Extraction of the Compton Form Factor H from DVCS Measurements in the Quark Sector
EExtraction of the Compton Form Factor H from DVCS Measurements in the Quark Sector H. MOUTARDE, on behalf of the CLAS group at Saclay
CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France
Abstract.
Working at twist 2 accuracy and assuming the dominance of the Generalized Parton Distribution H we study thehelicity-dependent and independent cross sections measured in Hall A, the beam spin asymmetries measured in Hall B atJefferson Laboratory and beam charge, beam spin and target spin asymmetries measured by Hermes. We extract the real andimaginary parts of the Compton Form Factor H , the latter being obtained with a 20–50 % uncertainty. We pay extra attentionto the estimation of systematic errors on the extraction of H . We discuss our results and compare to other extractions as wellas to the popular VGG model. Keywords:
Generalized Parton Distributions, Compton Form Factors, Deeply Virtual Compton Scattering.
PACS:
INTRODUCTION
The Deeply Virtual Compton Scattering (DVCS) process [1, 2] was early recognized as the cleanest way to accessGeneralised Parton Distributions (GPD) and has been so far a very active field of research. Theoretical developmentsare reviewed in Refs. [3, 4, 5, 6] and relevant experimental results are published in Refs. [7, 8, 9, 10, 11, 12, 13, 14,15, 16, 17, 18, 19].What we would like is to eventually produce a experimental 3d picture of the nucleon thanks to the knowledge ofGPDs. Even if getting a precise experimental knowledge of GPDs is a long-term program, we can already perform thefirst fits to get a flavour of the actual sensitivity of observables to GPDs and elaborate robust and efficient extractingmethods. We address these issues by studying recent JLab data, namely beam spin asymmetries [14] (BSA) andhelicity-dependent and independent cross sections [12]. These data offer the interesting features of a large kinematiccoverage and a fine kinematic binning. We also consider Hermes data : beam charge, beam spin and target spinasymmetries [15, 20].The first section of this paper describes the evaluation of the ep → ep γ cross sections and outlines our hypothesis. Inthe second part we explain our fitting strategy. In the third section we discuss the extracted values of H and comparethem to other similar studies. PRELIMINARY ANALYSISDVCS at leading twist
Four GPDs H , E , ˜ H and ˜ E appear at twist 2, but the cross sections depend on the Compton Form Factors (CFF) H , E , ˜ H and ˜ E . The convention of Ref. [21] is used to define the CFFs. The complex integration kernel yields a real andan imaginary part to a CFF. For example the CFF H satisfies : Re H = P (cid:90) + − dx H ( x , ξ , t ) (cid:18) ξ − x − ξ + x (cid:19) (1) Im H = π (cid:16) H ( ξ , ξ , t ) − H ( − ξ , ξ , t ) (cid:17) (2)where ξ = x B ( + t / ( Q )) / ( − x B + x B t / Q )) is the generalised Bjorken variable [21, 24], x B the standard Bjorkenvariable, Q the virtuality of the initial photon and t the square momentum transfer between initial and final protons. P denotes the principal value of the integral. a r X i v : . [ h e p - ph ] O c t he use of dispersion relations relating real and imaginary parts of a CFF had been discussed by Anikin and Teryaev[25, 26, 27], Diehl and Yvanov [28] and Goldstein and Liuti [29, 30, 31]. Dispersions relations are used in the model-dependent global fit of Kumeriˇcki and Müller [32]. However the unknown subtraction (the D-term [33]) and the limitedkinematic range of available data make this constraint rather weak in model-independent fitting. In this work we try tominimize the model-dependence of the extraction. Then we consider real and imaginary parts of CFFs as independentexcept when we use an explicit parametrisation of GPDs to compute CFFs. ep → ep γ observables at twist 2 The harmonic analysis of ep → ep γ cross-sections has so far relied on the 2002 work of Belitsky, Müller andKirchner [21] (BMK). In this formalism, the interference between the Bethe-Heitler (BH) and DVCS processes wastreated with a leading order approximation of the BH part. This assumption was removed by Belitsky and Müller inthe case of a spinless target [22] and for a (longitudinally polarised or unpolarised) spin 1/2 target [23] and by Guichonand Vanderhaegen in the case of a proton target [24]. In all the following, we will use the expressions from the latter.At last, the twist 2 approximation is motivated by the claim of early Q -scaling observed in Hall A [12]. Moreover the A LUDVCS asymmetry measured by Hermes [20] are small if not zero while they are supposed to vanish exactly at twist 2.
