# Border and skewness functions from a leading order fit to DVCS data

## Abstract

We propose new parameterizations for the border and skewness functions appearing in the description of 3D nucleon structure in the language of generalized parton distributions (GPDs). These parameterizations are constructed in a way to fulfill the basic properties of GPDs, like their reduction to parton density functions and elastic form factors. They also rely on the power behavior of GPDs in the \(x \rightarrow 1\) limit and the propounded analyticity property of Mellin moments of GPDs. We evaluate compton form factors (CFFs), the sub-amplitudes of the deeply virtual compton scattering (DVCS) process, at the leading order and leading twist accuracy. We constrain the restricted number of free parameters of these new parameterizations in a global CFF analysis of almost all existing proton DVCS measurements. The fit is performed within the PARTONS framework, being the modern tool for generic GPD studies. A distinctive feature of this CFF fit is the careful propagation of uncertainties based on the replica method. The fit results genuinely permit nucleon tomography and may give some insight into the distribution of forces acting on partons.

## 1 Introduction

Fifty years after the discovery of quarks at SLAC [1, 2], understanding of how partons form a complex object such as the nucleon still remains among the main challenges of nuclear and high energy physics. In the last twenty years we have witnessed a new liveliness in the field of QCD approaches to this problem due to the discovery of generalized parton distributions (GPDs) [3, 4, 5, 6, 7]. GPDs draw so much attention because of the wealth of new information they contain. Namely, GPDs allow for the so-called nucleon tomography [8, 9, 10], which is used to study a spacial distribution of partons in the plane perpendicular to the nucleon motion as a function of parton longitudinal momenta. Before, positions and longitudinal momenta of partons were studied without any connection through other yet less complex non-perturbative QCD objects: elastic form factors (EFFs) and parton distribution functions (PDFs). In addition, GPDs have another unique property, namely they are connected to the QCD energy-momentum tensor of the nucleon. This allows for an evaluation of the contribution of orbital angular momentum of quarks to the nucleon spin through the so-called Ji’s sum rule [4, 5]. This energy-momentum tensor may also help to define “mechanical properties” and describe the distribution of forces inside the nucleon [11, 12].

It was recognized from the beginning that deeply virtual compton scattering (DVCS) is one of the cleanest probes of GPDs. The first measurements of DVCS by HERMES [13] at DESY and by CLAS [14] at JLab have proved the usability of GPD formalism to interpret existing measurements, and have established a global experimental campaign for GPDs. Indeed, nowadays measurements of exclusive processes are among the main goals of experimental programs carried out worldwide by a new generation of experiments – those already running, like Hall A and CLAS at JLab upgraded to 12 GeV and COMPASS-II at CERN, and those foreseen in the future, like electron ion collider (EIC) and large hadron electron collider (LHeC). Such a vivid experimental status is complemented by a significant progress in the theoretical description of DVCS. In particular, such new developments like NLO [15, 16, 17, 18, 19, 20, 21, 22], finite-*t* and mass [23] corrections are now available. Except DVCS, a variety of other exclusive processes has been described to provide access to GPDs, in particular: timelike compton scattering [24], deeply virtual meson production [25], heavy vector meson production [26], double deeply virtual compton scattering [27, 28], two particles [29, 30] and neutrino induced exclusive reactions [31, 32, 33]. For some of those processes experimental data have been already collected, while other processes are expected to be probed in the future.

The phenomenology of GPDs is much more involved than that of EFFs and PDFs. It comes from the fact that GPDs are functions of three variables, entering observables in nontrivial convolutions with coefficient functions. In addition, GPDs are separately defined for each possible combination of parton and nucleon helicities, resulting in a plenitude of objects to be constrained at the same time. This fully justifies the need for a global analysis, where a variety of observables coming from experiments covering complementary kinematic ranges is simultaneously analyzed. So far, such analyzes have been done mainly for compton form factors (CFFs), being DVCS sub-amplitudes and the most basic GPD-sensitive quantities as one can unambiguously extract from the experimental data. Recent analyzes include local fits [34, 35], where CFFs are independently extracted in each available bin of data, and global fits [36], where CFFs parameterizations are constrained in the whole available phase-space. For a review of DVCS phenomenology we direct the reader to Ref. [37].

The aim of this analysis is the global extraction of CFFs from the available proton DVCS data obtained by Hall A, CLAS, HERMES and COMPASS experiments. We use the fixed-*t* dispersion relation technique [38] for the evaluation of CFFs at the leading order (LO) and leading twist (LT) accuracy. For a given CFF, the dispersion relation together with the analytical regularization techniques requires two components: (i) the GPD at \(\xi = 0\), and (ii) the skewness ratio at \(x=\xi \). Ansätze for those two quantities proposed in our analysis accumulate information encoded in available PDF and EFF parameterizations, and use theory developments like the \(x\rightarrow 1\) behavior of GPDs [39]. They allow to determine a border function [40, 41], being a GPD of reduced kinematic dependency \(x = \xi \), and the subtraction constant, directly related to the energy-momentum tensor of the nucleon.

Our original approach allows to utilize many basic properties of GPDs at the level of CFFs fits. We analyze PDFs, but also EFF and DVCS data, that is, we combine information coming from (semi-)inclusive, elastic and exclusive measurements. The analysis is characterized by a careful propagation of uncertainties coming from all those sources, which we achieved with the replica method. Obtained results allow for nucleon tomography, while the extracted subtraction constant may give some insight into the distribution of forces acting on partons inside the nucleon.

This work is done with PARTONS [42] that is the open-source software framework for the phenomenology of GPDs. It serves not only as the main component of the fit machinery, but it is also utilized to handle multithreading computations and MySQL databases to store and retrieve experimental data. PARTONS is also used for the purpose of comparing existing models with the results of this analysis.

This paper is organized as follows. Section 2 is a brief introduction of GPDs, DVCS and related observables, with details on the evaluation of CFFs given in Sect. 3. Ansätze for the border and skewness functions are introduced in Sect. 4. Sections 5 and 6 summarize our analyses of PDFs and EFFs, respectively. DVCS data used in this work are specified in Sect. 7. In Sect. 8 the propagation of uncertainties is discussed, while the results are given in Sect. 9. In Sect. 10 we summarize the content of this paper.

## 2 Theoretical framework

In this section a brief introduction to the GPD formalism is given. We emphasize the role of quark GPDs, as only those contribute to DVCS at LO. A deep understanding of the basic features of the contributing GPDs is crucial for constructing parameterizations of CFFs. More involved tools, like nucleon tomography, are important for the exploration of the partonic structure of the proton. This section also provides a foundation to DVCS description and illustrates the construction of observables used in our fits.

For brevity, we suppress in the following the dependence on the factorization and renormalization scales, \(\mu _{R}^{2}\) and \(\mu _{F}^{2}\), which in this analysis are identified with the hard scale of the process \(Q^{2}\). A detailed preface to the GPD formalism may be found in one of available reviews [43, 44, 45, 46].

### 2.1 Generalized parton distributions

*x*is the average longitudinal momentum of the active quark, \(\xi = -\varDelta ^+/(2P^+)\) is the skewness variable and \(t = \varDelta ^{2}\) is the square of four-momentum transfer to the hadron target, with the average hadron momentum

*P*obeying \(P^{2} = m^{2} - t/4\), where

*m*is the hadron mass. In this definition the usual convention is used, where the plus-component refers to the projection of any four-vector on a light-like vector

*n*.

*q*(

*x*) and \(\varDelta q(x)\) are the unpolarized and polarized PDFs, respectively. No similar relations exist for the GPDs \(E^{q}\) and \(\widetilde{E}^{q}\) that decouple from the forward limit. The relation to EFFs can be obtained by integrating GPDs over the partonic variable

*x*:

*q*to the Dirac, Pauli, axial and pseudoscalar EFFs, respectively.

*n*th Mellin moment of a given GPD is always an even polynomial in \(\xi \), of order \(n+1\) for the unpolarized GPDs and of order

*n*for the polarized GPDs. In particular:

*x*as a function of the position \(\mathbf {b}_{\perp }\) in the plane perpendicular to the nucleon motion. For unpolarized partons inside an unpolarized nucleon this density is expressed by:

*E*. The longitudinal polarization of partons distributed in a longitudinally polarized nucleon according to \(q(x, \mathbf {b}_{\mathbf {\perp }})\) can be studied with the Fourier transform of GPD \(\widetilde{H}\):

*E*:

*H*and \(\widetilde{H}\), and therefore we will not attempt to give any estimation on \(J^{q}\).

#### 2.1.1 Deeply virtual compton scattering

*l*,

*N*and \(\gamma \) denote lepton, nucleon and produced photon, respectively; the four-vectors of these states appear between parenthesis. Under specific kinematic conditions, the factorization theorem allows one to express the DVCS amplitude as a convolution of the hard scattering part, being calculable within the perturbative QCD approach, and GPDs, describing an emission of parton from the nucleon and its subsequent reabsorption, see Fig. 1. The factorization applies in the Bjorken limit and for \(-t/Q^{2} \ll 1\), where \(Q^{2}=-(k-k')^2\) is the virtuality of the virtual-photon mediating the exchange of four-momentum between lepton and proton at Born order.

