The effective bootstrap

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