The inversion formula and 6j symbol for 3d fermions
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Springer
Received: June 26, 2020 Accepted: August 6, 2020 Published: September 23, 2020
Soner Albayrak,a David Meltzerb and David Polanda a
Department of Physics, Yale University, 217 Prospect St, New Haven, CT 06511, U.S.A. b Walter Burke Institute for Theoretical Physics, Caltech, 1200 East Pasadena Blvd, Pasadena, CA 91125, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: In this work we study the 6j symbol of the 3d conformal group for fermionic operators. In particular, we study 4-point functions containing two fermions and two scalars and also those with four fermions. By using weight-shifting operators and harmonic analysis for the Euclidean conformal group, we relate these spinning 6j symbols to the simpler 6j symbol for four scalar operators. As one application we use these techniques to compute 3d mean field theory (MFT) OPE coefficients for fermionic operators. We then compute corrections to the MFT spectrum and couplings due to the inversion of a single operator, such as the stress tensor or a low-dimension scalar. These results are valid at finite spin and extend the perturbative large spin analysis to include non-perturbative effects in spin. Keywords: Conformal and W Symmetry, Conformal Field Theory, Nonperturbative Effects ArXiv ePrint: 2006.07374
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)148
JHEP09(2020)148
The inversion formula and 6j symbol for 3d fermions
Contents 1 Introduction
2
2 Review: scalar 6j symbol and the inversion formula
3 7 7 10
4 Mean field theory OPE coefficients 4.1 hψφφψi 4.2 hψ1 ψ2 ψ2 ψ1 i 4.3 hψψψψi
14 15 16 17
5 Analytic bootstrap for fermions 5.1 Spinning down the 6j symbol 5.2 OPE function and its decomposition 5.3 Applications and examples
17 18 24 26
6 Conclusion
33
A Conventions A.1 Review of embedding formalism A.2 Two and three point functions
34 34 35
B Partial waves and conformal blocks
36
C Two and three point pairings C.1 Two point pairings and plancherel measure C.2 Three point pairings
38 38 39
D Shadow coefficients, partial waves, and Euclidean inversion D.1 Shadow coefficients D.2 Bubble coefficients and partial wave normalization D.3 Partial wave expansion and Euclidean inversion formula
40 40 42 44
E Symmetries of 6j symbols
47
F OPE coefficients F.1 MFT coefficients for higher twist towers F.2 Corrections to OPE coefficients for [φψ]+ 0
48 48 50
G K coefficients
52
–1–
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3 Inversion formula and MFT for fermions 3.1 Conformal partial waves and 6j symbols 3.2 Shadow matrices and weight-shifting operators
1
Introduction
1
In [12] the 1d 6j symbol was calculated using the Euclidean definition directly. For a detailed analysis of tauberian theorems in CFTs see [15]. 3 For related work see [18–21]. 2
–2–
JHEP09(2020)148
Invariance under the Euclidean conformal group SO(d + 1, 1) is well-known to put strong constraints on the space and observables of quantum field theories [1, 2]. In conformal field theories (CFTs),
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