We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gauge theories in dimensions
d>2d>2.
In particular, we provide a simple and workable answer to the question of how to detect the gauge group in the bootstrap calculation. Our
recipe is based on the notion of decoupling operator, which has a simple (gauge) group theoretical origin, and is reminiscent of the null
operator of 2d2d
Wess-Zumino-Witten CFTs in higher dimensions. Using the decoupling operator we can efficiently detect the rank (i.e. color number) of gauge
groups, e.g., by imposing gap conditions in the CFT spectrum. We also discuss the physics of the equation of motion, which has interesting
consequences in the CFT spectrum as well. As an application of our recipes, we study a prototypical critical gauge theory, namely the
scalar QED which has a U(1)U(1)
gauge field interacting with critical bosons. We show that the scalar QED can be solved by conformal bootstrap, namely we have obtained its
kinks and islands in both d=3d=3
and d=2+\epsilond=2+ϵ
dimensions.