uncertainties of parton distribution functions

Sept 2003 Phystat 1

Uncertainties of Parton Distribution Functions

Daniel Stump

Michigan State University&

CTEQ

Sept 2003 Phystat 2

High energy particles interact through their quark and gluon constituents – the partons.

Asymptotic freedom : the parton cross sections can be approximated by perturbation theory.

Factorization theorem : Parton distribution functions in the nucleon are the link between the PQCD theory and measurements on nucleons.

Sept 2003 Phystat 3

Parton distribution functions are important.

Sept 2003 Phystat 4

The goals of QCD global analysis are

• to find accurate PDF’s;

• to know the uncertainties of the PDF’s;

• to enable predictions, including uncertainties.

Sept 2003 Phystat 5

The systematic study of uncertainties of PDF’s developed slowly. Pioneers…

J. Collins and D. Soper, CTEQ Note 94/01, hep-ph/9411214.

C. Pascaud and F. Zomer, LAL-95-05.

M. Botje, Eur. Phys. J. C 14, 285 (2000).

Today many groups and individuals are involved in this research.

Sept 2003 Phystat 6

CTEQ group at Michigan State (J. Pumplin, D. Stump, WK. Tung,

HL. Lai, P. Nadolsky, J. Huston, R. Brock) and others (J. Collins, S. Kuhlmann, F. Olness, J. Owens)

MRST group (A. Martin, R. Roberts, J. Stirling, R. Thorne)

Fermilab group (W. Giele, S. Keller, D. Kosower)

S. I. Alekhin

V. Barone, C. Pascaud, F. Zomer; add B. Portheault

HERA collaborationsZEUS – S. Chekanov et al; A. Cooper-SarkarH1 – C. Adloff et al

Current research on PDF uncertainties

Sept 2003 Phystat 7

Outline of this talk (focusing on CTEQ results)

• General comments; CTEQ6• Our treatment of experimental systematic errors• Compatibility of data sets• Uncertainty analysis• 2 case studies inclusive jet production in ppbar or pp strangeness asymmetry

Sept 2003 Phystat 8

… of short-distance processes using perturbative QCD (NLO)

The challenge of Global Analysis is to construct a set of PDF’s with good agreement between data and theory, for many disparate experiments.

Global Analysis

Sept 2003 Phystat 9

The program of Global Analysis is not a routine statistical analysis, because of systematic differences between experiments.

We must sometimes use physics judgement in this complex real-world problem.

Sept 2003 Phystat 10

Parametrization

At low Q0 , of order 1 GeV,

)( )1( ),( 2100 xPxxaQxf aa

P(x) has a few more parameters for increased flexibility.

~ 20 free shape parameters

Q dependence of f(x,Q) is obtained by solving the QCD evolution equations (DGLAP).


CTEQ6 -- Table of experimental data sets

H1 (a) 96/97 low-x e+p data ZEUS 96/97 e+p dataH1 (b) 98/99 high-Q e-p data D0 : d2/d dpT


Global Analysisdata from many disparate experiments


The Parton Distribution Functions


Different ways to plot the parton distributions

Linear

Logarithmic

Q2 = 10 (solid) and 1000 (dashed) GeV2


In order to show the large and small x regions simultaneously, we plot 3x5/3 f(x) versus x1/3. {Integral = momentum fraction}


Comparison of CTEQ6 and MRST2002

blue curves : CTEQ6Mblack dots : MRTS2002

gluon and u quark atQ2 = 10 GeV2


Our treatment of systematic errors


What is a systematic error?

“This is why people are so frightened of systematic errors, and most other textbooks avoid the subject altogether. You never know whether you have got them and can never be sure that you have not – like an insidious disease…

The good news, however, is that despite popular prejudices and superstitions, once you know what your systematic errors are, they can be handled with standard statistical methods.”

R. J. BarlowStatistics


Imagine that two experimental groups have measured a quantity , with the results shown.

OK, what is the value of ?

This is very analogous to what happens in global analysis of PDF’s. But in the case of PDF’s the systematic differences are only visible through the PDF’s.


We use 2 minimization with fitting of systematic errors.

Minimize 2 w. r. t. {a} optimal parameter values {a0}.

All this would be based on the assumption that

Di = Ti(a0) + i ri

drre

dP 2

2/2

For statistical errors define

error lstatistica :

valueal theoretic:

valuedata :

1

2

22

i

i

iN

i i

ii T

DTD

Ti = Ti(a1, a2, ..,, ad) a function of d theory parameters

(S. D.)


