Parton Distribution Functions
Robert Thorne
August 15, 2003
University of Cambridge
Royal Society Research Fellow
Lepton Photon Symposium 2003
The proton is described by QCD – the theory of the strong interactions. This makesan understanding of its structure a difficult problem.
However, it is also a very important problem – not only as a question in itself, butalso in order to search for and understand new physics.
Many important particle colliders use hadrons – HERA is an ep collider, the Tevatronis a pp collider, the LHC (large hadron collider) at CERN will be a pp collider. Anunderstanding of proton structure is essential in order to interpret the results.
Fortunately, when one has a relatively large scale in the process, in practice only> 1GeV2, the proton is essentially made up of the more fundamental constituents –quarks and gluons (partons), which are relatively independent.
Hence, the fundamental quantities one requires in the calculation of scatteringprocesses involving hadronic particles are the parton distributions.
These can be derived from, and then used within, the Factorization Theorem– separates processes into nonperturbative parts which can be determined fromexperiment, and perturbative parts which can be calculated as a power-series in thestrong coupling constant αS.
Lepton Photon Symposium 2003 1
e e
γ? Q2
x
P
CPi (x, αs(Q
2))
fi(x,Q2, αs(Q
2))
Hadron scattering with anelectron factorizes.
αs(Q2) – Strong Coupling
x = Q2
2mν– Momentum fraction of
Parton (ν=energy transfer)
Lepton Photon Symposium 2003 2
The cross-section for this process can be written in the factorized form
σ(ep→ eX) =∑
i
CPi (x, αs(Q
2))⊗ fi(x,Q2, αs(Q2))
where fi(x,Q2, αs(Q
2)) are the parton distributions, i.e the probability of finding aparton of type i carrying a fraction x of the momentum of the hadron.
Corrections to above formula of size Λ2QCD/Q
2 - Higher Twist.
The parton distributions are not easily calculable from first principles. However, theydo evolve with Q2 in a perturbative manner
dfi(x,Q2, αs(Q
2))
d lnQ2=∑
i
Pij(x, αs(Q2))⊗ fj(x,Q2, αs(Q
2))
where the splitting functions Pij(x,Q2, αs(Q
2)) are calculable order by order inperturbation theory.
Lepton Photon Symposium 2003 3
P
P
fi(xi, Q2, αs(Q
2))
CPij(xi, xj, αs(Q
2))
fj(xj, Q2, αs(Q
2))
The coefficient functionsCPi (x, αs(Q
2)) describing thehard scattering process areprocess dependent but arecalculable as a power-series.
CPi (x, αs(Q
2)) =∑
k
CP,ki (x)αks(Q
2).
Since the fi(x,Q2, αs(Q
2))are process-independent, i.e.universal, once they have beenmeasured at one experiment,one can predict many otherscattering processes.
Lepton Photon Symposium 2003 4
Global fits to determine parton distributions use all available data - largely ep→ eX(Structure Functions), and the most up-to-date QCD calculations to best determineparton distributions and their consequences. (Also → good determination of strongcoupling constant.)
Currently use NLO–in–αs(Q2), i.e.
CPi (x, αs(Q
2)) = αPS (Q2)(CP,0
i (x) + αS(Q2)CP,1
i (x)).
Pij(x, αs(Q2)) = αS(Q
2)P 0ij(x) + α2S(Q
2)P 1ij(x).
NNLO coefficient functions are known for some processes, e.g. structure functions,and NNLO splitting functions have considerable information (see later).
General procedure. Start parton evolution at low scale Q20 ∼ 1GeV2. Input partons
parameterized as, e.g.
xf(x,Q20) = a1(1− x)a2(1 + a3x
0.5 + a4x)xa5.
Evolve partons upwards using NLO DGLAP equations. Fit data for scales above2− 5GeV2.
Lepton Photon Symposium 2003 5
In principle 11 different parton distributions to consider
u, u, d, d, s, s, c, c, b, b, g
mc,mb À ΛQCD so heavy parton distributions determined perturbatively. Assumes = s. Leaves 6 independent combinations. Relate s to 1/2(u+ d) and use
uV = u− u, dV = d− d, sea = 2 ∗ (u+ d+ s), d− u, g.
Assuming isospin symmetry p→ n leads to
dp(x)→ un(x) up(x)→ dn(x).
Various sum rules constraining parton inputs and conserved order by order in αS forevolution, i.e. conservation of number of valence quarks.
∫ 1
0
uV (x) dx = 2
∫ 1
0
dV (x) dx = 1
Also conservation of momentum carried by partons – important constraint on thegluon, which is only probed indirectly.
∫ 1
0
x(
∑
i
(qi(x) + qi(x)) + g(x))
dx = 1.
Lepton Photon Symposium 2003 6
In determining partons need to consider that not only are there 6 different combinationsof partons, but also wide distribution of x from 0.75 to 0.00003. Need many differenttypes of experiment for full determination.
