Vector meson dominance and deep inelastic scattering at low and medium $Q^2$

We argue that deep inelastic scattering (DIS) at small values of $Q^2$ is an essentially nonperturbative process and can be described, partially at least, by the vector meson dominance (VMD) model. We show this by the straightforward calculation that the VMD model alone can successfully explain data on structure functions of DIS in a broad interval of $x(5\times {10}^{-2}÷{10}^{-4})$ for the region $Q^2\le 1{\text{GeV}}^2$. For a description of data at larger $Q^2$ we use the two-component ($\mathrm{VMD}+\text{perturbative}\mathrm{QCD}$) approach. We show that these two components can be separated if VMD is used in the aligned jet version. We take into account, in calculations of the VMD component of structure functions, the excited states of the $\rho$-meson and nondiagonal transitions between different members of the $\rho$-meson family. Amplitudes of these transitions arre obtained using a formalism of the light-front Bethe-Salpeter equation and the method of diffraction-scattering eigenstates. The perturbative QCD component is calculated using a framework of the color dipole model with the dipole cross section having a Regge-type energy dependence. We present results of the detailed comparison of our predictions with experimental data for structure functions of the nucleon. We also obtain approximate predictions for the structure functions in the region of very small $x$, up to ${10}^{-8}÷{10}^{-9}$, and show that the nonperturbative component at such values of $x$ is still relatively large and must be taken into account if $Q^2$ is about a few ${\mathrm{GeV}}^2$ or less.

Figure 1

The diagram representing schematically the contribution of $q\overline{q}$ channel in an imaginary part of the Compton forward scattering amplitude.

Figure 2

The schematic diagram for the Compton forward scattering amplitude in a nondiagonal VMD model.

Figure 3

The dependence of $M_{{\rho }_n}^2$ on the radial quantum number $n$ of the excited state. The dots are experimental values of vector meson masses of the $\rho$-family members (the values of masses for $n=3$ and for $n=4$ are poorly known). Lines correspond to two different parametrizations of mass spectrum [Eq. (53)].

Figure 4

The diagram used for a normalization of the BS wave function.

Figure 5

The total cross section of photoabsorption for the real photon (the dashed line). The solid line is the soft contribution. The experimental data points are taken from [51, 52, 53, 54, 55]. The experimental points in the interval $\sqrt{s}=40–210\mathrm{GeV}$ are taken from [51] (cosmic ray data).

Figure 6

The $Q^2$ dependence of the structure function $F_2$ for different values of $x$. In the left figure the data of each bin of fixed $x$ has been multiplied by $2^i$, where $i$ is a number of the bin, ranging from $i=8$ ($x=0.08$) to $i=\text{28}$ ($x=0.0000\text{63}$). The experimental points are taken from [45]. The results of calculations ($\text{soft}+\text{hard}$) are shown by dashed curves. The right figure contains the same results but the data of $x$ bins are shown, together with theoretical curves, for the bins with odd numbers only ($\mathrm{i}=9,11\dots$) and corresponding contributions of the soft component are shown separately, by solid curves.

Figure 7

The $x$ dependence of the structure function $F_L$ for different values of $Q^2({\mathrm{GeV}}^2)$. The experimental points are taken from [56] (H1 Collaboration). The results of calculations ($\text{soft}+\text{hard}$) are shown by dashed curves.

Figure 8

The $x$ dependence of the structure function $F_2$ for small values of $Q^2$ in the region of very small $x$. The experimental points are taken from [58, 59]. The results of calculations ($\text{soft}+\text{hard}$) are shown by dashed curves. On the left figure the data are scaled by powers of 1.5, $\mathrm{n}=1,2,3\dots$ (from bottom to top). The dotted lines are the GBW predictions [57]. On the right figure the data are scaled by powers of 1.5, $\mathrm{n}=1,2,3\dots$ (from bottom to top), and the corresponding contributions of the soft component are shown separately, by solid lines.

Figure 9

The $x$ dependence of the structure function $F_2$ for medium values of $Q^2$ in the region of very small $x$. The experimental points are taken from [60]. The results of calculations ($\text{soft}+\text{hard}$) are shown by dashed curves. On the left figure the data are scaled by powers of 1.5, $\mathrm{n}=1,2,3\dots$ (from bottom to top). The dotted lines are the GBW predictions [57]. On the right figure the data are scaled by powers of 1.5, $\mathrm{n}=2,4,6\dots$ (from bottom to top), and the corresponding contributions of the soft component are shown separately, by solid curves.

Figure 10

Horizontal axis: the number of vector mesons which is taken into account in a calculation of ${\sigma }_{\gamma p}^{(\text{soft})}(0,s)$. Vertical axis: the corresponding part of the full cross section. The curve saturates on the value $114\mu \mathrm{b}$ at $n=8$.