Status of Generalized Parton Distributions (GPD) Phenomenology Fits to Deeply Virtual Compton Scattering (DVCS) Data

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1 Status of Generalized Parton Distributions (GPD) Phenomenology Fits to Deeply Virtual Compton Scattering (DVCS) Data Krešimir Kumerički Physics Department University of Zagreb, Croatia International Workshop on Hadron Structure and Spectroscopy (IWHSS 2015) Suzdal, Russia, May 2015

2 Outline Introduction to GPDs Global fits (small x B ) Global fits (all data) Neural networks approach Conclusions 2 Krešimir Kumerički : GPD phenomenology

3 Attractiveness of GPDs (1/3) 1. Well-defined within the QCD [Müller 92, et al. 94, Ji, Radyushkin 96] dz F q (x, η, t = 2 z ) = 2π eixp+ P 2 q( z)γ + q(z) P 1 z + =0, z =0 x + η 2 P + GPD x η 2 P + Forward limit 0 xp + PDF xp η 2 P + 1 η 2 P + p + p + P = P 1 + P 2 ; = P 2 P 1 ; η = + P + (skewedness) 3 Krešimir Kumerički : GPD phenomenology

4 Attractiveness of GPDs (2/3) Decomposition into helicity conserving and non-conserving parts: F a = ū(p 2)γ + u(p 1 ) P + H a + ū(p 2)iσ +ν u(p 1 ) ν 2MP + E a a = q, g 2. Close contact to 3D quark-gluon hadron structure 1 1 [ ] dx x H q (x, η, 0) + E q (x, η, 0) = J q [Ji 97] 2 1 p xp b q(x, b ) = d 2 (2π) 2 eib H q (x, 0, 2 ) [Burkardt 00] 4 Krešimir Kumerički : GPD phenomenology

5 Attractiveness of GPDs (3/3) 3. Accessible to experiments Deeply virtual Compton scattering (DVCS) γ q 2 1 = Q2 q 2 2 = 0 γ P = P 1 + P 2, t = (P 2 P 1 ) 2 q = (q 1 + q 2 )/2 Generalized Bjorken limit: q 2 Q 2 /2 DVCS P 1 P 2 ξ = q2 2P q = x B 2 x B const We work at leading order accuracy where cross-section can be expressed in terms of four Compton form factors (CFFs) F {H(ξ, t, Q 2 ), E(ξ, t, Q 2 ), H(ξ, t, Q 2 ), Ẽ(ξ, t, Q 2 )} 5 Krešimir Kumerički : GPD phenomenology

6 Factorization of DVCS GPDs [Collins et al. 98] γ q 2 1 = Q2 q2 2 = 0 γ C a x + ξ x ξ 2 2 H a P 1 P 2 P = P 1 + P 2, t = (P 2 P 1 ) 2 q = (q 1 + q 2 )/2 q 2 Q 2 /2 ξ = q2 2P q const Compton form factor is a convolution: a H(ξ, t, Q 2 ) = dx C a (x, ξ, Q 2 /Q 2 0) H a (x, η = ξ, t, Q 2 0) a=ns,s(σ,g) 6 Krešimir Kumerički : GPD phenomenology

7 Curse of dimensionality It is relatively easy to find a coin lying somewhere on 100 meter string. It is very difficult to find it on a football field. 7 Krešimir Kumerički : GPD phenomenology

8 Curse of dimensionality It is relatively easy to find a coin lying somewhere on 100 meter string. It is very difficult to find it on a football field. When the dimensionality increases, the volume of the space increases so fast that the available data becomes sparse. Analogously, in contrast to PDFs(x), it is very difficult to perform truly model independent extraction of GPDs(x, ξ, t) Known GPD constraints don t decrease the dimensionality of the GPD domain space. As an intermediate step, one can attempt extraction of CFFs(x B, t) (Dependence on additional variable, photon virtuality Q 2, is in principle known given by evolution equations.) 7 Krešimir Kumerički : GPD phenomenology

