Parton Distribution Functions and Neural Networks Alberto Guffanti Albert-Ludwigs-Universität Freiburg PSI Villigen, March 11, 2010
What are Parton Distribution Functions? Consider a process with one hadron in the initial state σ D(x,Q 2 ) According to the Factorization Theorem we can write the cross section as � 1 � 1 � x ξ , ˆ � � d ξ s � ξ D a ( ξ, µ 2 ) d ˆ µ 2 , α s ( µ 2 ) d σ = σ a + O Q p 0 a A. Guffanti (Univ. Freiburg) PDF errors 2 / 40
What are Parton Distribution Functions? The initial condition cannot be computed in Perturbation Theory (Lattice? In principle yes, but ...) ... but the energy scale dependence is governed by DGLAP evolution equations ∂ ln Q 2 q NS ( ξ, Q 2 ) = P NS ( ξ, α s ) ⊗ q NS ( ξ, Q 2 ) � Σ � Σ � P qq � � � ∂ P qg ( ξ, Q 2 ) = ( ξ, Q 2 ) ( ξ, α s ) ⊗ g P gq P gg g ln Q 2 ... and the splitting functions P can be computed in PT and are known up to NNLO ( LO - Dokshitzer; Gribov, Lipatov; Altarelli, Parisi; 1977) ( NLO - Floratos, Ross, Sachrajda; Gonzalez-Arroyo, Lopez, Yndurain; Curci, Furmanski, Petronzio, 1981) ( NNLO - Moch, Vermaseren, Vogt; 2004) A. Guffanti (Univ. Freiburg) PDF errors 3 / 40
Why care about PDFs (and their uncertainties)? LHC parton kinematics 10 9 x 1,2 = (M/14 TeV) exp( ± y) Q = M 10 8 M = 10 TeV 7 10 6 M = 1 TeV 10 10 5 2 ) 2 (GeV M = 100 GeV 10 4 Q 3 10 y = 6 4 2 0 2 4 6 2 10 M = 10 GeV fixed HERA 10 1 target 10 0 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 x A. Guffanti (Univ. Freiburg) PDF errors 4 / 40
Why care about PDFs (and their uncertainties)? LHC parton kinematics 10 9 x 1,2 = (M/14 TeV) exp( ± y) Q = M 10 8 M = 10 TeV 100 σ ( gg → H ) [pb] 1.2 √ s = 14 TeV 7 10 1.1 1 1 MRST CTEQ Alekhin 0.9 6 M = 1 TeV 10 10 0.7 0.8 100 150 200 0.5 10 5 1.1 2 ) 2 (GeV 1.05 0.3 MRST 1 CTEQ M = 100 GeV 1 Alekhin 10 4 0.2 σ ( gg → H ) [pb] 0.95 Q √ s = 1 . 96 TeV 0.9 3 100 1000 10 0.1 y = 6 4 2 0 2 4 6 100 1000 100 120 140 160 180 200 M H [GeV] M H [GeV] 2 10 M = 10 GeV fixed HERA 10 1 [A. Djouadi and S. Ferrag, hep-ph/0310209] target 10 0 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 x A. Guffanti (Univ. Freiburg) PDF errors 4 / 40
Why care about PDFs (and their uncertainties)? LHC parton kinematics 10 9 x 1,2 = (M/14 TeV) exp( ± y) Q = M 10 8 M = 10 TeV 0.4 4 1.15 1.15 7 1.1 1.1 10 1.05 1.05 0.2 1 1 2 6 M = 1 TeV 0.95 0.95 10 0.9 0.9 100 150 200 100 150 200 0.1 10 5 1 2 ) 2 (GeV MRST MRST CTEQ CTEQ Alekhin 0.05 M = 100 GeV Alekhin 10 4 0.5 σ ( pp → HW ) [pb] σ ( p ¯ p → HW ) [pb] Q √ s = 1 . 96 TeV √ s = 14 TeV 0.03 3 10 0.3 y = 6 4 2 0 2 4 6 100 120 140 160 180 200 100 120 140 160 180 200 M H [GeV] M H [GeV] 2 10 M = 10 GeV fixed HERA 10 1 [A. Djouadi and S. Ferrag, hep-ph/0310209] target 10 0 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 x A. Guffanti (Univ. Freiburg) PDF errors 4 / 40
