Closed-form expression for cross-channel conformal blocks near the lightcone
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Springer
Received: October 26, 2019 Accepted: December 24, 2019 Published: January 10, 2020
Wenliang Li Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan
E-mail: [email protected] Abstract: In the study of conformal field theories, conformal blocks in the lightcone limit are fundamental to the analytic conformal bootstrap method. Here we consider the lightcone limit of 4-point functions of generic scalar primaries. Based on the nonperturbative pole structure in spin of Lorentzian inversion, we propose the natural basis functions for cross-channel conformal blocks. In this new basis, we find a closed-form expression for crossed conformal blocks in terms of the Kamp´e de F´eriet function, which applies to intermediate operators of arbitrary spin in general dimensions. We derive the general Lorentzian inversion for the case of identical external scaling dimensions. Our results for the lightcone limit also shed light on the complete analytic structure of conformal blocks in the lightcone expansion. Keywords: Conformal Field Theory, Conformal and W Symmetry, Nonperturbative Effects ArXiv ePrint: 1906.00707
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)055
JHEP01(2020)055
Closed-form expression for cross-channel conformal blocks near the lightcone
Contents 1
2 Nonperturbative poles in spin
3
3 Lightcone limit of crossed conformal blocks
5
4 Identical external scaling dimensions
6
5 Conclusion
8
1
Introduction
Many important physical systems are described by conformal field theories (CFTs), ranging from statistical physics to high energy physics. The conformal bootstrap program aims at classifying and solving conformal field theories using general principles [1, 2], such as conformal invariance, crossing symmetry, unitarity and analyticity. Although considerable results in 2d have been obtained for a long time [3], the conformal bootstrap in higher dimensions have made major progress only since the work of [4]. This modern numerical approach of the conformal bootstrap has led to nontrivial bounds on the parameter space of unitary CFTs. The impressive results include the precise determinations of the 3d Ising critical exponents [5–8]. We refer to [9] for a comprehensive review. In parallel, the analytic approach has also made notable progress after the revival of the d > 2 conformal bootstrap. By considering the lightcone limit of the crossing equation, it was shown in [10, 11] that the twist spectrum of a d > 2 CFT is additive at large spin: if the twist spectrum contains scalars of twist τ1 , τ2 , then there will be accumulation points at τ1 + τ2 + 2n with n = 0, 1, 2, · · · for the large spin sector. An earlier discussion in a more specific context can be found in [12]. The analytic conformal bootstrap can be formulated as an algebraic problem [13–15]. More recently, significant advances towards the nonperturbative regime have been made by upgrading the analytical toolkit from asymptotic
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