Progress On Studying Uncertainties Of Parton Distributions And Their Physical Predictions

Brief summary of traditional global QCD analysis What’s uncertain about parton distribution functions?

Studying uncertainties of parton distributions and their physical predictions: Ideal approaches in a world with perfect experiments Progress on a realistic global analysis of uncertainties • General approach and techniques � two complementary methods: (J. Pumplin, D. Stump)

• The Hessian method: orthonormal basis PDF sets, and a general master equation for calculating uncertainties of any quantity dependent on PDFs

• The Lagrange multiplier method: robust and optimized PDFs for quantifying uncertainties of specific physical variables

Results and physical applications • W/Z cross-sections at Tevatron/LHC/RHIC • W rapidity distribution, (W-mass measurement) • Parton Luminosities at Tevatron/LHC/RHIC/VLHC

(Higgs-, top-X-sections, high pT jets, ... etc.) Outlook

Global QCD Analysis in a Nutshell

Master Equation for QCD Parton Model

{ the Factorization Theorem

F�A(x;m

Q;M

Q) =X

a

faA(x;m

�) F̂�a (x;

Q

�;M

Q)+O((

�

Q)2)

FA

A

a

fAa

A

Fa^

H H

Sources of Uncertainties and Challenges:

� Experimental errors;

� Parametrization dependence;

� Higher-order corrections; Large Logarithms;

� Power-law (higher twist) corrections.

ExperimentalInput

Parton Dist. Fn.Non-Perturbative Parametrization at Q0

GLAP Evolution to Q

Hard Cross-sectionperturbative calculable

(may contain snLogn(M/Q))

(and uncertainties on errors!)

*, W, Z

q

g q

q

q

*, W, Z

g

q q

q

*, W, Z

(dir)

DIS

DY

Dir.Ph.

Jet Inc.

e N N

N N

p N N

k Np N

p N N

k Np N

p p

SLACBCDMS

NMC, E665H1, ZEUS

CDHS, CHARMCCFR

CHORUS

E605, E772NA51E866

CDF, D0

WA70, UA6E706

CDF, D0

CDF, D0

Experimental Input: Phyical Processes & Experiments

100

101

102

103

104

1/X

100

101

102

Q (GeV)

DIS

(fix

ed ta

rget

)H

ER

A (

’94)

DY

W-a

sym

met

ryD

irect

-γJe

ts

Th

e k

ine

ma

tic r

an

ge in

th

e (

x, Q

) p

lan

e o

f d

ata

po

ints

incl

ud

ed

in

a typ

ica

l glo

ba

l QC

D a

na

lysi

s –

CT

EQ

5

Global QCD Fit (CTEQ)

* Parametrization of the non-perturbative PDFs:(at Q0 = 1 GeV), e.g.

fi(x, Q0) = ai0x

ai1(1− x)a

i2(1 + ai

3xai4).

* The fitting is done by minimizing a global effec-tive “chi-square” function, χ2global,· overall figure of merit of the fit· regulates trade-off’s between the many expts.

χ2g =∑n

χ2n (n labels the experiments)

χ2n =

(1− Nn

σNn

)2+ wn

∑I

(NnDnI − TnI(a)

σDnI

)2

DnI : data point σDnI: combined error

TnI(a): theory value (dependent on {ai})for the Ith data point in experiment n.

Nn: Normalization para. for expt. n.wn: a “prior” based on physics considerations

(1: inclusion; 0: exclusion; or other).

* This effective χ2 function does not have thefull probabilistic significance of an ideal statisticalanalysis. (cf. uncertainty section.)

x

0

0.2

0.4

0.6

0.81

1.2

x f(x,Q)

10-4

10-3

10-2

10-1

.2.3

.4.5

.6.7

Q =

5 G

eV

.8

Ove

rvie

w o

f Par

ton

Dis

trib

utio

n F

unct

ions

of t

he P

roto

n

CT

EQ

5MG

luon

/ 15

d bar

u bar

s c u v d v (dba

r-u b

ar)

* 5

CTEQ5M �t to E866 :�pd

2�pp(x2)

0.0 0.1 0.2 0.3X2

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Xse

c(pd

)/2X

sec(

pp)

E866CTEQ5M

Fit to CDF W-lepton asymmetry:

0.5 1 1.5 2

Y

-0.1

0

0.1

CD

F W

-lept

on A

sym

met

ry

CTEQ5MCDF data

5010

015

020

025

030

035

040

0p T

(G

eV)

0

0.2

0.4

0.6

0.81

1.2

1.4

Incl

. Jet

: p

t7 * d

σ/dp

t

(Err

or b

ars:

sta

tistic

al o

nly)

14%

< C

orr.

