GPDs and fitting procedures for DVCS

Transcription

1 GPDs and fitting procedures for DVCS Krešimir Kumerički Physics Department University of Zagreb, Croatia Based mostly on: K.K., D. Müller, M. Murray, to appear in Phys. Part. Nucl. (2014), arxiv: K.K., D. Müller, A. Schäfer, JHEP 07 (2011) 073, arxiv: K.K., D. Müller, Nucl. Phys. B841 (2010) 1-58, arxiv: QCD Evolution Workshop, Santa Fe, New Mexico, U.S.A., May 12-16, 2014

2 Outline Introduction to GPDs Local fits Global fits (small x B ) Global fits (all DVCS data) Neural networks approach Outcomes 2 Krešimir Kumerički : GPDs and fitting procedures for DVCS

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 : GPDs and fitting procedures for DVCS

4 Attractiveness of GPDs (2/3) Decomposition into nucleon 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 H q (x, 0, 0) x 0 = q(x) 1 1 dx H q (x, η, t) = F q 1 (t) 2. Close contact to 3D quark-gluon hadron structure 1 1 [ ] dx x H q (x, η, t) + E q (x, η, t) = J q (t) [Ji 97] 2 1 d 2 q(x, b ) = (2π) 2 eib H q (x, 0, 2 ) [Burkardt 00] 4 Krešimir Kumerički : GPDs and fitting procedures for DVCS

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 : GPDs and fitting procedures for DVCS

6 Factorization of DVCS GPDs [Collins et al. 98] γ γ q 2 1 = Q2 q2 2 = 0 C a x + ξ 2 x ξ 2 H a P 1 P 2 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 : GPDs and fitting procedures for DVCS

7 Dispersion-relation access to GPDs at LO [Teryaev 05; K.K., Müller and Passek-K. 07, 08; Diehl and Ivanov 07] LO perturbative prediction is handbag amplitude 1 ( ) H(ξ, t, Q 2 ) LO 1 = dx ξ x iɛ 1 H(x, ξ, t, Q 2 ) ξ + x iɛ 1 giving access to GPD on the cross-over line η = x 1 π Im H(ξ = x, t, Q2 ) LO = H(x, x, t, Q 2 ) H( x, x, t, Q 2 ) while dispersion relation connects it to Re H Re H(ξ, t, Q 2 ) = 1 π PV 1 0 ( 1 dξ ξ ξ 1 ) ξ + ξ Im H(ξ, t, Q 2 )+C H (t, Q 2 ) 7 Krešimir Kumerički : GPDs and fitting procedures for DVCS

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. 8 Krešimir Kumerički : GPDs and fitting procedures for DVCS

9 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(ξ, t) (Dependence on additional variable, photon virtuality Q 2, is in principle known given by evolution equations.) 8 Krešimir Kumerički : GPDs and fitting procedures for DVCS

10 Fit types Fit type Data used pqcd order Target 1. Local fits fixed target LO CFFs 2. Global fits collider/fixed target ((N)N)LO CFFs/GPDs 3. Neural nets fixed target LO CFFs 9 Krešimir Kumerički : GPDs and fitting procedures for DVCS

11 Local fits to HERMES data Most complete set of asymmetries (14) is measured in 12 bins: bin no t [GeV 2 ] x B Q 2 [GeV 2 ] A C dσ e + dσ e = dσ e + + dσ e [ F 1 Re H t A cos(1φ) C 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 ] To express asymmetries in terms of CFFs we use formulas from [Belitsky, Müller, Kirchner 01, Belitsky, Müller 10]. 10 Krešimir Kumerički : GPDs and fitting procedures for DVCS

12 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: 1. Perform single-cff fit with each of 8 CFFs and see which one alone describes data best (it is Im H, by far). 2. Combine Im H with each of other seven CFFs and see which pair describes data best. 3. Proceed until there is either no improvement in data description or new CFFs are not extracted with any statistical significance 11 Krešimir Kumerički : GPDs and fitting procedures for DVCS

13 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: 1. Perform single-cff fit with each of 8 CFFs and see which one alone describes data best (it is Im H, by far). 2. Combine Im H with each of other seven CFFs and see which pair describes data best. 3. Proceed until there is either no improvement in data description or new CFFs are not extracted with any statistical significance It turns out that already 2nd step is the final one, 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 and fitting procedures for DVCS

