A periodic table of effective field theories. (English) Zbl 1377.81123
Summary: We systematically explore the space of scalar effective field theories (EFTs) consistent with a Lorentz invariant and local S-matrix. To do so we define an EFT classification based on four parameters characterizing 1) the number of derivatives per interaction, 2) the soft properties of amplitudes, 3) the leading valency of the interactions, and 4) the spacetime dimension. Carving out the allowed space of EFTs, we prove that exceptional EFTs like the non-linear sigma model, Dirac-Born-Infeld theory, and the special Galileon lie precisely on the boundary of allowed theory space. Using on-shell momentum shifts and recursion relations, we prove that EFTs with arbitrarily soft behavior are forbidden and EFTs with leading valency much greater than the spacetime dimension cannot have enhanced soft behavior. We then enumerate all single scalar EFTs in \(d < 6\) and verify that they correspond to known theories in the literature. Our results suggest that the exceptional theories are the natural EFT analogs of gauge theory and gravity because they are one-parameter theories whose interactions are strictly dictated by properties of the S-matrix.
MSC:
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 81U20 | \(S\)-matrix theory, etc. in quantum theory |
References:
| [1] | 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]. · Zbl 1207.81088 · doi:10.1016/j.nuclphysb.2005.02.030 |
| [2] | 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]. · doi:10.1103/PhysRevLett.94.181602 |
| [3] | N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP09 (2010) 016 [arXiv:0808.1446] [INSPIRE]. · Zbl 1291.81356 · doi:10.1007/JHEP09(2010)016 |
| [4] | Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys.B 435 (1995) 59 [hep-ph/9409265] [INSPIRE]. · Zbl 1049.81644 |
| [5] | Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys.B 425 (1994) 217 [hep-ph/9403226] [INSPIRE]. · Zbl 1049.81644 |
| [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]. · doi:10.1103/PhysRevLett.113.171601 |
| [7] | F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP07 (2014) 033 [arXiv:1309.0885] [INSPIRE]. · Zbl 1391.81198 · doi:10.1007/JHEP07(2014)033 |
| [8] | F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP07 (2015) 149 [arXiv:1412.3479] [INSPIRE]. · Zbl 1388.83196 · doi:10.1007/JHEP07(2015)149 |
| [9] | 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]. |
| [10] | Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett.105 (2010) 061602 [arXiv:1004.0476] [INSPIRE]. · doi:10.1103/PhysRevLett.105.061602 |
| [11] | Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Gravity as the Square of Gauge Theory, Phys. Rev.D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE]. |
| [12] | L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP12 (2013) 049 [arXiv:1308.2276] [INSPIRE]. · Zbl 1342.81159 · doi:10.1007/JHEP12(2013)049 |
| [13] | S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett.117 (2016) 241601 [arXiv:1609.00669] [INSPIRE]. · doi:10.1103/PhysRevLett.117.241601 |
| [14] | B. Basso, A. Sever and P. Vieira, Spacetime and Flux Tube S-Matrices at Finite Coupling for N =4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett.111 (2013) 091602 [arXiv:1303.1396] [INSPIRE]. · doi:10.1103/PhysRevLett.111.091602 |
| [15] | B. Basso, J. Caetano, L. Cordova, A. Sever and P. Vieira, OPE for all Helicity Amplitudes, JHEP08 (2015) 018 [arXiv:1412.1132] [INSPIRE]. · Zbl 1388.81277 · doi:10.1007/JHEP08(2015)018 |
