  The anomalous exponents governing the long wavelength behavior of the flat
phase of physical crystalline membranes are calculated within a self-consistent
screening approximation (SCSA) applied to second order expansion in 1/d_C (d_C
is the co-dimension), extending the seminal work of Le Doussal and Radzihovsky
[Phys. Rev. Lett. 69, 1209 (1992)]. In particular, the bending rigidity is
found to harden algebraically in the long wavelength limit with an exponent
eta=0.789..., which is used to extract the elasticity softening exponent
eta_u=0.422..., and the roughness exponent zeta=0.605.... The scaling relation
eta_u=2-2eta is proven to hold to all orders in SCSA. Further, applying the
SCSA to an expansion in 1/d_C, is found to be essential, as no solution to the
self-consistent equations is found in a two bubble level, which is the naive
second order expansion. Surprisingly, even though the expansion parameter for
physical membrane is 1/d_C=1, the SCSA applied to second order expansion
deviates only slightly from the first order, increasing zeta by mere 0.016.
This supports the high quality of the SCSA for physical crystalline membranes,
as well as improves the comparison to experiments and numerical simulations of
these systems. The prediction of SCSA applied to first order expansion for the
Poisson ratio is shown to be exact to all orders.
