Conformal four point functions and the operator product expansion

Abstract:We study the numerical bounds obtained using a conformal-bootstrap method -advocated in ref.[1] but never implemented so far -where different points in the plane of conformal cross ratios z andz are sampled. In contrast to the most used method based on derivatives evaluated at the symmetric point z =z = 1/2, we can consistently "integrate out" higher-dimensional operators and get a reduced simpler, and faster to solve, set of bootstrap equations. We test this "effective" bootstrap by studying the 3D Ising and O(n) vector models and bounds on generic 4D CFTs, for which extensive results are already available in the literature. We also determine the scaling dimensions of certain scalar operators in the O(n) vector models, with n = 2, 3, 4, which have not yet been computed using bootstrap techniques.

“…Operators with high scaling dimension are no longer suppressed and the remainder completely dominates the OPE. 8 More precisely, we have…”

Section: Comparison With Generalized Free Theories and Asymptotics Fomentioning

confidence: 99%

“…8 In this limit, the name remainder should actually be used for the finite sum of operators up to ∆ * . It was found in refs.…”

Section: Comparison With Generalized Free Theories and Asymptotics Fomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The effective bootstrap

Echeverri

Harling

Serone

2016

“…Similarly, the studies [24][25][26][27] have shown that it is even possible to analytically derive completely generic constraints on the spectrum of CFTs. Numerical works have been possible due to increased computer power in the last decades, and rely crucially on an increased understanding of conformal blocks -see [15,[28][29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

No unitary bootstrap for the fractal Ising model

Golden

Paulos

2015

We consider the conformal bootstrap for spacetime dimension 1 < d < 2. We determine bounds on operator dimensions and compare our results with various theoretical and numerical models, in particular with resummed -expansion and Monte Carlo simulations of the Ising model on fractal lattices. The bounds clearly rule out that these models correspond to unitary conformal field theories. We also clarify the d → 1 limit of the conformal bootstrap, showing that bounds can be -and indeed are -discontinuous in this limit. This discontinuity implies that for small = d − 1 the expected critical exponents for the Ising model are disallowed, and in particular those of the d − 1 expansion. Altogether these results strongly suggest that the Ising model universality class cannot be described by a unitary CFT below d = 2. We argue this also from a bootstrap perspective, by showing that the 2 ≤ d < 4 Ising "kink" splits into two features which grow apart below d = 2.

“…But a formal non perturbative development of the conformal bootstrap program for the O(N ) models was yet to be developed. With the explicit expressions of the conformal blocks in [43,44] and the subsequent works, it was possible to analyze the modern and conventional bootstrap numerically to find various bounds on the operator dimensions, central charges and coupling constants (i.e. the OPE coefficients) as discussed in [45][46][47][48] and so on.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

More on analytic bootstrap for O(N) models

Dey

Kaviraj

Sen

2016

This note is an extension of a recent work on the analytical bootstrapping of O(N ) models. An additonal feature of the O(N ) model is that the OPE contains trace and antisymmetric operators apart from the symmetric-traceless objects appearing in the OPE of the singlet sector. This in addition to the stress tensor (T µν ) and the φ i φ i scalar, we also have other minimal twist operators as the spin-1 current J µ and the symmetric-traceless scalar in the case of O(N ). We determine the effect of these additional objects on the anomalous dimensions of the corresponding trace, symmetric-traceless and antisymmetric operators in the large spin sector of the O(N ) model, in the limit when the spin is much larger than the twist. As an observation, we also verified that the leading order results for the large spin sector from the −expansion are an exact match with our n = 0 case. A plausible holographic setup for the special case when N = 2 is also mentioned which mimics the calculation in the CFT.