Treatment of the normalization error

N

i i

ii TDfffa

12

2N

2

norm

NN

2 1),(

In scattering experiments there is an overall normalization uncertainty from uncertainty of the luminosity. We define

where fN = overall normalization factor

Minimize 2 w. r. t. both {a} and fN.


A method for general systematic errors

j

K

jijiiii rraTD ˆ)(

10

K

jj

N

i i

j ijijis

TsD

1

2

12

2

2

Minimize 2 with respect to both shape parameters {a} and optimized systematic shifts {sj}.

quadratic penalty term

i : statistical error of Di

ij : set of systematic errors (j=1…K) of Di

Define


sexperiment

022

global )(, )( asaa

and minimize w.r.t {a}.

The systematic shifts {sj} are continually optimized[ s s0(a) ]

Because 2 depends quadratically on {sj} we can solve for the systematic shifts analytically, s s0(a).Then let,


So, we have accounted for …• Statistical errors• Overall normalization uncertainty (by fitting {fN,e})• Other systematic errors (analytically)

We may make further refinements of the fit with weighting factors

Default : we and wN,e = 1

The spirit of global analysis is compromise – the PDF’s should fit all data sets satisfactorily.If the default leaves some experiments unsatisfied, we may be willing to reduce the quality of fit to some experiments in order to fit better another experiment. (We use this sparingly!)

e e

ee

eee

fwfawfa

2N,

2

,NN2

N2global

)1(}){},({}){},({


How well does this fitting procedure work?

Quality


Comparison of the CTEQ6M fit to the H1 data in separate x bins. The data points include optimized shifts for systematic errors. The error bars are statistical only.


Comparison of the CTEQ6M fit to the inclusive jet data. (a) D0 cross section versus pT for 5 rapidity bins; (b) CDF cross section for central rapidity.


How large are the optimized normalization factors?

Expt f N

BCDMS 0.976H1 (a) 1.010H1 (b) 0.988ZEUS 0.997NMC 1.011CCFR 1.020E605 0.950D0 0.974CDF 1.004


We must always check that the systematic shifts are not unreasonably large.

j sj1 1.672 -0.673 -1.254 -0.445 0.006 -1.077 1.288 0.629 -0.40

10 0.21

j sj1 0.672 -0.813 -0.354 0.255 0.056 0.707 -0.318 1.059 0.61

10 0.2611 0.22

10 systematic shifts NMC data

11 systematic shifts ZEUS data


Comparison to NMC F2iiiii TD /)( βs

without systematic shifts


A study of compatibility


The PDF’s are not exactly CTEQ6 but very close –a no-name generic set of PDF’s for illustration purposes.

Table of Data Sets1 BCDMS F2p 339 366.1 1.082 BCDMS F2d 251 273.6 1.093 H1 (a) 104 97.8 0.944 H1 (b) 126 127.3 1.015 H1 (c ) 129 108.9 0.846 ZEUS 229 261.1 1.147 CDHSW F2 85 65.6 0.778 NMC F2p 201 295.5 1.479 NMC d/p 123 115.4 0.9410 CCFR F2 69 84.9 1.23

11 E605 119 94.7 0.8012 E866 pp 184 239.2 1.3013 E866 d/p 15 5.0 0.3314 D0 jet 90 62.6 0.7015 CDF jet 33 56.1 1.7016 CDHSW F3 96 76.4 0.8017 CCFR F3 87 26.8 0.3118 CDF W Lasy 11 8.7 0.79

N 2 2/N

Ntot = 2291

2global

= 2368.


The effect of setting all normalization constants to 1.

1 BCDMS F2p 186.52 BCDMS F2d 27.63 H1 (a) 7.34 H1 (b) 10.15 H1 (c ) 24.08 NMC F2p 4.011 E605 13.312 E866 pp 95.7

2

2 (opt. norm) = 2368.

2 (norm 1) = 2742.

2 = 374.0


Example 1. The effect of giving the CCFR F2 data set a heavy weight.

By applying weighting factors in the fitting function, we can test the “compatibility” of disparate data sets.

3 H1 (a) 8.3

7 CDHSW F2 6.3

8 NMC F2p 18.1

10 CCFR F2 19.7

12 E866 pp 5.5

14 D0 jet 23.5

2

2 (CCFR) = 19.72 (other) = +63.3

Giving a single data set a large weight is tantamount to determining the PDF’s from that data set alone. The result is a significant improvement for that data set but which does not fit the others.


