Fitting GPDs and CFFs various approaches. Physics Department University of Zagreb, Croatia

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1 Fitting GPDs and CFFs various approaches Krešimir Kumerički Physics Department University of Zagreb, Croatia Nuclear and Nucleon Structure Through Dileptons Production ECT Trento, Italy, October 2016

2 Outline Introduction DVCS Local fits Neural network fits Global fits Conclusion 2 Krešimir Kumerički : GPDs fitting

3 DVCS cross section I T DVCS 2 dσ T 2 = T BH 2 + T DVCS 2 + I. { e l c0 I + P 1 (φ)p 2 (φ) { c DVCS n=1 3 [ c I n cos(nφ) + sn I sin(nφ) ] }, n=1 [ c DVCS n cos(nφ) + sn DVCS sin(nφ) ] }, Choosing polarizations (and charges) we focus on particular harmonics: c I 1,unpol. [ F 1 Re H t 4M 2 p F 2 Re E + x ] B (F 1 + F 2 ) Re 2 x H B [Belitsky, Müller et. al 01 14] H(x B, t, Q 2 ),... four Compton form factors (CFFs) 3 Krešimir Kumerički : GPDs fitting

4 Experimental coverage (fixed target) Collab. Year Observables Kinematics No. of points xb Q 2 [GeV 2 ] t [GeV 2 ] total indep. HERMES 2001 A sin φ LU CLAS 2001 A sin φ LU CLAS 2006 A sin φ UL HERMES 2006 A cos φ C Hall A 2006 σ(φ), σ(φ) CLAS 2007 ALU(φ) A cos(0,1)φ C, A sin(φ φs ) UT,DVCS, sin(φ φs HERMES 2008 ) cos(0,1)φ AUT,I, cos(φ φs ) sin φ AUT,I < CLAS 2008 ALU(φ) HERMES 2009 A sin(1,2)φ LU,I, A sin φ LU,DVCS, A cos(0,1,2,3)φ C HERMES 2010 A sin(1,2,3)φ UL, HERMES 2011 A cos(0,1,2)φ LL cos(φ φs ) cos(0,1,2)φ ALT,I, sin(φ φs ) sin(1,2)φ ALT,I, cos(φ φs ) cos(0,1)φ ALT,BH+DVCS, sin(φ φs ) sin φ ALT,BH+DVCS HERMES 2012 A sin(1,2)φ LU,I, A sin φ LU,DVCS, A cos(0,1,2,3)φ C < < < < CLAS 2015 ALU(φ), AUL(φ), ALL(φ) CLAS 2015 σ(φ), σ(φ) Hall A 2015 σ(φ), σ(φ) Krešimir Kumerički : GPDs fitting

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6 Fit type Data used pqcd order Target 1. Local fits fixed target LO CFFs 2. Neural nets fixed target LO CFFs 3. Global fits collider/fixed target ((N)N)LO CFFs/GPDs 5 Krešimir Kumerički : GPDs fitting

7 Local fits to HERMES data Most complete set of asymmetries is measured in 12 bins: bin no t [GeV 2 ] x B Q 2 [GeV 2 ] A cos(1φ) C A C dσ e + dσ e dσ e + + dσ e [ F 1 Re H = t 4M 2 p T Interference (F) T DVCS 2 (F 2 ) + T 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 [K.K., Müller, Murray 13] (up to A DVCS 2 contamination taken into account) 6 Krešimir Kumerički : GPDs fitting

8 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. 7 Krešimir Kumerički : GPDs fitting

9 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. 8 Krešimir Kumerički : GPDs fitting

10 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. 9 Krešimir Kumerički : GPDs fitting

11 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 : GPDs fitting

12 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/ local fit of Im H + Re E local fit of Im H + Re H + Im H 8 Im H Re H Im H 4 0 Re E bin no bin no. 10 Krešimir Kumerički : GPDs fitting

13 Multilayer perceptron Essentially a least-squares fit of a complicated many-parameter function. f (x) = tanh( w i tanh( w j )) no theory bias 11 Krešimir Kumerički : GPDs fitting

14 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 = 120.2/ Krešimir Kumerički : GPDs fitting

