Neural Network Fragmentation Functions [V. Bertone et al. , Eur. - - PowerPoint PPT Presentation

Neural Network Fragmentation Functions

Valerio Bertone

NIKHEF and VU Amsterdam

REF 2017

November 15, 2017, Madrid

[V. Bertone et al., Eur. Phys. J. C 77 (2017) no.8, 516]

pT

|η| < 1.0 |η| < 0.8 √s = 7.0 TeV

pT

|η| < 1.0 √s = 1960.0 GeV

pT

|η| < 1.0 |η| < 0.8 √s = 2760 GeV

pT

|η| < 1.0 |η| < 0.8 |η| < 2.5 √s = 900 GeV

d’Enterria et al. [ArXiv:1311.1415]

Comparison data/theory for inclusive charged-hadron pT spectra: Large energy data tend to be overshot by predictions obtained with most of the current FF sets ⇒ gluon FF at large z?

Motivation

Why bother about fragmentation functions

Outline

Fitting methodology Theoretical framework and settings Experimental data Results Preliminary: charged hadron FFs Summary and outlook

Fitting methodology

Fitting methodology

The NNPDF approach

Monte Carlo sampling:

construct a set of data replicas which reproduces the statistical features of the

• riginal dataset,

clear statistical interpretation.

Neural network (NN) parametrisation:

flexible (redundant) functional form parametrised by a large set of parameters.

Genetic algorithm for the fit:

suitable exploration of the parameter space to avoid to fall into local minima of the figure of merit.

Determination of the best fit by cross-validation:

exploit the random distribution of the statistical fluctuations in a given MC replica to avoid over-learning.

So far, successfully used to extract PDFs.

Each NN has 37 free parameters. FFs are expressed as

The NNi(1) term ensures that fi(x) −

→

x→1 0

FFs are parametrised in terms of NN with architecture (2-5-3-1) as: fi(x) = NNi(x) − NNi(1)

Fitting methodology

A word on the parametrisation

g(x) = sign(x) ln(|x| + 1) Activation function:

ξ(j)

i

= g

(j-1)th layer

X

k

ξ(j−1)

k

ω(j)

ki − θ(j) i

!

Theoretical framework and settings

DGLAP evolution

FFs obey the standard collinear DGLAP evolution equations: Time-like splitting functions known up to NNLO

• A. Mitov and S. O. Moch [hep-ph/0604160]
• M. Gluck, E. Reya, and A.

Vogt [Phys.Rev. D48 (1993)]

µ2 ∂ ∂µ2 Dh

NS

= P +

NS ⊗ Dh NS

µ2 ∂ ∂µ2 ✓Dh

Σ

Dh

g

◆ = ✓Pqq Pqg Pgq Pgg ◆ ⊗ ✓Dh

Σ

Dh

g

◆ Numerical implementation up to NNLO in the APFEL code:

careful benchmark against in the the N-space MELA code, perfect agreement with QCDNUM (after a correction of a bug in the latter).

• V. Bertone, S. Carrazza, E. R. Nocera [arXiv:1501.00494]
• M. Botje [arXiv:1602.08383]
• V. Bertone, C. Carrazza, J. Rojo [arXiv:1310.1394]

Single inclusive annihilation

Single-Inclusive-Annihilation (SIA): The cross section factorises: dσh dz = Fh

L + Fh T ≡ Fh 2

= ˆ σh ⇥ C2,q ⊗ Dh

Σ + C2,g ⊗ Dh g + C2,NS ⊗ Dh NS

⇤ Identified hadron (π±, K±, p/p, …)

Single-Inclusive-Annihilation (SIA): The cross section factorises as: dσh dz = Fh

L + Fh T ≡ Fh 2

= ˆ σh ⇥ C2,q ⊗ Dh

Σ + C2,g ⊗ Dh g + C2,NS ⊗ Dh NS

⇤ Perturbative Wilson coefficients known up to NNLO in the massless scheme

• A. Mitov and S. O. Moch [hep-ph/0604160]

Single inclusive annihilation

Identified hadron (π±, K±, p/p, …)