Sys

. Err

. < 2

7%

Rat

io: P

rel.

data

/ N

LO Q

CD

(C

TE

Q5M

| C

TE

Q5H

J)

norm

. fac

or :

CT

EQ

5HJ:

1.04

CT

EQ

5M :

1

.00

CD

F

(10-1

4 nb

GeV

6 )

Dat

a / C

TE

Q5M

CT

EQ

5HJ

/ CT

EQ

5MC

DF

Dat

a (

Pre

l. )

CT

EQ

5HJ

CT

EQ

5M

Co

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ariso

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f C

DF

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oss

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ea

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ithN

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D c

alc

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tion

s w

ith C

TE

Q5

M a

nd

CT

EQ

5H

J P

DF

s

Inclusive Jet Production Data of D0

-- compared to NLO QCD

using CTEQ4HJ (CTEQ4M)

What's Uncertain about PDFs?

Quite a lot!

Strange and anti-strange Quarks

Details in the {u,d} quark sector: Up/Down differences and ratios (important for precision W/Z physics at colliders)

The Gluon Distribution (important for most SM and New Physics processes at very high energies)

Heavy Quark Distributions: Very little is known! Are heavy quarks “radiatively-generated"

exclusively; or are there intrinsic components of charm and/or bottom?

It is important to quantify the uncertainties of the parton distributions functions (PDF’s) and, more importantly, physical predictions which depend on PDF’s – not just the “bands”, but a systematic knowledge of the correlated variations.

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L

Percentage Uncertainties of Parton

Distributions (By M. Botje)

Sources of uncertainties:

� Experimental statistical and systematic errors;

� \input": various theoretical corrections;

� \analysis": phenom. analysis procedure;

� \Scale": PQCD renor. and fact. scale-dependence

Figure 9: The parton momentum densities xg, xS (both divided by a factor of 20),xuv and xdv versus x at Q2 = 10 GeV2. The full curves show the results from the QCDfit with the errors drawn as shaded bands. The dashed curves are from the CTEQ4 [8]parton distribution set.

25

Global Analyses using an Effective χ2 Function

• Stress inclusion of all relevant (global) experi-mental data to constraint the PDFs;• Intend to be logical & useful extension of con-ventional, practical global QCD analyses.

⇒ Use an effective χ2-analysis with function χ2global

Use a collection of DIS experiments

S. Alekhin, hep-ph/0011002

V. Barone, C. Pascaud and F. Zomer,Eur. Phys. J. C12, 243 (2000) [hep-ph/9907512].

M. Botje,Eur. Phys. J. C14, 285 (2000) [hep-ph/9912439].

Use full global analysis data set of CTEQ5,+ some key advances in methodology

J. Pumplin, D. Stump,R. Brock, D. Casey, J. Huston, J. Kalk, WKT

(MSU)

⇒ hep-ph/0008191, 0101032, 0101052 ⇐

For physical variable X:Std.Min. set S0 : X

Alternate hypotheses sets {Si

±} i=1, ..2n: X

Bird's Eye View of the MSU Study(Pumplin, Stump, WKT, et al.)

Choice of priors:NLO QCD ;

other theory models;Parametrization of

nonperturbative PDFs.

Inputs:

Output:

Experimental Inputs:(i) full global data set of CTEQ5;(ii) effective global 2 function:

"Sampling"2 complementary optimized systematic methods:

ai

aj

2 - contours

Error Estimates2 distinct steps:(i) calc. CL in each expt.;(ii) estimate overall uncertainty.

(i) Lagrange multiplier;(ii) Hessian matrix.

2-dim illustration of the neighborhood of the globalminimum in the 16-dim parton parameter space

LX

Comparison of data and CTEQ5m (\theory")

-4 -2 0 2 4x=Hm-tL�Σ

0

200

400

600

800

dN�d

x

all data

The histogram includes all data used in the �t exceptjet production.

The curve has no adjustable parameters; it's just

N exp(�x2)=p2�

where N is the number of data points. The area underthe curve (or histogram) is N .

Di�erences mi�ti are within the published measurementerrors.

At least globally the distribution of uctuations is Gaus-

sian with the right width.

First Test of whether the effective global 2 method makesany sense: look at the distribution of the fluctuations of

the 1300 data points included in the global fit.

