Theory of Small x Inclusive Photon Scattering, I

Abstract

In the early eighties, López, González-Arroyo and the present author proved that, if at a given $Q_0^2$ large enough for perturbative QCD to be valid, structure functions behave as a power of $x$ for $x\to 0$, then for all larger $Q^2$ one has $$F_2(x,Q^2)\simeq B_S[{\alpha }_s(Q^2){]}^{-d_{+}}{x}^{-\lambda }+B_{NS}[{\alpha }_s(Q^2){]}^{-D_{11}}{x}^{0.5},$$ $$F_G(x,Q^2)\simeq B_G[{\alpha }_s(Q^2){]}^{-d_{+}}{x}^{-\lambda }$$ $$R(x,Q^2)=\frac{r_0{\alpha }_s(Q^2)}{\pi },$$ with $D_{11},d_{+},B_G,r_0$ calculable in terms of $B_S,\lambda$. Moreover, it was suggested that the ``hard" part of the scattering cross section for real photons (Compton scattering) obeys a similar law, so that $${\sigma }_{\gamma p}\simeq B_{\gamma p}{s}^{\lambda }+A_{\gamma p}{\hat{\sigma }}^P,$$ with a value of $\lambda$ comparable to that in the expression for the structure functions, and where ${\hat{\sigma }}^P\sim {\log }^{2}s$ is a universal, Pomeron-type cross section, and $A_{\gamma p},B_{\gamma p}$ are constants. In the present paper it is shown that the recent HERA measurements may be described by these formulas, with a chi-sqared/d.o.f. substantially less than unity, and with values of the parameters compatible with those of the old fits of the '80s. Moreover, further discussions are presented both on the low $Q^2$ limit, and the transition between Compton and deep inelastic scattering, in particular in connection with possible saturation of the coupling constant ${\alpha }_s(Q^2)$ at small $Q^2$; and on the ultra high energy limit, and how one might test the so-called BFKL conjecture, $$\underset{\genfrac{}{}{0ex}{}{x\to 0}{Q^2\to \infty }}{\lim }{F}_2(x,Q^2)\sim x^{-c_0{\alpha }_s}.$$\hb With respect to the last we find some evidence against

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