Journal of High Energy Physics

, 2017:1 | Cite as

Double-soft behavior of the dilaton of spontaneously broken conformal invariance

  • Paolo Di Vecchia
  • Raffaele Marotta
  • Matin Mojaza
Open Access
Regular Article - Theoretical Physics


The Ward identities involving the currents associated to the spontaneously broken scale and special conformal transformations are derived and used to determine, through linear order in the two soft-dilaton momenta, the double-soft behavior of scattering amplitudes involving two soft dilatons and any number of other particles. It turns out that the double-soft behavior is equivalent to performing two single-soft limits one after the other. We confirm the new double-soft theorem perturbatively at tree-level in a D-dimensional conformal field theory model, as well as nonperturbatively by using the “gravity dual” of \( \mathcal{N}=4 \) super Yang-Mills on the Coulomb branch; i.e. the Dirac-Born-Infeld action on AdS5 × S 5.


Space-Time Symmetries Spontaneous Symmetry Breaking Scattering Amplitudes Conformal Field Models in String Theory 


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]
    H.B. Nielsen and S. Chadha, On How to Count Goldstone Bosons, Nucl. Phys. B 105 (1976) 445 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    K. Higashijima, Nambu-Goldstone theorem for conformal symmetry, in Toyonaka 1994, Group theoretical methods in physics, pp. 223–228.Google Scholar
  3. [3]
    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
  4. [4]
    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
  5. [5]
    S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial-vector current. II, Phys. Rev. 139 (1965) B1638 [INSPIRE].
  6. [6]
    S. Weinberg, Current-Commutator Theory of Multiple Pion Production, Phys. Rev. Lett. 16 (1966) 879 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    G. Mack, Partially conserved dilatation current, Nucl. Phys. B 5 (1968) 499 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    D.J. Gross and J. Wess, Scale invariance, conformal invariance and the high-energy behavior of scattering amplitudes, Phys. Rev. D 2 (1970) 753 [INSPIRE].ADSGoogle Scholar
  9. [9]
    F. Gürsey, On a conform-invariant spinor wave equation, Nuovo Cim. 3 (1956) 988.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J. Wess, The Conformal Invariance in Quantum Field Theory, Nuovo Cim. 18 (1960) 1086.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    H.A. Kastrup, On the physical interpretation and representation-theoretic analysis of the conformal transformations of space and time, Annalen Phys. 464 (1962) 388 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    T. Fulton, F. Rohrlich and L. Witten, Conformal invariance in physics, Rev. Mod. Phys. 34 (1962) 442 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    C.G. Callan Jr., Broken scale invariance in scalar field theory, Phys. Rev. D 2 (1970) 1541 [INSPIRE].ADSGoogle Scholar
  14. [14]
    S.R. Coleman and R. Jackiw, Why dilatation generators do not generate dilatations?, Annals Phys. 67 (1971) 552 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    R.H. Boels and W. Wormsbecher, Spontaneously broken conformal invariance in observables, arXiv:1507.08162 [INSPIRE].
  16. [16]
    Y.-t. Huang and C. Wen, Soft theorems from anomalous symmetries, JHEP 12 (2015) 143 [arXiv:1509.07840] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    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
  18. [18]
    F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    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
  20. [20]
    O. Antipin, M. Mojaza and F. Sannino, Light Dilaton at Fixed Points and Ultra Light Scale Super Yang-Mills, Phys. Lett. B 712 (2012) 119 [arXiv:1107.2932] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    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].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    R.F. Dashen and M. WEinstein, Soft pions, chiral symmetry and phenomenological lagrangians, Phys. Rev. 183 (1969) 1261 [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    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
  25. [25]