Single-Inclusive-Annihilation (SIA): The cross section factorises: dσh dz = Fh

L + Fh T ≡ Fh 2

= ˆ σh ⇥ C2,q ⊗ Dh

Σ + C2,g ⊗ Dh g + C2,NS ⊗ Dh NS

⇤ Only a limited set of combinations of FFs can be constrained by SIA cross sections

Single inclusive annihilation

Identified hadron (π±, K±, p/p, …)

Single inclusive annihilation

Single-Inclusive-Annihilation (SIA): The cross section factorises: dσh dz = Fh

L + Fh T ≡ Fh 2

= ˆ σh ⇥ C2,q ⊗ Dh

Σ + C2,g ⊗ Dh g + C2,NS ⊗ Dh NS

⇤ Numerical implementation up to NNLO in the APFEL code:

partial benchmark against the DSS implementation.

• D. P

. Anderle, F. Ringer, M. Stratmann [arXiv:1510.05854]

Identified hadron (π±, K±, p/p, …)

5 independent FFs for each hadronic species h:

inclusive SIA data only constrains three FF combinations, heavy-quark FFs constrained directly by tagged SIA data.

Parametrisation scale:

substantial heavy-quark intrinsic component, heavy-quark FFs parametrised on the same footing as the light FFs.

Settings

Physical parameters:

αs(MZ) = 0.118, αem(MZ) = 1/127, mc = 1.51 GeV, mb = 4.92 GeV

Kinematic cuts: Q0 = 5 GeV (> mc, mb) Each FF is parametrised by a Neural Net (architecture 2-5-3-1).

zmin ≤ z ≤ zmax, zmin = ⇢ 0.02 for √s = MZ 0.075

• therwise

, zmax = 0.9

• Dh

u+, Dh s++d+, Dh c+, Dh b+, Dh g

5 independent FFs for each hadronic species h:

inclusive SIA data only constrains three FF combinations, heavy-quark FFs constrained directly by tagged SIA data.

Parametrisation scale:

substantial heavy-quark intrinsic component, heavy-quark FFs parametrised on the same footing as the light FFs.

Settings

Physical parameters:

αs(MZ) = 0.118, αem(MZ) = 1/127, mc = 1.51 GeV, mb = 4.92 GeV

Kinematic cuts: Q0 = 5 GeV (> mc, mb) Each FF is parametrised by a Neural Net (architecture 2-5-3-1).

zmin ≤ z ≤ zmax, zmin = ⇢ 0.02 for √s = MZ 0.075

• therwise

, zmax = 0.9

• Dh

u+, Dh s++d+, Dh c+, Dh b+, Dh g

contributions ∝ Mh/ sz2 and ln(z)

5 independent FFs for each hadronic species h:

inclusive SIA data only constrains three FF combinations, heavy-quark FFs constrained directly by tagged SIA data.

Parametrisation scale:

substantial heavy-quark intrinsic component, heavy-quark FFs parametrised on the same footing as the light FFs.

Settings

Physical parameters:

αs(MZ) = 0.118, αem(MZ) = 1/127, mc = 1.51 GeV, mb = 4.92 GeV

Kinematic cuts: Q0 = 5 GeV (> mc, mb) Each FF is parametrised by a Neural Net (architecture 2-5-3-1).

zmin ≤ z ≤ zmax, zmin = ⇢ 0.02 for √s = MZ 0.075

• therwise

, zmax = 0.9

• Dh

u+, Dh s++d+, Dh c+, Dh b+, Dh g

contributions ∝ ln(1 - z)

Experimental data

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

Dataset

Only SIA cross sections (normalised and absolute) included.

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

Dataset

Only SIA cross sections (normalised and absolute) included.

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

PETRA (DESY)

• Phys. Lett. B94 (1980) 444

Z.Phys. C17 (1983) 5-15 Z.Phys. C42 (1989) 189

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

Dataset

Only SIA cross sections (normalised and absolute) included.

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

PETRA (DESY) KEK

• Phys. Rev. Lett. 111 (2013) 062002
• Phys. Rev. D92 (2015), no. 9 092007
• Phys. Lett. B345 (1995) 335-342
• Phys. Lett. B94 (1980) 444

Z.Phys. C17 (1983) 5-15 Z.Phys. C42 (1989) 189

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

Dataset

Only SIA cross sections (normalised and absolute) included.

