Journal of High Energy Physics

, 2019:195 | Cite as

Soft bootstrap and supersymmetry

  • Henriette Elvang
  • Marios HadjiantonisEmail author
  • Callum R. T. Jones
  • Shruti Paranjape
Open Access
Regular Article - Theoretical Physics


The soft bootstrap is an on-shell method to constrain the landscape of effective field theories (EFTs) of massless particles via the consistency of the low-energy S-matrix. Given assumptions on the on-shell data (particle spectra, linear symmetries, and low-energy theorems), the soft bootstrap is an efficient algorithm for determining the possible consistency of an EFT with those properties. The implementation of the soft bootstrap uses the recently discovered method of soft subtracted recursion. We derive a precise criterion for the validity of these recursion relations and show that they fail exactly when the assumed symmetries can be trivially realized by independent operators in the effective action. We use this to show that the possible pure (real and complex) scalar, fermion, and vector exceptional EFTs are highly constrained. Next, we prove how the soft behavior of states in a supermultiplet must be related and illustrate the results in extended supergravity. We demonstrate the power of the soft bootstrap in two applications. First, for the \( \mathcal{N} \) = 1 and \( \mathcal{N} \) = 2 1 nonlinear sigma models, we show that on-shell constructibility establishes the emergence of accidental IR symmetries. This includes a new on-shell perspective on the interplay between \( \mathcal{N} \) = 2 supersymmetry, low-energy theorems, and electromagnetic duality. We also show that \( \mathcal{N} \) = 2 supersymmetry requires 3-point interactions with the photon that make the soft behavior of the scalar O(1) instead of vanishing, despite the underlying symmetric coset. Second, we study Galileon theories, including aspects of supersymmetrization, the possibility of a vector-scalar Galileon EFT, and the existence of higher-derivative corrections preserving the enhanced special Galileon symmetry. The latter is addressed both by soft bootstrap and by application of double-copy/KLT relations applied to higher-derivative corrections of chiral perturbation theory.


Effective Field Theories Scattering Amplitudes Sigma Models Supersymmetric Effective Theories 


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.


