W and Z production at the Tevatron

1
Deep Inelastic Scattering (DIS) Parton
Distribution Functions and low-x physics Goa Sept
2008 A.M.Cooper-Sarkar Oxford What have we
learnt from DIS in the last 30 years? QPM,
QCD Parton Distribution Functions, as Low-x
physics
2
PDFs were first investigated in deep inelastic
lepton-hadron scatterning -DIS
Leptonic tensor - calculable
2
Lµ? Wµ?
ds
Hadronic tensor- constrained by Lorentz invariance
E
q k k, Q2 -q2 Px p q , W2 (p
q)2 s (p k)2 x Q2 / (2p.q) y
(p.q)/(p.k) W2 Q2 (1/x 1) Q2 s x y
Ee
Ep
s 4 Ee Ep Q2 4 Ee E sin2?e/2 y (1 E/Ee
cos2?e/2) x Q2/sy
The kinematic variables are
measurable
3

Completely generally the double differential
cross-section for e-N scattering
d2?(eN) Y F2(x,Q2) - y2
FL(x,Q2) Y_xF3(x,Q2), Y 1 (1-y)2
dxdy
Leptonic part
hadronic part
F2, FL and xF3 are structure functions which
express the dependence of the cross-section on
the structure of the nucleon The Quark-Parton
model interprets these structure functions as
related to the momentum distributions of quarks
or partons within the nucleon AND the measurable
kinematic variable x Q2/(2p.q) is interpreted
as the FRACTIONAL momentum of the incoming
nucleon taken by the struck quark
(xPq)2x2p2q22xp.q 0 for massless quarks
and p20 so x Q2/(2p.q) The FRACTIONAL
momentum of the incoming nucleon taken by the
struck quark is the MEASURABLE quantity x
4

Consider electron muon scattering
ds 2pa2 s 1 (1-y)2 , for elastic eµ
dy
Q4
isotropic
non-isotropic

ds 2pa2 ei2 s 1 (1-y)2 , so for elastic
electron quark scattering, quark charge ei e
Q4
dy
d2s 2pa2 s 1 (1-y)2 Si ei2(xq(x) xq(x))
so for eN, where eq has c. of m. energy2
equal to xs, and q(x) gives probability that such
a quark is in the Nucleon
dxdy
Q4
Now compare the general equation to the QPM
prediction to obtain the results
F2(x,Q2) Si ei2(xq(x) xq(x)) Bjorken
scaling this depends only on x
FL(x,Q2) 0 - spin ½ quarks
xF3(x,Q2) 0 - only ? exchange
5
Compare to the general form of the cross-section
for n/n scattering via W/- FL (x,Q2)
0 xF3(x,Q2) 2Six(qi(x) - qi(x))
Valence F2(x,Q2) 2Six(qi(x) qi(x))
Valence and Sea And there will be a
relationship between F2eN and F2nN Also NOTE n,n
scattering is FLAVOUR sensitive
Consider n,n scattering neutrinos are
handed ds(n) GF2 x s ds(n) GF2 x s
(1-y)2
dy
dy
p
p
For n q (left-left)
For n q (left-right)
d2s(n) GF2 s Si xqi(x) (1-y)2xqi(x)
dxdy
p
For nN
d2s(n) GF2 s Si xqi(x) (1-y)2xqi(x)
dxdy
p
For nN
Clearly there are antiquarks in the nucleon
3 Valence quarks plus a flavourless qq Sea
m-