15 Neural Net HERMES fit - BSA/BCA 13 Krešimir Kumerički : GPDs fitting

16 H(ξ,t) H(ξ,t) H(ξ,t) E(ξ,t) Neural Net HERMES fit - CFFs t = 0.3 GeV 2 NNet14 prelim. 1 NNet14 prelim. 3 t = 0.12 GeV 2 t = 0.3 GeV 2 t = 0.12 GeV 2 t = 0.3 GeV 2 t = 0.12 GeV t = 0.3 GeV 2 t = 0.12 GeV ξ =x B /(2 x B ) ξ =x B /(2 x B ) Good agreement with local fits, as it should be. 14 Krešimir Kumerički : GPDs fitting

17 Hybrid GPD models for global fits Sea quarks and gluons modelled using SO(3) partial wave expansion in conformal GPD moment space + Q 2 evolution. Valence quarks model CFFs directly (ignoring Q 2 evolution): [ 4 Im H(ξ, t) = π 9 Hu val (ξ, ξ, t) Hd val (ξ, ξ, t) + 2 ] 9 Hsea (ξ, ξ, t) ( ) 2x α(t) ( ) 1 x b H(x, x, t) = n r 2 α 1 ( ) 1 + x 1 + x p. 1 1 x t 1+x M 2 Re H determined by dispersion relations 15 free parameters in total for H, H, E, Ẽ. 15 Krešimir Kumerički : GPDs fitting

18 Model KM09a KM09b KM10 KM10a KM10b KMS11 KMM12 KM15 free params. {3}+(3)+5 {3}+(3)+6 {3}+15 {3}+10 {3}+15 NNet {3}+15 {3}+15 χ 2 /d.o.f. 32.0/ / / / / / / /275 F 2 {85} {85} {85} {85} {85} {85} {85} σ DVCS (45) (45) dσ DVCS/dt (56) (56) A sin φ LU A sin φ LU,I cos 0φ AC 6 6 A cos φ C σ sin φ,w σ cos 0φ,w σ cos φ,w σ cos φ,w /σ cos 0φ,w 4 4 A sin φ UL cos 0φ ALL 4 14 A cos φ LL 10 sin(φ φs ) cos φ AUT,I 4 4 [K.K., Müller, et al ] These models are publicly available (google for gpd page ) 16 Krešimir Kumerički : GPDs fitting

19 2015 CLAS cross-sections (1/2) Restriction to kinematics where leading-order framework should be valid: t/q 2 < 0.25 with Q 2 > 1.5 GeV 2, means using 48 out of measured 110 x B Q 2 t bins. dσ =dσ +dσ KM15 prelim. KMM12 CLAS 2015 x B = 0.335, Q 2 = 2.78, t = -0.2 x B = 0.335, Q 2 = 2.78, t = x B = 0.335, Q 2 = 2.78, t = σ =dσ dσ x B = 0.335, Q 2 = 2.78, t = x B = 0.335, Q 2 = 2.78, t = x B = 0.335, Q 2 = 2.78, t = φ [deg] φ [deg] φ [deg] χ 2 /npts = /1014 for dσ and 936.1/1012 for σ 17 Krešimir Kumerički : GPDs fitting

20 φ-space vs. harmonics (1/2) φ-space figures and perfect χ 2 are not revealing the whole story Instead to σ(φ) it is favourable to work with harmonics like σ sin nφ,w 1 π π π dw sin(nφ) σ(φ), with specialy weighted Fourier integral measure dw 2πP 1(φ)P 2 (φ) π π dφ P 1(φ)P 2 (φ) dφ, thus cancelling strongly oscillating factors 1/(P 1 (φ)p 2 (φ)) in Bethe-Heitler and interference terms in dσ. Series of such weighted harmonic terms converges then faster with increasing n than normal Fourier series. 18 Krešimir Kumerički : GPDs fitting

21 φ-space vs. harmonics (2/2) How many harmonics to extract? One approach: 1. Fit harmonic expansion σ(φ) = c 0 + c 1 cos φ + + s 1 sin φ + to randomly chosen subset of data in a bin, and calculate χ 2 error for description of the rest of data (so called cross-validation procedure) 2. Increase the number of harmonics until χ 2 /dof starts to worsen 19 Krešimir Kumerički : GPDs fitting