H1 F e+p2 (x,Q2) 1996-97 moderate Q2 and 1996-97 high Q2, and F e−p
2 (x,Q2) 1998-99
high Q2 small x. ZEUS F e+p2 (x,Q2) 1996-97 small x wide range of Q2. (1999-2000)
NMC Fµp2 (x,Q2), Fµd
2 (x,Q2), (Fµn2 (x,Q2)/Fµp
2 (x,Q2)), E665 Fµp2 (x,Q2), Fµd
2 (x,Q2)medium x.
BCDMS Fµp2 (x,Q2), Fµd
2 (x,Q2), SLAC Fµp2 (x,Q2), Fµd
2 (x,Q2) large x.
CCFR (NuTeV) Fν(ν)p2 (x,Q2), F
ν(ν)p3 (x,Q2) large x , singlet, valence.
E605 (E866) pN → µµ+X large x sea.
E866 Drell-Yan asymmetry u, d d− u.
CDF W-asymmetry u/d ratio at high x.
CDF D0 Inclusive jet data high x gluon.
CCFR (NuTev) Dimuon data constrains strange sea.
Lepton Photon Symposium 2003 7
Large x.
Quark distributions are determined mainly by structure functions. Dominated bynon-singlet valence distributions.
Simple evolution of non-singlet distributions and conversion to structure function
dfNS(x,Q2)
d lnQ2= PNS(x, αs(Q
2))⊗ fNS(x,Q2)
FNS2 (x,Q2) = CNS(x, αs(Q
2))⊗ fNS(x,Q2, αs(Q2)).
So evolution of high x structure functions good test of theory and of αS(Q2).
However - perturbation theory involves contributions to coefficient function ∼αnS(Q
2) ln2n−1(1 − x) and higher twist known to be enhanced as x → 1. Henceto avoid contamination of NLO theory make cut
W 2 = Q2(1/x− 1) +m2p ≤ 10− 15GeV2.
Lepton Photon Symposium 2003 8
0.2
0.3
10 102
x=0.35
Q2 (GeV2)
F2p
0.1
0.2
10 102
x=0.45
Q2 (GeV2)
F2p
0.05
0.1
10 102
x=0.55
Q2 (GeV2)
F2p
0.02
0.04
10 102
x=0.65
Q2 (GeV2)
F2p
SLAC (×1.025)
BCDMS (×0.98)
Description of large x BCDMS and SLACmeasurements of F p
2 .
Determines αS(M2Z).
Lepton Photon Symposium 2003 9
Small x.
The extension to very low x has been made in the past decade by HERA. In thisregion there is very great scaling violation of the partons from the evolution equationsand also interplay between the quarks and gluons.
At each subsequent order in αS each splitting function and coefficient function obtainsan extra power of ln(1/x) (some accidental zeros in Pgg), i.e.
Pij(x, αs(Q2)), CP
i (x, αs(Q2)) ∼ αms (Q
2) lnm−1(1/x),
and hence the convergence at small x is questionable.
The global fits usually assume that this turns out to be unimportant in practice, andproceed regardless. The fit is good, but could be improved.
Small x predictions somewhat uncertain. Very active area of research (later).
Lepton Photon Symposium 2003 10
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
100
101
102
103
104
105
106
107
108
109
fixedtarget
HERA
x1,2
= (M/14 TeV) exp(±y)Q = M
LHC parton kinematics
M = 10 GeV
M = 100 GeV
M = 1 TeV
M = 10 TeV
66y = 40 224
Q2
(GeV
2 )
x
Small x parton distributions thereforeinteresting within QCD.
Also vital for understanding the standardproduction processes at the LHC, andperhaps some of the more exotic ones.
Lepton Photon Symposium 2003 11
High-x Sea quarks determined by Drell-Yan data (assuming knowledge of valencequarks). Recent suggested discrepancy by fit to E866/NuSea collaboration. Implieslarger high-x valence quarks.
0.3 0.4 0.5 0.6 0.7 0.8x
1
0.75
1
1.25
dσex
p /dx 1 /
dσM
RST
2001
/dx 1
MRST2001uncertaintypd fitpp fit
0.05 0.1 0.15 0.2 0.25x
2
0.75
1
1.25
dσex
p /dx 2 /
dσM
RST
2001
/dx 2
pd → µ+µ− X
pp → µ+µ− X
+/- 6.5% normalization uncertainty
a)
b)
+/- 6.5% normalization uncertainty
E866 pd data and MRST2001 (xF > 0.45)
10-6
10-5
10-4
10-3
10-2
5 6 7 8 9 10 20M
3 d2 σ/dM
dxF
M (GeV)
xF=0.45-0.50 (/28)
xF=0.50-0.55 (/29)
xF=0.55-0.60 (/210)
xF=0.60-0.65 (/211)
xF=0.65-0.70 (/212)
xF=0.70-0.75 (/213)xF=0.75-0.80 (/214)
Not observed by MRST or by CTEQ.
Lepton Photon Symposium 2003 12
10-5 .001 0.01 0.05 0.1 .2 .3 .4 .5 .6 .7
x
-2×10-3
0
2×10-3
4×10-3
6×10-3
8×10-3
S- (
= x
s- (x,Q
)) d
x/dz
Momentum Asymmetry
(scale: linear in z = x1/3
)
NuTeV measure R− =σνNC−σ
νNC
σνCC−σν
CC.
