Parton

Distributionsand

their

Uncertainties

JonPumplin

DPF2002{W

illiamfburg5/25/02

CTEQ6PDFanalysis(J.Pumplin,D.Stump,W.K.Tung,

J.Huston,H.Lai,P.Nadolsky[hep-ph/0201195])

•

includenew

datasets

•

includecorrelatedsystematicexperimentalerrors

•

evaluateuncertaintiesoftheresult:

{

EigenvectorPDFsetstomapuncertainties

{

Lagrangemultiplierresults

•

UniversalPDFinterface:LesHouchesAccord

•

Results:

{

W

andZproduction

{

parton-partonluminosities

{

gluonandquarkdistributions

•

Measuresofuncertainty

{

Measurementof

αs

{

Statisticalbootstraps

•

Outlook

1

Overview

ofQCD

GlobalAnalysis

FA

A

H

a

fA aA

Fa ^

H

Experim

entalInput

Parton D

istributions:N

onperturbative parametrization at Q

0D

GLA

P E

volution to Q

Hard S

cattering:(P

erturbativelyC

alculable)=

FλA

(x,mQ,MQ

)=

∑afaA

(x,mµ

)⊗

^Fλa

(x,Qµ,MQ

)+

O

((�Q

)2)

Sourcesofuncertainty:

1.Experimentalerrorsincludedin

χ

2

2.Unknownexperimentalerrors

3.Parametrizationdependence

4.Higher-ordercorrections&

LargeLogarithms

5.PowerLaw

corrections(\highertwist")

FundamentaldiÆculties:

1.Goodexperimentsrununtilsystematicerrors

dominate;andthemagnitudeofsystematic

errorsinvolvesguesswork.

2.Systematicerrorsofthetheoryandtheir

correlationscannotevenbeguessed.

2

*, W, Zq

gq

qq

*, W, Z

g qqq

*, W, Z

(dir)

DIS

DY

Dir.P

h.

Jet Inc.

e N N N N

p N Nk Np N

p N Nk Np Np p

SLA

CB

CD

MS

NM

C, E

665H

1, ZE

US

CD

HS, C

HA

RM

CC

FR

CH

OR

US

E605, E

772N

A51

E866

CD

F, D0

WA

70, UA

6E

706

CD

F, D0

CD

F, D0

Experim

ental Input

3

Kinematicregioncoveredbydata

A

widevarietyofdataaretiedtogetherbythe

DGLAP

renormalizationgroupevolutionequation.

Consistency{orlackthereof{betweenthe

experimentscanbeobservedonlybyapplyingQCD

totiethem

togetherinaglobal�t.

Allexperimentsthatusehadronsintheinitialstate

{Tevatron,LHC,andnon-acceleratorexperiments{

requirethepartondistributionsfortheiranalysis.4

SelectionofData

CTEQ5

CTEQ6

#

sys

#

sys

BCDMS

µp

168

no

BCDMS

µp

339

yes

BCDMS

µd

156

no

BCDMS

µd

251

yes

H1

ep172

no

H1a

ep•

104

yes

H1b

ep•

126

yes

ZEUS

ep

186

no

ZEUS

ep•

229

yes

NMC

µp

104

no

NMC

µp

201

yes

NMC

µp/µn

123

no

NMC

µp/µn

123

yes

CCFR

F

2

νN

87

no

CCFR

F

2

νN

159

yes

CCFR

F

3

νN

87

no

CCFR

F

3

νN

87

no

E605

pp

DY

119

no

E605

pp

119

no

NA51

pd/pp

DY

1

no

NA51

pd/pp

1

no

E866

pd/pp

DY

15

no

E866

pd/pp

15

no

CDF

W

11

no

CDF

W

11

no

CDF

jet

33

yes

CDF

jet

33

yes

D�jet

24

yes

D�Jet

•90

yes

New

Data

(Directphotondataarenotusedbecauseof

uncontrolledsystematic\

kT

"e�ects,whichneed

resummation)

5

CTEQ6Globalanalysis

Inputfrom

Experiment:

•∼

2000datapointswith

Q>

2GeV

from

e

,

µ

,

ν

DIS;leptonpairproduction(DY);lepton

asymmetryin

W

production;high

pT

inclusive

jets;

αs (M

Z

)from

LEP

Inputfrom

Theory:

•

NLO

QCD

evolutionandhardscattering

•

Parametrizeat

Q

0:

A

0

xA

1

(1 −

x

)

A

2

(1+

A

3

xA

4)

•s

=

�s

=

0

.4(�u

+

�d)/

2at

Q

0;nointrinsic

b

or

c

Constructe�ective

χ

2global= ∑

expts

χ

2n

:

•χ

2globalincludestheknownsystematicerrors

•

Minimizing

χ

2globalyields\BestFit"PDFs.

