The Structure of the Proton

1
The Structure of the Proton

• A.M.Cooper-Sarkar
• Feb 6th 2003
• RSE

• Parton Model
• QCD as the theory of strong interactions
• Parton Distribution Functions
• Extending QCD calculations across the kinematic
  plane understanding small-x, high density,
  non-perturbative regions

2
Leptonic tensor - calculable
2
LmnWmn
dF
Hadronic tensor- constrained by Lorentz invariance
Et
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

d2s(eN) Y F2(x,Q2) - y2
FL(x,Q2) K Y_xF3(x,Q2), YK 1 K (1-y)2
dxdy
for charged lepton hadron scattering
F2, FL and xF3 are structure functions The
Quark Parton Model interprets them
ds 2pa2 ei2 xs 1 (1-y)2 , for elastic eq
Q4
dy
d2s 2pa2 s 1 (1-y)2 Gi ei2(xq(x) xq(x))
dxdy
Q4

for eN
non-isotropic
isotropic
(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
Now compare the general equation to the QPM
prediction F2(x,Q2) Gi ei2(xq(x)
xq(x)) Bjorken scaling FL(x,Q2)
0 - spin ½ quarks xF3(x,Q2) 0
- only ( exchange
4
Consider n,n scattering neutrinos are
handed ds(n) GF2 x s ds(n) GF2 x s
(1-y)2
Compare to the general form of the cross-section
for n/n scattering via W/- FL (x,Q2)
0 xF3(x,Q2) 2Gix(qi(x) - qi(x))
Valence F2(x,Q2) 2Gix(qi(x) qi(x))
Valence and Sea And there will be a
relationship between F2eN and F2nN NOTE n,n
scattering is FLAVOUR sensitive
dy
dy
p
p
For n q (left-left)
For n q (left-right)
d2s(n) GF2 s Gi xqi(x) (1-y)2xqi(x)
dxdy
p
For nN
d2s(n) GF2 s Gi 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
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So in n,n scattering the sums over q, q ONLY
contain the appropriate flavours BUT- high
statistics n,n data are taken on isoscalar
targets e.g. Fe Y (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)
6
QCD improves the Quark Parton Model
I F2(ltN) dx 0.5 where did the momentum go?
What if
or
x
x
Pqq
Pgq
y
y
Before the quark is struck?
Pqg
Pgg
y gt x, z x/y
Note q(x,Q2) ?s lnQ2, but ?s(Q2)1/lnQ2, so ?s
lnQ2 is O(1), so we must sum all terms
?sn lnQ2n Leading Log Approximation x decreases
from
?s? ?s(Q2)
xi1
xi
xi-1
target to probe xi-1gt xi gt xi1.
pt2 of quark relative to proton increases from
target to probe pt2i-1 lt pt2i lt pt2 i1 Dominant
diagrams have STRONG pt ordering
The DGLAP equations
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Bjorken scaling is broken ln(Q2)
Note strong rise at small x
Terrific expansion in measured range across the
x, Q2 plane throughout the 90s HERA data Pre
HERA fixed target mp,mD NMC,BDCMS, E665 and n,n
Fe CCFR
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• Valence distributions evolve slowly
• Sea/Gluon distributions evolve fast
• Parton Distribution Functions PDFs are extracted
  by MRST, CTEQ, ZEUS, H1
• Parametrise the PDFs at Q20 (low-scale)

Note scale of xg, xS

• xuv(x) Auxau (1-x)bu (1 eu vx gu x)
• xdv(x) Adxad (1-x)bd (1 ed vx gd x)
• xS(x) Asx-ls (1-x)bs (1 es vx gs x)
• xg(x) Agx-lg(1-x)bg (1 eg vx gg x)
• Some parameters are fixed through sum rules-
• others by model choices- typically 15 parameters
• Use QCD to evolve these PDFs to Q2 gt Q20
• Construct the measurable structure functions in
  terms of PDFs for 1500 data points across the
  x,Q2 plane
• Perform c2 fit