R− = 12 − sin2 θW − (1− 7
3 sin2 θW ) [S
−][V −]
.
[S−] = 0.002 reduces NuTeV anomaly from3σ to 1.5σ.
10-5
.001 0.01 0.05 0.1 .2 .3 .4 .5 .6 .7
x
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
s- (x,Q
) dx
/dz
class Aclass Bclass C
Strangeness Asymmetry Q2 = 10 GeV
2
(scale: linear in z = x1/3
)
s(x) and s(x) distributions can now beprobed separately using NuTeV dimuon data
ν + s→ µ−+ c(µ+), ν + s→ µ+ + c(µ−).
Examinied in detail by CTEQ.
→ s(x) < s(x) at quite small x.
∫
(s(x) − s(x)) dx = 0, (zero strangenessnumber).
→∫
x(s(x)− s(x)) dx = [S−] > 0.
0 < [S−] < 0.004.
Lepton Photon Symposium 2003 13
MRST also look at effect of isospin violation.
R− =1
2− sin2 θW + (1− 7
3sin2 θW )
[δUv]− [δDv]
2[V −].
[δUv] = [Upv ]− [Dn
v ], [δDv] = [Dpv]− [Un
v ].
upv(x) = dnv(x) + κf(x), dpv(x) = unv(x)− κf(x).
κ = −0.2 → a similar reduction of the NuTeV anomaly, i.e. ∆sin2θW ∼ −0.002.Larger (more negative) κ allowed.
0
50
100
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
∆χ2
κ
Valence quarks
Lepton Photon Symposium 2003 14
High-x Gluon distribution.
0 100 200 300 400 500 600 7000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
gg
qg
frac
tion
ET (GeV)
Best determination from inclusive jetmeasurements by D0 and CDF atTevatron. Measure dσ/dETdη for centralrapidity CDF or in bins of rapidity D0.
At central rapidity xT = 2ET/√s,
and measurements extend up to ET ∼400GeV, i.e. xT ∼ 0.45, and down toET ∼ 60GeV, i.e. xT ∼ 0.06.
Gluon-gluon fusion dominates, butg(x, µ2) falls off more quickly as x → 1than q(x, µ2) so there is a transitionfrom gluon-gluon fusion at small xT , togluon-quark to quark-quark at high xT .However, even at the highest xT gluon-quark contributions are significant.
Jet photoproduction at HERA will beanother constraint in the future.
Lepton Photon Symposium 2003 15
Results.
Above procedure completely determines parton distributions at present. Total fitreasonably good, e.g CTEQ6. αS(M
2Z) fixed at 0.118. Total χ2 = 1954/1811.
Data set No. of χ2
data ptsH1 ep 230 228
ZEUS ep 229 263BCDMS µp 339 378BCDMS µd 251 280NMC µp 201 305
E605 (Drell-Yan) 119 95D0 Jets 90 65CDF Jets 33 49
For MRST αS(M2Z) = 0.119. Compromise between different data sets. Total
χ2 = 2328/2097 – but errors treated differently, and different data sets and cuts.
Same sort of conclusion for other global fits (H1, ZEUS, Alekhin, GKK) (with ratherfewer data).
Some areas where theory perhaps needs to be improved. (See later.)
Lepton Photon Symposium 2003 16
0
1
2
10-4
10-3
10-2
10-1
1x
xf(x
,Q2 )
MRST2001 partons Q2 = 10 GeV2
g/10
u
ds
c
0
1
2
10-4
10-3
10-2
10-1
1x
xf(x
,Q2 )
Q2 = 104 GeV2
g/10
ud
sc
MRST2001 partons.
CTEQ etc. (generally) verysimilar.
Lepton Photon Symposium 2003 17
Parton Uncertainties – currently an issue attracting a lot of work. Number ofapproaches.
Hessian (Error Matrix) approach first used by H1 and ZEUS, recently extended byCTEQ.
χ2 − χ2min ≡ ∆χ2 =∑
i,j
Hij(ai − a(0)i )(aj − a(0)j )
We can then use the standard formula for linear error propagation.
(∆F )2 = ∆χ2∑
i,j
∂F
∂ai(H)−1ij
∂F
∂aj,
This has been used to find partons with errors by Alekhin and H1, each with restricteddata sets.
Simple method problematic due to extreme variations in ∆χ2 in different directionsin parameter space - particularly with more parameters (more data). → numericalinstability.
Solved (helped) by finding and rescaling eigenvectors of H leading to diagonal form∆χ2 =
∑
i z2i . First used by CTEQ. Now used in slightly weaker form by MRST and
ZEUS.
Lepton Photon Symposium 2003 18
In full global fit art in choosing “correct” ∆χ2 given complication of errors. Ideally∆χ2 = 1, but unrealistic.