Example 1b. The effect of giving the CCFR F2 data weight 0, i.e., removing the data set from the global analysis.

3 H1 (a) 8.3

6 ZEUS 6.9

8 NMC F2p 10.1

10 CCFR F2 40.0

2

2 (CCFR) = +40.0

2 (other) = 17.4

Imagine starting with the other data sets, not including CCFR. The result of adding CCFR is that 2

global of the other sets increases by 17.4 ; this must be an acceptable increase of 2 .


2 BCDMS F2d 15.13 H1 (a) 12.44 H1 (b) 4.36 ZEUS 27.57 CDHSW F2 19.28 NMC F2p 8.0

10 CCFR F2 54.514 D0 jet 22.016 CDHSW F3 11.017 CCFR F3 5.9

Example 5. Giving heavy weight to H1 and BCDMS

2 for all data sets 2

2(H & B) = 38.72(other) = +149.9


Lessons from these reweighting studies

• Global analysis requires compromises – the PDF model that gives the best fit to one set of data does not give the best fit to others. This is not surprising because there are systematic differences between the experiments.

• The scale of acceptable changes of 2 must be large. Adding a new data set and refitting may increase the 2‘s of other data sets by amounts >> 1.


Clever ways to test the compatibility of disparate data sets

• Plot 2 versus 2

J Collins and J Pumplin (hep-ph/0201195)

• The Bootstrap MethodEfron and Tibshirani, Introduction to the Bootstrap (Chapman&Hall)Chernick, Bootstrap Methods (Wiley)


Uncertainty Analysis

(I) Methods


We continue to use 2global as figure of merit. Explore the

variation of 2global in the neighborhood of the minimum.

0

22

2

1

aa

H

The Hessian method

( = 1 2 3 … d)

a1

a2

the standard fit, minimum 2

nearby points are also acceptable


Classical error formula for a variable X(a)

a

XH

a

XX

1

,

22

Obtain better convergence using eigenvectors of H

2

1

)()(2

d

SXSXX

S(+) and S() denote PDF sets displaced from the

standard set, along the directions of the th

eigenvector, by distance T = (2) in parameter space.(available in the LHAPDF format : 2d alternate sets)

“Master Formula”


Minimization of F [w.r.t {a} and ] gives the best fit

for the value X(a min,) of the variable X.

Hence we obtain a curve of 2global versus X.

The Lagrange Multiplier Method

… for analyzing the uncertainty of PDF-dependent predictions.

The fitting function for constrained fits

aXaaF 2global,

: Lagrange multiplier

controlled by the parameter


The question of tolerance

X : any variable that depends on PDF’s

X0 : the prediction in the standard set

2(X) : curve of constrained fits

For the specified tolerance ( 2 = T2 ) there is a corresponding range of uncertainty, X.

What should we use for T?


Estimation of parameters in Gaussian error analysis would have

T = 1

We do not use this criterion.


Aside: The familiar ideal example

Consider N measurements {i} of a quantity with normal errors {i}

true iii r

Estimate by minimization of 2,

i i

i iiN

i i

i2

2

combined1

2

22

/1

/

)()(

The mean of combined is true , the SD is

2/12/1

iic

.1)()( 22 ccc

The proof of this theorem is straightforward. It does not apply to our problem because of systematic errors.

and

π

edP

/-r

2

22

( = / N )


Add a systematic error to the ideal model…

rr iiii~

true

Estimate by minimization of 2

2

12

22 )(

),( ss

sN

i i

ii

( s : systematic shift, : observable )

Then, letting , again )](,[)( 022 s

. 1)()( 22 ccc

/1

/2

2

combinedii

ii i

2

22

/1

1)(

ii

cand

(for simplicity suppose i =

( = 2/N + 2 )


Reasons• We keep the normalization factors fixed as we vary the point in parameter space. The criterion

2 = 1 requires that the systematic shifts be

continually optimized versus {a}. • Systematic errors may be nongaussian.

• The published “standard deviations” ij may be

inaccurate.• We trust our physics judgement instead.

Still we do not apply the criterion 2 = 1 !


To judge the PDF uncertainty, we return to the individual experiments.

Lumping all the data together in one

variable – 2global – is too constraining.

Global analysis is a compromise. All data sets should be fit reasonably well -- that is what we check. As we

vary {a}, does any experiment rule out the displacement from the standard set?


