Parton Distributions of the Virtual Photon

TL;DR

The paper addresses extending real-photon parton distributions to virtual photons by using a dispersion relation over qq̄ fluctuations and a vector-meson dominance plus anomalous decomposition. It derives a general expression for $f_a^{γ⋆}(x,Q^2,P^2)$ and introduces several closed-form approximations (P0, P0', P_eff, P_int) that map virtuality effects onto existing real-photon PDFs, enabling practical SaS parametrizations. Comparisons with jet-rate data illustrate distinct $P^2$-dependent behaviors and demonstrate the potential to discriminate between prescriptions with future measurements. A public code accompanies the framework, enabling evaluation of $f_a^{γ⋆}$ and $F_2^{γ⋆}$ across $0≤P^2≤Q^2$ for phenomenological studies.

Abstract

We propose a generic ansatz for the extension of parton distributions of the real photon to those of the virtual photon. Alternatives and approximations are studied that allow closed-form parametrizations.

Figures (2)

• Figure 1: The u-quark parton distribution $xu(x,Q^2 = 10 \, \mathrm{GeV}^2)/\alpha_{\mathrm{em}}$. Top full line for $P^2 = 0$, below comparison of five alternatives for $P^2 = 0.25$ and 1 GeV$^2$. Ordered roughly from top to bottom, dashed is $P_0$, dotted is $P_0'$, dash-dotted is $P_{\mathrm{eff}}$, large dots is $P_{\mathrm{int}}$ and full is the integral in eq. (\ref{['decompvirt']}).
• Figure 2: The fall-off of parton distributions with virtuality $P$ (we have chosen $P$ as $x$ scale rather than $P^2$, so as to better show the small-$P$ region), normalized to the value at $P^2 = 0$, $\mathcal{I}(P^2)/\mathcal{I}(0)$, for $Q^2 = 20.25$ GeV$^2$. Here $\mathcal{I}$, defined by eq. (\ref{['Idef']}), is the colour-factor-weighted sum of parton distributions in the range $0.1 < x < 0.75$. Curves are labelled as in Fig. 1.