We derive the three-loop order renormalization group equations that describe the flat phase of polymerized membranes within the modified minimal subtraction scheme, following the pioneering one-loop order computation of Aronovitz and Lubensky [Phys. Rev. Lett. 60, 2634 (1988)] and the recent two-loop order one of Coquand, Mouhanna, and Teber [Phys. Rev. E 101, 062104 (2020)]. We analyze the fixed points of these equations and compute the associated field anomalous dimension $\eta$ at three-loop order. Our results display a marked proximity with those obtained using nonperturbative techniques and reexpanded in powers of $\epsilon =4-D$. Moreover, the three-loop order value that we get for $\eta$ at the stable fixed point, $\eta =0.8872$, in $D=2$, is compatible with known theoretical results and within the range of accepted numerical values.

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• Critical phenomena
• Membrane structures
• Renormalization group
• Series expansions & exact enumeration