^{*}

^{3}.

We compute the second moments of pion and rho parton distribution functions in lattice QCD with

The pion is routinely investigated on the lattice, although, to the best of our knowledge, disconnected quark contributions, e.g., to quark momentum fractions, were not included so far. Since the pion is the pseudo-Goldstone boson of dynamical chiral symmetry breaking its quark structure could differ substantially from that of other mesons and if so, the flavor singlet sea quark contribution is a natural place for such a difference to show up. In contrast to the pion, the quark structure of the

For the pion experimental data exists primarily from two classes of experiments, namely Drell-Yan reactions with (secondary) pion beams, e.g.,

In contrast to the pion case, there exists very little relevant experimental data for the

The study of mesonic structure using lattice QCD has, by now, a history of over three decades. Traditionally, such calculations focused on moments on PDFs and distribution amplitudes (DAs). While earlier simulations

The reach of such calculations of moments of PDFs and DAs is limited, primarily because for higher moments the problems caused by operator mixing become untraceable. Therefore, in recent years ever more attention has focused on coordinate space methods

In this article we directly calculate the second moments of the pion and rho PDFs by evaluating operators that contain a covariant derivative. The same method would not be directly applicable to higher moments, since one would face the problem of mixing with lower-dimensional operators. Sparked by the presentation in Ref.

This article is structured as follows. We set the stage with a general discussion of PDF properties and their connection to DIS structure functions in Sec.

The cross section of deep inelastic scattering can be written as a product of a leptonic and a hadronic part. The hadronic tensor is given by

The hadronic tensor can be factorized into a hard scattering kernel, which can be calculated perturbatively, and in PDFs containing the nonperturbative information. The PDFs related to the structure functions in Eq.

In general one finds three quark and three gluon PDFs analogous to Eqs.

In order to see the connection between PDFs and structure functions let us consider the operator product expansion

For any general operator

In general one finds six towers of twist

See Eq. (33) in Ref.

In perturbation theory and to leading twist accuracy the structure functions are directly related to the PDFs (see, e.g., Refs.

Next, we perform a Lorentz decomposition for the forward matrix elements of the operators

In the structure functions the sum of quark and antiquark contributions

We can also relate the moments of the PDFs to the reduced matrix elements. By substituting Eq.

In analogy to the relations in Eqs.

To calculate the second moment of the structure functions introduced in the last section we analyzed a subset of the lattice gauge ensembles generated within the Coordinated Lattice Simulations (CLS) effort

To avoid freezing of the topological charge and large autocorrelation times for the very fine lattices we use open boundary conditions for most of our simulations

The two- and three-point functions introduced in Sec.

CLS gauge ensembles analyzed in this work labeled by their identifier and sorted by the inverse coupling

In order to obtain physically meaningful results, the bare operators introduced in Eqs.

On the lattice, the continuous Euclidean O(4) symmetry is reduced to that of its finite hypercubic subgroup H(4). Therefore, symmetry imposes much weaker constraints on the mixing of operators under renormalization. In order to avoid mixing as far as possible, in particular mixing with lower-dimensional operators, we use operators from suitably chosen multiplets that possess a definite

Our final results will be given in the

Renormalization factors

In order to calculate the DIS structure functions on the lattice one has to compute two- and three-point correlation functions in the forward limit:

For the pion case we first define the matrix elements

Similarly, one can show that the spectral decomposition of the three-point function in Eq.

For the rho mesons we define, in analogy to the pseudoscalar case

Inserting two complete sets of states into the three-point function

In the three-point functions

For the rho meson case we perform the analysis analogously. However, in particular for ensembles with small quark masses and large volumes, one would in this situation expect a contribution from (possibly multiple) two-pion states, which can have even smaller energy than the “ground-state” rho meson itself. Despite the fact that we do not find any trace of these two-pion states in our numerical analysis, we cannot claim to have this problem fully under control; cf. the discussion of this delicate issue in Sec.

Instead of performing a fit to three-point functions, one can equivalently fit to ratios of two- and three-point functions. As discussed in Ref.

In an infinite volume, above the particle creation threshold, a continuum of states would contribute to the spectral decomposition of the rho meson. In particular in Eq.

For particles of integer spin and at zero momentum (i.e., in the center-of-mass frame), the full symmetry group on the lattice is the octahedral group

For half-integer spin one would have to consider the corresponding double covers of

Little groups and decomposition of angular momentum 1 in irreducible representations for all momentum sectors

The connection between the finite volume energy spectrum of two-pion states and infinite volume scattering phase shifts has been established by Lüscher in his seminal articles

Using Eq.

