NNPDF: Neural Networks, Monte Carlo techniques and Parton Distribution Functions

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NNPDF: Neural Networks, Monte Carlo techniques and Parton Distribution Functions Alberto Guffanti Albert-Ludwigs-Universität Freiburg on behalf of the NNPDF Collaboration: R. D. Ball, L.

1 NNPDF: Neural Networks, Monte Carlo techniques and Parton Distribution Functions Alberto Guffanti Albert-Ludwigs-Universität Freiburg on behalf of the NNPDF Collaboration: R. D. Ball, L. Del Debbio (Edinburgh, S. Forte, J. Rojo(Milano, J. I. Latorre, F. Cerutti (Barcelona, M. Ubiali (Aachen, V. Bertone and AG (Freiburg HiX 3 rd International Workshop on Nucleon Structure at Large Bjorken Jefferson Lab, Newport News, Virginia October 13-15,

2 What are Parton Distribution Functions? Consider a process with one hadron in the initial state s D(,Q According to the Factorization Theorem we can write the cross section as dσ = 1 ( dξ a ξ D a(ξ, µ d ˆσ a ξ, ŝ ( 1 µ, α s(µ + O Q p A. Guffanti (Univ. Freiburg NNPDF / 3

3 What are Parton Distribution Functions? The absolute value of PDFs at a given and Q cannot be computed in QCD Perturbation Theory (Lattice? In principle yes, but but the scale dependence is governed by DGLAP evolution equations ln Q qns (ξ, Q = P NS (ξ, α s q NS (ξ, Q ( ( ( Σ ln Q (ξ, Q Pqq P = qg Σ (ξ, α g P gq P s gg g (ξ, Q... 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; 4 A. Guffanti (Univ. Freiburg NNPDF 3 / 3

4 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 ( Epectation values are Functional integrals F[f (] = Df F[f (]P[f (] A. Guffanti (Univ. Freiburg NNPDF 4 / 3

5 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 ( Epectation values are Functional integrals F[f (] = Df F[f (]P[f (] Determine a function from a finite set of data points A. Guffanti (Univ. Freiburg NNPDF 4 / 3

6 Solution Standard Approach Introduce a simple functional form with enough free parameters Fit parameters minimizing χ. q(, Q = α (1 β P(; λ 1,..., λ n. A. Guffanti (Univ. Freiburg NNPDF 5 / 3

7 Solution Standard Approach Introduce a simple functional form with enough free parameters Fit parameters minimizing χ. q(, Q = α (1 β P(; λ 1,..., λ n. 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 NNPDF 5 / 3

8 1 NMC mp/mn E65 pp D-Y ep 6 BCDMS F mp Shortcomings of the Standard approach What is the meaning of a one-σ uncertainty? Standard χ = 1 criterion is too restrictive to account for large discrepancies among eperiments. [Collins & Pumplin, 1] 3 H1 F 4 NMC F mp 5 Zeus F ep 7 BCDMS F md 8 CCFR F na A. Guffanti (Univ. Freiburg NNPDF 6 / 3

9 1 NMC mp/mn E65 pp D-Y ep 6 BCDMS F mp Shortcomings of the Standard approach What is the meaning of a one-σ uncertainty? Standard χ = 1 criterion is too restrictive to account for large discrepancies among eperiments. [Collins & Pumplin, 1] 3 H1 F 4 NMC F mp 5 Zeus F ep 7 BCDMS F md 8 CCFR F na Introduce a TOLERANCE criterion, i.e. take the envelope of uncertainties of eperiments to determine the χ to use for the global fit (CTEQ. distance BCDMSp BCDMSd H1a H1b ZEUS Eigenvector 4 NMCp NMCr CCFR CCFR3 E65 CDFw E866 Djet CDFjet A. Guffanti (Univ. Freiburg NNPDF 6 / 3