The amplitudes \({\mathcal {T}}_{{\mathrm {DVCS}}}\) and \({\mathcal {I}}\) may be expressed by combinations of CFFs, which are convolutions of GPDs with the hard scattering part of the interaction. CFFs are the most basic quantities that one can unambiguously extract from the experimental data. The way of how CFFs enter the final amplitudes depends on the beam and target helicity states, which provides a welcome experimental filter to distinguish between many possible CFFs and justifies the need of measuring many observables. For brevity we skip the formulas showing how \({\mathcal {T}}_{{\mathrm {DVCS}}}\) and \({\mathcal {I}}\) depend on CFFs. They can be found in Ref. [56]. The evaluation of CFFs is discussed in Sect. 3.

#### 2.1.2 Observables

*U*standing for “Unpolarized” and

*L*standing for “Longitudinally polarized”. We also analyze data for “Transversely polarized targets”, which are distinguished by the subscript

*T*. These data are provided for two moments, \(\sin (\phi -\phi _{S})\) and \(\cos (\phi -\phi _{S})\), which are distinguished by the corresponding labels in the superscripts, as for instance in \(A^{-, \sin (\phi -\phi _{S})}(x_{\mathrm {Bj}}, t, Q^{2}, \phi )\). Furthermore there are observables probing only the beam charge dependency (subscript

*C*), and those combining cross sections measured with various beam charges to drop either the DVCS or interference contribution (subscripts \(\mathrm {I}\) and \({\mathrm {DVCS}}\), respectively).

*t*-distribution of the DVCS cross section integrated over \(\phi \). Within the LO formalism one can relate this observable to the transverse extension of partons in the proton. However, this requires the following assumptions, which are expected to hold at small \(x_{\mathrm {Bj}}\): (i) dominance of the imaginary part of CFF related to GPD

*H*, (ii) negligible skewness effect at \(\xi = x\), (iii) exponential

*t*dependence of the GPD

*H*at fixed

*x*. Since the DVCS

*t*-distribution is usually not exactly exponential, in particular because GPDs for valence and sea quarks may have different

*t*-dependencies, we evaluate \(b(x_{\mathrm {Bj}}, Q^{2})\) by probing DVCS

*t*-distribution in several equidistant points ranging from \(|t| = 0.1~{\mathrm {GeV}}^{2}\) to \(|t| = 0.5~{\mathrm {GeV}}^{2}\) and perform a linear regression on the logarithmized results. The chosen range of

*t*is typical for the existing measurements of \(b(x_{\mathrm {Bj}}, Q^{2})\) by COMPASS [57] and HERA experiments [58, 59, 60].

## 3 Compton form factors

### 3.1 *Imaginary part*

*q*in units of the positron charge

*e*and \(G^{q (+)}\) is the singlet (

*C*-even) combination of GPDs:

#### 3.1.1 Real part

At LO the real part of a given CFF \({\mathcal {G}}\) can be evaluated by probing the corresponding GPD *G* in two ways, that is, by integrating over one of two lines laying in the (\(x, \xi \))-plane. This duality is a consequence of the polynomiality property required by the Lorentz invariance of GPDs, see Sect. 2.

*x*values of the involved GPDs are probed at fixed \(\xi \):

*t*dispersion relation [38] and it involves the integral probing GPDs at \(\xi = x\):

#### 3.1.2 Subtraction constant

*D*-term form factor, \(D^{q}(t)\), in the following way:

*D*-term [61]. It was originally introduced to restore the polynomiality property in the first models based on double distributions [62], but later it has been recognized as an important element of the GPD phenomenology. Because the

*D*-term vanishes outside the ERBL region \(|x| < |\xi |\), it is not observed in the limit of \(\xi =0\), and it can be only studied in the “skewed” case of \(\xi \ne 0\).

*D*-term in terms of Gegenbauer polynomials,

*j*and can be analytically continued to \(j=-1\), if \( G^{q (+)}(x,x, t) - G^{q (+)}(x, 0, t)\) has a proper analytic behavior, as described in [64]. Such an analytical continuation can be written as:

*f*around zero as needed to make the integral convergent, and one treats the compensating terms to be convergent as well.

^{1}We will make such an assumption, and calculate the subtraction constant as:

## 4 Ansatz

We present in this study a global extraction of CFFs. According to the terminology used within the GPD community, “global” refers to constraining parameters of an assumed CFF functional forms from various measurements on a wide kinematic range. On the contrary in local extractions, CFFs are extracted as a set of disconnected values in bins of \(\xi \) and *t* (see for instance Refs. [34, 35]). We restrict our analysis to the LO approximation and we neglect any contribution coming from higher-twist effects and kinematic (target mass and finite-*t*) corrections. We adopt the description of cross sections in terms of DVCS and BH amplitudes by Guichon and Vanderhaeghen, used for phenomenology for instance in Refs. [71, 72] and publicly available in the open-source PARTONS framework [42]. We point out, that the limited phase space covered by available data, the precision of those data and the plenitude of involved dependencies force us to keep the parameterizations as simple as possible. Otherwise significant correlations appear between fitted parameters, which somehow obscures the interpretation of obtained results.

The Ansatz introduced in this section is explicitly given for a factorization scale that one may recognize as the reference scale \(Q_{0}^{2}\) at which the model is defined. To include the factorization scale dependence in our fit, that is for the comparison with experimental data of \(Q^{2} \ne Q_{0}^{2}\), we consider the so-called forward evolution, i.e. the one followed by PDFs. The usage of the genuine GPD evolution equations requires the knowledge of GPDs in the full range of *x* independently on \(\xi \), while in this analysis only the GPDs at \(x = \xi \) are considered. It was checked however with the GK GPD model [73, 74, 75], that the difference between the two evolution schemes is small for \(x = \xi \), unless \(Q^{2} \gg Q_{0}^{2}\).

### 4.1 Decomposition into valence and sea contributions

#### 4.1.1 CFFs \({\mathcal {H}}\) and \(\widetilde{{\mathcal {H}}}\)

Data used in this analysis are primarily sensitive to the CFFs \({\mathcal {H}}\) and \(\widetilde{{\mathcal {H}}}\). The LO and LT formalism allows us to evaluate those CFFs with Eq. (34) for the imaginary part and with Eq. (38) for the real part. The subtraction constant, which appears in the dispersion relation, is evaluated with Eq. (49), making use of the analytic regularization prescription given by Eq. (48). All together, only the GPDs \(H^{q}\) and \(\widetilde{H}^{q}\) at \(\xi =0\) and \(\xi =x\) are needed.

*q*(

*x*), for the GPD \(H^{q}\) or a parameterization of the polarized PDF, \(\varDelta q(x)\), for the GPD \(\widetilde{H}^{q}\). The profile function, \(f_{G}^{q}(x)\), fixes the interplay between the

*x*and

*t*variables, and it is given by:

*x*region and by \(C_{G}^{q}(1-x)x\) in the high

*x*region. The terms proportional to \(B_{G}^{q}\) and \(C_{G}^{q}\) were found to work best in the analysis of EFF data (see Sect. 6), where all combinations of \((1-x)^{i}\) and \((1-x)^{j}x^{k}\) polynomials with \(i, j, k = 1, \ldots , 5\) were examined. We note that \(A_{G}^{q}\log (1/x)\) can not be directly multiplied by a polynomial of

*x*, which is imposed by a need of keeping \(G^{q}(x, 0, t) / x^{-(\delta + A_{G}^{q}t)}\) analytic at \(x=0\). To keep the distance between the active quark and the spectator system finite, see Eq. (23), we require to have \(C_{H}^{q_{\mathrm {val}}} = -A_{H}^{q_{\mathrm {val}}}\).

The profile function given by Eq. (55) is more flexible than that used in the GK model [73, 74, 75], \(f_{G, {\mathrm {GK}}}^{q}(x) = A_{G}^{q} + B_{G}^{q}\log (1/x)\), and in the VGG model [76, 78, 79, 80], \(f_{G, {\mathrm {VGG}}}^{q}(x) = A_{G}^{q}\log (1/x)(1-x)\). In particular, it should be flexible enough to take into account a different interplay between the *x* and *t* variables in the valence and sea regions if required by experimental data. We note that \(f_{G, {\mathrm {DK}}}^{q}(x) = A_{G}^{q}\log (1/x)(1-x)^{3} + B_{G}^{q}(1-x)^{3} + C_{G}^{q}(1-x)^{2}x\) used in Refs. [47, 81] to fit EFF data can not be used in this analysis because of the aforementioned issue with the analyticity caused by \(A_{G}^{q}\log (1/x)(1-x)^{3}\) term.

