Fine Spectrum from Crude Analytic Bootstrap

The magnetic line defect in the $O(N)$ model gives rise to a non-trivial one-dimensional defect conformal field theory of theoretical and experimental value. This model is considered here in $d=4-\varepsilon$ and the full spectrum of defect operators with dimensions close to one, two and three at order $\varepsilon$ is presented. The spectrum of several classes of operators of dimension close to four and operators of large charge are also discussed. Analytic bootstrap techniques are used extensively, and efficient tools to deal with the unmixing of nearly degenerate operators are developed. Integral identities are also incorporated, and it is shown that they lead to constraints on some three-point function coefficients and anomalous dimensions to order $\varepsilon^2$.