Abstract
We apply numerical conformal bootstrap techniques to the four-point function of a Weyl spinor in 4d non-supersymmetric CFTs. We find universal bounds on operator dimensions and OPE coefficients, including bounds on operators in mixed symmetry representations of the Lorentz group, which were inaccessible in previous bootstrap studies. We find discontinuities in some of the bounds on operator dimensions, and we show that they arise due to a generic yet previously unobserved “fake primary” effect, which is related to the existence of poles in conformal blocks. We show that this effect is also responsible for similar discontinuities found in four-fermion bootstrap in 3d, as well as in the mixed-correlator analysis of the 3d Ising CFT. As an important byproduct of our work, we develop a practical technology for numerical approximation of general 4d conformal blocks.
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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].
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].
D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, Rev. Mod. Phys. 91 (2019) 15002 [arXiv:1805.04405] [INSPIRE].
V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. El-Showk et al., Solving the 3d Ising model with the conformal bootstrap II. C-minimization and precise critical exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving out the space of 4D CFTs, JHEP 05(2012) 110 [arXiv:1109.5176] [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].
S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, JHEP 03 (2018) 148 [arXiv:1605.08087] [INSPIRE].
D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
D. Mazac, Analytic bounds and emergence of AdS 2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP 02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP 02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
F. Caracciolo and V.S. Rychkov, Rigorous limits on the interaction strength in quantum field theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].
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. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D conformal field theories with global symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].
A. Vichi, Improved bounds for CFT’s with global symmetries, JHEP 01 (2012) 162 [arXiv:1106.4037] [INSPIRE].
F. Caracciolo, A. Castedo Echeverri, B. von Harling and M. Serone, Bounds on OPE coefficients in 4D conformal field theories, JHEP 10 (2014) 020 [arXiv:1406.7845] [INSPIRE].
H. Iha, H. Makino and H. Suzuki, Upper bound on the mass anomalous dimension in many-flavor gauge theories: a conformal bootstrap approach, PTEP 2016 (2016) 053B03 [arXiv:1603.01995] [INSPIRE].
Y. Nakayama, Bootstrap bound for conformal multi-flavor QCD on lattice, JHEP 07 (2016) 038 [arXiv:1605.04052] [INSPIRE].
D. Poland and A. Stergiou, Exploring the minimal 4D \( \mathcal{N}=1 \) SCFT, JHEP 12 (2015) 121 [arXiv:1509.06368] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions with global symmetries, JHEP 01 (2018) 036 [arXiv:1705.03484] [INSPIRE].
A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, JHEP 05 (2019) 098 [arXiv:1705.04278] [INSPIRE].
A. Dymarsky et al., The 3d stress-tensor bootstrap, JHEP 02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Deconstructing conformal blocks in 4D CFT, JHEP 08 (2015) 101 [arXiv:1505.03750] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
L. Iliesiu et al., Fermion-scalar conformal blocks, JHEP 04 (2016) 074 [arXiv:1511.01497] [INSPIRE].
A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Seed conformal blocks in 4D CFT, JHEP 02 (2016) 183 [arXiv:1601.05325] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP 07 (2016) 018 [arXiv:1603.05551] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP 02 (2018) 096 [arXiv:1612.08987] [INSPIRE].
G.F. Cuomo, D. Karateev and P. Kravchuk, General bootstrap equations in 4D CFTs, JHEP 01 (2018) 130 [arXiv:1705.05401] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
P. Kravchuk, Casimir recursion relations for general conformal blocks, JHEP 02 (2018) 011 [arXiv:1709.05347] [INSPIRE].
G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].
E. Elkhidir, D. Karateev and M. Serone, General three-point functions in 4D CFT, JHEP 01 (2015) 133 [arXiv:1412.1796] [INSPIRE].
D. Simmons-Duffin, A Semidefinite program solver for the conformal bootstrap, JHEP 06(2015) 174 [arXiv:1502.02033] [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].
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 mixed correlators in the 3D ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
Y. Nakayama and T. Ohtsuki, Conformal bootstrap dashing hopes of emergent symmetry, Phys. Rev. Lett. 117 (2016) 131601 [arXiv:1602.07295] [INSPIRE].
W.E. Caswell, Asymptotic behavior of nonabelian gauge theories to two loop order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].
T. Banks and A. Zaks, On the phase structure of vector-like gauge theories with massless fermions, Nucl. Phys. B 196 (1982) 189 [INSPIRE].
T. DeGrand, Lattice tests of beyond standard model dynamics, Rev. Mod. Phys. 88 (2016) 015001 [arXiv:1510.05018] [INSPIRE].
F.F. Hansen et al., Phase structure of complete asymptotically free SU(N c) theories with quarks and scalar quarks, Phys. Rev. D 97 (2018) 065014 [arXiv:1706.06402] [INSPIRE].
S. Weinberg, Minimal fields of canonical dimensionality are free, Phys. Rev. D 86 (2012) 105015 [arXiv:1210.3864] [INSPIRE].
E. Elkhidir and D. Karateev, Scalar-fermion analytic bootstrap in 4D, arXiv:1712.01554 [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves: further mathematical results, arXiv:1108.6194 [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].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Harmonic analysis and mean field theory, arXiv:1809.05111 [INSPIRE].
V.K. Dobrev et al., Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory, Lect. Notes Phys. 63 (1977) 1 [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [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].
S. Giombi, V. Kirilin and E. Skvortsov, Notes on spinning operators in fermionic CFT, JHEP 05 (2017) 041 [arXiv:1701.06997] [INSPIRE].
G. Mack, All unitary ray representations of the conformal group SU(2, 2) with Positive Energy, Commun. Math. Phys. 55 (1977) 1.
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision Islands in the Ising and O(N) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
D. Li, D. Meltzer and A. Stergiou, Bootstrapping mixed correlators in 4D \( \mathcal{N}=1 \) SCFTs, JHEP 07 (2017) 029 [arXiv:1702.00404] [INSPIRE].
A.L. Fitzpatrick et al., Covariant Approaches to Superconformal Blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton, U.S.A. (1992).
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Karateev, D., Kravchuk, P., Serone, M. et al. Fermion conformal bootstrap in 4d. J. High Energ. Phys. 2019, 88 (2019). https://doi.org/10.1007/JHEP06(2019)088
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DOI: https://doi.org/10.1007/JHEP06(2019)088