Lagrange Multiplier Method: (specific but robust)Let X be a physical quantity, then minimize Ψ =χ2g + λX to probe the neighborhood of the mini-mum.

ai

2-dim (i,j) renderingof d-dim PDF parameter space

contours of c2global

LM method

MC sampling

X: physicsvariable

LX

aj

• Sample variation ofχ2g as a function of X

by {SmX}, m = 1,2 . . .;

• Obtain PDF sets{S±

X} which extremizethe variation of X fora given tolerance ∆χ2

of χ2global.21 21.5 22 22.5

W nb

1200

1230

1260

1290

1320

1350

2gl

obal

W production at the Tevatron

W

T2

• Uncertainty of X: ∆X = 12(X(S

+X )− X(S−

X)),Apply to, e.g. σW/Z, σTop, σHiggs, . . . , at the Teva-

tron, RHIC and LHC.

Error Estimate: 1. Use all information provided by each individual experi-ments to calculate the (90%) CL w.r.t. that experiment.Example, H1 expt. (with full correlated error matrix):

21 21.5 22 22.5W nb Tevatron

0.98

1.02

1.06

1.1

H1

2N

This 2 functiondoes have statis-tical significance!The dashed hori-zontal line is the90% CL level.The error bar w.r.t.the H1 expt. is shownby the red line with arrows.

2. Make the 90% error bar calc. for all expts.

Tevatron

Wnb

20

21

22

23

24

25

BC

DM

Sp

BC

DM

Sd

H1

ZE

US

NM

Cp

NM

Cr

NM

Crx

CC

FR2

CC

FR3

E60

5

NA

51

CD

Fw

E86

6

D0j

et

CD

Fjet

W

In view of the problem with interpretation of the absolute 2 for some real experiments, use relative w.r.t. the best estimate solution S0.

Optimized Sampling of PDF Parameter Space– “Alternate Hypotheses” in uncertainty study

Hessian Matrix: (general but approx.)Diagonalize the Hessian matrix calculated from χ2g ,then move along each of the eigenvectors, i, to getup/down PDF sets {S±

i }.

(a)Original parameter basis

(b)Orthonormal eigenvector basis

zk

Tdiagonalization and

rescaling bythe iterative method

ul

ai

2-dim (i,j) rendition of d-dim (~16) PDF parameter space

Hessian eigenvector basis sets

ajul

p(i)

s0s0

contours of constant 2global

ul: eigenvector in the l-direction p(i): point of largest ai with tolerance T

s0: global minimum p(i)zl

· ⇒ PDF sets {S±i }, i = 1, . . . , n, spanning the

full PDF parameter space in the neighborhood ofthe global mimimum;

· The uncertainty of any physical variable X canbe calculated as: (the master eq.)

∆X =T

2t

√∑i

(X(S+i )− X(S−i ))

2

7KH�+HVVLDQ�0HWKRG�RI�TXDQWLI\LQJ�XQFHUWDLQWLHV�E\�D�FRPSOHWH�VHW�RI�RUWKRQRUPDO HLJHQYHFWRU�3')V

Global landscape:- physics* gradients vary widely:steep / flatness measured by eigen values of Hessian: ~ 106

Local bumpiness: unphysical various reasons: theory model is NOT smooth!;finite steps, MC methods, ...

Challenges to Numerical Calculation of the Hessian Matrix

in 16-dimensional PDF parameter space

ai

aj

It is well known that general purpose minimization programs do not provide reliable error estimates for global analysis. (Step size choices not designed to deal with these problems)

Eigenvalues of the Hessian Matrixfor two different choices of parametrization

of the PDF parameter space

Iterative Method to generate Eigenvectors:(and dramatically improve numerical reliability)

ai0

aj0

Physical parameters

aiN = zi

ajN = zj

Nth iteration:Eigenvectororthonormalbasis

ain

ajn

nth iteration

the χ2 = const. ellipsoid

jijij

i aXH

aXX

∂∂

�∂∂∆=∆ − )( 122 χ

2222 )]()([)( −+� −=� ∂

∂∆=∆ ii iii

SXSXzXX χ

Si+Si

−

Sj+

Sj−

* This iterative method has been shown by Botje and Zomer to greatly improve their own global analyses, and it is now being considered for adoption in MINUIT (as an option) by its author F. James.

Considerations in Estimating the Tolerance T of χglobal2

(∆χglobal2 < T2)

• ∆χglobal2 = 1 has no statistical significance in this

context. Most experiments only give a combined (“effective”) systematic error; Combining χ2 ’s of 15 experiments on very diverse processes and accuracies is an extremely dicey business.

