Abstract
We study a general class of functionals providing an analytic handle on the conformal bootstrap equations in one dimension. We explicitly identify the extremal functionals, corresponding to theories saturating conformal bootstrap bounds, in two regimes. The first corresponds to functionals that annihilate the generalized free fermion spectrum. In this case, we analytically find both OPE and gap maximization functionals proving the extremality of the generalized free fermion solution to crossing. Secondly, we consider a scaling limit where all conformal dimensions become large, equivalent to the large AdS radius limit of gapped theories in AdS2. In this regime we demonstrate analytically that optimal bounds on OPE coefficients lead to extremal solutions to crossing arising from integrable field theories placed in large AdS2. In the process, we uncover a close connection between asymptotic extremal functionals and S-matrices of integrable field theories in flat space and explain how 2D S-matrix bootstrap results can be derived from the 1D conformal bootstrap equations. These points illustrate that our formalism is capable of capturing non-trivial solutions of CFT crossing.
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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].
D. Poland and D. Simmons-Duffin, The conformal bootstrap, Nature Phys. 12 (2016) 535 [INSPIRE].
S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
G. Mack, Duality in quantum field theory, Nucl. Phys. B 118 (1977) 445 [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics, Springer (2016) [arXiv:1601.05000] [INSPIRE].
D. Simmons-Duffin, The Conformal Bootstrap, in proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., 1-26 June 2015, J. Polchinski, P. Vieira and O. DeWolfe eds., World Scientific (2017), pp. 1-74 [arXiv:1602.07982] [INSPIRE].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [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, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
L.F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].
L.F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
M.F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127 [INSPIRE].
D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
D. Mazáč, Analytic bounds and emergence of AdS 2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazáč and M.F. Paulos, The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation, arXiv:1811.10646 [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal Bootstrap in Mellin Space, Phys. Rev. Lett. 118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].
S. El-Showk and M.F. Paulos, Bootstrapping Conformal Field Theories with the Extremal Functional Method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
M. Creutz, Rigorous bounds on coupling constants in two-dimensional field theories, Phys. Rev. D 6 (1972) 2763 [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part II: two dimensional amplitudes, JHEP 11 (2017) 143 [arXiv:1607.06110] [INSPIRE].
M. Mezei, S.S. Pufu and Y. Wang, A 2d/1d Holographic Duality, arXiv:1703.08749 [INSPIRE].
V. de Alfaro, S. Fubini and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].
M. Billó, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri and R. Pellegrini, Line defects in the 3d Ising model, JHEP 07 (2013) 055 [arXiv:1304.4110] [INSPIRE].
D. Gaiotto, D. Mazáč and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
S. Giombi, R. Roiban and A.A. Tseytlin, Half-BPS Wilson loop and AdS 2 /CFT 1, Nucl. Phys. B 922 (2017) 499 [arXiv:1706.00756] [INSPIRE].
M. Beccaria, S. Giombi and A.A. Tseytlin, Non-supersymmetric Wilson loop in \( \mathcal{N}=4 \) SYM and defect 1d CFT, JHEP 03 (2018) 131 [arXiv:1712.06874] [INSPIRE].
S. Giombi and S. Komatsu, Exact Correlators on the Wilson Loop in \( \mathcal{N}=4 \) SYM: Localization, Defect CFT and Integrability, JHEP 05 (2018) 109 [Erratum JHEP 11 (2018) 123] [arXiv:1802.05201] [INSPIRE].
L. Bianchi, L. Griguolo, M. Preti and D. Seminara, Wilson lines as superconformal defects in ABJM theory: a formula for the emitted radiation, JHEP 10 (2017) 050 [arXiv:1706.06590] [INSPIRE].
L. Bianchi, M. Preti and E. Vescovi, Exact Bremsstrahlung functions in ABJM theory, JHEP 07 (2018) 060 [arXiv:1802.07726] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, Long-range critical exponents near the short-range crossover, Phys. Rev. Lett. 118 (2017) 241601 [arXiv:1703.03430] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, A scaling theory for the long-range to short-range crossover and an infrared duality, J. Phys. A 50 (2017) 354002 [arXiv:1703.05325] [INSPIRE].
M.F. Paulos, S. Rychkov, B.C. van Rees and B. Zan, Conformal Invariance in the Long-Range Ising Model, Nucl. Phys. B 902 (2016) 246 [arXiv:1509.00008] [INSPIRE].
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
J.M. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
D.J. Gross and V. Rosenhaus, A line of CFTs: from generalized free fields to SYK, JHEP 07 (2017) 086 [arXiv:1706.07015] [INSPIRE].
S. Pal, Unitarity and universality in nonrelativistic conformal field theory, Phys. Rev. D 97 (2018) 105031 [arXiv:1802.02262] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
J. Qiao and S. Rychkov, Cut-touching linear functionals in the conformal bootstrap, JHEP 06 (2017) 076 [arXiv:1705.01357] [INSPIRE].
D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE Convergence in Conformal Field Theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].
R. Reemtsen and J.-J. Rückmann, Semi-infinite programming. Volume 25, Springer Science & Business Media (1998).
R. Rattazzi, S. Rychkov and A. Vichi, Central Charge Bounds in 4D Conformal Field Theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].
S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, JHEP 03 (2018) 148 [arXiv:1605.08087] [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].
I. Arefeva and V. Korepin, Scattering in two-dimensional model with Lagrangian L=(1/γ)[(1/2)(∂ μ u)2 +m 2(cosu−1)], Pisma Zh. Eksp. Teor. Fiz. 20 (1974) 680[INSPIRE].
A.B. Zamolodchikov, Exact two-particle S-matrix of quantum sine-Gordon solitons, Commun. Math. Phys. 55 (1977) 183 [Pisma Zh. Eksp. Teor. Fiz. 25 (1977) 499] [INSPIRE].
P. Dorey, Exact S-matrices, in proceedings of the Eotvos Summer School in Physics: Conformal Field Theories and Integrable Models, Budapest, Hungary, 13-18 August 1996, pp. 85-125 [hep-th/9810026] [INSPIRE].
B. Gabai, D. Mazáč, A. Shieber, P. Vieira and Y. Zhou, No Particle Production in Two Dimensions: Recursion Relations and Multi-Regge Limit, arXiv:1803.03578 [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
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Mazáč, D., Paulos, M.F. The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices. J. High Energ. Phys. 2019, 162 (2019). https://doi.org/10.1007/JHEP02(2019)162
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DOI: https://doi.org/10.1007/JHEP02(2019)162