The defect b-theorem under bulk RG flows

It is known that for RG flows confined to a two-dimensional defect, where the bulk maintains its conformal nature, the coefficient of the Euler density in the defect's Weyl anomaly (termed b) cannot increase as the flow progresses from the ultraviolet to the infrared, a principle known as the b-theorem. In this paper, we investigate whether this theorem still holds when the bulk, instead of being critical, also undergoes an RG flow. To address this question, we examine two distinct and perturbatively tractable examples. Our analysis reveals that a straightforward extension of the b-theorem to these cases of RG flows fails.