About current extractions methods
The methods to extract GPDs from measurements fall today into 4 groups : local fits
The real and imaginary parts of CFFs are the free parameters. A given kinematic bin ( x B , t , Q ) is consideredindependantly of the others. The model dependence is low but some assumptions on the kinematics may benecessary to avoid underconstrained problems [38, 39, 40, 41]. It does not allow any extrapolations and inparticular not those necessary to obtain spatial information in the transverse plane [42] or the quark orbital angularmomentum in the nucleon [1]. global fits CFFs or GPDs have model-dependent expressions, the free parameters of which are fitted. See for example[32] for recent fit of DVCS data on unpolarized targets or [43, 44, 45] for recent studies of Deeply Virtual MesonProduction data. hybrid fits
This is a combination of the two previous methods, used in [46] and detailed in the present text. neural networks fits
Work is in progress but preliminary results were recently presented [47]. H -dominance hypothesis When we consider experiments on unpolarised proton targets we can neglect E , ˜ H and ˜ E ( H -dominance). Thecontribution of these GPDs to helicity-dependent cross-sections is indeed kinematically suppressed [46] and we cancheck (for instance thanks to the VGG model [34, 35, 36, 37]) that Im E and Im ˜ H are usually smaller than Im H . Atlast for small t and ξ we expect ˜ H / H to be close to ∆ q / q i.e. .Thus assuming H -dominance we may hope to extract information on H from BSAs or helicity-dependent crosssections with a systematic error of 20 % to 50 % , this approximation being better at small t . The advantage of thisapproach is the dramatic decrease of the number of degrees of freedom involved in fits. Guidal indeed showed [38] inthe same kinematic region that it is not possible to extract sensible information about the real and imaginary parts ofall CFFs by direct fits of helicity-dependent and independent cross sections. A direct test of the H -dominance assumption with the VGG model gives an upper bound of 25 %. This is comparable to the typical statisticaluncertainty on BSAs. ITTING STRATEGIES
The possibility to study GPDs in DVCS rests on factorisation theorems [48] which require a small value of | t | / Q .In the following, we restrict ourselves to kinematic configurations for which | t | / Q < / aposteriori . As stated in the first section, we perform two kinds of fits : Local fits Re H and Im H are the free parameters of the fits. Global fit
We use a "dual model"-like parametrization ; it consists basically in a simultaneous expansion in Gegen-bauer and Legendre polynomials. See Ref. [46] for details on the fitting Ansatz.We estimate the systematic errors associated to our H -dominance hypothesis by first fitting data setting the subdom-inant GPDs to 0, then fitting the same data setting the subdominant GPDs to their VGG value, and computing thedifference. We estimate the systematic uncertainty related to the global fitting Ansatz by varying the number of poly-nomials involved in the expansion.A careful fitting strategy is elaborated to handle the high number of free parameters involved in the fits and makesure that the numeric implementation is under control. We adopt an iterative (5 steps) fitting procedure. See Ref. [46]for full details. RESULTSExtraction of Im H and Re H The Fig. 1 displays our results for Im H and Re H . Both local fits and global fit give results with comparableaccuracy for Im H , but as expected the results of the global fits are smoother. This is especially true concerning Re H : in this case the local fits suffer from large fluctuations of Re H with values which fall outside the plot range.However, we could not reliably extract values of the CFFs for the larger values of ξ with the global fit.The results for local and global fits are almost always compatible, which is a strong consistency check. Both rely onthe assumptions of twist 2 accuracy and of H -dominance. On one hand, local fits suffer from numerical fluctuations(the 2-parameter local fits are not constrained enough on some bins) but are almost model-independent. On the otherhand, global fits are smoother, but suffer from oscillations. That both methods give the same results indicates thatfluctuations and oscillations are reasonably controlled in the bins for which results are displayed. Since, in both cases,the total error bars have the same size, we conclude that our estimation of systematic uncertainties on the global fitAnsatz is realistic even if not perfect.All fits keep data satisfying | t | / Q < /
2. For local fits, changing the maximal value of | t | / Q amounts to droppingpoints. For global fits, the whole results may be changed, but the good agreement between the results of both types offits, and the slow Q -evolution of the extracted CFFs, indicate that this restricted kinematic region is suitable for ananalysis in the GPD framework.Our results are explicitly given in Ref. [46]. Error bars are dominated by systematic effects. Typically we obtain arelative accuracy of 20 to 50 % on Im H , which is quite satisfactory under the assumption of H -dominance and giventhe statistical accuracy of JLab data. On the contrary, Re H is still largely undetermined, and is never extracted with aprecision better than 50 %. Discussion
The Fig. 2 compares our results on Hall A kinematics to twist 2 model-independent extractions [38, 40] of Guidaland two extractions with the BMK formalism [12, 32]. Firstly, the use of the GV expressions creates importantdeviations to the extraction of Ref. [12]. Since the extracted combinations of GPDs are not the same, we will notmake the argument more quantitative. Another discrepancy occur when comparing to the results of the global fit ofthe GPD H of Kumeriˇcki and Müller [32]. Secondly, we obtained results in good agreement with Ref. [38]. Notehowever the surprisingly big shift between the results of Ref. [38] and Ref. [40] induced by the inclusion of target spinasymmetries. These points need to be elucidated but we can already mention the following points : • Large contributions of ˜ H and ˜ E are necessary in order to fit JLab Hall A data with the parametrization ofKumeriˇcki and Müller. Other DVCS data on unpolarized targets can be included in this global fit under the (a)(b)(c)(d) I m H I m H I m H I m H Local Fits Global Fit) (GeV Q ) (GeV Q R e HR e HR e HR e H Local Fits Global Fit) (GeV Q ) (GeV Q -7.5-5-2.502.557.5 -7.5-5-2.502.557.5 -7.5-5-2.502.557.5 -7.5-5-2.502.557.5 -7.5-5-2.502.557.5 -7.5-5-2.502.557.5 -7.5-5-2.502.557.5 -7.5-5-2.502.557.5 FIGURE 1. Q -behaviour (1 < Q < ) of the extracted values of Im H (left) and Re H (right) of local fits and global fiton Hall B kinematics : 0 . < − t < . (a), 0 . < − t < . (b), 0 . < − t < . (c), and 0 . < − t < . GeV (d).The error bars include both statistics and systematics. Im H ranges between 0 and 10 and Re H ranges between -7.5 and +7.5. Theblack full circles correspond to x B =0.125, red squares to x B =0.175, green up triangles to x B =0.250, blue down triangles to x B =0.360and magenta open circles to x B =0.491. H -dominance assumption. • The shift between the two extractions of