the structure of the proton a.m.cooper-sarkar feb 6 th 2003 rse parton model qcd as the theory of...
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The Structure of the ProtonA.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
d~
2
LW
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
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 themd = ei
2 xs [ 1 + (1-y)2] , for elastic eqQ4 dy
d2= s [ 1 + (1-y)2] i ei2(xq(x) + xq(x))
dxdy Q4
for eNisotropic
non-isotropic
Now 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
Compare to the general form of the cross-section for / scattering via W+/-
FL (x,Q2) = 0
xF3(x,Q2) = 2ix(qi(x) - qi(x))
Valence
F2(x,Q2) = 2ix(qi(x) + qi(x))
Valence and Sea
And there will be a relationship between F2
eN and F2
NOTE scattering is FLAVOUR sensitive
-
d
uW+
W+ can only hit quarks of charge -e/3 or antiquarks -2e/3
p)~ (d + s) + (1- y)2 (u + c)
p) ~ (u + c) (1- y)2 + (d + s)
Consider , scattering: neutrinos are handed
d()= GF2 x s d() = GF
2 x s (1-y)2
dy dy For q (left-left) For q (left-right)
d2() = GF2 s i [xqi(x) +(1-y)2xqi(x)]
dxdy For N
d2() = GF2 s i [xqi(x) +(1-y)2xqi(x)]
dxdy For N
Clearly there are antiquarks in the
nucleon
3 Valence quarks plus a flavourless
qq Sea
q = qvalence +qsea q = qsea qsea= qsea
)(9
4)(
9
1)(
9
1)(
9
4()(2 ccssdduuxplF
)()(),(3 vv duxdduuxNxF
So in scattering the sums over q, q ONLY contain the appropriate flavours BUT-
high statistics data are taken on isoscalar targets e.g. Fe (p + n)/2=N
d in proton = u in neutron
u in proton = d in neutron
A TRIUMPH(and 20 years of understanding the c c contribution)
3)(1
0
3 dxdudxx
xFvv
)(),(2 ccssdduuxNF
1
0 2 ?1),( dxNF
)(
5
8)(
5
2
18
5)(2 ccssdduuxlNF
)(18
5)( 22 NFlNF
GLS sum rule
Total momentum of quarks
QCD improves the Quark Parton Model
What if
or
Before the quark is struck?
F2(N) dx ~ 0.5 where did the momentum go?
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
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
•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)
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
Note scale of xg,
xS
Not
e er
ror
band
s on
PD
Fs
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/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
(d and u in the sea are NOT equal, dv/uv 0 as x 1)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
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
sensitivity to sin2W, MZ and electroweak couplings vi, ai for u and d type quarks- (with electron beam polarization)
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? )
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
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 +- X, via q q * +-, gives information on the Sea
Asymmetry between pp +- X and pn +- X gives more information on d - u difference
W PRODUCTION- p p W+(W-) X, via u d W+, d u W- gives more information on u, d differences
PROMPT - p N X, via g q 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 jet + X, via g g, g q, g q subprocesses gives more information on the gluon –
for ET > 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
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
2 = i j [ FiQCD – Fi MEAS] Vij
-1 [ FjQCD – Fj
MEAS]
Vij = ij(iSTAT)2 +
SYS jSYS
Where iSYS is the correlated error on point i due to systematic error source
2 = i [ FiQCD – si
SYS- Fi MEAS]2 + s2
(iSTAT) 2
s are fit parameters of zero mean and unit variance
modify the measurement/prediction by each source of systematic uncertainty
HOW to APPLY this ?
OFFSET method
1. Perform fit without correlated errors
2. Shift measurement to upper limit of one of its systematic uncertainties (s = +1)
3. Redo fit, record differences of parameters from those of step 1
4. Go back to 2, shift measurement to lower limit (s= -1)
5. Go back to 2, repeat 2-4 for next source of systematic uncertainty
6. Add all deviations from central fit in quadrature
HESSIAN method
Allow fit to determine the optimal values of s
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)O
ffset method
Hessian m
ethod
Q2=2.5 Q2=7
Q2=20 Q2=200
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
model error
MODEL EXPERIMENTAL - OFFSET
MODEL EXPERIMENTAL - HESSIAN
You win some and you lose some!
The change in parameters under model changes is frequently outside the 2=1 criterion of the central fit
e.g. the effect of using different data sets on the value of s
Are some data sets incompatible? PDF fitting is a compromise, CTEQ suggest 2=50 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
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.
Value of s and shape of gluon are correlated
s increases harder gluon
dF2 = s(Q2) [ Pqq F2 + 2 i ei2 Pqg xg]
dlnQ2 2
NLOQCD fit results
Determinations of s
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?
Ways to measure 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
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
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
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 -s, s = d ln F2
MRST PDF fit
xg(x) ~ x -g
xS(x) ~ x -s
at low x
For Q2 >~ 5 GeV2
g > s
g
s
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
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) – 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 = r+1 ≥ 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, p=q=r
LL(1/x) is STRONGLY ordered in ln(1/x) and can be disordered in pt
s p(Q2)(ln 1/x)r, p=r
BFKL summation at LL(1/x)
xg(x) ~ x -
= s CA ln2 ~ 0.5, for s ~ 0.25
steep gluon even at moderate Q2
But this is considerable softened at NLL(1/x)
( way beyond scope !)
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
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
Current measurements of FL and F2charm at small x are not yet accurate enough to
distinguish different approaches
FLF2 charm
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
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
Linear DGLAP evolution doesn’t work for Q2 < 1 GeV2, WHAT does? – REGGE ideas?
Reg
ge r
egio
npQ
CD
reg
ion
Small x is 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
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?
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
=0(1 – exp( –r2/2R0(x)2)), R0(x)2 ~(x/x0)~1/xg(x)
r/R0 small large Q2, x ~ r2~ 1/Q2
r/R0 large small Q2, x ~ 0 saturation of the dipole cross-section
GBW dipole model
(*p)
But(*p) = 42 F2 is generalQ2
(p) is finite for real photons , Q2=0. At high Q2, F2 ~flat (weak lnQ2 breaking) and (*p) ~ 1/Q2
(at small x)
Now
there is HE
RA
data right across the transition region
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- RHIC, diffractive interactions – 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
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. The HERA data has stimulated new theoretical approaches in all these areas.