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A next-to-leading Luescher formula

Diego Bombardelli

TL;DR

This paper introduces next-to-leading Luescher-like corrections for excited-state energies in integrable theories by modeling excited states as momentum-dependent defects that twist the vacuum. It derives explicit leading and next-to-leading finite-size corrections from the S-matrix, and provides rapidity corrections within an excited-state TBA framework, validated against NLIE results in several relativistic models. The authors also conjecture AdS5/CFT4 generalizations, linking the approach to the Konishi operator and highlighting a practical path to high-loop finite-size corrections. Overall, the work bridges Luescher-type methods with NLIE/TBA formalisms and sets the stage for systematic finite-volume analyses in both relativistic and AdS/CFT contexts.

Abstract

We propose a next-to-leading Luescher-like formula for the finite-size corrections of the excited states energies in integrable theories. We conjecture the expressions of the corrections for both the energy and the particles' rapidities by interpreting the excited states as momenta-dependent defects. We check the resulting formulas in some simple relativistic model and conjecture those for the AdS5/CFT4 case.

A next-to-leading Luescher formula

TL;DR

This paper introduces next-to-leading Luescher-like corrections for excited-state energies in integrable theories by modeling excited states as momentum-dependent defects that twist the vacuum. It derives explicit leading and next-to-leading finite-size corrections from the S-matrix, and provides rapidity corrections within an excited-state TBA framework, validated against NLIE results in several relativistic models. The authors also conjecture AdS5/CFT4 generalizations, linking the approach to the Konishi operator and highlighting a practical path to high-loop finite-size corrections. Overall, the work bridges Luescher-type methods with NLIE/TBA formalisms and sets the stage for systematic finite-volume analyses in both relativistic and AdS/CFT contexts.

Abstract

We propose a next-to-leading Luescher-like formula for the finite-size corrections of the excited states energies in integrable theories. We conjecture the expressions of the corrections for both the energy and the particles' rapidities by interpreting the excited states as momenta-dependent defects. We check the resulting formulas in some simple relativistic model and conjecture those for the AdS5/CFT4 case.

Paper Structure

This paper contains 11 sections, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main idea
  3. Lüscher corrections
  4. Leading order
  5. Next-to-leading order
  6. Rapidities' corrections
  7. Comparison with NLIE
  8. Gross-Neveu model
  9. Sine-Gordon and $O(4)$ $\sigma$-model
  10. Conjectures for AdS/CFT
  11. Summary and Outlook

Figures (2)

  • Figure 1: On the left the momentum-dependent defect is circling the compactified space. On the right it is located in (Euclidean) time, and it acts as an operator on the periodic Hilbert space of the rotated channel. The red point is a scattering between virtual particles, the blue one between a virtual particle and the defect.
  • Figure 2: One-particle (left) and two-particle (right) eigenvalues of the defect operator. The horizontal thick lines represent mirror particles, whose flavors are denoted by greek indexes. The vertical thin lines with latin indexes are physical particles.