Have we seen anything beyond (N)NLO DGLAP at HERA?

The evidence from HERA for parton saturation, and other low-x effects beyond the conventional DGLAP formalism, is recalled and critically reviewed in the light of new data and analyses presented at the conference. In the mid-90’s the original surprise of the HERA Neutral Current ep scattering data was the strong rise of the structure function F2 at low-x. This was taken to imply a strong rise of the gluon density at low-x which was widely interpreted as implying the possibility of gluon saturation and the need for non-linear terms in the parton evolution equations. Even somewhat more conservative interpretations suggested the need to go beyond the DGLAP formalism at small-x, resumming ln(1/x) as in the BFKL formalism. However, at low-x linear NLO DGLAP evolution itself predicts a rise in F2, and in the gluon and sea PDFs, provided that Q is large enough. One can begin parton evolution at a low Q input scale, Q20, using flat (or even valence-like) gluon and sea-quark input shapes in x and the DGLAP Q 2 evolution will generate a steep low-x rise of the gluon and sea at larger Q ≫ Q20. The real surprise seen in the data of the late 90’swas that steep shapes were already observed at rather low Q. Traditionally values of Q20 ∼ 4GeV 2 were used, but the data already show a steep rise of F2 at low-x for Q values, Q ∼ 1GeV, see Fig. 1 left-hand-side. To interpret these data in terms of conventional NLO DGLAP evolution we clearly need a low starting scale and thus we are forced into using perturbative QCD at a scale for which αs(Q) is quite largeαs(1.0) ∼ 0.35. Even if this is considered to be acceptable, we also need to use flexible input parton shapes, which can reproduce the steepness of the data. Surprisingly enough this does NOT imply that both the gluon and the sea input are already steep at Q ∼ 1GeV. The sea input is indeed steep, but the gluon input is valence-like, with a tendency to be negative at low-x!see Fig. 1 right-hand-side. (Essentially the gluon evolution must be fast in order that upward evolution can produce the extreme steepness of high-Q data, however this also implies that downward evolution is fast and this results in the valence-like gluon at low-Q). Thus when statements are made that HERA has established that the low-x gluon is steep one must remember that this is only true for higher Q, Q > ∼ 10GeV , within the DGLAP formalism. However this formalism seems to work to much lower Q. Let us examine how the gluon and sea PDFs are extracted from the measurements. At low-x, the sea PDF is extracted fairly directly since, F2(x,Q) ∼ xq(x,Q). However the gluon PDF is extracted from the scaling violations, ∂F2/∂ln(Q) ∼ Pqgxg(x,Q), such that the measurement is related to a convolution of the splitting function Pqg and the gluon distribution. Thus if the correct splitting function is NOT that of the conventional DGLAP formalism, or if a more complex non-linear realtionship is needed, then a turn over of the data ∂F2/∂ln(Q) at low-Q and lowx may not imply a turn over of the gluon distribution. It was suggested that measurements of other gluon related quantities could help to shed light on this question and the longitudinal structure function, FL, and the heavy quark structure functions, F cc̄ 2 , F bb̄ 2 , are obvious candidates. All of these quantities have now been measured (see talks of K. Papageorgiou and P. Thompson in these proceedings) and, within present experimental uncertainties, they can be explained by the conventional NLO DGLAP formalism (with the heavy quark results shedding more light on the complexities of general-mass-variable-flavour number schemes than on the gluon PDF). Other measurements of more exclusive quantities can also give information on the correctness of the conventional formalism at low-x. For example HERA forward jet mesaurements (see talk of A. Savin in these proceedings). DGLAP evolution would suppress the forward jet cross-section, for