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