W can only hit quarks of charge -e/3 or
antiquarks -2e/3
n
W
u
d
s(np) (d s) (1- y)2 (u c) s(np) (u
c) (1- y)2 (d s)
q qvalence qsea q qsea qsea
qsea
6
So in n, n scattering the sums over q, qbar ONLY
contain the appropriate flavours BUT- high
statistics n, n data are taken on isoscalar
targets e.g. Fe (p n)/2N d in proton u
in neutron u in proton d in neutron
GLS sum rule
Total momentum of quarks
A TRIUMPH
(and 20 years of understanding the c c
contribution)
7
BUT Bjorken scaling is broken F2 does not
depend only on x it also depends on Q2 ln(Q2)
Particularly strongly at small x
8
QCD improves the Quark Parton Model
What if
or
x
x
Pqq
Pgq
y
y
Before the quark is struck?
Pqg
Pgg
y gt x, z x/y
So F2(x,Q2) Si ei2(xq(x,Q2) xq(x,Q2)) in LO
QCD The theory predicts the rate at which the
parton distributions (both quarks and gluons)
evolve with Q2- (the energy scale of the probe)
-BUT it does not predict their shape
The DGLAP parton evolution equations
9
What if higher orders are needed?
Note q(x,Q2) as lnQ2, but as(Q2)1/lnQ2, so as
lnQ2 is O(1), so we must sum all terms
asn lnQ2n Leading Log Approximation x decreases
from
?s? ?s(Q2)
target to probe xi-1gt xi gt xi1.
Pqq(z) P0qq(z) as P1qq(z) as2 P2qq(z)
LO NLO NNLO
pt2 of quark relative to proton increases from
target to probe pt2i-1 lt pt2i lt pt2 i1 Dominant
diagrams have STRONG pt ordering
F2 is no longer so simply expressed in terms of
partons - convolution with coefficient functions
is needed but these are calculable in QCD
FL is no longer zero.. And it depends on the gluon
10
How do we determine Parton Distribution Functions
? Parametrise the parton distribution functions
(PDFs) at Q20 (1-7 GeV2)- Use NLO QCD DGLAP
equations to evolve these PDFs to Q2
gtQ20 Construct the measurable structure functions
and cross-sections by convoluting PDFs with
coefficient functions make predictions for
2000 data points across the x,Q2 plane-
Perform ?2 fit to the data
Formalism NLO DGLAP MSbar factorisation Q02 functi
onal form _at_ Q02 sea quark (a)symmetry etc.
fi (x,Q2) ? ? fi (x,Q2)
aS(MZ )
Data DIS (SLAC, BCDMS, NMC, E665, CCFR, H1,
ZEUS, CCFR, NuTeV ) Drell-Yan (E605, E772, E866,
) High ET jets (CDF, D0) W rapidity asymmetry
(CDF) etc.
LHAPDFv5
11
The DATA the main contribution is DIS data
Terrific expansion in measured range across the
x, Q2 plane throughout the 90s HERA data Pre
HERA fixed target ?p,?D NMC, BDCMS, E665 and
?,?bar Fe CCFR We have to impose appropriate
kinematic cuts on the data so as to remain in the
region when the NLO DGLAP formalism is valid

