Two-loop AdS₅ × S⁵ superstring: Testing asymptotic Bethe ansatz and finite size corrections

We continue the investigation of two-loop string corrections to the energy of a folded string with a spin S in AdS₅ and an angular momentum J in S⁵, in the scaling limit of large J and S with ℓ = πJ/√Λln S = fixed. We compute the generalized scaling function at two-loop order f₂(ℓ) both for small and large values of ℓ matching the predictions based on the asymptotic Bethe ansatz. In particular, in the small l expansion, we derive an exact integral form for the ℓ-dependent coefficient of Catalan's constant term in f2(ℓ). Also, by resumming a certain subclass of multi-loop Feynman diagrams we obtain an exact expression for the leading ln part of f(ℓ, Λ) which is valid to any order in the α' ∼ 1/√Λ expansion. At large ℓ the string energy has a BMN-like expansion and the first few leading coefficients are expected to be protected, i.e. to be the same at weak and strong coupling. We provide a new example of this non-renormalization for the term which is generated at two loops in string theory and at one-loop in gauge theory (sub-sub-leading in 1/J). We also derive a simple algebraic formula for the term of maximal transcendentality in f₂(l) expanded at large ℓ. In the second part of the paper we initiate the study of 2-loop finite size corrections to the string energy by formally compactifying the spatial world-sheet direction in the string action expanded near long fastspinning string. We observe that the leading finite-size corrections are of 'Casimir' type coming from terms containing at least one massless propagator. We consider in detail the one-loop order (reproducing the leading Landau-Lifshitz model prediction) and then focus on the two-loop contributions to the 1/ln S term (for J = 0). We find that in a certain regularization scheme used to discard power divergences the two-loop coefficient of the 1/ln S term appears to vanish.