  1. [1]
    S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
  2. [2]
    C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
  3. [3]
    D.V. Volkov, Phenomenological Lagrangians, Fiz. Elem. Chast. Atom. Yadra 4 (1973) 3 [INSPIRE].MathSciNetGoogle Scholar
  4. [4]
    H. Georgi, On-shell effective field theory, Nucl. Phys. B 361 (1991) 339 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342 (1995) 189 [hep-ph/9304230] [INSPIRE].
  6. [6]
    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
  7. [7]
    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
  8. [8]
    C. Cheung, K. Kampf, J. Novotny, C.-H. Shen and J. Trnka, A Periodic Table of Effective Field Theories, JHEP 02 (2017) 020 [arXiv:1611.03137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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
  11. [11]
    K. Risager, A Direct proof of the CSW rules, JHEP 12 (2005) 003 [hep-th/0508206] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    H. Elvang, D.Z. Freedman and M. Kiermaier, Proof of the MHV vertex expansion for all tree amplitudes in N = 4 SYM theory, JHEP 06 (2009) 068 [arXiv:0811.3624] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    T. Cohen, H. Elvang and M. Kiermaier, On-shell constructibility of tree amplitudes in general field theories, JHEP 04 (2011) 053 [arXiv:1010.0257] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    C. Cheung, C.-H. Shen and J. Trnka, Simple Recursion Relations for General Field Theories, JHEP 06 (2015) 118 [arXiv:1502.05057] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    L. Susskind and G. Frye, Algebraic aspects of pionic duality diagrams, Phys. Rev. D 1 (1970) 1682 [INSPIRE].ADSGoogle Scholar
  16. [16]
    A. Galli and E. Galli, Implications of certain algebraic aspects of dual resonance models, Phys. Rev. D 2 (1970) 1081 [INSPIRE].ADSGoogle Scholar
  17. [17]
    A. Galli and E. Galli, Further comments on the relations between dual models and chiral symmetry breaking, Phys. Rev. D 4 (1971) 1253 [INSPIRE].ADSGoogle Scholar
  18. [18]
    Z. Yin, The Infrared Structure of Exceptional Scalar Theories, arXiv:1810.07186 [INSPIRE].
  19. [19]
    D. Simmons-Duffin, The Conformal Bootstrap, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., June 1–26, 2015, pp. 1–74 (2017) [] [arXiv:1602.07982] [INSPIRE].
  20. [20]
    D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    I. Low and A.V. Manohar, Spontaneously broken space-time symmetries and Goldstone’s theorem, Phys. Rev. Lett. 88 (2002) 101602 [hep-th/0110285] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    K. Hinterbichler and A. Joyce, Hidden symmetry of the Galileon, Phys. Rev. D 92 (2015) 023503 [arXiv:1501.07600] [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    F. Farakos, C. Germani and A. Kehagias, On ghost-free supersymmetric galileons, JHEP 11 (2013) 045 [arXiv:1306.2961] [INSPIRE].
  24. [24]
    H. Elvang, M. Hadjiantonis, C.R.T. Jones and S. Paranjape, On the Supersymmetrization of Galileon Theories in Four Dimensions, Phys. Lett. B 781 (2018) 656 [arXiv:1712.09937] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    R. Klein, E. Malek, D. Roest and D. Stefanyszyn, No-go theorem for a gauge vector as a spacetime Goldstone mode, Phys. Rev. D 98 (2018) 065001 [arXiv:1806.06862] [INSPIRE].ADSGoogle Scholar
  26. [26]
    K. Kampf, J. Novotny and J. Trnka, Tree-level Amplitudes in the Nonlinear σ-model, JHEP 05 (2013) 032 [arXiv:1304.3048] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    H. Lüo and C. Wen, Recursion relations from soft theorems, JHEP 03 (2016) 088 [arXiv:1512.06801] [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
  29. [29]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press (2015) [INSPIRE].
  30. [30]
    J.M. Henn and J.C. Plefka, Scattering Amplitudes in Gauge Theories, Lect. Notes Phys. 883 (2014) 1 [INSPIRE].
  31. [31]
    L.J. Dixon, A brief introduction to modern amplitude methods, in Proceedings, 2012 European School of High-Energy Physics (ESHEP 2012), La Pommeraye, Anjou, France, June 06–19, 2012, pp. 31–67 (2014) [] [arXiv:1310.5353] [INSPIRE].
  32. [32]
    H. Elvang, C.R.T. Jones and S.G. Naculich, Soft Photon and Graviton Theorems in Effective Field Theory, Phys. Rev. Lett. 118 (2017) 231601 [arXiv:1611.07534] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1., Phys. Rev. 122 (1961) 345 [INSPIRE].
  34. [34]
    C. Cheung and C.-H. Shen, private communications.Google Scholar
  35. [35]