1. Q2 cut Q2 gt few GeV2 so that perturbative QCD
   is applicable- as(Q2) small
2. W2 cut to avoid higher twist terms- usual
   formalism is leading twist
3. x cut to avoid regions where ln(1/x) resummation
   (BFKL) and non-linear effects may be necessary.

12
Need to extend the formalism?

Optical theorem
2
The handbag diagram- QPM
Im
QCD at LL(Q2) Ordered gluon ladders (asn lnQ2
n) NLL(Q2) one rung disordered asn lnQ2 n-1
?
BUT what about completely disordered Ladders?
at small x there may be a need for BFKL ln(1/x)
resummation?
And what about Higher twist diagrams ? Are they
always subdominant? Important at high x, low Q2
13
The strong rise in the gluon density at small-x
leads to speculation that there may be a need for
non-linear equations?- gluons recombining gg?g
Strong coupling
Non-linear fan diagrams form part of possible
higher twist contributions at low x
14
The CUTS
In practice it has been amazing how low in Q2 the
standard formalism still works- down to Q2 1
GeV2 cut Q2 gt 2 GeV2 is typical It has also
been surprising how low in x down to x 10-5
no x cut is typical Nevertheless there are doubts
as to the applicability of the formalism at such
low-x.. (See later) there could be ln(1/x)
corrections and/or non-linear high density
corrections for x lt 5 10 -3
15
Higher twist terms can be important at low-Q2 and
high-x ? this is the fixed target region
(particularly SLAC- and now JLAB data).
Kinematic target mass corrections and dynamic
contributions 1/Q2
X? 2x/(1 v(14m2x2/Q2))
Fit with F2F2LT (1 D2(x)/Q2)
Fits establish that higher twist terms are not
needed if W2 gt 15 GeV2 typical W2 cut
16
The form of the parametrisation Parametrise the
parton distribution functions (PDFs) at Q20 (1-7
GeV2)
Parameters Ag, Au, Ad are fixed through momentum
and number sum rules other parameters may be
fixed by model choices- Model choices ?Form of
parametrization at Q20, value of Q20,, flavour
structure of sea, cuts applied, heavy flavour
scheme ? typically 15-22 parameters Use QCD to
evolve these PDFs to Q2 gtQ20 Construct the
measurable structure functions by convoluting
PDFs with coefficient functions make predictions
for 2000 data points across the x,Q2
plane Perform ?2 fit to the data
xuv(x) Auxau (1-x)bu (1 eu vx ?u x) xdv(x)
Adxad (1-x)bd (1 ed vx ?d x) xS(x)
Asx-?s (1-x)bs (1 es vx ?s x) xg(x)
Agx-?g(1-x)bg (1 eg vx ?g x)
These parameters control the low-x shape
These parameters control the middling-x shape
These parameters control the high-x shape
Alternative form for CTEQ xf(x) A0xA1(1-x)A2
eA3x (1eA4x)A5
The fact that so few parameters allows us to fit
so many data points established QCD as the THEORY
OF THE STRONG INTERACTION and provided the first
measurements of ?s (as one of the fit parameters)
17
The form of the parametrisation at Q20
Ultimately we may get this from lattice QCD, or
other models- the statistical model is quite
successful (Soffer et al). But we can make some
guesses at the basic form xa (1-x)b .. at one
time (20 years ago?) we thought we understood
it! --------the high x power from counting rules
----(1-x)2ns-1 - ns spectators
valence
(1-x)3, sea (1-x)7, gluon (1-x)5 --------the
low-x power from Regge low-x corresponds to
high centre of mass energy for the virtual boson
proton collision (x Q2 / (2p.q)) -----Regge
theory gives high energy cross-sections as s
(a-1) -----------which gives x dependence x
(1-a), where a is the intercept of the Regge