| [16] | E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys.252 (2004) 189 [hep-th/0312171] [INSPIRE]. · Zbl 1105.81061 · doi:10.1007/s00220-004-1187-3 |
| [17] | R. Roiban, M. Spradlin and A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev.D 70 (2004) 026009 [hep-th/0403190] [INSPIRE]. |
| [18] | F. Cachazo and D. Skinner, Gravity from Rational Curves in Twistor Space, Phys. Rev. Lett.110 (2013) 161301 [arXiv:1207.0741] [INSPIRE]. |
| [19] | F. Cachazo, L. Mason and D. Skinner, Gravity in Twistor Space and its Grassmannian Formulation, SIGMA10 (2014) 051 [arXiv:1207.4712] [INSPIRE]. · Zbl 1296.81160 |
| [20] | N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in Twistor Space, JHEP03 (2010) 110 [arXiv:0903.2110] [INSPIRE]. · Zbl 1271.81169 · doi:10.1007/JHEP03(2010)110 |
| [21] | L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP07 (2014) 048 [arXiv:1311.2564] [INSPIRE]. · doi:10.1007/JHEP07(2014)048 |
| [22] | E. Casali, Y. Geyer, L. Mason, R. Monteiro and K.A. Roehrig, New Ambitwistor String Theories, JHEP11 (2015) 038 [arXiv:1506.08771] [INSPIRE]. · Zbl 1388.81502 · doi:10.1007/JHEP11(2015)038 |
| [23] | Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett.115 (2015) 121603 [arXiv:1507.00321] [INSPIRE]. · doi:10.1103/PhysRevLett.115.121603 |
| [24] | Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP03 (2016) 114 [arXiv:1511.06315] [INSPIRE]. · Zbl 1388.81906 · doi:10.1007/JHEP03(2016)114 |
| [25] | N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP03 (2010) 020 [arXiv:0907.5418] [INSPIRE]. · Zbl 1271.81098 · doi:10.1007/JHEP03(2010)020 |
| [26] | N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, Cambridge University Press, (2012). · Zbl 1365.81004 |
| [27] | N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP10 (2014) 030 [arXiv:1312.2007] [INSPIRE]. · Zbl 1468.81075 · doi:10.1007/JHEP10(2014)030 |
| [28] | N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, JHEP12 (2014) 182 [arXiv:1312.7878] [INSPIRE]. · doi:10.1007/JHEP12(2014)182 |
| [29] | S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev.137 (1965) B1022. · doi:10.1103/PhysRev.137.B1022 |
| [30] | 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]. · doi:10.1103/PhysRevLett.114.221602 |
| [31] | J.A. Cronin, Phenomenological model of strong and weak interactions in chiral U(3) × U(3), Phys. Rev.161 (1967) 1483 [INSPIRE]. · doi:10.1103/PhysRev.161.1483 |
| [32] | S. Weinberg, Dynamical approach to current algebra, Phys. Rev. Lett.18 (1967) 188 [INSPIRE]. · doi:10.1103/PhysRevLett.18.188 |
| [33] | S. Weinberg, Nonlinear realizations of chiral symmetry, Phys. Rev.166 (1968) 1568 [INSPIRE]. · doi:10.1103/PhysRev.166.1568 |
| [34] | K. Hinterbichler and A. Joyce, Hidden symmetry of the Galileon, Phys. Rev.D 92 (2015) 023503 [arXiv:1501.07600] [INSPIRE]. |
| [35] | 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]. · doi:10.1103/PhysRevLett.116.041601 |
| [36] | G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys.10 (1974) 363 [INSPIRE]. · doi:10.1007/BF01807638 |
| [37] | 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]. |
| [38] | J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett.B 37 (1971) 95 [INSPIRE]. |
| [39] | E. Witten, Global Aspects of Current Algebra, Nucl. Phys.B 223 (1983) 422 [INSPIRE]. · doi:10.1016/0550-3213(83)90063-9 |
| [40] | B. Henning, X. Lu, T. Melia and H. Murayama, 2, 84, 30, 993, 560, 15456, 11962, 261485, …: Higher dimension operators in the SM EFT,arXiv:1512.03433[INSPIRE]. |
| [41] | 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]. |
