Conventional DGLAP

1. There are various reasons to worry that conventional LO and NLO ln(Q2) summations – as embodied in the DGLAP equations may be inadequate • It was a surprise to see F2 steep at small x - even for very very low Q2, Q2 ~ 1 GeV2 • Should perturbative QCD work? αs is becoming large - αs at Q2 ~ 1 GeV2 is ~ 0.4 • There hasn’t been enough lever arm in Q2 for evolution, but even the starting distribution is steep- the HUGE rise at low-x makes us think • there should beln(1/x) resummation(BFKL) as well as the traditional ln(Q2) DGLAP resummation- BFKL predicted F2(x,Q2) ~ x –λs, with λs=0.5, even at low Q2 • and/or there should be non-linear high density corrections for x < 5 10 -3

2. Conventional DGLAP

3. In fact when HERA low-x data were first published the gluon went from being flat at low-x to steep at low-x But then when the HERA data proved to still be steep even at very low-Q2 the DGLAP fits started to produce gluons which are turning over again at low-x. The gluon evolves very fast- in order to evolve so fast upwards it also has to evolve fast downwards

4. We need other gluon sensitive measurements at low x, like FL or F2charm, F2beauty….BUT FL looked pretty conventional --until recently could be described with usual NLO DGLAP formalism But see latest measurements at lower Q2 We are learning more about heavy quark treatments than about the gluon, so far

5. We need other gluon sensitive measurementslike FL: in NLO DGLAP FL is given by And at low-x this becomes gluon dominated And compare to alternative theoretical predictions: White and Thorne (WT) which has NLL ln1/x resummation included Dipole Models which can accommodate non-linear effects/ saturation eg IIM colour glass condensate Now there are HERA measurements on FL from 2007: Compare to various NLO DGLAP fits But this is not conclusive So…

6. But there are other ways of looking for unconventional ‘beyond DGLAP’ behaviour Look at the hadron final states..lack of pt ordering has its consequences.Forward jets with xj» x and ktj2 ~ Q2 are suppressed for DGLAP evolution but not for kt disordered BFKL evolution But this has only served to highlight the fact that the conventional calculations of jet production were not very well developed. There has been much progress on more sophisticated calculations e.g DISENT, NLOJET, rather than ad-hoc calculations (LEPTO-MEPS, ARIADNE CDM …) The data do not agree with DGLAP at LO or NLO, or with LEPTO-MEPS..but agree with ARIADNE. ARIADNE is not kt ordered but it is not a convincing BFKL calculation either………

7. Forward Jets DISENT vs data Results from 2007 Comparison to LO and NLO conventional calculations NLO below data, especially at small xBj but theoretical uncertainty is large

8. Q2 = 2GeV2 xg(x) The negative gluon predicted at low x, low Q2 from NLO DGLAPremains at NNLO (worse) The corresponding FL is NOT negative at Q2 ~ 2 GeV2 – but has peculiar shape Including ln(1/x) resummation in the calculation of the splitting functions (BFKL `inspired’)can improve the shape - and the c2 of the global fit improves Back to considering inclusive quantities

9. The use of non-linear evolution equations also improves the shape of the gluon at low x, Q2 The gluon becomes steeper (high density) and the sea quarks less steep Non-linear effects gg  g involve the summation of FAN diagrams – higher twist Q2 = 1.4 GeV2 xg xuv xu xd xc xs Non linear DGLAP End lecture -6

10. Small x is high W2, x=Q2/2p.q Q2/W2 s(g*p) ~ (W2) α-1 – Regge prediction for high energy cross-sections αis the intercept of the Regge trajectory α=1.08 for the SOFT POMERON Such energy dependence is well established from the SLOW RISE of all hadron-hadron cross-sections - including s(gp) ~ (W2) 0.08 for real photon- proton scattering For virtual photons, at small x s(g*p) = 4p2α F2 Linear DGLAP evolution doesn’t work for Q2 < 1 GeV2, WHAT does? – REGGE ideas? q px2 = W2 p Regge region pQCD region Q2 →s~ (W2)α-1→ F2 ~ x 1-α = x -l so a SOFT POMERON would imply l = 0.08 gives only a very gentle rise of F2 at small x For Q2 > 1 GeV2 we have observed a much stronger rise…..

11. gentle rise σ(γ*p) much steeper rise F2 ~ x -λs, λs = d ln F2 d ln 1/x So is there a HARD POMERON corresponding to this steep rise? A QCD POMERON, α(Q2) – 1 = l(Q2) A BFKL POMERON, α – 1 = l= 0.5 A mixture of HARD and SOFT Pomerons to explain the transition Q2 = 0 to high Q2? What about the Froissart bound ? – the rise MUST be tamed – non-linear effects? The slope of F2 at small x , F2 ~x -l , is equivalent to a rise of s(g*p) ~ (W2)l which is only gentle for Q2 < 1 GeV2

13. Dipole models provide another way to model the transition Q2=0 to high Q2 At low x, g* Y qq and the LONG LIVED (qq) dipole scatters from the proton σ(γ*p) Now there is HERA data right across the transition region The dipole-proton cross section depends on the relative size of the dipole r~1/Q to the separation of gluons in the target R0 s =s0(1 – exp( –r2/2R0(x)2)), R0(x)2 ~(x/x0)l~1/xg(x) Buts(g*p) = 4pa2 F2 is general Q2 (at small x) r/R0 small → large Q2, x σ~ r2~ 1/Q2, F2 flat Bjorken scaling r/R0 large → small Q2, x s ~ s0→ saturation of the dipole cross-section s(gp) is finite for real photons , Q2=0. At high Q2, F2 ~flat (weak lnQ2 breaking) and s(g*p) ~ 1/Q2 GBW dipole model

14. x < 0.01 σ= σ0 (1 – exp(-1/t)) Involves only t=Q2R02(x) t = Q2/Q02 (x/x0)l And INDEED, for x<0.01, s(g*p) depends only on t, not on x, Q2 separately Q2 > Q2s Q2 < Q2s x > 0.01 tis a new scaling variable, applicable at small x It can be used to define a `saturation scale’ , Q2s = 1/R02(x) . x -l~ x g(x), gluon density - such that saturation extends to higher Q2 as x decreases Some understanding of this scaling, of saturation and of dipole models is coming from work on non-linear evolution equations applicable at high density– Colour Glass Condensate, JIMWLK, Balitsky-Kovchegov. There can be very significant consequences for high energy cross-sections e.g. neutrino cross-sections – also predictions for heavy ions- RHIC, diffractive interactions – Tevatron, HERA and the LHC- even some understanding of soft hadronic physics

15. The Pomeron also makes less indirect appearances in HERA data in diffractive events, which comprise ~10% of the total. The proton stays more or less intact, and a Pomeron, with fraction X_P of the proton’s momentum, is hit by the exchanged boson. One can picture partons within the Pomeron, having fraction beta of the Pomeron momentum One can define diffractive structure functions, which broadly factorize in to a Pomeron flux (function of x_P, t) and a Pomeron structure function (function of beta, Q2). The Pomeron flux has been used to measure Pomeron Regge intercept – which seems marginally harder than that of the soft Pomeron The Pomeron structure functions indicate a large component of hard gluons

16. But this is not the only view of difraction. These data have also been interpreted in terms of dipole models CDM fits to ZEUS diffractive data

17. Ther is more evidence from diffractive Vector meson production and DVCS End lecture 6? DVCS also seems to show a form of geometric scaling

18. extras

20. White and Thorne have an NLL BFKL calculation accounting for running coupling AND heavy quark effects – this has various attractive features……