•

Variationof

χ

2globalinneighborhoodofthe

minimum

de�nesuncertaintylimits.

•

Estimateuncertaintyasregionofparameter

spacewhere

χ

2

<χ

2(BestFit)+

T

2

with

T≈

10.

(Quitedi�erentfrom

Gaussianstatisticsbecauseof

unknowncorrelatedsystematicerrorsintheoryand

experiments{asmeasuredbyinconsistencybetween

experiments).

6

CommentonParametrization

For

d

val ,

uval ,or

g

,weuse

xf

(x,Q

0)=

A0

xA

1

(1 −x

)

A

2

eA

3

x

(1+

eA

4x

)

A

5

Thiscorrespondsto

ddx

ln( x

f

)=

A1

x−

A

2

1−x

+

c

3+

c

4

x

1+

c

5

x

i.e.,weadda1:1Pad�eform

tothesingulartermsof

thetraditional

A

0

xA

1

(1 −x

)

A

2

parametrization.

A

suÆciently

exibleparametrizationisimportant;

butforconvergence,theremustnotbetoomany

\

atdirections."Forthatreason,someofthe

parametersarefrozenforsome

avors.

(TomeasureasetofcontinuousPDFfunctionsat

Q

0

onthe

basisofa�nitesetofdatapointswouldappeartobean

ill-posedmathematicalproblem.However,thisdiÆcultyisnot

sosevereasmightbeexpectedsincetheactualpredictionsof

interestthatarebasedonthePDFsarediscretequantities.In

particular,�ne-scalestructurein

x

inthePDFsat

Q

0

tendto

besmoothedoutbyevolutionin

Q

.Theycorrespondto

at

directionsin

χ

2

space,sotheyarenotaccuratelymeasured;but

theyhavelittlee�ectontheapplicationsofinterest.)

7

MSU/CTEQ

uncertaintymethodsa

i

aj

2 - contours

2-dim illustration of the

neighborhood of the globalm

inimum

in the 16-dim parton

parameter space

LX

...

•

HessianMatrixMethod:

eigenvectorsof

errormatrixyield40sets

{S ±i }

thataredisplaced

\up"or\down"by�

χ

2

=

100from

thebest�t.

Geterrorbysum

ofsquaresandconstruct

extremePDFsforanyobservable;orsimplylook

atextremesfrom

the40sets.

•

LagrangeMultiplierMethod:

Track

χ

2

as

functionof

F

(e.g.

σW

)byminimizing

χ

2

+

λF

.

Yieldsspecial-purposePDFsthatgiveextremes

of

σW

,or〈y〉

forrapiditydistributionof

W

,or

σfor

t �t

production;or

σt

�t ( √s

=

14TeV)/

σt

�t ( √s

=

2TeV),or

MW

mass

measurementerror,...

8

Hessian(ErrorMatrix)method

Classicalerrorformulae

�

χ

2

=∑ij

(ai −

a

(0)

i

)(H

)

ij (aj −

a

(0)

j

)

(�

F

)2=

�

χ2 ∑ij

∂F

∂ai

(H−

1)

ij∂F

∂aj

Hessianmatrix

H

isinverseoferrormatrix.

Directapplicationfailsbecauseofextreme

di�erencesinvariationof

χ

2fordi�erentdirections

inthespaceof�ttingparameters(\steep"and

\

at"directions),asshownbyahugerangeof

eigenvaluesof

H

:

Eigenvalues of H

essian matrix

9

Convergenceproblemsaresolvedbyaniterative

methodthat�ndsandrescalestheeigenvectorsof

H

,leadingtoadiagonalform

�

χ

2

=

∑i

z

2i

(�

F

)2

= ∑i (

F

(S

(+)

i

)

−F

(S

(−

)

i

) )

2

where

S

(+)

i

and

S

(+)

i

arePDFsetsthataredisplaced

alongtheeigenvectordirections.Theiterative

procedureisavailableinFORTRAN

at

http://www.pa.msu.edu/ ∼

pumplin/iterate/

10

χ

2

andSystematicErrors

Thesimplestde�nition

χ

20

=

N∑i

=

1

( Di −

Ti )

2σ

2i

D

i

=

data

Ti

=

theory

σi

=

\expt.error"

isoptimalforrandom

Gaussianerrors,

Di

=

Ti

+

σi r

i

with

P

(r

)=

e −r

2/

2

√

2

π.

