Description and interpretation of DVCS measurements

Having in mind the well-known limitations of certain models of generalized parton distributions (GPDs), we show that the allegedly universal GPDs, describing both deeply virtual Compton scattering (DVCS) and deeply virtual meson production (DVMP) data, fail to describe measurements of deep inelastic scattering. We present a new global DVCS fit that describes reasonably the world data set, which includes now also new measurements from Hall A and CLAS collaborations. We also explicitly illustrate that Compton form factors (CFFs) cannot be unambiguously extracted from photon electroproduction measurements off unpolarized proton alone and we argue that it is more reliable to interpret such measurements in terms of certain CFF combinations, where still some care is needed in order to estimate the propagated error.


Introduction
The deeply virtual Compton scattering (DVCS) process off a nucleon target is considered to be the theoretically cleanest process allowing access to generalized parton distributions (GPDs) from experimental measurements. Motivated by this, deeply virtual Compton scattering has been studied in the last decade quite extensively by HERA and JLab collaborations. Thereby, cross section measurements on an unpolarized proton have been performed in collider experiments H1 and ZEUS as well as now in fixed target experiments by the CLAS and Hall A collaborations, while HERMES provided an almost (over-)complete set of asymmetry measurements. On the other hand, considering possible photon and proton helicities, there are twelve DVCS amplitudes where, loosely spoken, the four photon helicity conserved amplitudes or Compton form factors (CFFs) H, E, H, E appear in the 1/Q 2 expansion as leading contributions. In the leading order and leading twist approximation these CFFs are given in terms of quark GPDs H, E, H, E, where ξ ≈ x B /(2 − x B ), x B is Bjorken's scaling variable, t is the square of the momentum transfer, and the factorization scale has been equated to the (negative) photon virtuality q 2 = −Q 2 . The proton where α = e 2 /4π ≈ 1/137 is the electromagnetic fine structure constant, y = Q 2 /x B s − M 2 with s = (p 1 + q 1 ) 2 , and 2 = 4x 2 B M 2 /Q 2 is an abbreviation. The electroproduction amplitude, entering as square of its absolute value into the cross section, |T | 2 = |T BH | 2 + I + |T VCS | 2 , I = 2 eT BH T † VCS (3) is the sum of the charge-odd Bethe-Heitler T BH amplitude and the charge-even virtual Compton scattering amplitude T VCS amplitude. The three different contributions might be decomposed in Fourier harmonics, e.g., for an unpolarized nucleon |T BH | 2 = e 6 x 2 B y 2 (1 + 2 ) 2 t P 1 (φ)P 2 (φ) 2 n=0 c BH n,unp cos(nφ) , where the product of (rescaled) electron propagators 1/P 1 (φ)P 2 (φ) implies an additional φ dependence. The coefficients c BH n,unp depend on the electromagnetic proton form factors for which we use the Kelly parametrization [2]. For the interference term we have I = ±e 6 x B y 3 tP 1 (φ)P 2 (φ) where the + (−) sign refers to electron (positron) scattering and c VCS n,unp cos(nφ) + s VCS 1,unp sin(φ) for the squared DVCS amplitude. The harmonics of the interference term are given in terms of linear CFF combinations while those of the squared DVCS term are given in terms of bilinear (linear in both CFFs and their complex conjugates) combinations. Projecting first the experimental data onto (weighted) harmonics, see [3], is a standard filter procedure we utilize first in both global GPD model fitting and local CFF extraction.
The main emphasis of the present contribution is to spell out some phenomenological inconsistencies in the description of DVCS data which are not very much clarified in the literature. In Sec. 2 we show that universality of the GPD framework is violated in a certain model class and we provide a description of the world data DVCS set that is obtained by global fitting. In Sec. 3 we spell out that the interpretation of some results that arise from so-called quasi model-independent extraction of CFFs from CLAS cross section measurements might be misleading. Finally, we conclude.

