OPE Convergence in Conformal Field Theory

By Duccio Pappadopulo, S. Rychkov, J. Espin, R. Rattazzi. 2012

282 citations recorded by the source corpus.

We clarify questions related to the convergence of the OPE and conformal block decomposition in unitary Conformal Field Theories (for any number of spacetime dimensions). In particular, we explain why these expansions are convergent in a finite region. We also show that the convergence is exponentially fast, in the sense that the operators of dimension above ∆ contribute to correlation functions at most exp(−a∆). Here the constant a > 0 depends on the positions of operator insertions and we compute it explicitly. August 2012 ar X iv :1 20 8. 64 49 v1 [ he pth ] 3 1 A ug 2 01 2

Subfields: Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/83b241039476ec34cc37e827f31355b27a80da38

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Citations in the corpus

The ϵ ?> -expansion from conformal field theory (2015, background): “6 Nevertheless, there have been notable analytic results for the large N [40–42], large Δ [43, 44], large l [45–51], or SUSY [52, 53] cases.”
Convexity and liberation at large spin (2012, methodology): “These methods of bounding solutions to the bootstrap equations are easily described in Euclidean space, and the consistency of the procedure follows from the fact that the contribution of the operators with high dimension to the OPE in Euclidean space is exponentially suppressed [27].”
A semidefinite program solver for the conformal bootstrap (2015, background): “where r∗ = 3 − 2 √ 2 is the radius of the point z = z = 1 2 in the radial coordinates of [76, 77].”
Radial Coordinates for Conformal Blocks (2013, methodology): “The example of scalar exchanged primaries and their first two descendant levels was considered in [27].”
Random statistics of OPE coefficients and Euclidean wormholes (2020, background): “a well-behaved geometry whose partition function probes OPE coefficients with three heavy operators (other asymptotic formulae for OPE coefficients with one or two heavy states do exist though [28, 29]).”
Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$c-Minimization and Precise Critical Exponents (2012, background): “By the exponential decoupling theorem of [9], low-lying operators will dominate the constraints of crossing symmetry, and it is more practical to start the search by exploring the low dimensions first.”
The S-matrix bootstrap. Part I: QFT in AdS (2016, result): “This asymptotic behavior is also compatible with the general results of [46] but it is stronger.”
A spacetime derivation of the Lorentzian OPE inversion formula (2017, methodology): “Note that this corresponds to starting with the standard Euclidean configuration described in [13, 14] with ρ = ρ, and then applying a large Lorentzian relative boost between the points 1, 2 and 3, 4.”
Entanglement Entropy at Large Central Charge (2013, background): “It would be interesting to explore whether ap can be bounded some other way to demonstrate this suppression, perhaps along the lines of [50].”
Looking for a bulk point (2015, methodology): “In radial quantization, this corresponds to the configuration shown in figure 6, where ρ = eτE+iφ is the radial coordinate introduced in [34, 39].”
Holographic Quantum Matter (2016, methodology): “Detailed reviews of the OPE are available elsewhere [238; 588], and here we only use a few basic results to examine its consequences for the twopoint correlator of the current operator at short times.”
Weight shifting operators and conformal blocks (2017, unknown): “where r is the radial coordinate of [141, 142] and h∞,ρ is some relatively easily computable function, then we can write the conformal block as a sum over residues [14, 105].”
TASI Lectures on the Conformal Bootstrap (2016, background): “One can prove that as z → 0, the sum on the right-hand side is dominated by operators of dimension ∆ ∼ 1/√z [38].”
Light-ray operators in conformal field theory (2018, unknown): “This expansion is convergent whenever χ, χ ∈ C\[1,∞) [71].”
Bounding the space of holographic CFTs with chaos (2016, background): “This sum is infinite, but convergent for |z| < 1, |z| < 1 independently [56, 81].”
Bootstrapping mixed correlators in the 3D Ising model (2014, background): “The convergence region F includes a finite open neighborhood of the point z = z = 1 2 [42].”
The lightcone bootstrap and the spectrum of the 3d Ising CFT (2016, methodology): “regime [70], numerical bootstrap methods should only be sensitive to low-dimension operators.”
Analyticity in spin in conformal theories (2017, background): “In particular, the first singularities being at ρ = ±1, the OPE converges whenever ρ, ρ̄ are both within the unit disc; for rigorous estimates we refer to [24].”