Guidal is attributed to a ˜ H contribution. • Using VGG and the MIT Bag GPD model [49] we obtained a refined estimate of the systematic uncertaintycaused by H -dominance. We obtained larger error bars for the extracted CFF on Hall A kinematics, and errorbars compatible with previous estimates on Hall B kinematics.These facts indicate that JLab Hall A data can certainly not be understood under the restricted set of assumptions : twist2 + H -dominance. Confirming an explanation involving GPDs other than H will require the inclusion of observablesmore sensitive to these GPDs. Moreover it was already stressed in Ref. [46] that the twist 2 assumption need to bechecked further. Interestingly the three different methods described in Refs. [32, 38, 40, 46] give results in goodagreement on Hall B data. The reason of the distinction (discrepancy for Hall A data, agreement for Hall B data) needto be elucidated to make progress on GPD extraction in the valence region. Let us remind that JLab Hall A and Hall Bdata are compatible. CONCLUSIONS
Working at leading twist, and assuming H -dominance, we extracted Im H and Re H with 20–50 % accuracy on Im H .The good agreement between the results of local and global fits is a strong consistency check. Moreover values of Im H on Hall B kinematics extracted by three different groups with three different methods are in good agreement. The -0.4 -0.3 -0.2 -0.1) t (GeV ) t (GeV ) t (GeV -5-4-3-2-10345678 R e H I m H Global Fit EPJA 39, 5 (2009) PRL 97, 262002 (2006) -0.4 -0.3 -0.2 -0.1 -0.4 -0.3 -0.2 -0.1 arXiv:1003.0307 -0.4 -0.3 -0.2 -0.1
With HT (arXiv:0904.0458)Without HT (arXiv:0904.0458) -0.4 -0.3 -0.2 -0.1 -5-4-3-2-101 -0.4 -0.3 -0.2 -0.1 -5-4-3-2-101 -0.4 -0.3 -0.2 -0.1 -5-4-3-2-101
FIGURE 2. Im H (up) and Re H (down) vs t (0 . < − t < . ) on Hall A kinematics ( x B =0.36 and Q = 2.3 GeV ). Im H ranges between 3 and 8, and Re H between -5 and 1. We compare our results (left column) to those of Guidal [38] (middle column)and Muñoz-Camacho et al. [12] (right column). In the latter column H + x B − x B (cid:16) + F F ˜ H (cid:17) − t M F F E is plotted, and not H . Theerror bars include both statistical and systematic uncertainties. The green (resp. blue) curve is the result of the model-dependentglobal fit of Kumeriˇcki and Müller [32] with (resp. without) ˜ H . The full black square is the result of Ref. [40]. comparison of local and global fits and the weak Q -dependency of the results also validates a posteriori the restrictionto kinematic configurations with | t | / Q < /
2. Note however that the interpretation of JLab Hall A measurements isdifficult and probably requires to go beyond the common H -dominance approximation.Our results are dominated by systematic uncertainties. The systematic uncertainties related to H -dominance willdecrease in the future using additional BSA measurements [50] on unpolarised and longitudinally polarised protontarget which will put stronger constraints on the global fits. These future measurements will also help clarifying theinterpretation of Hall A helicity-independent cross sections. ACKNOWLEDGMENTS
The author would like to thank the CLAS group at Saclay, P. Guichon, M. Guidal, K. Kumeriˇcki, D. Müller andK. Passek-Kumeriˇcki for many fruitful and stimulating discussions. The author also thank the organizers of the 12thInternational Conference on Meson-Nucleon Physics and the Structure of the Nucleon held in College of Williamand Mary (May 31-June 4, 2010), Williamsburg, Virginia. This work was supported in part by the Commissariat àl’Energie Atomique and the GDR n° 3034 Physique du Nucleon.
