### Abstract

We continue the investigation of two-loop string corrections to the energy of a folded string with a spin S in AdS_{5} and an angular momentum J in S^{5}, 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 f_{2}(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 f_{2}(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.

Original language | English (US) |
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Article number | 045402 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 44 |

Issue number | 4 |

DOIs | |

State | Published - Jan 28 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

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## Cite this

_{5}× S

^{5}superstring: Testing asymptotic Bethe ansatz and finite size corrections.

*Journal of Physics A: Mathematical and Theoretical*,

*44*(4), [045402]. https://doi.org/10.1088/1751-8113/44/4/045402