The effective bootstrap

Journal of High Energy Physics, Sep 2016

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 and \( \overline{z} \) are sampled. In contrast to the most used method based on derivatives evaluated at the symmetric point \( z=\overline{z}=1/2 \), 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.

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The effective bootstrap

Received: June The e ective bootstrap Alejandro Castedo Echeverri 0 1 2 Benedict von Harling 0 1 2 Marco Serone 0 1 2 SISSA 0 1 2 0 Strada Costiera 11 , I-34151 Trieste , Italy 1 Notkestrasse 85 , 22607 Hamburg , Germany 2 Via Bonomea 265, I-34136 Trieste , Italy We study the numerical bounds obtained using a conformal-bootstrap method | advocated in ref. [1] but never implemented so far | where di erent points in the plane of conformal cross ratios z and z 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 \e ective" 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. Conformal and W Symmetry; Conformal Field Theory Convergence of the OPE Bootstrapping with multiple points 3.1 Choice of points Comparison with generalized free theories and asymptotics for z ! 1 Remainder for CFTs with O(n) symmetry 3D Ising and O(n) models A closer look at the spectrum of 3D O(n) models 1 Introduction 2.1 2.2 4.1 4.3 Results 4.2 4D CFTs 2 3 4 5 6 1 Details of the implementation Conclusions Introduction There has recently been a great revival of interest in the conformal bootstrap program [2, 3] after ref. [4] observed that its applicability extends to Conformal Field Theories (CFTs) in d > 2 dimensions. Since ref. [4], considerable progress has been achieved in understanding CFTs in d 2 dimensions, both numerically and analytically. Probably the most striking progress has been made in the numerical study of the 3D Ising model, where amazingly precise operator dimensions and OPE coe cients have been determined [5{7]. Essentially all numerical bootstrap studies so far have used the constraints imposed by crossing symmetry on 4-point correlators evaluated at a speci c value of the conformal cross-ratios, u = v = 1=4, or equivalently in z-coordinates at z = z = 1=2 [8]. This is the point of best convergence for the combined conformal block expansions in the s and t channels. Taking higher and higher derivatives of the bootstrap equations evaluated at this point has proven to be very e ective and successful in obtaining increasingly better bounds. We will denote this method in the following as the \derivative method". A drawback of the derivative method | both in its linear [4, 6, 9] or semi-de nite [10, 11] programming incarnations | is the need to include a large number of operators in the bootstrap equations. This makes any, even limited, analytical understanding of the obtained results quite di cult. A possible approximation scheme is in fact available: ref. [12] has determined the rate of convergence of the Operator Product Expansion (OPE), on which the bootstrap equations are based. This allows us to extract the maximal error from neglecting operators with dimensions larger than some cuto in the bootstrap equations and thus to consistently truncate them. These truncated bootstrap equations can then be evaluated at di erent points in the z-plane. This method, which we denote as the \multipoint method", has { 1 { and study the resulting bounds. It is important to emphasize that the method of ref. [ 1 ] combines what are in principle two independent ideas: i) multipoint bootstrap and ii) truncation of the bootstrap equations. One could study i) without ii), or try to analyze ii) without i). We will not consider these other possibilities here. We begin in section 2 with a brief review of the results of refs. [ 1, 12, 13 ] on the convergence of the OPE. We use generalized free theories as a toy laboratory to test some of the results obtained in ref. [12]. We then generalize the results of ref. [12] for CFTs with an O(n) global symmetry. We write the bootstrap equations and set the stage for our numerical computations in section 3. Our results are then presented in section 4. For concreteness, we study bounds on operator dimensions and the central charge in 3D and 4D CFTs, with and without an O(n) global symmetry (with no supersymmetry). For these bounds, extensive results are already available in the literature (see e.g. refs. [5{7, 10, 14{22]). In particular, we focus our attention on the regions where the 3D Ising and O(n) vector models have been identi ed. We show how the results depend on the number N of points in the z-plane at which we evaluate the bootstrap equations and the cut-o on the dimension of operators in the bootstrap equations. Using values for the dimension of the operator in O(n) vector models available in the literature and a t extrapolation procedure, we then determine the dimensions of the second-lowest O(n) singlet and symmetric-traceless operators S0 and T 0 for n = 2; 3; 4. To our knowledge, these have not been obtained before using bootstrap techniques. Our (...truncated)


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Alejandro Castedo Echeverri, Benedict von Harling. The effective bootstrap, Journal of High Energy Physics, 2016, pp. 97, Volume 2016, Issue 9, DOI: 10.1007/JHEP09(2016)097