Why care about PDFs (and their uncertainties)? Errors on PDFs are in some cases the dominating theoretical error on precision observables Ex. σ ( Z 0 ) at the LHC: δ PDF ∼ 3 % , δ NNLO ∼ 2 % [J. Campbell, J. Huston and J. Stirling, (2007)] A. Guffanti (Univ. Freiburg) PDF errors 5 / 40
Why care about PDFs (and their uncertainties)? Errors on PDFs are in some cases the dominating theoretical error on precision observables Ex. σ ( Z 0 ) at the LHC: δ PDF ∼ 3 % , δ NNLO ∼ 2 % [J. Campbell, J. Huston and J. Stirling, (2007)] Errors on PDFs might reduce sensitivity to New Physics Ex. Extra Dimensions discovery in dijet cross section at the LHC: Mc=2TeV Mc=4TeV Mc=6TeV Mc=8TeV Standard Model zone 2 XDs 4XDs 6XDs [S. Ferrag (ATLAS), hep-ph/0407303] A. Guffanti (Univ. Freiburg) PDF errors 5 / 40
Problem Faithful estimation of errors on PDFs Single quantity: 1- σ error Multiple quantities: 1- σ contours Function: need an "error band" in the space of functions ( i.e. the probability density P [ f ] in the space of functions f ( x ) ) Expectation values are Functional integrals � �F [ f ( x )] � = D f F [ f ( x )] P [ f ( x )] A. Guffanti (Univ. Freiburg) PDF errors 6 / 40
Problem Faithful estimation of errors on PDFs Single quantity: 1- σ error Multiple quantities: 1- σ contours Function: need an "error band" in the space of functions ( i.e. the probability density P [ f ] in the space of functions f ( x ) ) Expectation values are Functional integrals � �F [ f ( x )] � = D f F [ f ( x )] P [ f ( x )] Determine an infinite-dimensional object (a function) from a finite set of data points ... mathematically ill-defined problem. A. Guffanti (Univ. Freiburg) PDF errors 6 / 40
Solution Standard Approach Introduce a simple functional form with enough free parameters q ( x , Q 2 ) = x α ( 1 − x ) β P ( x ; λ 1 , ..., λ n ) . Fit parameters minimizing χ 2 . A. Guffanti (Univ. Freiburg) PDF errors 7 / 40
Solution Standard Approach Introduce a simple functional form with enough free parameters q ( x , Q 2 ) = x α ( 1 − x ) β P ( x ; λ 1 , ..., λ n ) . Fit parameters minimizing χ 2 . Open problems : Error propagation from data to parameters and from parameters to observables is not trivial . Theoretical bias due to the chosen parametrization is difficult to assess. A. Guffanti (Univ. Freiburg) PDF errors 7 / 40
Shortcomings of the Standard approach What is the meaning of a one- σ uncertainty? Standard ∆ χ 2 = 1 criterion is too restrictive 2 1 3 4 to account for large discrepancies among 5 experiments. 1 NMC m p/ m n 6 2 E605 pp D-Y 8 3 H1 F 2 ep � 7 m p 4 NMC F 2 5 Zeus F 2 ep m p 6 BCDMS F 2 7 BCDMS F m d [Collins & Pumplin, 2001] 2 n A 8 CCFR F 2 A. Guffanti (Univ. Freiburg) PDF errors 8 / 40
