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A general methodology for updating PDF sets with LHC data

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.

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A general methodology for updating PDF sets with LHC data

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  1. A general methodology for updating PDF sets with LHC data Francesco de Lorenzi*, Ronan McNulty (University College Dublin) * now at Iowa State (ATLAS) 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. Ronan McNulty EWWG 05.04.11

  3. Global Fits S Parametrisation Tevatron LO/NLO/ NNLO HERA Consistency of data LHC Heavy Flavour Fixed Target Tensions Tolerances DGLAP Factorisation Theory PDF Ronan McNulty EWWG 05.04.11

  4. What do you get for your money?(An end-user perspective) PDF Hessian A set of parameters: (λ1,λ2,λ3…λn) A covariance matrix: These correspond to distances along a set of orthonormal vectors defined at the minimum of the global chisquared. For any function, F, depending on λ, the best value is: F(λ1,λ2,λ3…λn) and the uncertainty is Ronan McNulty EWWG 05.04.11

  5. What do you get for your money?(An end-user perspective) PDF Hessian 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)2σi2δdv2= Σ(ddv/dλi) …. etc. Thus PDFs and uncertainties defined through λi 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. Ronan McNulty EWWG 05.04.11

  6. How does new data affect PDF? The global PDF fit gives us λ10+-δ10, λ20+- δ20… λn0+- δn0 and predictsσZth(λ10,λ20,λ30…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 terms are non-zero. So improved values for λi with smaller (though correlated) errors Ronan McNulty EWWG 05.04.11

  7. The old function relationship still holds (though the basis is no longer orthonormal) 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) duv duv 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) Ronan McNulty EWWG 05.04.11

  8. Constraining the PDFs • F can be s, u V, d V, g Before the fit After the fit δi2 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 in the PDFs with a given luminosity. The central value though, will depend on what you measure. Ronan McNulty EWWG 05.04.11

  9. truth VS fitted (each eigenvalues) 3 truth -3 3 fitted Ronan McNulty EWWG 05.04.11 Francesco De Lorenzi 9

  10. Pull = (truth – fitted)/error_fitted -5 5 Ronan McNulty EWWG 05.04.11 Francesco De Lorenzi 10

  11. Expected improvement of MSTW08 PDFs with 1fb-1 of LHCb data Ronan McNulty EWWG 05.04.11

  12. What do you get for your money?(An end-user perspective) PDF NNPDF N equal probablity replicas: uvi,dvi,gi,si etc. (i=1,N) The ensemble defines the probability phase space. For any function, F, depending on u,d,s…g, there will be 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. Ronan McNulty EWWG 05.04.11

  13. How does new data affect PDF? 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. Ronan McNulty EWWG 05.04.11

  14. Expected improvement of NNPDF20 PDFs with 1fb-1 of LHCb data Ronan McNulty EWWG 05.04.11

  15. Improvements for various PDF sets with 1fb-1 of LHCb electroweak data Ronan McNulty EWWG 05.04.11

  16. How does new data affect PDF? 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. 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. Ronan McNulty EWWG 05.04.11

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