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HERA KINEMATIC PLANE. Accessible Kinematic Plane now almost completely covered Measurements extend to cover high y, high x and very high Q 2 Probe distances to ~ 1/1000 th of proton size. Q 2 = xys. Tevatron. COMPASS.

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  • Accessible Kinematic Plane now almost completely covered

  • Measurements extend to cover high y, high x and very high Q2

  • Probe distances to ~ 1/1000th of proton size


Q2 = xys








The data show that F2 depends more and more steeply on Q2 as x falls. These logarithmic scaling violations are predicted by QCD. The driver is gluon emission from the quark lines - the gluons in turn spilt into quark-antiquark pairs, which in turn radiate gluons - and so on, ad infinitum. At each branching, the energy is shared, so the result is to throw more and more partons to lower and lower x - the “steep rise in F2” which is one of the most

significant discoveries of HERA.


Scaling and its violations

(non) – dependence on Q2

Elastic scattering off pointlike

and free partons

→ does not depend on Q2

‘a point is a point’


Result of emission of gluons

from partons inside proton

Scaling violations

Depletion at high x

→ quarks emit gluons

Increase at low x

→ quarks having emitted gluons

Effect increases with αslog Q2


Interpretation: DGLAP evolution

F2(x,Q2) can in principle be calculated on the Lattice

→ Some results emerged in the last few years

Standard analysis assumes that F2(x,Q2) not calculable

However: evolution with Q2 calculable in pQCD

Dokshitzer, Gribov, Lipatov, Altarelli, Parisi (DGLAP):

Parton Density Functions (PDFs)

qi(x,Q2) … Density of quark i at given x, Q2

g(x,Q2) … Density of gluons at given x, Q2

Pij(x/z) … Splitting functions

Quark-Parton Model (QPM)

…in DIS scheme






Splitting FunctionsPij(z)

Probability of parton i going into parton j with momentum fraction z

Calculable in pQCD as expansions in αS

In Leading Order Pij(z) take simple forms


Fit to DGLAP equations

I) Rewrite DGLAP equations

a) Simplify notation



b) Sum i) over q and q separately



Nf … number of flavors

c) Define: Valence quark density

Singlet quark density

← u,u,d


d) Rewrite DGLAP equations

Valence quark density decouples from g(x,Q2)

Only evolves via gluon emission depending on Pqq

II) DGLAP equations govern evolution with Q2

Do not predict x dependence:

Parameterize x-dependence at a given Q2 = Q20 = 4 – 7 GeV2

55 parameters

High x behaviour: valence quarks

Low x behaviour


III) Sum rules and simplifying assumptions

Valence distributions 2 valence up-quarks

1 valence down quarks

Symmetric sea

Treatment of heavy flavors (different treatments available…)


Above mHF: generate dynamically via DGLAP evolution

Momentum sum rule: proton momentum conserved

Effect number of parameters:

55 (parameters) – 3 (sum rules) – 13 (symmetric sea) – 22(heavy flavors) = 17

Difficult fits, involving different data sets with systematic errors…


Several groups perform global fits

CTEQ: currently CTEQ6

MRS: currently MRST2001

GRV: currently GRV98

Experiments: H1, ZEUS

Overall good agreement between fits

Despite some different assumptions

Results of fits I

Fit quality: excellent everywhere!

→ no significant deviations

Evolution with Q2: 5 orders of magnitude

QCDs greatest success!!!

No deviations at high Q2:

→ no new physics:

no contact interactions

no leptoquarks

Fit includes data with low Q2: αS(Q2) large

→ surprise

→ expected to work only for Q2 ≥ 10 GeV2


Results of fits II

Gluon density

Quark and gluon densities

Inferred from QCD fit

not probed directly by γ

Errors of order 4% at Q2 = 200 GeV2


Valence quarks

Strong coupling constant

Based on NLO pQCD

including terms of αS2

Scale error reduced with NNLO

not yet available


Other interpretations

DGLAP formalism

Standard approach: Equations to NLO

Include all terms O(αS2)

Calculation of NNLO corrections

First results by the MRST group

Effects seem small, but will reduce uncertainties

Collinear Factorization

DGLAP also resums terms proportional (αS log Q2)n

corresponds to gluon ladder with kT ordered gluons

kT,n >> kT,n-1 … >> kT,0

struck parton collinear with incoming proton

Does not resum terms proportional to (αS log 1/x)n

→ Is this ok at small x?


BFKL formalism


Y Balitskii, V Fadin, L Lipatov, E Kuraev

Resums terms proportional to (αs log 1/x)n

gluons in ladder not kT ordered, but ordered in x

x1 >> x2 … >> xn

Predicts x, but not Q2dependence

kT Factorization

results in kT unintegrated gluon distributions g(x,kT2,Q2)





CCFM formalism

S Catani, M Ciafaloni, F Fiorani, G Marchesini

Resums terms proportional to (αs log 1/x)n and (αs log 1/(1-x))n

gluons in ladder now ordered in angle

kT Factorization

results in kT unintegrated gluon distributions g(x,kT2,Q2)

Easier to implement in MC programs, e.g. CASCADE

Low x: approaches BFKL

High x: approaches DGLAP


Asymmetric sea

FNAL fixed target experiment E-866

Measurement of Drell-Yan production with H2 and D2 targets

p N →μ+ μ- X

…with x = x1 – x2

Sea not flavor symmetric!!!

Explanations: Meson clouds

Chiral model



Longitudinal Structure Function FL from NC DIS

Need to vary y, keeping x, Q2 fixed

→ vary s

Disentangle F2(x,Q2) and FL(x,Q2)

Data from SLAC and CERN: e/μ scattering on fixed targets with different beam energies

Measurement of R(x,Q2): Ratio of longitudinal and transverse cross section


Measurements at high x > 0.1

but low Q2 < 80 GeV2


Rfit … fit to empirical function

RQCD … prediction based on

PDFs from F2data

RQCD+TM … same as above, corrected

for target mass effects

Differences between data and QCD

higher twist effects?

decrease as 1/Q2