The low x Structure Function Data. Introduction. Brian Foster Bristol/DESY. Corfu Summer School, 4.9.01. Low x physics at HERA. Other probes of QCD dynamics @ HERA. Diffraction and its connection with low x DIS. Summary & Outlook. For low x , HERA ~ only game in town.
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The lowx Structure Function Data
Introduction
Brian Foster
Bristol/DESY
Corfu Summer
School, 4.9.01
Other probes of QCD dynamics @ HERA
Diffraction and its connection with lowx DIS
Summary & Outlook
Brian Foster  Corfu lectures
For low x, HERA ~ only game in town.
Brian Foster  Corfu lectures
Factorization  hard processes can be regarded as convolution
of “subprocess” cross section with probability to find
participating partons in target & probe  subsequent
hadronisation ~ independent process
For DIS can (normally) consider
virtual photon as dfunction
=> s ~ f swhere
s is subprocess cross section
f is parton dist. function,
satisfying f/ m ~ f P
(m = renormalisation scale,
Pis a splitting function)
Brian Foster  Corfu lectures
where are AP splitting functions
In general Ps are perturbative expansions to particular
orders, keeping terms most important for particular regions:
Leading lnQ2 terms come, in axial gauge, from evolution
along parton chain strongly ordered in transverse momenta,
LO DGLAP sums up
terms  NLO sums
terms which arise when two adjacent
kts become comparable, losing factorlnQ2.
Brian Foster  Corfu lectures
In small x region, leading terms in ln 1/x must be summed
independent of Q2. This is done by the BFKL equation.
LO
terms arise from strong x ordering
Generally, however, QCD coherence angular ordering 
work in unintegrated f(x,kt2,m2)  2 hard scales more
complicated CCFM evolution equation. DGLAP/BFKL two
limits of angular ordering. DGLAP, q kt/kl, q grows since kt
grows; in BFKL, q grows because kl x falls.
Brian Foster  Corfu lectures
Kinematics
e(k)
e'(k')
2
Q
Q2 = xys
g
*(q)
2
W
xP
p(P)
s = k+P=energy in the ep c.m.s.
Q2 = (kk')2 = q2 =virtuality of the exchanged
x = Q2/(2P•q)=fraction of proton momentum
carried by the struck quark
y = (P•q)/(P•k) =fraction of beam lepton energy
transferred to the photon
W 2 = ys ~ Q2/xenergy in the *p c.m.s.
Brian Foster  Corfu lectures
To reach lowest possibleQ2, some tricks needed!
As well as exquisite
understanding of
detector 
Brian Foster  Corfu lectures
ZEUS BPT dataAt lowQ2 , F2falls likeQ2
Brian Foster  Corfu lectures
Since =>
sr =F
sr ~F2 for small y,
æ
sr ~F2  fory 1, so =
F2fit 
ç

F
F
F
(x,Q2)
ç
L
L
L
è
(x,Q2)
2
Brian Foster  Corfu lectures
Brian Foster  Corfu lectures
Brian Foster  Corfu lectures
Brian Foster  Corfu lectures
Brian Foster  Corfu lectures
Brian Foster  Corfu lectures
There are many parameterisations of the structure function
data on the market  some more deeply based on physics
others rather just convenient functional forms.
e.g. DL fit ;
wheree0 is “hard Pomeron”
SinceW 2 ~ Q2/x , and ,
Regge theory, which governs highenergys s relevant for lowx
Brian Foster  Corfu lectures
Another model exploits the “doublelogarithmic” limit of QCD:
Haidt, coming from a different direction, uses
BallForte fit
Brian Foster  Corfu lectures
Erdmann uses
Brian Foster  Corfu lectures
Several on the market  MRST, CTEQ essentially global NLO
QCD fits to all DIS data (HERA & fixed target) plus other
relevant channels; GRV attempts to generate structure functions
by evolution from “valencelike” gluon at very low Q . All give
excellent fits to the data, with many free parameters.
GRV’98
CTEQ
Deviation of exp. data from CTEQ fit
Brian Foster  Corfu lectures
Although using a subset (DIS) of the data, recently “homegrown”
pdfs appeared which give errors on fitted pdfs as well as the
correlations  e.g. Botje
Botje
Brian Foster  Corfu lectures
NNLO estimates for
the splitting fns. now
becoming available 
(Van Neerven&Vogt)
MRS, CTEQ groups
using them.
Some strange effects!
“Premature”
(K.Ellis, DIS2000)
Brian Foster  Corfu lectures
Thorne achieves
interesting improvements
by incorporating
ln(1/x) terms in splitting
fns after NNLO BFKL
using running coupling
BFKL eq.
Brian Foster  Corfu lectures
ZEUS has very precise F2
data over 6 orders of
magnitudein (x, Q2).
What can it tell us?
Look at the log. derivative
since ~ LO gluon  most
sensitive to lowxdynamics
 fit x bins with form
F2 = A+B(ln Q2)+C(ln Q2)2
Plot derivative as fn. of x&
Q2in bins of constantW
Brian Foster  Corfu lectures
Errors on F2
syst. +stat. in
quadrature
(correlations
ignored.
Brian Foster  Corfu lectures
There is no turnover at constant
Q2
One can look at
3D surface of log. Slopes.
The fundamental point is that the precision and kinematic
range of the data is now opening up qualitatively new areas of
study. The question is  what does it mean?
Brian Foster  Corfu lectures
As x falls, as we have seen, the gluon radiation drives a strong
increase in parton density and hence increase inF2.
At some point, the number of partons becomes so large
that they cannot “fit” inside the proton and their
wavefunctions overlap  this is known as parton saturation.
Brian Foster  Corfu lectures
Recently, great deal of interest in dipole models & saturation.
