the structure of the proton a.m.cooper-sarkar april 2003 parton model- qcd as the theory of strong...
TRANSCRIPT
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
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
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
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
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
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)
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
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
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
HERA –II data will enable accurate PDF extractions without need of nuclear corrections
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
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
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
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.
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?
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
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 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
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
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 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
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
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
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.