The Structure of the Proton
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Transcript of The Structure of the Proton
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The Structure of the ProtonA.M.Cooper-Sarkar
April 2003
• Parton Model- QCD as the theory of strong interactions
• Parton Distribution Functions
• Extending QCD calculations across the kinematic plane – understanding
1. Small-x
2. High density
3. Low Q2
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d~
2
L W
Ee
E
Ep
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
s = 4 Ee Ep
Q2 = 4 Ee Esin2e/2y = (1 – E/Ee cos2e/2)x = Q2/sy
The kinematic variables are measurable
Leptonic tensor - calculable
Hadronic tensor- constrained by
Lorentz invariance
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d2(e±N) = [ Y+ F2(x,Q2) - y2 FL(x,Q2) Y_xF3(x,Q2)], Y = 1 (1-y)2
dxdy
F2, FL and xF3 are structure functions –
The Quark Parton Model interprets them
d2= s [ 1 + (1-y)2] i ei2(xq(x) + xq(x))
dxdy Q4
Compare the general equation to the QPM prediction
F2(x,Q2) = i ei2(xq(x) + xq(x)) – Bjorken scaling
FL(x,Q2) = 0 - spin ½ quarks
xF3(x,Q2) = 0 - only exchange
(xP+q)2=x2p2+q2+2xp.q ~ 0
for massless quarks and p2~0 so
x = Q2/(2p.q)
The FRACTIONAL momentum of the incoming nucleon taken by the struck
quark is the MEASURABLE quantity x
4
22
Q
sfor charged lepton hadron scattering
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QCD improves the Quark Parton Model
What if
or
Before the quark is struck?
Pqq Pgq
Pqg Pgg
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 fromtarget to probe
xi-1> xi > xi+1….pt
2 of quark relative to proton increases from target to probe
pt2
i-1 < pt2
i < pt2 i+1
Dominant diagrams have STRONG pt ordering
ss(Q2)
The DGLAP equations
xi+1
xi
xi-1
x xy y
y > x, z = x/y
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Terrific expansion in measured range across the x, Q2 plane throughout the 90’s
HERA data
Pre HERA fixed target p,D NMC, BDCMS, E665 and , Fe CCFR
Bjorken scaling is broken – ln(Q2)N
ote
stro
ng r
ise
at s
mal
l x
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xuv(x) =Auxau (1-x)bu (1+ u x + u x)
xdv(x) =Adxad (1-x)bd (1+ d x + d x)
xS(x) =Asx-s (1-x)bs (1+ s x + s x)
xg(x) =Agx-g(1-x)bg (1+ g x + g x)
Some parameters are fixed through sum rules - others by model choices-
typically ~15 parameters
•Use QCD to evolve these PDFs to Q2 > Q20
•Construct the measurable structure functions in terms of PDFs for ~1500 data points across the x,Q2 plane
•Perform 2 fit
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)
How are such fits done?
Parametrise the parton distribution functions (PDFs) at Q20 (low-scale)
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These days we assume the validity of the picture to measure parton distribution functions PDFs which
are transportable to other hadronic processes
• Parton Distribution Functions PDFs are extracted by MRST, CTEQ, ZEUS, H1
• Valence distributions evolve slowly
• Sea/Gluon distributions evolve fast
But where is the information coming from?
F2(e/p)~ 4/9 x(u +u) +1/9x(d+d)
F2(e/D)~5/18 x(u+u+d+d)
u is dominant , valence dv, uv only accessible at high x
Valence information at small x only from xF3(Fe) xF3(N) = 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 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 + (ve
2+ae2)(vi
2+ai2) PZ
2
Bi(Q2) = – 2 ei ai ae PZ + 4ai ae vi ve PZ2
PZ2 = Q2/(Q2 + M2
Z) 1/sin2W
a new valence structure function xF3 measurable from
low to high x- on a pure proton target
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CC processes give flavour information
d2(e-p) = GF2 M4
W [x (u+c) + (1-y)2x (d+s)] dxdy 2x(Q2+M2
W)2
d2(e+p) = GF2 M4
W [x (u+c) + (1-y)2x (d+s)] dxdy 2x(Q2+M2
W)2
MW informationuv at high x dv at high x
Measurement of high-x dv on a pure proton target
(even Deuterium needs corrections, does dv/uv 0, as x 1? )
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Valence PDFs from ZEUS data alone-
NC and CC e+ and e- beams high x
valence dv from CC e+, uv from CC e-
and NC e+/-
Valence PDFs from a GLOBAL fit to all DIS data high x valence from CCFR xF3(,Fe) data and NMC F2(p)/F2(D) ratio
HERA –II data will enable accurate PDF extractions without need of nuclear corrections
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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
HIGH ET INCLUSIVE JET PRODUCTION – p p jet + X, via g g, g q, g q subprocesses gives more information on the gluon –
But in 1996 an excess of jets with ET > 200 GeV in CDF data appeared to indicate new physics beyond the Standard Model BUT a modification of the PDFs with a harder high-$x$ gluon (which still gave a ‘reasonable fit’ to other data) could explain it
An excess of high-Q2 events in HERA data (1997) was initially taken as evidence for lepto quarks- but a modification to the high-$x$ u quark PDF could explain it
Currently the anomalous NuTeV measurement of sin2θW can be explained by dropping the assumption s = s in the sea?