0.09 0.1 0.11 0.12 0.13 0.14 0.15alpha S
0.09 0.1 0.11 0.12 0.13 0.14 0.15�Χ2�1 ranges from GA
BCDMSp
BCDMSd
H1a
H1b
ZEUS
NMCp
NMCr
CCFR2
CCFR3
E605
CDFw
E866
D0jet
CDFjet
200
220
240
260
280
0.116 0.118 0.12 0.122
χ2 − n
o. p
ts Total (2097 pts)
0
20
40
60
0.116 0.118 0.12 0.122
D0 jet (82 pts)
CDF1B jet (31 pts)
Total jet (113 pts)
60
80
100
120
140
0.116 0.118 0.12 0.122
χ2 − n
o. p
ts E605 (136 pts)
0
20
40
60
80
0.116 0.118 0.12 0.122
BCDMS F2µp (167 pts)
BCDMS F2µd (155 pts)
-40
-20
0
20
40
0.116 0.118 0.12 0.122
αs(MZ2)
χ2 − n
o. p
ts NMC F2µp (126 pts)
NMC F2µd (126 pts)
SLAC F2 ep (53 pts)
SLAC F2 ed (54 pts)
-40
-20
0
20
0.116 0.118 0.12 0.122
αs(MZ2)
CCFR F2νN (74 pts)
CCFR xF3νN (105 pts)
H1 (400 pts)
ZEUS (272 pts)
Many approaches use ∆χ2 ∼ 1. CTEQ choose ∆χ2 ∼ 100 (conservative?). MRSTchoose ∆χ2 ∼ 20 for 1− σ error.
Lepton Photon Symposium 2003 19
Results for Alekhin partons (left) at Q2 = 9GeV2 with uncertainties (solid lines),(dashed lines – CTEQ5M, dotted lines – MRST01), and CTEQ Hessian approach forluminosity uncertainty (right).
102
√s (GeV)
Luminosity function at TeV RunII
Fra
ctio
nal U
ncer
tain
ty
0.1
0.1
0.1
-0.1
-0.1
-0.2
0.2
Q-Q --> W+ (W-)
Q-Q --> γ* (Z)
G-G
0
0
0
^
≈
≈ ≈
≈−
−
0.3
0.4
200 4005020
±
Lepton Photon Symposium 2003 20
Other Approaches.
Statistical Approach (Giele, Keller and Kosower) constructs an ensemble ofdistributions labelled by F each with probability P ({F}). Can incorporate fullinformation about measurements and their error correlations in the calculation ofP ({F}). Calculate by summing over Npdf different distributions with unit weight butdistributed according to P ({F}). (Npdf can be made as small as 100). Mean µO anddeviation σO of observable O then given by
µO =∑
{F}
O({F})P ({F}), σ2O =∑
{F}
(O({F})− µO)2P ({F}).
Currently uses only proton DIS data sets in order to avoid complicated uncertaintyissues such as shadowing effects for nuclear targets. Demands strict consistencybetween data sets. It is difficult to find many compatible DIS experiments. Fermi2001partons determined by only H1(94), BCDMS, E665 data sets.
Good principle if theory and data good enough.
Some good predictions, e.g. σW and σZ at Tevatron. Some unusual parameterscompared to other sets, e.g. low αS(M
2Z), very hard dV (x) at high x.
Lepton Photon Symposium 2003 21
In the offset method the best fit and parameters a0 are obtained using only uncorrelatederrors. The quality of fit is then estimated by adding in quadrature. Systematic errorsare determined (effectively) by letting each source of systematic error vary by 1 − σand adding the deviations in quadrature. Used by ZEUS. Effective ∆χ2 > 1.
ZEUS
10-2
10-1
1
0 0.2 0.4 0.6 0.8 1
xqV
Q2=1 GeV2
10-2
10-1
1
0 0.2 0.4 0.6 0.8 1
20 GeV2
ZEUS NLO QCD fit
10-2
10-1
1
0 0.2 0.4 0.6 0.8 1
200 GeV2
tot. error
xuV
xdV
10-2
10-1
1
0 0.2 0.4 0.6 0.8 1
x
2000 GeV2
uncorr. error
ZEUS
10-2
10-1
1
0 0.2 0.4 0.6 0.8 1
xqV
Q2=1 GeV2
10-2
10-1
1
0 0.2 0.4 0.6 0.8 1
20 GeV2
ZEUS ONLY fit
10-2
10-1
1
0 0.2 0.4 0.6 0.8 1
200 GeV2
tot. error
xuV
xdV
10-2
10-1
1
0 0.2 0.4 0.6 0.8 1
x
2000 GeV2
uncorr. error
Valence partons extracted by ZEUS from global fit and fit to own data alone (withsome input assumptions). Potential for real constraint in future.
Lepton Photon Symposium 2003 22
Can also look at uncertainty on a givenphysical quantity using Lagrange Multipliermethod, first suggested by CTEQ andconcentrated on by MRST. Minimize
Ψ(λ, a) = χ2global(a) + λF (a).
Gives best fits for particular values ofquantity F (a) without relying on Gaussianapprox for χ2. Uncertainty then determinedby deciding allowed range of ∆χ2.
CTEQ obtain for αS = 0.118
∆σW (LHC) ≈ ±4% ∆σW(Tev) ≈ ±4∆σH(LHC) ≈ ±5%.
MRST use a wider range of data, and if ∆χ2 ∼ 50 find for αS = 0.119
∆σW (Tev) ≈ ±1.2% ∆σW(LHC) ≈ ±2%∆σH(Tev) ≈ ±4% ∆σH(LHC) ≈ ±2%.
