Lara De Nardo BNL October 8, 2007 QCD fits to the g 1 world data and uncertainties THE HERMES...

Lara De Nardo BNL October 8, 2007

QCD fits to the gQCD fits to the g11 world data and world data and uncertaintiesuncertainties

THE HERMES EXPERIENCETHE HERMES EXPERIENCE

Lara De NardoLara De NardoTRIUMF/DESYTRIUMF/DESY

Global Analysis WorkshopOctober 8, 2007


OutlineOutline

Definition of g1

Recent g1 data from HERMES

Fits to g1 data

Some ingredients in fits

The 2

Statistical uncertainties

Systematic uncertainties

Conclusions


param.kin.fact.measuredparam.

kinematicfactors

From the Measured Inclusive From the Measured Inclusive Asymetries to gAsymetries to g11

)2,(

22

2)2,(

||2

2

28

4

24

1

21

1)2,(1

Qxgy

QxAQx

unpol

y

Q

yy

Qxg

The measured quantity is the asymmetry

From the asymmetry, with the additional information on beam energy, unpol and g2 each experiment provides a value of g1

A first step in the analysis of world data consists in trying to make all data sets compatible wrt the choice of g2 and unpol

NB: unpol depends on R and F2, so the chosen F2 has to be compatible with the chosen R


gg11 World Data World Data

Lara De Nardo PacSpin2007

AA11 World Data World Data

21

121 1 A

F

gA

1

1

F

g


HERMES resultsHERMES results0.0041 < x < 0.9

0.18 GeV2 < Q2 < 20 GeV2

Correction for smearing and radiative effects introduces statistical correlationsStatistical uncertainties are diagonal elements of covariance matrix

Phys.Rev.D75(2007)012007Phys.Rev.D75(2007)012007


gg11 QCD fits QCD fitsg1 QCD fits are models for q(x,Q2) obtained by fitting inclusive world data on g1 :

nqnq

nqn n

pppp ,.....,..,,.........,....., 11

1 1

1get

2

2112 )(

data

datafit

data

gg

minimize

Calculate g1fit at the Q2 of all data points: )),((),( 22

1 iii

fit CQxqQxg

DGLAP: iii Pq

Q

q

2 to go from Q20 to Q2

data

),....,(),( 120

inq

iii ppfQxq

Start from model at initial Q2=Q20:

minimize jdatafit

datai

datafit ggjigg ))(,cov()( 11112

(for HERMES data)

, qpNS, qn

NS, Goruv, dv, q, G (need assumption on the sea)


The ingredients of a fitThe ingredients of a fit

Higher twists (LSS06)

Inclusion of semi-inclusive data (dFS) or (AAC)

Statistical Uncertainties

Systematic Uncertainties

Higher twists (LSS06)

Inclusion of semi-inclusive data (dFS) or (AAC)

Statistical Uncertainties

Systematic Uncertainties

Many fits on the ‘market’ look at one or another problem, and it is sometimes very difficult to compare fits from different groups, as they are based on different assumptions.

0LLA


LSS’06LSS’06

4

4

2

22

1

),(),(

QQ

QxhQxg HT

exp2

1

21

21

exp

21

21

),(

),(),(

),(

),(

QxF

QxgQxg

QxF

Qxg HTLT

),(),( 20

20 QxxfxAQxfx MRST

iiii

Initial parameterization:

Higher twist terms included in the fit:

Gsduf vv ,,,

New data:Low Q2 CLAS dataCOMPASS data (large Q2)

4

42

2

22

12

1 ),(),(),(Q

MQxh

Q

MQxgQxg TMCpQCDLT

)(),( in

ip xhxh

i=1,…,5: 10 parameters+6 for PDs (=8-2 for sum rules)

E.Leader et al.,Phys.Rev.D75(2007)074027


LSS’06: Higher TwistsLSS’06: Higher TwistsOther groups (BB) claim to have not found such a strong signal for HT

H.Boettcher, private comm.

The values are close, but the conclusions are different!Diff. depend on the size of the error bars and on their meaning

LSS’06: Strong HT signal


Statistical UncertaintiesStatistical UncertaintiesAt least two groups (BB and AAC) report statistical uncertainties inflated by They cannot be directly compared to those of other groups.