14 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 : GPDs and fitting procedures for DVCS

15 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 13 Krešimir Kumerički : GPDs and fitting procedures for DVCS

16 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. Stable and fast computer code for evolution and fitting 3. Moments are equal to matrix elements of local operators and are thus directly accessible on the lattice 14 Krešimir Kumerički : GPDs and fitting procedures for DVCS

17 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. Stable and fast computer code for evolution and fitting 3. Moments are equal to matrix elements of local operators and are thus directly accessible on the lattice To model η-dependence we use SO(3) partial wave expansion of crossed process γ γ p p where scattering angle θ corresponds to 1 η. 14 Krešimir Kumerički : GPDs and fitting procedures for DVCS

18 l-pw model only leading SO(3) partial wave H j (ξ, t, µ 2 0) = α a(t)=α a(0)+0.15t ( 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 a (t) = j + 1 α(0) j + 1 α(t) ) ( 1 t ) pa M0 a2... corresponding in forward case to PDFs of form Σ(x) = N Σ x α Σ(0) (1 x) 7 ; G(x) = N G x α G(0) (1 x) 5 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 15 Krešimir Kumerički : GPDs and fitting procedures for DVCS

19 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 16 Krešimir Kumerički : GPDs and fitting procedures for DVCS

20 dσ(γ p γp)/dt [nb/gev 2 ] t [GeV 2 ] Example of fit result H1, Q 2 = 4 GeV 2 H1, Q 2 = 8 GeV 2 σ(γ p γp) [nb] 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 ] 17 Krešimir Kumerički : GPDs and fitting procedures for DVCS

21 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 = 0.001, t = 0.2 GeV 2 (a) Q 2 [GeV 2 ] H G (x, x, t, Q 2 ) at N P LO x = 0.001, t = 0.2 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] 18 Krešimir Kumerički : GPDs and fitting procedures for DVCS

22 Extending global analysis to fixed target data Hybrid models at LO 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) 19 Krešimir Kumerički : GPDs and fitting procedures for DVCS

23 Re H determined by dispersion relations 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. 20 Krešimir Kumerički : GPDs and fitting procedures for DVCS ) 2

24 H1 (2007), ZEUS (2008) 21 Krešimir Kumerički : GPDs and fitting procedures for DVCS

25 HERMES (2008) 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 22 Krešimir Kumerički : GPDs and fitting procedures for DVCS

26 CLAS (2007) BSA. (Only data with t 0.3 GeV 2 used for fits.) 23 Krešimir Kumerički : GPDs and fitting procedures for DVCS

27 Hall A (2006) Fit to unpolarized cross section dσ/(dx B dq 2 dtdφ) KM10 fit needs unusually large Re H. Re H 24 Krešimir Kumerički : GPDs and fitting procedures for DVCS

28 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 have at the moment 25 Krešimir Kumerički : GPDs and fitting procedures for DVCS

29 Polarized target (II) Surprisingly large sin(2φ) harmonic of A UL cannot be described within this leading twist framework 26 Krešimir Kumerički : GPDs and fitting procedures for DVCS

30 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) 27 Krešimir Kumerički : GPDs and fitting procedures for DVCS

31 Neural networks Essentially a least-squares fit of a complicated many-parameter function. f (x) = tanh( w i tanh( w j )) no theory bias 28 Krešimir Kumerički : GPDs and fitting procedures for DVCS

32 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 and fitting procedures for DVCS

33 Neural Net HERMES fit - BSA/BCA 30 Krešimir Kumerički : GPDs and fitting procedures for DVCS

34 Neural Net HERMES fit - CFFs 31 Krešimir Kumerički : GPDs and fitting procedures for DVCS

35 Prediction for COMPASS II BCSA BCSA = dσ µ + dσ µ dσ µ + + dσ µ (E µ = 160 GeV) 32 Krešimir Kumerički : GPDs and fitting procedures for DVCS

36 Simulation of EIC capabilities Parton densities from combined fit to HERA collider and EIC pseudo-data [Aschenauer, Fazio, K.K and Müller 13] 33 Krešimir Kumerički : GPDs and fitting procedures for DVCS

37 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 : GPDs and fitting procedures for DVCS

