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Large spin corrections in ${\cal N}=4$ SYM sl(2): still a linear integral equation

Diego Bombardelli, Davide Fioravanti, Marco Rossi

TL;DR

This work develops an interval-based non-linear integral equation (NLIE) framework for the $sl(2)$ sector of ${\cal N}=4$ SYM to study large-spin corrections. The key insight is that nonlinear finite-range terms vanish faster than any inverse logarithm as $s\to\infty$, reducing the problem to a linear BES-like description that captures both leading $\ln s$ and subleading $O(s^0)$ terms. All-loop generalization shows that the counting function splits into a one-loop part and a higher-loop part, with the high-loop density obeying a linear BES equation whose forcing encodes universal scaling through the function $f(g)$ and a subleading correction $E^{extra}(g,s)$. The results yield explicit expressions for the energy $E(g,s)$ and even charges $Q_r(g,s)$, matching known BES predictions at leading order and providing new subleading information, thereby enabling a non-perturbative handle on large-spin dynamics and potential access to strong coupling regimes when wrapping effects are under control.

Abstract

Anomalous dimension and higher conserved charges in the $sl(2)$ sector of ${\cal N}=4$ SYM for generic spin $s$ and twist $L$ are described by using a novel kind of non-linear integral equation (NLIE). The latter can be derived under typical situations of the SYM sectors, i.e. when the scattering need not depend on the difference of the rapidities and these, in their turn, may also lie on a bounded range. Here the non-linear (finite range) integral terms, appearing in the NLIE and in the dimension formula, go to zero as $s\to \infty$. Therefore they can be neglected at least up to the $O(s^0)$ order, thus implying a linear integral equation (LIE) and a linear dimension/charge formula respectively, likewise the 'thermodynamic' (i.e. infinite spin) case. Importantly, these non-linear terms go faster than any inverse logarithm power $(\ln s)^{-n}$, $n>0$, thus extending the linearity validity.

Large spin corrections in ${\cal N}=4$ SYM sl(2): still a linear integral equation

TL;DR

This work develops an interval-based non-linear integral equation (NLIE) framework for the sector of SYM to study large-spin corrections. The key insight is that nonlinear finite-range terms vanish faster than any inverse logarithm as , reducing the problem to a linear BES-like description that captures both leading and subleading terms. All-loop generalization shows that the counting function splits into a one-loop part and a higher-loop part, with the high-loop density obeying a linear BES equation whose forcing encodes universal scaling through the function and a subleading correction . The results yield explicit expressions for the energy and even charges , matching known BES predictions at leading order and providing new subleading information, thereby enabling a non-perturbative handle on large-spin dynamics and potential access to strong coupling regimes when wrapping effects are under control.

Abstract

Anomalous dimension and higher conserved charges in the sector of SYM for generic spin and twist are described by using a novel kind of non-linear integral equation (NLIE). The latter can be derived under typical situations of the SYM sectors, i.e. when the scattering need not depend on the difference of the rapidities and these, in their turn, may also lie on a bounded range. Here the non-linear (finite range) integral terms, appearing in the NLIE and in the dimension formula, go to zero as . Therefore they can be neglected at least up to the order, thus implying a linear integral equation (LIE) and a linear dimension/charge formula respectively, likewise the 'thermodynamic' (i.e. infinite spin) case. Importantly, these non-linear terms go faster than any inverse logarithm power , , thus extending the linearity validity.

Paper Structure

This paper contains 13 sections, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A new approach: the NLIE on the interval(s)
  3. An example: the $XXX_{-\frac{1}{2}}$ spin chain
  4. Determination of $F_0$ when $s\rightarrow +\infty$
  5. The leading order density equation from the NLIE
  6. Determination of $G_0$ when $s\rightarrow \infty$
  7. Evaluation of the charges when $s\rightarrow \infty$
  8. All loops
  9. Equations for $F$, $F_0$ and $F^H$.
  10. The NLIE in the $s\rightarrow +\infty$ limit
  11. An alternative derivation of the anomalous dimension
  12. Summary
  13. Numerical evaluation of the non-linear integrals

Figures (2)

  • Figure 1: Comparison between numerical evaluations (dots) of $\delta Z_0(u=1;s)$ and their best fit as $a_1/s+a_2/s^2+a_3/s^3$ (line), with $a_1\simeq0.001$, $a_2\simeq-0.105$, $a_3\simeq-8.137$.
  • Figure 2: Comparison between numerical evaluations (dots) of $\delta E_0(s)$ and their best fit as $b_1/s+b_2/s^2+b_3/s^3$ (line), with $b_1\simeq0.675$, $b_2\simeq-4.667$, $b_3\simeq-39.390$.