The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT
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Springer
Received: July 20, 2020 Accepted: August 17, 2020 Published: September 17, 2020
Junyu Liu,a,b David Meltzer,a David Polandc and David Simmons-Duffina a
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, U.S.A. b Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, U.S.A. c Department of Physics, Yale University, New Haven, CT 06520, U.S.A.
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We study the spectrum and OPE coefficients of the three-dimensional critical O(2) model, using four-point functions of the leading scalars with charges 0, 1, and 2 (s, φ, and t). We obtain numerical predictions for low-twist OPE data in several charge sectors using the extremal functional method. We compare the results to analytical estimates using the Lorentzian inversion formula and a small amount of numerical input. We find agreement between the analytic and numerical predictions. We also give evidence that certain scalar operators lie on double-twist Regge trajectories and obtain estimates for the leading Regge intercepts of the O(2) model. Keywords: Conformal and W Symmetry, Conformal Field Theory ArXiv ePrint: 2007.07914
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)115
JHEP09(2020)115
The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT
Contents 1 Large-scale bootstrap: analytics and numerics
1
2 Numerical computations in the O(2) model 2.1 Bootstrap setup 2.2 Numerical spectrum 2.3 Predictions for scalar CFT data and the sharing effect
3 3 5 6 8 8 10 11 11 12 13 15 17
4 Inversion formula for the O(2) model 4.1 Generating functions 4.2 Double-twist sums
19 19 22
5 Comparing numerical and analytic predictions 5.1 Charge 0± , 1, 2, 30**(+-), 1, 2, 3 5.2 Charge 4 predictions 5.3 Ward identity checks 5.4 Leading scalar predictions 5.5 Regge intercepts 5.6 Crossing symmetry and the dDisc
24 24 31 31 33 35 38
6 Future directions 6.1 Numerical bootstrap 6.2 Analytic bootstrap 6.3 Additional applications
42 42 43 44
A Summary of notation
45
B Integrals of hypergeometric functions
45
–i–
JHEP09(2020)115
3 Analytic predictions using the inversion formula 3.1 The inversion formula 3.2 Expanding conformal blocks in the inversion formula 3.2.1 SL2 expansion for the Weyl-reflected block 3.2.2 SL2 expansion for G 3.2.3 Dimensional reduction for G 3.3 Double-twist improvement (DTI) 3.4 Exact vs. approximate generating function 3.5 The twist Hamiltonian
47 47 49 50 52 53 55
D Conformal block expansions in three dimensions D.1 SL2 expansion D.2 Dimensional reduction D.3 Connections
55 55 57 58
E Some comparisons of computational performance E.1 Comparison using different expansions E.2 Comparison using different values of z0 E.3 Effects of the twist Hamiltonian E.4 DTI versus non-DTI
58 58 59 60 61
F Conformal block conventions
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G Technical details about SL2 sums G.1 Asympt
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