22 φ-space vs. harmonics (2/2) How many harmonics to extract? One approach: 1. Fit harmonic expansion σ(φ) = c 0 + c 1 cos φ + + s 1 sin φ + to randomly chosen subset of data in a bin, and calculate χ 2 error for description of the rest of data (so called cross-validation procedure) 2. Increase the number of harmonics until χ 2 /dof starts to worsen Highest extractable harmonics in 2015 cross-section data: CLAS Hall A sine cosine sine cosine σ w 0.9 ± ± ± ± 0.3 dσ w 0.3 ± ± ± ± 0.7 (So σ w = s 1 sin φ and dσ w = c 0 + c 1 cos φ is enough.) 19 Krešimir Kumerički : GPDs fitting

23 2015 CLAS cross-sections (2/2) Q 2 =1.63 x B =0.18 Q 2 =1.64 x B =0.21 Q 2 =1.88 x B =0.21 Q 2 =1.79 x B =0.24 Q 2 =2.12 x B = Q 2 =1.63 x B =0.18 Q 2 =1.64 x B =0.21 Q 2 =1.88 x B =0.21 Q 2 =1.79 x B =0.24 Q 2 =2.12 x B =0.24 σ sinφ,w dσ cos0φ,w Q 2 =2.35 x B =0.28 Q 2 =2.58 x B =0.30 Q 2 =2.78 x B =0.34 Q 2 =2.97 x B =0.36 Q 2 =3.18 x B = Q 2 =2.35 x B =0.28 Q 2 =2.58 x B =0.30 Q 2 =2.78 x B =0.34 Q 2 =2.97 x B =0.36 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 =0.185 Q 2 =1.64 x B =0.214 Q 2 =1.88 x B =0.215 Q 2 =1.79 x B =0.244 Q 2 =2.12 x B =0.245 χ 2 /npts = 62.2/48 for dσcos φ,w (O.K. but not so perfect as in φ-space) 0.01 dσ cosφ,w Q 2 =2.35 x B =0.275 Q 2 =2.58 x B =0.305 Q 2 =2.78 x B =0.335 Q 2 =2.97 x B =0.365 Q 2 =3.18 x B = t [GeV 2 ] 20 Krešimir Kumerički : GPDs fitting

24 2015 CLAS cross-sections (2/2) Q 2 =1.63 x B =0.18 Q 2 =1.64 x B =0.21 Q 2 =1.88 x B =0.21 Q 2 =1.79 x B =0.24 Q 2 =2.12 x B = Q 2 =1.63 x B =0.18 Q 2 =1.64 x B =0.21 Q 2 =1.88 x B =0.21 Q 2 =1.79 x B =0.24 Q 2 =2.12 x B =0.24 σ sinφ,w dσ cos0φ,w Q 2 =2.35 x B =0.28 Q 2 =2.58 x B =0.30 Q 2 =2.78 x B =0.34 Q 2 =2.97 x B =0.36 Q 2 =3.18 x B = Q 2 =2.35 x B =0.28 Q 2 =2.58 x B =0.30 Q 2 =2.78 x B =0.34 Q 2 =2.97 x B =0.36 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 =0.185 Q 2 =1.64 x B =0.214 Q 2 =1.88 x B =0.215 Q 2 =1.79 x B =0.244 Q 2 =2.12 x B =0.245 χ 2 /npts = 62.2/48 for dσcos φ,w (O.K. but not so perfect as in φ-space) 0.01 dσ cosφ,w Q 2 =2.35 x B = Q 2 =2.58 x B = Q 2 =2.78 x B = t [GeV 2 ] Q 2 =2.97 x B = Q 2 =3.18 x B = Krešimir Kumerički : GPDs fitting In addition to χ 2 /d.o.f. use: pull 1 N N f (x i ) y i i=1 y i

