Auxiliary fields approach to shift-symmetric theories: The ${\phi }^4$ derivative theory and the crumpled-to-flat transition of membranes at two-loop order

We introduce a technique relying on the use of auxiliary fields in order to eliminate explicit field-derivatives that plague the high orders renormalization group treatment of shift-symmetric (derivative) theories. This technique simplifies drastically the computation of fluctuations in such theories. This is illustrated by deriving the two-loop renormalization group equations—and the three-loop anomalous dimension—of the ${\mathit{\phi }}^4$ derivative theory in $D=4-\epsilon$, which is also relevant to describe the crumpled-to-flat transition of polymerized membranes. Some features of this transition are provided.