Abstract
We derive closed-form expressions for all ingredients of the Zamolodchikov-like recursion relation for general spinning conformal blocks in 3-dimensional conformal field theory. This result opens a path to efficient automatic generation of conformal block tables, which has immediate applications in numerical conformal bootstrap program. Our derivation is based on an understanding of null states and conformally-invariant differential operators in momentum space, combined with a careful choice of the relevant tensor structures bases. This derivation generalizes straightforwardly to higher spacetime dimensions d, provided the relevant Clebsch-Gordan coefficients of Spin (d) are known.
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References
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].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, Rev. Mod. Phys.91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions, JHEP03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D Fermions with global symmetries, JHEP01 (2018) 036 [arXiv:1705.03484] [INSPIRE].
D. Karateev, P. Kravchuk, M. Serone and A. Vichi, Fermion conformal bootstrap in 4d, JHEP06 (2019) 088 [arXiv:1902.05969] [INSPIRE].
A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, JHEP05 (2019) 098 [arXiv:1705.04278] [INSPIRE].
A. Dymarsky et al., The 3d stress-tensor bootstrap, JHEP02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
C. Behan, PyCFTBoot: a flexible interface for the conformal bootstrap, Commun. Comput. Phys.22 (2017) 1 [arXiv:1602.02810] [INSPIRE].
Y. Nakayama and T. Ohtsuki, Conformal bootstrap dashing hopes of emergent symmetry, Phys. Rev. Lett.117 (2016) 131601 [arXiv:1602.07295] [INSPIRE].
M.F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127 [INSPIRE].
M. Go and Y. Tachikawa, autoboot: a generator of bootstrap equations with global symmetry, JHEP06 (2019) 084 [arXiv:1903.10522] [INSPIRE].
G.F. Cuomo, D. Karateev and P. Kravchuk, General bootstrap equations in 4D CFTs, JHEP01 (2018) 130 [arXiv:1705.05401] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves: further mathematical results, arXiv:1108.6194 [INSPIRE].
M. Hogervorst, H. Osborn and S. Rychkov, Diagonal limit for conformal blocks in d dimensions, JHEP08 (2013) 014 [arXiv:1305.1321] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D ising model, JHEP11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev.D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
W. Landry, scalar blocks, https://gitlab.com/bootstrapcollaboration/scalar blocks.
P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP02 (2018) 096 [arXiv:1612.08987] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
L. Iliesiu et al., Fermion-scalar conformal blocks, JHEP04 (2016) 074 [arXiv:1511.01497] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP07 (2016) 018 [arXiv:1603.05551] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP07 (2016) 057 [arXiv:1603.05552] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
P. Kravchuk, Casimir recursion relations for general conformal blocks, JHEP02 (2018) 011 [arXiv:1709.05347] [INSPIRE].
R. Erramilli, L. Iliesiu, P. Kravchuk, W. Landry, D. Poland and D. Simmons-Duffin, to appear.
Al.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys.73 (1987) 1088.
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving out the space of 4D CFTs, JHEP05 (2012) 110 [arXiv:1109.5176] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Harmonic analysis and mean field theory, JHEP10 (2019) 217 [arXiv:1809.05111] [INSPIRE].
G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys.55 (1977) 1.
M. Gillioz, Momentum-space conformal blocks on the light cone, JHEP10 (2018) 125 [arXiv:1807.07003] [INSPIRE].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Shocks, superconvergence and a stringy equivalence principle, arXiv:1904.05905 [INSPIRE].
M. Gillioz, X. Lu and M.A. Luty, Scale anomalies, states and rates in conformal field theory, JHEP04 (2017) 171 [arXiv:1612.07800] [INSPIRE].
M. Gillioz, X. Lu and M.A. Luty, Graviton scattering and a sum rule for the c anomaly in 4D CFT, JHEP09 (2018) 025 [arXiv:1801.05807] [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys.2 (1998) 783 [hep-th/9712074] [INSPIRE].
J. Slovák, Natural operators on conformal manifolds, in the proceedings of Differential geometry and its applications, August 24–28, Silesian University, Opava, Czech Republic (1993).
D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskiĭ, Quantum theory of angular momentum, World Scientific, Singapore (1988).
S.M. Chester and S.S. Pufu, Towards bootstrapping QED 3, JHEP08 (2016) 019 [arXiv:1601.03476] [INSPIRE].
S.M. Chester, L.V. Iliesiu, M. Mezei and S.S. Pufu, Monopole operators in U(1) Chern-Simons-Matter theories, JHEP05 (2018) 157 [arXiv:1710.00654] [INSPIRE].
Z. Li, Solving QED3 with conformal bootstrap, arXiv:1812.09281 [INSPIRE].
Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, \( \mathcal{N} \) = 1 superconformal blocks for general scalar operators, JHEP08 (2014) 049 [arXiv:1404.5300] [INSPIRE].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping SCFTs with four supercharges, JHEP08 (2015) 142 [arXiv:1503.02081] [INSPIRE].
I. Buric, V. Schomerus and E. Sobko, Superconformal blocks: general theory, arXiv:1904.04852 [INSPIRE].
K. Sen and M. Yamazaki, Polology of superconformal blocks, arXiv:1810.01264 [INSPIRE].
P. Kravchuk, unpublished work.
L. Iliesiu et al., The conformal bootstrap at finite temperature, JHEP10 (2018) 070 [arXiv:1802.10266] [INSPIRE].
L. Iliesiu, M. Koloğlu and D. Simmons-Duffin, Bootstrapping the 3d Ising model at finite temperature, arXiv:1811.05451 [INSPIRE].
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Erramilli, R.S., Iliesiu, L.V. & Kravchuk, P. Recursion relation for general 3d blocks. J. High Energ. Phys. 2019, 116 (2019). https://doi.org/10.1007/JHEP12(2019)116
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DOI: https://doi.org/10.1007/JHEP12(2019)116