• Basic assumption of 15 acceptable experiments, in spite of the fact that some are nominally inconsistent in the strict statistical sense, implies T >> 1.

• Basic assumption of compatibility of the 15 experiments, in spite of the fact that some are nominally incompatible in the strict statistical sense, implies T >> 1. We can “measure” the acceptable T for compatibility.

• Quantitative estimate of T from comparison of sample PDFs (the Alternate Hypotheses) obtained from Hessian and Lagrange methods to the individual experiments. o Evaluate the 90% CL range for each experiment based

on available error information (using χ2 normalized to standard fit, if necessary)

o Combine these ranges into one estimated tolerance for Τ Bottom Line:

10 < T < 15

Two

ext

rem

e p

art

on

dis

trib

utio

ns

with

T =

10

at Q

= 2

Ge

V a

nd

Q

= 1

00

G

eV

(in

da

she

d li

ne

s). T

he

un

cert

ain

ty b

an

d fo

rT

< 1

0 is

sh

ad

ed

.(T

de

fine

s th

e tole

ranc

e:

2 glo

ba

l < T

2)

Glu

onU

p qu

ark

Ra

tio o

f p

art

on

dis

trib

utio

n to

Be

st F

it d

istr

ibu

tion

for

two

ext

rem

e T

= 1

0 c

ase

s a

t Q

= 1

0 G

eV.

Th

e r

egio

n a

llow

ed

by

T <

10

is s

ha

de

d.

Glu

on

Up

-qu

ark

Predicted rapidity distribution for W production at the Tevatron

Six curves represent the alternate hypotheses of one± 2 for the physical variables t, <y>, and <y2>.

Predicted rapidity distribution for W production at the Tevatron-- deviation from the best estimate solution S0

Six curves represent the alternate hypotheses of one± 2 for the physical variables t, <y>, and <y2>.

Predicted rapidity distribution for W production at the Tevatron-- deviation from the best estimate solution S0

Three (of the six) curves for each case representing the"extremes" for the physical variables t, <y>, and <y2>.

Compare the results obtained by theLagrange and Hessian methods

Since the LM method is fully robust (no approx.), the agreement proves the efficacy of the the Hessian method.

Correlated uncertainties in W/Z productionCross-section at the Tevatron

CDF

D0

Ellipse : Allowed region with Tolerance = 10.

** CDF with the same luminosity input as D0

**

102 103

Sqrt (s^)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fra

ctio

nal U

ncer

tain

ties

at Tevatron Run IIFractional Luminosity Uncertainties

W, Z

GGQQbar γQQbar W+

QQbar ZGQ γGQ W-

102

Sqrt (s^)

0

0.1

0.2

0.3

Fra

ctio

nal U

ncer

tain

ties

at Tevatron Run IIFractional Luminosity Uncertainties

W, Z 115 GeV

GGQQbar γQQbar W+

QQbar ZGQ γGQ W-

102 103

sqrt (s^)

0

0.05

0.1

0.15

0.2

0.25

Fra

ctio

nal U

ncer

tain

ties

W, Z 115 GeV

Luminosity Uncertainties at LHC

GGQQbar γQQbar W+

GQ W+

GQ Z

101 102

sqrt (s^)

0

0.1

0.2

0.3

0.4

0.5

0.6

Fra

ctio

nal U

ncer

tain

ties

W, Z

Luminosity Uncertainties at RHIC 500 GeV

GGQQbar γQQbar W+

GQ W+

GQ W-

101 102

sqrt (s^)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fra

ctio

nal U

ncer

tain

ties

W, Z

Luminosity Uncertainties at RHIC 200 GeV

GGQQbar γQQbar W+

QQbar W-

GQ γGQ W+

Outlook

This is only the very beginning of studying uncertainties in global QCD analysis in a quantitative manner It should be regarded more as a demonstration of

principles. There is a lot of room for collaboration among

theorists and experimentalists

Several data sets are about to be updated (CCFR, H1, ZEUS, ... ) � Specific results of this study will likely change soon.

Many other sources of uncertainties in the overall global analysis have not yet been incorporated: Theoretical uncertainties due to higher-order PQCD

corrections and resummation; Uncertainties introduced by the choice of parametrization

(This has been partially explored by us.) The Hessian eigenvectors provide a systematic way to distinguish “flat/steep” directions in parameter space; and to suggest ways to improve the choice of parametrization.

Continued progress in this venture is of vital importance for our understanding of the parton structure of hadrons (fundamental physics of its own right), for precision SM physics studies at future colliders, and for New Physics searches.