| [42] | K. Kampf, J. Novotny and J. Trnka, Tree-level Amplitudes in the Nonlinear σ-model, JHEP05 (2013) 032 [arXiv:1304.3048] [INSPIRE]. · Zbl 1392.81139 · doi:10.1007/JHEP05(2013)032 |
| [43] | S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press, (2013). · Zbl 1264.81010 |
| [44] | 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]. |
| [45] | C. Cheung, On-Shell Recursion Relations for Generic Theories, JHEP03 (2010) 098 [arXiv:0808.0504] [INSPIRE]. · Zbl 1271.81102 · doi:10.1007/JHEP03(2010)098 |
| [46] | T. Cohen, H. Elvang and M. Kiermaier, On-shell constructibility of tree amplitudes in general field theories, JHEP04 (2011) 053 [arXiv:1010.0257] [INSPIRE]. · Zbl 1250.81072 · doi:10.1007/JHEP04(2011)053 |
| [47] | C. Cheung, C.-H. Shen and J. Trnka, Simple Recursion Relations for General Field Theories, JHEP06 (2015) 118 [arXiv:1502.05057] [INSPIRE]. · Zbl 1388.81257 · doi:10.1007/JHEP06(2015)118 |
| [48] | B. Feng, K. Zhou, C. Qiao and J. Rao, Determination of Boundary Contributions in Recursion Relation, JHEP03 (2015) 023 [arXiv:1411.0452] [INSPIRE]. · Zbl 1388.81741 · doi:10.1007/JHEP03(2015)023 |
| [49] | Q. Jin and B. Feng, Recursion Relation for Boundary Contribution, JHEP06 (2015) 018 [arXiv:1412.8170] [INSPIRE]. · Zbl 1388.81914 · doi:10.1007/JHEP06(2015)018 |
| [50] | Q. Jin and B. Feng, Boundary Operators of BCFW Recursion Relation, JHEP04 (2016) 123 [arXiv:1507.00463] [INSPIRE]. · Zbl 1388.81084 · doi:10.1007/JHEP04(2016)123 |
| [51] | C. Cheung, A. de la Fuente and R. Sundrum, 4D Scattering Amplitudes and Asymptotic Symmetries from 2D CFT, JHEP01 (2017) 112 [arXiv:1609.00732] [INSPIRE]. · Zbl 1373.81319 · doi:10.1007/JHEP01(2017)112 |
| [52] | H. Lüo and C. Wen, Recursion relations from soft theorems, JHEP03 (2016) 088 [arXiv:1512.06801] [INSPIRE]. · doi:10.1007/JHEP03(2016)088 |
| [53] | P. Benincasa and F. Cachazo, Consistency Conditions on the S-matrix of Massless Particles, arXiv:0705.4305 [INSPIRE]. |
| [54] | D.A. McGady and L. Rodina, Higher-spin massless S-matrices in four-dimensions, Phys. Rev.D 90 (2014) 084048 [arXiv:1311.2938] [INSPIRE]. |
| [55] | S. Weinberg, Infrared photons and gravitons, Phys. Rev.140 (1965) B516. · doi:10.1103/PhysRev.140.B516 |
| [56] | Y.-J. Du and H. Lüo, On single and double soft behaviors in NLSM, JHEP08 (2015) 058 [arXiv:1505.04411] [INSPIRE]. · doi:10.1007/JHEP08(2015)058 |
| [57] | I. Low, Double Soft Theorems and Shift Symmetry in Nonlinear σ-models, Phys. Rev.D 93 (2016) 045032 [arXiv:1512.01232] [INSPIRE]. |
| [58] | F. Cachazo, S. He and E.Y. Yuan, New Double Soft Emission Theorems, Phys. Rev.D 92 (2015) 065030 [arXiv:1503.04816] [INSPIRE]. |
| [59] | T. Klose, T. McLoughlin, D. Nandan, J. Plefka and G. Travaglini, Double-Soft Limits of Gluons and Gravitons, JHEP07 (2015) 135 [arXiv:1504.05558] [INSPIRE]. · Zbl 1388.83281 · doi:10.1007/JHEP07(2015)135 |
| [60] | P. Di Vecchia, R. Marotta and M. Mojaza, Double-soft behavior for scalars and gluons from string theory, JHEP12 (2015) 150 [arXiv:1507.00938] [INSPIRE]. · Zbl 1388.81970 · doi:10.1007/JHEP12(2015)150 |
| [61] | D. Nandan, J. Plefka and W. Wormsbecher, Collinear limits beyond the leading order from the scattering equations, arXiv:1608.04730 [INSPIRE]. · Zbl 1377.81221 |
| [62] | G. Chen and Y.-J. Du, Amplitude Relations in Non-linear σ-model, JHEP01 (2014) 061 [arXiv:1311.1133] [INSPIRE]. · Zbl 1390.81194 · doi:10.1007/JHEP01(2014)061 |
| [63] | F. Cachazo, P. Cha and S. Mizera, Extensions of Theories from Soft Limits, JHEP06 (2016) 170 [arXiv:1604.03893] [INSPIRE]. · Zbl 1388.81203 · doi:10.1007/JHEP06(2016)170 |