*t*-dependence of the skewness function, are fixed in a way to avoid singularities in the evaluation of the subtraction constant. Namely, to use the analytic regularization prescription at fixed

*t*one has:

*a*and

*f*(

*x*) were introduced in Eq. (48) and \(\delta \) describes the behavior of PDFs at \(x \rightarrow 0\):

We stress that our Ansatz for the skewness function is explicitly defined at \(x=\xi \) and it can not be generalized to the case of \(x\ne \xi \) without a non-trivial modification. The form of the skewness function has been selected because of the following reasons: (i) for sufficiently small *x* and *t*, the skewness function coincides with a constant value given by \(a_{G}^{q}\). Such a behavior was predicted for HERA kinematics [82] and it was used in one of the first extractions of GPD information [83] from H1 data [84]. These data suggest \(a_{H}^{q} \approx 1\). (ii) In the limit of \(x \rightarrow 1\) the skewness function is driven by \(1/(1-x^{2})^{2}\). This form has been deduced from Ref. [39], where the power behavior of GPDs in the limit of \(x \rightarrow 1\) was studied within the pQCD approach. (iii) The subdominant *t*-dependence in Eq. (58) has been inspired by the skewness function evaluated from GK [73, 74, 75] and VGG [76, 78, 79, 80] GPD models, both being based on the one component double distribution modeling scheme [62]. In those models the *t*-dependence of the skewness function is dominated by \(b_{G}^{q} + c_{G}^{q} \log (1 + x)\) term. In our Ansatz we multiply it by \((1 - x)\) to avoid any *t*-dependence at \(x \rightarrow 1\), which is imposed by Ref. [39], where it was shown that GPDs should not depend on *t* in this limit, regardless the value of \(\xi \).

#### 4.1.2 CFFs \({\mathcal {E}}\) and \(\widetilde{{\mathcal {E}}}\)

For *E* and \(\widetilde{E}\) we use a simplified treatment justified by the poor sensitivity of the existing measurements on the corresponding CFFs. Moreover, the forward limit of those GPDs has not been measured, resulting in a need of fixing more parameters than for the GPDs *H* and \(\widetilde{H}\), if an analog modeling was to be adopted.

*t*-dependence is introduced analogously as for the GPDs

*H*and \(\widetilde{H}\), i.e.:

*t*-dependence is neglected. An additional \(\xi \)-dependence coming from twist-three and twist-four distribution amplitudes of the nucleon is denoted by

*f*(

*x*). In the present analysis, where only the leading twist is considered, it is assumed that \(f(x)/f(0) = 1\).

#### 4.1.3 Inequalities

*H*and for the valence contribution to GPD \(\widetilde{H}\). It is due to a low sensitivity to those contributions and a limited phase space covered by the available data.

*e*(

*x*), which presently is not the case, as we fit EFF and DVCS data separately. In addition, for the GPDs \(\widetilde{H}\) and

*E*we consider only the valence contribution, while the sea sector may have a significant impact on (72) in the range of small \(x_{\mathrm {Bj}}\). The validity of (72) has however been checked a posteriori, that is after the \(\chi ^2\)-minimization. The result of this test shows that the inequality is violated for most of the replicas for very small \(\mathbf {b}_{\mathbf {\perp }}^2\). To correct this one should change the simplified Ansatz used for GPD

*E*and preferably fit its free parameters simultaneously with those for other GPDs, which is beyond the scope of this analysis.

#### 4.1.4 Approximations and summary

For the GPD *H* and valence quarks, all parameters of the profile function given by Eq. (55) are fitted to EFF data, where we fixed \(C_{H}^{u_{\mathrm {val}}} = -A_{H}^{u_{\mathrm {val}}}\) and \(C_{H}^{d_{\mathrm {val}}} = -A_{H}^{d_{\mathrm {val}}}\). The unconstrained parameter in the skewness function given by Eq. (58) is fitted to DVCS data, where it is assumed that \(a_{H}^{u_{\mathrm {val}}} = a_{H}^{d_{\mathrm {val}}} \equiv a_{H}^{q_{\mathrm {val}}}\). For sea quarks all parameters for both profile and skewness functions are fitted to DVCS data. Due to a limited sensitivity to the sea sector, we assume the symmetry with respect to the change of quark flavor, i.e. one has \(a_{H}^{u_{\mathrm {sea}}} = a_{H}^{d_{\mathrm {sea}}} = a_{H}^{s} \equiv a_{H}^{q_{\mathrm {sea}}}\), \(A_{H}^{u_{\mathrm {sea}}} = A_{H}^{d_{\mathrm {sea}}} = A_{H}^{s} \equiv A_{H}^{q_{\mathrm {sea}}}\), etc. Sea components are not yet fully symmetric in our fit, as we do not impose the flavor symmetry in PDFs, see Sect. 5. Due to the lack of precision of axial EFF data, for the GPD \(\widetilde{H}\) all parameters for both profile and skewness functions are fitted to DVCS data. As the contribution coming from \(\varDelta q_{\mathrm {sea}}\) is subdominant, we neglect it entirely. For valence quarks, similarly to the GPD *H*, we allow the profile function to be different for \(u_{\mathrm {val}}\) and \(d_{\mathrm {val}}\), while for the skewness function one has \(a_{\widetilde{H}}^{u_{\mathrm {val}}} = a_{\widetilde{H}}^{d_{\mathrm {val}}} \equiv a_{\widetilde{H}}^{q_{\mathrm {val}}}\). For *E* and \(\widetilde{E}\) only the valence quarks are considered. For the GPD *E* all free parameters for both profile function and forward limit given by Eq. (64) are fitted to EFF data. For the GPD \(\widetilde{E}\) the normalization parameter \(N_{\widetilde{E}}\) is fitted to DVCS data. In total, we fit 9 parameters to EFF data (see Table 2) and 13 parameters are constrained by DVCS data (see Table 5).

## 5 Analysis of PDFs

The analytic regularization prescription introduced in Eq. (48) requires the function \(q(x)/x^{-\delta }\) to be non-zero and analytic at \(x = 0\). However, the numeric evaluation of such a function and its derivatives near \(x = 0\) is difficult for typical PDF sets, like those published by NNPDF [85] and CTEQ [86] groups, because of several numerical issues. Namely, interpolations in PDF grids and extrapolations outside those grids for small *x*, and a delicate cancellation of the numerator and denominator of \(q(x)/x^{-\delta }\) make the evaluation numerically unstable. The problem can be avoided with functional parameterizations of PDFs, which can be used for a straightforward evaluation of \(q(x)/x^{-\delta }\) and its derivatives. In addition, such parameterizations allow for a significant reduction of computation time and a precise determination of \(\delta \).

constrained for each quark flavor in a fit to “NNPDF30_lo_as_0118_nf_3” set [85] for unpolarized PDFs and to “NNPDFpol11_100” set [87] for polarized PDFs. These fits are performed for a grid of nearly 10,000 points equidistantly distributed in the ranges of \(10^{-4}< x < 0.9\) and \(1~{\mathrm {GeV}}^{2}< Q^{2} < 20~{\mathrm {GeV}}^{2}\). We have selected the sets by NNPDF group as: (i) they are provided for both *q*(*x*) and \(\varDelta q(x)\), (ii) for *q*(*x*) they are provided at LO and three active flavors, (iii) they provide reliable estimation of uncertainties extracted through the neural network approach and the replica method. The PDF sets are handled with LHAPDF interface [88].

A consistent fit to all replicas of used NNPDF sets allows us to reproduce the original uncertainties of PDFs. Figure 3 demonstrates the agreement between the original sets and our fits. The comparison is satisfactory and in particular the central values of fitted PDFs stay within the original uncertainty bands.

## 6 Analysis of Dirac and Pauli form factors

*x*and

*t*variables for the GPDs

*H*and

*E*, however only for the valence quarks. More precisely, we fix the parameters of \(f_{H/E}^{q_{\mathrm {val}}}(x)\) defined in Eq. (55) by using the Ansatz for \(H^{q_{\mathrm {val}}}(x, 0, t)\) given in Eq. (54) and that for \(E^{q_{\mathrm {val}}}(x, 0, t)\) given in Eq. (67). We achieve this by studying proton, \(F_{i}^{p}(t)\), and neutron, \(F_{i}^{n}(t)\), Dirac and Pauli form factors:

- the magnetic form factor normalized to both magnetic moment, \(\mu _{p}\) or \(\mu _{n}\), and dipole form factor:where \(i=p, n\) and$$\begin{aligned} G_{M, N}^{i}(t) = \frac{G_{M}^{i}(t)}{\mu _{i}G_{D}(t)} , \end{aligned}$$(79)with \(M_D^2 = 0.71~{\mathrm {GeV}}^2\).$$\begin{aligned} G_{D}(t) = \frac{1}{\left( 1 - t/M_D^2 \right) ^{2}} , \end{aligned}$$(80)
- the ratio of electric and normalized magnetic form factorswhere \(i=p, n\).$$\begin{aligned} R^{i}(t) = \frac{\mu _{i}G_{E}^{i}(t)}{G_{M}^{i}(t)} . \end{aligned}$$(81)
- the squared charge radius of neutron:$$\begin{aligned} r_{nE}^{2} = 6 \frac{dG_{E}^{n}(t)}{dt}\bigg |_{t = 0} . \end{aligned}$$(82)

EFF data used in this analysis. For \(R^{n}\) we use data coming from Ref. [81], which are evaluated from those specified in this table

*i*. The total uncertainty, which is also used to evaluate the \(\chi ^{2}\) value in the fit to EFF data, reads:

*i*. The generator of random numbers following a specified normal distribution, \(f(x | \mu , \sigma )\), is denoted by \({\mathrm {rnd}}_{j}(\mu , \sigma )\), where

*j*is both the identifier of a given replica and a unique random seed.