25 2006 vs 2015 Hall A cross-sections σ sinφ, w Q 2 = x B = Q 2 = x B = Q 2 = x B = σ sinφ, w Q 2 = x B = Q 2 = x B = Q 2 = x B = t [GeV 2 ] t [GeV 2 ] t [GeV 2 ] KM15 KMM12 dσ cos0φ, w KM15 KMM12 dσ cos0φ, w dσ cosφ, w dσ cosφ, w t [GeV 2 ] t [GeV 2 ] t [GeV 2 ] Improvement of global χ 2 /d.o.f / / Krešimir Kumerički : GPDs fitting

26 H1 (2007), ZEUS (2008) 22 Krešimir Kumerički : GPDs fitting

27 2007 CLAS beam spin asymmetry ALU(90 ) ALU(90 ) ALU(90 ) Q 2 = x B = Q 2 = x B = Q 2 = x B = t [GeV 2 ] Q 2 = x B = Q 2 = x B = Q 2 = x B = t [GeV 2 ] Only data with t 0.3 GeV 2 used for fits. Q 2 = x B = KM15 Q 2 = x B = Q 2 = x B = t [GeV 2 ] 23 Krešimir Kumerički : GPDs fitting

28 Longitudinally polarized target HERMES (2010) Surprisingly large sin(2φ) harmonic of A UL cannot be described within this leading twist framework 24 Krešimir Kumerički : GPDs fitting

29 χ 2 /npts and pull values Collaboration Observable n pts KMM12 KM15 χ 2 /n pts pull χ 2 /n pts pull ZEUS σ DVCS ZEUS,H1 dσ DVCS /dt HERMES cos 0φ A C HERMES A cos φ C HERMES A sin φ LU,I CLAS A sin φ LU CLAS A sin φ LU CLAS σ sin φ,w CLAS dσ cos 0φ,w CLAS dσ cos φ,w Hall A σ sin φ,w Hall A dσ cos 0φ,w Hall A dσ cos φ,w Hall A σ sin φ,w Hall A dσ cos 0φ,w Hall A dσ cos φ,w HERMES,CLAS A sin φ UL HERMES cos 0φ A LL HERMES A sin(φ φ S ) cos φ UT,I CLAS A sin φ UL CLAS cos 0φ A LL CLAS A cos φ LL Krešimir Kumerički : GPDs fitting

30 CFFs from various fits t = 0.28 GeV 2 Q 2 2 GeV Im H 10 Re H Im H KM15 prelim. KMM12 GK07 NNet 14 Moutarde 09 Boër/Guidal 14 Re E x B x B 26 Krešimir Kumerički : GPDs fitting

31 What is constrained by data? Although models over the years describe data in similar way they are actually quite different Krešimir Kumerički : GPDs fitting

32 What is constrained by data? (II) Comparing Im H and D-term for a typical JLab kinematics: 28 Krešimir Kumerički : GPDs fitting

33 What is constrained by data? (II) Comparing Im H and D-term for a typical JLab kinematics: DDVCS would help here! 28 Krešimir Kumerički : GPDs fitting

34 Going beyond first approximations Published DVCS data is well described within leading order, leading twist and other approximations. Some available corrections: NLO evolution and NLO coefficient functions finite-t and target mass corrections [Braun et al. 14] twist-3 GPDs massive charm [Noritzsch 03] small-x resummation [Ivanov et al. 16] can presently be absorbed in redefinition of GPDs (think DVCS scheme factorization), but have to be taken into account when working with more processes (striving towards universal GPDs) and with more precise future data. 29 Krešimir Kumerički : GPDs fitting

35 Summary Global fits of all proton DVCS data using flexible hybrid models are in healthy shape (some tension for first cos(φ) harmonic of JLab cross sections). Constraints coming from DDVCS would be most welcome: DDVCS is cheaper and complementary to experiments with large Q 2 lever arm. 30 Krešimir Kumerički : GPDs fitting

36 Summary Global fits of all proton DVCS data using flexible hybrid models are in healthy shape (some tension for first cos(φ) harmonic of JLab cross sections). Constraints coming from DDVCS would be most welcome: DDVCS is cheaper and complementary to experiments with large Q 2 lever arm. The End 30 Krešimir Kumerički : GPDs fitting