Advertisement

Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM

  • Freddy Cachazo
  • Song He
  • Ellis Ye Yuan
Open Access
Regular Article - Theoretical Physics

Abstract

The tree-level S-matrix of Einstein’s theory is known to have a representation as an integral over the moduli space of punctured spheres localized to the solutions of the scattering equations. In this paper we introduce three operations that can be applied on the integrand in order to produce other theories. Starting in d + M dimensions we use dimensional reduction to construct Einstein-Maxwell with gauge group U(1) M . The second operation turns gravitons into gluons and we call it “squeezing”. This gives rise to a formula for all multi-trace mixed amplitudes in Einstein-Yang-Mills. Dimensionally reducing Yang-Mills we find the S-matrix of a special Yang-Mills-Scalar (YMS) theory, and by the squeezing operation we find that of a YMS theory with an additional cubic scalar vertex. A corollary of the YMS formula gives one for a single massless scalar with a ϕ4 interaction. Starting again from Einstein’s theory but in d + d dimensions we introduce a “generalized dimensional reduction” that produces the Born-Infeld theory or a special Galileon theory in d dimensions depending on how it is applied. An extension of Born-Infeld formula leads to one for the Dirac-Born-Infeld (DBI) theory. By applying the same operation to Yang-Mills we obtain the U(N ) non-linear sigma model (NLSM). Finally, we show how the Kawai-Lewellen-Tye relations naturally follow from our formulation and provide additional connections among these theories. One such relation constructs DBI from YMS and NLSM.

Keywords

Scattering Amplitudes Classical Theories of Gravity 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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    R. Roiban, M. Spradlin and A. Volovich, Tree-level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    H. Elvang and Y.-t. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE].
  4. [4]
    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].ADSGoogle Scholar
  5. [5]
    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].ADSCrossRefGoogle Scholar
  6. [6]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    D. Fairlie and D. Roberts, Dual models without tachyonsa new approach, unpublished Durham preprint (1972).Google Scholar
  8. [8]
    D.E. Roberts, Mathematical structure of dual amplitudes, Ph.D. Thesis, Durham University, Durham U.K. (1972).Google Scholar
  9. [9]
    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
  10. [10]
    D.J. Gross and P.F. Mende, String theory beyond the Planck scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    E. Witten, Parity invariance for strings in twistor space, Adv. Theor. Math. Phys. 8 (2004) 779 [hep-th/0403199] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    P. Caputa and S. Hirano, Observations on open and closed string scattering amplitudes at high energies, JHEP 02 (2012) 111 [arXiv:1108.2381] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    P. Caputa, Lightlike contours with fermions, Phys. Lett. B 716 (2012) 475 [arXiv:1205.6369] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    Y. Makeenko and P. Olesen, The QCD scattering amplitude from area behaved Wilson loops, Phys. Lett. B 709 (2012) 285 [arXiv:1111.5606] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    F. Cachazo, Fundamental BCJ relation in \( \mathcal{N}=4 \) SYM from the connected formulation, arXiv:1206.5970 [INSPIRE].
  16. [16]
    L. Dolan and P. Goddard, The polynomial form of the scattering equations, JHEP 07 (2014) 029 [arXiv:1402.7374] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    C. Kalousios, Massless scattering at special kinematics as Jacobi polynomials, J. Phys. A 47 (2014) 215402 [arXiv:1312.7743] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  18. [18]
    L. Dolan and P. Goddard, Proof of the formula of Cachazo, He and Yuan for Yang-Mills tree amplitudes in arbitrary dimension, JHEP 05 (2014) 010 [arXiv:1311.5200] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S.G. Naculich, Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, JHEP 09 (2014) 029 [arXiv:1407.7836] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    F. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills scattering amplitudes from scattering equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor strings in four dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    N. Berkovits, Infinite tension limit of the pure spinor superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A.A. Tseytlin, Born-Infeld action, supersymmetry and string theory, hep-th/9908105 [INSPIRE].
  26. [26]
    M.A. Luty, M. Porrati and R. Rattazzi, Strong interactions and stability in the DGP model, JHEP 09 (2003) 029 [hep-th/0303116] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].ADSMathSciNetGoogle Scholar
  28. [28]
    F. Cachazo and Y. Geyer, Atwistor stringinspired formula for tree-level scattering amplitudes in \( \mathcal{N}=8 \) SUGRA, arXiv:1206.6511 [INSPIRE].
  29. [29]
    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].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The momentum kernel of gauge and gravity theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Scattering amplitudes in \( \mathcal{N}=2 \) Maxwell-Einstein and Yang-Mills/Einstein supergravity, JHEP 01 (2015) 081 [arXiv:1408.0764] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    J.A. Cronin, Phenomenological model of strong and weak interactions in chiral U(3) × U(3), Phys. Rev. 161 (1967) 1483 [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    S. Weinberg, Dynamical approach to current algebra, Phys. Rev. Lett. 18 (1967) 188 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    S. Weinberg, Nonlinear realizations of chiral symmetry, Phys. Rev. 166 (1968) 1568 [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    K. Kampf, J. Novotny and J. Trnka, Tree-level amplitudes in the nonlinear σ-model, JHEP 05 (2013) 032 [arXiv:1304.3048] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    K. Kampf and J. Novotny, Unification of Galileon dualities, JHEP 10 (2014) 006 [arXiv:1403.6813] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
  39. [39]
    F. Cachazo, Resultants and gravity amplitudes, arXiv:1301.3970 [INSPIRE].
  40. [40]
    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
  41. [41]
    R. Boels, K.J. Larsen, N.A. Obers and M. Vonk, MHV, CSW and BCFW: field theory structures in string theory amplitudes, JHEP 11 (2008) 015 [arXiv:0808.2598] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    R.C. Myers, Non-Abelian phenomena on D-branes, Class. Quant. Grav. 20 (2003) S347 [hep-th/0303072] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [INSPIRE].
  44. [44]
    Y.-X. Chen, Y.-J. Du and B. Feng, On tree amplitudes with gluons coupled to gravitons, JHEP 01 (2011) 081 [arXiv:1011.1953] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    S. Stieberger and T.R. Taylor, Graviton as a pair of collinear gauge bosons, Phys. Lett. B 739 (2014) 457 [arXiv:1409.4771] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  46. [46]
    R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    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
  48. [48]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, P. Tourkine and P. Vanhove, Scattering equations and string theory amplitudes, Phys. Rev. D 90 (2014) 106002 [arXiv:1403.4553] [INSPIRE].ADSGoogle Scholar
  49. [49]
    S. Weinberg, Photons and gravitons in S-matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. 135 (1964) B1049 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    Z. Bern, A. De Freitas and H.L. Wong, On the coupling of gravitons to matter, Phys. Rev. Lett. 84 (2000) 3531 [hep-th/9912033] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    Y.-J. Du, B. Feng and C.-H. Fu, BCJ relation of color scalar theory and KLT relation of gauge theory, JHEP 08 (2011) 129 [arXiv:1105.3503] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal basis for gauge theory amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonUnited States
  3. 3.Department of Physics & AstronomyUniversity of WaterlooWaterlooCanada

Personalised recommendations