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

PETRA (DESY) KEK SLAC

• Phys. Rev. Lett. 111 (2013) 062002
• Phys. Rev. D92 (2015), no. 9 092007
• Phys. Lett. B345 (1995) 335-342
• Phys. Lett. B94 (1980) 444

Z.Phys. C17 (1983) 5-15 Z.Phys. C42 (1989) 189

• Phys. Rev. D88 (2013) 032011
• Phys. Rev. Lett. 61 (1988) 1263
• Phys. Rev. D69 (2004) 072003

X.Q. Lu, Ph.D. thesis, Johns Hopkins U., 1986

10.5 12 14 22 29 34 44 58 91.2 0.01 0.1 1

√s [GeV] z π±

TASSO

π±

BELLE TOPAZ

π±

BABAR TPC SLD

π±

ALEPH DELPHI OPAL

0.01 0.1

z K±

Dataset

Only SIA cross sections (normalised and absolute) included. PETRA (DESY) KEK SLAC LEP (CERN) 428 data points (after cuts)

• Phys. Rev. Lett. 111 (2013) 062002
• Phys. Rev. D92 (2015), no. 9 092007
• Phys. Lett. B345 (1995) 335-342
• Z. Phys. C66 (1995) 355-366
• Eur. Phys. J. C5 (1998) 585-620
• Z. Phys. C63 (1994) 181-196.
• Phys. Lett. B94 (1980) 444

Z.Phys. C17 (1983) 5-15 Z.Phys. C42 (1989) 189

• Phys. Rev. D88 (2013) 032011
• Phys. Rev. Lett. 61 (1988) 1263
• Phys. Rev. D69 (2004) 072003

X.Q. Lu, Ph.D. thesis, Johns Hopkins U., 1986

Only SIA cross sections (normalised and absolute) included. We have fitted FFs also to K± and p/p data.

Dataset

385 data points (after cuts) 360 data points (after cuts)

1 1 1 1 0.01 0.1 1

z K±

0.01 0.1 1

z p/p

• 10

20 30 40 50 60 70 80 90 100

√s [GeV]

10 20 30 40 50 60 70 80 90 100

√s [GeV]

10 20 30 40 50 60 70 80 90 100

√s [GeV]

10 20 30 40 50 60 70 80 90 100

√s [GeV]

1

π± π± π± π±

ALEPH DELPHI OPAL

0.01 0.1 1

z K±

0.01 0.1 1

z p/p

Results: the NNFF1.0 sets

Fit quality

Fit quality increasingly better going from LO to NNLO: substantial from LO to NLO, more moderate from NLO to NNLO. NNLO corrections are anyway beneficial (particularly for pions).

Fit quality

Fit quality increasingly better going from LO to NNLO: substantial from LO to NLO, more moderate from NLO to NNLO. NNLO corrections are anyway beneficial (particularly for pions). Tension between BELLE and BABAR for kaons and protons:

• pposite trend upon inclusion of

higher-order corrections.

Fit quality

Fit quality increasingly better going from LO to NNLO: substantial from LO to NLO, more moderate from NLO to NNLO. NNLO corrections are anyway beneficial (particularly for pions). Tension between BELLE and BABAR for kaons and protons:

• pposite trend upon inclusion of

higher-order corrections. Anomalously small χ2 for BELLE: possible underestimate of the uncorrelated systematic uncertainty.

Fit quality

Possible tension also between DELPHI inclusive and the other experiments at MZ:

• pposite trend upon inclusion of higher-order corrections.

Fit quality increasingly better going from LO to NNLO: substantial from LO to NLO, more moderate from NLO to NNLO. NNLO corrections are anyway beneficial (particularly for pions). Tension between BELLE and BABAR for kaons and protons:

• pposite trend upon inclusion of

higher-order corrections. Anomalously small χ2 for BELLE: possible underestimate of the uncorrelated systematic uncertainty.

