Lara De Nardo BNL QCD fits to the g QCD fits to the g 1 1 world data and world data and uncertainties uncertainties THE HERMES EXPERIENCE THE HERMES EXPERIENCE Lara De Nardo Lara De Nardo TRIUMF/DESY TRIUMF/DESY Global Analysis Workshop October 8, 2007
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QCD fits to the g 1 world data and uncertainties THE HERMES EXPERIENCE
QCD fits to the g 1 world data and uncertainties THE HERMES EXPERIENCE. Lara De Nardo TRIUMF/DESY. Global Analysis Workshop October 8, 2007. Outline. Definition of g 1 Recent g 1 data from HERMES Fits to g 1 data Some ingredients in fits The c 2 Statistical uncertainties - PowerPoint PPT Presentation
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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
Lara De Nardo BNL October 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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
gg11 World Data World Data
Lara De Nardo PacSpin2007
AA11 World Data World Data
21
121 1 A
F
gA
1
1
F
g
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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)
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
Evolution of Statistical UncertaintiesEvolution of Statistical Uncertainties
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
For PamelaFor Pamela
Lara De Nardo BNL October 8, 2007
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.
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
Evolution of Statistical UncertaintiesEvolution of Statistical Uncertainties
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
Lara De Nardo BNL October 8, 2007
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
Lara De Nardo BNL October 8, 2007
One can get even more fancy and consider the experimental systematic uncertainties:
Si can also be calculated analytically at each step