Experimental uncertainties in the parton distributions on Higgs production

Stan BentvelsenMichiel BotjeJob Tijssen

Parton parameterizations For proper error estimation on

hadronic cross sections we need the uncertainty on the PDF’s: Propagation of uncertainties on

experimental data to the fitted PDF’s Statistical uncertainties and

(correlated) systematic effects Uncertainties in the theoretical

description of the fit procedure Flavour thresholds, s Scales uncertainties Nuclear effects Higher twist, …

A number of groups have published the PDF fits with propagated experimental uncertainties: Botje (Eur Phys J C14 (dec 1999)) CTEQ (J. Pumplin et al, hep-ph/0201195) MRST (A. Martin et al, hep-ph/0211080)

Alekhin (S. Alekhin, hep-ph/0011002) Fermi2001 (Giele et al, hep-ph/0104052)

Theoretical uncertainties not treated here

PDF’s cannot be calculated from theory and are obtained from QCD DGLAP evolution fits to data.

DIS data from fixed target and HERA

Jet cross sections pp colliders Drell-Yan processes

The hadronic cross section

The uncertainty due to the gluon distribution (error propagation)

Have to take all correlations of the gluon distribution into account

Higgs production SM Higgs production

dominated by gluon fusion:

Zeroth width approximation

PDF uncertainties In the past:

Errors associated with parton densities were often determined from the spread of different parton distribution sets.

By no means representation of experimental and theoretical uncertainties

Relatively recently: Error propagation in global

QCD fits (increasing accuracy HERA/Tevatron data)

Based on: Least squares minimization

and linear error propagation (Monte Carlo integration

techniques)

Available pdf data sets Botje (Dec 1999)

Q02 = 4 GeV2, 28 free parameters

X>10-3

Q2> 3 GeV2

W2>7 GeV2

Total 1578 data points, 2min=1537

Structure function data only (no pp jet data, no W± asymmetry) Not including ‘latest’ HERA structure functions

CTEQ (Dec 2002) Q0

2 = 1.3 GeV2

20 (effectively independent) free parameters Q2> 4 GeV2

Total 1757 data points, 2min =1980

MRST (Nov 2002) Q0

2 = 1 GeV2

15 (effectively independent) free parameters, Q2> 2 GeV2

Total 2097 data points, 2min ~2267

All parameterizations use NLO DGLAP evolution inMS-scheme.CTEQ and MRST also providepdf’s in DIS scheme, as well asleading order (event generators)

‘CTEQ6’ seriesGluon distribution

somewhat harder wrt CTEQ5

‘MRST02’ seriesGluon distribution slightly

harder wrt MRST2001

BotjeNeeds update with latest data

Correlations between the mutual pdf’s important. The largest origin of the correlations are the momentum sum rules

Uncertainty on gluon- and quark integrals separately much larger than on the sum of the two (Q2=4 GeV2):

Also large correlation of gluon distribution and value of s. The value of s is kept fixed in the QCD evolution, at values obtained

from precision e+e- collisions. As consistency checks the fits are repeated for varying s Quoted obtained errors on s from these checks range between 1 – 6 %

In this study uncertainties on s are ignored

Correlations

Error estimates on PDF’s Input parameters pi from least squares 2 minimalization

Covariance matrix of input parameters pi obtained from expansion around minimim 2

Two methods to propagate the experimental systematic uncertainties:(Botje: hep-ph/0110123) Covariance matrix method (Hessian method)

CTEQ, MRS, H1, … Rigorous statistical technique

Assume errors are gaussian distributed, use linear approximation Exact in 1st order approximation

Offset method Botje, ZEUS. Offset data by systematic error, redo fit, add deviations in quadrature

Gives a conservative error of uncertainties

Using uncertainties Covariance of any F and G:

Botje: Store covariance matrix Vij

p, parton densities, and all derivatives q/pi in tables

Error propagation done by EPDFLIB library User supplies FORTRAN function with definition F and G in terms of pdf’s

as well as derivatives F/q and G/q. EPDFLIB calculates <FG>

CTEQ, MRS: Diagonalize the covariance matrix Vij

p using ‘rotated’ parameters zi Uncertainty on F and G simplifies to

In order to sample quadratic behavior 2 accurately, pdf sets are determined for both zi+z, zi-z : (F+

