${\gamma }^{*}\gamma$ and ${\gamma }^{*}p$ scattering in improved holographic QCD

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 $\sigma (\gamma p\to X)$ and $\sigma (\gamma \gamma \to X)$. The data is gathered from several sources including HERA and LEP. The kinematical range considered is wide with a photon virtuality $Q^2\le 400{\mathrm{GeV}}^2$, Bjorken variable $x<0.01$ and $\sqrt{s}>4\mathrm{GeV}$. This is done by considering a $U(1)$ 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 $\sigma (\gamma \gamma \to X)$ with a ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2=1.07$. For a joint fit of $F_2$, $F_L$ and $\sigma (\gamma p\to X)$ we find ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2=1.40$. A joint fit of all observables for both processes ${\gamma }^{*}\gamma$ and ${\gamma }^{*}p$ gives a ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2=1.38$. The gravitational couplings of the $U(1)$ 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.

Figure 1

Tree level Witten diagram representing spin $J$ exchange in $12\to 34$ scattering. The $n_1$ and $n_3$ labels denote the incoming/outgoing photon polarizations, for forward scattering $n_1=n_3$.

Figure 2

Tree level Witten diagram representing spin $J$ exchange in a $2\to 2$ scattering process between bulk photons. We will consider a kinematical regime dominated by the exchange of spin $J$ fields in the graviton Regge trajectory.

Figure 3

Predicted proton structure function $F_2(x,Q^2)$ vs experimental points. The curves were obtained using the values of Table 1.

Figure 4

Predicted proton structure function $F_L(x,Q^2)$ vs experimental points. The curves were obtained using the values of Table 1.

Figure 5

Predicted $\sigma (\gamma p\to X)$ vs experimental points. The curve was obtained using the values from Table 1.

Figure 6

Predicted $\sigma (\gamma \gamma \to X)$ vs experimental points. The curve was obtained using the parameter values in Table 2.

Figure 7

Predicted $F_2^{\gamma }(x,Q^2)$ vs experimental points. The curves were obtained using the values from Table 2.

Figure 8

Comparison between the holographic gluon PDFs with the ones of CT18NNLO and NNPDF for $Q^2=5,10,20,100{\mathrm{GeV}}^2$.