  Undoped graphene (Gr) sheets at low temperatures are known, via Random Phase
Approximation (RPA) calculations, to exhibit unusual van der Waals (vdW)
forces. Here we show that graphene is the first known system where effects
beyond the RPA make qualitative changes to the vdW force. For large
separations, $D \gtrsim 10$nm where only the $\pi_z$ vdW forces remain, we find
the Gr-Gr vdW interaction is substantially reduced from the RPA prediction. Its
$D$ dependence is very sensitive to the form of the long-wavelength many-body
enhancement of the velocity of the massless Dirac fermions, and may provide
independent confirmation of the latter via direct force measurements.
