Unitarity and the QCD-improved dipole picture

As a consequence of QCD factorization theorems, a wide variety inclusive and exclusive cross sections may be formulated in terms of a universal colour dipole cross section at small-$x$. It is well known that for small transverse-size dipoles this cross section is related to the leading-log gluon density. Using the measured pion-proton cross section as a guide, we suggest a reasonable extrapolation of the dipole cross section to the large transverse-size region. We point out that the observed magnitude and small-$x$ rise of the gluon density from conventional fits implies that unitarity corrections are required in, or close to, the HERA kinematic region, even for small `perturbative' dipoles. This means that the usual perturbative leading twist description, for moderate virtualities, $1 < Q^2 < 10$ GeV$^2$, has rather large `higher-twist' corrections at small $x$. In addition, for these virtualities, we also find sizeable contributions from large non-perturbative dipoles ($b \gsim 0.4$ fm) to $F_2$, and also to $F_L$. This also leads to deviations from the standard leading-twist DGLAP results, at small-x and moderate $Q^2$.