Lepton Photon Symposium 2003 23
MRST also allow αS to be free.
-10
-5
0
5
10
-4 -3 -2 -1 0 1 2 3 4Per cent change in W cross section
χ2 increase in global analysis as theW and H cross sections are varied at the TEVATRON
50
100
150
200250
•P
•Q
• R
•S
Per
cent
cha
nge
in H
cro
ss s
ectio
n
+
50*
100*
•P*
•Q*
• R*
•S*
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4Per cent change in W cross section
χ2 increase in global analysis as theW and H cross sections are varied at the LHC
50
100
150
200
•A
•B
• C
•D
Per
cent
cha
nge
in H
cro
ss s
ectio
n
+
50*
100*
•A*
•B*
•C*
•D*
χ2-plots for W and Higgs (120GeV) production at the Tevatron and LHC αS free(blue) and fixed (red) at αS = 0.119.
Lepton Photon Symposium 2003 24
Same general procedure usedby CTEQ to look at effect ofnew physics in contact term
±(2π/Λ2)(qLγµqL)(qLγµqL).
Curves show fit to D0 jet datafor Λ = 1.6, 2.0, 2.4,∞ TeV,A = −1.
Λ > 1.6, TeV.
Lepton Photon Symposium 2003 25
Hence, the estimation of uncertainties due to experimental errors has many differentapproaches and different types and amount of data actually fit. Overall conclude thatuncertainty due to experimental errors only more than few % for quantities determinedby high x gluon and very high x down quark.
Values of αs(M2Z) and its error from different NLO QCD fits with different error
tolerances. Reasonable agreement in general – but some outliers.
CTEQ6 ∆χ2 = 100 αs(M2Z) = 0.1165± 0.0065(exp)
ZEUS ∆χ2eff = 50 αs(M
2Z) = 0.1166± 0.0049(exp)
±0.0018(model)
±0.004(theory)
MRST01 ∆χ2 = 20 αs(M2Z) = 0.1190± 0.002(exp)
±0.003(theory)
H1 ∆χ2 = 1 αs(M2Z) = 0.115± 0.0017(exp)
+ 0.0009− 0.0005 (model)
±0.005(theory)
Alekhin ∆χ2 = 1 αs(M2Z) = 0.1171± 0.0015(exp)
±0.0033(theory)
GKK CL αs(M2Z) = 0.112± 0.001(exp)
Theory errors highly correlated.
Lepton Photon Symposium 2003 26
Different approaches lead to similar accuracy of measured quantities, but can lead todifferent central values. Must consider effect of assumptions made during fit.
-10
-5
0
5
10
-4 -3 -2 -1 0 1 2 3 4Per cent change in W cross section
χ2 increase in global analysis as theW and H cross sections are varied at the TEVATRON
+
50
100
150
200250
•P
•Q
•R
•S
• CTEQ6
• MRST2002
Per
cent
cha
nge
in H
cro
ss s
ectio
nUncertainty of gluon from Hessian method
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
10-4
10-3
10-2
10-1
1
Ratio of xg(x,Q2)/xg(x,Q2,MRST2001C) at Q2=5 GeV2
x
Hessian uncertainty
CTEQ6M
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
10-4
10-3
10-2
10-1
1
Ratio of xg(x,Q2)/xg(x,Q2,MRST2001C) at Q2=100 GeV2
x
Cuts made on data, data sets fit, parameterization for input sets, form of strange sea,heavy flavour prescription, assumption of no isospin violation, strong coupling ......
Many can be as important as experimental errors on data used (or more so).
Lepton Photon Symposium 2003 27
Results from LHC/LP Study Working Group (Bourilkov).
Table 1: Cross sections for Drell-Yan pairs (e+e−) with PYTHIA 6.206, rapidity < 2.5.The errors shown are the PDF uncertainties.
PDF set Comment xsec [pb] PDF uncertainty %81 < M < 101 GeV
CTEQ6 LHAPDF 1065 ± 46 4.4MRST2001 LHAPDF 1091 ± ... 3Fermi2002 LHAPDF 853 ± 18 2.2
Comparison of σW ·Blν for MRST2002 and Alekhin partons.
PDF set Comment xsec [nb] PDF uncertaintyAlekhin Tevatron 2.73 ± 0.05 (tot)MRST2002 Tevatron 2.59 ± 0.03 (expt)CTEQ6 Tevatron 2.54 ± 0.10 (expt)Alekhin LHC 215 ± 6 (tot)MRST2002 LHC 204 ± 4 (expt)CTEQ6 LHC 205 ± 8 (expt)
In both cases differences (mainly) due to detailed constraint (by data) on quarkdecomposition.
