Strong-coupling expansion of cusp anomaly and gluon amplitudes from quantum open strings in AdS₅ × S⁵

An important "observable" of planar N = 4  SYM theory is the scaling function f (λ) that appears in the anomalous dimension of large spin twist 2 operators and also in the cusp anomaly of light-like Wilson loop. The non-trivial relation between the anomalous dimension and the Wilson loop interpretations of f (λ) is well-understood on the perturbative gauge theory side of the AdS/CFT duality. In the first part of this paper we present the dual string-theory counterpart of this relation, i.e., the equivalence between the closed-string and the open-string origins of f (λ). We argue that the coefficient of the log S term in the energy of the closed string with large spin S in AdS₅ should be equal to the coefficient in the logarithm of expectation value of the null cusp Wilson loop, to all orders in λ^{- 1 / 2} expansion. The reason is that the corresponding minimal surfaces happen to be related by a conformal transformation (and an analytic continuation). As a check, we explicitly compute the leading 1-loop string sigma model correction to the cusp Wilson loop, reproducing the same subleading coefficient in f (λ) as found earlier in the spinning closed string case. The same function f (λ) appears also in the resummed form of the 4-gluon amplitude as discussed at weak coupling by Bern, Dixon and Smirnov and recently found at the leading order at strong coupling by Alday and Maldacena (AM). Here we attempt to extend the latter approach to a subleading order in λ^{- 1 / 2} by computing the IR singular part of the 1-loop string correction to the corresponding T-dual Wilson loop. We discuss explicitly the 1-cusp case and comment on apparent problems with the dimensional regularization proposal of AM when directly applied order by order in strong coupling (string inverse tension) expansion.