Values of parameters fitted to EFF data together with estimated uncertainties coming from those data and used PDF parameterizations

Parameter | Mean | Data unc. | Unpol. PDF unc. |
---|---|---|---|

\(A_{H/E}^{u_{\mathrm {val}}}\) | 0.99 | 0.01 | 0.08 |

\(B_{H}^{u_{\mathrm {val}}}\) | \(-\) 0.50 | 0.02 | 0.14 |

\(A_{H/E}^{d_{\mathrm {val}}}\) | 0.70 | 0.02 | 0.08 |

\(B_{H}^{d_{\mathrm {val}}}\) | 0.47 | 0.07 | 0.24 |

\(\alpha \) | 0.69 | 0.01 | 0.03 |

\(B_{E}^{u_{\mathrm {val}}}\) | \(-\) 0.69 | 0.04 | 0.18 |

\(C_{E}^{u_{\mathrm {val}}}\) | \(-\) 0.92 | 0.04 | 0.09 |

\(B_{E}^{d_{\mathrm {val}}}\) | \(-\) 0.54 | 0.06 | 0.20 |

\(C_{E}^{d_{\mathrm {val}}}\) | \(-\) 0.73 | 0.06 | 0.22 |

## 7 DVCS data sets

Table 3 summarizes DVCS data used in this analysis. Currently, only proton data are used, while those sparse ones for neutron targets are foreseen to be included in the future. We note, that recent Hall A data [117, 118] published for unpolarized cross sections, \(d^{4}\sigma _{UU}^{-}\), are not used in the present analysis since it was not possible to correctly describe them with our fitting Ansatz, and their inclusion was causing a bias on the final extraction of GPD information. This specific question will be addressed in a future study. However we keep the recent Hall A data for differences of cross sections, \(\varDelta d^{4}\sigma _{LU}^{-}\). In addition, we skip all DVCS data published by HERA experiments since they should be dominated by gluons, and would probably lead to misleading conclusions in our analysis which is adapted to the quark sector. We will back to these problems in Sect. 9.

DVCS data used in this analysis

No. | Collab. | Year | References | Observable | Kinematic dependence | No. of points used / all | |
---|---|---|---|---|---|---|---|

1 | HERMES | 2001 | [13] | \(A_{LU}^{+}\) | \(\phi \) | 10/10 | |

2 | 2006 | [119] | \(A_{C}^{\cos i \phi }\) | \(i = 1\) | | 4/4 | |

3 | 2008 | [120] | \(A_{C}^{\cos i \phi }\) | \(i = 0, 1\) | \(x_{\mathrm {Bj}}\) | 18/24 | |

\(A_{UT, {\mathrm {DVCS}}}^{\sin (\phi -\phi _{S})\cos i \phi }\) | \(i = 0\) | ||||||

\(A_{UT, \mathrm {I}}^{\sin (\phi -\phi _{S})\cos i \phi }\) | \(i = 0, 1\) | ||||||

\(A_{UT, \mathrm {I}}^{\cos (\phi -\phi _{S})\sin i \phi }\) | \(i = 1\) | ||||||

4 | 2009 | [121] | \(A_{LU, \mathrm {I}}^{\sin i \phi }\) | \(i = 1, 2\) | \(x_{\mathrm {Bj}}\) | 35/42 | |

\(A_{LU, {\mathrm {DVCS}}}^{\sin i \phi }\) | \(i = 1\) | ||||||

\(A_{C}^{\cos i \phi }\) | \(i = 0, 1, 2, 3\) | ||||||

5 | 2010 | [122] | \(A_{UL}^{+, \sin i \phi }\) | \(i = 1, 2, 3\) | \(x_{\mathrm {Bj}}\) | 18/24 | |

\(A_{LL}^{+, \cos i\phi }\) | \(i = 0, 1, 2\) | ||||||

6 | 2011 | [123] | \(A_{LT, {\mathrm {DVCS}}}^{\cos (\phi -\phi _{S})\cos i \phi }\) | \(i = 0, 1\) | \(x_{\mathrm {Bj}}\) | 24/32 | |

\(A_{LT, {\mathrm {DVCS}}}^{\sin (\phi -\phi _{S})\sin i \phi }\) | \(i = 1\) | ||||||

\(A_{LT, \mathrm {I}}^{\cos (\phi -\phi _{S})\cos i \phi }\) | \(i = 0, 1, 2\) | ||||||

\(A_{LT, \mathrm {I}}^{\sin (\phi -\phi _{S})\sin i \phi }\) | \(i = 1, 2\) | ||||||

7 | 2012 | [124] | \(A_{LU, \mathrm {I}}^{\sin i \phi }\) | \(i = 1, 2\) | \(x_{\mathrm {Bj}}\) | 35/42 | |

\(A_{LU, {\mathrm {DVCS}}}^{\sin i \phi }\) | \(i = 1\) | ||||||

\(A_{C}^{\cos i \phi }\) | \(i = 0, 1, 2, 3\) | ||||||

8 | CLAS | 2001 | [14] | \(A_{LU}^{-, \sin i \phi }\) | \(i = 1, 2\) | – | 0/2 |

9 | 2006 | [125] | \(A_{UL}^{-, \sin i \phi }\) | \(i = 1, 2\) | – | 2/2 | |

10 | 2008 | [126] | \(A_{LU}^{-}\) | \(\phi \) | 283/737 | ||

11 | 2009 | [127] | \(A_{LU}^{-}\) | \(\phi \) | 22/33 | ||

12 | 2015 | [128] | \(A_{LU}^{-}\), \(A_{UL}^{-}\), \(A_{LL}^{-}\) | \(\phi \) | 311/497 | ||

13 | 2015 | [129] | \(d^{4}\sigma _{UU}^{-}\) | \(\phi \) | 1333/1933 | ||

14 | Hall A | 2015 | [117] | \(\varDelta d^{4}\sigma _{LU}^{-}\) | \(\phi \) | 228/228 | |

15 | 2017 | [118] | \(\varDelta d^{4}\sigma _{LU}^{-}\) | \(\phi \) | 276/358 | ||

16 | COMPASS | 2018 | [57] | | – | 1/1 | |

SUM: | 2600/3970 |

## 8 Uncertainties

In this analysis all the uncertainties are evaluated with the replica method, which for a sufficiently large set of replicas allows one to accurately reproduce the probability distribution of a given problem. We distinguish four types of uncertainties on the extracted parameterizations of CFFs. They origin from: (i) DVCS data, (ii) unpolarized PDFs, (iii) polarized PDFs) and (iv) EFFs. Each type of uncertainty is estimated independently, as described in the following.

*i*, which comes with statistical, \({\varDelta }_{i}^{\mathrm {stat}}\), systematic, \({\varDelta }_{i}^{\mathrm {sys}}\), and normalization, \({\varDelta }_{i}^{\mathrm {norm}}\), uncertainties. The latter one appears whenever the observable is sensitive to either beam or target polarization, or to both of them. In such cases the polarization describes the analyzing power for the extraction of those observable and the normalization uncertainties are related to the measurement or other determination of the involved polarizations. The total uncertainty, which is also used in the fit of CFFs to evaluate the \(\chi ^{2}\) value, is evaluated according to Eq. (84).

The uncertainties coming from unpolarized and polarized PDFs are estimated by propagating our parameterizations of replicas by NNPDF group, see Sect. 5. Namely, we repeat our fit to DVCS data separately for each PDF replica, that is 100 times for unpolarized PDFs and 100 times for polarized PDFs. A similar method is used to evaluate the uncertainties coming from EFF parameterizations. We repeat our fit to DVCS data for each replica obtained in the analysis of EFF data, see Sect. 6.

As a result of this analysis we obtain a set of 401 replicas, where each of them represents a possible realization of CFF parameterizations. For a given kinematic point the mean and uncertainties can be then estimated by taking the mean and the standard deviation of values evaluated from those replicas.

## 9 Results

### 9.1 *Performance*

Values of the \(\chi ^{2}\) function per data set. For a given data set, cf. Table 3, given are: \(\chi ^{2}\) value, the number of experimental points *n*, and the ratio between these two numbers

No. | Collab. | Year | References | \(\chi ^{2}\) | | \(\chi ^{2}/n\) |
---|---|---|---|---|---|---|

1 | HERMES | 2001 | [13] | 9.8 | 10 | 0.98 |

2 | 2006 | [119] | 2.9 | 4 | 0.72 | |

3 | 2008 | [120] | 24.2 | 18 | 1.35 | |

4 | 2009 | [121] | 40.1 | 35 | 1.15 | |

5 | 2010 | [122] | 40.3 | 18 | 2.24 | |

6 | 2011 | [123] | 14.5 | 24 | 0.60 | |

7 | 2012 | [124] | 25.4 | 35 | 0.73 | |

8 | CLAS | 2001 | [14] | – | 0 | – |

9 | 2006 | [125] | 0.9 | 2 | 0.47 | |

10 | 2008 | [126] | 371.1 | 283 | 1.31 | |

11 | 2009 | [127] | 36.4 | 22 | 1.66 | |

12 | 2015 | [128] | 351.4 | 311 | 1.13 | |

13 | 2015 | [129] | 937.9 | 1333 | 0.70 | |

14 | Hall A | 2015 | [117] | 220.2 | 228 | 0.97 |

15 | 2017 | [118] | 258.8 | 276 | 0.94 | |

16 | COMPASS | 2018 | [57] | 10.7 | 1 | 10.67 |

Values of the parameters fitted to DVCS data together with estimated uncertainties coming from those data, (un-)polarized PDFs and EFFs. Two last columns indicate the limits in which the minimization routine was allowed to vary the corresponding parameters. In addition, exemplary values of \(b_{G}^{q}\) and \(c_{G}^{q}\) parameters evaluated at \(Q^{2} = 2~{\mathrm {GeV}}^2\) from Eqs. (62) and (63) are given