Description of the data

10-4 10-3 10-2 10-1 100 101 0.2 0.4 0.6 0.8

z

dσ/dz [GeV] BELLE

NNLO theory π± K± p/p

• 0.2

0.4 0.6 0.8

z

(1/σtot) dσ/dz BABAR

0.8 1 1.2 BELLE data/theory (NNLO) 0.8 1 1.2 BELLE 0.8 1 1.2 BELLE 0.8 1 1.2 BABAR (prompt) 0.8 1 1.2 BABAR (conventional) 0.2 0.4 0.6 0.8 1 0.8 1 1.2

z

BABAR (prompt)

Data/Theory comparison for BELLE and BABAR using NNFF1.0 at NNLO: the bands indicate the 1-σ uncertainty. Very good description in the region not excluded by the kinematic cuts (shaded areas). Different trend of the data at low z for kaons and particularly for protons: possible reason of the worsening of the χ2.

Data/Theory comparison for the experiments at MZ using NNFF1.0 at NNLO. Very good description in the region allowed by the kinematic cuts. Often also the data excluded by the cuts are well described. The predictions for pions for DELPHI overshoot the data:

• rigin of the worse χ2 as

compared to the other experiments at MZ.

10-2 10-1 100 101 102

(1/σtot) dσ/dz ALEPH

NNLO theory π± K± p/p

• (1/σtot) dσ/dz

DELPHI

10-2 10-1 100 101 102 0.01 0.1

z

(1/σtot) dσ/dz OPAL

0.01 0.1

z

(1/σtot) dσ/dz SLD

0.6 1 1.4 data/theory (NNLO) ALEPH (inclusive) 0.6 1 1.4 ALEPH (inclusive) 0.6 1 1.4 ALEPH (inclusive) 0.6 1 1.4 DELPHI (inclusive) 0.6 1 1.4 DELPHI (inclusive) 0.6 1 1.4 DELPHI (inclusive) 0.6 1 1.4 OPAL (inclusive) 0.6 1 1.4 OPAL (inclusive) 0.6 1 1.4 SLD (inclusive) 0.6 1 1.4 SLD (inclusive) 0.01 0.1 1 0.6 1 1.4

z

SLD (inclusive)

Description of the data

Fragmentation functions

Perturbative stability (Pions)

3 6 9 12

zDu+

π±

(z,Q)

2 4 6 8

zDc+

π±

(z,Q) zDd++s+

π± (z,Q)

Q = 10 GeV

LO NLO NNLO

zDg

π±

(z,Q) zDb+

π±

(z,Q)

1 2

Ratio to LO

1 2 1 2

Ratio to NLO

1 2 0.1 1

z

0.1 1

z

0.1 1

z

Stabilisation going from LO to NNLO, LO uncertainties slightly larger: poorer theoretical description.

Fragmentation functions

Perturbative stability (Kaons)

Same for kaons…

0.5 1 1.5 2

zDu+

K±

(z,Q)

0.5 1 1.5 2

zDc+

K±

(z,Q) zDd++s+

K± (z,Q)

Q = 10 GeV

LO NLO NNLO

zDg

K±

(z,Q) zDb+

K±

(z,Q)

1 2

Ratio to LO

1 2 1 2

Ratio to NLO

1 2 0.1 1

z

0.1 1

z

0.1 1

z

Fragmentation functions

Perturbative stability (Protons)

…and for protons.

0.5 1 1.5 2

zDu+

p/p

• (z,Q)

0.25 0.5 0.75 1

zDc+

p/p

• (z,Q)

zDd++s+

p/p

• (z,Q)

Q = 10 GeV

LO NLO NNLO

zDg

p/p

• (z,Q)

zDb+

p/p

• (z,Q)

1 2

Ratio to LO

1 2 1 2

Ratio to NLO

1 2 0.1 1

z

0.1 1

z

0.1 1

z

Fragmentation functions

Comparison with other FF sets

More sizeable differences between NNFF1.0 and JAM16: the d+ + s+ and gluon distributions well beyond 1-σ, generally smaller uncertainties of the JAM16 distributions (despite similar dataset).