10 NC NC r r 1 NMC mp/mn E65 pp D-Y ep 6 BCDMS F mp 3 NC NC r r 1 NC NC 4 r r NC 5 r 6 NC NC r r (CTEQ (MRST (MRST (CTEQ Shortcomings of the Standard approach What is the meaning of a one-σ uncertainty? Standard χ = 1 criterion is too restrictive to account for large discrepancies among eperiments. [Collins & Pumplin, 1] 3 H1 F 4 NMC F mp 5 Zeus F ep 7 BCDMS F md 8 CCFR F na Introduce a TOLERANCE criterion, i.e. take the envelope of uncertainties of eperiments to determine the χ to use for the global fit (CTEQ. distance BCDMSp BCDMSd H1a H1b ZEUS Eigenvector 4 NMCp NMCr CCFR CCFR3 E65 CDFw E866 Djet CDFjet MSTW 8 NLO PDF fit Make it DYNAMICAL, i.e. determine χ separately for each hessian eigenvector (MSTW. χ global Tolerance T = H1 ep 97- σ H1 ep 97- σ NuTeV ν N µµx NMC µd F CCFR ν N µµx NuTeV ν N µµx E866/NuSea pd/pp DY E866/NuSea pd/pp DY NuTeV ν N F NuTeV ν N µµx NuTeV ν N µµx NuTeV ν N µµx BCDMS µd F D II W lν asym. BCDMS µp F BCDMS µd F ZEUS ep 95- σ H1 ep 97- σ SLAC ed F BCDMS µd F ZEUS ep 95- σ H1 ep 97- σ E866/NuSea pd/pp DY E866/NuSea pd/pp DY NuTeV ν N F E866/NuSea pp DY D II W lν asym. NMC µd F NuTeV ν N F H1 ep 97- σ E866/NuSea pd/pp DY CCFR ν N µµx CCFR ν N µµx NuTeV ν N µµx E866/NuSea pd/pp DY D II W lν asym. H1 ep 97- σ H1 ep 97- σ NuTeV ν N µµx NuTeV ν N F Eigenvector number A. Guffanti (Univ. Freiburg NNPDF 6 / 3

11 Shortcomings of the Standard approach What determines PDF uncertainties? Uncertainties in standard fits often increase when adding new data to the fit. Need of etending the parametriztion in order to accomodate the new data Smaller high- gluon (and slightly smaller αs results in larger small- gluon now shown at NNLO. Ratio to MSTW 8 NNLO Gluon at Q = GeV MSTW 8 NNLO MRST 6 NNLO Larger small- uncertainty due to etrat free parameter. [R. Thorne, PDF4LHC] PDF4LHCMSTW 4 A. Guffanti (Univ. Freiburg NNPDF 7 / 3

12 Shortcomings of the standard approace What determines PDF uncertainties? Parmetrization bias? Small- gluon - Status prior to NNPDF CTEQ6.6 MRST1E 3 g (, Q A. Guffanti (Univ. Freiburg NNPDF 8 / 3

13 Shortcomings of the standard approace What determines PDF uncertainties? Parmetrization bias? Small- gluon - NNPDF1. released NNPDF1. CTEQ6.6 MRST1E g (, Q A. Guffanti (Univ. Freiburg NNPDF 8 / 3

14 Shortcomings of the standard approace What determines PDF uncertainties? Parmetrization bias? Small- gluon - Status NNPDF. CT MSTW8 g (, Q A. Guffanti (Univ. Freiburg NNPDF 8 / 3

15 NNPDF Methodology Main Ingredients Monte Carlo determination of errors No need to rely on linear propagation of errors Possibility to test for the impact of non gaussianly distributed errors Possibility to test for non-gaussian behaviour in fitted PDFs (1 σ vs. 68% CL Neural Networks Provide an unbiased parametrization Stopping based on Cross Validation Ensures proper fitting avoiding overlearning A. Guffanti (Univ. Freiburg NNPDF 9 / 3

