parton distribution functions - fermilabis a pp„collider, the lhc (large hadron collider) at cern...

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)


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.


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.


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.


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.


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.


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.


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).


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).


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.


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.


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.


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


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

qq

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.


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.)


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.


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.


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.


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

±


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.


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.


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%.


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


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.


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.


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.


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


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).


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.


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.


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.


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.


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.


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).


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.


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.


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.


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.


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.


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.


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.


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


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


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

)


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

)


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

)



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


H1 set of parton parameters from GKKapproach. Red curve Gaussian approxand blue line MRST value. Green curvefor αS is LEP result.


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


χ2 against αS(M2Z) for CTEQ for two choices of defintion of NLO coupling.


CTEQ6 fit to D0 jet data.


Variation in CTEQ6 gluon along most sensitive eigenvalue direction.


Variation in CTEQ6 jet predictions for variations in each of the 20 eigenvectordirections.


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


parton distribution functions - fermilabis a pp„collider, the lhc (large hadron collider) at cern...

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