$\gamma^* \gamma$ and $\gamma^* p$ scattering in IHQCD

Abstract

We perform joint fits using measurements of the proton structure functions $F_2(x,Q^2)$ and $F_L(x,Q^2)$, of the photon structure function $F_2^\gamma(x,Q^2)$ and of the total cross-sections $\sigma\left( \gamma p \rightarrow X\right)$ and $\sigma \left( \gamma \gamma \rightarrow X\right)$. The data is gathered from several sources including HERA and LEP. The kinematical range considered is wide with a photon virtuality $Q^2 \leq400 \, {\rm GeV}^2$, Bjorken variable $x<0.01$ and $\sqrt{s} > 4 \, {\rm GeV}$. This is done by considering a $U\left(1\right)$ vector gauge field minimally coupled to the graviton Regge trajectory in improved holographic QCD. We find good agreement with the data for a joint fit of $F_2^\gamma$ and $\sigma\left(\gamma \gamma \rightarrow X\right)$ with a $\chi_{d.o.f.}^2 = 1.07$. For a joint fit of $F_2$, $F_L$ and $\sigma\left( \gamma p \rightarrow X\right)$ we find $\chi_{d.o.f.}^2 = 1.40$. A joint fit of all observables for both processes $\gamma^* \gamma$ and $\gamma^* p$ gives a $\chi_{d.o.f.}^2 = 1.38$. The gravitational couplings of the $U\left(1\right)$ vector gauge field and of the proton with the Reggeons of the graviton Regge trajectory are given as an output of the fitting procedure. An approximate relation between the structure functions, valid at NLO QCD, is used to estimate the parton distribution function of the gluon inside the proton.