Abstract
We study 2-point correlation functions for scalar operators in position space through holography including bulk cubic couplings as well as higher curvature couplings to the square of the Weyl tensor. We focus on scalar operators with large conformal dimensions. This allows us to use the geodesic approximation for propagators. In addition to the leading order contribution, captured by geodesics anchored at the insertion points of the operators on the boundary and probing the bulk geometry thoroughly studied in the literature, the first correction is given by a Witten diagram involving both the bulk cubic coupling and the higher curvature couplings. As a result, this correction is proportional to the VEV of a neutral operator Ok and thus probes the interior of the black hole exactly as in the case studied by Grinberg and Maldacena [13]. The form of the correction matches the general expectations in CFT and allows to identify the contributions of TnOk (being Tn the general contraction of n energy-momentum tensors) to the 2-point function. This correction is actually the leading term for off-diagonal correlators (i.e. correlators for operators of different conformal dimension), which can then be computed holographically in this way.
Highlights
Which in the end becomes the geodesic length
We study 2-point correlation functions for scalar operators in position space through holography including bulk cubic couplings as well as higher curvature couplings to the square of the Weyl tensor
We focus on scalar operators with large conformal dimensions
Summary
One can check that if ∆i = ∆j, the only solution to these equations is zI = z0 and μi = μj = 0, νi = νj = 0 This corresponds to “straight” geodesics running radially at fixed τ so that the resulting configuration of joining two of them would have a “wedge” and would result in a non-smooth curve — unless the insertions are opposite located in the circle, so that the two “straight geodesics” are aligned. At low temperatures (more precisely, at small T |x|), and in d = 4, this was considered, using the background in [20, 21], in [3]; where it was argued that the leading correction only changes the coefficients by effectively shifting. The correction term δAd is of the order α 3, and for d reads δA4
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