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Two-loop AdS_5 x S^5 superstring: testing asymptotic Bethe ansatz and finite size corrections

S. Giombi, R. Ricci, R. Roiban, A. A. Tseytlin

TL;DR

This work computes two-loop quantum corrections to the energy of fast-spinning folded strings in $AdS_5\times S^5$ within the scaling limit $\ell=\frac{\pi J}{\sqrt{\lambda}\,\ln S}$, testing the all-loop ABA predictions and initiating a systematic study of finite-size (Casimir-like) effects on the worldsheet. Using the AdS light-cone gauge and a generalized cusp background, the authors derive the two-loop generalized scaling function $f_2(\ell)$, obtaining exact small-$\ell$ Catalan-constant coefficients, an all-loop leading-log resummation that matches ABA, and a large-$\ell$ $\pi^2$-coefficient that confirms non-renormalization of leading terms. They extend the analysis to 2-loop finite-size corrections on a cylinder, showing that only massless modes contribute to the leading $1/\ln^2 S$ effects and that regularization choices crucially affect the final result, with implications for wrapping corrections and LL-model correspondence. Overall, the paper provides strong consistency checks between string-theoretic two-loop computations and integrability-based predictions, while highlighting technical subtleties in finite-size regularization and its physical interpretation.

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 ell=pi J/(lambda^(1/2) ln S)=fixed. We compute the generalized scaling function at two-loop order f_2(ell) both for small and large values of ell matching the predictions based on the asymptotic Bethe ansatz. In particular, in the small ell expansion, we derive an exact integral form for the ell-dependent coefficient of the Catalan's constant term in f_2(ell). Also, by resumming a certain subclass of multi-loop Feynman diagrams we obtain an exact expression for the leading (ln ell) part of f(lambda^(1/2), ell) which is valid to any order in the alpha'~1/lambda^(1/2) expansion. At large ell the string energy has a BMN-like expansion and the first few leading coefficients are expected to be the same at weak and at 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(ell) expanded at large ell. 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 fast-spinning 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.

Two-loop AdS_5 x S^5 superstring: testing asymptotic Bethe ansatz and finite size corrections

TL;DR

This work computes two-loop quantum corrections to the energy of fast-spinning folded strings in within the scaling limit , testing the all-loop ABA predictions and initiating a systematic study of finite-size (Casimir-like) effects on the worldsheet. Using the AdS light-cone gauge and a generalized cusp background, the authors derive the two-loop generalized scaling function , obtaining exact small- Catalan-constant coefficients, an all-loop leading-log resummation that matches ABA, and a large- -coefficient that confirms non-renormalization of leading terms. They extend the analysis to 2-loop finite-size corrections on a cylinder, showing that only massless modes contribute to the leading effects and that regularization choices crucially affect the final result, with implications for wrapping corrections and LL-model correspondence. Overall, the paper provides strong consistency checks between string-theoretic two-loop computations and integrability-based predictions, while highlighting technical subtleties in finite-size regularization and its physical interpretation.

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 ell=pi J/(lambda^(1/2) ln S)=fixed. We compute the generalized scaling function at two-loop order f_2(ell) both for small and large values of ell matching the predictions based on the asymptotic Bethe ansatz. In particular, in the small ell expansion, we derive an exact integral form for the ell-dependent coefficient of the Catalan's constant term in f_2(ell). Also, by resumming a certain subclass of multi-loop Feynman diagrams we obtain an exact expression for the leading (ln ell) part of f(lambda^(1/2), ell) which is valid to any order in the alpha'~1/lambda^(1/2) expansion. At large ell the string energy has a BMN-like expansion and the first few leading coefficients are expected to be the same at weak and at 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(ell) expanded at large ell. 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 fast-spinning 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.

Paper Structure

This paper contains 33 sections, 187 equations, 4 figures.

Table of Contents

  1. Introduction
  2. AdS light-cone action and the generalized cusp solution
  3. Generalized scaling function from the partition function
  4. Two-loop partition function
  5. Bosonic and fermionic double-bubble
  6. Bosonic Sunset
  7. Fermionic Sunset
  8. Tadpole contribution
  9. Generalized scaling function
  10. $\ln^2 2$ and $\ln 2$ terms at small $\hat{\nu}$
  11. The Catalan constant term in the small $\ell$ expansion of ${\rm f}_2$
  12. Leading logarithms at small $\ell$: all-loop resummation
  13. The $\pi^2$ term in the large $\ell$ expansion of ${\rm f}_2$
  14. Non-renormalization of the leading terms in the large $\ell$ expansion
  15. String finite size corrections: computations on $\mathbb{R} \times S^1$ worldsheet
  16. ...and 18 more sections

Figures (4)

  • Figure 1: The 2-loop 1PI topologies: "sunset" and "double-bubble". The propagators here are either bosonic or fermionic.
  • Figure 2: The 2-loop tadpole topology. The non-vanishing graphs have the internal line corresponding to a $\tilde{\phi}$-propagator while the propagators in the loops can be either bosonic or fermionic.
  • Figure 3: Multi-loop maximally reducible diagrams contributing to the leading logarithmic terms in the string free energy. The loops are made of the light $S^5$ fluctuations and the propagators correspond to the constant mode of the AdS fluctuation $\tilde{\phi}$.
  • Figure 4: The plots of the massive 2-loop integral ${\rm I}[1,1/2,1/2]$ (blue) and $\frac{1}{L^2}+\frac{{\rm K}}{4\pi^2}$ (red) as function of $L$. ${\rm I}[1,1/2,1/2]$ quickly approaches the continuum limit $\frac{{\rm K}}{4\pi^2}$.