16 NNPDF Methodology... in a Nutshell Generate N rep Monte-Carlo replicas of the eperimental data (sampling of the probability density in the space of data Fit a set of Parton Distribution Functions on each replica (sampling of the probability density in the space of PDFs Epectation values for observables are Monte Carlo integrals F[f i (, Q ] = 1 N rep ( F f (net(k i (, Q N rep k=1... the same is true for errors, correlations, etc. A. Guffanti (Univ. Freiburg NNPDF / 3

17 NNPDF Methodology Monte Carlo replicas generation Generate artificial data according to distribution [ N (art (k O i = (1 + r (k N σ sys N O (ep i + p=1 r (k p where r i are univariate gaussian random numbers σ i,p + r (k i,s σi s Validate Monte Carlo replicas against eperimental data (statistical estimators, faithful representation of errors, convergence rate increasing N rep ] O( replicas needed to reproduce correlations to percent accuracy A. Guffanti (Univ. Freiburg NNPDF 11 / 3

18 Proper Fitting avoiding Overlearning Let s see how proper fitting works in a toy model A. Guffanti (Univ. Freiburg NNPDF 1 / 3

19 Proper Fitting avoiding Overlearning Let s see how proper fitting works in a toy model A. Guffanti (Univ. Freiburg NNPDF 1 / 3

20 Proper Fitting avoiding Overlearning Let s see how proper fitting works in a toy model A. Guffanti (Univ. Freiburg NNPDF 1 / 3

21 Proper Fitting avoiding Overlearning Let s see how proper fitting works in a toy model A. Guffanti (Univ. Freiburg NNPDF 1 / 3

22 Proper Fitting avoiding Overlearning Let s see how proper fitting works in a toy model Need a redundant parametrization to avoid parametrization bias. Need a way of stopping the fit before overlearning sets in to avoid fitting statistical noise. A. Guffanti (Univ. Freiburg NNPDF 1 / 3

23 Neural Networks... a suitable basis of functions We use Neural Networks as functions to represent PDFs at the starting scale We employ Multilayer Feed-Forward Neural Networks trained using a Genetic Algorithm Activation determined by weights and thresholds ξ i = g ω ij ξ j θ i 1, g( = 1 + e β j A. Guffanti (Univ. Freiburg NNPDF 13 / 3

24 Neural Networks... a suitable basis of functions We use Neural Networks as functions to represent PDFs at the starting scale We employ Multilayer Feed-Forward Neural Networks trained using a Genetic Algorithm Activation determined by weights and thresholds ξ i = g ω ij ξ j θ i 1, g( = 1 + e β j E.: 1--1 NN: ξ (3 1 (ξ(1 1 = 1 θ (3 ( 1 ω e 1+e θ( 1 ξ(1 1 ω(1 11 ω ( 1 1+e θ( ξ(1 1 ω(1 1 A. Guffanti (Univ. Freiburg NNPDF 13 / 3

25 Neural Networks... a suitable basis of functions We use Neural Networks as functions to represent PDFs at the starting scale We employ Multilayer Feed-Forward Neural Networks trained using a Genetic Algorithm Activation determined by weights and thresholds ξ i = g ω ij ξ j θ i 1, g( = 1 + e β j E.: 1--1 NN: ξ (3 1 (ξ(1 1 = 1 θ (3 ( 1 ω e 1+e θ( 1 ξ(1 1 ω(1 11 ω ( 1 1+e θ( ξ(1 1 ω(1 1 They provide a parametrization which is redundant and robust against variations A. Guffanti (Univ. Freiburg NNPDF 13 / 3

26 Neural Networks Stopping criterion Stopping criterion based on Training-Validation separation Divide the data in two sets: Training and Validation Minimize the χ of the data in the Training set Compute the χ for the data in the Validation set When Validation χ stops decreasing, STOP the fit A. Guffanti (Univ. Freiburg NNPDF 14 / 3

27 Neural Networks Stopping criterion Stopping criterion based on Training-Validation separation Divide the data in two sets: Training and Validation Minimize the χ of the data in the Training set Compute the χ for the data in the Validation set When Validation χ stops decreasing, STOP the fit #E tr and E val - rep E tr E val #E tr and E val - rep E tr E val # iterations # iterations A. Guffanti (Univ. Freiburg NNPDF 14 / 3

