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

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