In testing the goodness of fit, we keep the normalization factors (i.e., optimized luminosity shifts) fixed as we vary the shape parameters.

End result

1 norms fixed

2

e.g., ~100 for ~2000 data points.

This does not contradict the 2 = 1 criterion used by

other groups, because that refers to a different 2 in which the normalization factors are continually

optimized as the {a} vary.


Some groups do use the criterion of 2 = 1 for PDF error analysis.

Often they are using limited data sets – e.g., an experimental group using only their own data. Then the

2 = 1 criterion may underestimate the uncertainty implied by systematic differences between experiments.

An interesting compendium of methods, by R. Thorne

CTEQ6 2 = 100 (fixed norms)

ZEUS 2 = 50 (effective)

MRST01 2 = 20

H1 2 = 1

Alekhin 2 = 1

GKK not using


Uncertainties of Parton Distributions

(II) Results


Estimate the uncertainty on the predicted cross section for ppbar W+X at the Tevatron collider.

global 2

local 2’s


Each experiment defines a “prediction” and a “range”.This figure shows the 2 = 1 ranges.


This figure shows broader ranges for each experiment based on the “90% confidence level” (cumulative distribution function of the rescaled 2).


The final result is an uncertainty range for the prediction of W.

Survey of wBlpredictions (by R. Thorne) …

PDF set energy wBlnb

PDF uncert

Alekhin Tevatron 2.73 0.05

MRST2002 Tevatron 2.59 0.03

CTEQ6 Tevatron 2.54 0.10

Alekhin LHC 215. 6.

MRST2002 LHC 204. 4.

CTEQ6 LHC 205. 8.


How well can we determine the value of S( MZ ) from Global Analysis?

For each value of S, find the best global fit. Then look at the 2 value for each experiment as a function of S.


Each experiment defines a “prediction” and a “range”.This figure shows the 2 = 1 ranges.

Particle data group (shaded strip) is 0.1170.002.

The fluctuations are larger than expected for normal statistics. The vertical lines have 2

global=100,s(MZ)=0.11650.0065


Uncertainties of the PDF’s themselves (only interesting to the model builders)

Gluon and U quark at Q2 = 10 GeV2.


Comparing “alternate sets”

Gluon at Q2 = 10 GeV2 U quark at Q2 = 10 GeV2

red – CTEQ6.1 blue – Fermi2002 (H1, BCDMS, E665)


CTEQ error band with MRST2002 superimposedQ

2 = 10

GeV

2


Uncertainties of LHC parton-parton luminosities

212121,

)ˆ()()()ˆ( dxdxsxxsxfxfCsLum jiji

ij

Provides simple estimates of PDF uncertainties at the LHC.


Outlook

• Necessary infrastructure for hadron colliders

• Tools exist to study uncertainties .

• This physics is data driven -- HERA II and Fermilab Run 2 will contribute.

• Ready for the LHC


Cases


Inclusive jet production and the search for new physics

(hep-ph/0303013)

Inclusive jet cross section : D0 data and 40 alternate PDF sets

Fractional differences


Is there room for new physics from Run Ib?

Contact interaction model with = 1.6, 2.0, 2.4 TeV


The inclusive jet cross section versus pT for 3 rapidity bins at the LHC. Predictions of all 40 eigenvector basis sets are superimposed.


Strangeness asymmetry

The NuTeV Collaboration has measured the cross sections for -Fe and -Fe to X. A significant fraction of the CS comes from s and bar sbar interactions.

We have added this data into the global fit to determine

1

0

1

0

)(][ and )}()({)(

)(][ and )()()(

),( and ),(

dxxSSxsxsxxS

dxxssxsxsxs

QxsQxs


Figure 1.Typical strangeness asymmetry s(x) and the associated momentum asymmetry S(x). The axes are chosen such that both large and small x regions are adepquately represented, and that the area iunder each curve equals the correponding integral.

[S-] valuesA : 0.312 x 103

B : 0.160 x 103

C : 0.103 x 103


Figure 2.Correlation between 2 values and [S]

Red: dimuon cross section

Blue: other data sensitive to ssbar (F3)


Figure 3.Comparison of the s(x) and S(x) functions for three PDF sets:

our central fit “B” (dot-dash)BPZ (blue)NuTeV (red)

uncertainties of parton distribution functions

Documents

systematic differences

lowx e p data zeus

highq ep data d0

set of pdfs

accurate pdfs

routine statistical

parton cross sections

e p datah1 b