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We determine the generalized form factors, which correspond to the second Mellin moment (i.e., the first

The understanding of hadron structure has greatly evolved over the last decades. The collected knowledge is parametrized by a large number of functions. Generalized parton distributions (GPDs) are one set of such functions. They parametrize, e.g., the transverse coordinate distribution of partons in a fast moving hadron and contain information on how these distributions depend on the parton or hadron spin direction. Pinning down all these multivariable functions experimentally is unrealistic at present. Therefore, lattice QCD has to substitute some of the missing experimental data. With this article we contribute to the effort of various lattice groups to provide some of these needed results

From the experimental point of view, GPDs play a similarly important role for the description of exclusive hadronic reactions as parton distribution functions (PDFs) do for inclusive reactions. The most extensively studied channel is deeply virtual Compton scattering (DVCS), i.e., Compton scattering with a highly virtual incoming photon and a correspondingly large, spacelike momentum transfer

The theoretical understanding of GPDs and their moments, the generalized form factors (GFFs), has already a long history and is presented in the seminal work of Refs.

We remark that recently new methods have been proposed to obtain information on parton distribution functions (PDFs), distribution amplitudes (DAs), transverse momentum dependent PDFs (TMDPDFs) and GPDs that is complementary to the computation of Mellin moments with respect to Bjorken-

This paper is organized as follows. In Sec.

The starting point is the off-forward nucleon matrix element,

In physical terms (for

As time is analytically continued to imaginary time to enable the numerical evaluation on the lattice, the light cone loses its meaning. The operator product expansion (OPE) relates, however, Mellin moments of GPDs to local matrix elements that are amenable to lattice calculation. For

In principle one can determine Mellin moments of GPDs for any

In the continuum we can decompose the matrix elements,

As has already been mentioned above, in the forward limit (

Furthermore, in this limit

The five GFFs

On the lattice, the GFFs are extracted from combinations of hadronic two- and three-point correlation functions in Euclidean space-time. The two-point function reads

The definition of the operator

A detailed description of the renormalization procedure, that consists of first nonperturbatively matching from the lattice to the

The renormalization factors used to translate our bare lattice data to the

Relative error of the GFFs for the flavor combination

In the following we demonstrate the extraction procedure for the vector GFFs. The axial and tensor GFFs are treated analogously. We start by expanding Eq.

The right-hand side of Eq.

If a row vanishes, then it does not restrict the GFF, and we remove it from the system of equations.

For a given ensemble this system of equations has to be solved separately for each virtuality to yield the GFFs as functions of

Our analysis is based on the large set of gauge configurations produced by the QCDSF and the RQCD (Regensburg QCD) Collaborations using the standard Wilson gauge action with two mass-degenerate nonperturbatively improved clover fermions; see Table

Parameters of the

We parametrize our two-point correlation functions with a two-exponential fit ansatz,

Overview of the nucleon energies for our ensembles. We compare the energies

The top panel shows the correlated

For the lattice calculations of three-point functions we use the sequential source method where we set the outgoing nucleon momentum

As explained above, for every current

The determination of the form factors is carried out in two ways. The first method consists of two steps: First we extract the ground state nucleon matrix elements

Once the fit parameters

Ideally,

Fit results using the single step minimization method. We show ensemble VIII at the virtuality

Individual operator contributions to the fits shown in Fig.

A comparison of the two fit methods shows that the results are consistent within errors for all GFFs and for all ensembles. The single step method, however, results in somewhat smaller statistical errors and a smoother

Comparison of single step and two step fit methods for the axial GFFs for ensemble VI. The right panels show

For some of our ensembles we have three-point function data for different source-sink separations. This allows us to analyze the influence of excited states on the GFFs. Our analysis is based on ensemble IV with five source-sink separations in the range

For the tensor and axial GFFs we find that within statistical errors the

The vector GFFs vs

Below we show results for the nucleon GFFs on a subset of the ensembles listed in Table

Results for the vector GFFs,

The vector and axial GFFs vs

Results for the axial GFFs are shown in the right panel of Fig.

Continuing with the tensor GFFs, we show results for

The tensor GFFs

The other two GFFs,

Strong anticorrelations between

The GFFs

Since it is not clear up to what values of

From top to bottom

Chiral fit A vs

Again, we study the effect of the uncertainties of the renormalization constants using the strategy described in Appendix

Results for

Results for

Within the errors, our values agree with the isovector results of Ref.

We use our lattice results for the vector GFFs

We fit the GFFs for ensemble VI to the

The pole mass

This arbitrariness means it is difficult to obtain reliable, parametrization independent results for the moment

The

We discovered that some integrated quantities have a much milder

Dependence of

Probability (weighted with

The half

We see the probability of a transversely polarized

We have calculated all quark GFFs, corresponding to operators with one derivative, of the nucleon GPDs at leading twist-2. Our lattice calculation includes the dominating connected contributions and neglects contributions from disconnected diagrams. The available gauge ensembles cover a wide range of quark masses and volumes. However, the three available lattice spacings only vary from 0.081 fm down to 0.060 fm. Within errors, all GFFs show a mild dependence on the quark mass, lattice spacing and volume.

We have compared two different fitting strategies for the GFFs and found that the direct fit method appears to be more reliable. With this method the number of fit parameters is reduced to the relevant degrees of freedom. We recommend to use this method in future studies. The final results for the GFFs are shown in Figs.

We have also studied the total angular momentum and the transverse spin density of quarks in the nucleon. Both quantities can be extracted from fits to our GFF data. For the total angular momentum we obtain a similar estimate in the isovector case as ETMC in Ref.

The ensembles were generated by RQCD and QCDSF primarily on the QPACE computer

In this study we use 16 linear combinations of operators for the tensor GFFs. The first eight from the

The renormalization factors are products of perturbative and nonperturbative parts,

The nonperturbative renormalization factors

About ten well-decorrelated configurations are often sufficient.

of our gauge configurations to Landau gauge and calculate (in momentum space) the quark propagatorNext the vertex function

Our estimates for the renormalization factors carry an uncertainty which has to be propagated into the GFFs. We do this in a very naive but conservative way by carrying out the whole analysis both using the central values of the renormalization factors and adding the error of these factors to their central values. The difference between these two sets of results is then due to the uncertainty of the renormalization. This procedure is applied to all ensembles and to all the available virtualities