Abstract
Exceptional theories are a group of one-parameter scalar field theories with (enhanced) vanishing soft limits in the S-matrix elements. They include the nonlinear sigma model (NLSM), Dirac-Born-Infeld scalars and the special Galileon theory. The soft behavior results from the shift symmetry underlying these theories, which leads to Ward identities generating subleading single soft theorems as well as novel Berends-Giele recursion relations. Such an approach was first applied to NLSM in refs. [1, 2], and here we use it to systematically study other exceptional scalar field theories. In particular, using the subleading single soft theorem for the special Galileon we identify the Feynman vertices of the corresponding extended theory, which was first discovered using the Cachazo-He-Yuan representation of scattering amplitudes. Furthermore, we present a Lagrangian for the extended theory of the special Galileon, which has a rich particle content involving biadjoint scalars, Nambu-Goldstone bosons and Galileons, as well as additional flavor structure.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Low and Z. Yin, Ward identity and scattering amplitudes for nonlinear σ-models, Phys. Rev. Lett. 120 (2018) 061601 [arXiv:1709.08639] [INSPIRE].
I. Low and Z. Yin, The infrared structure of Nambu-Goldstone bosons, JHEP 10 (2018) 078 [arXiv:1804.08629] [INSPIRE].
F.E. Low, Scattering of light of very low frequency by systems of spin 1/2, Phys. Rev. 96 (1954) 1428 [INSPIRE].
M. Gell-Mann and M.L. Goldberger, Scattering of low-energy photons by particles of spin 1/2, Phys. Rev. 96 (1954) 1433 [INSPIRE].
F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].
T.H. Burnett and N.M. Kroll, Extension of the low soft photon theorem, Phys. Rev. Lett. 20 (1968) 86 [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].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
D.J. Gross and R. Jackiw, Low-energy theorem for graviton scattering, Phys. Rev. 166 (1968) 1287 [INSPIRE].
R. Jackiw, Low-energy theorems for massless bosons: photons and gravitons, Phys. Rev. 168 (1968) 1623 [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.B. Treiman, E. Witten, R. Jackiw and B. Zumino, Current algebra and anomalies, World Scientific, Singapore (1986).
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [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].
T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New symmetries of massless QED, JHEP 10 (2014) 112 [arXiv:1407.3789] [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a new soft graviton theorem, arXiv:1404.4091 [INSPIRE].
E. Casali, Soft sub-leading divergences in Yang-Mills amplitudes, JHEP 08 (2014) 077 [arXiv:1404.5551] [INSPIRE].
A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [INSPIRE].
Y. Hamada and G. Shiu, Infinite set of soft theorems in gauge-gravity theories as Ward-Takahashi identities, Phys. Rev. Lett. 120 (2018) 201601 [arXiv:1801.05528] [INSPIRE].
R.F. Dashen and M. WEinstein, Soft pions, chiral symmetry and phenomenological lagrangians, Phys. Rev. 183 (1969) 1261 [INSPIRE].
S. Weinberg, Current-commutator theory of multiple pion production, Phys. Rev. Lett. 16 (1966) 879.
S. Weinberg, Summing soft pions, Phys. Rev. D 2 (1970) 674 [INSPIRE].
L. Susskind and G. Frye, Algebraic aspects of pionic duality diagrams, Phys. Rev. D 1 (1970) 1682 [INSPIRE].
J.R. Ellis, The Adler zero condition and current algebra, Nucl. Phys. B 21 (1970) 217 [INSPIRE].
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].
C. Cheung et al., A periodic table of effective field theories, JHEP 02 (2017) 020 [arXiv:1611.03137] [INSPIRE].
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].
K. Kampf, J. Novotny and J. Trnka, Tree-level amplitudes in the nonlinear σ-model, JHEP 05 (2013) 032 [arXiv:1304.3048] [INSPIRE].
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].
K. Hinterbichler and A. Joyce, Hidden symmetry of the Galileon, Phys. Rev. D 92 (2015) 023503 [arXiv:1501.07600] [INSPIRE].
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].
M.P. Bogers and T. Brauner, Lie-algebraic classification of effective theories with enhanced soft limits, JHEP 05 (2018) 076 [arXiv:1803.05359] [INSPIRE].
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].
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].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
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].
F. Cachazo, S. He and E.Y. Yuan, New double soft emission theorems, Phys. Rev. D 92 (2015) 065030 [arXiv:1503.04816] [INSPIRE].
F. Cachazo, P. Cha and S. Mizera, Extensions of theories from soft limits, JHEP 06 (2016) 170 [arXiv:1604.03893] [INSPIRE].
Y.-J. Du and H. Lüo, On single and double soft behaviors in NLSM, JHEP 08 (2015) 058 [arXiv:1505.04411] [INSPIRE].
Y.-J. Du and H. Lüo, Leading order multi-soft behaviors of tree amplitudes in NLSM, JHEP 03 (2017) 062 [arXiv:1611.07479] [INSPIRE].
