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Off-diagonal distributions fixed by diagonal partons at small $x$ and $ξ$

A. G. Shuvaev, K. J. Golec-Biernat, A. D. Martin, M. G. Ryskin

TL;DR

The paper demonstrates that off-diagonal (skewed) parton distributions H(x,ξ) at small x and ξ are uniquely determined by the conventional diagonal partons through the conformal-moment framework and a Shuvaev transform. It provides explicit expressions showing H_q and H_g can be constructed from diagonal quark and gluon distributions, with t-dependence factored by a form factor. The analysis yields power-law predictions H_i(x,ξ) ∼ ξ^{-λ_i} with x/ξ scaling and reveals stronger skewed effects for singlet quarks than for gluons, boosting diffractive vector-meson production cross sections by factors ~R_g^2. These results enable incorporating off-diagonal data into global parton analyses to better constrain small-x behavior, with controlled ξ → 0 uncertainties.

Abstract

We show that the off-diagonal (or skewed) parton distributions are completely determined at small $x$ and $ξ$ by the (conventional) diagonal partons. We present predictions which can be used to estimate the off-diagonal distributions at small $x$ and $ξ$ at any scale.

Off-diagonal distributions fixed by diagonal partons at small $x$ and $ξ$

TL;DR

The paper demonstrates that off-diagonal (skewed) parton distributions H(x,ξ) at small x and ξ are uniquely determined by the conventional diagonal partons through the conformal-moment framework and a Shuvaev transform. It provides explicit expressions showing H_q and H_g can be constructed from diagonal quark and gluon distributions, with t-dependence factored by a form factor. The analysis yields power-law predictions H_i(x,ξ) ∼ ξ^{-λ_i} with x/ξ scaling and reveals stronger skewed effects for singlet quarks than for gluons, boosting diffractive vector-meson production cross sections by factors ~R_g^2. These results enable incorporating off-diagonal data into global parton analyses to better constrain small-x behavior, with controlled ξ → 0 uncertainties.

Abstract

We show that the off-diagonal (or skewed) parton distributions are completely determined at small and by the (conventional) diagonal partons. We present predictions which can be used to estimate the off-diagonal distributions at small and at any scale.

Paper Structure

This paper contains 4 sections, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Off-diagonal distributions in terms of conformal moments
  3. Predictions of the off-diagonal distributions for small $x$ and $\xi$
  4. Discussion

Figures (4)

  • Figure 1: A schematic diagram showing the variables for the off-diagonal parton distribution $H (x, \xi)$ where $x_{1,2} = x \pm \xi$.
  • Figure 2: Predictions at small $x$ and $\xi$ for the ratio of off-diagonal ($H (x, \xi)$) to diagonal $(H_q (\bar{x}, 0) = f_q (\bar{x}), H_g (\bar{x}, 0) = \bar{x} f_g (\bar{x}))$ parton distributions. The diagonal partons are taken to have the form $x f (x) = N x^{- \lambda}$. Plots (a), (b) and (c) show the quark singlet, gluon and quark non-singlet ratios taking $\bar{x} = x + \xi$ as the argument of the diagonal partons. Plot (d) shows the gluon ratio again but versus a linear scale and with argument $\bar{x} = 2 \xi$ so as to display the $x$ behaviour of $H_g (x, \xi)$.
  • Figure 3: The off-diagonal to diagonal ratio, $R$, at small $x$ and $\xi$ versus the power $\lambda$ which specifies the $x^{- \lambda}$ behaviour of the input diagonal parton as in (\ref{['eq:a20']}). Note that the quark singlet ratio has been divided by 2. The vertical arrows indicate the values of $\lambda$ found in a global parton analysis MRST at $Q^2 = 4$ and 100 GeV$^2$.
  • Figure 4: Meson ($M$) Regge exchange indicating the structure of the off-diagonal contribution of (\ref{['eq:a28']}).