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

Neural Network Fragmentation Functions [V. Bertone et al. , Eur. Phys. J. C 77 (2017) no.8, 516] Valerio Bertone NIKHEF and VU Amsterdam REF 2017 November 15, 2017, Madrid

Motivation Why bother about fragmentation functions Comparison data/theory for inclusive charged-hadron p T spectra: √ s = 7 . 0 TeV √ s = 1960 . 0 GeV d’Enterria et al. [ArXiv:1311.1415] | η | < 1 . 0 | η | < 0 . 8 | η | < 1 . 0 p T p T √ s = 2760 GeV √ s = 900 GeV | η | < 1 . 0 | η | < 1 . 0 | η | < 0 . 8 | η | < 0 . 8 | η | < 2 . 5 p T p T Large energy data tend to be overshot by predictions obtained with most of the current FF sets ⇒ gluon FF at large z ?

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 original 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 .

Fitting methodology A word on the parametrisation FFs are parametrised in terms of NN with architecture (2-5-3-1) as: ! (j-1)th layer ξ ( j ) ξ ( j − 1) ω ( j ) ki − θ ( j ) X = g i k i k Activation function: g ( x ) = sign( x ) ln( | x | + 1) Each NN has 37 free parameters. FFs are expressed as f i ( x ) = NN i ( x ) − NN i (1) The NN i (1) term ensures that f i ( x ) − x → 1 0 →

Theoretical framework and settings

DGLAP evolution FFs obey the standard collinear DGLAP evolution equations: µ 2 ∂ P + ∂ µ 2 D h NS ⊗ D h = NS NS ✓ D h ✓ D h ◆ ✓ P qq ◆ ◆ µ 2 ∂ P qg Σ Σ ⊗ = ∂ µ 2 P gq P gg D h D h g g 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)] Numerical implementation up to NNLO in the APFEL code: V. Bertone, C. Carrazza, J. Rojo [arXiv:1310.1394] careful benchmark against in the the N -space MELA code, V. Bertone, S. Carrazza, E. R. Nocera [arXiv:1501.00494] perfect agreement with QCDNUM (after a correction of a bug in the latter). M. Botje [arXiv:1602.08383]

Single inclusive annihilation Single-Inclusive-Annihilation (SIA): Identified hadron ( π ± , K ± , p/p, …) The cross section factorises: d σ h F h L + F h T ≡ F h = 2 dz σ h C 2 ,q ⊗ D h Σ + C 2 ,g ⊗ D h g + C 2 , NS ⊗ D h ⇥ ⇤ = ˆ 0 NS

Single inclusive annihilation Single-Inclusive-Annihilation (SIA): Identified hadron ( π ± , K ± , p/p, …) The cross section factorises as: d σ h F h L + F h T ≡ F h = 2 dz σ h C 2 ,q ⊗ D h Σ + C 2 ,g ⊗ D h g + C 2 , NS ⊗ D h ⇥ ⇤ = ˆ 0 NS Perturbative Wilson coefficients known up to NNLO in the massless scheme A. Mitov and S. O. Moch [hep-ph/0604160]

Single inclusive annihilation Single-Inclusive-Annihilation (SIA): Identified hadron ( π ± , K ± , p/p, …) The cross section factorises: d σ h F h L + F h T ≡ F h = 2 dz σ h C 2 ,q ⊗ D h Σ + C 2 ,g ⊗ D h g + C 2 , NS ⊗ D h ⇥ ⇤ = ˆ 0 NS Only a limited set of combinations of FFs can be constrained by SIA cross sections

Single inclusive annihilation Single-Inclusive-Annihilation (SIA): Identified hadron ( π ± , K ± , p/p, …) The cross section factorises: d σ h F h L + F h T ≡ F h = 2 dz σ h C 2 ,q ⊗ D h Σ + C 2 ,g ⊗ D h g + C 2 , NS ⊗ D h ⇥ ⇤ = ˆ 0 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]

Settings Physical parameters : α s ( M Z ) = 0 . 118 , α em ( M Z ) = 1 / 127 , m c = 1 . 51 GeV , m b = 4 . 92 GeV Parametrisation scale : Q 0 = 5 GeV ( > m c , m b ) substantial heavy-quark intrinsic component, heavy-quark FFs parametrised on the same footing as the light FFs. 5 independent FFs for each hadronic species h : D h u + , D h s + + d + , D h c + , D h b + , D h � g inclusive SIA data only constrains three FF combinations, heavy-quark FFs constrained directly by tagged SIA data . Each FF is parametrised by a Neural Net (architecture 2-5-3-1). Kinematic cuts : ⇢ 0 . 02 for √ s = M Z z min = z max = 0 . 9 z min ≤ z ≤ z max , , 0 . 075 otherwise

Settings Physical parameters : α s ( M Z ) = 0 . 118 , α em ( M Z ) = 1 / 127 , m c = 1 . 51 GeV , m b = 4 . 92 GeV Parametrisation scale : Q 0 = 5 GeV ( > m c , m b ) substantial heavy-quark intrinsic component, heavy-quark FFs parametrised on the same footing as the light FFs. 5 independent FFs for each hadronic species h : D h u + , D h s + + d + , D h c + , D h b + , D h � g inclusive SIA data only constrains three FF combinations, heavy-quark FFs constrained directly by tagged SIA data . Each FF is parametrised by a Neural Net (architecture 2-5-3-1). contributions ∝ M h / sz 2 and ln( z ) Kinematic cuts : ⇢ 0 . 02 for √ s = M Z z min = z max = 0 . 9 z min ≤ z ≤ z max , , 0 . 075 otherwise

