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 $x\rightarrow 0$, then for all larger $Q^2$ one has $$F_2(x,Q^2)\simeq B_S[α_s(Q^2)]^{-d_+}x^{-λ} +B_{NS}[α_s(Q^2)]^{-D_{11}}x^{0.5},$$ $$F_G(x,Q^2)\simeq B_G[α_s(Q^2)]^{-d_+}x^{-λ}$$ $$R(x,Q^2)=\frac{r_0α_s(Q^2)}π,$$ with $D_{11},\,d_+,\,B_G,\,r_0$ calculable in terms of $B_S,\,λ$. Moreover, it was suggested that the ``hard" part of the scattering cross section for real photons (Compton scattering) obeys a similar law, so that $$σ_{γp}\simeq B_{γp}s^λ+A_{γp}\hatσ^P,$$ with a value of $λ$ comparable to that in the expression for the structure functions, and where $\hatσ^P\sim\log^2s$ is a universal, Pomeron-type cross section, and $A_{γp},\, B_{γp}$ 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 $α_s(Q^2)$ at small $Q^2$; and on the ultra high energy limit, and how one might test the so-called BFKL conjecture, $$\lim_{x\rightarrow 0\atop Q^2\rightarrow \infty}F_2(x,Q^2)\sim x^{-c_0α_s}.$$\hb With respect to the last we find some evidence against