38 GPD page and server Durham-like CFF/GPD server page 35 Krešimir Kumerički : GPDs and fitting procedures for DVCS

39 New directions (instead of Summary) Improving GPD models Adding deeply virtual meson production data and going to NLO [Müller, Lautenschlager, Schäfer 13] Including higher twists [Braun, Manashov et al.] Global neural network fits 36 Krešimir Kumerički : GPDs and fitting procedures for DVCS

40 New directions (instead of Summary) Improving GPD models Adding deeply virtual meson production data and going to NLO [Müller, Lautenschlager, Schäfer 13] Including higher twists [Braun, Manashov et al.] Global neural network fits The end. 36 Krešimir Kumerički : GPDs and fitting procedures for DVCS

41 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits Mapping asymmetries to CFFs 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. 37 Krešimir Kumerički : GPDs and fitting procedures for DVCS

42 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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. 38 Krešimir Kumerički : GPDs and fitting procedures for DVCS

43 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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). 39 Krešimir Kumerički : GPDs and fitting procedures for DVCS

44 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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) 39 Krešimir Kumerički : GPDs and fitting procedures for DVCS

45 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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) 39 Krešimir Kumerički : GPDs and fitting procedures for DVCS

46 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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. 39 Krešimir Kumerički : GPDs and fitting procedures for DVCS

47 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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 40 Krešimir Kumerički : GPDs and fitting procedures for DVCS

48 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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 40 Krešimir Kumerički : GPDs and fitting procedures for DVCS

49 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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 41 Krešimir Kumerički : GPDs and fitting procedures for DVCS

50 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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 41 Krešimir Kumerički : GPDs and fitting procedures for DVCS

51 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits 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] 42 Krešimir Kumerički : GPDs and fitting procedures for DVCS

52 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits Fit results - LO For consistency, we don t take standard PDFs, but fit GPDs to DIS data. This determines N sea, N G, α sea (0) and α G (0), leaving only M sea 0, s sea and s G for DVCS data χ 2 values: model α s χ 2 /d.o.f. DIS χ 2 /d.o.f. DVCS χ 2 t /n.o.p. χ2 W /n.o.p. χ2 Q 2 /n.o.p. l, dipole LO 49.7/ / / / /16 l, exp. LO 49.7/ / /56 79./ /16 nl, dipole LO 49.7/ / /56 27./ /16 nl, exp. LO 49.7/ / / / /16 Σ, dipole LO 49.7/ / / /29 16./16 Σ, exp. LO 49.7/ /98 51./ / /16 l, dipole LO 321./ / / /16 Parameter values: model α s N sea α sea (0) (M sea ) 2 s sea α G (0) s G B sea b eff BCA [GeV 2 ] [GeV 2 ] [GeV 2 ] l, dipole LO l, exp. LO nl, dipole LO nl, exp. LO Σ, dipole LO Σ, exp. LO (boldface numbers = bad fits) 43 Krešimir Kumerički : GPDs and fitting procedures for DVCS

53 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits Fit results - NLO χ 2 values: model α s χ 2 /d.o.f DIS χ 2 /d.o.f DVCS χ 2 t /n.o.p χ2 W /n.o.p χ2 Q 2 /n.o.p l NLO(MS) 71.6/ / / / /16 l NLO(CS) 71.6/ / / /29 17./16 nl NLO(MS) 71.6/ / / / /16 nl NLO(CS) 71.6/ / / / /16 Σ NLO(MS) 71.6/ /98 60./ / /16 Σ NLO(CS) 71.6/ / / / /16 Parameter values: model α s N sea α sea (0) (M sea ) 2 s sea α G (0) s G B sea b eff BCA l NLO(MS) l NLO(CS) nl NLO(MS) nl NLO(CS) Σ NLO(MS) Σ NLO(CS) (boldface numbers = bad fits) s sea,g small skewness ratio r Krešimir Kumerički : GPDs and fitting procedures for DVCS

54 App: Local mapping fits App: Neural Nets App: Conformal GPDs App: Fits Parameter values KMM12 KM Mv = Mv = rv = rv = bv = bv = C = C = MC = MC = tmv = tmv = trv = trv = tbv = tbv = rpi = rpi = Mpi = Mpi = M02S = M02S = SECS = SECS = THIS = THIS = SECG = SECG = THIG = THIG = Krešimir Kumerički : GPDs and fitting procedures for DVCS