9 Experimental coverage Coming soon: JLab12, COMPASS,... EIC 8 Krešimir Kumerički : GPD phenomenology

10 Modelling GPDs Double distribution representation, Radyushkin ansatz 1 1 y ( ) q(y, t) z H(x, η, t) = dy dxδ(x y zη) 1 y Π +D-term 1 y 0 1+y Π(z) = (1 z 2 ) b [Vanderhaeghen, Guichon, Guidal 99; Goloskokov, Kroll 05] Light-front wave function overlap [Goldstein, Hernandez, Liuti 11; Müller, Hwang 14] Conformal partial wave expansion [Müller et al 06, 07]; dual representation [Polyakov, Shuvaev 02] 9 Krešimir Kumerički : GPD phenomenology

11 Modelling GPDs in moment space Instead of considering momentum fraction dependence H(x,...)... it is convenient to make a transform into complementary space of conformal moments j: H q j (η,...) Γ(3/2)Γ(j+1) 2 j+1 Γ(j+3/2) 1 1 dx η j C 3/2 j (x/η) H q (x, η,...) They are analogous to Mellin moments in DIS: x j C 3/2 j (x) C 3/2 j (x) Gegenbauer polynomials 10 Krešimir Kumerički : GPD phenomenology

12 Advantages of conformal moments 1. The evolution equations are most simple: There is no mixing among moments at LO, and in special (CS) scheme not even at NLO 2. Powerful analytic methods of complex j plane are available (similar to complex angular momentum of Regge theory) 3. Stable and fast computer code for evolution and fitting 4. Moments are equal to matrix elements of local operators and are thus directly accessible on the lattice 11 Krešimir Kumerički : GPD phenomenology

13 l-pw model only leading SO(3) partial wave H j (η, t, µ 2 0) = α Σ,G (t)=α Σ,G (0)+5t ( N Σ F Σ (t) B ( 1 + j α Σ (0), 8 ) N G F G(t) B ( 1 + j α G (0), 6 ) F Σ,G (t) = j + 1 α(0) j + 1 α(t)... corresponding in forward case to PDFs of form ( ) 1 t M Σ,G 0 Σ(x) = N Σ x α Σ(0) (1 x) 7 ; G(x) = N G x α G(0) (1 x) 5 2 ) 2 M G 0 = 0.7 GeV is fixed by the J/ψ production data Free parameter (for DVCS): M Σ 0 For small ξ (small x Bj ) valence quarks are less important Σ sea 12 Krešimir Kumerički : GPD phenomenology

14 Inclusion of subleading PW flexible models ( N H j (η, t)= sea F sea (t) B ( 1 + j α sea (0), 8 ) ) N G F G(t) B ( 1 + j α G (0), 6 ) + }{{} skewness r 1.6 (too large) ( ) partial waves, η- s G ssea subleading dependence! }{{} < 0 negative skewness nl-pw addition of second PW needed for good fits two new parameters: s (2) sea and s (2) G nnl-pw addition of third PW (doesn t improve fits much but makes possible positive gluon GPDs at small Q 2 ) s (4) sea and s (4) G 13 Krešimir Kumerički : GPD phenomenology

15 1 (N)NLO corrections 0.8 K P abs(q 2 = 2.5 GeV 2 ) = 0 P=2 (NNLO), CS P=1 (NLO), CS P=1 (NLO), MS Thick lines: hard gluon N G = 0.4 α G (0)=α Σ (0) + 5 Thin lines: soft gluon N G = α G (0)=α Σ (0) ξ K P abs H (P) H (P 1) 14 Krešimir Kumerički : GPD phenomenology

16 Fit to HERA collider data dσ DVCS dt 4πα2 (2W 2 +Q 2 ) 2 W 2 W 2 +Q 2 H 2 dσ(γ p γp)/dt [nb/gev 2 ] 10 1 H1, Q 2 = 4 GeV 2 H1, Q 2 = 8 GeV t [GeV 2 ] σ(γ p γp) [nb] 10 1 H1, W = 82 GeV ZEUS, W = 89 GeV Q 2 [GeV 2 ] σ(γ p γp) [nb] 10 5 ZEUS, Q 2 = 9.6 GeV F W [GeV] Q 2 [GeV 2 ] 15 Krešimir Kumerički : GPD phenomenology