28 ] NNPDF. Dataset [ GeV T / p / M Q NNPDF. dataset NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H DYE65 DYE886 CDFWASY CDFZRAP DZRAP CDFRKT DRCON data points -1 1 (for comparison MSTW8 includes 699 data points Deep Inelastic Scattering F d /F p NMC-pd F p NMC SLAC BCDMS F d SLAC BCDMS σ ± HERA-I comb. NC ZEUS (HERA-II σ ± HERA-I comb. CC ZEUS (HERA-II F L H1 σ ν, σ ν CHORUS dimuon prod. NuTeV Drell-Yan & Vector Boson prod. dσ DY /dm dy E65 dσ DY /dm d F E866 W asymm. CDF Z rap. distr. D/CDF Inclusive jet prod. Incl. σ (jet CDF (k T - Run II Incl. σ (jet D (cone - Run II A. Guffanti (Univ. Freiburg NNPDF 15 / 3

29 NNPDF. Technical improvements The fit is carried at NLO QCD, in the Zero-Mass Variable Flavour Number Scheme Fast DGLAP evolution based on higher-order interpolating polynomials Improved treatment of normalization errors (t method For details see [R. D. Ball et al., arxiv:91.76] Improvements in training/stopping Target Weighted Training Improved stopping for avoiding under-/over-learning All details given in [R. D. Ball et al., arxiv:.447] A. Guffanti (Univ. Freiburg NNPDF 16 / 3

30 NNPDF. FastKernel NLO computation of hadronic observables too slow for parton global fits. MSTW8 and CTEQ include Drell-Yan NLO as (local K factors rescaling the LO cross section K-factor depends on PDFs and it is not always a good approimation. * NNPDF. includes full NLO calculation of hadronic observables. * Use available fastnlo interface for jet inclusive cross-sections.[hep-ph/6985] * Built up our own FastKernel computation of DY observables. 1,1 d 1 1 d fa( 1 f b ( C ab N 1 1 ( 1, fa( 1,α f b (,β d 1 d I (α,β ( 1, C ab ( 1,, α,β=1,1, DGLAP evol. and double convolution improved Use high-orders polynomial interpolation Precompute all Green Functions A truly NLO analysis A. Guffanti (Univ. Freiburg NNPDF 17 / 3

31 NNPDF. Parametrization Parton Distributions Combination NN architechture Singlet (Σ( = (37 pars Gluon (g( = (37 pars Total valence (V ( u V ( + d V ( = (37 pars Non-singlet triplet (T 3 ( = (37 pars Sea asymmetry ( S ( d( ū( = (37 pars Total Strangeness (s + ( (s( + s(/ = (37 pars Strange valence (s ( (s( s(/ = (37 pars 59 parameters Standard fits have 5 parameters in total No change in the parametrization from NNPDF1.... despite substantial enlargement of the dataset A. Guffanti (Univ. Freiburg NNPDF 18 / 3

32 NNPDF. General features of the fit Distribution of χ for sets χ tot 1.1 E ± σ E.3 ±. E tr ± σ E tr.9 ±.11 E val ± σ Eval.35 ±.1 TL ± σ TL ± 657 χ (k ± σ χ 1.9 ± NMC-pd NMC SLACp SLACd BCDMSp BCDMSd HERA1-NCep HERA1-NCem HERA1-CCep HERA1-CCem CHORUSnu CHORUSnb FLH8 NTVnuDMN NTVnbDMN Z6NC Z6CC DYE65 DYE886p DYE886r CDFWASY CDFZRAP DZRAP CDFRKT DRCON (k χ distribution for MC replicas.6.5 E tr distribution for MC replicas.5.4 Distribution of training lenghts (k χ tr (k E Training lenght [GA generations] A. Guffanti (Univ. Freiburg NNPDF 19 / 3