I,F-I)

Store set of 2Np pdf’s for systematic uncertainties. Uncertainty on F corresponds to:

pi: free fit paramete

rs

CTEQ: sum over 40 setsMRST: sum over 30 sets

Error definition with tolerance Uncertainty on quantity F given by

Deviation from 2=1 by CTEQ and MRS groups, by the factor:

CTEQ: produce pdf sets with tolerance T2=100 MRST: produce pdf sets with tolerance T2=50

Rather arbitrary definition to get the standard deviations of a quantity Motivated by investigation probabilities of individual data sets

Botje: produce sets for statistical and systematic errors separately Tolerance T2=1 for statistical uncertainty Added in quadrature to systematic uncertainty

Tolerance T2= 2

Example: valence distributions

BotjeCTEQMRST

Up-valence distributionAs function of log10(x) at Q2=10

GeV2

Relative uncertainties large at very small and very large values of x

Region around x=10-2 where the three sets are not compatible at 1

Relativeuncertainty

Distributions normalized to

MRST

MRSTCTEQBotje

Up-valence at high Q2 value At larger Q2 values the

uncertainties tend to get smaller

Up-valence distribution at Q2=106 GeV2

Relativeuncertainty

Distributions normalized to

MRST

MRSTCTEQBotje

Uncertainty on gluon distribution

Gluon distribution at two scales

Relativeuncertainty

Distributions normalized to

MRST

MRSTCTEQBotje

BotjeCteqMRS

Gluon distribution at Q2=10 GeV2

Larger uncertainties (note the scale)Botje deviates from MRST/CTEQ at low x(cf data cut at x>10-3)Very typical small uncertainty around x~0.2, rapid increase for larger xMRST smallest uncertainties

Gluon at large Q

Relativeuncertainty

Distributions normalized to

MRST

MRSTCTEQBotje

Gluon distribution at Q2=106 GeV2

Uncertainties at small values of xare getting very small

Higgs production cross section

Relativeuncertainty

Distributions normalized to

MRST

MRSTCTEQBotje

Higgs production cross sectionas function of log10(MH)Or rather the ‘gluon-gluon’ luminosityUncertainties remarkable smallAt Mh=100:

Botje: 5.6%, CTEQ: 4.6%, MRST: 2.2%At Mh=1000:

Botje: ~10%, CTEQ: ~10%, MRST: 5%

Log10(Mh)

Higgs production uncertainty Full check by interfacing to HiGlu package with pdf sets

NLO ggHiggs production in MS-scheme Matches the PDF sets scheme evolution

Cross section ratio NLO to LO given by K-factor (1.5-1.7) Pdf uncertainty very similar for LO and NLO

At TevaTron the uncertainties forthis process are larger

Log10(Mh)

BornNLO

Log10(Mh)

Cm energy=2 TeV

Rapidity distribution At large rapidity the

uncertainty is largest, and is somewhat larger at central rapidity

Select range in rapidity -1 < y < 1

Redo the error analysis on gg:

Uncertainty increases From 4-6% To 6-9%Mh=115 GeV

y

All y-1<y<1

Rapidity y

WW production Other luminosity functions readily be

obtained Example of W+ and W- production at

LHC Uncertainty fairly constant over range

s MRS smallest uncertainty, -1-2% Botje and CTEQ in range 4-5%

qqW+

qqW-

Correlation between Higgs

and W production,

~0.6

Conclusions Propagation of experimental uncertainties to pdf are

getting available Correlations and systematic experimental uncertainties are

important and are taken into account

Definition of the uncertainty on pdf’s not straightforward cf ‘tolerances’ MRST/CTEQ

Uncertainty on ggH process remarkably small All are below 6% for mH=115 GeV, upto 2%.

Cleary theoretical uncertainties –not treated here, are important

And probably more important