Note error bands on PDFs
9

• 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)
• These days we assume the validity of the picture
  to measure PDFs which are transportable to other
  hadronic processes
• But where is the information coming from?
• F2(e/mp) 4/9 x(u u) 1/9x(dd)
• F2(e/mD)5/18 x(uudd)
• u is dominant , valence dv, uv only accessible at
  high x
• (d and u in the sea are NOT equal, dv/uv Y 0 as
  x Y 1)
• Valence information at small x only from xF3(nFe)
• xF3(nN) x(uv dv) - BUT Beware Fe target!
• HERA data is just ep xS, xg at small x
• xS directly from F2
• xg indirectly from scaling violations dF2 /dlnQ2

Fixed target p/D data- Valence and Sea
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• HERA at high Q2 Y Z0 and W/- become as important
  as
• exchange Y NC and CC cross-sections comparable
• for NC processes
• F2 3i Ai(Q2) xqi(x,Q2) xqi(x,Q2)
• xF3 3i 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 measurable
  from
• low to high x- on a pure proton target
• Y sensitivity to sin2?W, MZ and electroweak
  couplings vi, ai for u and d type quarks- (with
  electron beam polarization)

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CC processes give flavour information
d2s(ep) GF2 M4W x (uc) (1-y)2x (ds)
d2s(e-p) GF2 M4W x (uc) (1-y)2x (ds)
dxdy
2px(Q2M2W)2
dxdy
2px(Q2M2W)2
uv at high x
dv at high x
MW information
Measurement of high x dv on a pure proton target
(even Deuterium needs corrections, does dv/uv Y
0, as x Y 1? )
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Valence PDFs from ZEUS data alone- NC and CC e
and e- beams Y high x valence dv from CC e, uv
from CC e- and NC e/-
Valence PDFs from a GLOBAL fit to all DIS data Y
high x valence from CCFR xF3(n,nFe) data and NMC
F2(mp)/F2(mD) ratio
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Parton distributions are transportable to other
processes Accurate knowledge of them is essential
for calculations of cross-sections of any process
involving hadrons. Conversely, some processes
have been used to get further information on the
PDFs E.G DRELL YAN
p N Y mm- X, via q q Y ( Y mm-, gives
information on the Sea Asymmetry between pp Y
mm- X and pn Y mm- X gives more information on
d - u difference W PRODUCTION- p p Y W(W-) X,
via u d Y W, d u Y W- gives more information on
u, d differences PROMPT g - p N Y g X, via g
q Y g q gives more information on the gluon (but
there are current problems concerning intrinsic
pt of initial partons) HIGH ET INCLUSIVE JET
PRODUCTION p p Y jet X, via g g, g q, g q
subprocesses gives more information on the gluon
for ET gt 200 GeV an excess of jets in CDF data
appeared to indicate new physics beyond the
Standard Model BUT a modification of the u PDF
which still gave a reasonable fit to other data
could explain it Cannot search for physics within
(Higgs) or beyond (Supersymmetry) the Standard
Model without knowing EXACTLY what the Standard
Model predicts Need estimates of the PDF
uncertainties
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c2 3i FiQCD Fi MEAS2
(siSTAT)2(siSYS)2 si2
Errors on the fit parameters evaluated from )c2
1, can be propagated back to the PDF shapes to
give uncertainty bands on the predictions for
structure functions and cross-sections THIS IS
NOT GOOD ENOUGH Experimental errors can be
correlated between data points- e.g.
Normalisations BUT there are more subtle cases-
e.g. Calorimeter energy scale/angular resolutions
can move events between x,Q2 bins and thus change
the shape of experimental distributions c2 3i
3j FiQCD Fi MEAS Vij-1 FjQCD FjMEAS
Vij dij(siSTAT)2 3l DilSYS DjlSYS Where
)i8SYS is the correlated error on point i due to
systematic error source 8 c2 3i FiQCD 38
slDilSYS- Fi MEAS2 3 sl2
(siSTAT) 2
s8 are fit parameters of zero mean and unit
variance Y modify the measurement/prediction by
each source of systematic uncertainty HOW to
APPLY this ?
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• OFFSET method
• Perform fit without correlated errors
• Shift measurement to upper limit of one of its
  systematic uncertainties (sl 1)
• Redo fit, record differences of parameters from
  those of step 1
• Go back to 2, shift measurement to lower limit
  (sl -1)
• Go back to 2, repeat 2-4 for next source of
  systematic uncertainty
• Add all deviations from central fit in quadrature
• HESSIAN method
• Allow fit to determine the optimal values of sl
• If we believe the theory why not let it calibrate
  the detector?
• In a global fit the systematic uncertainties of
  one experiment will correlate to those of another
  through the fit
• We must be very confident of the theory/model