Model predictions versus global fitting
Based on Regge theory arguments, one might assume that in collider experiments only one CFF contributes, giving access to the unpolarized quark GPD H q combination, e.g., in LO approximation by means of the convolution formula (1). The situation is different for fixed target kinematics, in particular for the JLab experiments with a 6 GeV electron beam and unpolarized or longitudinally polarized proton target. Additionally to model predictions, global fitting attempts were undertaken to extract the values of CFFs. It is clear from the outset that also in the case of a complete or overcomplete measurement the extraction of such information is a challenging task that can only be resolved within some physical or model-motivated assumptions. The common assumption most authors adopt is the dominance of leading twist (LT) in a perturbative leading order (LO) description, i.e., the number of complex-valued Compton form factors F ∈ {H, E, H, E, · · · } reduces from twelve to four. This restriction might be not justified outside the DVCS kinematics, e.g., −t ∼ Q 2 /2 or so.
The new CLAS beam spin asymmetry A LU and longitudinally polarized target spin asymmetries A UL and A LL are measured in five {x B , Q 2 }-bins versus t [4] The Hall A collaboration provided high precision cross section measurements with a polarized electron beam and unpolarized proton target [5], which were superseded more recently for five −t/GeV 2 ∈ {0.17, 0.23, 0.28, 0.33, 0.37} values within five kinematical settings [6], where last two settings have subset of the same data re-binned for measuring the x B -dependence. Somewhat less precise measurements of cross-section, but covering much larger kinematic space were published by CLAS [7].

Model predictions
It is known since more than a decade that certain GPD models such as Radyushkin's double distribution ansatz (RDDA) or the leading SO(3)-partial wave expansion of conformal GPD moments fail to globally describe the DVCS data at leading twist and leading order of perturbation theory. While the failure of the latter model is well-accepted, problems of the former one are often forgotten or hidden in the literature, see, e. g., Ref. [8], where a good description of DVCS data by RDDA-based model has been reported. It is rather popular to decorate the RDDA with t-dependence in such a manner, that the t-dependence is entirely contained in the non-forward parton density q(x, t). The model bias in such a parametrization is that the profile function is t-independent and, thus, for valence quarks it can be adjusted from form factor measurements where one relies on the functional ansatz. To complete polynomiality in the charge even sector, one might add a so-called D-term. Such an ansatz is the basis of older models, where partially a factorized t-dependence was utilized, the GK model [9], and it is also implemented in the VGG code [10]. The phenomenological problem with such models is that the skewness effect at small x B is too large and so their predictions substantially overshoot the DVCS cross section measurements in collider kinematics, as pointed out for some time already in Ref. [11]. Thus, it is at first surprising that the GK model correctly describes the H1 and ZEUS DVCS data. However, the reason for that is the reduction to n F = 3 of the flavour content of the PDF parametrization, adopted from CTEQ6 [12], see convolution formula (1). Such exclusion of the charm quark contribution from the model leads then to the large underestimate of the deep inelastic structure function F 2 (x B , Q 2 ). This is displayed by the short dashed line in the left panel of Fig. 1 to be compared with the LO CTEQ6L1 (dash-dotted), NLO CTEQ6M (dashed), and our LO parametrization (solid). Hence, we conclude that the universality of the GPD framework in the GK model (and other GPD models based on RDDA) is not implemented.
On the other hand the GK model describes the asymmetry measurements of HERMES that were obtained within the missing mass technique, except for the beam spin asymmetry A sin(1φ) LU , which is a key observable, compare full squares with short dashed line in the upper right panel of Fig. 1. On the other hand the recoil detector data of the HERMES collaboration (empty squares), which are consistent with the missing mass technique data, are described rather well by the GK model. Unfortunately, the two different data sets are not combined and the missing mass technique data suffer from a larger experimental uncertainty. As we will see in the next section the GK model quantitatively overestimates the beam spin asymmetry and the beam spin dependent cross section of JLab measurements.