Withsystematicerrors,

Di

=

Ti (a

)+

αi r

stat,i

+

K∑k

=

1

rk β

ki .

The�ttingparametersare

{aλ }

(theoreticalmodel)and

{rk }

(correctionsforsystematicerrors).

Publishedexperimentalerrors:

•αi

isthe`standarddeviation'oftherandom

uncorrelated

error.

•βki

isthe`standarddeviation'ofthe

k

th(completely

correlated!)systematicerroron

Di .

11

Totakeintoaccountthesystematicerrors,wede�ne

χ′2( a

λ ,rk )

=

N∑i

=

1 (Di − ∑

krk β

ki −

Ti )

2

α

2i

+ ∑k

r

2k ,

andminimizewithrespectto

{rk }

.Theresultis

r̂k

= ∑k′ (A

−

1 )kk′ B

k′,

(systematicshift)

where

Akk′

=

δkk′+

N∑i

=

1

βki β

k′i

α

2i

Bk

=

N∑i

=

1

βki ( D

i −Ti )

α

2i

.

The

r̂k 's

dependonthePDFmodelparameters

{aλ }

.Wecan

solveforthem

explicitlysincethedependenceisquadratic.

Wethenminimizetheremaining

χ

2( a

)withrespecttothe

modelparameters

{aλ }

.

•{aλ }

determine

fi (x

,Q

20 ).

•{r̂k }

arearetheoptimal\corrections"forsystematic

errors;i.e.,systematicshiftstobeappliedtothedata

pointstobringthedatafrom

di�erentexperimentsinto

compatibility,withintheframeworkofthetheoretical

model.

12

ComparisontoData

ComparisonoftheCTEQ6M

�ttodatawith

correlatedsystematicerrors.

dataset

Ne

χ

2eχ

2e/N

e

BCDMSp

339

377.6

1.114

BCDMSd

251

279.7

1.114

H1a

104

98.59

0.948

H1b

126

129.1

1.024

ZEUS

229

262.6

1.147

NMC

F2p

201

304.9

1.517

NMC

F2d/p

123

111.8

0.909

D�

jet

90

69.0

0.766

CDFjet

33

48.57

1.472

Otherdatasets:

CCFR

ν

DIS

(150/156)

E605

Drell-Yan

(95/119)

E866

Drell-Yan

(6/15)

CDF

W-leptonasymmetry

(10/11)

13

CTEQ6M

�ttoZEUS

dataatlow

x

510

50100

5001000

Q2�G

eV2�

1

1.5 2

2.5

F2�x,Q2��offset

x�0.000161

x�0.000253

x�0.0004

x�0.000632 x�

0.0008

x�0.00102

x�0.0013

x�0.00161

x�0.0021

x�0.00253

x�0.0032

x�0.005

x�0.008

ZE

US

datalow

xvalues

Thedatapointsincludetheestimatedcorrections

forsystematicerrors.Thatistosay,thecentralvalues

plottedhavebeenshiftedbyanamountthatisconsistentwith

theestimatedsystematicerrors,wherethesystematicerror

parametersaredeterminedusingotherexperimentsviathe

global�t.

Theerrorbarsarestatisticalerrorsonly.

14

CTEQ6M

�ttoZEUS

dataathigh

x

10100

100010000

Q2�G

eV2�

0

0.25

0.5

0.75 1

1.25

1.5

1.75

F2�x,Q2��offset

x�0.013x�

0.021

x�0.032

x�0.05

x�0.08

x�0.13

x�0.18

x�0.25

x�0.4

x�0.65

ZE

US

datahigh

xvalues

Thedatapointsincludetheestimatedcorrections

forsystematicerrors.

Theerrorbarsarestatisticalerrorsonly.

15

�4

�2

02

4�

i

0 20 40 60 80

100

N

ZE

US

(a)Histogram

ofresidualsfortheZEUSdata.The

curveisaGaussianofwidth1.

�4

�2

02

4�D

i �T

i ��Αi

0 20 40 60 80

N

ZE

US

(b)A

similarcomparisonbutwithoutthecorrections

forsystematicerrorsonthedatapoints.

16

�4

�2

02

4�

i

0 20 40 60 80N

NM

CF2p

(a)Histogram

ofresidualsfortheNMC

data.

�4

�2

02

4�D

i �T

i ��Αi

0 20 40 60 80

N

NM

CF2p

(b)A

similarcomparisonbutwithoutthecorrections

forsystematicerrorsonthedatapoints.