17 Resulting small-x H(x, x, t) xh sea (x, x, t, Q 2 ) at N P LO P = 0 P = 1 MS P = 1 CS P = 2 CS x = 01, t = GeV 2 (a) Q 2 [GeV 2 ] H G (x, x, t, Q 2 ) at N P LO x = 01, t = GeV 2 (b) Q 2 [GeV 2 ] P=0: LO; P=1: NLO; P=2: NNLO The whole procedure is extended to meson production [Müller, Lautenschlager, Passek-Kumerički, Schäfer 13] 16 Krešimir Kumerički : GPD phenomenology

18 Extending global analysis to fixed target data Hybrid models at LO (1st just for unpolarized target) Sea quarks and gluons modelled like just described (conformal moments + SO(3) partial wave expansion + Q 2 evolution). Valence quarks model (ignoring Q 2 evolution): [ 4 Im H(ξ, t) = π H(x, x, t) = n r 2 α ( 2x 1 + x 9 Hu val (ξ, ξ, t) Hd val (ξ, ξ, t) + 2 ] 9 Hsea (ξ, ξ, t) ) α(t) ( ) 1 x b 1 ( ) 1 + x p. 1 1 x t 1+x M 2 Fixed: n (from PDFs), α(t) (eff. Regge), p (counting rules) α val (t) = t/gev 2 (ρ, ω) 17 Krešimir Kumerički : GPD phenomenology

19 Re H determined by dispersion relations [Teryaev 05] Re H(ξ, t) = 1 π PV 1 0 dξ ( 1 ξ ξ 1 ξ + ξ Typical set of free parameters: ) Im H(ξ, t) M0 sea, s sea (2,4), s (2,4) G sea quarks and gluons H r val, M val, b val valence H r val, M val, b val valence H C, M C subtraction constant (H, E) r π, M π pion pole Ẽ Global fit to 175 data points turns out fine: KM10 model: χ 2 /d.o.f. = 135.9/160. C ( 1 t M 2 C s sea,g = strengths of subleading partial waves. LO evolution is included. 18 Krešimir Kumerički : GPD phenomenology ) 2

20 H1 (2007), ZEUS (2008) dσ/dt [nb/gev 2 ] σ [nb] H1 W=82 GeV Q 2 =8, 15.5, 25 GeV t [GeV 2 ] ZEUS (idem) : H1 (idem) : W [GeV] ZEUS W=104 GeV Q 2 =3.2 GeV t [GeV 2 ] KM15 prelim. KMM12 ZEUS 2008 ZEUS (idem) Q 2 [GeV 2 ] 19 Krešimir Kumerički : GPD phenomenology

21 Harmonic analysis of ep epγ cross section dσ(ep epγ) T BH 2 + T DVCS 2 + I T BH 2 K BH 2 { = cn BH P 1 (φ)p 2 (φ) n=0 3 { T DVCS 2 = K DVCS cn DVCS I = n=0 Q ek I P 1 (φ)p 2 (φ) cos(nφ) + sn BH sin(nφ) } } cos(nφ) + sn DVCS sin(nφ) 3 } {c In cos(nφ) + sin sin(nφ) c I n F 1(t) Re H(ξ, t), F 1 (t) Im H(ξ, t),... c DVCS n H(ξ, t)e(ξ, t),... Just a few leading harmonics important at twist-2 level Sometimes convenient to work with weighted harmonics of cross-section: dσ cos(nφ),w π dx B dq 2 dt π n=0 dσ P 1 (φ)p 2 (φ)dφ cos(nφ) dx B dq 2 dtdφ For fits, it would be really nice to have estimates of experimental uncertainties for harmonics (especially systematic ones)! 20 Krešimir Kumerički : GPD phenomenology

22 HERMES (2012) BCA dσ e + dσ e dσ e + + dσ e BSA dσ e dσ e dσ e + dσ e cos 0φ AC + A A cos 1φ C sin 1φ LU cos φ Re H sin φ Im H A cosφ C KM15 prelim. KMM12 HERMES 2012 A cos0φ C A sinφ LU,I t [GeV 2 ] x B Q 2 [GeV 2 ] 21 Krešimir Kumerički : GPD phenomenology

23 CLAS (2007) BSA. (Only data with t GeV 2 used for fits.) CLAS (Girod:2007aa) 0.4 A sinφ LU 0.4 Q 2 = 2.28, x B = 38 Q 2 = 2.61, x B = 4 Q 2 = 2.93, x B = 43 A sinφ LU Q 2 = 1.68, x B = 45 Q 2 = 1.93, x B = 45 Q 2 = 2.19, x B = Krešimir Kumerički : GPD phenomenology 2 2 2