    C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, J. Trnka and C. Wen, Vector Effective Field Theories from Soft Limits, Phys. Rev. Lett. 120 (2018) 261602 [arXiv:1801.01496] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    C. de Rham and G. Gabadadze, Selftuned Massive Spin-2, Phys. Lett. B 693 (2010) 334 [arXiv:1006.4367] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
  38. [38]
    K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Supergravity and the S Matrix, Phys. Rev. D 15 (1977) 996 [INSPIRE].ADSGoogle Scholar
  40. [40]
    M.T. Grisaru and H.N. Pendleton, Some Properties of Scattering Amplitudes in Supersymmetric Theories, Nucl. Phys. B 124 (1977) 81 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    H. Elvang, D.Z. Freedman and M. Kiermaier, Solution to the Ward Identities for Superamplitudes, JHEP 10 (2010) 103 [arXiv:0911.3169] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    R.H. Boels and W. Wormsbecher, Spontaneously broken conformal invariance in observables, arXiv:1507.08162 [INSPIRE].
  43. [43]
    P. Di Vecchia, R. Marotta, M. Mojaza and J. Nohle, New soft theorems for the gravity dilaton and the Nambu-Goldstone dilaton at subsubleading order, Phys. Rev. D 93 (2016) 085015 [arXiv:1512.03316] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    M. Bianchi, A.L. Guerrieri, Y.-t. Huang, C.-J. Lee and C. Wen, Exploring soft constraints on effective actions, JHEP 10 (2016) 036 [arXiv:1605.08697] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    N. Bobev, H. Elvang and T.M. Olson, Dilaton effective action with N = 1 supersymmetry, JHEP 04 (2014) 157 [arXiv:1312.2925] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    N. Craig, H. Elvang, M. Kiermaier and T. Slatyer, Massive amplitudes on the Coulomb branch of N = 4 SYM, JHEP 12 (2011) 097 [arXiv:1104.2050] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    H. Elvang and M. Kiermaier, Stringy KLT relations, global symmetries and E 7(7) violation, JHEP 10 (2010) 108 [arXiv:1007.4813] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  49. [49]
    J. Broedel and L.J. Dixon, R 4 counterterm and E 7(7) symmetry in maximal supergravity, JHEP 05 (2010) 003 [arXiv:0911.5704] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    I. Low and Z. Yin, The Infrared Structure of Nambu-Goldstone Bosons, JHEP 10 (2018) 078 [arXiv:1804.08629] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  51. [51]
    B. Zumino, Supersymmetry and Kähler Manifolds, Phys. Lett. B 87 (1979) 203 [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge, U.K. (2012) [INSPIRE].CrossRefzbMATHGoogle Scholar
  53. [53]
    M. Bando, T. Kuramoto, T. Maskawa and S. Uehara, Structure of Nonlinear Realization in Supersymmetric Theories, Phys. Lett. B 138 (1984) 94 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].
  55. [55]
    A. Laddha and P. Mitra, Asymptotic Symmetries and Subleading Soft Photon Theorem in Effective Field Theories, JHEP 05 (2018) 132 [arXiv:1709.03850] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    M. Heydeman, J.H. Schwarz and C. Wen, M5-Brane and D-brane Scattering Amplitudes, JHEP 12 (2017) 003 [arXiv:1710.02170] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Galileons as Wess-Zumino Terms, JHEP 06 (2012) 004 [arXiv:1203.3191] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    C. de Rham and A.J. Tolley, DBI and the Galileon reunited, JCAP 05 (2010) 015 [arXiv:1003.5917] [INSPIRE].CrossRefGoogle Scholar
  59. [59]
    J. Beltran Jimenez and L. Heisenberg, Derivative self-interactions for a massive vector field, Phys. Lett. B 757 (2016) 405 [arXiv:1602.03410] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  60. [60]
    A. Padilla, D. Stefanyszyn and T. Wilson, Probing Scalar Effective Field Theories with the Soft Limits of Scattering Amplitudes, JHEP 04 (2017) 015 [arXiv:1612.04283] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    J. Broedel and L.J. Dixon, Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators, JHEP 10 (2012) 091 [arXiv:1208.0876] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    G. Chen and Y.-J. Du, Amplitude Relations in Non-linear σ-model, JHEP 01 (2014) 061 [arXiv:1311.1133] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    K. Kampf, J. Novotny and J. Trnka, Recursion relations for tree-level amplitudes in the SU(N) nonlinear σ-model, Phys. Rev. D 87 (2013) 081701 [arXiv:1212.5224] [INSPIRE].ADSGoogle Scholar
  65. [65]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2019

Authors and Affiliations

  1. 1.Leinweber Center for Theoretical Physics, Randall Laboratory of Physics, Department of PhysicsUniversity of MichiganAnn ArborU.S.A.

Personalised recommendations