NPAR

(These inflated uncertainties do not correspond to what is normally understood as statistical uncertainty, obtained as the standard deviation of the distribution of results derived by fitting a large number of MC data sets resembling the experimental data sets, but with each data point fluctuating independently according to the experimental statistical uncertainty)

2

pi

pj

22=1 defines the 1 uncertainty for single parameters

2~NPAR is the 1 uncertainty for the NPAR parameters to be simultaneously located inside the hypercontour(2.ne.1 normally used for unknown systematics, see CTEQ, MRST….)

NPAR=number of parameters


Evolution of Statistical UncertaintiesEvolution of Statistical Uncertainties

Statistical uncertainties are given by:

Calculable exactly at Q20 since the functional form of q is known at Q2

0.

)cov(),(),(),()( ,2222

jijij i

q ppQxdp

qdQx

dp

qdQx

kdp

d

Gdt

ddt

d

qdt

dNS NSqP 1

GPP 32

GPP 54

The derivatives of the distributions evolve just like the distributions!

iiidp

GdP

dp

dP

dp

Gd

dt

d

54

iiidp

GdP

dp

dP

dp

d

dt

d

32

X-space



iiidp

GdP

dp

dP

dp

Gd

dt

d

54

iiidp

GdP

dp

dP

dp

d

dt

d

32

Only at Q20G does not depend on the parameters!

G acquires a dependence on the parameters in Q2

The same is true for (it depends on the G parameters through the evolution)The NS evolves independently from the other distributionsFor details on the unc. calculations in Mellin space see BB paper (Nucl.Phys.B636(2002)225)

Parameter of (x,Q2

0)

At initial Q2=Q20


Normalization UncertaintiesNormalization UncertaintiesAccount for the (substantial) syst. unc. common to an entire data set for one experiment by adding it incoherently in quadrature to the uncertainty in each data point.Done with a 2 penalty term, see BB:

Norm. unc. quoted by expt.

Fitted normalization

02

iN

2,11

2

,1

21

2

,1,1,1,1

/1

/1

datak

n

j

theorj

i

n

k

datak

datak

theork

theork

i

ggN

ggggN

data

data

The normalizations can also be calculated analytically at each step, without increasing the number of parameters in the fit:

exp

1 12

,1

2

,1,1

2

22 1n

i

n

jdataji

theorj

dataji

i

i

data

gN

ggN

N

N


One can also consider the experimental systematic uncertainties:

Si can also be calculated analytically at each step

dataksysi

datak

datak gSgg ,1,1,1

Systematic uncertaintiesSystematic uncertainties

exp

1 1

22

,1

2

,1,1,12n

i

n

jidata

j

theorj

datajsysi

dataj

data

Sg

ggSg

02

iS

2,11

2

,1

1

2

,1,1,1,1

/1

/

dataj

n

j

theorjsys

n

k

datak

theorksys

datak

theork

i

gg

ggggS data

data


Putting it all togetherPutting it all together

We finally get the 2 including Covariance (for HERMES data)

Normalizations

Systematic parameters

exp

1 12

22,1

,1,11,1

,1,12 1n

i

n

j i

ii

i

theorkdata

ksysidatakjk

i

theorjdata

jsysidataj

data

N

NS

N

ggSgCov

N

ggSg

Also in this case the parameters Si and Ni can be calculated analytically at each step, but their expression is more complicated


Results Stability Results Stability Up to three minima have recently been seen with similar values of 2.

Test the stability and accuracy of the methods using MonteCarlo pseudo-data generated from a chosen set of polarised parton distributions compare then with the fit results.

