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Structure Functions for small DIS $x$

Hrachya M. Babujian, Angela Foerster, Michael Karowski

TL;DR

This work analyzes structure functions in a set of exactly solvable 1+1D QFTs to understand small-x behavior. By exploiting exact form factors and S-matrices, the authors demonstrate a universal factorization W(p,q) ≈ f(x) g(q^2) for asymptotically free models (O(N) sigma-model and SU(N) chiral Gross-Neveu), with f(x) ∝ 1/x and an accompanying logarithmic factor, confirming Balog and Weisz's conjecture. In contrast, the sine-Gordon, sinh-Gordon, and Z(N) models exhibit power-law or linear small-x behavior, with explicit exponents determined by coupling parameters; the sine-Gordon case shows especially striking power-law behavior that resonates with high-energy DIS trends. The results suggest a potential bridge between 2D integrable field theories and the longitudinal sector of 4D QCD, motivating further exploration of Balog-Weisz-type universalities beyond two dimensions. Overall, the paper provides concrete, model-dependent calculations of g(q^2) and universal x-dependent factors across a diversity of integrable theories, illustrating how exact non-perturbative data informs DIS-like observables in low dimensions.

Abstract

Structure Functions for small DIS (deep inelastic scattering) $x$ for integrable models are investigated, in particular, for the $O(N)$~$σ$-model and $SU(N)$ chiral Gross-Neveu model, which are asymptotically free. We get the universal behavior $x^{-1}\ln^{-2}x$ at small Bjorken variable $x$ and confirm a Balog Weisz conjecture. For a the second group of models, the Sine-Gordon, sinh-Gordon and $Z(N)$, we find power behavior $x^{-λ}$. The special behavior of the structure function for the Sine-Gordon model is probably crucial for future investigation in 4D QCD.

Structure Functions for small DIS $x$

TL;DR

This work analyzes structure functions in a set of exactly solvable 1+1D QFTs to understand small-x behavior. By exploiting exact form factors and S-matrices, the authors demonstrate a universal factorization W(p,q) ≈ f(x) g(q^2) for asymptotically free models (O(N) sigma-model and SU(N) chiral Gross-Neveu), with f(x) ∝ 1/x and an accompanying logarithmic factor, confirming Balog and Weisz's conjecture. In contrast, the sine-Gordon, sinh-Gordon, and Z(N) models exhibit power-law or linear small-x behavior, with explicit exponents determined by coupling parameters; the sine-Gordon case shows especially striking power-law behavior that resonates with high-energy DIS trends. The results suggest a potential bridge between 2D integrable field theories and the longitudinal sector of 4D QCD, motivating further exploration of Balog-Weisz-type universalities beyond two dimensions. Overall, the paper provides concrete, model-dependent calculations of g(q^2) and universal x-dependent factors across a diversity of integrable theories, illustrating how exact non-perturbative data informs DIS-like observables in low dimensions.

Abstract

Structure Functions for small DIS (deep inelastic scattering) for integrable models are investigated, in particular, for the ~-model and chiral Gross-Neveu model, which are asymptotically free. We get the universal behavior at small Bjorken variable and confirm a Balog Weisz conjecture. For a the second group of models, the Sine-Gordon, sinh-Gordon and , we find power behavior . The special behavior of the structure function for the Sine-Gordon model is probably crucial for future investigation in 4D QCD.
Paper Structure (38 sections, 4 theorems, 129 equations, 5 figures)

This paper contains 38 sections, 4 theorems, 129 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Structure functions
  3. DIS kinematic variables
  4. New variables:
  5. $\Lambda_{0}$ and $u_{r-1}^{(0)}$ in terms of $x$ and $\kappa$ for small $x$:
  6. Factorization
  7. Results for various models
  8. The sinh-Gordon model S-matrix
  9. The $Z(N)$ model S-matrix
  10. The function $g(q^{2})$
  11. The $O(N)$ $\sigma$-model
  12. $O(N)$ current and pseudo-potential:
  13. Three-intermediate-particle $(r=3)$ approximation:
  14. The $SU(N)$ chiral Gross-Neveu model
  15. The $SU(N)$ Noether current:
  16. ...and 23 more sections

Key Result

Lemma 1

We set $p=\binom{m}{0}$ or $\theta=0$, write $\underline{\theta }=\underline{\hat{\theta}}+W,\theta_{r}$ and $\underline{\alpha}=\alpha _{1},\dots,\alpha_{r}=\underline{\hat{\alpha}},\alpha_{r}$. For Lorentz scalar operators $\mathcal{O}$ and $\mathcal{O}^{\prime}$ the form factor equations (fi)-(fv and finally

Figures (5)

  • Figure 1: Plots of $g(q^{2})$ versus $-q^{2}/m^{2}$ for $O(N)$, where $N=$ 3 (red), 4 (yellow), 5 (green) and 6 (blue)
  • Figure 2: Plots of $g(q^{2})$ versus $-q^{2}/m^{2}$ for $SU(N)$, where $N=$ 2 (red), 3 (yellow), 5 (green) and 8 (blue)
  • Figure 3: Plots of $g(q^{2})$ versus $-q^{2}/m^{2}$ for Sine-Gordon, where $\nu=$ 1.2 (red), 1.8 (yellow), 3 (green) and 6 (blue)
  • Figure 4: Plot of $g(q^{2})$ versus $-q^{2}/m^{2}$ for Sinh-Gordon, where $\nu=-1/2$
  • Figure 5: Plot of $g(q^{2})$ versus $-q^{2}/m^{2}$ for $Z(3)$

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • Lemma 7