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The eikonal spin-dependent Odderon and gluon Sivers function of a proton, and its small-$x$ evolution

Sanjin Benić, Adrian Dumitru, Florian Hechenberger, Tomasz Stebel

Abstract

The matrix element in the proton of the eikonal Odderon operator, with a helicity flip, has been shown to correspond to the dipole gluon Sivers function. We employ a three quark light-front model of the proton to determine the Sivers function at moderately small $x_0 \sim 0.1$ and transverse momentum $k_\perp \lesssim 1$~GeV. The model light-cone (LC) wave function predicts the properties of $x f_{1T}^{\perp g}(x,k_\perp)$ such as its overall magnitude, the position of its peak in $k_\perp$, and its behavior at small $k_\perp$. We then compute numerically the BFKL anomalous dimension characterizing the power-law tail at $k_\perp \gtrsim 1.5$~GeV of the gluon Sivers function at small LC momentum fractions, $α_s \log x_0/x = 1$: $x f_{1T}^{\perp g}(x,k_\perp) \sim k_\perp^{-3.3}$.

The eikonal spin-dependent Odderon and gluon Sivers function of a proton, and its small-$x$ evolution

Abstract

The matrix element in the proton of the eikonal Odderon operator, with a helicity flip, has been shown to correspond to the dipole gluon Sivers function. We employ a three quark light-front model of the proton to determine the Sivers function at moderately small and transverse momentum ~GeV. The model light-cone (LC) wave function predicts the properties of such as its overall magnitude, the position of its peak in , and its behavior at small . We then compute numerically the BFKL anomalous dimension characterizing the power-law tail at ~GeV of the gluon Sivers function at small LC momentum fractions, : .
Paper Structure (8 sections, 38 equations, 3 figures)

This paper contains 8 sections, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The spin dependent Odderon(s)
  3. Eikonal triple gluon coupling to the proton
  4. BFKL evolution of the helicity flip Odderon
  5. Numerical results
  6. Summary and discussion
  7. LC constituent quark model wave function
  8. Dirac and Pauli electromagnetic form factors

Figures (3)

  • Figure 1: (a) The forward Odderon $k_\perp^2\mathcal{O}_S(k_\perp)/g^6$ in transverse momentum space. The dashed line shows the best fit via eq. \ref{['eq:k2odderon_log_fit']}. (b) and (c) Gluon Sivers function $x f_{1T}^\perp (x,k_\perp)$ obtained from the helicity flip, forward Odderon for $\alpha_s =0.25$. The dashed black line in the lower inset emphasizes the log-type divergence at low-$k_\perp$, see also \ref{['eq:k2odderon_log_fit']} .
  • Figure 2: (a) Radial part of the helicity flip Odderon amplitude at different rapidities, $Y=0, 1, 2, 4$. The inset is a double logarithmic version of the plot to better show the small-$r_\perp$ regime. (b): high-$k_\perp$ tail of the momentum-space Odderon amplitude $k_\perp \mathcal{O}_S(k_\perp)$, which is proportional to $xf_{1T}^{\perp g}(x,k_\perp^2)$, at $Y=4$. The dashed line indicates the $\sim1/k_\perp^{3.3}$ power-law asymptotics.
  • Figure 3: The Dirac (left) and Pauli (right) electromagnetic form factors obtained from the three-quark LCwf compared to a modern fit (solid black line) to experimental data Ye:2017gyb.