Shortcomings of the Standard approach What is the meaning of a one- σ uncertainty? Standard ∆ χ 2 = 1 criterion is too restrictive 2 1 3 4 to account for large discrepancies among 5 experiments. 1 NMC m p/ m n 6 2 E605 pp D-Y 8 3 H1 F 2 ep � 7 m p 4 NMC F 2 5 Zeus F 2 ep m p 6 BCDMS F 2 7 BCDMS F m d [Collins & Pumplin, 2001] 2 n A 8 CCFR F 2 Introduce a TOLERANCE criterion, i.e. take Eigenvector 4 40 30 BCDMSp BCDMSd the envelope of uncertainties of experiments NMCp CCFR2 CCFR3 CDFjet 20 ZEUS NMCr CDFw E605 E866 D0jet H1a H1b 10 to determine the ∆ χ 2 to use for the global fit distance 0 � 10 (CTEQ). � 20 � 30 A. Guffanti (Univ. Freiburg) PDF errors 8 / 40
Shortcomings of the Standard approach What is the meaning of a one- σ uncertainty? Standard ∆ χ 2 = 1 criterion is too restrictive 2 1 3 4 to account for large discrepancies among 5 experiments. 1 NMC m p/ m n 6 2 E605 pp D-Y 8 3 H1 F 2 ep � 7 m p 4 NMC F 2 5 Zeus F 2 ep m p 6 BCDMS F 2 7 BCDMS F m d [Collins & Pumplin, 2001] 2 n A 8 CCFR F 2 Introduce a TOLERANCE criterion, i.e. take Eigenvector 4 40 30 BCDMSp BCDMSd the envelope of uncertainties of experiments NMCp CCFR2 CCFR3 CDFjet 20 ZEUS NMCr CDFw E605 E866 D0jet H1a H1b 10 to determine the ∆ χ 2 to use for the global fit distance 0 � 10 (CTEQ). � 20 � 30 MSTW 2008 NLO PDF fit 20 global E866/NuSea pd/pp DY 2 2 E866/NuSea pd/pp DY X d F E866/NuSea pp DY χ 15 µ asym. 3 NC µ µ BCDMS NC NC N xF ∆ σ r → asym. NC 2 σ r 2 r H1 ep 97-00 2 N σ r d F H1 ep 97-00 d F NC X σ H1 ep 97-00 Make it DYNAMICAL , i.e. determine ∆ χ 2 Tolerance T = d F ν H1 ep 97-00 σ r µ X l ν NuTeV ν NuTeV µ µ H1 ep 97-00 µ µ → 10 µ X X l ν BCDMS NMC → µ → II W + 100 (CTEQ) NMC µ µ µ → N → µ → II W ν N N N CCFR NuTeV ν ∅ ν D + 50 (MRST) NuTeV ν NuTeV ∅ 5 D separately for each hessian eigenvector 0 E866/NuSea pd/pp DY -5 X X E866/NuSea pd/pp DY 2 N xF 3 µ µ X 3 p F µ E866/NuSea pd/pp DY µ µ N xF X → → - 50 (MRST) NC → µ µ d F 2 BCDMS µ E866/NuSea pd/pp DY ν N N (MSTW). σ r µ NC SLAC ed F 2 NuTeV asym. ν NC ν -10 H1 ep 97-00 N NuTeV ν → µ σ r CCFR r NuTeV - 100 (CTEQ) X CCFR ν N BCDMS ZEUS ep 95-00 H1 ep 97-00 σ µ µ ν N F 2 → NuTeV NC ν N r → l ν ν ZEUS ep 95-00 σ NuTeV -15 NuTeV II W ∅ D -20 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 11 12 13 14 15 16 17 18 19 20 10 11 12 13 14 15 16 17 18 19 20 Eigenvector number A. Guffanti (Univ. Freiburg) PDF errors 8 / 40
Shortcomings of the standard approach What determines PDF uncertainties? Uncertainties in standard fits often increase when adding new data to the fit. Related to the need of extending the parametriztion in order to accomodate the new data Smaller high- x gluon (and slightly smaller α S ) results in larger small- x gluon – now shown at NNLO. Gluon at Q 2 = 10 4 GeV 2 1.1 Ratio to MSTW 2008 NNLO 1.08 MSTW 2008 NNLO 1.06 MRST 2006 NNLO 1.04 1.02 1 0.98 0.96 0.94 0.92 0.9 -5 -3 10 10 -4 10 10 -2 10 -1 1 x Larger small- x uncertainty due to extrat free parameter. [R. Thorne, PDF4LHC] PDF4LHCMSTW 24 A. Guffanti (Univ. Freiburg) PDF errors 9 / 40
THE NNPDF METHODOLOGY [R. D. Ball, L. Del Debbio, S. Forte, J. I. Latorre, A. Piccione, J. Rojo, M. Ubiali and AG] A. Guffanti (Univ. Freiburg) PDF errors 10 / 40
Recommend
More recommend
