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Quantum corrections to spinning strings in AdS(5)xS(5) and Bethe ansatz: a comparative study

Sakura Schafer-Nameki, Marija Zamaklar, Konstantin Zarembo

TL;DR

The paper tests the quantum-corrected spectrum of macroscopic strings in $AdS_5\times S^5$ by comparing the one-loop string energy shift to finite-size corrections from the quantum string Bethe ansatz. While a $\zeta$-function regularization makes the string expansion finite and yields agreement with the Bethe ansatz up to three orders in $1/\mathcal{J}^2$, a nonperturbative discrepancy emerges in the large winding number limit, supported by numerical evidence. The work highlights subtle aspects of regularization, anomaly contributions in Bethe equations, and the ongoing interplay between string quantization and integrable spin-chain techniques in AdS/CFT. It suggests that understanding potential $1/\sqrt{\lambda}$ corrections or alternative regularizations is essential for a complete reconciliation of string theory and Bethe-ansatz predictions.

Abstract

We analyze quantum corrections to rigid spinning strings in AdS(5)xS(5). The one-loop worldsheet quantum correction to the string energy is compared to the finite-size correction from the quantum string Bethe ansatz. Expanding the summands of the string theory energy shift in the parameter λ/J^2 and subsequently resumming them yields a divergent result. However, upon zeta-function regularization this result agrees with the Bethe ansatz at the first three orders. We also perform an analogous computation in the limit of large winding number, which results in a disagreement with the string Bethe ansatz prediction. A similar mismatch is observed numerically. We comment on the possible origin of this discrepancy.

Quantum corrections to spinning strings in AdS(5)xS(5) and Bethe ansatz: a comparative study

TL;DR

The paper tests the quantum-corrected spectrum of macroscopic strings in by comparing the one-loop string energy shift to finite-size corrections from the quantum string Bethe ansatz. While a -function regularization makes the string expansion finite and yields agreement with the Bethe ansatz up to three orders in , a nonperturbative discrepancy emerges in the large winding number limit, supported by numerical evidence. The work highlights subtle aspects of regularization, anomaly contributions in Bethe equations, and the ongoing interplay between string quantization and integrable spin-chain techniques in AdS/CFT. It suggests that understanding potential corrections or alternative regularizations is essential for a complete reconciliation of string theory and Bethe-ansatz predictions.

Abstract

We analyze quantum corrections to rigid spinning strings in AdS(5)xS(5). The one-loop worldsheet quantum correction to the string energy is compared to the finite-size correction from the quantum string Bethe ansatz. Expanding the summands of the string theory energy shift in the parameter λ/J^2 and subsequently resumming them yields a divergent result. However, upon zeta-function regularization this result agrees with the Bethe ansatz at the first three orders. We also perform an analogous computation in the limit of large winding number, which results in a disagreement with the string Bethe ansatz prediction. A similar mismatch is observed numerically. We comment on the possible origin of this discrepancy.

Paper Structure

This paper contains 19 sections, 125 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Quantum corrections in string theory
  3. Energy shift
  4. Perturbative expansion
  5. Bethe ansatz
  6. Classical limit
  7. Quantum corrections
  8. Mode expansion
  9. Perturbative expansion and comparison to string theory
  10. Limit of large winding number
  11. Bethe ansatz calculation
  12. Numerical evaluation of energy shifts
  13. Conclusions
  14. Contribution of $\mathfrak{sl}(2)$ modes
  15. Perturbative expansion of modes
  16. ...and 4 more sections

Figures (5)

  • Figure 1: Graph of the quartic polynomial $-P(z)$ (the ordering of the zeroes is $a<b<c$).
  • Figure 2: Energy shifts $(\delta E)\times {\mathcal{J}}^2$ for ${\mathcal{J}} = 5...50 \, , \, m=3\, , \, k= -2$, Bethe vs. semi-classical string.
  • Figure 3: Energy shifts $(\delta E)\times {\mathcal{J}}^2$ for ${\mathcal{J}} = 3 \, , \, m=2\, , \, -k=(40...1)$. The upper curve is the string calculation. The lower curve is the prediction of the Bethe ansatz.
  • Figure 4: Profiles of the summands for $k=400$, $k=100$, $k=20$ and $k=5$, respectively, with $({\mathcal{J}}= 3$, $m=2)$.
  • Figure 5: Profiles of the $\omega$ frequencies for $k=400, ({\mathcal{J}}= 3$, $m=2$). The plot on the right hand side zooms into the plot on the left hand side.