Parameter | Mean | Data unc. | Unpol. PDF unc. | Pol. PDF unc. | EFF unc. | Limit | |
---|---|---|---|---|---|---|---|

min | max | ||||||

\(a_{H}^{q_{\mathrm {val}}}\) | 0.81 | 0.04 | 0.17 | 0.02 | \(<0.01\) | 0.2 | 2.0 |

\(a_{H}^{q_{\mathrm {sea}}}\) | 0.99 | 0.01 | 0.02 | \(<0.01\) | \(<0.01\) | 0.2 | 2.0 |

\(a_{\widetilde{H}}^{q}\) | 1.03 | 0.04 | 0.30 | 0.24 | 0.01 | 0.2 | 2.0 |

\(N_{\widetilde{E}}\) | \(-\,0.46\) | 0.10 | 0.09 | 0.06 | 0.01 | \(-\,10\) | 10 |

\(A_{H}^{q_{\mathrm {sea}}}\) | 2.56 | 0.23 | 0.30 | 0.09 | 0.03 | 0.1 | 10 |

\(B_{H}^{q_{\mathrm {sea}}}\) | \(-\,5\) | At the limit | \(-\,5\) | 20 | |||

\(C_{H}^{q_{\mathrm {sea}}}\) | 34 | 27 | 49 | 14 | 3 | \(-\,5\) | 200 |

\(A_{\widetilde{H}}^{u_{\mathrm {val}}}\) | 0.77 | 0.12 | 0.30 | 0.23 | 0.07 | 0.1 | 10 |

\(B_{\widetilde{H}}^{u_{\mathrm {val}}}\) | \(-\,0.02\) | 0.26 | 0.75 | 0.24 | 0.15 | \(-\,5\) | 20 |

\(C_{\widetilde{H}}^{u_{\mathrm {val}}}\) | \(-\,0.92\) | 0.07 | 0.44 | 0.24 | 0.04 | \(-\,5\) | 200 |

\(A_{\widetilde{H}}^{d_{\mathrm {val}}}\) | 0.64 | 0.24 | 0.30 | 0.28 | 0.05 | 0.1 | 10 |

\(B_{\widetilde{H}}^{d_{\mathrm {val}}}\) | \(-\,1.19\) | 0.45 | 0.91 | 0.98 | 0.22 | \(-\,5\) | 20 |

\(C_{\widetilde{H}}^{d_{\mathrm {val}}}\) | \(-\,0.55\) | 0.24 | 0.26 | 0.27 | 0.10 | \(-\,5\) | 200 |

\(b_{H}^{u_{\mathrm {val}}}\) | \(-\,0.36\) | 0.10 | 0.15 | 0.04 | 0.01 | – | – |

\(c_{H}^{u_{\mathrm {val}}}\) | 11.2 | 3.1 | 2.7 | 1.1 | 0.3 | – | – |

\(b_{H}^{d_{\mathrm {sea}}}\) | \(-\,0.222\) | 0.062 | 0.090 | 0.022 | 0.006 | – | – |

\(c_{H}^{d_{\mathrm {sea}}}\) | 14 | 4 | 15 | 1 | 1 | – | – |

In this analysis a set of 401 replicas is obtained that can be used to estimate mean values and uncertainties of the fitted parameters. We summarize this in Table 5, where we also indicate ranges in which the minimization routine was allowed to vary the fitted parameters. As one can see from this table, the parameter \(B_{H}^{q_{\mathrm {sea}}}\) is found to be at the lower limit of the allowed range. Without this limit the fit would end with a much smaller value of \(B_{H}^{q_{\mathrm {sea}}}\) compensated by a higher value of \(A_{H}^{q_{\mathrm {sea}}}\), which we consider to be unphysical. The problem with \(B_{H}^{q_{\mathrm {sea}}}\) is a consequence of the sparse data covering low-\(x_{\mathrm {Bj}}\) region, but it may be also a sign of the breaking of the LO description in the range dominated by sea quarks and gluons.

In addition to the fitted parameters, in Table 5 we also show typical values of \(b_{G}^{q}\) and \(c_{G}^{q}\) evaluated from Eqs. (62) and (63), respectively. These values indicate that the *t*-dependence in the skewness function is subdominant, which we consider to be an expected feature.

*t*-slope

*b*coming from the measurements by COMPASS [57], ZEUS [59] and H1 [58, 60] experiments. We remind that for this observable only the COMPASS point is used in this analysis. As one can judge from the figure, the agreement between our fit and experimental data is lost for small \(x_{\mathrm {Bj}}\), i.e. for the range where sea quarks and gluons dominate. We only leave the COMPASS point to have a coverage in the intermediate range of \(x_{\mathrm {Bj}}\), where valence quarks may still contribute significantly. While it is tempting to improve the agreement for small \(x_{\mathrm {Bj}}\) by adding extra terms to \(f_{H}^{q_{\mathrm {sea}}}(x)\), like those proportional to \((1-x)^n\) with large

*n*, we refrain from doing that, treating the disagreement as a possible manifestation of NLO effects. The included comparison with GK [73, 74, 75] and VGG [76, 78, 79, 80] GPD models shows that GK manages to reproduce the trend dictated by the HERA data reasonably well. This is not a full surprise, as this model was constrained in the low-\(x_{\mathrm {Bj}}\) region, however by deeply virtual meson production (DVMP) data. It is worth pointing out, that the lowest possible order of contribution to DVMP starts with \(\alpha _{S}^1\), while for DVCS one has \(\alpha _{S}^{0}\). It is beyond the scope of this paper, but in the future multichannel analyses of exclusive processes the issue of using the same order of pQCD calculations may become important.

#### 9.1.1 Subtraction constant

*t*at \(Q^{2} = 2~{\mathrm {GeV}}^{2}\) and as a function of \(Q^{2}\) at \(t = 0\) in Fig. 10. One may observe a large uncertainty coming from PDF parameterizations, however in general small values of the subtraction constant are preferred. We also observe a general trend for the saturation of the subtraction constant at large values of \(Q^{2}\).

#### 9.1.2 Compton form factors

#### 9.1.3 Nucleon tomography

Exemplary distributions of \(u(x, b_{\perp })\) and \(\varDelta u_{\mathrm {val}}(x, b_{\perp })\)observables corresponding to Eqs. (17) and (18), respectively, are shown in Fig. 15. These plots illustrate how parton densities (longitudinal polarization of those partons) are distributed inside the unpolarized (longitudinally polarized) nucleon, however without the possibility of showing the corresponding uncertainties.

*x*. A firm conclusion however is not possible at this moment because of the large uncertainties.

## 10 Summary

In this paper we proposed new parameterizations for the border and skewness functions. Together with the assumption about the analyticity properties of the Mellin moments of GPDs, those two ingredients allowed the evaluation of DVCS CFFs with the LO and LT accuracy. The evaluation was done with the dispersion relation technique and it included a determination of the DVCS subtraction constant, which is related to the QCD energy-momentum tensor.

In order to build and constrain our parameterizations we utilized many basic properties of GPDs, like their relation to PDFs and EFFs, the positivity bounds, the power behavior in the limit of \(x \rightarrow 1\) and even the polynomiality property allowing the evaluation of the subtraction constant by comparing two equivalent ways of CFF computation. Our parameterizations provide a genuine access to GPDs at (*x*, 0, *t*) kinematics and therefore they can be used for nucleon tomography.

We performed the analysis of PDFs and obtained a set of functional parameterizations allowing the reproduction of original values and uncertainties. A small number of free parameters appearing in our approach was constrained by EFF and DVCS data. We considered all proton DVCS data, however not all of them entered the final analysis because of the used kinematic cuts and the initial data exploration. Our work was done within the PARTONS project that provides a modern platform for the study of GPDs and related topics.

The quality of our fits is quite good. The fit to EFF data returns \(\chi ^{2} = 129.6\) for 178 data points and 9 free parameters, while that to DVCS data returns \(\chi ^{2} = 2346.3\) for 2600 data points and 13 free parameters. The good performance proves that our parameterizations, including the assumed analyticity property, are not contradicted by the used experimental data. We consider the unsurprising discrepancy between our results and the HERA data as a possible manifestation of gluon effects, and the surprising discrepancy with unpolarized cross sections published by Hall A as a possible need for higher twist or kinematic corrections.

This analysis is characterized by a careful propagation of uncertainties coming from PDFs parameterizations, EFF and DVCS data, which we achieved by using the replica method. The first successful step towards the reduction of model uncertainties has been done already by selecting PDFs parameterizations based on the neural network technique. We plan to extend the usage of neural networks to other sectors of our future analyses for further reduction of model uncertainties.

## Footnotes

## Notes

### Acknowledgements

The authors would like to thank M. Burkardt, M. Garçon, F.-X. Girod, B. Pasquini, M.V. Polyakov, A.V. Radyushkin, K.M. Semenov-Tian-Shansky and F. Yuan for fruitful discussions and valuable inputs. This work was supported in part by the Commissariat à l’Energie Atomique et aux Energies Alternatives and by the Grant no. 2017/26/M/ST2/01074 of the National Science Centre, Poland. The computing resources of Świerk Computing Centre (CIŚ) are greatly acknowledged.

