Abstract
We study the scattering of lumps in the 2+1-dimensional Ising CFT, indirectly, by analytically continuing its spectrum using the Lorentzian inversion formula. We find evidence that the intercept of the model is below unity: j* ≈ 0.8, indicating that scattering is asymptotically transparent corresponding to a negative Lyapunov exponent. We use as input the precise spectrum obtained from the numerical conformal bootstrap. We show that the truncated spectrum allows the inversion formula to reproduce the properties of the spin-two stress tensor to 10−4 accuracy and we address the question of whether the spin-0 operators of the model lie on Regge trajectories. This hypothesis is further supported by analytics in the large-N O(N) model. Finally, we show that anomalous dimensions of heavy operators decrease with energy at a rate controlled by (j* − 1), implying regularity of the heavy spectrum.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
L.F. Alday and A. Zhiboedov, Conformal Bootstrap With Slightly Broken Higher Spin Symmetry, JHEP 06 (2016) 091 [arXiv:1506.04659] [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
U. Amaldi, An ISR Discovery: The Rise of the Proton-Proton Cross-Section, in 60 Years of CERN Experiments and Discoveries, World Scientific (2015), pp. 257–286 [DOI].
J. Murugan, D. Stanford and E. Witten, More on Supersymmetric and 2d Analogs of the SYK Model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
P. Caputa, T. Numasawa and A. Veliz-Osorio, Out-of-time-ordered correlators and purity in rational conformal field theories, PTEP 2016 (2016) 113B06 [arXiv:1602.06542] [INSPIRE].
R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, The Pomeron and gauge/string duality, JHEP 12 (2007) 005 [hep-th/0603115] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
M.J. Menon and P.V.R.G. Silva, A study on analytic parametrizations for proton–proton cross-sections and asymptotia, J. Phys. G 40 (2013) 125001 [arXiv:1305.2947] [INSPIRE].
L.F. Alday, J. Henriksson and M. van Loon, Taming the ϵ-expansion with large spin perturbation theory, JHEP 07 (2018) 131 [arXiv:1712.02314] [INSPIRE].
S. Albayrak, D. Meltzer and D. Poland, More Analytic Bootstrap: Nonperturbative Effects and Fermions, JHEP 08 (2019) 040 [arXiv:1904.00032] [INSPIRE].
M. Hogervorst, Dimensional Reduction for Conformal Blocks, JHEP 09 (2016) 017 [arXiv:1604.08913] [INSPIRE].
J. Liu, D. Meltzer, D. Poland and D. Simmons-Duffin, The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT, JHEP 09 (2020) 115 [arXiv:2007.07914] [INSPIRE].
D. Simmons-Duffin, Projectors, Shadows, and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
M.S. Costa, T. Hansen and J.a. Penedones, Bounds for OPE coefficients on the Regge trajectory, JHEP 10 (2017) 197 [arXiv:1707.07689] [INSPIRE].
S. Kundu, A Generalized Nachtmann Theorem in CFT, JHEP 11 (2020) 138 [arXiv:2002.12390] [INSPIRE].
L.F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017 [arXiv:1711.02031] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Harmonic Analysis and Mean Field Theory, JHEP 10 (2019) 217 [arXiv:1809.05111] [INSPIRE].
C. Cardona and K. Sen, Anomalous dimensions at finite conformal spin from OPE inversion, JHEP 11 (2018) 052 [arXiv:1806.10919] [INSPIRE].
C. Sleight and M. Taronna, Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications, Fortsch. Phys. 66 (2018) 1800038 [arXiv:1804.09334] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].
B. Basso and G.P. Korchemsky, Anomalous dimensions of high-spin operators beyond the leading order, Nucl. Phys. B 775 (2007) 1 [hep-th/0612247] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, M.T. Walters and J. Wang, Eikonalization of Conformal Blocks, JHEP 09 (2015) 019 [arXiv:1504.01737] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
S. Ferrara, A.F. Grillo and R. Gatto, Manifestly conformal-covariant expansion on the light cone, Phys. Rev. D 5 (1972) 3102 [INSPIRE].
S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, Covariant expansion of the conformal four-point function, Nucl. Phys. B 49 (1972) 77 [INSPIRE].
S. Ferrara, R. Gatto and A.F. Grillo, Properties of Partial Wave Amplitudes in Conformal Invariant Field Theories, Nuovo Cim. A 26 (1975) 226 [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
L.F. Alday, J. Henriksson and M. van Loon, An alternative to diagrams for the critical O(N) model: dimensions and structure constants to order 1/N2, JHEP 01 (2020) 063 [arXiv:1907.02445] [INSPIRE].
L.N. Lipatov, Small-x physics in perturbative QCD, Phys. Rept. 286 (1997) 131 [hep-ph/9610276] [INSPIRE].
L. Cornalba, M.S. Costa and J. Penedones, Eikonal Methods in AdS/CFT: BFKL Pomeron at Weak Coupling, JHEP 06 (2008) 048 [arXiv:0801.3002] [INSPIRE].
S. Caron-Huot, When does the gluon reggeize?, JHEP 05 (2015) 093 [arXiv:1309.6521] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
A.N. Vasiliev, Y.M. Pismak and Y.R. Khonkonen, 1/n Expansion: Calculation of the exponent ν in the order 1/n3 by the Conformal Bootstrap Method, Theor. Math. Phys. 50 (1982) 127 [INSPIRE].
K. Lang and W. Rühl, The Critical O(N) σ-model at dimensions 2 < d < 4: A List of quasiprimary fields, Nucl. Phys. B 402 (1993) 573 [INSPIRE].
T. Leonhardt and W. Rühl, The Minimal conformal O(N) vector sigma model at d = 3, J. Phys. A 37 (2004) 1403 [hep-th/0308111] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions, Nucl. Phys. B 767 (2007) 327 [hep-th/0611123] [INSPIRE].
S. Caron-Huot and J. Sandor, Power corrections to the Regge limit in the fishnet model, in preparation.
B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean Bootstrap, JHEP 10 (2019) 270 [arXiv:1808.03212] [INSPIRE].
S. Caron-Huot and Y.-Z. Li, Helicity basis for three-dimensional conformal field theory, Journal of High Energy Physics 2021 (2021) 41 [arXiv:2102.08160].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].
W. Li, Lightcone expansions of conformal blocks in closed form, JHEP 06 (2020) 105 [arXiv:1912.01168] [INSPIRE].
J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin, d-dimensional SYK, AdS Loops, and 6j Symbols, JHEP 03 (2019) 052 [arXiv:1808.00612] [INSPIRE].
C. Cardona, S. Guha, S.K. Kanumilli and K. Sen, Resummation at finite conformal spin, JHEP 01 (2019) 077 [arXiv:1811.00213] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2007.11647
Rights and permissions
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.
About this article
Cite this article
Caron-Huot, S., Gobeil, Y. & Zahraee, Z. The leading trajectory in the 2+1D Ising CFT. J. High Energ. Phys. 2023, 190 (2023). https://doi.org/10.1007/JHEP02(2023)190
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2023)190