Lepton Photon Symposium 2003 28
0
0.2
0.4
0.6
0.8
10-3
10-2
10-1
1
0
0.2
0.4
0.6
0.8
10-3
10-2
10-1
1
0
0.2
0.4
0.6
0.8
10-3
10-2
10-1
1
0
0.2
0.4
0.6
0.8
10-3
10-2
10-1
1
0
2
4
6
10-3
10-2
10-1
1
xP(x
)
xuv
xUxU
xdv
xD
x
xD
H1
Col
labo
ratio
n
x
xg
H1 PDF 2000: Q2 = 4 GeV2
Fit to H1 dataexperimental errorsmodel uncertainties
Fit to H1 + BCDMS dataparton distribution
Also demonstrated bymost recent H1 fit (toown data alone) wheremodel error dominates.
Again shows constraintnow achieved by HERAdata alone – with someassumptions.
Lepton Photon Symposium 2003 29
Problems in the fit.
Variations from different approaches partially due to inadequacy of theory .
Failings of NLO QCD indicated by some areas where fit quality could be improved.
Good fit to HERA data, but some problems at highest Q2 at moderate x, i.e. indF2/d lnQ
2.
Want more gluon in the x ∼ 0.01 range, and/or larger αS(M2Z).
Possible sign of required ln(1/x) corrections.
Lepton Photon Symposium 2003 30
MRST(2001) NLO fit , x= 0.008 - 0.032
0.5
1
1.5
2
2.5
3
1 10 102
103
F 2p (x,Q
2 ) +
0.2
5(8-
i)
Q2 (GeV2)
x=8.0×10-3
x=1.0×10-2
x=1.3×10-2
x=1.6×10-2
x=1.75×10-2
x=2.0×10-2
x=2.5×10-2
x=3.2×10-2
MRST 2001
Comparison of MRST(2001) F2(x,Q2) with HERA, NMC and E665 data (left) and
of CTEQ6 F2(x,Q2) and H1 data.
Lepton Photon Symposium 2003 31
ZEUS
-2
0
2
4
6
10-4
10-3
10-2
10-1
Q2=1 GeV2
ZEUS NLO QCD Fit
xg
xS
-2
0
2
4
6
10-4
10-3
10-2
10-1
2.5 GeV2
xS
xg
0
10
20
10-4
10-3
10-2
10-1
7 GeV2
tot. error(αs free)
xS
xg
xf
0
10
20
10-4
10-3
10-2
10-1
20 GeV2
tot. error(αs fixed)
uncorr. error(αs fixed)
xS
xg
0
10
20
30
10-4
10-3
10-2
10-1
200 GeV2
xS
xg
0
10
20
30
10-4
10-3
10-2
10-1
2000 GeV2
x
xS
xg
Data require gluon to be negative atlow Q2, e.g. MRST Q2
0 = 1GeV2.Needed by all data (e.g Tevatron jets)not just low Q2 low x data.
→ FL(x,Q2) dangerously small at
smallest x,Q2.
Other groups find similar problemswith gluon and/or FL(x,Q
2) at lowx, e.g. ZEUS.
Lepton Photon Symposium 2003 32
MRST 2002 and D0 jet data, αS(MZ)=0.1197 , χ2= 85/82 pts
0
0.5
1
0 50 100 150 200 250 300 350 400 450
0.0 < | η | < 0.5
0
0.5
1
0 50 100 150 200 250 300 350 400 450
0.5 < | η | < 1.0
0
0.5
1
0 50 100 150 200 250 300 350 400 450
1.0 < | η | < 1.5
(Dat
a -
The
ory)
/ T
heor
y
-0.5
0
0.5
1
0 50 100 150 200 250 300 350 400 450
1.5 < | η | < 2.0
ET (GeV)
Difficult to reconcile fit to jets andrest of data.
MRST find a reasonable fit to jetdata, but need to use the largesystematic errors.
Better for CTEQ6 largely due todifferent cuts on other data. Usuallyworse for other partons (jets notin fits). General tension betweenHERA and NMC data and jets.
In general different data competeover the gluon and αS(M
2Z).
Lepton Photon Symposium 2003 33
Theoretical Errors
Hence it is vital to consider theoretical corrections. These include ....
- higher orders (NNLO)
- small x (αns lnn−1(1/x))
- large x (αns ln2n−1(1− x))
- low Q2 (higher twist)
In order to investigate true theoretical error must consider large and small xresummations, and/or use what we already know about NNLO.
Coefficient functions known at NNLO. Singular limits x → 1, x → 0 known forNNLO splitting functions as well as limited moments (Larin, Nogueira, van RitenbergVermaseren, Retey). Complete soon. Approximate NNLO splitting functions devisedby van Neerven and Vogt.
Improve quality of fit very slightly (MRST). Not much improvement at small x.Lowered value of αS(M
2Z) = 0.1155 (from 0.119), determined mainly by high x data.
Alekhin finds αS(M2Z) = 0.1143 at NNLO.
Lepton Photon Symposium 2003 34
14
15
16
17
18
19
20
21
22
23
24
NLONNLO
LO
LHC Z(x10)
W
σ . B
l (
nb)
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
NNLONLO
LO
TevatronCDF(e,µ) D0(e)
Z(x10)
W
CDF(e) D0(e,µ)
σ . B
l (
nb)
Reasonable stability order by order for(quark-dominated) W and Z cross-sections.
However, changes of order 4%.Much bigger than uncertainty due toexperimental errors.