REFERENCES
1. X.-D. Ji.,
Phys. Rev. Lett. , 610 (1997).2. M. Diehl, Th. Gousset, B. Pire and J.P. Ralston, Phys. Lett. B , 193 (1997).3. K. Goeke, M.V. Polyakov and M. Vanderhaeghen,
Prog. Part. Nucl. Phys. , 401 (2001).4. M. Diehl, Phys. Rept. , 41 (2003).. A.V. Belitsky and A.V. Radyushkin,
Phys. Rept. , 1 (2005).6. S. Boffi and B. Pasquini,
Riv. Nuovo Cim. , 387 (2007).7. A. Airapetian et al. , Phys. Rev. Lett. , 182001 (2001).8. C. Adloff et al. , Phys. Lett. B , 47 (2001).9. S. Stepanyan et al. , Phys. Rev. Lett. , 182002 (2001).10. S. Chekanov et al. , Phys. Lett. B , 46 (2003).11. S. Chen et al. , Phys. Rev. Lett. , 072002 (2006).12. C. Muñoz Camacho et al. , Phys. Rev. Lett. , 262002 (2006).13. F.D. Aaron et al. , Phys. Lett. B , 796 (2008).14. F.X. Girod et al. , Phys. Rev. Lett. , 162002 (2008).15. A. Airapetian et al. , JHEP , 066 (2008).16. A. Airapetian et al. , JHEP , 083 (2009).17. A. Airapetian et al. , JHEP , 019 (2010).18. A. Airapetian et al. , Nucl. Phys. B , 1 (2010).19. A. Airapetian et al. , Phys. Rev. C , 035202 (2010).20. D. Zeiler et al. , arXiv:0810.5007 [hep-ex].21. A.V. Belitsky, D. Mueller and A. Kirchner, Nucl. Phys. B , 323 (2002).22. A.V. Belitsky and D. Mueller,
Phys. Rev. D , 014017 (2009).23. A.V. Belitsky and D. Mueller, arXiv:1005.5209 [hep-ph].24. P.A.M. Guichon and M. Vanderhaeghen, Analytic ee’ γ cross section , in Atelier DVCS, Laboratoire de Physique Corpusculaire,Clermont-Ferrand, June 30 - July 01, 2008 .25. O.V. Teryaev, arXiv:hep-ph/0510031.26. I.V. Anikin and O.V. Teryaev, arXiv:0710.4211 [hep-ph].27. I.V. Anikin and O.V. Teryaev,
Phys. Rev. D , 056007 (2007).28. M. Diehl and D.Yu. Ivanov, Eur. Phys. J. C , 919 (2007).29. G.R. Goldstein and S. Liuti, Phys. Rev. D , 071501 (2009).30. G.R. Goldstein and S. Liuti, arXiv:0908.2215 [hep-ph].31. G.R. Goldstein and S. Liuti, arXiv:1006.0213 [hep-ph].32. K. Kumeriˇcki and D. Mueller, Nucl. Phys. B , 1 (2010).33. M.V. Polyakov and C. Weiss,
Phys. Rev. D , 114017 (1999).34. P.A.M. Guichon and M. Vanderhaeghen, Prog. Part. Nucl. Phys. , 125 (1998).35. M. Vanderhaeghen, P.A.M. Guichon and M. Guidal, Phys. Rev. Lett. , 5064 (1998).36. M. Vanderhaeghen, P.A.M. Guichon and M. Guidal, Phys. Rev. D , 094017 (1999).37. M. Guidal, M.V. Polyakov, A.V. Radyushkin and M. Vanderhaeghen, Phys. Rev. D , 054013 (2005).38. M. Guidal, Eur. Phys. J. A , 319 (2008).39. M. Guidal and H. Moutarde, Eur. Phys. J. A , 71 (2009).40. M. Guidal, Phys. Lett. B , 156 (2010).41. M. Guidal, arXiv:1005.4922 [hep-ph].42. M. Burkardt,
Phys. Rev. D , 071503 (2000), Erratum-ibid. D66, 119903 (2002).43. S. Goloskokov and P. Kroll, Eur. Phys. J. C. , 281 (2005).44. S. Goloskokov and P. Kroll, Eur. Phys. J. C. , 367 (2008).45. S. Goloskokov and P. Kroll, Eur. Phys. J. C. , 137 (2010).46. H. Moutarde, Phys. Rev D , 094021 (2009).47. K. Kumeriˇcki and D. Müller, arXiv:1008.2762 [hep-ph].48. J.C. Collins and A. Freund, Phys. Rev. D , 074009 (1999).49. X.-D. Ji, W. Melnitchouk and X. Song, Phys. Rev. D , 5511 (1997).50. V. Kubarovsky, Deeply Virtual Exclusive Reactions with CLAS , in