Closed-form expression for cross-channel conformal blocks near the lightcone

118

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.

Section: Double-twist Improvement (Dti)mentioning

confidence: 99%

The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT

Liu

Meltzer

Poland

et al. 2020

118

“…Refs. [78,79,80,27,17,76,81,82] -especially because the p-adic analog of these methods yields, promisingly, a closed-form expression for the crossed channel decomposition of the four-point exchange diagram on the Bruhat-Tits tree [72].…”

Section: Spectral Decomposition Of Ads Diagramsmentioning

confidence: 99%

“…Recently, progress has also been made in obtaining relations obeyed by the decomposition coefficients of four-point exchange diagrams in the crossed channel[77,78,79,80,27,17,76,81,82] but in this paper we will restrict our discussion to only direct channel decomposition 6. In AdS 3 /CFT 2 , various limits of Virasoro blocks, obtained by taking particular heavy/light limits of dimensions of external operators, are also interpreted in terms of lengths of bulk geodesics and as geodesic diagrams in defect geometries[62,93,94,95,96]).…”

mentioning

confidence: 99%

Propagator identities, holographic conformal blocks, and higher-point AdS diagrams

Jepsen

Parikh

2019

Conformal blocks are the fundamental, theory-independent building blocks in any CFT, so it is important to understand their holographic representation in the context of AdS/CFT. We describe how to systematically extract the holographic objects which compute higherpoint global (scalar) conformal blocks in arbitrary spacetime dimensions, extending the result for the four-point block, known in the literature as a geodesic Witten diagram, to five-and six-point blocks. The main new tools which allow us to obtain such representations are various higher-point propagator identities, which can be interpreted as generalizations of the well-known flat space star-triangle identity, and which compute integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators in negatively curved spacetime. Using the holographic representation of the higher-point conformal blocks and higher-point propagator identities, we develop geodesic diagram techniques to obtain the explicit directchannel conformal block decomposition of a broad class of higher-point AdS diagrams in a scalar effective bulk theory, with closed-form expressions for the decomposition coefficients. These methods require only certain elementary manipulations and no bulk integration, and furthermore provide quite trivially a simple algebraic origin of the logarithmic singularities of higher-point tree-level AdS diagrams. We also provide a more compact repackaging in terms of the spectral decomposition of the same diagrams, as well as an independent discussion on the closely related but computationally simpler framework over p-adics which admits comparable statements for all previously mentioned results.

“…Unlike the parity-even case, parity-odd families have non-zero off-diagonal components: . To calculate the scalar coefficients dp J,0 i we will use the Lorentzian inversion formula [9,12,16], combined with either dimensional reduction of the 3d block [16,32] or resummations of the lightcone expansion [20]. Using dimensional reduction we find for the n = 0 doubletwist operators:…”

Section: Jhep09(2020)148mentioning

confidence: 99%

“…ψψO 6 come at subleading order. 20 By dividing by the MFT coefficients given in (4.18), we obtain the anomalous dimension at large spin:…”

Section: Jhep09(2020)148mentioning

confidence: 99%

The inversion formula and 6j symbol for 3d fermions

Albayrak

Meltzer

Poland

2020

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.