trajectory- different for singlet (no overall
flavour) F2 x0 and non-singlet (flavour-
valence-like) xF3x0.5 The shapes of F2 and xF3
even looked as if they followed these shapes
pre HERA
18
But at what Q2 would these be true? Valence
distributions evolve slowly but sea and gluon
distributions evolve fast. We input
non-perturbative ideas about shapes at a low
scale, but we are just parametrising our
ignorance. It turns out that we need the
arbitrary polynomial In any case the further you
evolve in Q2 the less the parton distributions
look like the low Q2 inputs and the more they are
determined by QCD evolution
Valence distributions evolve slowly Sea/Gluon
distributions evolve fast
(Some people dont use a starting parametrization
at all- but let neural nets learn the shape of
the data- NNPDF)
19

• Where is the information coming from?
• Originally- pre HERA
• Fixed target e/µ p/D data from NMC, BCDMS, E665,
  SLAC
• F2(e/?p) 4/9 x(u ubar) 1/9x(ddbar) 4/9 x(c
  cbar) 1/9x(ssbar)
• F2(e/?D)5/18 x(uubarddbar) 4/9 x(c cbar)
  1/9x(ssbar)
• Also use ?, ?bar fixed target data from CCFR( now
  also NuTeV/Chorus) (Beware Fe target needs
  nuclear corrections)
• F2(?,?bar N) x(u ubar d dbar s sbar c
  cbar)
• xF3(?,?bar N) x(uv dv ) (provided s sbar)
• Valence information for 0lt x lt 1
• Can get 4 distributions from this e.g. u, d,
  ubar, dbar but need assumptions
• like qqbar for all flavours, sbar 1/4
  (ubardbar) and heavy quark treatment.
• Note gluon enters only indirectly via DGLAP
  equations for evolution

Assuming u in proton d in neutron
strong-isospin
20
Flavour structure
Historically an SU(3) symmetric sea was
assumed uuvusea, ddvdsea usea ubar dsea
dbar s sbar K and ccbar0 Measurements of
F2µn uv 4dv 4/3K
F2µp 4uv dv 4/3K Establish no valence
quarks at small-x F2µn/F2µp ?1 But F2µn/F2µp
?1/4 as x ? 1 Not to 2/3 as it would for
dv/uv1/2, hence it look s as if dv/uv ?0 as x
?1 i.e the dv momentum distribution is softer
than that of uv- Why? Non-perturbative physics
--diquark structures? How accurate is this? Could
dv/uv ?1/4 (Farrar and Jackson)?
21
Flavour structure in the sea
dbar ?ubar in the sea Consider the Gottfried
sum-rule (at LO) ? dx (F2p-F2n) 1/3 ?dx
(uv-dv) 2/3?dx(ubar-dbar) If ubardbar then the
sum should be 0.33 the
measured value from NMC 0.235 0.026 Clearly
dbar gt ubarwhy? low Q2 non-perturbative
effects,
Pauli blocking,
p ?np,pp0,?p-
m-
sbar?(ubardbar)/2, in fact sbar
(ubardbar)/4 Why? The mass of the strange quark
is larger than that of the light quarks Evidence
neutrino opposite sign dimuon production
rates And even s?sbar? Because of p??K
n
W
c?s µ?
s
22
So what did HERA bring?
Low-x within conventional NLO DGLAP Before the
HERA measurements most of the predictions for
low-x behaviour of the structure functions and
the gluon PDF were wrong HERA ep neutral current
(?-exchange) data give much more information on
the sea and gluon at small x..
xSea directly from F2, F2 xq xGluon from
scaling violations dF2 /dlnQ2 the relationship
to the gluon is much more direct at small-x,
dF2/dlnQ2 Pqg xg
23
Low-x
t ln Q2/?2
Gluon splitting functions become singular
At small x, small zx/y
as 1/ln Q2/?2
A flat gluon at low Q2 becomes very steep AFTER
Q2 evolution AND F2 becomes gluon dominated
F2(x,Q2) x -?s, ?s?g - e
xg(x,Q2) x -?g
And yet people didnt expect this. See later
24
High Q2 HERA data-still to be fully exploited

