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Double Field Theory (DFT) is an attempt to make the O(d,d) T-duality symmetryof string theory manifest, already before reducing on a d-torus. It is knownthat supergravity can be formulated in an O(D,D) covariant way, and remarkablythis remains true to the first order in $\alpha'$. We set up a systematic wayto analyze O(D,D) invariants, working order by order in fields, which we carryout up to order $\alpha'^3$. At order $\alpha'$ we recover the known Riemannsquared invariant, while at order $\alpha'^2$ we find no independent invariant.This is compatible with the $\alpha'$ expansion in string theory. However, atorder $\alpha'^3$ we show that there is again no O(D,D) invariant, incontradiction to the fact that all string theories have a quartic Riemanninvariant with coefficient proportional to $\zeta(3)$. We conclude that DFT andsimilar frameworks cannot capture the full $\alpha'$ expansion in stringtheory.

O(D,D) and the string $\alpha'$ expansion: An obstruction