33 -5-5 NNPDF. Partons - Comparison to other global fits 4 3 NNPDF. CTEQ6.6 MSTW NNPDF. CTEQ6.6 MSTW NNPDF. CTEQ6.6 MSTW g (, Q 1 g (, Q.3. (, Q Σ NNPDF. CTEQ6.6 MSTW NNPDF. CTEQ6.6 MSTW NNPDF. CTEQ6.6 MSTW 8 (, Q T V (, Q.8.6 (, Q S (, Q s NNPDF. CTEQ6.6 MSTW 8 (, Q s NNPDF. CTEQ6.6 MSTW 8 (, Q s NNPDF. CTEQ6.6 MSTW A. Guffanti (Univ. Freiburg NNPDF / 3

34 NNPDF. Results - Partons - A couple of upshots Reduction of uncertainties with respect to older NNPDF sets due to inclusion of new data (, Q T NNPDF. NNPDF1. NNPDF A. Guffanti (Univ. Freiburg NNPDF 1 / 3

35 NNPDF. Results - Partons - A couple of upshots Reduction of uncertainties with respect to older NNPDF sets due to inclusion of new data (, Q T NNPDF. NNPDF1. NNPDF Uncertainties on PDFs competitive with results from other groups... (, Q T NNPDF. CTEQ6.6 MSTW A. Guffanti (Univ. Freiburg NNPDF 1 / 3

36 NNPDF. Results - Partons - A couple of upshots Reduction of uncertainties with respect to older NNPDF sets due to inclusion of new data (, Q T NNPDF. NNPDF1. NNPDF Uncertainties on PDFs competitive with results from other groups... (, Q T NNPDF. CTEQ6.6 MSTW but still retain unbiasedness in regions where there are little or no eperimental constraints (, Q s NNPDF. CTEQ6.6 MSTW A. Guffanti (Univ. Freiburg NNPDF 1 / 3

37 PDF Uncertainties and Correlations A practitioner s guide to NNPDF predictions Central Value F = 1 N set N set F[q (k ] k=1 Standard Deviation ( 1 N set ( 1/ σ F = F[{q (k }] F[{q}] N Nset k=1 Correlation ρ cos[ϕ(f, G] = < F G > rep < F > rep < G > rep < F > rep < F > rep < G > rep < G > rep A. Guffanti (Univ. Freiburg NNPDF / 3

38 ρ dbar s ρ dbar s σ Z PDF induced correlations E.: Top-quark studies within the NNPDF framework [J. Rojo and AG, arxiv:8.4671] It is easy to compute correlations among PDFs and observables PDF-ttbar cross-section correlation PDF-single-top cross-section correlation 1 g u ubar d 1 g u ubar d.5 sbar cb.5 sbar cb e or pairs of observbles W e W 166 Z σ [pb] tt 16 σ [pb] tt 16 [pb] tt σ σ W +[nb] σ W -[nb] [nb] A. Guffanti (Univ. Freiburg NNPDF 3 / 3

39 Reweighting PDFs Assessing the impact of new data on PDF fits Inspired by Giele and Keller [hep-ph/983393] The N rep replicas of a NNPDF fit give the probability density in the space of PDFs Epectation values for observables are Monte Carlo integrals F[f i (, Q ] = 1 N rep ( F f (net(k i (, Q N rep k=1 (... the same is true for errors, correlations, etc. We can assess the impact of including new data in the fit updating the probability density distribution. A. Guffanti (Univ. Freiburg NNPDF 4 / 3

40 Reweighting PDFs Assessing the impact of new data on PDF fits According to Bayes Theorem we have P new({f } = N χp(χ {f }P init({f }, P(χ {f } = [χ (y, {f }] n dat 1 e χ (y,{f } Monte Carlo integrals are now weighted sums where the weights are N rep ( F[f i (, Q ] = w k F f (net(k i (, Q w k = k=1 [χ (y, f k ] ndat 1 e χ (y,f k Nrep i=1 [χ (y, f i ] n dat 1 e χ (y,f i A. Guffanti (Univ. Freiburg NNPDF 5 / 3

