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Spinning superstrings at two loops: strong-coupling corrections to dimensions of large-twist SYM operators

R. Roiban, A. A. Tseytlin

TL;DR

The authors compute the two-loop strong-coupling correction to the dimensions of large-twist SYM operators by analyzing a folded spinning string in AdS_5×S^5 with fixed charges in a semiclassical scaling limit. They develop a general framework using the world-sheet partition function with fixed E−S and J, perform a detailed sigma-model loop expansion around the classical solution, and extract energy corrections up to two loops. The results confirm the predicted leading log structure from the O(6) sigma-model and Alday–Maldacena, including matching coefficients for the ℓ^2 log ℓ and ℓ^2 log^2 ℓ terms, while highlighting the need to resum logarithms to interpolate to weak coupling. The work also sets up a systematic approach for higher-loop corrections and discusses the role of charge renormalization in shaping the strong-coupling expansion.

Abstract

We consider folded spinning strings in AdS_5xS^5 (with one spin component S in AdS_5 and J in S^5) corresponding to the Tr(D^S Z^J) operators in the sl(2) sector of the N=4 SYM theory in the special scaling limit in which both the string mass M ~ \sqrt λ\ln S and J are sent to infinity with their ratio fixed. Expanding in the parameter \el= J/M we compute the 2-loop string sigma model correction to the string energy and show that it agrees with the expression proposed by Alday and Maldacena in arxiv:0708.0672. We suggest that a resummation of the logarithmic \el^2 \ln^n \el terms is necessary in order to establish an interpolation to the weakly coupled gauge theory results. In the process, we set up a general framework for the calculation of higher loop corrections to the energy of multi-spin string configurations. In particular, we find that in addition to the direct 2-loop term in the string energy there is a contribution from lower loop order due to a finite ``renormalization'' of the relation between the parameters of the classical solution and the fixed spins, i.e. the charges of the SO(2,4) x SO(6) symmetry.

Spinning superstrings at two loops: strong-coupling corrections to dimensions of large-twist SYM operators

TL;DR

The authors compute the two-loop strong-coupling correction to the dimensions of large-twist SYM operators by analyzing a folded spinning string in AdS_5×S^5 with fixed charges in a semiclassical scaling limit. They develop a general framework using the world-sheet partition function with fixed E−S and J, perform a detailed sigma-model loop expansion around the classical solution, and extract energy corrections up to two loops. The results confirm the predicted leading log structure from the O(6) sigma-model and Alday–Maldacena, including matching coefficients for the ℓ^2 log ℓ and ℓ^2 log^2 ℓ terms, while highlighting the need to resum logarithms to interpolate to weak coupling. The work also sets up a systematic approach for higher-loop corrections and discusses the role of charge renormalization in shaping the strong-coupling expansion.

Abstract

We consider folded spinning strings in AdS_5xS^5 (with one spin component S in AdS_5 and J in S^5) corresponding to the Tr(D^S Z^J) operators in the sl(2) sector of the N=4 SYM theory in the special scaling limit in which both the string mass M ~ \sqrt λ\ln S and J are sent to infinity with their ratio fixed. Expanding in the parameter \el= J/M we compute the 2-loop string sigma model correction to the string energy and show that it agrees with the expression proposed by Alday and Maldacena in arxiv:0708.0672. We suggest that a resummation of the logarithmic \el^2 \ln^n \el terms is necessary in order to establish an interpolation to the weakly coupled gauge theory results. In the process, we set up a general framework for the calculation of higher loop corrections to the energy of multi-spin string configurations. In particular, we find that in addition to the direct 2-loop term in the string energy there is a contribution from lower loop order due to a finite ``renormalization'' of the relation between the parameters of the classical solution and the fixed spins, i.e. the charges of the SO(2,4) x SO(6) symmetry.

Paper Structure

This paper contains 14 sections, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of computation of quantum corrections to string energy
  3. Partition function with fixed charges
  4. Partition function of a world-sheet theory with fixed charges
  5. Sigma model loop expansion
  6. Classical solution and fluctuation action
  7. Solution and $AdS_5 \times S^5$ superstring action in Poincaré coordinates
  8. Classical value of the action
  9. Fluctuation Lagrangian
  10. Quantum corrections to the effective action
  11. One loop
  12. Two loops
  13. Quantum corrections to the string energy
  14. Resumming the logarithms: towards interpolation to gauge-theory results

Figures (1)

  • Figure 1: Topologies of possible two-loop diagrams; each line denotes either a bosonic or a fermionic propagator.