17

�2

�1

01

23

r �

1 2 3 4

1 2 3 4Z

EU

Sshifts

SystematicshiftsfortheZEUSdata

(10systematicerrors)

�2

�1

01

23

r �

1 2 3 4

1 2 3 4N

MC

shifts

SystematicshiftsfortheNMC

data

(11systematicerrors)

18

CDF

inclusivejetcrosssection

-50 0 50

100(D-T)/T

CT

EQ

4M

Statistical E

rrors o

nly

-50 0 50

100C

TE

Q4H

J

-50 0 50

100

50100

150200

250300

350400G

eVJet T

ransverse E

nerg

y

MR

ST

100200

300400

pT�G

eV�

�0.2

�0.1 0

0.1

0.2

0.3

0.4�corrected

data�

theory��theory

CD

Finclusive

jet

Recallthattheseinclusivejetcrosssectionmeasurements

providedthe�rstmajorstimulustothestudyofPDF

uncertainties{inparticular,theuncertaintiesassociatedwith

choicesmadeintheform

ofparametrizationsat

Q

0.

19

CDF

Inclusivejets{systematicerrors

-20 0 20(a) H

igh

PT H

adro

n resp

on

se

-20 0 20(b

) Lo

w P

T Had

ron

respo

nse

-20 0 20(c) E

nerg

y Scale S

tability

-20 0 20(d

) Frag

men

tation

-20 0 20(e) U

nd

erlying

Even

t

-20 0 20(f) N

eutral P

ion

Resp

on

se

-20 0 20

100200

300400

(g)C

alorim

eter Reso

lutio

n

Percentage change in cross section

-20 0 20

100200

300400

(h) N

orm

alization

Tran

sverse En

ergy (G

eV)

kr̂k

1

−

0

.511

2

0

.816

3

0

.022

4

1

.347

5

−

1

.307

6

0

.089

7

−

0

.222

8

xxx

20

W

rapiditydistributions

Ourmethodsallow

ustocalculatetheextreme

predictionsduetoPDF

uncertaintyforwhatever

quantityisofexperimentalinterest.

Forexample,extremesof

σW

,〈y〉,〈y

2〉

for

W

productionatFNAL

{relevantfor

MW

measurement:

Samecurvesaftersubtractingcentralvalues...

21

Uncertaintyofthegluondistribution

Uncertaintybands(envelopeofpossible�ts)forthe

gluondistributionat

Q

2

=

10GeV2.

Thecurvescorrespondto

CTEQ5M1(solid)

CTEQ5HJ(dashed)

MRST2001(dotted)

Ironically,thedi�erencesbetweentheseis

comparabletotheestimateduncertainty!

Theuncertaintiesofquarkdistributions(notshown)are

smallerthanthisgluonuncertainty,becausetheDIS

measurementsaresensitivetothesquareofthequarkchargein

leadingorder.TheuncertaintiesofallPDFsdecreasewith

increasing

Q

{\convergentevolution"

22

M

easurem

entof

αs

We�ndthattheCTEQ6analysisisnicelyconsistent

withtheWorldAveragedeterminationof

αs (M

Z

).

Butitisnotpreciseenoughtoimprovethatvalue.

0.090.1

0.110.12

0.130.14

0.15Α

s �MZ�

0.090.1

0.110.12

0.130.14

0.15PD

G2000

summ

ary

AverageH

adronicJetse�

e

e�e�

eventshap

Fragmentation

Zw

idthepeventshap

PolarizedD

IS

DIS

taudecays

Lattice

Ydecay

23

χ

2

versus

αS

(MZ

)forindividualdatasetsinCTEQ6

0.1080.118

0.1280 5 10 15 20

CD

Fjet�33�

0.1080.118

0.128

0 2 4 6 8 10C

DFw�11�

0.1080.118

0.128�

2 0 2 4 6E

866�15�

0.1080.118

0.128

0 20 40 60D

0jet�90�

0.1080.118

0.128�

5 0 5 10 15C

CFR

3�87�

0.1080.118

0.1280 5 10 15 20 25

E605�119�

0.1080.118

0.128

0 1 2 3 4N

A51�1�

0.1080.118

0.128

0 10 20 30N

MC

r�123�

0.1080.118

0.128�

5 0 5 10 15N

MC

rx�13�

0.1080.118

0.128

0 10 20 30C

CFR

2�69�

0.1080.118

0.128�

10 0 10 20 30 40H

1b�126�

0.1080.118

0.128�

10 0 10 20 30 40 50Z

EU

S�229�

0.1080.118

0.128

0 20 40 60N

MC

p�201�

0.1080.118

0.128

0 20 40 60B

CD

MSp�339�

0.1080.118

0.128�

20 0 20 40 60B

CD

MSd�251�

0.1080.118

0.128�

5 0 5 10 15 20 25 30H

1a�104�

24

Measurmentof

αS

(MZ

):

Ifassume�

χ2

=

1criterionineachexperiment,the

experimentsareinconsistent.