I. Low, Double soft theorems and shift symmetry in nonlinear σ-models, Phys. Rev. D 93 (2016) 045032 [arXiv:1512.01232] [INSPIRE].
I. Low, Adler’s zero and effective Lagrangians for nonlinearly realized symmetry, Phys. Rev. D 91 (2015) 105017 [arXiv:1412.2145] [INSPIRE].
I. Low, Minimally symmetric Higgs boson, Phys. Rev. D 91 (2015) 116005 [arXiv:1412.2146] [INSPIRE].
S. Weinberg, The quantum theory of fields. Volume 2: modern applications, Cambridge University Press, Cambridge U.K. (2013).
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
R.G. Leigh, Dirac-Born-Infeld action from Dirichlet σ-model, Mod. Phys. Lett. A 4 (1989) 2767 [INSPIRE].
A.A. Tseytlin, Born-Infeld action, supersymmetry and string theory, hep-th/9908105 [INSPIRE].
T. Brauner and H. Watanabe, Spontaneous breaking of spacetime symmetries and the inverse Higgs effect, Phys. Rev. D 89 (2014) 085004 [arXiv:1401.5596] [INSPIRE].
M. Fasiello and A.J. Tolley, Cosmological stability bound in massive gravity and bigravity, JCAP 12 (2013) 002 [arXiv:1308.1647] [INSPIRE].
C. de Rham, M. Fasiello and A.J. Tolley, Galileon duality, Phys. Lett. B 733 (2014) 46 [arXiv:1308.2702] [INSPIRE].
K. Kampf and J. Novotny, Unification of galileon dualities, JHEP 10 (2014) 006 [arXiv:1403.6813] [INSPIRE].
J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Semi-abelian Z-theory: NLSM + ϕ 3 from the open string, JHEP 08 (2017) 135 [arXiv:1612.06446] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, R. Monteiro and D. O’Connell, Algebras for Amplitudes, JHEP 06 (2012) 061 [arXiv:1203.0944] [INSPIRE].
C. Cheung and C.-H. Shen, Symmetry for flavor-kinematics duality from an action, Phys. Rev. Lett. 118 (2017) 121601 [arXiv:1612.00868] [INSPIRE].
S. Mizera and B. Skrzypek, Perturbiner methods for effective field theories and the double copy, JHEP 10 (2018) 018 [arXiv:1809.02096] [INSPIRE].
C. Cheung, C.-H. Shen and C. Wen, Unifying relations for scattering amplitudes, JHEP 02 (2018) 095 [arXiv:1705.03025] [INSPIRE].
K. Zhou and B. Feng, Note on differential operators, CHY integrands and unifying relations for amplitudes, JHEP 09 (2018) 160 [arXiv:1808.06835] [INSPIRE].
M. Bollmann and L. Ferro, Transmuting CHY formulae, JHEP 01 (2019) 180 [arXiv:1808.07451] [INSPIRE].
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].
C. Cheung, G.N. Remmen, C.-H. Shen and C. Wen, Pions as gluons in higher dimensions, JHEP 04 (2018) 129 [arXiv:1709.04932] [INSPIRE].
T. Griffin, K.T. Grosvenor, P. Hořava and Z. Yan, Scalar field theories with polynomial shift symmetries, Commun. Math. Phys. 340 (2015) 985 [arXiv:1412.1046] [INSPIRE].
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95.
E. Witten, Global aspects of current algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
H. Elvang, M. Hadjiantonis, C.R.T. Jones and S. Paranjape, Soft bootstrap and supersymmetry, JHEP 01 (2019) 195 [arXiv:1806.06079] [INSPIRE].
L. Rodina, Scattering amplitudes from soft theorems and infrared behavior, Phys. Rev. Lett. 122 (2019) 071601 [arXiv:1807.09738] [INSPIRE].
D. Liu, I. Low and Z. Yin, Universal imprints of a pseudo-Nambu-Goldstone Higgs boson, Phys. Rev. Lett. 121 (2018) 261802 [arXiv:1805.00489] [INSPIRE].
D. Liu, I. Low and Z. Yin, Universal relations in composite Higgs models, arXiv:1809.09126 [INSPIRE].
B. Finelli, G. Goon, E. Pajer and L. Santoni, The effective theory of shift-symmetric cosmologies, JCAP 05 (2018) 060 [arXiv:1802.01580] [INSPIRE].
C. Deffayet, X. Gao, D.A. Steer and G. Zahariade, From k-essence to generalised Galileons, Phys. Rev. D 84 (2011) 064039 [arXiv:1103.3260] [INSPIRE].
C. Deffayet and D.A. Steer, A formal introduction to Horndeski and Galileon theories and their generalizations, Class. Quant. Grav. 30 (2013) 214006 [arXiv:1307.2450] [INSPIRE].
J.A. Cronin, Phenomenological model of strong and weak interactions in chiral U(3) × U(3), Phys. Rev. 161 (1967) 1483 [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1810.07186
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yin, Z. The infrared structure of exceptional scalar theories. J. High Energ. Phys. 2019, 158 (2019). https://doi.org/10.1007/JHEP03(2019)158
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2019)158