Predictions of new JLab results and a new global fit
To fit globally the world data set of DVCS measurements from HERA and JLab collaborations, we employ a rather simple hybrid model. It is based on a double partial wave expansion for the seaquark and gluon GPD, including the LO QCD evolution, and a dispersion integral representation for the valence quark content that does not evolve. Details of the model can be found in [14], and last published incarnation of the model (KMM12) [17] was obtained by a standard least-squares fitting procedure to the statistically independent set of 95 DVCS data points 1 [13] are confronted with the GK model (shortdashed), based on the next-to-leading order PDF parametrization CTEQ6M (dashed), the leading order PDF parametrization CTEQ6L1 (dot-dashed) as well as our DIS fit (solid) from Ref. [14]. Right: Beam spin (upper panel) and beam charge (lower panel) asymmetry from the HERMES collaboration: measurements of A sin(φ) LU,+ with the recoil detector (empty squares) [15] and obtained from measurements of A sin(φ) LU,I and A cos(φ) C within the missing mass technique (filled squares) [16] versus the GK model prediction (short-dashed) and various fits, KM10 (longdashed), KM10a (dashed), KM10b (shorter dashed), KMM12 (dash-dotted), KMM12b (shorter dash-dotted), and KM15 (solid), to the HERMES data (filled squares) as solid curves.
where most problematic data was (i) HERMES data with longitudinally polarized target (and to some extent beam), and (ii) the very precise Hall A cross-section data (especially t-dependence of the first cosine harmonic).
On Table 1 we list all data used in fits, as well as χ 2 /n pts values for particular datasets, for the KMM12 model, and for the new KM15 model to be discussed below. To give additional information about compatibility of the fit with the data, we list also the values of the "pull" of each dataset, defined for dataset consisting of N measurements as where f (x i ) is model value for a given kinematics x i , while y i and ∆y i are the corresponding experimental value and uncertainty, respectively. For a perfect model, pull (11) [14,17]. Parameters of the old KMM12 fit [17] are also given for comparison.   One notices that the cross-section went down a bit and the strong t-dependence of the first cosine harmonic (dσ cos φ,w , lowest panels) seen in 2006 data is significantly milder now. All this made possible the decrease of skewness parameters r val andr val , i.e., the decrease of H andH, as well as the decrease of the pion pole contribution, thus improving the fit of most of the other observables, see Table. 1. In particular, description of HERMES A sin φ LU,I is significantly improved, and overshooting (large positive pull) of CLAS A sin φ LU is not present any more. Looking at longitudinally polarized target data, χ 2 of A sin φ UL is also improved, while very large χ 2 contribution of HERMES A cos 0φ LL is mostly due to a single outlier point so we don't consider this a problem. First cosine harmonic of the cross section remains the most problematic observable to describe. As discussed above, in the Hall A case the tension is relaxed but still exists, and although χ 2 value for new CLAS measurement is fine, the pull of 6.36 is unacceptably large, showing consistent overshooting of the data.

Local versus global extraction of CFFs
It is a standard problem in the phenomenology of scattering processes that if only an incomplete measurement is performed one can extract only certain combinations of squared helicity amplitudes at given kinematics. In the photon leptoproduction the interference between Bethe-Heitler bremsstrahlung process and deeply virtual Compton scattering offers various possibilities to access CFFs. To utilize the azimuthal angular dependence in the separation procedure, it is of advantage to EPJ Web of Conferences perform first a Fourier analysis of the azimuthal angular dependence. In the following it is assumed that the eight sub-CFFs, appearing in LO and LT approximation (1) or, equivalently, in the hand-bag approach, dominate the twist-two associated first and zeroth asymmetry harmonics, where the latter one might be more contaminated by higher twist contributions than the former ones. Various methods for local extraction have been tested -we will discuss here only two: map of random numbers and a so-called quasi model-independent fitter code that is based on reference GPD models.