    I. Low, Double Soft Theorems and Shift Symmetry in Nonlinear σ-models, Phys. Rev. D 93 (2016) 045032 [arXiv:1512.01232] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    Y.-J. Du and H. Lüo, On single and double soft behaviors in NLSM, JHEP 08 (2015) 058 [arXiv:1505.04411] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M. Campiglia, L. Coito and S. Mizera, Can scalars have asymptotic symmetries?, arXiv:1703.07885 [INSPIRE].
  30. [30]
    H. Lüo and C. Wen, Recursion relations from soft theorems, JHEP 03 (2016) 088 [arXiv:1512.06801] [INSPIRE].CrossRefGoogle Scholar
  31. [31]
    A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
  32. [32]
    M. Ademollo et al., Soft Dilations and Scale Renormalization in Dual Theories, Nucl. Phys. B 94 (1975) 221 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    J.A. Shapiro, On the Renormalization of Dual Models, Phys. Rev. D 11 (1975) 2937 [INSPIRE].ADSGoogle Scholar
  34. [34]
    P. Di Vecchia, R. Marotta and M. Mojaza, Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string, JHEP 05 (2015) 137 [arXiv:1502.05258] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    P. Di Vecchia, R. Marotta and M. Mojaza, Subsubleading soft theorems of gravitons and dilatons in the bosonic string, JHEP 06 (2016) 054 [arXiv:1604.03355] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  36. [36]
    P. Di Vecchia, R. Marotta and M. Mojaza, Soft behavior of a closed massless state in superstring and universality in the soft behavior of the dilaton, JHEP 12 (2016) 020 [arXiv:1610.03481] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  37. [37]
    G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    C.G. Callan Jr., S.R. Coleman and R. Jackiw, A new improved energy - momentum tensor, Annals Phys. 59 (1970) 42 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer, (1997),
  40. [40]
    W.D. Goldberger, B. Grinstein and W. Skiba, Distinguishing the Higgs boson from the dilaton at the Large Hadron Collider, Phys. Rev. Lett. 100 (2008) 111802 [arXiv:0708.1463] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    A. Schwimmer and S. Theisen, Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching, Nucl. Phys. B 847 (2011) 590 [arXiv:1011.0696] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    H. Elvang, D.Z. Freedman, L.-Y. Hung, M. Kiermaier, R.C. Myers and S. Theisen, On renormalization group flows and the a-theorem in 6d, JHEP 10 (2012) 011 [arXiv:1205.3994] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    H. Elvang and T.M. Olson, RG flows in d dimensions, the dilaton effective action and the a-theorem, JHEP 03 (2013) 034 [arXiv:1209.3424] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    M.E. Shaposhnikov and F.V. Tkachov, Quantum scale-invariant models as effective field theories, arXiv:0905.4857 [INSPIRE].
  46. [46]
    R. Armillis, A. Monin and M. Shaposhnikov, Spontaneously Broken Conformal Symmetry: Dealing with the Trace Anomaly, JHEP 10 (2013) 030 [arXiv:1302.5619] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    F. Gretsch and A. Monin, Perturbative conformal symmetry and dilaton, Phys. Rev. D 92 (2015) 045036 [arXiv:1308.3863] [INSPIRE].ADSMathSciNetGoogle Scholar
  48. [48]
    F. Englert, C. Truffin and R. Gastmans, Conformal Invariance in Quantum Gravity, Nucl. Phys. B 117 (1976) 407 [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    S.B. Treiman, E. Witten, R. Jackiw and B. Zumino, Current Algebra And Anomalies, World Scientific, Singapore (1985),
  50. [50]
    D.J. Gross and R. Jackiw, Construction of covariant and gauge invariant t* products, Nucl. Phys. B 14 (1969) 269 [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    S. Weinberg, The Quatum Theory of Fields, Volume II, Modern Applications, Cambridge University Press, (1996).Google Scholar
  52. [52]
    A.L. Guerrieri, Y.-t. Huang, Z. Li and C. Wen, On the exactness of soft theorems, arXiv:1705.10078 [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Paolo Di Vecchia
    • 1
    • 2
  • Raffaele Marotta
    • 3
  • Matin Mojaza
    • 4
  1. 1.The Niels Bohr InstituteUniversity of CopenhagenCopenhagen ØDenmark
  2. 2.Nordita, KTH Royal Institute of Technology and Stockholm UniversityStockholmSweden
  3. 3.Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte S. Angelo ed. 6NapoliItaly
  4. 4.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutPotsdamGermany

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