Ourerrorestimate( T

=

10)is

αS

(MZ

)=

0

.1165

±0

.0065

Thiscorrespondstosomewhatconservative

assumptions{perhapstobethoughtofasan

e�ective\2

σ

"limit.Henceitiscomparabletothe

MRST

limitbasedon

T

=

5.

0.090.1

0.110.12

0.130.14

0.15alpha

S

0.090.1

0.110.12

0.130.14

0.15�Χ

2�1

rangesfrom

GA

BC

DM

Sp

BC

DM

Sd

H1a

H1b

ZE

US

NM

Cp

NM

Cr

CC

FR2

CC

FR3E605CD

Fw

E866

D0jet

CD

Fjet

MR

STvalue

25

SimilarsituationforW

andZ

crosssections

W

henastrict�

χ

2

=

1criterionwasappliedto

self-consistentsubsetsoftheexperiments,the

subsetswerenotconsistentwitheachother.

Thetrueerroristhereforeconsiderablylargerthan

�

χ

2

=

1wouldimply.

26

New

waystomeasureconsistencyof�t

(WorkinprogresswithJohnCollins)

Keyidea:Inadditiontothe

Hypothesis-testingcriterion�

χ

2∼√

2

N

weusethestronger

Parameter-�ttingcriterion�

χ

2∼

1

Theparametersherearerelativeweightsassignedto

variousexperiments,ortoresultsobtainedusing

variousexperimentalmethods.Examples:

•

Plotminimum

χ

2i

vs.

χ

2tot −

χ

2i

,where

χ2i

isone

oftheexperiments,oralldataonnuclei,orall

dataatlow

Q

2,...

or•

PlotbothasfunctionofLagrangemultiplier

u

where

(1−

u

)χ

2i

+

(1+

u

)(χ

2tot −

χ

2i

)isthe

quantityminimized.

Canobtainquantitativeresultsby�ttingtoamodel

withasinglecommonparameter

p

:

χ

2i

=

A

+

(p

sin

θ )

2

⇒p

=

0±

sin

θ

χ

2not

i

=

B

+

(p−

S

cos

θ )

2

⇒p

=

S±

cos

θ

Thesedi�erby

S±

1,i.e.,by

S

\standarddeviations"

27

NM

C D

2/H2

NM

C D

2/H2

S =

2.6

BC

DM

S D

2B

CD

MS

D2

S =

7.6

Fitsto8oftheexperimentsintheCTEQ5analysis

Expt

1

2

3

4

5

6

7

8

S

2

.7

3

.3

3

.3

4

.2

5

.3

7

.6

7

.4

8

.3

tan

φ0

.56

0

.54

0

.99

0

.86

0

.71

1

.14

0

.65

0

.39

28

Application:Uncertaintiesof

luminosityfunctionsatLHC

102

103

√s (GeV

)

Luminosity function at LH

C

Fractional Uncertainty

0.1

0.1

0.1

-0.1

-0.1

-0.2

0.2

Q-Q

--> W

+

Q-Q

--> W

- G-G

000 ≈≈≈ ≈

−−

±

50200

500

^

Notethatonecomponentofthe

uncertaintyinpredictingtheHiggs

productioncrosssectionatLHC

is

anuncertaintyof ∼

8%

duetoPDF

uncertainty.

29

Outlook

•

Partondistributionsoftheprotonareincreasinglywell

measured.

•

Usefultoolsareinplacetoestimatetheuncertaintyof

PDFsandtopropagatethoseuncertaintiestophysical

predictions.

•

TheLesHouchesAccordinterfacemakesiteasytohandle

thelargenumberofPDFsolutionsthatareneededto

characterizeuncertainties.(hep-ph/0204316)

•

Workonre�ningtheknowledgeofthe\Tolerance

Parameter"

T

isunderway

{

Collins&

Pumplin[hep-ph/0105207]

{

Statisticalbootstrapmethods

•

Improvementsinthetreatmentofheavyquarke�ectsare

inprogress.

•

FermilabrunIIdataandHERA

IIdatawillprovidethe

nextmajorexperimentalstepsforward.

PartonDistributionFunctionsareamajoravenue

towardunderstandingthefundamental

nonperturbativephysicsoftheproton.Theyarealso

acrucialprerequisiteforprecisionStandardModel

studiesandNew

Physicssearchesathadroncolliders

andexperimentswithhadrontargets.

30