Mapping of random numbers
The mathematical problem to extract CFFs locally, i.e., at fixed values of x B , Q 2 , and t, from experimental measurements, can be simply understood as a map from the space of observables into the space of CFFs [3]. For asymmetry measurements with both kinds of leptons available this map of random numbers has been described in Ref. [17]. For cross section measurements with a polarized electron beam and/or polarized nucleon target it can be abstractly formulated as a set of linear and quadratic equations, where the n-dimensional vector X contains the experimental measurements of (weighted) harmonics at given kinematics, F is a n-dimensional vector of sub-CFFs (12) under the assumption that the remaining 24 − n dimensional sub-CFF vector G can be considered as fixed. The constant vector contains contributions from the Bethe-Heitler term, interference term, and/or squared VCS amplitude, while the linear coefficient matrix c −1 (F 1 , F 2 ; G) arises from the interference term and/or squared VCS amplitude. The (multi-valued) solution of the (quadratic) equations can be found by diagonalization.
Another local extraction method relies on the finding that apart from some power suppressed contributions, which are considered small, only certain linear and bilinear CFF combinations enter the interference term and the squared VCS amplitude, respectively. Strictly spoken, for each kinematical point we have different functional forms of these combinations which also depend on the center-ofmass energy or energy loss variable y. However, in the leading twist approximation the center-of-mass energy dependence approximately separates and Eq. (13) might be written as where now F contains the full set of (sub-)CFFs and k −1 and K are matrices of dimensions n. For cross section measurements with an unpolarized nucleon target and a polarized electron beam [6,7] the information which can be extracted are the first (weighted) harmonic of the beam helicity dependent cross section ∆(φ) = 1 2 (dσ ↑ − dσ ↓ )/dx B Q 2 dtdφ, and zeroth and first (weighted) harmonics of the unpolarized cross section Σ(φ) = 1 2 (dσ ↑ + dσ ↓ )/dx B Q 2 dtdφ, allowing us to map directly the experimental data to the imaginary and real part of the CFF combination C I unp (F ) and the bilinear form C VCS unp (F , F * ). For precise definition of various symbols see Ref. [1].
In the practical application of the strategy (13) one might straightforwardly calculate the sub-CFFs F (X; F 1 , F 2 |G), i.e., their mean and variance, as function of the experimental measurements, the form factor parametrization, and some given values of the remaining CFFs G, whose fixed values can be varied in some model-dependent or theory-inspired region. Thus, the uncertainty of the extracted sub-CFFs F (X; F 1 , F 2 |G) contains essentially two main sources: the experimental uncertainty of the DVCS and the form factor measurements, including also the uncertainties of QED corrections, and the estimated uncertainty of the unknown CFF contributions. The strategy (14) has the advantages that within this assumption the uncertainties of the extracted linear and bilinear CFF combinations are governed only by the experimental uncertainties. However, some care is needed to have control over the uncertainties that are induced by the assumed approximation. A consistency check between the two methods is provided by evaluating the set of (approximate) equalities which have to be (almost) independent on the values of G.

Quasi model-independent method based on reference GPD models
Another strategy, based on the method of least squares fitting, was suggested in Ref. [28] and utilized to locally extract GPD information in LO and LT approximation (1) from experimental measurements [29][30][31]. Here, the parameters are eight multipliers, where their values are normalized to reference GPD predictions of the VGG code, reading here in our notation as where F VGG are predicted by the VGG code via the convolution formula (1) and the multipliers a m H , · · · , a e E are the eight new model-independent fitting parameters. Further reduction might be made by setting some CFFs to zero, e.g., the imaginary part of the helicity non-conserved CFF E whose smallness is suggested by the GPD findings in the DVMP analyses of π + electroproduction [32]. Often the method was/is employed to an incomplete set of observables, i.e., the fitting problem is ill-posed and to reach convergency one constrains the seven parameters |a m F | 5 or so, including the more unreliable model estimates |a e F | 5 for the real part of CFFs. This method has been recently extended by generating an ensemble of fits with randomly distributed start values, where it is hoped that in this statistical manner also ill-posed fitting problems can be managed [33].