3 6 9 12

zDu+

π±

(z,Q)

2 4 6 8

zDc+

π±

(z,Q) zDd++s+

π± (z,Q)

Q = 10 GeV, NLO

NNFF1.0 1σ DEHSS 68% CL JAM 1σ

zDg

π±

(z,Q) zDb+

π±

(z,Q)

1 2

Ratio to NNFF1.0

1 2 1 2

Ratio to NNFF1.0

1 2 0.1 1

z

0.1 1

z

0.1 1

z

Compare our results against the most recent FF sets: DEHSS

• D. de Florian et al. [arXiv:1410.6027]

JAM16

• N. Sato et al. [arXiv:1609.00899]

Only available for pions and kaons at NLO. Pions: fair agreement between NNFF1.0 and DEHSS: differences u+ and gluon distributions at the 1-σ level.

0.5 1 1.5 2

zDu+

K±

(z,Q)

0.5 1 1.5 2

zDc+

K±

(z,Q) zDd++s+

K± (z,Q)

Q = 10 GeV, NLO

NNFF1.0 1σ DEHSS 68% CL JAM 1σ

zDg

K±

(z,Q) zDb+

K±

(z,Q)

1 2

Ratio to NNFF1.0

1 2 1 2

Ratio to NNFF1.0

1 2 0.1 1

z

0.1 1

z

0.1 1

z

Fragmentation functions

Comparison with other FF sets

larger differences in the uncertainties, particularly for the gluon distributions. Compare our results against the most recent FF sets: DEHSS

• D. de Florian et al. [arXiv:1410.6027]

JAM16

• N. Sato et al. [arXiv:1609.00899]

Only available for pions and kaons at NLO. Kaons: more substantial differences: fair agreement only for b+.

Charged hadron FFs

A brief overview

Many experiments provide data for charged hadron production:

this data includes, not only pions, kaons and protons, but also heavier (and less abundant) charged hadrons.

Restricting to SIA experiments, data is available from:

TASSO, TPC, ALEPH, DELPHI, OPAL, SLD.

Some experiments measure also the longitudinal cross sections:

ALEPH, DELPHI, OPAL.

Predictions for the longitudinal cross section start at O(αs):

this data provides a strong handle on the gluon distribution. As a consequence it is not possible to go beyond NLO (i.e. O(αs2)) yet.

Charged hadron FFs

Comparison to the LHC data

Strong sensitivity to the gluon distribution. Very significant impact of the longitudinal data:

dramatic reduction of the uncertainty, better agreement with CMS data.

LHC (and Tevatron) data is expected to have a big impact on FFs.

1 2 3 4 5 6 7 1 10 100 E d3σ/d3p [Theory / Data] pT [GeV] CMS charged particle differential cross section at 2.76 TeV for |η|<1 Data NNFF1.0 NNFF1.0 (no FL) DSS07

Summary

NNFF1.0 sets based on the NNPDF approach:

based on SIA data, provided at LO, NLO, and NNLO, for π±, K± and p/p, faithful uncertainty estimate (validated by closure tests), remarkable perturbative convergence, acceptable agreement with other determinations.

Same approach adopted for the charged hadron FFs.

Outlook

Proton-proton collision data from the LHC and Tevatron, SIDIS data from HERMES and COMPASS, heavy-quark mass corrections to SIA.

Backup slides

Motivation

Why bother about fragmentation functions

Laeder et al. [ArXiv:1103.5979]

Extraction of the longitudinally polarised parton distribution functions:

• J. J. Ethier et al. [ArXiv:1705.05889]

In the presence of semi-inclusive DIS data the strange quark distribution is very sensitive to the choice of the FF set used in the analysis. Even fitting PDFs and FFs simultaneously does no lead to a definitive answer.

Charged hadron FFs

Preliminary results

5 10 15 20 25 30

zDh±

Σ (z,Q2)

Q2=M2

Z

NNPDF DSS

0.4 1 1.6 0.01 0.1 z

NNPDF/DSS

(5×) zDh±

g (z,Q2)

0.01 0.1 z 2 4

zDh±

c+ (z,Q2)

0.4 1 1.6

NNPDF/DSS

2 4

zDh±

b+ (z,Q2)

0.01 0.1 1 0.4 1 1.6 z

NNPDF/DSS

• E. Nocera [ArXiv:1709.03400]

Charged hadron FFs at the Z-boson mass scale: Significant differences w.r.t. DSS, particularly for the gluon.