Settings Physical parameters : α s ( M Z ) = 0 . 118 , α em ( M Z ) = 1 / 127 , m c = 1 . 51 GeV , m b = 4 . 92 GeV Parametrisation scale : Q 0 = 5 GeV ( > m c , m b ) substantial heavy-quark intrinsic component, heavy-quark FFs parametrised on the same footing as the light FFs. 5 independent FFs for each hadronic species h : D h u + , D h s + + d + , D h c + , D h b + , D h � g inclusive SIA data only constrains three FF combinations, heavy-quark FFs constrained directly by tagged SIA data . Each FF is parametrised by a Neural Net (architecture 2-5-3-1). contributions ∝ ln(1 - z ) Kinematic cuts : ⇢ 0 . 02 for √ s = M Z z min = z max = 0 . 9 z min ≤ z ≤ z max , , 0 . 075 otherwise

Experimental data

Dataset Only SIA cross sections (normalised and absolute) included. π ± π ± π ± π ± π ± π ± π ± π ± K ± K ± 91.2 91.2 58 58 44 44 √ s [GeV] √ s [GeV] 34 34 29 29 22 22 14 14 12 12 10.5 10.5 TASSO TASSO BABAR BABAR ALEPH ALEPH BELLE BELLE TPC TPC DELPHI DELPHI SLD SLD OPAL OPAL TOPAZ TOPAZ 0.01 0.01 0.1 0.1 1 1 0.01 0.01 0.1 0.1 z z z z

Dataset Only SIA cross sections (normalised and absolute) included. PETRA (DESY) π ± π ± π ± π ± π ± π ± π ± π ± K ± K ± Phys. Lett. B94 (1980) 444 91.2 91.2 Z.Phys. C17 (1983) 5-15 Z.Phys. C42 (1989) 189 58 58 44 44 √ s [GeV] √ s [GeV] 34 34 29 29 22 22 14 14 12 12 10.5 10.5 TASSO TASSO BABAR BABAR ALEPH ALEPH BELLE BELLE TPC TPC DELPHI DELPHI SLD SLD OPAL OPAL TOPAZ TOPAZ 0.01 0.01 0.1 0.1 1 1 0.01 0.01 0.1 0.1 z z z z

Dataset Only SIA cross sections (normalised and absolute) included. PETRA (DESY) π ± π ± π ± π ± π ± π ± π ± π ± K ± K ± Phys. Lett. B94 (1980) 444 91.2 91.2 Z.Phys. C17 (1983) 5-15 Z.Phys. C42 (1989) 189 58 58 KEK Phys. Rev. Lett. 111 (2013) 062002 44 44 Phys. Rev. D92 (2015), no. 9 092007 √ s [GeV] √ s [GeV] 34 34 Phys. Lett. B345 (1995) 335-342 29 29 22 22 14 14 12 12 10.5 10.5 TASSO TASSO BABAR BABAR ALEPH ALEPH BELLE BELLE TPC TPC DELPHI DELPHI SLD SLD OPAL OPAL TOPAZ TOPAZ 0.01 0.01 0.1 0.1 1 1 0.01 0.01 0.1 0.1 z z z z

Dataset Only SIA cross sections (normalised and absolute) included. PETRA (DESY) π ± π ± π ± π ± π ± π ± π ± π ± K ± K ± Phys. Lett. B94 (1980) 444 91.2 91.2 Z.Phys. C17 (1983) 5-15 Z.Phys. C42 (1989) 189 58 58 KEK Phys. Rev. Lett. 111 (2013) 062002 44 44 Phys. Rev. D92 (2015), no. 9 092007 √ s [GeV] √ s [GeV] 34 34 Phys. Lett. B345 (1995) 335-342 29 29 SLAC 22 22 Phys. Rev. D88 (2013) 032011 Phys. Rev. Lett. 61 (1988) 1263 Phys. Rev. D69 (2004) 072003 14 14 X.Q. Lu, Ph.D. thesis, Johns Hopkins U., 1986 12 12 10.5 10.5 TASSO TASSO BABAR BABAR ALEPH ALEPH BELLE BELLE TPC TPC DELPHI DELPHI SLD SLD OPAL OPAL TOPAZ TOPAZ 0.01 0.01 0.1 0.1 1 1 0.01 0.01 0.1 0.1 z z z z

Dataset Only SIA cross sections (normalised and absolute) included. PETRA (DESY) π ± π ± π ± π ± K ± Phys. Lett. B94 (1980) 444 91.2 Z.Phys. C17 (1983) 5-15 Z.Phys. C42 (1989) 189 58 KEK Phys. Rev. Lett. 111 (2013) 062002 44 Phys. Rev. D92 (2015), no. 9 092007 √ s [GeV] 34 Phys. Lett. B345 (1995) 335-342 29 SLAC 22 Phys. Rev. D88 (2013) 032011 Phys. Rev. Lett. 61 (1988) 1263 Phys. Rev. D69 (2004) 072003 14 X.Q. Lu, Ph.D. thesis, Johns Hopkins U., 1986 12 10.5 LEP (CERN) Z. Phys. C66 (1995) 355-366 TASSO BABAR ALEPH Eur. Phys. J. C5 (1998) 585-620 BELLE TPC DELPHI Z. Phys. C63 (1994) 181-196. SLD OPAL TOPAZ 428 data points (after cuts) 0.01 0.1 1 0.01 0.1 z z