• HERA data have also provided information at high
  Q2 ? Z0 and W/- become as important as ?
  exchange ? NC and CC cross-sections comparable
• For NC processes
• F2 ?i Ai(Q2) xqi(x,Q2) xqi(x,Q2)
• xF3 ?i Bi(Q2) xqi(x,Q2) - xqi(x,Q2)
• Ai(Q2) ei2 2 ei vi ve PZ (ve2ae2)(vi2ai2)
  PZ2
• Bi(Q2) 2 ei ai ae PZ 4ai ae vi ve
  PZ2
• PZ2 Q2/(Q2 M2Z) 1/sin2?W
• ?a new valence structure function xF3 due to Z
  exchange is measurable from low to high x- on a
  pure proton target ? no heavy target corrections-
  no assumptions about strong isospin

25
CC processes give flavour information
d2?(ep) GF2 M4W x (uc) (1-y)2x (ds)
d2?(e-p) GF2 M4W x (uc) (1-y)2x (ds)
dxdy
2?x(Q2M2W)2
dxdy
2?x(Q2M2W)2
uv at high x
dv at high x
MW information
Measurement of high-x dv on a pure proton target
d is not well known because u couples more
strongly to the photon. Historically information
has come from deuterium targets but even
Deuterium needs binding corrections. Open
questions does u in proton d in neutron?,

does dv/uv ? 0, as x ? 1?
26
How has our knowledge evolved?
27
The u quark
LO fits to early fixed-target DIS data
28
Rise from HERA data
29
The story about the gluon is more complex
Gluon
30
Gluon
HERA steep rise of F2 at low x
31
Gluon
More recent fits with HERA data- steep rise even
for low Q2 1 GeV2
Tev jet data
Does gluon go negative at small x and low Q? see
MRST/W PDFs
32
PDF comparison 2008 The latest HERAPDF0.1 gives
very small experimental errors and modest model
errors
33
Modern analyses assess PDF uncertainties within
the fit Clearly errors assigned to the data
points translate into errors assigned to the fit
parameters -- and these can be propagated to any
quantity which depends on these parameters the
parton distributions or the structure functions
and cross-sections which are calculated from
them lt ?2F gt Sj Sk ? F Vjk ? F
? pj ? pk The errors
assigned to the data are both statistical and
systematic and for much of the kinematic plane
the size of the point-to-point correlated
systematic errors is 3 times the statistical
errors This must be treated carefully in the ?2
definition
34
Some data sets incompatible/only marginally
compatible? One could restrict the data sets to
those which are sufficiently consistent that
these problems do not arise (H1, GKK,
Alekhin) But one loses information since partons
need constraints from many different data sets
no-one experiment has sufficient kinematic range
/ flavour information (though HERA comes
close) To illustrate the ?2 for the MRST
global fit is plotted versus the variation of a
particular parameter (as ). The individual ?2e
for each experiment is also plotted versus this
parameter in the neighbourhood of the global
minimum. Each experiment favours a different
value of. as PDF fitting is a compromise. Can
one evaluate acceptable ranges of the parameter
value with respect to the individual experiments?
35
CTEQ6 look at eigenvector combinations of their
parameters rather than the parameters themselves.
They determine the 90 C.L. bounds on the
distance from the global minimum from ? P(?e2,
Ne) d?e2 0.9 for each experiment
illustration for eigenvector-4
Distance
This leads them to suggest a modification of the
?2 tolerance, ??2 1, with which errors are
evaluated such that ??2 T2, T 10. Why?
Pragmatism. The size of the tolerance T is set by
considering the distances from the ?2 minima of
individual data sets from the global minimum for
all the eigenvector combinations of the
parameters of the fit. All of the worlds data
sets must be considered acceptable and compatible
at some level, even if strict statistical
criteria are not met, since the conditions for
the application of strict statistical criteria,
namely Gaussian error distributions are also not
met. One does not wish to lose constraints on the
PDFs by dropping data sets, but the level of
inconsistency between data sets must be reflected
in the uncertainties on the PDFs.
36
Recent development Combining ZEUS and H1 data
sets

