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We propose a roadmap for bootstrapping conformal field theories (CFTs)described by gauge theories in dimensions $d>2$. In particular, we provide asimple and workable answer to the question of how to detect the gauge group inthe bootstrap calculation. Our recipe is based on the notion of\emph{decoupling operator}, which has a simple (gauge) group theoreticalorigin, and is reminiscent of the null operator of $2d$ Wess-Zumino-Witten CFTsin higher dimensions. Using the decoupling operator we can efficiently detectthe rank (i.e. color number) of gauge groups, e.g., by imposing gap conditionsin the CFT spectrum. We also discuss the physics of the equation of motion,which has interesting consequences in the CFT spectrum as well. As anapplication of our recipes, we study a prototypical critical gauge theory,namely the scalar QED which has a $U(1)$ gauge field interacting with criticalbosons. We show that the scalar QED can be solved by conformal bootstrap,namely we have obtained its kinks and islands in both $d=3$ and $d=2+\epsilon$dimensions.

A roadmap for bootstrapping critical gauge theories: decoupling operators of conformal field theories in $d>2$ dimensions