| [64] | J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Abelian Z-theory: NLSM amplitudes and α′-corrections from the open string, arXiv:1608.02569 [INSPIRE]. · Zbl 1380.83251 |
| [65] | A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP07 (2014) 151 [arXiv:1308.0589] [INSPIRE]. · Zbl 1333.81273 · doi:10.1007/JHEP07(2014)151 |
| [66] | T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New Symmetries of Massless QED, JHEP10 (2014) 112 [arXiv:1407.3789] [INSPIRE]. · doi:10.1007/JHEP10(2014)112 |
| [67] | T. He, P. Mitra and A. Strominger, 2D Kac-Moody Symmetry of 4D Yang-Mills Theory, JHEP10 (2016) 137 [arXiv:1503.02663] [INSPIRE]. · Zbl 1390.81630 · doi:10.1007/JHEP10(2016)137 |
| [68] | D. Kapec, M. Pate and A. Strominger, New Symmetries of QED, arXiv:1506.02906 [INSPIRE]. · Zbl 1383.81351 |
| [69] | A. Strominger, Magnetic Corrections to the Soft Photon Theorem, Phys. Rev. Lett.116 (2016) 031602 [arXiv:1509.00543] [INSPIRE]. · Zbl 1356.81175 · doi:10.1103/PhysRevLett.116.031602 |
| [70] | 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.A269 (1962) 21. · Zbl 0106.41903 |
| [71] | R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond.A270 (1962) 103. · Zbl 0101.43605 |
| [72] | G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett.105 (2010) 111103 [arXiv:0909.2617] [INSPIRE]. · doi:10.1103/PhysRevLett.105.111103 |
| [73] | A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP07 (2014) 152 [arXiv:1312.2229] [INSPIRE]. · Zbl 1392.81215 · doi:10.1007/JHEP07(2014)152 |
| [74] | T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP05 (2015) 151 [arXiv:1401.7026] [INSPIRE]. · Zbl 1388.83261 · doi:10.1007/JHEP05(2015)151 |
| [75] | D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP08 (2014) 058 [arXiv:1406.3312] [INSPIRE]. · doi:10.1007/JHEP08(2014)058 |
| [76] | V. Lysov, S. Pasterski and A. Strominger, Low’s Subleading Soft Theorem as a Symmetry of QED, Phys. Rev. Lett.113 (2014) 111601 [arXiv:1407.3814] [INSPIRE]. · doi:10.1103/PhysRevLett.113.111601 |
| [77] | T.T. Dumitrescu, T. He, P. Mitra and A. Strominger, Infinite-Dimensional Fermionic Symmetry in Supersymmetric Gauge Theories, arXiv:1511.07429 [INSPIRE]. · Zbl 1388.81084 |
| [78] | F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev.110 (1958) 974 [INSPIRE]. · Zbl 0082.42802 · doi:10.1103/PhysRev.110.974 |
| [79] | T.H. Burnett and N.M. Kroll, Extension of the low soft photon theorem, Phys. Rev. Lett.20 (1968) 86 [INSPIRE]. · doi:10.1103/PhysRevLett.20.86 |
| [80] | C.D. White, Factorization Properties of Soft Graviton Amplitudes, JHEP05 (2011) 060 [arXiv:1103.2981] [INSPIRE]. · Zbl 1296.83030 · doi:10.1007/JHEP05(2011)060 |
| [81] | F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE]. · Zbl 1392.81215 |
| [82] | G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Aspects of Galileon Non-Renormalization, JHEP11 (2016) 100 [arXiv:1606.02295] [INSPIRE]. · Zbl 1390.83279 · doi:10.1007/JHEP11(2016)100 |
| [83] | 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]. · Zbl 1328.81152 · doi:10.1007/s00220-015-2461-2 |
| [84] | K. Hinterbichler and A. Joyce, Goldstones with Extended Shift Symmetries, Int. J. Mod. Phys.D 23 (2014) 1443001 [arXiv:1404.4047] [INSPIRE]. · Zbl 1315.81093 · doi:10.1142/S0218271814430019 |
| [85] | K. Kampf and J. Novotny, Unification of Galileon Dualities, JHEP10 (2014) 006 [arXiv:1403.6813] [INSPIRE]. · doi:10.1007/JHEP10(2014)006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