24 Including data with polarized target KMM12: χ 2 /n d.o.f. = 124.1/80, strictly speaking not a good fit, but best what we had until 2015 A sin(φ φ S )cosφ UT,I A sin(φ φ S ) UT,DVCS t [GeV 2 ] KM15 prelim. KMM12 HERMES x B Q 2 [GeV 2 ] 23 Krešimir Kumerički : GPD phenomenology

25 Longitudinally polarized target Surprisingly large sin(2φ) harmonic of A UL cannot be described within this leading twist framework A sin 1φ UL A sin 2φ UL A cos0φ LL A cos1φ LL Krešimir Kumerički : GPD phenomenology

26 2015 CLAS asymmetries (1/2) CLAS 2015 (Pisano:2015iqa) -- BSA CLAS 2015 (Pisano:2015iqa) -- TSA A sinφ LU A sinφ UL Q 2 = , x B = Q 2 = , x B = Q 2 = , x B = Q 2 = , x B = A sinφ LU A sinφ UL 0.4 Q 2 = , x B = Q 2 = , x B = t [GeV 2 ] 0.4 Q 2 = , x B = Q 2 = , x B = t [GeV 2 ] A sinφ LU KM15 prelim. KMM12 CLAS 2015 A sinφ UL KM15 prelim. KMM12 CLAS 2015 Q 2 = , x B = t [GeV 2 ] Q 2 = , x B = t [GeV 2 ] 25 Krešimir Kumerički : GPD phenomenology

27 2015 CLAS asymmetries (2/2) CLAS 2015 (Pisano:2015iqa) -- BTSA0 CLAS 2015 (Pisano:2015iqa) -- BTSA1 A cos0φ LL A cos0φ LL A cos0φ LL Q 2 = , x B = Q 2 = , x B = Q 2 = , x B = Q 2 = , x B = t [GeV 2 ] KM15 prelim. KMM12 CLAS 2015 Q 2 = , x B = t [GeV 2 ] A cosφ LL A cosφ LL A cosφ LL Q 2 = , x B = Q 2 = , x B = Q 2 = , x B = Q 2 = , x B = Q 2 = , x B = t [GeV 2 ] KM15 prelim. KMM12 CLAS t [GeV 2 ] 26 Krešimir Kumerički : GPD phenomenology

28 2015 CLAS cross-sections (1/2) dσ =dσ +dσ KM15 prelim. KMM12 CLAS 2015 x B = 35, Q 2 = 2.78, t = - x B = 35, Q 2 = 2.78, t = -6 x B = 35, Q 2 = 2.78, t = σ =dσ dσ x B = 35, Q 2 = 2.78, t = x B = 35, Q 2 = 2.78, t = -6 x B = 35, Q 2 = 2.78, t = φ [deg] φ [deg] φ [deg] 27 Krešimir Kumerički : GPD phenomenology

29 2015 CLAS cross-sections (2/2) Q 2 =1.63 x B =8 Q 2 =1.64 x B =1 Q 2 =1.88 x B =1 Q 2 =1.79 x B =4 Q 2 =2.12 x B = Q 2 =1.63 x B =8 Q 2 =1.64 x B =1 Q 2 =1.88 x B =1 Q 2 =1.79 x B =4 Q 2 =2.12 x B =4 σ sinφ,w dσ cos0φ,w Q 2 =2.35 x B =8 Q 2 =2.58 x B =0 Q 2 =2.78 x B =4 Q 2 =2.97 x B =6 Q 2 =3.18 x B = Q 2 =2.35 x B =8 Q 2 =2.58 x B =0 Q 2 =2.78 x B =4 Q 2 =2.97 x B =6 Q 2 =3.18 x B = σ sinφ,w dσ cos0φ,w t [GeV 2 ] t [GeV 2 ] dσ cosφ,w Q 2 =1.63 x B =85 Q 2 =1.64 x B =14 Q 2 =1.88 x B =15 Q 2 =1.79 x B =44 Q 2 =2.12 x B = dσ cosφ,w Q 2 =2.35 x B =75 Q 2 =2.58 x B =05 Q 2 =2.78 x B =35 Q 2 =2.97 x B =65 Q 2 =3.18 x B = t [GeV 2 ] Krešimir Kumerički : GPD phenomenology

