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Motivation Structure Functions Parton Distributions Conclusions
Neural Networks and the Structure of the Proton
Andrea Piccione
Milano, 12 Ottobre 2005
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The NNPDF Collaboration
Luigi Del Debbio1, Stefano Forte2,Jose I. Latorre3, A. P.4 and Joan Rojo3
1 Theory Division, CERN2 Dipartimento di Fisica, Universita di Milano and INFN, Sezione di Milano
3 Departament d’Estructura i Constituents de la Materia, Universitat de Barcelona4 Dipartimento di Fisica Teorica, Universita di Torino and INFN, Sezione di Torino
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The name of the game
What do we need for the LHC?
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The name of the game
What do we need for the LHC?
[Djouadi and Ferrag 2003]
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The name of the game
Deep Inelastic Scattering
I The cross section
d2σ
dxdQ2=
4πα2
Q4
»[1 + (1− y)2]F1 +
1− y
x(F2 − 2xF1)
–
I The structure function
F2(x , Q2) = x
"nfX
q=1
e2q Cq ⊗ qq(x , Q2) + 2nf Cg ⊗ g(x , Q2)
#
I Parton distribution evlution is described by DGLAP equations
Q2 d
dQ2q(x , Q2) =
αs(Q2)
2π(P ⊗ q)(x , Q2)
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The name of the game
Deep Inelastic Scattering and QCD
I The cross section
d2σ
dxdQ2=
4πα2
Q4
»[1 + (1− y)2]F1 +
1− y
x(F2 − 2xF1)
–
I The structure function
F2(x , Q2) = x
"nfX
q=1
e2q Cq ⊗ qq(x , Q2) + 2nf Cg ⊗ g(x , Q2)
#
I Parton distribution evlution is described by DGLAP equations
Q2 d
dQ2q(x , Q2) =
αs(Q2)
2π(P ⊗ q)(x , Q2)
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The name of the game
The problem
I For a single quantity → 1 sigma error
I For a pair of numbers → 1 sigma ellipse
I For a function → We need an “error band” in the space offunctions (i.e. the probability density P [f ] in the space offunctions f (x))
Expectation values → Functional integrals
〈F [f (x)]〉 =
ZDfF [f (x)]P [f (x)]
Determine an infinite-dimensional object (a function) from finiteset of data points → Mathematically ill-posed problem
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The name of the game
The problem
I For a single quantity → 1 sigma error
I For a pair of numbers → 1 sigma ellipse
I For a function → We need an “error band” in the space offunctions (i.e. the probability density P [f ] in the space offunctions f (x))
Expectation values → Functional integrals
〈F [f (x)]〉 =
ZDfF [f (x)]P [f (x)]
Determine an infinite-dimensional object (a function) from finiteset of data points → Mathematically ill-posed problem
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
Ways out
The standard approach
1. Choose a simple functional form with enough free parameters
q(x , Q20 ) = xα(1− x)βP(x ; λ1, . . . , λn)
2. Fit parameters by minimizing χ2
Problem:
I Which is the theoretical bias due to the parametrization?
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
Ways out
The standard approach
1. Choose a simple functional form with enough free parameters
q(x , Q20 ) = xα(1− x)βP(x ; λ1, . . . , λn)
2. Fit parameters by minimizing χ2
Problem:
I Which is the theoretical bias due to the parametrization?
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
Ways out
The NNPDF approach
I Determination of the Structure Functions:this is the easiest case, since no evolution is required, but onlydata fitting. A good application to test the technique → Done
I Determination of the Parton Distributions:the real stuff → Working on it ...
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The NNPDF approach
What are Neural Networks?Neural networks are algorithms providing robust, universal,unbiased approximants to incomplete or noisy data
I Activation function:
ξ(l)i = g
0@nl−1Xj=1
ω(l−1)ij ξ
(l−1)j − θ
(l)i
1Ag(x) =
1
1 + e−x
I Parameters: weights ω(l)ij and
thresholds θ(l)i
I Assumption: smoothness and
convergence criterium
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The NNPDF approach
What are Neural Networks?Neural networks are algorithms providing robust, universal,unbiased approximants to incomplete or noisy data
I Activation function:
ξ(l)i = g
0@nl−1Xj=1
ω(l−1)ij ξ
(l−1)j − θ
(l)i
1Ag(x) =
1
1 + e−x
I Parameters: weights ω(l)ij and
thresholds θ(l)i
I Assumption: smoothness and
convergence criterium
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The NNPDF approach
General strategy
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
Results
Credits
I S. Forte, L. Garrido, J. I. Latorre and A. P., “Neural networkparametrization of deep-inelastic structure functions,”JHEP05 (2002) 062 [arXiv:hep-ph/0204232]
I L. Del Debbio, S. Forte, J. I. Latorre, A. P. and J. Rojo[NNPDF Collaboration], “Unbiased determination of theproton structure function F p
2 with faithful uncertaintyestimation”, JHEP03 (2005) 080 [arXiv:hep-ph/0501067]
Source code, driver program and graphical web interface for F2
plots and numerical computations available @
http://sophia.ecm.ub.es/f2neural
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
Results
Fit of F p2 (x , Q2) [NNPDF 2005]
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The NNPDF approach
Same strategy, but much more complex!
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The NNPDF approach
Same strategy, but much more complex!
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The NNPDF approach
Same strategy, but much more complex!
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
The NNPDF approach
Same strategy, but much more complex!
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
Results
Non-Singlet
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
Outlook
I NN fit of other structure functions (e. g. F3(x , Q2))
I Construct full set of NNPDF parton distributions from allavailable data
I Assess impact of uncertainties of PDFs for relevantobservables at LHC
I Make formalism compatible with standard interfaces(LHAPDF, PDFLIB) → NNPDF partons available for use inMonte Carlo generators
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton
Motivation Structure Functions Parton Distributions Conclusions
Outlook
I NN fit of other structure functions (e. g. F3(x , Q2))
I Construct full set of NNPDF parton distributions from allavailable data
I Assess impact of uncertainties of PDFs for relevantobservables at LHC
I Make formalism compatible with standard interfaces(LHAPDF, PDFLIB) → NNPDF partons available for use inMonte Carlo generators
Andrea Piccione Milano, 12 Ottobre 2005
Neural Networks and the Structure of the Proton