The QCD Scale Parameter from the Photon Structure Function

TL;DR

The paper addresses how to determine the QCD scale parameter $\Lambda_{\overline{\textrm{MS}}}$ from the photon structure function by separating the PSF into perturbative (PQCD) and non-perturbative (NP) contributions. It combines the operator product expansion and renormalization group analysis to compute the PQCD moments, which are inverted via Mellin transform to obtain $F_2^\gamma(x,Q^2,P^2)$, while modeling NP with vector meson dominance, dominated by the $\rho$ meson, and extrapolating to $P^2=0$. The NP term is related to the vector meson structure via $F^{\gamma}_{2,NP}(x,Q^2,P^2) \approx \frac{\alpha\pi}{\gamma_\rho^2}\frac{F^{\rho}_2(x,Q^2)}{(1+P^2/M_\rho^2)^2}$ and is anchored to $F_2^\pi$ through isospin relations. A $\chi^2$-minimization of moments across a wide $Q^2$ range yields $\Lambda_{\overline{\textrm{MS}}}=365.1^{+43.5}_{-53.1}$ MeV and $\alpha_S(M_Z,\Lambda_{\overline{\textrm{MS}}})=0.1146^{+0.0021}_{-0.0028}$, compatible with PDG values, supporting the viability of the PSF-based extraction with avenues for refinement by including more vector mesons and richer data sets.

Abstract

Photon structure function has been a solid platform in testing strong interaction along with nucleon structure function. Strong Interaction has the property that it is perturbatively calculable at high energy but becomes non-perturbative at low energy. This nature makes QCD hard to handle theoretically in factorizing these two regions. The fundamental dimensional parameter, so called the QCD scale parameter, $Λ_{\overline{\textrm{MS}}}$, is one of key players to factorize two energy regions. In this work, we extract the QCD scale parameter from the photon structure function by separating the perturbative QCD and non-perturbative QCD. In the process we use the vector dominance model for the non-perturbative energy region of the photon structure function.

Figures (5)

• Figure 1: $ee \rightarrow eeX$ via quarks to produce the hadron $X (=q\bar{q})$. The substructure of this process contains the two photon process $\gamma\gamma\rightarrow q\bar{q}$ at tree level eventually $q\bar{q}\rightarrow X$.
• Figure 2: Box Diagram for two photon process. The scattering amplitude for this box diagram has no internal structure at tree level. However, the quantum fluctuation due to the QCD interaction produces very complicated inner structure inside the box, namely, some complicated gluon connections with the quark lines that form the box.
• Figure 3: $F_2,_{NP}(x,Q^2)$ based on VMD for a few values of $Q^2$
• Figure 4: Logarithmic Error of $F_2,_{NP}(x,Q^2)$
• Figure 5: The total vPSF for $Q^2=5$ GeV$^2$ and $P^2=0.35$ GeV$^2$ compared to PLUTO results