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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2 = i [ FiQCD – Fi
MEAS]2
(iSTAT)2+(i
SYS)2 = i2
Errors on the fit parameters evaluated from 2 = 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
Must modify the formulation of the χ2
2 = i j [ FiQCD – Fi MEAS] Vij
-1 [ FjQCD – Fj
MEAS]
Vij = ij(iSTAT)2 + i
SYS jSYS
Where iSYS is the correlated error on point i due to systematic error source
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Model Assumptions –
•Value of Q20, form of the parametrization
•Kinematic cuts on Q2, W2, x
•Data sets included …….
Are some data sets incompatible?
PDF fitting is a compromise
e.g. the effect of using different data sets on the value of s
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Comparison of ZEUS and H1 gluon distributions –
Yellow band (total error) of H1 comparable to red band (total error) of ZEUS
Comparison of ZEUS and H1 valence distributions.
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Pqq(z) = P0qq(z) + s P1qq(z) +s2 P2qq(z)
LO NLO NNLO
Theoretical Assumptions- Need to extend the formalism?
What if
Optical theorem2 Im
The handbag diagram- QPM
QCD at LL(Q2)
Ordered gluon ladders (s
n lnQ2 n)
NLL(Q2) one rung disordered s
n lnQ2 n-1
Or higher twist diagrams?
low Q2, high x
Eliminate with a W2 cut
BUT what about completely disordered
Ladders?
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Clues from the gluon distribution Knowledge increased dramatically in the 90’s
Scaling violations dF2/dlnQ2 in DIS
High ET jets in hadroproduction- Tevatron
BGF jets in DIS * g q q
Prompt
HERA charm production * g c c
For small x scaling violation data from HERA are most accurate
Pre HERAPost HERA
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xg(x,Q2) ~ x -g
At small x,
small z=x/y
Gluon splitting functions become singular
t = ln Q2/2
s ~ 1/ln Q2/2
Gluon becomes very steep at small x AND F2 becomes gluon dominated
F2(x,Q2) ~ x -s, s=g -
,)/1ln(
)/ln(12 2
1
0
0
x
ttg
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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
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The steep behaviour of the gluon is deduced
from the DGLAP QCD formalism –
BUT the
steep behaviour of the Sea can be measured from
F2 ~ x -s, s = 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= 0,1,2……for 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 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 = q +1 r 0 at Next to Leading log Q2 (NLLQ2)
These are the DGLAP summations
LL(Q2) is STRONGLY ordered in pt.
But if ln(1/x) is large we should consider p = r q 1 at Leading Log 1/x (LL(1/x))
p = r +1 q 1 at Next to Leading Log (NLL(1/x)) These are the 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 -λ
λ = s CA ln2 ~ 0.5
steep gluon even at moderate Q2
Disordered gluon ladders
But NLL(1/x) softens this somewhat
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Furthermore if the gluon density becomes large there maybe non-linear effects
Gluon recombination g g g
~ s22/Q2
may compete with gluon evolution g g g
~ s
where is the gluon density
~ xg(x,Q2) –no.of gluons per ln(1/x)R2 nucleon size
Non-linear evolution equations – GLR
d2xg(x,Q2) = 3s xg(x,Q2) – s2 81 [xg(x,Q2)]2
dlnQ2dln1/x 16Q2R2
s s2 2/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!
Colour Glass Condensate, JIMWLK, BK
Higher twist
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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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Current measurements of FL and F2charm at small x are not yet accurate enough to
distinguish different approaches
FLF2 charm
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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
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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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
Q2 = 2 Q2=10 Q2=100 GeV2
Non linear
DGLAP
xg
xu
xs
xuv
xd
xc
Such diagrams form part of possible higher twist contributions at low x there maybe further clues from lower Q2 data?
xg
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Linear DGLAP evolution doesn’t work for Q2 < 1 GeV2, WHAT does? – REGGE ideas?
Reg
ge r
egio
npQ
CD
reg
ion
Small x high W2 (x=Q2/2p.q Q2/W2 )
(*p) ~ (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
(p) ~ (W2) 0.08 for real photon- proton scattering
For virtual photons, at small x (*p) = 42 F2
Q2
~ (W2)-1 F2 ~ x 1- = x - so a SOFT POMERON would imply = 0.08 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
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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
(*p) gent
le r
ise
muc
h st
eepe
r ri
se
GBW dipole
QCD improved dipole
Regge region pQCD generated slope
So is there a HARD POMERON corresponding to this steep rise?
A QCD POMERON, (Q2) – 1 = (Q2)
A BFKL POMERON, – 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?
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Dipole models provide a 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
σ =σ0(1 – exp( –r2/2R0(x)2)), R0(x)2 ~(x/x0)~1/xg(x)
r/R0 small large Q2, x σ~ r2~ 1/Q2
r/R0 large small Q2, x σ ~ σ0 saturation of the dipole cross-section
GBW dipole model
(*p)
Butσ(*p) = 42 F2 is generalQ2
At high Q2, F2 ~flat (weak lnQ2 breaking) and σ(*p) ~ 1/Q2
As Q2 0, σ(γp) is finite for real photons scattering.
(at small x)
Now
there is HE
RA
data right across the transition region
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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 at RHIC,
diffractive interactions at the Tevatron and HERA, even some understanding of soft hadronic physics
= 0 (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
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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
Historicallythese data established the Quark-Parton Model and the Theory of QCD, providing measurements of the value of s(MZ
2) and evidence for the running of s(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 at low Q2. The HERA data has stimulated new theoretical approaches in all these areas.