30 2006 vs 2015 Hall A cross-sections σ sinφ,w Q 2 = 1.50 x B = 6 Q 2 = 1.90 x B = 6 Q 2 = 2.30 x B = 6 σ sinφ,w Q 2 = 1.57 x B = 8 Q 2 = 1.94 x B = 7 Q 2 = 2.32 x B = t [GeV 2 ] 0.4 t [GeV 2 ] t [GeV 2 ] KMM12 KM15 prelim. dσ cos0φ,w dσ cos0φ,w 6 3 KMM12 KM15 prelim dσ cosφ,w 1 0 dσ cosφ,w t [GeV 2 ] 0.4 t [GeV 2 ] 0.4 t [GeV 2 ] 29 Krešimir Kumerički : GPD phenomenology

31 χ 2 /npts values npts KMM12 KM15 prelim H1ZEUS X_DVCS : (35) HERMES ALUI : ( 5) (one 2.7 sigma outlier) HERMES BCA : (10) CLAS BSA : ( 4) HRM/CLS AUL : ( 3) HERMES ALL : ( 3) (one 3.7 sigma outlier) HERMES AUTI : ( 3) CLAS BSDw_s1: (24) CLAS BSSw_c0: (24) CLAS BSSw_c1: (24) Hall A BSDw_s1: ( 5) Hall A BSSw_c0: ( 5) Hall A BSSw_c1: ( 5) (corr. syst. not included) === TOTAL : (150) Krešimir Kumerički : GPD phenomenology

32 Neural nets multilayer perceptron Essentially a least-squares fit of a complicated many-parameter function. f (x) = tanh( w i tanh( w j )) no theory bias 31 Krešimir Kumerički : GPD phenomenology

33 Preliminary neural Net HERMES fit Fit to all HERMES DVCS data with two types of neural nets (x B, t) (7 neurons) (Im H, Re H, Im H): χ 2 /n pts = 135.4/144 (x B, t) (7 neurons) (Im H, Re E): χ 2 /n pts = 12/ Krešimir Kumerički : GPD phenomenology

34 Neural Net HERMES fit - BSA/BCA 33 Krešimir Kumerički : GPD phenomenology

35 H(ξ,t) H(ξ,t) H(ξ,t) E(ξ,t) Neural Net HERMES fit - CFFs t = GeV 2 NNet14 prelim. 1 NNet14 prelim. 3 t = 2 GeV 2 t = GeV 2 t = 2 GeV 2 t = GeV 2 t = 2 GeV t = GeV 2 t = 2 GeV ξ =x B /(2 x B ) ξ =x B /(2 x B ) 34 Krešimir Kumerički : GPD phenomenology

36 Prediction for COMPASS-II BCSA BCSA = dσ µ + dσ µ dσ µ + + dσ µ (E µ = 160 GeV) 35 Krešimir Kumerički : GPD phenomenology

37 Comparison to others [Guidal 08, Guidal and Moutarde 09], seven CFF fit (blue squares), [Guidal 10] H, H CFF fit (green diamonds), [Moutarde 09] H GPD fit (red circles). Also reasonable agreement with [Goloskokov and Kroll]. 36 Krešimir Kumerički : GPD phenomenology

38 KM models are available at WWW Google for gpd page get binary code for cross sections % xs.exe xs.exe ModelID Charge Polarization Ee Ep xb Q2 t phi returns cross section (in nb) for scattering of lepton of energy Ee on unpolarized proton of energy Ep. Charge=-1 is for electron. ModelID is one of 0 debug, always returns 42, 1 KM09a - arxiv: fit without Hall A, 2 KM09b - arxiv: fit with Hall A, 3 KM10 - preliminary hybrid fit with LO sea evolution, from Trento presentation, 4 KM10a - preliminary hybrid fit with LO sea evolution, without Hall A data 5 KM10b - preliminary hybrid fit with LO sea evolution, with Hall A data xb Q2 t phi -- usual kinematics (phi is in Trento convention) % xs.exe Krešimir Kumerički : GPD phenomenology