Scattering phase shifts

Scattering phase shifts (assuming that only the

In Fig.

Comparison of the Breit-Wigner (red) and Gounaris-Sakurai (blue) parametrizations of the phase shift. Here we use the pion mass and (naively measured) rho mass from D200 as input. The rho-pi-pi coupling constant is set to the phenomenological value

As pointed out in Ref.

In Fig.

The pion form factor obtained using Eq.

That being said, we want to stress that the analysis provided above is actually only valid for unsmeared currents. Obviously, the situation might be less critical for the smeared currents that we use in our simulation.

To compute the reduced matrix elements introduced, e.g., in Eq.

The fits are performed using a constant fit window of

Rho masses for all ensembles analyzed from a double exponential fit (open boundaries) or a single exponential fit (periodic boundaries) to the two-point function correlator. The triangles depict the results in the rest frame (

The observables studied in this article are affected by disconnected quark loops. Often one can circumvent this problem by considering isovector current insertions, where the up and down quark disconnected loops cancel each other identically in the limit of exact isospin symmetry. This is not a viable solution in this case, since the connected part also vanishes for the isovector currents. Unfortunately, the disconnected contributions are notorious for having a large statistical error. However, as will be discussed later in this section, this is not true in general.

In cases where the disconnected contributions are zero within the error one might be tempted to simply drop them. However, in situations where the statistical error is large (for instance the flavor singlet operators), their inclusion can shift the mean and, even more important, can increase the error for the final result substantially. I.e., they have to be included, if one wants to provide reliable error estimates for phenomenological applications. Nevertheless, we perform a second analysis in these cases, where we solely use the connected part, which allows us to compare to other lattice results for connected contributions.

In principle, one would like to add the connected and disconnected contributions already at the correlation function level. However, this is not feasible since the connected and disconnected parts are calculated in different ways. The connected part is calculated with the stochastic propagator estimation presented in Appendix

Next, we extract the ground-state contribution from the ratios defined in Sec.

In Figs.

In summary this implies that the fits not only take into account the data points shown in Figs.

Extraction of renormalized values for

Extraction of renormalized values for

Summary of the occurrence of the individual fit parameters in the ansatz

As discussed above, we perform the extraction of the ground-state matrix elements separately for the connected (left column) and disconnected (middle and right column) contributions. While analyzing the disconnected contribution we found that the considerable noise on the light and strange quark loops is highly correlated for all included ensembles, cf. Table

For the disconnected contribution we have data points for a large number of combinations of final times

As described in Sec.

In Figs.

The final results for the reduced matrix elements are given in Table

Lattice spacing dependence of the extrapolations for the flavor singlet (

Extrapolation for the flavor singlet (

Results obtained from the extrapolations in Figs.

Using Eq.

The connected-only results are presented as a comparison option for other studies neglecting disconnected contributions.

reduces the errors significantly. The reader should be aware of the fact that we use (in both cases) the flavor nonsinglet renormalization constants, which is only an approximation, cf. the discussion in Sec.For the flavor nonsinglet contributions given in Table

Estimated results for the first moments of the structure functions

In this article we have presented the computation of the first moments for the structure functions

Despite the fact that we for the first time presented comprehensive results including disconnected contributions we have reduced the statistical error considerably. This can be seen by comparing the error of the connected contribution alone with earlier studies. However, to determine the phenomenologically important moments of the structure functions (at leading twist), one needs the flavor singlet combination, where the statistical error is still large. Future studies will have to aim at a further reduction of these statistical errors. Once this is achieved, a nonperturbative calculation of the singlet renormalization factors and the inclusion of mixing with gluonic operators might also be worthwhile.

Support of this project was granted by the German DFG (SFB/TRR 55). In addition this project has received funding from the European Unions Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 813942. We are grateful to Gunnar S. Bali, Lorenzo Barca, Sara Collins, Vladimir Braun, Meinulf Göckeler, and Christoph Lehner for the various fruitful discussions and to Wolfgang Söldner for providing parts of the intermediate results to be published in Ref.

We define the light-cone coordinates used in Sec.

For the polarization vectors we use a dimensionless definition. They obey the general transversality condition

To avoid mixing as far as possible we use operators from suitably chosen multiplets that possess a definite

Using the common sequential source method

First of all we factorize the three-point correlation function into two largely independent parts denoted as the spectator

Sketch of a generic meson three-point function in the forward and backward directions. The source time slice is

In addition to connected contributions to the three-point function, which we treat in Appendix

Sketch of a generic meson disconnected three-point function. The source time slice is

In Fig.

Extrapolation for the flavor singlet operator (