In the NNPDF procedure applied to PDFs the parametrisation is: fi(x) = xαi(1 − x)βiNNi(x) Preprocessing function The preprocessing function:

helps implement physical constraints (e.g. fi(1) = 0 and integrability), determines the behaviour in the extrapolation regions, facilitates the task of the neural network making the fit easier.

The values of αi and βi are iteratively determined from data. For the fits of FFs we remove the preprocessing functions and use: fi(x) = NNi(x) − NNi(1) no need to iterate to determine αi and βi. the NN defines the behaviour also in the extrapolation regions.

Fitting methodology

The NNPDF approach

• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 2.5

• 10
• 5

5 10 x g0(x) g1(x)

ξ(j)

i

= g

(j-1)th layer

X

k

ξ(j−1)

k

ω(j)

ki − θ(j) i

!

Saturating function g0(x) = 1 1 + e−x Non-saturating function g1(x) = sign(x) ln(|x| + 1) Removing the preprocessing functions requires a proper choice of the activation function of the neural networks:

Fitting methodology

The NNPDF approach

θ(1)

1

θ(1)

2

θ(1)

3

θ(2)

1

θ(2)

2

θ(3)

1

ω(1)

11

ω(1)

12

ω(1)

13

ω(2)

11

ω(2)

12

ω(2)

21

ω(2)

22 ω(2) 31

ω(2)

32

ω(3)

11

ω(3)

21

Input Hidden Hidden Output

x Ni(x)

Saturating function g0(x) = 1 1 + e−x Non-saturating function g1(x) = sign(x) ln(|x| + 1) Removing the preprocessing functions requires a proper choice of the activation function of the neural networks:

θ(1)

1

θ(1)

2

θ(1)

3

θ(2)

1

θ(2)

2

θ(3)

1

ω(1)

11

ω(1)

12

ω(1)

13

ω(2)

11

ω(2)

12

ω(2)

21

ω(2)

22 ω(2) 31

ω(2)

32

ω(3)

11

ω(3)

21

Input Hidden Hidden Output

x Ni(x)

ξ(j)

i

= g

(j-1)th layer

X

k

ξ(j−1)

k

ω(j)

ki − θ(j) i

!

Fitting methodology

The NNPDF approach

xDg(x,Q2)

Data region

Saturating function g0(x) = 1 1 + e−x Non-saturating function g1(x) = sign(x) ln(|x| + 1)

ξ(j)

i

= g

(j-1)th layer

X

k

ξ(j−1)

k

ω(j)

ki − θ(j) i

!

Removing the preprocessing functions requires a proper choice of the activation function of the neural networks:

Fitting methodology

The NNPDF approach

xDg(x,Q2)

Data region

θ(1)

1

θ(1)

2

θ(1)

3

θ(2)

1

θ(2)

2

θ(3)

1

ω(1)

11

ω(1)

12

ω(1)

13

ω(2)

11

ω(2)

12

ω(2)

21

ω(2)

22 ω(2) 31

ω(2)

32

ω(3)

11

ω(3)

21

Input Hidden Hidden Output

x Ni(x)

Saturating function g0(x) = 1 1 + e−x Non-saturating function g1(x) = sign(x) ln(|x| + 1)

ξ(j)

i

= g

(j-1)th layer

X

k

ξ(j−1)

k

ω(j)

ki − θ(j) i

!

Removing the preprocessing functions requires a proper choice of the activation function of the neural networks: Our choice for the FFs

Fitting methodology

The NNPDF approach

xDg(x,Q2)

Data region

θ(1)

1

θ(1)

2

θ(1)

3

θ(2)

1

θ(2)

2

θ(3)

1

ω(1)

11

ω(1)

12

ω(1)

13

ω(2)

11

ω(2)

12

ω(2)

21

ω(2)

22 ω(2) 31

ω(2)

32

ω(3)

11

ω(3)

21

Input Hidden Hidden Output

x Ni(x)

Closure tests

How do we know whether our fitting strategy is reliable?