39 GPD page and server Durham-like CFF/GPD server prototype page 38 Krešimir Kumerički : GPD phenomenology

40 New directions Improved global LO fits with all unpolarized and polarized proton data. Adding deeply virtual meson production and going NLO [Müller, Lautenschlager, Schäfer 13] Including higher twists [Braun, Manashov, Müller, Pirnay 14] Global neural network fits 39 Krešimir Kumerički : GPD phenomenology

41 New directions Improved global LO fits with all unpolarized and polarized proton data. Adding deeply virtual meson production and going NLO [Müller, Lautenschlager, Schäfer 13] Including higher twists [Braun, Manashov, Müller, Pirnay 14] Global neural network fits The End 39 Krešimir Kumerički : GPD phenomenology

42 Local fits App: Neural Nets App: Conformal GPDs Local fits to HERMES data Most complete set of asymmetries is measured in 12 bins: A cos(1φ) C dσ A BH 2 + A DVCS 2 (F 2 ) + A Interference (F) A C dσ e + dσ e dσ e + + dσ e [ F 1 Re H = t 4M 2 p A Interference (F) A DVCS 2 (F 2 ) + A BH 2 F 2 Re E + x ] B 2 (F 1 + F 2 ) Re H... and similarly for other observables almost linear relation between observables and CFFs (up to A DVCS 2 contamination taken into account) 40 Krešimir Kumerički : GPD phenomenology

43 Local fits App: Neural Nets App: Conformal GPDs Mapping Inverting these relations gives mapping from the set of observables... A sin(1φ) LU,I A cos(1φ) LL,+ sin(ϕ) cos(0φ) AUT,DVCS cos(ϕ) cos(0φ) ALT,BH+DVCS A cos(1φ) C A cos(0φ) LL,+ sin(ϕ) cos(0φ) AUT,I cos(ϕ) cos(0φ) ALT,I A cos(0φ) C sin(ϕ) cos(1φ) AUT,I sin(ϕ) sin(1φ) ALT,I A sin(1φ) UL,+ cos(ϕ) sin(1φ) AUT,I cos(ϕ) cos(1φ) ALT,I... to the set of eight real and imaginary parts of CFFs (sometimes called sub-cffs or just CFFs)... ( F = Im H, Re H, Im E, Re E, Im H, Re H, ) Im Ẽ, Re Ẽ... where error propagation is straightforward. 41 Krešimir Kumerički : GPD phenomenology

44 Local fits App: Neural Nets App: Conformal GPDs Mapping results (Here compared also to standard local least-squares fit) m linearized map one to one map fit to 14 asymmetries m m m bin e e e e bin Only Im H, Re H and Im H are reliably constrained. 42 Krešimir Kumerički : GPD phenomenology

45 Local fits App: Neural Nets App: Conformal GPDs Method of stepwise regression Constraints from data are too weak to constrain simultaneously all eight {Im F, Re F} CFFs Let us take smaller number of CFFs, choosing only those which are reliably extracted. Stepwise regression algorithm: Starting from zero, add, one by one, CFFs that best improve description of data until improvement becomes statistically insignificant. 43 Krešimir Kumerički : GPD phenomenology

46 Local fits App: Neural Nets App: Conformal GPDs Method of stepwise regression Constraints from data are too weak to constrain simultaneously all eight {Im F, Re F} CFFs Let us take smaller number of CFFs, choosing only those which are reliably extracted. Stepwise regression algorithm: Starting from zero, add, one by one, CFFs that best improve description of data until improvement becomes statistically insignificant. Algorithm stops after just 2 CFFs, and there are two equally good pairs of CFFs: 1. (Im H, Re H) with χ 2 /n d.o.f. = 102.3/120, and 2. (Im H, Re E) with χ 2 /n d.o.f. = 103.0/ Krešimir Kumerički : GPD phenomenology

47 Local fits App: Neural Nets App: Conformal GPDs Stepwise regression results Scenario 1: Fit of Im H, Re H and Im H. χ 2 /n d.o.f. = 148.8/144. (In good agreement with [Guidal 10]) Scenario 2: Fit of Im H and Re E. χ 2 /n d.o.f. = 134.2/ Krešimir Kumerički : GPD phenomenology

