Advertisement

Extensions of theories from soft limits

  • Freddy Cachazo
  • Peter Cha
  • Sebastian Mizera
Open Access
Regular Article - Theoretical Physics

Abstract

We study a variety of field theories with vanishing single soft limits. In all cases, the structure of the soft limit is controlled by a larger theory, which provides an extension of the original one by adding more fields and interactions. Our main example is the U(N ) non-linear sigma model in its CHY representation. Its extension is a theory in which the NLSM Goldstone bosons interact with a cubic biadjoint scalar. Other theories we study and extend are the special Galileon and Born-Infeld theory, including its maximally supersymmetric version in four dimensions, the DBI-Volkov-Akulov theory. In all the cases, we propose the CHY representation of the complete tree-level S-matrix of the extended theories. In fact, CHY formulas are the key technique for studying the single soft limit behavior of the original theories. As a byproduct, we show that the tree-level S-matrix of the extended NLSM theory can be constructed using a very compact BCFW-like recursion relation, where physical poles are at most linear in the deformation parameter.

Keywords

Effective field theories Scattering Amplitudes Sigma Models 

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]
    S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev. 137 (1965) B1022 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    L. Susskind and G. Frye, Algebraic aspects of pionic duality diagrams, Phys. Rev. D 1 (1970) 1682 [INSPIRE].ADSGoogle Scholar
  3. [3]
    C. Cheung, K. Kampf, J. Novotny and J. Trnka, Effective field theories from soft limits of scattering amplitudes, Phys. Rev. Lett. 114 (2015) 221602 [arXiv:1412.4095] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    C. Cheung, K. Kampf, J. Novotny, C.-H. Shen and J. Trnka, On-shell recursion relations for effective field theories, Phys. Rev. Lett. 116 (2016) 041601 [arXiv:1509.03309] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    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
  7. [7]
    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
  8. [8]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    F. Cachazo, S. He and E.Y. Yuan, Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    R. Roiban, M. Spradlin and A. Volovich, A googly amplitude from the B model in twistor space, JHEP 04 (2004) 012 [hep-th/0402016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    F. Cachazo and D. Skinner, Gravity from rational curves in twistor space, Phys. Rev. Lett. 110 (2013) 161301 [arXiv:1207.0741] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    F. Cachazo, L. Mason and D. Skinner, Gravity in twistor space and its Grassmannian formulation, SIGMA 10 (2014) 051 [arXiv:1207.4712] [INSPIRE].MathSciNetMATHGoogle Scholar
  14. [14]
    D.V. Volkov and V.P. Akulov, Is the neutrino a Goldstone particle?, Phys. Lett. B 46 (1973) 109 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    S. He, Z. Liu and J.-B. Wu, Scattering equations, twistor-string formulas and double-soft limits in four dimensions, arXiv:1604.02834 [INSPIRE].
  16. [16]
    E. Bergshoeff, F. Coomans, R. Kallosh, C.S. Shahbazi and A. Van Proeyen, Dirac-Born-Infeld-Volkov-Akulov and deformation of supersymmetry, JHEP 08 (2013) 100 [arXiv:1303.5662] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    H. Lüo and C. Wen, Recursion relations from soft theorems, JHEP 03 (2016) 088 [arXiv:1512.06801] [INSPIRE].CrossRefGoogle Scholar
  18. [18]
    M. Gell-Mann and M. Levy, The axial vector current in beta decay, Nuovo Cim. 16 (1960) 705 [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    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
  20. [20]
    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
  21. [21]
    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    Z. Bern, L.J. Dixon and D.A. Kosower, On-shell recurrence relations for one-loop QCD amplitudes, Phys. Rev. D 71 (2005) 105013 [hep-th/0501240] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    K. Risager, A direct proof of the CSW rules, JHEP 12 (2005) 003 [hep-th/0508206] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    K. Hinterbichler, Theoretical aspects of massive gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    G.R. Dvali, G. Gabadadze and M. Porrati, 4D gravity on a brane in 5D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    K. Kampf and J. Novotny, Unification of Galileon dualities, JHEP 10 (2014) 006 [arXiv:1403.6813] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    K. Hinterbichler and A. Joyce, Hidden symmetry of the Galileon, Phys. Rev. D 92 (2015) 023503 [arXiv:1501.07600] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    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].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    Y.-J. Du, B. Feng, C.-H. Fu and Y. Wang, Note on soft graviton theorem by KLT relation, JHEP 11 (2014) 090 [arXiv:1408.4179] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    R. Kallosh, Volkov-Akulov theory and D-branes, hep-th/9705118 [INSPIRE].
  33. [33]
    W.-M. Chen, Y.-T. Huang and C. Wen, New fermionic soft theorems for supergravity amplitudes, Phys. Rev. Lett. 115 (2015) 021603 [arXiv:1412.1809] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    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
  35. [35]
    Eulerian number wikipedia webpage, https://en.wikipedia.org/wiki/Eulerian number.
  36. [36]
    F. Cachazo, Resultants and gravity amplitudes, arXiv:1301.3970 [INSPIRE].
  37. [37]
    Z.-W. Liu, Soft theorems in maximally supersymmetric theories, Eur. Phys. J. C 75 (2015) 105 [arXiv:1410.1616] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    A.J. Larkoski, D. Neill and I.W. Stewart, Soft theorems from effective field theory, JHEP 06 (2015) 077 [arXiv:1412.3108] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    A.A. Tseytlin, Vector field effective action in the open superstring theory, Nucl. Phys. B 276 (1986) 391 [Erratum ibid. B 291 (1987) 876] [INSPIRE].
  40. [40]
    P. Benincasa and F. Cachazo, Consistency conditions on the S-matrix of massless particles, arXiv:0705.4305 [INSPIRE].
  41. [41]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
  42. [42]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
  43. [43]
    A. Strominger, On BMS invariance of gravitational scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    F. Cachazo, S-matrix theory, talk at the Amplitudes 2015 Conference, http://amp15.itp.phys.ethz.ch/talks/Cachazo.pdf, Zurich Switzerland (2015).
  45. [45]
    G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, JHEP 03 (2015) 033 [arXiv:1502.00010] [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of Physics & AstronomyUniversity of WaterlooWaterlooCanada

Personalised recommendations