41 Reweighting PDFs Proof-of-concept: Inclusive Jet data, reweighting vs. refitting NNPDF. (DIS+DYP NNPDF. (DIS+DYP + rw JET NNPDF. Use DIS+DY-fit as prior probability distribution Add Tevatron Inclusive Jet data through refitting and through reweighting Reweighting and refitting yield statistically equivalent results g (, Q Abs.er.(g (, Q NNPDF. (DIS+DYP NNPDF. (DIS+DYP + rw JET NNPDF A. Guffanti (Univ. Freiburg NNPDF 6 / 3

42 Reweighting PDFs Reweighting real data: W lepton asymmetry In the NNPDF. fit we only included CDF W asymmetry data 1. NNPDF. 1 NNPDF. + WLA(e,binA,D We evaluated W electron asymmetry with NNPDF replicas set using DYNNLO [Catani et al., arxiv:93.1]... and included D W electron asymmetry data points through reweighting. Main impact on reduction of middle- Valence uncertainty. V (, Q V (, Q NNPDF. NNPDF. + WLA(e,binA,D No need of refitting A. Guffanti (Univ. Freiburg NNPDF 7 / 3

43 ] NNPDF.1 Dataset [ GeV T / p / M Q NNPDF.1 dataset NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H ZEUSFC H1FC DYE65 DYE886 CDFWASY CDFZRAP DZRAP CDFRKT DRCON data points -1 1 Deep Inelastic Scattering F d /F p NMC-pd F p NMC, SLAC, BCDMS F d SLAC, BCDMS σ ± NC HERA-I, ZEUS (HERA-II σ ± CC HERA-I, ZEUS (HERA-II F L H1 σ ν, σ ν CHORUS dimuon prod. NuTeV F c ZEUS (99,3,8,9 F c H1 (1,9, Drell-Yan & Vector Boson prod. dσ DY /dm dy E65 dσ DY /dm d F E866 W asymm. CDF Z rap. distr. D/CDF Inclusive jet prod. Incl. σ (jet CDF (k T - Run II Incl. σ (jet D (cone - Run II A. Guffanti (Univ. Freiburg NNPDF 8 / 3

44 -5-5 NNPDF.1 Heavy Flavour treatment - FONLL We adopt the FONLL-A General Mass-Variable Flavour Number Scheme (M. Cacciari, M. Greco and P. Nason, (1998 Preliminary results for partons (S. Forte, P. Nason E. Laenen and J. Rojo, ( 4 CTEQ6.6 6 CTEQ CTEQ6.6 3 MSTW8nlo NNPDF. NNPDF.1 (prel 5 MSTW8nlo NNPDF. NNPDF.1 (prel 1. 1 MSTW8nlo NNPDF. NNPDF.1 (prel 4.8 g (, Q 1 (, Q Σ 3 (, Q V T Small- and medium- gluon and Singlet are most sensitive, Non-Singlet combinations mostly unaffected Effect of improved heavy flavour treatment is within one-σ, both on PDFs and LHC Standard Candles Fied-(3- and 4-Flavour Number Scheme PDFs will also be released A. Guffanti (Univ. Freiburg NNPDF 9 / 3

45 Conclusions The NNPDF methodology based on using Monte Carlo techniques and Neural Networks is well suited to address problems of standard fits. NNPDF. is the first global NNPDF fit Eact inclusion of NLO corrections No sign of strong tension among different datasets NNPDF sets are available within the LHAPDF interface. Reweighting technique allows to study impact of new data without refitting Net steps: Improved treatment of Heavy Flavour contributions (FONLL Inclusion of higher order contributions (NNLO QCD/EW effects Study the impact of theoretical uncertainties (α S, quark masses... Inclusion of resummation (small-/large- effects A. Guffanti (Univ. Freiburg NNPDF 3 / 3

Parton Distribution Functions for the LHC

Parton Distribution Functions for the LHC Present Status & Uncertainty determination Alberto Guffanti Niels Bohr International Academy & Discovery Center Niels Bohr Institute - Copenhagen Ph. D. course