Amplitudes Field Theories in Higher Dimensions 

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]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    R. Roiban, M. Spradlin and A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    F. Cachazo and Y. Geyer, ATwistor StringInspired Formula For Tree-Level Scattering Amplitudes in N = 8 SUGRA, arXiv:1206.6511 [INSPIRE].
  4. [4]
    F. Cachazo and D. Skinner, Gravity from Rational Curves in Twistor Space, Phys. Rev. Lett. 110 (2013) 161301 [arXiv:1207.0741] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimension, arXiv:1307.2199 [INSPIRE].
  6. [6]
    F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and KLT Orthogonality, arXiv:1306.6575 [INSPIRE].
  7. [7]
    H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    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].ADSMathSciNetGoogle Scholar
  9. [9]
    F. Cachazo, S. He and E.Y. Yuan, Scattering in Three Dimensions from Rational Maps, JHEP 10 (2013) 141 [arXiv:1306.2962] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A. Hodges, A simple formula for gravitational MHV amplitudes, arXiv:1204.1930 [INSPIRE].
  12. [12]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Gravity and Yang-Mills Amplitude Relations, Phys. Rev. D 82 (2010) 107702 [arXiv:1005.4367] [INSPIRE].ADSGoogle Scholar
  13. [13]
    J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude II. Amplitude and Hypergeometric Function Structure, Nucl. Phys. B 873 (2013) 461 [arXiv:1106.2646] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    A. Hodges, New expressions for gravitational scattering amplitudes, Journal of High Energy Physics 1307 (2013) [arXiv:1108.2227] [INSPIRE].
  17. [17]
    M. Kiermaier, Gravity as the square of gauge theory, in Amplitudes 2010, Queen Mary, University of London, U.K., May 2010.Google Scholar
  18. [18]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    C.-H. Fu, Y.-J. Du and B. Feng, An algebraic approach to BCJ numerators, JHEP 03 (2013) 050 [arXiv:1212.6168] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Gravity as the Square of Gauge Theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].ADSGoogle Scholar
  21. [21]
    L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    C. Beasley and E. Witten, Residues and world sheet instantons, JHEP 10 (2003) 065 [hep-th/0304115] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    D.J. Gross and P.F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  3. 3.Department of Physics & AstronomyUniversity of WaterlooWaterlooCanada

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