This fairly good convergence is largelyguaranteed because the quarks are fitdirectly to data.
Lepton Photon Symposium 2003 35
0
0.1
0.2
0.3
0.4
0.5
10-5
10-4
10-3
10-2
10-1
1
F L(x
,Q2 )
Q2=2 GeV2
NNLO (average)NNLO (extremes)NLOLO
0
0.1
0.2
0.3
0.4
0.5
10-5
10-4
10-3
10-2
10-1
1
Q2=5 GeV2
0
0.1
0.2
0.3
0.4
0.5
10-5
10-4
10-3
10-2
10-1
1x
F L(x
,Q2 )
Q2=20 GeV2
0
0.1
0.2
0.3
0.4
0.5
10-5
10-4
10-3
10-2
10-1
1x
Q2=100 GeV2
More danger in gluon dominatedquantities, e.g. FL(x,Q
2).
Hence the convergence from orderto order is uncertain.
Lepton Photon Symposium 2003 36
Alternative approach.
In order to investigate real quality of fit and regions with problems vary kinematic cutson data.
Procedure – change W 2cut, Q
2cut and xcut, re-fit and see if quality of fit to remaining
data improves and/or input parameters change dramatically. Continue until quality offit and partons stabilize.
For W 2cut raising from 12.5GeV2 to 15GeV2 sufficient.
Raising Q2cut from 2GeV2 in steps there is a slow continuous and significant
improvement for higher Q2 up to > 10GeV2 (cut 560 data points) – suggestsany corrections mainly higher orders not higher twist.
Raising xcut from 0 to 0.005 (cut 271 data points) continuous improvement. At eachstep moderate x gluon becomes more positive.
→ MRST2003 conservative partons. Should be most reliable method of partondetermination (∆χ2 = −70 for remaining data), but only applicable for restrictedrange of x, Q2. → αS(M
2Z) = 0.1165± 0.004.
Lepton Photon Symposium 2003 37
Gluon outside conservative range very negative, and dF2(x,Q2)/d lnQ2 incorrect,
(NNLO much more stable than NLO). Theory corrections could cure this (quiteplausible). Empirical resummation corrections improve global fit, e.g.
Pgg → ....+3.86α4Sx
(
ln3(1/x)
6− ln2(1/x)
2
)
,
Pqg → ....+ 5.12αSNf α
4S
3πx
(
ln3(1/x)
6− ln2(1/x)
2
)
.
Saturation corrections do not help at NLO or NNLO.
Cuts suggestive of possible/probable theoretical errors for small x and/or small Q2.
Much explicit work on ln(1/x)-resummation in structure functions and partondistributions - RT, Ciafaloni, Colferai, Salam and and Stasto, Altarelli, Ball andForte, .......
(Also work on connecting the partons to alternative approaches at small x, e.g.Golec-Biernat, Wusthoff (dipole models), Donnachie, Landshoff (pomerons), ....)
Can suggest improvements to fit and changes in predictions.
Lepton Photon Symposium 2003 38
FL LO , NLO and NNLO
0
0.1
0.2
0.3
0.4
0.5
10-5
10-4
10-3
10-2
10-1
1
F L(x
,Q2 )
Q2=2 GeV2
NLO fitNNLO fitresum fitLO fit
0
0.1
0.2
0.3
0.4
0.5
10-5
10-4
10-3
10-2
10-1
1
Q2=5 GeV2
0
0.1
0.2
0.3
0.4
0.5
10-5
10-4
10-3
10-2
10-1
1x
F L(x
,Q2 )
Q2=20 GeV2
0
0.1
0.2
0.3
0.4
0.5
10-5
10-4
10-3
10-2
10-1
1x
Q2=100 GeV2
Comparison of prediction forFL(x,Q
2) at LO, NLO and NNLOusing MRST partons and also aln(1/x)-resummed prediction RT.
Accurate and direct measurementsof FL(x,Q
2) and other quantitiesat low x and/or Q2 (predictedrange and accuracy of FL(x,Q
2)measurements at HERA III shownon the figure) would be a great helpin determining whether NNLO issufficient or whether resummed (orother) corrections are necessary, orhelpful for maximum precision.
Lepton Photon Symposium 2003 39
Conclusions
One can determine the parton distributions by performing global fits to all up-to-datedata over wide range of parameter space. The fit quality using NLO QCD is fairlygood.
Various ways of looking at uncertainties due to errors on data alone. No totallypreferred approach – all have pros and cons. Uncertainties rather small using allapproaches – ∼ 1− 5% except in certain regions of parameter space.
Uncertainty from input assumptions e.g. cuts on data, data used, ..., comparable andpotentially larger. Can shift central values of predictions on/using partons significantly.
Errors from higher orders/resummation potentially large in some regions of parameterspace, and due to correlations between partons feed into all regions. Cutting out lowx and/or Q2 allows much improved fit to remaining data, and altered partons. NNLOappears to be much more stable than NLO.
Theory often the dominant source of uncertainty at present. Systematic studyneeded. Much progress – NNLO, resummations ..., but much still to do. Both fortheory and in obtaining useful new data (HERA III ?). Very busy and important areaof research.