## References

- 1.E.D. Bloom et al., Phys. Rev. Lett.
**23**, 930 (1969). https://doi.org/10.1103/PhysRevLett.23.930 ADSCrossRefGoogle Scholar - 2.M. Breidenbach, J.I. Friedman, H.W. Kendall, E.D. Bloom, D.H. Coward, H.C. DeStaebler, J. Drees, L.W. Mo, R.E. Taylor, Phys. Rev. Lett.
**23**, 935 (1969). https://doi.org/10.1103/PhysRevLett.23.935 ADSCrossRefGoogle Scholar - 3.D. Müller, D. Robaschik, B. Geyer, F.M. Dittes, J. Hořejši, Fortschr. Phys.
**42**, 101 (1994). https://doi.org/10.1002/prop.2190420202 CrossRefGoogle Scholar - 4.X.D. Ji, Phys. Rev. Lett.
**78**, 610 (1997). https://doi.org/10.1103/PhysRevLett.78.610 ADSCrossRefGoogle Scholar - 5.X.D. Ji, Phys. Rev. D
**55**, 7114 (1997). https://doi.org/10.1103/PhysRevD.55.7114 ADSCrossRefGoogle Scholar - 6.A.V. Radyushkin, Phys. Lett. B
**385**, 333 (1996). https://doi.org/10.1016/0370-2693(96)00844-1 ADSCrossRefGoogle Scholar - 7.A.V. Radyushkin, Phys. Rev. D
**56**, 5524 (1997). https://doi.org/10.1103/PhysRevD.56.5524 ADSCrossRefGoogle Scholar - 8.M. Burkardt, Phys. Rev. D
**62**, 071503 (2000). https://doi.org/10.1103/PhysRevD.62.071503, https://doi.org/10.1103/PhysRevD.66.119903 (**Erratum: Phys. Rev. D66, 119903 (2002)**) - 9.M. Burkardt, Int. J. Mod. Phys. A
**18**, 173 (2003). https://doi.org/10.1142/S0217751X03012370 ADSCrossRefGoogle Scholar - 10.M. Burkardt, Phys. Lett. B
**595**, 245 (2004). https://doi.org/10.1016/j.physletb.2004.05.070 ADSCrossRefGoogle Scholar - 11.K. Goeke, J. Grabis, J. Ossmann, M.V. Polyakov, P. Schweitzer, A. Silva, D. Urbano, Phys. Rev. D
**75**, 094021 (2007). https://doi.org/10.1103/PhysRevD.75.094021 ADSCrossRefGoogle Scholar - 12.M.V. Polyakov, P. Schweitzer, Int. J. Mod. Phys. A
**33**(26), 1830025 (2018). https://doi.org/10.1142/S0217751X18300259 ADSCrossRefGoogle Scholar - 13.A. Airapetian et al., Phys. Rev. Lett.
**87**, 182001 (2001). https://doi.org/10.1103/PhysRevLett.87.182001 ADSCrossRefGoogle Scholar - 14.S. Stepanyan et al., Phys. Rev. Lett.
**87**, 182002 (2001). https://doi.org/10.1103/PhysRevLett.87.182002 ADSCrossRefGoogle Scholar - 15.X.D. Ji, J. Osborne, Phys. Rev. D
**57**, 1337 (1998). https://doi.org/10.1103/PhysRevD.57.1337 ADSCrossRefGoogle Scholar - 16.X.D. Ji, J. Osborne, Phys. Rev. D
**58**, 094018 (1998). https://doi.org/10.1103/PhysRevD.58.094018 ADSCrossRefGoogle Scholar - 17.L. Mankiewicz, G. Piller, E. Stein, M. Vanttinen, T. Weigl, Phys. Lett. B
**425**, 186 (1998). https://doi.org/10.1016/S0370-2693(98)00190-7, https://doi.org/10.1016/S0370-2693(99)00883-7 (**Erratum: Phys. Lett. B461, 423 (1999)**) - 18.A.V. Belitsky, D. Mueller, L. Niedermeier, A. Schafer, Phys. Lett. B
**474**, 163 (2000). https://doi.org/10.1016/S0370-2693(99)01283-6 ADSCrossRefGoogle Scholar - 19.A. Freund, M.F. McDermott, Phys. Rev. D
**65**, 091901 (2002). https://doi.org/10.1103/PhysRevD.65.091901 ADSCrossRefGoogle Scholar - 20.A. Freund, M.F. McDermott, Phys. Rev. D
**65**, 074008 (2002). https://doi.org/10.1103/PhysRevD.65.074008 ADSCrossRefGoogle Scholar - 21.A. Freund, M. McDermott, Eur. Phys. J. C
**23**, 651 (2002). https://doi.org/10.1007/s100520200928 ADSCrossRefGoogle Scholar - 22.B. Pire, L. Szymanowski, J. Wagner, Phys. Rev. D
**83**, 034009 (2011). https://doi.org/10.1103/PhysRevD.83.034009 ADSCrossRefGoogle Scholar - 23.V.M. Braun, A.N. Manashov, B. Pirnay, Phys. Rev. Lett.
**109**, 242001 (2012). https://doi.org/10.1103/PhysRevLett.109.242001 ADSCrossRefGoogle Scholar - 24.E.R. Berger, M. Diehl, B. Pire, Eur. Phys. J. C
**23**, 675 (2002). https://doi.org/10.1007/s100520200917 ADSCrossRefGoogle Scholar - 25.L. Favart, M. Guidal, T. Horn, P. Kroll, Eur. Phys. J. A
**52**(6), 158 (2016). https://doi.org/10.1140/epja/i2016-16158-2 ADSCrossRefGoogle Scholar - 26.D.Yu. Ivanov, A. Schafer, L. Szymanowski, G. Krasnikov, Eur. Phys. J. C
**34**(3), 297 (2004). https://doi.org/10.1140/epjc/s2004-01712-x, https://doi.org/10.1140/epjc/s10052-015-3298-8 (**Erratum: Eur. Phys. J. C75(2), 75 (2015)**) - 27.M. Guidal, M. Vanderhaeghen, Phys. Rev. Lett.
**90**, 012001 (2003). https://doi.org/10.1103/PhysRevLett.90.012001 ADSCrossRefGoogle Scholar - 28.A.V. Belitsky, D. Mueller, Phys. Rev. Lett.
**90**, 022001 (2003). https://doi.org/10.1103/PhysRevLett.90.022001 ADSCrossRefGoogle Scholar - 29.R. Boussarie, B. Pire, L. Szymanowski, S. Wallon, JHEP
**02**, 054 (2017). https://doi.org/10.1007/JHEP02(2017)054 ADSCrossRefGoogle Scholar - 30.A. Pedrak, B. Pire, L. Szymanowski, J. Wagner, Phys. Rev. D
**96**(7), 074008 (2017). https://doi.org/10.1103/PhysRevD.96.074008 ADSCrossRefGoogle Scholar - 31.B.Z. Kopeliovich, I. Schmidt, M. Siddikov, Phys. Rev. D
**86**, 113018 (2012). https://doi.org/10.1103/PhysRevD.86.113018 ADSCrossRefGoogle Scholar - 32.B. Pire, L. Szymanowski, J. Wagner, Phys. Rev. D
**95**(11), 114029 (2017). https://doi.org/10.1103/PhysRevD.95.114029 ADSCrossRefGoogle Scholar - 33.B. Pire, L. Szymanowski, J. Wagner, Phys. Rev. D
**95**(9), 094001 (2017). https://doi.org/10.1103/PhysRevD.95.094001 ADSCrossRefGoogle Scholar - 34.R. Dupre, M. Guidal, M. Vanderhaeghen, Phys. Rev. D
**95**(1), 011501 (2017). https://doi.org/10.1103/PhysRevD.95.011501 ADSCrossRefGoogle Scholar - 35.V.D. Burkert, L. Elouadrhiri, F.X. Girod, Nature
**557**(7705), 396 (2018). https://doi.org/10.1038/s41586-018-0060-z ADSCrossRefGoogle Scholar - 36.K. Kumerički, D. Müller, EPJ Web Conf.
**112**, 01012 (2016). https://doi.org/10.1051/epjconf/201611201012 CrossRefGoogle Scholar - 37.K. Kumericki, S. Liuti, H. Moutarde, Eur. Phys. J. A
**52**(6), 157 (2016). https://doi.org/10.1140/epja/i2016-16157-3 ADSCrossRefGoogle Scholar - 38.O.V. Teryaev, in
*11th international conference on elastic and diffractive scattering: towards high energy frontiers: the 20th anniversary of the Blois workshops, 17th Rencontre de Blois (EDS 05) Chateau de Blois, Blois, France, May 15–20*(2005)Google Scholar - 39.F. Yuan, Phys. Rev. D
