a general methodology for updating pdf sets with lhc data
DESCRIPTION
A general methodology for updating PDF sets with LHC data. Francesco de Lorenzi*, Ronan McNulty (University College Dublin) * now at Iowa State (ATLAS). Concept. It would be nice to know the impact that your experimental measurement has on the PDFs. - PowerPoint PPT PresentationTRANSCRIPT
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Ronan McNulty EWWG 05.04.11 1
A general methodology for updating PDF sets with LHC data
Francesco de Lorenzi*, Ronan McNulty (University College Dublin)
* now at Iowa State (ATLAS)
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Ronan McNulty EWWG 05.04.11 2
Concept
• It would be nice to know the impact that your experimental measurement has on the PDFs.
• Since each PDF set defines a probability density function (either multinomial distribution or sampling), in principle one can update this using your measurement.
• NB: This technique assumes that the PDFs can be described using standard statistical technique.
• NB: This is often not true (issue of tolerances).
• Consequently this methodology is NOT a replacement for Global Fits, but is indicative of the likely improvements your data will bring.
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Ronan McNulty EWWG 05.04.11 3
S
Global Fits
HERA
Tevatron
Fixed Target
LHC
DGLAP
Factorisation
Theory
LO/NLO/NNLO
HeavyFlavour
Parametrisation
Consistencyof data
Tensions
Tolerances
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Ronan McNulty EWWG 05.04.11 4
For any function, F, depending on λ, the best value is: F(λ1,λ2,λ3…λn)
and the uncertainty is
What do you get for your money?(An end-user perspective)
PDFHessian
A set of parameters: (λ1,λ2,λ3…λn)A covariance matrix:
These correspond to distances along a set of orthonormal vectors definedat the minimum of the global chisquared.
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Ronan McNulty EWWG 05.04.11 5
PDFs are of form (e.g.):
So uv=uv(λ1,λ2,…λn), dv=dv(λ1,λ2…λn), g=g(λ1,λ2,…λn)
And δuv2= Σ(duv/dλi)2σi
2 δdv2= Σ(ddv/dλi) …. etc.
Thus PDFs and uncertainties defined through λi
What do you get for your money?(An end-user perspective)
PDFHessian
with η(λi) ε (λi) (λi)
But also observables e.g. σZ(λ1,λ2…λn) with
or differential cross-sections, or ratios of cross-sections.
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Ronan McNulty EWWG 05.04.11 6
How does new data affect PDF?The global PDF fit gives us λ1
0+-δ10, λ2
0+- δ20… λn
0+- δn0 and
predicts σZth(λ1
0,λ20,λ3
0…λn0) and now we measure σZ.+- δZ .
We can fit for new values of λi by minimising:
measurement prediction constraint
And taking second derivative of χ2 we get a covariance matrix:
where δi<= δi0
and in general off-diagonal termsare non-zero.
So improved values for λi with smaller (though correlated) errors
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Ronan McNulty EWWG 05.04.11 7
While observables like σZ(λ1,λ2,λ3…λn) with δσ=
or differential cross-sections, or ratios of cross-sections, are predicted. (Compare to two slides back)
PDFs are of form (e.g.):
So uv=uv(λ1,λ2,…λn), dv=dv(λ1,λ2…λn), g=(λ1,λ2,…λn)
And δuv= δdv=…. etc.
Thus PDFs and uncertainties defined through λi
with η(λi) ε (λi) (λi)
duvduv
The old function relationship still holds (though the basis is nolonger orthonormal)
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Ronan McNulty EWWG 05.04.11 8
Constraining the PDFs
• F can be , u V, d V, g
Before the fit After the fit
Uncertainties obtained adding in quadrature eigenvector deviation from central value
V is the new covariance matrix of the fit
Note: Before taking any data, it’s possible to estimate the improvement inthe PDFs with a given luminosity. The central value though, will depend on what you measure.
δi2
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Ronan McNulty EWWG 05.04.11 9
truth VS fitted (each eigenvalues)tr
uth
fitted-3 3
3
9Francesco De Lorenzi7/29/2009
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Ronan McNulty EWWG 05.04.11 10
Pull = (truth – fitted)/error_fitted
-5 510Francesco De Lorenzi7/29/2009
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Expected improvement of MSTW08 PDFs with 1fb-1 of LHCb data
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What do you get for your money?(An end-user perspective)
PDFNNPDF
N equal probablity replicas: uvi,dv
i,gi,si etc. (i=1,N)The ensemble defines the probability phase space.
For any function, F, depending on u,d,s…g, there willbe N possible values for F.
The ‘best value’ of F from mean:
The uncertainty from RMS.
Again, F can be a PDF, or an observable.
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Ronan McNulty EWWG 05.04.11 13
Simplistic prescription: • The data will be compatible with some
replicas, and incompatible with others.• Evaluate with • Remove incompatible replicas. (Prob<1%) • The spread of replicas reduces; the
uncertainty is smaller; the mean may shift.
How does new data affect PDF?
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Expected improvement of NNPDF20 PDFs with 1fb-1 of LHCb data
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Improvements for various PDF sets with 1fb-1 of LHCb electroweak data
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Sophisticated prescription (NNPDF):• Take each PDF replica as a Bayesian Prior.• Evaluate Chi2 compared to this replica• Probability is• Each replica is weighted by this probability• Take mean and variance of weighted replicas.
How does new data affect PDF?
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Ronan McNulty EWWG 05.04.11 17
Summary• The NNPDF (sophisticated) prescription has been
used to show affect of LHCb, ATLAS and CMS data. (See Maria’s talk)
• Similar simple techniques can also be used to update the Hessian methods (shown here).
CAVEAT: You are assuming that the global fits obey standard error propagation.
Past experience has shown this is not always the case.