• Not just statistical improvement. Each experiment
  can be used to calibrate the other since they
  have rather different sources of experimental
  systematics
• Before combination the systematic errors are 3
  times the statistical for Q2lt 100
• After combination systematic errors are lt
  statistical
• ? very consistent data input HERAPDFs use ??21

37
Diifferent uncertainty estimates on the gluon
persist as Q2 increases
Q210
Note CTEQ in general bigger uncertaintiesbut NOT
for low-x gluon
CTEQ6.5
MSTW08
Q210000
38
The general trend of PDF uncertainties for
conventional NLO DGLAP is that The u quark is
much better known than the d quark The valence
quarks are much better known than the gluon/sea
at high-x The valence quarks are poorly known at
small-x but they are not important for physics in
this region The sea and the gluon are well known
at low-x but only down to x10-4 The sea is
poorly known at high-x, but the valence quarks
are more important in this region The gluon is
poorly known at high-x And it can still be very
important for physics
39
End lecture 1

• Next PDFs for the LHC
• Be-(a)ware of low-x physics

40
The Standard Model is not as well known as you
might think
Knowledge from HERA ?the LHC- transport PDFs to
hadron-hadron cross-sections using QCD
factorization theorem for short-distance
inclusive processes
The central rapidity range for W/Z production AT
LHC is at low-x (5 10-4 to 5 10-2)
41
W/Z production have been considered as good
standard candle processes with small theoretical
uncertainty. PDF uncertainty is THE dominant
contribution and most PDF groups quote
uncertainties lt5 (but note HERAPDF 1-2)

PDF set sW BW?l? (nb) sW- BW?l? (nb) sz Bz?ll (nb)
ZEUS-2005 11.870.45 8.740.31 1.970.06
MRST01 11.610.23 8.620.16 1.950.04
HERAPDF 12.130.13 9.130.15 2.010.025
CTEQ65 12.470.47 9.140.36 2.030.07
CTEQ61 11.610.56 8.540.43 1.890.09
BUT the central values differ by more than some
of the uncertainty estimates.
42
Look at predictions for W/Z rapidity
distributions Pre- and Post-HERA
Why such an improvement?
Pre HERA
Post HERA
Its due to the improvement in the low-x gluon
At the LHC the q-qbar which make the boson are
mostly sea-sea partons at low-x And at Q2MZ2
the sea is driven by the gluon
43
This is post HERA but just one experiment
This is post HERA using the new (2008) HERA
combined PDF fit
However there is still the possibility of trouble
with the formalism at low-x
44
Moving on to BSM physics Example of how PDF
uncertainties matter for BSM physics Tevatron
jet data were originally taken as evidence for
new physics--
i
These figures show inclusive jet cross-sections
compared to predictions in the form (data -
theory)/ theory Something seemed to be going on
at the highest E_T And special PDFs like
CTEQ4/5HJ were tuned to describe it better- note
the quality of the fits to the rest of the data
deteriorated. But this was before uncertainties
on the PDFs were seriously considered
45
Today Tevatron jet data are considered to lie
within PDF uncertainties. (Example from CTEQ
hep-ph/0303013) We can decompose the
uncertainties into eigenvector combinations of
the fit parameters-the largest uncertainty is
along eigenvector 15 which is dominated by the
high x gluon uncertainty
46
Such PDF uncertainties on the jet cross sections
compromise the potential for discovery of any
physics effects which can be written as a contact
interaction E.G. Dijet cross section potential
sensitivity to compactification scale of extra
dimensions (Mc) reduced from 6 TeV to 2 TeV.
Mc 2 TeV, no PDF error
Mc 2 TeV, with PDF error
Mc 6 TeV, no PDF error
47
BEWARE of different sort of new physics
LHC is a low-x machine (at least for the early
years of running) Low-x information comes from
evolving the HERA data Is NLO (or even NNLO)
DGLAP good enough? The QCD formalism may need
extending at small-x BFKL ln(1/x)
resummation High density non-linear effects
etc. (Devenish and Cooper-Sarkar, Deep Inelastic
Scattering, OUP 2004, Section 6.6.6 and Chapter
9 for details!)
48
Before the HERA measurements most of the
predictions for low-x behaviour of the structure
functions and the gluon PDF were wrong Now it
seems that the conventional NLO DGLAP formalism
works TOO WELL _ there should be ln(1/x)
corrections and/or non-linear high density
corrections for x lt 5 10 -3
49
Low-x
t ln Q2/?2
Gluon splitting functions become singular
At small x, small zx/y
as 1/ln Q2/?2
A flat gluon at low Q2 becomes very steep AFTER
Q2 evolution AND F2 becomes gluon dominated
F2(x,Q2) x -?s, ?s?g - e
xg(x,Q2) x -?g
50
So it was a surprise to see F2 steep at small x
- for low Q2, Q2 1 GeV2 Should perturbative
QCD work? as is becoming large - as at Q2 1
GeV2 is 0.4
51
Need to extend formalism at small x? The
splitting functions Pn(x), n 0,1,2for LO,
NLO, NNLO etc Have contributions Pn(x) 1/x
an ln n (1/x) bn ln n-1 (1/x) . These
splitting functions are used in evolution
dq/dlnQ2 ?s dy/y P(z) q(y,Q2) And thus give
rise to contributions to the PDF ?s p (Q2) (ln
Q2)q (ln 1/x) r DGLAP sums- LL(Q2) and NLL(Q2)
etc STRONGLY ordered in pt. But if ln(1/x)
is large we should consider Leading Log 1/x
(LL(1/x)) and Next to Leading Log (NLL(1/x)) -
BFKL summations LL(1/x) is STRONGLY ordered in
ln(1/x) and can be disordered in pt BFKL
summation at LL(1/x) ? xg(x) x -? ? as CA
ln2 0.5 p ? steep gluon even at moderate Q2