Comparison of results
Various strategies to locally extract CFFs from the almost over-complete DVCS asymmetry measurements of the HERMES collaboration were studied in Ref. [17]. Namely, one-to-one mapping, stepwise regression, and least squares fitting methods. The quasi model-independent method, see Sec. 3.2, utilizes all available harmonics, i.e., together with the stronger hypothesis that all observables are describable in the hand-bag approach noise is used to stabilize the fitting procedure. The results for { mH, m H, eH} of Ref. [33] are in good agreement with our findings [17]. Note that in Ref. [17] all local CFF values were reported, including the negative one for H at x B = 0.08, Q 2 = 1.9 GeV 2 , and −t = 0.03 GeV 2 , however, we concluded that only the sub-CFFs mH, eH, m H, and maybe mE are accessible from HERMES measurements. The new CLAS beam spin asymmetry A LU and longitudinally polarized target spin asymmetry A UL and A LL measurements [4] were also analyzed with the quasi model-independent tool, shortly described in Sec. 3.2, were results for the imaginary parts of CFFs H and H were presented in Fig.  25 of Ref. [4]. In contrast to the HERMES analysis, there are only three asymmetries that are sensitive to two linear combinations (5) and (6) of the imaginary part of CFFs and one to the real part of Eq. (6), where in addition the normalization of asymmetries, i.e., the unpolarized cross section that is dominated by the Bethe-Heilter amplitude, is nonlinearly governed by all CFFs. We emphasise that CLAS data in the DVCS region do not indicate (apart from noise) the appearance of higher harmonics and that there might be some tension among longitudinal target spin asymmetry measurements from HERMES and CLAS collaborations. The results from the quasi model-independent approach have naturally large error bars, which provide a statistical representation of both the propagated experimental uncertainties and the model-dependent constraints on the non-extractable degree of freedom. As we would have expected, the locally extracted results in the DVCS kinematics are predicted from our older KMM12b and new KM15 fit (solid curves), shown for three DVCS bins in Fig. 3 while the GK (dashed) model prediction for GPD H is too large. Note that all models have a rather similar GPD H prediction. The t-dependence of GPD H comes out flatter than for GPD H, which has a natural explanation in GPD models that incorporate form factor constraint. Namely, from the common GPD modeling and the fact that the axial nucleon form factor possesses a flatter t-dependence than the Pauli form factor, the above interpretation can be considered a model-dependent GPD prediction and only by means of the GPD framework one might loosely link the CFF H to the axial-charge distribution, quantified by the axial form factor.
As explained in Sec. 3.1, the information which can be extracted from the new Hall A [6] and CLAS [7] cross section measurements are the imaginary and real parts of the CFF combination C I unp (F ) and the bilinear form C VCS unp (F , F * ), see Eq. (15), while extraction of sub-CFFs is an ill-posed problem. We utilized the method (15), where uncertainties were added in quadrature and propagated via the transformation of the covariance matrix, see [17]. Note that in the propagation of experimental uncertainties we ignored the fact that systematic ones are asymmetric; however, for the purpose of an illustration at twist-two level this is not an important issue. Our results, shown as filled squares in the upper row of Fig. 4 coincide for the imaginary and real part of the linear combination C I unp and for the bilinear form C VCS with those that were obtained by the Hall A collaboration by fits to the φ-dependent cross sections, see Fig. 22 of Ref. [6], where the net uncertainties are dominated by the systematic errors. The same method applied to the previous Hall A data [5], shown by empty squares, reveals some differences between the data sets (here we assume that these CFF combinations are smooth functions that vary only slightly within the small kinematical difference). However, as we will explain now, the adopted approximation implies an additional uncertainty and so the net errors of these results might be underestimated.