Lepton Photon Symposium 2003 40
0.6
0.8
1
1.2
10-3
10-2
10-1 x
Ratio MRST(cons.)/MRST2002at Q2 = 10 GeV2
g
us
d
c
0.6
0.8
1
1.2
10-3
10-2
10-1 x
Ratio MRST(cons.)/MRST2002at Q2 = 104 GeV2
du
gs
c
0.8
0.9
1
1.1
10-4
10-3
10-2
10-1
x
Ratio MRST(NNLO,cons.)/MRST2002(NNLO)at Q2 = 10 GeV2
g
us
d
c
0.8
0.9
1
1.1
10-4
10-3
10-2
10-1
x
Ratio MRST(NNLO,cons)/MRST2002(NNLO)at Q2 = 104 GeV2
d
u
gs
c
Lepton Photon Symposium 2003 41
MRST(2001) NLO fit , x = 0.00005 - 0.00032
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
1 10
F 2p (x,Q
2 ) +
0.2
5(8-
i)
Q2 (GeV2)
x=5.3×10-5
x=7.8×10-5
x=1.0×10-4
x=1.3×10-4
x=1.7×10-4
x=2.1×10-4
x=2.5×10-4
x=3.2×10-4
MRST2002
MRST(abs.)
H1 96/97+98/99
ZEUS 96/97 (×0.98)
NMCE665
MRST fit with shadowingcorrections extraploated to Q2 ≤5GeV2
Lepton Photon Symposium 2003 42
2.50
2.55
2.60
2.65
2.70
2.75
2.80
W @ Tevatron
NLO
Q2
cut = 7 GeV2
Q2
cut = 10 GeV2
NNLO
xcut = 0 0.0002 0.001 0.0025 0.005 0.01
σ W .
Blν
(nb
)
MRST NLO and NNLO partons
0.5
0.6
0.7
0.8
0.9
1.0
1.1
H @ Tevatron
NLO
Q2
cut = 7 GeV2
Q2
cut = 10 GeV2
NNLO
xcut = 0 0.0002 0.001 0.0025 0.005 0.01
σ Hig
gs
(pb
)
MRST NLO and NNLO partons
14
16
18
20
22
24
W @ LHC
NLO
Q2
cut = 7 GeV2
Q2
cut = 10 GeV2
NNLO
xcut = 0 0.0002 0.001 0.0025 0.005 0.01
σ W .
Blν
(nb
)
MRST NLO and NNLO partons
28
30
32
34
36
38
40
42
44
46
H @ LHC
NLO
Q2
cut = 7 GeV2
Q2
cut = 10 GeV2
NNLO
xcut = 0 0.0002 0.001 0.0025 0.005 0.01
σ Hig
gs
(pb
)
MRST NLO and NNLO partons
Lepton Photon Symposium 2003 43
Table 2: Cross sections for Drell-Yan pairs (e+e−) with PYTHIA 6.206. The errorsshown are the statistical errors of the Monte-Carlo generation.
PDF set Comment xsec81 < M < 101 GeV
CTEQ5L PYTHIA internal 1516 ± 5 pbCTEQ5L PDFLIB 1536 ± 5 pbCTEQ6 LHAPDF 1564 ± 5 pbMRST2001 LHAPDF 1591 ± 5 pbFermi2002 LHAPDF 1299 ± 4 pb
M > 1000 GeVCTEQ5L PYTHIA internal 6.58 ± 0.02 fbCTEQ5L PDFLIB 6.68 ± 0.02 fbCTEQ6 LHAPDF 6.76 ± 0.02 fbMRST2001 LHAPDF 7.09 ± 0.02 fbFermi2002 LHAPDF 7.94 ± 0.03 fb
Lepton Photon Symposium 2003 44
H1 set of parton parameters from GKKapproach. Red curve Gaussian approxand blue line MRST value. Green curvefor αS is LEP result.
Lepton Photon Symposium 2003 45
Aproximate NNLO splitting functions devised by van Neerven and Vogt.
0
1000
2000
3000
10-3
10-2
10-1
1-3000
0
3000
6000
9000
10-3
10-2
10-1
1
-4000
-2000
0
2000
10-3
10-2
10-1
1
xP(2) (x)qq
Nf = 4
oldnew
xP(2) (x)qg
x
xP(2) (x)gq
x
xP(2) (x)gg
-20000
-10000
0
10000
10-3
10-2
10-1
1
Lepton Photon Symposium 2003 46
χ2 against αS(M2Z) for CTEQ for two choices of defintion of NLO coupling.
Lepton Photon Symposium 2003 47
CTEQ6 fit to D0 jet data.
Lepton Photon Symposium 2003 48
Variation in CTEQ6 gluon along most sensitive eigenvalue direction.
Lepton Photon Symposium 2003 49
Variation in CTEQ6 jet predictions for variations in each of the 20 eigenvectordirections.
Lepton Photon Symposium 2003 50
Variation in χ2 against [S−] for NuTeV dimuon data (red) and all data sensitive tostrangeness asymmetry (blue).
-0.2 0 0.2 0.4@S-D x 100
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
Χ2�Χ B2
Lepton Photon Symposium 2003 51