**69**, 051501 (2004). https://doi.org/10.1103/PhysRevD.69.051501 ADSCrossRefGoogle Scholar - 40.A.V. Radyushkin, Phys. Rev. D
**83**, 076006 (2011). https://doi.org/10.1103/PhysRevD.83.076006 ADSCrossRefGoogle Scholar - 41.A.V. Radyushkin, Int. J. Mod. Phys. Conf. Ser.
**20**, 251 (2012). https://doi.org/10.1142/S2010194512009300 CrossRefGoogle Scholar - 42.B. Berthou et al., Eur. Phys. J. C
**78**(6), 478 (2018). https://doi.org/10.1140/epjc/s10052-018-5948-0 ADSCrossRefGoogle Scholar - 43.A.V. Belitsky, A.V. Radyushkin, Phys. Rep.
**418**, 1 (2005). https://doi.org/10.1016/j.physrep.2005.06.002 ADSCrossRefGoogle Scholar - 44.M. Diehl, Phys. Rep.
**388**, 41 (2003). https://doi.org/10.1016/j.physrep.2003.08.002, https://doi.org/10.3204/DESY-THESIS-2003-018 - 45.X. Ji, Ann. Rev. Nucl. Part. Sci.
**54**, 413 (2004). https://doi.org/10.1146/annurev.nucl.54.070103.181302 ADSCrossRefGoogle Scholar - 46.S. Boffi, B. Pasquini, Rivista Nuovo Cimento
**30**, 387 (2007). https://doi.org/10.1393/ncr/i2007-10025-7 ADSCrossRefGoogle Scholar - 47.M. Diehl, T. Feldmann, R. Jakob, P. Kroll, Eur. Phys. J. C
**39**, 1 (2005). https://doi.org/10.1140/epjc/s2004-02063-4 ADSCrossRefGoogle Scholar - 48.B. Pire, J. Soffer, O. Teryaev, Eur. Phys. J. C
**8**, 103 (1999). https://doi.org/10.1007/s100529901063 ADSCrossRefGoogle Scholar - 49.M. Diehl, T. Feldmann, R. Jakob, P. Kroll, Nucl. Phys. B
**596**, 33 (2001). https://doi.org/10.1016/S0550-3213(00)00684-2, https://doi.org/10.1016/S0550-3213(01)00183-3 (**Erratum: Nucl. Phys. B605, 647 (2001)**) - 50.P.V. Pobylitsa, Phys. Rev. D
**65**, 077504 (2002). https://doi.org/10.1103/PhysRevD.65.077504 ADSCrossRefGoogle Scholar - 51.P.V. Pobylitsa, Phys. Rev. D
**65**, 114015 (2002). https://doi.org/10.1103/PhysRevD.65.114015 ADSCrossRefGoogle Scholar - 52.P.V. Pobylitsa, Phys. Rev. D
**66**, 094002 (2002). https://doi.org/10.1103/PhysRevD.66.094002 ADSCrossRefGoogle Scholar - 53.P.V. Pobylitsa, Phys. Rev. D
**67**, 034009 (2003). https://doi.org/10.1103/PhysRevD.67.034009 ADSCrossRefGoogle Scholar - 54.P.V. Pobylitsa, Phys. Rev. D
**67**, 094012 (2003). https://doi.org/10.1103/PhysRevD.67.094012 ADSCrossRefGoogle Scholar - 55.P.V. Pobylitsa, Phys. Rev. D
**70**, 034004 (2004). https://doi.org/10.1103/PhysRevD.70.034004 ADSCrossRefGoogle Scholar - 56.A.V. Belitsky, D. Müller, Y. Ji, Nucl. Phys. B
**878**, 214 (2014). https://doi.org/10.1016/j.nuclphysb.2013.11.014 ADSCrossRefGoogle Scholar - 57.R. Akhunzyanov, et al., Phys. Lett. B (2018). arXiv:1802.02739 [hep-ex]
- 58.A. Aktas et al., Eur. Phys. J. C
**44**, 1 (2005). https://doi.org/10.1140/epjc/s2005-02345-3 CrossRefGoogle Scholar - 59.S. Chekanov et al., JHEP
**05**, 108 (2009). https://doi.org/10.1088/1126-6708/2009/05/108 ADSCrossRefGoogle Scholar - 60.F.D. Aaron et al., Phys. Lett. B
**681**, 391 (2009). https://doi.org/10.1016/j.physletb.2009.10.035 ADSCrossRefGoogle Scholar - 61.M.V. Polyakov, C. Weiss, Phys. Rev. D
**60**, 114017 (1999). https://doi.org/10.1103/PhysRevD.60.114017 ADSCrossRefGoogle Scholar - 62.A.V. Radyushkin, Phys. Lett. B
**449**, 81 (1999). https://doi.org/10.1016/S0370-2693(98)01584-6 ADSCrossRefGoogle Scholar - 63.M.V. Polyakov, Phys. Lett. B
**555**, 57 (2003). https://doi.org/10.1016/S0370-2693(03)00036-4 ADSMathSciNetCrossRefGoogle Scholar - 64.M.V. Polyakov, K.M. Semenov-Tian-Shansky, Eur. Phys. J. A
**40**, 181 (2009). https://doi.org/10.1140/epja/i2008-10759-2 ADSCrossRefGoogle Scholar - 65.K. Kumericki, D. Mueller, K. Passek-Kumericki, Nucl. Phys. B
**794**, 244 (2008). https://doi.org/10.1016/j.nuclphysb.2007.10.029 ADSCrossRefGoogle Scholar - 66.K. Kumericki, D. Mueller, K. Passek-Kumericki, Eur. Phys. J. C
**58**, 193 (2008). https://doi.org/10.1140/epjc/s10052-008-0741-0 ADSCrossRefGoogle Scholar - 67.A.V. Radyushkin, Phys. Rev. D
**87**(9), 096017 (2013). https://doi.org/10.1103/PhysRevD.87.096017 ADSMathSciNetCrossRefGoogle Scholar - 68.A.V. Radyushkin, Phys. Rev. D
**88**(5), 056010 (2013). https://doi.org/10.1103/PhysRevD.88.056010 ADSCrossRefGoogle Scholar - 69.D. Müller, M.V. Polyakov, K.M. Semenov-Tian-Shansky, JHEP
**03**, 052 (2015). https://doi.org/10.1007/JHEP03(2015)052 ADSCrossRefGoogle Scholar - 70.D. Müller, K.M. Semenov-Tian-Shansky, Phys. Rev. D
**92**(7), 074025 (2015). https://doi.org/10.1103/PhysRevD.92.074025 ADSCrossRefGoogle Scholar - 71.H. Moutarde, Phys. Rev. D
**79**, 094021 (2009). https://doi.org/10.1103/PhysRevD.79.094021 ADSCrossRefGoogle Scholar - 72.P. Kroll, H. Moutarde, F. Sabatie, Eur. Phys. J. C
**73**(1), 2278 (2013). https://doi.org/10.1140/epjc/s10052-013-2278-0 ADSCrossRefGoogle Scholar - 73.S.V. Goloskokov, P. Kroll, Eur. Phys. J. C
**42**, 281 (2005). https://doi.org/10.1140/epjc/s2005-02298-5 ADSCrossRefGoogle Scholar - 74.S.V. Goloskokov, P. Kroll, Eur. Phys. J. C
**53**, 367 (2008). https://doi.org/10.1140/epjc/s10052-007-0466-5 ADSCrossRefGoogle Scholar - 75.S.V. Goloskokov, P. Kroll, Eur. Phys. J. C
**65**, 137 (2010). https://doi.org/10.1140/epjc/s10052-009-1178-9 ADSCrossRefGoogle Scholar - 76.K. Goeke, M.V. Polyakov, M. Vanderhaeghen, Prog. Part. Nucl. Phys.
**47**, 401 (2001). https://doi.org/10.1016/S0146-6410(01)00158-2 ADSCrossRefGoogle Scholar - 77.A.V. Belitsky, D. Mueller, A. Kirchner, Nucl. Phys. B
**629**, 323 (2002). https://doi.org/10.1016/S0550-3213(02)00144-X ADSCrossRefGoogle Scholar - 78.M. Vanderhaeghen, P.A.M. Guichon, M. Guidal, Phys. Rev. Lett.
**80**, 5064 (1998). https://doi.org/10.1103/PhysRevLett.80.5064 ADSCrossRefGoogle Scholar - 79.M. Vanderhaeghen, P.A.M. Guichon, M. Guidal, Phys. Rev. D
**60**, 094017 (1999). https://doi.org/10.1103/PhysRevD.60.094017 ADSCrossRefGoogle Scholar - 80.M. Guidal, M.V. Polyakov, A.V. Radyushkin, M. Vanderhaeghen, Phys. Rev. D
**72**, 054013 (2005). https://doi.org/10.1103/PhysRevD.72.054013 ADSCrossRefGoogle Scholar - 81.M. Diehl, P. Kroll, Eur. Phys. J. C
**73**(4), 2397 (2013). https://doi.org/10.1140/epjc/s10052-013-2397-7 ADSCrossRefGoogle Scholar - 82.L.L. Frankfurt, A. Freund, M. Strikman, Phys. Rev. D
**58**, 114001 (1998). https://doi.org/10.1103/PhysRevD.58.114001, https://doi.org/10.1103/PhysRevD.59.119901 (**Erratum: Phys. Rev. D59, 119901 (1999)**) - 83.K. Kumerički, D. Mueller, Nucl. Phys. B
**841**, 1 (2010). https://doi.org/10.1016/j.nuclphysb.2010.07.015 ADSCrossRefGoogle Scholar - 84.F.D. Aaron et al., Phys. Lett. B
**659**, 796 (2008). https://doi.org/10.1016/j.physletb.2007.11.093 ADSCrossRefGoogle Scholar - 85.R.D. Ball et al., JHEP
**04**, 040 (2015). https://doi.org/10.1007/JHEP04(2015)040 ADSCrossRefGoogle Scholar - 86.P.M. Nadolsky, H.L. Lai, Q.H. Cao, J. Huston, J. Pumplin, D. Stump, W.K. Tung, C.P. Yuan, Phys. Rev. D
**78**, 013004 (2008). https://doi.org/10.1103/PhysRevD.78.013004 ADSCrossRefGoogle Scholar - 87.E.R. Nocera, R.D. Ball, S. Forte, G. Ridolfi, J. Rojo, Nucl. Phys. B
**887**, 276 (2014). https://doi.org/10.1016/j.nuclphysb.2014.08.008 ADSCrossRefGoogle Scholar - 88.A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, G. Watt, Eur. Phys. J. C
**75**, 132 (2015). https://doi.org/10.1140/epjc/s10052-015-3318-8 ADSCrossRefGoogle Scholar - 89.J. Arrington, W. Melnitchouk, J.A. Tjon, Phys. Rev. C
**76**, 035205 (2007). https://doi.org/10.1103/PhysRevC.76.035205 ADSCrossRefGoogle Scholar - 90.B.D. Milbrath et al., Phys. Rev. Lett.