L.T.
Breit, mom.
prest
In principle offers unification of inclusive DIS, diffraction
+
1 g
exchange
2 g octet
exchange
2 g singlet
exchange
Diffraction
Inclusive F2
Brian Foster  Corfu lectures
Example of this type of model: GolecBiernat & Wuesthoff predicts
Q2s0
Brian Foster  Corfu lectures
The GolecBiernat&Wusthoff model does a reasonable
qualitative job  but so does QCD, and/or a variety of simple
parameterisations.
Brian Foster  Corfu lectures
Although one can make QCD fit the logarithmic slopes, the
resultant pdfs, as we saw earlier, are strange to say the least!
ZEUS
prel..
ZEUS
prel..
Brian Foster  Corfu lectures
But the predictedFLis even stranger!
ZEUS
prel..
Brian Foster  Corfu lectures
The agreement of the data with dipole models and the saturation
concept is intriguing  are we seeing the first departure from
linear evolution in QCD? Clearly premature to draw this
conclusion  NLO QCD can also reproduce the data to the same
level  although at the cost of producing pdfs that are very
difficult to interpret in a sensible way.
The fundamental point is that the precision and kinematic range
of the data is now opening up qualitatively new areas of study.
Perhaps we are seeing a qualitatively new behaviour of QCD 
but we can’t be certain. One of the problems is that the
interesting “critical line” is down atQ2 ~ 1 GeV2  we need to
measure at low x but higher Q2.  needs a higher energy than
HERA can achieve.
Brian Foster  Corfu lectures
In diffraction, proton stays intact
In great majority of DIS events,
proton breaks up into hadrons +
“remnant” in forward direction
We saw that, in dipole models, there was an intimate connection
between DIS & diffraction. Is this borne out by the data?
Brian Foster  Corfu lectures
The most basic measurement is the total cross section for
diffraction. Does it agree with our expectations?
}
GB
&W
No. It has sameW2dependence asstot  W0.4.
Contradicts optical theoremstot ~ Wa => sdiff ~ W2a; and
ifstot ~ g, sdiff ~g2; and Regge, from Pomeron traj. stot~ W0.16
Brian Foster  Corfu lectures
What about the structure functions? The analogue to F2 is F2D
Brian Foster  Corfu lectures
Note steep rise inW
dependence ofs 
indicative of hard
processes becoming
dominant.
The intimate link between diffraction & nondiffractive DIS
via dipole models & saturation also clearly applicable to vector
meson production.
Brian Foster  Corfu lectures
Inset shows
fit with Wd.
For theJ/y, the charm mass seems to be large enough to
provide a hard scale even atQ2 = 0.
Brian Foster  Corfu lectures
Fit to
measured
cross sections
with Wd.
For ther, theQ2provides a hard scale.
Brian Foster  Corfu lectures
Fit to
measured
cross sections
with Wd.
For all vector mesons , Q2 + M2seems to be a common hard
scale.
Brian Foster  Corfu lectures
For Q2 > 5 GeV2, lVM ~ 2* lDIS
Compare the W dependence for VM production and DIS.
Brian Foster  Corfu lectures
But the J/y wave
function needs to be
modelled so that
model dependence
enters extraction.
What else can we learn from VMs? Since J/y seems to always
be in the pQCD realm, we can in principle learn about proton
gluon distribution.
Diffraction  vector mesons
Brian Foster  Corfu lectures
By fitting crosssection t dependence we can look at
Pomeron trajectory.
Diffraction  vector mesons
Brian Foster  Corfu lectures
What is the appropriate hard scale in VM production?
Diffraction  vector mesons
Brian Foster  Corfu lectures
VM production at high t.
Diffraction  vector mesons
Brian Foster  Corfu lectures
J/y
f/r
f
Production s ratios BFKL prediction (Forshaw et al.)
y/r
r
2g
BFKL
Diffraction  vector mesons
Brian Foster  Corfu lectures
Simplest final state in diffraction
Deeply virtual Compton scattering
Measures Re part of a QCD amplitude
Measures “skewed” parton distributions 
generalisation on normal proton pdf’s.
Brian Foster  Corfu lectures
Data cf. QEDC only QEDC & DVCS MC
Deeply virtual Compton scattering
DVCS process clearly necessary  extract crosssection
Brian Foster  Corfu lectures
Now cross section measured, can go onto to look at
interference etc.
Deeply virtual Compton scattering
Brian Foster  Corfu lectures
The agreement of the data with dipole models and the saturation
concept is intriguing  are we seeing the first departure from
linear evolution in QCD? Clearly premature to draw this
conclusion  one of the problems is that the interesting “critical
line” is down at Q2~ 1 GeV2  we want to measure at low x
but higher Q2.
This I guess will have to wait
for THERA, LHC ep option,
….? Is there something else
we can do “now”? Yes,
possibly. Running HERA
with nuclei rather than p
gives access to highdensity
of partons at low x.
Brian Foster  Corfu lectures
Many open questions 
and of course
this is not only important
to those
interested in QCD!
If we want to useW,Z
production at LHC as
lumi. monitor, we had better
understand smallxat HERA!
h
Brian Foster  Corfu lectures
The quality & precision of the HERA data are driving studies
of lowx physics.
Watching the Herculean labours of the F2 experts extracting the
9697 result tells me that we are nearing the end of the road for
improved precision in the standard inclusive F2 at low x 
from now one attention will turn to semiinclusive(particularly
F2charm) and rare processes
Much theoretical help required (as always) to tell us where/how
to look
The connection between diffraction and DIS is certainly a very
interesting one that can throw much light on lowx physics.
It may well justify a “HERAIII” programme  but all
this will depend on TESLA!
Brian Foster  Corfu lectures