? Disordered gluon ladders But NLL(1/x) changes
this somewhat (many experts here)
52
The steep behaviour of the gluon is deduced from
the DGLAP QCD formalism BUT the steep
behaviour of the low-x Sea can be measured
from F2 x -?s, ?s d ln F2
d ln 1/x

• Small x is high W2, xQ2/2p.q Q2/W2. At
  small x
• ?(?p) 4p2a F2/Q2
• F2 x ?s ? ? (?p) (W2)?s
• But s(gp) (W2) a-1 is the Regge prediction
  for high energy cross-sections where a is the
  intercept of the Regge trajectory
  a 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

Obviously the virtual-photon proton cross-section
only obeys the Regge prediction for Q2 lt 1.
Does the steeper rise of ? (?p) require a HARD
POMERON? --The BFKL Pomeron with alpha1.5 ?
What about the Froissart bound?
53

• Furthermore if the gluon density becomes large
  there maybe non-linear effects
• Gluon recombination g g ? g
• ? as2?2/Q2
• may compete with gluon evolution g ? g g
• ? as ?
• where ? is the gluon density
• xg(x,Q2) no.of gluons per ln(1/x)

Colour Glass Condensate, JIMWLK, BK
Strong coupling
nucleon size
?R2
Non-linear evolution equations GLR d2xg(x,Q2)
3as xg(x,Q2) as2 81 xg(x,Q2)2
dlnQ2dln1/x
p
16Q2R2
as ?
as2 ?2/Q2
The non-linear term slows down the evolution of
xg(x,Q2) and thus tames the rise at small x The
gluon density may even saturate (-respecting the
Froissart bound)
Extending the conventional DGLAP equations across
the x, Q2 plane Plenty of debate about the
positions of these lines!
54
Do the data NEED unconventional explanations ? In
practice the NLO DGLAP formalism works well down
to Q2 1 GeV2 BUT below Q2 5 GeV2 the gluon
is no longer steep at small x in fact its
becoming negative! xS(x) x ?s, xg(x) x
?g ?g lt ?s at low Q2, low x So far, we only
used F2 xq dF2/dlnQ2 Pqg
xg Unusual behaviour of dF2/dlnQ2 may come
from unusual gluon or from unusual Pqg-
alternative evolution?. Non-linear effects? We
need other gluon sensitive measurements at low x,
like FL or F2charm.
Valence-like gluon shape
55
But charm is not so simple to calculate Heavy
quark treatments differ
Massive quarks introduce another scale into the
process, the approximation mq20 cannot be
used Zero Mass Variable Flavour Number Schemes
(ZMVFNs) traditional c0 until Q2 4mc2, then
charm quark is generated by g? c cbar splitting
and treated as massless-- disadvantage incorrect
to ignore mc near threshold Fixed Flavour Number
Schemes (FFNs) If W2 gt 4mc2 then c cbar can be
produced by boson-gluon fusion and this can be
properly calculated - disadvantage ln(Q2/mc2)
terms in the cross-section can become large-
charm is never considered part of the proton
however high the scale is. General Mass variable
Flavour Schemes (GMVFNs) Combine correct
threshold treatment with resummation of
ln(Q2/mc2) terms into the definition of a charm
quark density at large Q2 Arguments as to correct
implementation but should look like FFN at low
scale and like ZMVFN at high scale. Additional
complications for W exchange s?c threshold.
56
We are learning more about heavy quark treatments
than about the gluon, so far
57
And now we have actually measured FL!
FL looks pretty conventional- can be described
with usual NLO DGLAP formalism But see later
(Thorne and White)
58
No smoking gun for something new at low-xso
lets look more exclusively Lets look at jet
production First lets just see what jets can do
for us in a regular NLO DGLAP fit
There is a decrease in gluon PDF uncertainty
from using jet data in PDF fits Direct
Measurement of the Gluon Distribution ZEUS-jets
PDF fit
59
Before jets
After jets
And correspondingly the contribution of the
uncertainty on as(MZ) to the uncertainty on the
PDFs is much reduced
Nice measurement of as(MZ) 0.1183 0.0028
(exp) 0.0008 (model) From simultaneous fit of
as(MZ) PDF parameters
And use of jet data can help to tie down both
alphas itself and alphas related uncertainties
on the gluon PDF
60
(No Transcript)
61
No smoking gun for seomthing new at low-xso
lets look more exclusively Now lets look at
forward jets
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 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 Monte-Carlo calculations (LEPTO-MEPS,
ARIADNE CDM ) The data do not agree with DGLAP
at LO or NLO, or with the Monte-Carlo
LEPTO-MEPS..but agree with ARIADNE. ARIADNE is
not kt ordered but it is not a rigorous BFKL
calculation either
62
Forward Jets
DISENT vs data
Comparison to LO and NLO conventional calculations
NLO below data, especially at small xBj but
theoretical uncertainty is large
63
Forward Jets
Comparison to ARIADNE and LEPTO

• Lepto doesnt suffice
• Ariadne default
• overestimates high Etjet,
• overestimates high ?jet
• (proton remnant)
• Ariadne tuned is good