**80**, 452 (1998). https://doi.org/10.1103/PhysRevLett.80.452, https://doi.org/10.1103/PhysRevLett.82.2221 (**Erratum: Phys. Rev. Lett. 82, 2221 (1999)**) - 91.T. Pospischil et al., Eur. Phys. J. A
**12**, 125 (2001). https://doi.org/10.1007/s100500170046 ADSCrossRefGoogle Scholar - 92.O. Gayou et al., Phys. Rev. C
**64**, 038202 (2001). https://doi.org/10.1103/PhysRevC.64.038202 ADSCrossRefGoogle Scholar - 93.O. Gayou et al., Phys. Rev. Lett.
**88**, 092301 (2002). https://doi.org/10.1103/PhysRevLett.88.092301 ADSCrossRefGoogle Scholar - 94.V. Punjabi et al., Phys. Rev. C
**71**, 055202 (2005). https://doi.org/10.1103/PhysRevC.71.055202, https://doi.org/10.1103/PhysRevC.71.069902 (**Erratum: Phys. Rev. C71, 069902 (2005)**) - 95.G. MacLachlan et al., Nucl. Phys. A
**764**, 261 (2006). https://doi.org/10.1016/j.nuclphysa.2005.09.012 ADSCrossRefGoogle Scholar - 96.A.J.R. Puckett et al., Phys. Rev. Lett.
**104**, 242301 (2010). https://doi.org/10.1103/PhysRevLett.104.242301 ADSCrossRefGoogle Scholar - 97.M. Paolone et al., Phys. Rev. Lett.
**105**, 072001 (2010). https://doi.org/10.1103/PhysRevLett.105.072001 ADSCrossRefGoogle Scholar - 98.G. Ron et al., Phys. Rev. C
**84**, 055204 (2011). https://doi.org/10.1103/PhysRevC.84.055204 ADSCrossRefGoogle Scholar - 99.X. Zhan et al., Phys. Lett. B
**705**, 59 (2011). https://doi.org/10.1016/j.physletb.2011.10.002 ADSCrossRefGoogle Scholar - 100.H. Anklin et al., Phys. Lett. B
**428**, 248 (1998). https://doi.org/10.1016/S0370-2693(98)00442-0 ADSCrossRefGoogle Scholar - 101.G. Kubon et al., Phys. Lett. B
**524**, 26 (2002). https://doi.org/10.1016/S0370-2693(01)01386-7 ADSCrossRefGoogle Scholar - 102.H. Anklin et al., Phys. Lett. B
**336**, 313 (1994). https://doi.org/10.1016/0370-2693(94)90538-X ADSCrossRefGoogle Scholar - 103.J. Lachniet et al., Phys. Rev. Lett.
**102**, 192001 (2009). https://doi.org/10.1103/PhysRevLett.102.192001 ADSCrossRefGoogle Scholar - 104.B. Anderson et al., Phys. Rev. C
**75**, 034003 (2007). https://doi.org/10.1103/PhysRevC.75.034003 ADSCrossRefGoogle Scholar - 105.C. Herberg et al., Eur. Phys. J. A
**5**, 131 (1999). https://doi.org/10.1007/s100500050268 ADSCrossRefGoogle Scholar - 106.D.I. Glazier et al., Eur. Phys. J. A
**24**, 101 (2005). https://doi.org/10.1140/epja/i2004-10115-8 ADSCrossRefGoogle Scholar - 107.B. Plaster et al., Phys. Rev. C
**73**, 025205 (2006). https://doi.org/10.1103/PhysRevC.73.025205 ADSCrossRefGoogle Scholar - 108.I. Passchier et al., Phys. Rev. Lett.
**82**, 4988 (1999). https://doi.org/10.1103/PhysRevLett.82.4988 ADSCrossRefGoogle Scholar - 109.H. Zhu et al., Phys. Rev. Lett.
**87**, 081801 (2001). https://doi.org/10.1103/PhysRevLett.87.081801 ADSCrossRefGoogle Scholar - 110.G. Warren et al., Phys. Rev. Lett.
**92**, 042301 (2004). https://doi.org/10.1103/PhysRevLett.92.042301 ADSCrossRefGoogle Scholar - 111.E. Geis et al., Phys. Rev. Lett.
**101**, 042501 (2008). https://doi.org/10.1103/PhysRevLett.101.042501 ADSCrossRefGoogle Scholar - 112.J. Bermuth et al., Phys. Lett. B
**564**, 199 (2003). https://doi.org/10.1016/S0370-2693(03)00725-1 ADSCrossRefGoogle Scholar - 113.D. Rohe. Private communication with M. Diehl and P. KrollGoogle Scholar
- 114.S. Riordan et al., Phys. Rev. Lett.
**105**, 262302 (2010). https://doi.org/10.1103/PhysRevLett.105.262302 ADSCrossRefGoogle Scholar - 115.R. Schiavilla, I. Sick, Phys. Rev. C
**64**, 041002 (2001). https://doi.org/10.1103/PhysRevC.64.041002 ADSCrossRefGoogle Scholar - 116.J. Beringer et al., Phys. Rev. D
**86**, 010001 (2012). https://doi.org/10.1103/PhysRevD.86.010001 ADSCrossRefGoogle Scholar - 117.M. Defurne et al., Phys. Rev. C
**92**(5), 055202 (2015). https://doi.org/10.1103/PhysRevC.92.055202 ADSCrossRefGoogle Scholar - 118.M. Defurne, Nat. Commun.
**8**(1), 1408 (2017). https://doi.org/10.1038/s41467-017-01819-3 ADSCrossRefGoogle Scholar - 119.A. Airapetian et al., Phys. Rev. D
**75**, 011103 (2007). https://doi.org/10.1103/PhysRevD.75.011103 ADSCrossRefGoogle Scholar - 120.A. Airapetian et al., JHEP
**06**, 066 (2008). https://doi.org/10.1088/1126-6708/2008/06/066 CrossRefGoogle Scholar - 121.A. Airapetian et al., JHEP
**11**, 083 (2009). https://doi.org/10.1088/1126-6708/2009/11/083 CrossRefGoogle Scholar - 122.A. Airapetian et al., JHEP
**06**, 019 (2010). https://doi.org/10.1007/JHEP06(2010)019 ADSCrossRefGoogle Scholar - 123.A. Airapetian, Phys. Lett. B
**704**, 15 (2011). https://doi.org/10.1016/j.physletb.2011.08.067 ADSCrossRefGoogle Scholar - 124.A. Airapetian et al., JHEP
**07**, 032 (2012). https://doi.org/10.1007/JHEP07(2012)032 ADSCrossRefGoogle Scholar - 125.S. Chen et al., Phys. Rev. Lett.
**97**, 072002 (2006). https://doi.org/10.1103/PhysRevLett.97.072002 ADSCrossRefGoogle Scholar - 126.F.X. Girod et al., Phys. Rev. Lett.
**100**, 162002 (2008). https://doi.org/10.1103/PhysRevLett.100.162002 ADSCrossRefGoogle Scholar - 127.G. Gavalian et al., Phys. Rev. C
**80**, 035206 (2009). https://doi.org/10.1103/PhysRevC.80.035206 ADSCrossRefGoogle Scholar - 128.S. Pisano et al., Phys. Rev. D
**91**(5), 052014 (2015). https://doi.org/10.1103/PhysRevD.91.052014 ADSCrossRefGoogle Scholar - 129.H.S. Jo et al., Phys. Rev. Lett.
**115**(21), 212003 (2015). https://doi.org/10.1103/PhysRevLett.115.212003 ADSCrossRefGoogle Scholar - 130.F. James, M. Roos, Comput. Phys. Commun.
**10**, 343 (1975). https://doi.org/10.1016/0010-4655(75)90039-9 ADSCrossRefGoogle Scholar - 131.R. Brun, F. Rademakers, Nucl. Instrum. Methods
**A389**, 81 (1997). https://doi.org/10.1016/S0168-9002(97)00048-X ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}