Double Asymptotic Scaling at HERA
  Perturbative QCD predicts that at sufficiently large $Q^2$ and small $x$ nucleon structure functions should exhibit scaling in the two variables $\sqrt{\ln\smallfrac{1}{x}\ln\ln Q^2}$ and $\sqrt{\ln\smallfrac{1}{x}\big/\ln\ln Q^2}$, provided only that the small-$x$ behaviour of the input to the perturbative QCD evolution is sufficiently soft. We derive these asymptotic results by writing the gluonic Altarelli--Parisi equation at small $x$ as a two--dimensional wave equation, which propagates the gluon distribution from its boundaries into the asymptotic region. We then show that the existing experimental data on $F_2^p(x,Q^2)$ from HERA provide a remarkable confirmation of both of these scaling predictions. The so--called `hard' pomeron, which does not scale, may thus be excluded by more than three standard deviations, at least in the presently accessible kinematical regime. We propose that existing and future data from HERA should be binned in the two scaling variables, in order to search for scaling violations. 
R. D. Ball and S. Forte