48 Local fits App: Neural Nets App: Conformal GPDs Function fitting by a neural net Theorem: Given enough neurons, any smooth function f (x 1, x 2, ) can be approximated to any desired accuracy. Single hidden layer is sufficient (but not always most efficient). 45 Krešimir Kumerički : GPD phenomenology

49 Local fits App: Neural Nets App: Conformal GPDs Function fitting by a neural net Theorem: Given enough neurons, any smooth function f (x 1, x 2, ) can be approximated to any desired accuracy. Single hidden layer is sufficient (but not always most efficient). With simple training of neural nets to data there is a danger of overfitting (a.k.a. overtraining) 45 Krešimir Kumerički : GPD phenomenology

50 Local fits App: Neural Nets App: Conformal GPDs Function fitting by a neural net Theorem: Given enough neurons, any smooth function f (x 1, x 2, ) can be approximated to any desired accuracy. Single hidden layer is sufficient (but not always most efficient). With simple training of neural nets to data there is a danger of overfitting (a.k.a. overtraining) 45 Krešimir Kumerički : GPD phenomenology

51 Local fits App: Neural Nets App: Conformal GPDs Function fitting by a neural net Theorem: Given enough neurons, any smooth function f (x 1, x 2, ) can be approximated to any desired accuracy. Single hidden layer is sufficient (but not always most efficient). With simple training of neural nets to data there is a danger of overfitting (a.k.a. overtraining) Solution: Divide data (randomly) into two sets: training sample and validation sample. Stop training when error of validation sample starts increasing. 45 Krešimir Kumerički : GPD phenomenology

52 Local fits App: Neural Nets App: Conformal GPDs Toy fitting example Fit to data generated according to function (which we pretend not to know). Fit with 1. Standard Minuit fit with ansatz f (x) = x a (1 x) b 2. Neural network fit 46 Krešimir Kumerički : GPD phenomenology

53 Local fits App: Neural Nets App: Conformal GPDs Toy fitting example Fit to data generated according to function (which we pretend not to know). Fit with 1. Standard Minuit fit with ansatz f (x) = x a (1 x) b 2. Neural network fit 46 Krešimir Kumerički : GPD phenomenology

54 Local fits App: Neural Nets App: Conformal GPDs Modelling conformal moments of GPDs (I) How to model η-dependence of GPD s H j (η, t)? Idea: consider crossed t-channel process γ γ p p When crossing back to DVCS channel we have: cos θ cm 1 η 47 Krešimir Kumerički : GPD phenomenology

55 Local fits App: Neural Nets App: Conformal GPDs Modelling conformal moments of GPDs (I) How to model η-dependence of GPD s H j (η, t)? Idea: consider crossed t-channel process γ γ p p When crossing back to DVCS channel we have: cos θ cm 1 η... and dependence on θ cm in t-channel is given by SO(3) partial wave decomposition of γ γ scattering H(η,...) = H (t) (cos θ cm = 1 η,...) = J (2J+1)f J (...)d J 0,ν(cos θ) d0,ν J Wigner SO(3) functions (Legendre, Gegenbauer,... ) ν = 0, ±1 depending on hadron helicities 47 Krešimir Kumerički : GPD phenomenology

56 Local fits App: Neural Nets App: Conformal GPDs Modelling conformal moments of GPDs (II) OPE expansion of both H and H (t) leads to H j (η, t) = η j+1 H (t) j (cos θ = 1 η, s(t) = t) and t-channel partial waves are modelled as: γ γ j+1 h J,j (1 t/m 2 ) p m(j) 1 m(j) t 1 J α(t) 1 H j (η, t) = h J,j J α(t) J p p 1 ( 1 t M 2 (J) ) p η j+1 J d J 0,ν Similar to dual parametrization [Polyakov, Shuvaev 02] 48 Krešimir Kumerički : GPD phenomenology

57 Local fits App: Neural Nets App: Conformal GPDs Checking the evolution code We checked agreement in the forward limit with QCD Pegasus code for PDF evolution [Vogt 04] 10 1 LO gluon 10 0 NLO gluon 10-1 NNLO gluon 10-2 LO singlet Q NLO singlet Q 10-3 NNLO singlet Q 10-4 LO C=1 NS Q NLO C=1 NS Q 10-5 LO C=-1 NS Q NLO C=-1 NS Q GeParD vs GeV^ Krešimir Kumerički : GPD phenomenology