1) Assume underlying FFs are known (e.g. HKNS07). 2) Generate pseudo-data with given statistical and correlated systematics. 3) Perform a fit and compare to the “truth”.

If needed, use the closure tests to tune the fitting algorithm. Levels of closure tests: NNPDF Collaboration [arXiv:1410.8849]

level 0:

data point central values equal to the HKNS07 “true” values, uncertainties assumed equal to the experimental ones, we must find χ2 ~ 0 and that uncertainty on predictions tends to zero.

level 2:

data points obtained as random fluctuations with exp. covariance matrix about the “truth”, generate Monte Carlo replicas of this data, fit a PDF set to each Monte Carlo replica, we must find χ2 ~ 1 and that HKNS07 “true” FFs are within the 1-σ band.

Closure tests: “level 0”

χ2 (global) = 0.00027

We find:

Predictions coincide with the data central values. Prediction uncertainties shrink to zero. FFs in the data region very close to the “truth”. Uncertainties blow up in the extrapolation region.

kinematic cut

xDg(x,Q2)

Predictions coincide with the data central values. Prediction uncertainties shrink to zero. FFs in the data region very close to the “truth”. Uncertainties blow up in the extrapolation region.

Closure tests: “level 0”

χ2 (global) = 0.00027

We find: Functional uncertainty

kinematic cut

xDg(x,Q2)

Closure tests: “level 2”

We find:

χ2 (global) = 0.99307

“True” FFs do fall within the 1-σ band of the fitted FFs (in the data region), Faithful representation of the experimental uncertainty.

xDu(x,Q2) xDg(x,Q2)

kinematic cut kinematic cut

χ2 to data and PDF uncertainty on predictions tend to zero as the training length increases. LEVEL 0 closure test works!

Methodological Improvements

Closure Tests: LEVEL 0

50

Fits produced with increasing (fixed) training length. All fits on the same dataset (NNPDF2.3 dataset).

χ2 tends to χ2MSTW (0.96) and the MSTW “true PDFs” are 68% of times within the 1-σ error band. LEVEL 2 closure test works!

Methodological Improvements

Closure Tests: LEVEL 2

51

Fits produced with increasing (fixed) training length. All fits on the same dataset (NNPDF2.3 dataset).

Methodological Improvements

Closure Tests

Both LEVEL 0 and LEVEL 2 gave a positive result.

PDF errors in LEVEL 0 closure test reflect the functional uncertainty. PDF errors in LEVEL 2 closure test reflect the data uncertainty.

In the data region: data uncertainty ≫ functional uncertainty In the extrapolation region: data uncertainty ~ functional uncertainty. Prove of the reliability of the NNPDF methodology.

0.6 1 1.4 TASSO12 data/theory (NNLO) π± TASSO14 TASSO22 0.6 1 1.4 TASSO34 TASSO44 TPC (inclusive) 0.2 1 1.8 TPC (uds tagged) TPC (c tagged) TPC (b tagged) 0.6 1 1.4 TOPAZ DELPHI (uds tagged) DELPHI (b tagged) 0.6 1 1.4 0.01 0.1

z

SLD (uds tagged) 0.01 0.1

z

SLD (c tagged) 0.01 0.1 1

z

SLD (b tagged)

0.2 1 1.8 TASSO12 data/theory (NNLO) p/p

• TASSO14

TASSO22 0.2 1 1.8 TASSO30 TASSO34 TPC (inclusive) 0.2 1 1.8 TOPAZ DELPHI (uds tagged) DELPHI (b tagged) 0.2 1 1.8 0.01 0.1

z

SLD (uds tagged) 0.01 0.1

z

SLD (c tagged) 0.01 0.1 1

z

SLD (b tagged)

0.2 1 1.8 TASSO12 data/theory (NNLO) K± TASSO14 0.2 1 1.8 TASSO22 TASSO34 TPC (inclusive) 0.2 1 1.8 TOPAZ DELPHI (uds tagged) DELPHI (b tagged) 0.2 1 1.8 0.01 0.1

z

SLD (uds tagged) 0.01 0.1

z

SLD (c tagged) 0.01 0.1 1

z

SLD (b tagged)