xg(x)
Q27
Q22.5
Offset method
Q220
Q2200
Hessian method
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• Model Assumptions theoretical assumptions
  later!
• Value of Q20, form of the parametrization
• Kinematic cuts on Q2, W2, x
• Data sets included .
• Changing model assumptions changes parameters
• Y model error
• sMODEL n sEXPERIMENTAL - OFFSET
• sMODEL o sEXPERIMENTAL - HESSIAN
• You win some and you lose some!
• The change in parameters under model changes is
  frequently outside the )c21 criterion of the
  central fit
• e.g. the effect of using different data sets on
  the value of as

Are some data sets incompatible? Y PDF fitting is
a compromise, CTEQ suggest )c250 may be a more
reasonable criterion for error estimation size
of errors determined by the Hessian method rises
to the size of errors determined by the Offset
method
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Comparison of ZEUS (offset) and H1(Hessian) gluon
distributions Yellow band (total error) of H1
comparable to red band (total error) of ZEUS
Comparison of ZEUS and H1 valence distributions.
ZEUS plus fixed target data H1 data alone-
Experimental errors alone by OFFSET and HESSIAN
methods resp.
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Determinations of ?s
NLOQCD fit results
Value of ?s and shape of gluon are correlated ?s
increases ? harder gluon dF2 ?s(Q2) Pqq q
F2 2 3i ei2 Pqg q xg
dlnQ2
2p
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Theoretical Assumptions- Need to extend the
formalism?
What if
Optical theorem
2
Im
The handbag diagram- QPM
QCD at LL(Q2) Ordered gluon ladders (asn lnQ2
n) NLL(Q2) one rung disordered asn lnQ2 n-1
Pqq(z) P0qq(z) ?s P1qq(z) ?s2 P2qq(z)
LO NLO NNLO
BUT what about completely disordered Ladders?
Or higher twist diagrams?
low Q2, high x
Eliminate with a W2 cut
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Ways to measure the gluon distribution
Knowledge increased dramatically in the 90s
Pre HERA
Post HERA
Scaling violations dF2/dlnQ2 in DIS High ET jets
in hadroproduction- Tevatron BGF jets in DIS g g
Y q q Prompt g HERA charm production g g Y c c
For small x scaling violation data from HERA are
most accurate
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t ln Q2/72
Gluon splitting functions become singular
At small x, small zx/y
?s 1/ln Q2/72
Gluon becomes very steep at small x AND F2
becomes gluon dominated F2(x,Q2) x -ls,
lslg - ,
xg(x,Q2) x -lg
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Still it was a surprise to see F2 still steep at
small x - even for Q2 1 GeV2 should
perturbative QCD work? ?s is becoming large - ?s
at Q2 1 GeV2 is 0.32
23
MRST PDF fit xg(x) x -lg xS(x) x -ls at low
x For Q2 gt 5 GeV2 lg gt ls
lg
ls
The steep behaviour of the gluon is deduced from
the DGLAP QCD formalism BUT the steep behaviour
of the Sea is measured from F2 x -ls, ls d ln
F2
Perhaps one is only surprised that the onset of
the QCD generated rise appears to happen at Q2
1 GeV2 not Q2 5 GeV2
d ln 1/x
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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 dq/dlnQ2 ?s I 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 Conventionally we sum p q r 0 at
Leading Log Q2 - (LL(Q2)) p q1 r 0 at
Next to Leading log Q2 (NLLQ2) DGLAP
summations But if ln(1/x) is large we should
consider p r q 1 at Leading Log 1/x
(LL(1/x)) p r1 q 1 at Next to Leading Log
(NLL(1/x)) - BFKL summations

• LL(Q2) is STRONGLY ordered in pt.
• At small x it is also STRONGLY ordered in x
  Double Leading Log Approximation
• ?s p (Q2) (ln Q2)q(ln 1/x)r, pqr
• LL(1/x) is STRONGLY ordered in ln(1/x) and can be
  disordered in pt Y
• ?s p(Q2)(ln 1/x)r, pr
• BFKL summation at LL(1/x) Y
• xg(x) x -l
• l ?s CA ln2 0.5, for ?s 0.25
• steep gluon even at moderate Q2
• But this is considerable softened at NLL(1/x)
• (Y way beyond scope !)