64
jet calculations which go up to O(as3 ) can
describe the data- including detailed
correlations between x and pt, or x and azimuthal
angle
SO we started looking at more complex jet
production processes.
65
Q2 2GeV2
xg(x)
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 c2 of the global fit
improves
Back to considering inclusive quantities
New work in 2007 by White and Thorne
66
White and Thorne have an NLL BFKL calculation
accounting for running coupling AND heavy quark
effects this has various attractive features
Plus improved ?2 for global PDF fit Other groups
working on NLL BFKL are Ciafaloni, Colferai,
Salam, Stasto Altarelli, Ball, Forte But there
are no corresponding global fits
67
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
Q2 1.4 GeV2
xg
xuv
xu
xd
xc
xs
Non linear
DGLAP
There is some phenomenology which supports the
use of non-linear evolution equations as well,
but it is not so well developed Part of our
problem is that as well move to low-x we are also
moving to lower Q2 and the strongly coupled
region..
68
Small x is high W2, xQ2/2p.q Q2/W2 s(gp)
(W2) a-1 Regge prediction for high energy
cross-sections a is the intercept of the Regge
trajectory a1.08 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(gp) 4p2a F2
Linear DGLAP evolution doesnt work for
Q2 lt 1 GeV2, WHAT does? REGGE ideas?
q
px2 W2
p
Regge region
pQCD region
Q2
? s (W2)a-1 ? F2 x 1-a x -l so
a SOFT POMERON would imply l 0.08 gives
only a very gentle rise of F2 at small x For Q2 gt
1 GeV2 we have observed a much stronger rise..
69
QCD improved dipole
GBW dipole
gentle rise
F((p)
Regge region
pQCD generated slope
So is there a HARD POMERON corresponding to this
steep rise? A QCD POMERON, a(Q2) 1 l(Q2) A
BFKL POMERON, a 1 l 0.5 A mixture of HARD
and SOFT Pomerons to explain the transition Q2
0 to high Q2?
much steeper rise
The slope of F2 at small x , F2 x -l , is
equivalent to a rise of s(gp) (W2)l which is
only gentle for Q2 lt 1 GeV2
70
Do we understand the rise of hadron-hadron
cross-sections at all? Could there always have
been a hard Pomeron- is this why the effective
Pomeron intercept is 1.08 rather than 1.00? Does
the hard Pomeron mix in more strongly at higher
energies? What about the at the LHC?
Recent work by Caldwell suggests that a double
Pomeron model fits gamma p best If this is the
case in ?-p, it could be so in p-p But we
havent noticed yet because the hard Pomeron is
not strongly coupled at low energies At the LHC
we could notice this But what about the Froissart
bound ? the rise MUST be tamed eventually
non-linear effects/saturation may be
necessary Predictions for the p-p cross-section
at LHC energies are not so certain
71
Dipole models provide another way to model the
transition Q20 to high Q2 At low x, ? ? qqbar
and the LONG LIVED (qqbar) dipole scatters from
the proton
?(?p)
Now there is HERA data right across the
transition region
The dipole-proton cross section depends on the
relative size of the dipole r1/Q to the
separation of gluons in the target R0
s s0(1 exp( r2/2R0(x)2)), R0(x)2
(x/x0)?1/xg(x)
But s(gp) 4pa2 F2 is general
Q2
(at small x)
r/R0 large ?small Q2, x s s0 ? saturation of
the dipole cross-section
r/R0 small ? large Q2, x s r2 1/Q2
s(gp) is finite for real photons , Q20. At high
Q2, F2 flat (weak lnQ2 breaking) and s(gp)
1/Q2
GBW dipole model
72
x lt 0.01
s s0 (1 exp(-1/t)) Involves only tQ2R02(x),
t Q2/Q02 (x/x0)? And INDEED, for xlt0.01,
s(?p) depends only on t, not on x, Q2 separately
Q2 gt Q2s
Q2 lt Q2s
x gt 0.01
t is a new scaling variable, applicable at small
x It can be used to define a saturation scale ,
Q2s 1/R02(x) . x -? x g(x), gluon density-
such that saturation extends to higher Q2 as x
decreases (Qs is low 1-2 GeV2 at HERA) 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 (LHC
and neutrino), predictions for heavy ions- RHIC,
diffractive interactions etc.
73
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 XP of the protons
momentum, is hit by the exchanged boson. One can
picture partons within the Pomeron, having
fraction ß of the Pomeron momentum One can define
diffractive structure functions, which broadly
factorize in to a Pomeron flux (function of xP,
t) and a Pomeron structure function (function of
ß, Q2). The Pomeron flux has been used to measure
Pomeron Regge intercept which come out
marginally harder than that of the soft
Pomeron The Pomeron structure functions indicate
a large component of hard gluons in the Pomeron
74
But this is not the only view of difraction.
These data have also been interpretted in terms
of dipole models
ColourDipoleModel fits to ZEUS diffractive data
If the total cross-section is given by s ? d2r
dz ?(?(z,r)?2 sdipole(W) Then the diffractive
cross-section can be written as s ? d2r dz
?(?(z,r)?2 sdipole2(W) The fact that the ratio of
sdiff/stot is observed to be constant implies a
constant sdipole which could indicate saturation
75
Ther is more evidence for hard Pomeron behaviour
from diffractive Vector meson production and DVCS
These processes are elastic ?p ? V p Hence if
stot Im Aelastic W 2(a0)-1) Then selastic
?Aelastic?2 W 4(a(0)-1) A soft Pomeron
a(0)1.08 would give selastic W 0.32 A hard
Pomeron a(0)1.3 would give s elastic W1.2 The
experimental measurements vary from soft to hard
depending on whether there is a hard scale in the
process We can plot the effective Pomeron
intercepts vs Q2MV2
DVCS also seems to show a form of geometric
scaling
76

• Summary
• What have we learnt from DIS in the last 30
  -35years?
• Verified the basic idea of QPM
• Established QCD as the theory of the strong
  interaction
• Measurement of essential parameters Parton
  Distribution Functions, as(MZ ) and the running
  of as
• Low-x physics-
• QCD beyond DGLAP..BFKL..non-linear..CGC..dipole
  models..diffraction etc
• The LHC could discover a different kind of new
  physics