LSS’06


Symmetric Sea Symmetric Sea Assumption: new results Assumption: new results

from COMPASSfrom COMPASS

The estimated v is 2.5stat away from the symmetric sea scenario


ConclusionsConclusionsAt the moment still other data (like for AAC or semi-inclusive asymmetries for De Florian et al.) has to come in aid of QCD fits in order to pin down the gluon distribution

For more precise data on G from scaling violations of g1, proposed e-p colliders e-LIC and eRHIC

The latest QCD fits look at various aspects of q (AAC:gluon, LSS:HT…)It would be nice to have one comprehensive analysis with all these features:

HT calculation

Statistical error band calculation explicitely state which 2 choice was made and possibly provide results with the two choices

Propagation of systematic uncertainties

Fit NS to test the Bjorken Sum Rule

Fit s

……

0LLA


For PamelaFor Pamela


QCD fits to the gQCD fits to the g11 world data and uncertainties world data and uncertainties

Lara De NardoLara De NardoTRIUMF/DESYTRIUMF/DESY

In my talk I will mainly focus on the calculation of uncertainties in polarised parton distributions obtained by QCD fits to g1 world data.I will discuss how some published results on statistical uncertainties cannot be readily compared with results from other papers, because of different interpretations of these uncertainties; from this it follows that sometimes opposite conclusions are reached based on similar results.


Statistical UncertaintiesStatistical UncertaintiesAt least two groups (BB and AAC) report statistical uncertainties inflated by They cannot be directly compared to those of other groups.

NPAR

(These inflated uncertainties do not correspond to what is normally understood as statistical uncertainty, obtained as the standard deviation of the distribution of results derived by fitting a large number of MC data sets resembling the experimental data sets, but with each data point fluctuating independently according to the experimental statistical uncertainty)

2

pi

pj

22=1 defines the 1 uncertainty for single parameters

2~NPAR is the 1 uncertainty for the NPAR parameters to be simultaneously located inside the hypercontour(normally used for unknown systematics, see CTEQ, MRST….)

NPAR=number of parameters



Statistical uncertainties are given by:

Calculable exactly at Q20 since the functional form of q is known at Q2

0.

)cov(),(),(),()( ,2222

jijij i

q ppQxdp

qdQx

dp

qdQx

kdp

d

Gdt

ddt

d

qdt

dNS NSqP 1

GPP 32

GPP 54

The derivatives of the distributions evolve just like the distributions!

iiidp

GdP

dp

dP

dp

Gd

dt

d

54

iiidp

GdP

dp

dP

dp

d

dt

d

32

X-space



iiidp

GdP

dp

dP

dp

Gd

dt

d

54

iiidp

GdP

dp

dP

dp

d

dt

d

32

Only at Q20G does not depend on the parameters!

G acquires a dependence on the parameters in Q2

The same is true for (it depends on the G parameters through the evolution)The NS evolves independently from the other distributionsFor details on the unc. calculations in Mellin space see BB paper (Nucl.Phys.B636(2002)225)

Parameter of (x,Q2

0)

At initial Q2=Q20


Normalization UncertaintiesNormalization UncertaintiesAccount for the (substantial) syst. unc. common to an entire data set for one experiment by adding it incoherently in quadrature to the uncertainty in each data point.Done with a 2 penalty term, see BB:

Norm. unc. quoted by expt.

Fitted normalization

02

iN

2,11

2

,1

21

2

,1,1,1,1

/1

/1

datak

n

j

theorj

i

n

k

datak

datak

theork

theork

i

ggN

ggggN

data

data

The normalizations can also be calculated analytically at each step, without increasing the number of parameters in the fit:

exp

1 12

,1

2

,1,1

2

22 1n

i

n

jdataji

theorj

dataji

i

i

data

gN

ggN

N

N


One can get even more fancy and consider the experimental systematic uncertainties:

Si can also be calculated analytically at each step

dataksysi

datak

datak gSgg ,1,1,1

Systematic uncertaintiesSystematic uncertainties

exp

1 1

22

,1

2

,1,1,12n

i

n

jidata

j

theorj

datajsysi

dataj

data

Sg

ggSg

02

iS

2,11

2

,1

1

2

,1,1,1,1

/1

/

dataj

n

j

theorjsys

n

k

datak

theorksys

datak

theork

i

gg

ggggS data

data

Lara De Nardo BNL October 8, 2007 QCD fits to the g 1 world data and uncertainties THE HERMES...

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Transcript of Lara De Nardo BNL October 8, 2007 QCD fits to the g 1 world data and uncertainties THE HERMES...