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

1. Should perturbative QCD work? αs is becoming large - αs at Q2 ~ 1 GeV2 is ~ 0.4

2. 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

3. there should be ln(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

4. and/or there should be non-linear high density corrections for x < 5 10 -3

Conventional DGLAP

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

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 Q2We are learning more about

heavy quark treatments than about the gluon, so far

We need other gluon sensitive measurements like FL: in NLO DGLAP FL is given by

And at low-x this becomes gluon dominated

Now there are HERA measurements on FL from 2007: Compare to various

NLO DGLAP fits

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

But this is not conclusive So…

Look at the hadron final states..lack of pt ordering has its consequences. Forward jets with xj » x and ktj 2 ~ 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………

But there are other ways of looking for unconventional ‘beyond DGLAP’ behaviour

NLO below data, especially at small xBj

but theoretical uncertainty is large

Forward JetsForward Jets

DISENT vs data Results from 2007

Comparison to LO and NLO conventional calculations

xg(x)

Q2 = 2GeV2

The negative gluon predicted at low x, low Q2 from NLO DGLAP remains 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 2 of the global fit improves

Back to considering inclusive quantities

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

Non linear

DGLAP

xg

xu

xs

xuv

xd

xc

End lecture -6

Linear DGLAP evolution doesn’t work for Q2 < 1 GeV2, WHAT does? – REGGE ideas?

Reg

ge r

egio

npQ

CD

reg

ion

Small x is high W2, x=Q2/2p.q Q2/W2

(*p) ~ (W2) α-1 – Regge prediction for high energy cross-sections

α is 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 (p) ~ (W2) 0.08 for real photon- proton scattering

For virtual photons, at small x (*p) = 42α F2 Q2

→~ (W2)α-1 → F2 ~ x 1-α = x - so a SOFT POMERON would imply = 0.08 gives only a very gentle rise

of F2 at small x

For Q2 > 1 GeV2 we have observed a much stronger rise…..

px2 = W2

q

p

The slope of F2 at small x , F2 ~x - , is equivalent to a rise of (*p) ~ (W2)which is only gentle for Q2 < 1 GeV2

σ(γ*p) gent

le r

ise

muc

h st

eepe

r ri

seF2 ~ 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 = (Q2)

A BFKL POMERON, α – 1 = = 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?

Dipole models provide another way to model the transition Q2=0 to high Q2

At low x, * qq and the LONG LIVED (qq) dipole scatters from the proton

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

=0(1 – exp( –r2/2R0(x)2)), R0(x)2 ~(x/x0)~1/xg(x)

r/R0 small → large Q2, x σ~ r2~ 1/Q2, F2 flat

Bjorken scaling

r/R0 large → small Q2, x ~ 0 → saturation of the dipole cross-section

GBW dipole model

σ(γ*p)

But(*p) = 42 F2 is generalQ2

(p) is finite for real photons , Q2=0. At high Q2, F2 ~flat (weak lnQ2 breaking) and (*p) ~ 1/Q2

(at small x)

Now

there is HE

RA

data right across the transition region

is a new scaling variable, applicable at small x

It can be used to define a `saturation scale’ , Q2s = 1/R0

2(x) x -~ 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

σ = σ0 (1 – exp(-1/))

Involves only

=Q2R02(x)

= Q2/Q02 (x/x0)

And INDEED, for x<0.01, (*p) depends only on , not on x, Q2

separately

x < 0.01

x > 0.01

Q2 < Q2s

Q2 > Q2s

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

CDM fits to ZEUS diffractive data

But this is not the only view of difraction. These data have also been interpreted in terms of dipole models

Ther is more evidence from diffractive Vector meson production and DVCS

DVCS also seems to show a form of

geometric scaling

End lecture 6?

extras

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