Abstract
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.
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References
H.B. Nielsen and S. Chadha, On How to Count Goldstone Bosons, Nucl. Phys. B 105 (1976) 445 [INSPIRE].
K. Higashijima, Nambu-Goldstone theorem for conformal symmetry, in Toyonaka 1994, Group theoretical methods in physics, pp. 223–228.
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].
S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev. 137 (1965) B1022 [INSPIRE].
S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial-vector current. II, Phys. Rev. 139 (1965) B1638 [INSPIRE].
S. Weinberg, Current-Commutator Theory of Multiple Pion Production, Phys. Rev. Lett. 16 (1966) 879 [INSPIRE].
G. Mack, Partially conserved dilatation current, Nucl. Phys. B 5 (1968) 499 [INSPIRE].
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].
F. Gürsey, On a conform-invariant spinor wave equation, Nuovo Cim. 3 (1956) 988.
J. Wess, The Conformal Invariance in Quantum Field Theory, Nuovo Cim. 18 (1960) 1086.
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].
T. Fulton, F. Rohrlich and L. Witten, Conformal invariance in physics, Rev. Mod. Phys. 34 (1962) 442 [INSPIRE].
C.G. Callan Jr., Broken scale invariance in scalar field theory, Phys. Rev. D 2 (1970) 1541 [INSPIRE].
S.R. Coleman and R. Jackiw, Why dilatation generators do not generate dilatations?, Annals Phys. 67 (1971) 552 [INSPIRE].
R.H. Boels and W. Wormsbecher, Spontaneously broken conformal invariance in observables, arXiv:1507.08162 [INSPIRE].
Y.-t. Huang and C. Wen, Soft theorems from anomalous symmetries, JHEP 12 (2015) 143 [arXiv:1509.07840] [INSPIRE].
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].
F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].
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].
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].
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].
R.F. Dashen and M. WEinstein, Soft pions, chiral symmetry and phenomenological lagrangians, Phys. Rev. 183 (1969) 1261 [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
K. Kampf, J. Novotny and J. Trnka, Tree-level Amplitudes in the Nonlinear σ-model, JHEP 05 (2013) 032 [arXiv:1304.3048] [INSPIRE].
I. Low, Double Soft Theorems and Shift Symmetry in Nonlinear σ-models, Phys. Rev. D 93 (2016) 045032 [arXiv:1512.01232] [INSPIRE].
Y.-J. Du and H. Lüo, On single and double soft behaviors in NLSM, JHEP 08 (2015) 058 [arXiv:1505.04411] [INSPIRE].
A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
M. Campiglia, L. Coito and S. Mizera, Can scalars have asymptotic symmetries?, arXiv:1703.07885 [INSPIRE].
H. Lüo and C. Wen, Recursion relations from soft theorems, JHEP 03 (2016) 088 [arXiv:1512.06801] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
M. Ademollo et al., Soft Dilations and Scale Renormalization in Dual Theories, Nucl. Phys. B 94 (1975) 221 [INSPIRE].
J.A. Shapiro, On the Renormalization of Dual Models, Phys. Rev. D 11 (1975) 2937 [INSPIRE].
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].
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].
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].
G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].
C.G. Callan Jr., S.R. Coleman and R. Jackiw, A new improved energy - momentum tensor, Annals Phys. 59 (1970) 42 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer, (1997), https://doi.org/10.1007/978-1-4612-2256-9.
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].
A. Schwimmer and S. Theisen, Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching, Nucl. Phys. B 847 (2011) 590 [arXiv:1011.0696] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
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].
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].
M.E. Shaposhnikov and F.V. Tkachov, Quantum scale-invariant models as effective field theories, arXiv:0905.4857 [INSPIRE].
R. Armillis, A. Monin and M. Shaposhnikov, Spontaneously Broken Conformal Symmetry: Dealing with the Trace Anomaly, JHEP 10 (2013) 030 [arXiv:1302.5619] [INSPIRE].
F. Gretsch and A. Monin, Perturbative conformal symmetry and dilaton, Phys. Rev. D 92 (2015) 045036 [arXiv:1308.3863] [INSPIRE].
F. Englert, C. Truffin and R. Gastmans, Conformal Invariance in Quantum Gravity, Nucl. Phys. B 117 (1976) 407 [INSPIRE].
S.B. Treiman, E. Witten, R. Jackiw and B. Zumino, Current Algebra And Anomalies, World Scientific, Singapore (1985), https://doi.org/10.1142/0131.
D.J. Gross and R. Jackiw, Construction of covariant and gauge invariant t* products, Nucl. Phys. B 14 (1969) 269 [INSPIRE].
S. Weinberg, The Quatum Theory of Fields, Volume II, Modern Applications, Cambridge University Press, (1996).
A.L. Guerrieri, Y.-t. Huang, Z. Li and C. Wen, On the exactness of soft theorems, arXiv:1705.10078 [INSPIRE].
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Di Vecchia, P., Marotta, R. & Mojaza, M. Double-soft behavior of the dilaton of spontaneously broken conformal invariance. J. High Energ. Phys. 2017, 1 (2017). https://doi.org/10.1007/JHEP09(2017)001
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DOI: https://doi.org/10.1007/JHEP09(2017)001