p
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• Furthermore if the gluon density becomes large
  there maybe non-linear effects
• Gluon recombination g g Y g
• s ?s2r2/Q2
• may compete with gluon evolution g Y g g
• s ?s r
• where D is the gluon density
• r xg(x,Q2) no.of gluons per ln(1/x)

Colour Glass Condensate, JIMWLK, BK
nucleon size
BR2
Non-linear evolution equations GLR d2xg(x,Q2)
3?s xg(x,Q2) ?s2 81 xg(x,Q2)2
Higher twist
p
dlnQ2dln1/x
16Q2R2
as r
as2 r2/Q2
The non-linear term slows down the evolution of
xg 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!
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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! We only measure
F2 xq dF2/dlnQ2 Pqg xg Unusual
behaviour of dF2/dlnQ2 may come from unusual
gluon or from unusual Pqg- alternative
evolution? We need other gluon sensitive
measurements at low x Like FL, or F2charm
Valence-like gluon shape
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FL
F2 charm
Current measurements of FL and F2charm at small x
are not yet accurate enough to distinguish
different approaches
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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
Are there more defnitive signals for BFKL
behaviour? In principle yes, in the hadron final
state, from the lack of pt ordering However,
there have been many suggestions and no
definitive observations- We need to improve the
conventional calculations of jet production
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Q2 1.4 GeV2
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 Y g
involve the summation of FAN diagrams
xg
xuv
xu
xd
xc
xs
Q2 2
Q210
Q2100 GeV2
Such diagrams form part of possible higher twist
contributions at low x Y there maybe further
clues from lower Q2 data?
xg
Non linear
DGLAP
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Small x is high W2, xQ2/2p.q . Q2/W2 s(gp)
(W2) ?-1 Regge prediction for high energy
cross-sections ? is the intercept of the Regge
trajectory ?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(gp) 4p2? F2
Linear DGLAP evolution doesnt work for
Q2 lt 1 GeV2, WHAT does? REGGE ideas?
q
px2 W2
p
Regge region
pQCD region
Q2
Y s (W2)?-1 Y F2 x 1-? x -l so
a SOFT POMERON would imply l 0.08 Y only a
very gentle rise of F2 at small x For Q2 gt 1 GeV2
we have observed a much stronger rise Y
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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, ?(Q2) 1 l(Q2) A
BFKL POMERON, ? 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?
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
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Dipole models provide a way to model the
transition Q20 to high Q2 At low x, g Y qq and
the LONG LIVED (qq) dipole scatters from the
proton
F((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)l1/xg(x)
But s(gp) 4pa2 F2 is general
Q2
(at small x)
r/R0 small Y large Q2, x F r2 1/Q2
r/R0 large Y small Q2, x s s0 Y saturation of
the dipole cross-section
s(gp) is finite for real photons , Q20. At high
Q2, F2 flat (weak lnQ2 breaking) and s(gp)
1/Q2
GBW dipole model
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x lt 0.01
F F0 (1 exp(-1/t)) Involves only t Q2R02(x)
Y t Q2/Q02 (x/x0)l And INDEED, for xlt0.01,
s(gp) 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 -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 and
HERA, even some understanding of soft hadronic
physics
34
Summary
Measurements of Nucleon Structure Functions are
interesting in their own right- telling us about
the behaviour of the partons which must
eventually be calculated by non-perturbative
techniques- lattice gauge theory etc. They are
also vital for the calculation of all hadronic
processes- and thus accurate knowledge of them
and their uncertainties is vital to investigate
all NEW PHYSICS Historically these data
established the Quark-Parton Model and the Theory
of QCD, providing measurements of the value of
as(MZ2) and evidence for the running of
as(Q2) There is a wealth of data available now
over 6 orders of magnitude in x and Q2 such that
conventional calculations must be extended as we
move into new kinematic regimes at small x, at
high density and into the non-perturbative
region. The HERA data has stimulated new
theoretical approaches in all these areas.