On the other hand we might map the experimental data to the CFF H(E, H) and the bilinear form C VCS (F * , F ) by utilizing the "exact" formulae for the restricted set of four CFFs. As expected, the extracted mean values for the linear combination generates for the real part of the linear combination and for the bilinear form a larger variation of the extracted mean values inside the gray rectangles, which reflects the importance of power suppressed contributions. Again the differences are much more pronounced for the extracted values of the bilinear CFF coefficient C VCS unp . Taking into account this additional uncertainty, we can conclude that our previous KM10b ( [35], dashed curves) and KMM12b (dash-dotted curves) fits in which only the relative φ-modulation of the superseded dataset [5] was utilized predicts the locally extracted values of the new data [6] while the new global KM15 fit (solid curves) is entirely consistent with the locally extracted values. Compared to the older data set (empty squares), the GK model prediction (short dashed curves and dotted curves that adds a D-term contribution from Ref. [34] ) gets more disfavored with respect to the new data set (filled squares). Clearly, if one asks for the values of the CFF H the uncertainties depend much on the constraints for the remaining CFFs. This is illuminated in the first two panels of the lower row in Fig. 4 where due to the variation (18) the mean values of H(E = 0, H = 0) (filled squares) can move in the region of the rectangles. The findings for mH would be consistent with the values from the RDDA based GK model, however, taking values for H and E from the GK model (circles) or the KM15 fit (stars) we realize that H is consistent with the KM15 fit and to some extent with our model predictions. In the right panel of the lower row we show the correlation between the real and imaginary part of the CFF H within the 1-σ band and with remaining CFFs fixed by the GK model (dashed curves) or the KM15 fit (solid curves). The model values for H are again displayed as filled circle and star and one realizes that the GK model violates this constraint while the KM15 fit respects it.
The CLAS collaboration used the quasi model-independent tool and tried to extract the CFF H from the cross section measurements [7], where only the two CFFs H and H were taken as four fitting parameters. Apart from noise effects this is essentially a three parameter fit problem, see Eq. (15), where, however, four constrained parameters were used. We utilized a similar method, shown for the {x B = 0.185, Q 2 = 1.63 GeV 2 }-bin in Fig. 5 by the empty circles and squares. In the lower panel row we show that the resulting values, in particular for mH (left panel), might depend on the initial values and we might reproduce the CLAS result (see empty rhombi) if we take for the initial values the GK model predictions, however, we might find another solution if we utilize the KM15 result (empty squares) for the start values. Compared to our mapping procedure, providing us H(E, H) (filled squares and rectangles), the uncertainties from the fitting procedures are underestimated. The reason is that the unconstrained degrees of freedom tend to be on the boundary, see t = −0.11 GeV 2 example in the lower right panel, and thus the error propagation is not reliable. Such large values for H can also be considered as inconsistent with the findings from the new CLAS asymmetry measurements, described above. Plugging our CFF fit results into the CFF combinations, see upper panel row, provides results that are consistent with our mapping procedure, however, the uncertainties from the fits might still be underestimated. Our model predictions and fit results are consistent with the CFF values from the map. In contrast to the CLAS statement, we conclude that the locally extracted H values disfavour a RDDA based GPD model (here the GK model).

Conclusions
We compared model predictions with the new DVCS measurements from the CLAS and Hall A collaborations where we utilize a (weighted) Fourier transform as filter to strengthen the discriminating power of predictions. We presented a new global DVCS fit KM15 showing that superseding 2006 Hall A cross section data with new one relieves some tensions, but description of first cosine harmonic of cross section remains imperfect. We showed that our global fit predictions KM10b and KMM12b, which ignored the normalization of the first Hall A cross section measurements, predict the new measurements reasonably well while a RDDA based model, e.g., the GK model, becomes more disfavored by the new data sets. These conclusions are partially in disagreement with the first analyses of the data by collaborations themselves. In particular, the new cross section measurements support the previous findings that even in fixed target kinematics such models provide a GPD H that is too large. We further provided within our simple GPD ansatz from previous fits a good description of the new global DVCS data set and showed that the locally extracted CFF values are consistent with the global description. This analysis includes the GPD H which is rather compatible with RDDA based model predictions with respect to normalization and t-dependence which comes out flatter than for GPD H. Due to the new cross section measurements the indication that the GPD H has a rather small skewness effect becomes more reliable.