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We introduce a technique relying on the use of auxiliary fields in order toeliminate explicit field-derivatives that plague the high ordersrenormalization group treatment of shift-symmetric, derivative, theories. Thistechnique simplifies drastically the computation of fluctuations in suchtheories. This is illustrated by deriving the two-loop renormalization groupequations and the three-loop anomalous dimension of the $\varphi^4$ derivativetheory in $D=4-\epsilon$, which is also relevant to describe thecrumpled-to-flat transition of polymerized membranes. Some features of thistransition are provided.

physical review. d/physical review. d.

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