Search

We present results on the momentum fraction carried by quarks and gluons in the pion and the kaon. We employ three gauge ensembles generated with $N_f=2+1+1$ Wilson twisted-mass clover-improved fermions with physical quark masses. We perform, for the first time, a continuum extrapolation directly at the physical pion. We find that the total momentum fraction carried by quarks is $\langle x \rangle_{q, R}^{\pi}= 0.575(79)$ and $\langle x \rangle_{q,R}^{K} = 0.683(50)$ and by gluons $\langle x \rangle_{g, R}^{\pi}=0.402(53)$ and $\langle x \rangle_{g, R}^{K}=0.422(67)$ in the pion and in the kaon, respectively, in the $\overline{\mathrm{MS}}$ scheme and at the renormalization scale of 2 GeV. Having computed both the quark and gluon contributions in the continuum limit, we verify the momentum sum, finding 0.984(89) for the pion and 1.13(11) for the kaon., Comment: arXiv admin note: substantial text overlap with arXiv:2405.08529

In this work, we present a lattice QCD calculation of the Mellin moments of the twist-2 axial-vector generalized parton distribution (GPD), $\widetilde{H}(x,\xi,t)$, at zero skewness, $\xi$, with multiple values of the momentum transfer, $t$. Our analysis employs the short-distance factorization framework on ratio-scheme renormalized quasi-GPD matrix elements. The calculations are based on an $N_f=2+1+1$ twisted mass fermions ensemble with clover improvement, a lattice spacing of $a = 0.093$ fm, and a pion mass of $m_\pi = 260$ MeV. We consider both the iso-vector and iso-scalar cases, utilizing next-to-leading-order perturbative matching while omitting the disconnected contributions and gluon mixing in the iso-scalar case. For the first time, we determine the Mellin moments of $\widetilde{H}$ up to the fifth order. From these moments, we discuss the quark helicity and orbital angular momentum contributions to the nucleon spin, as well as the spin-orbit correlations of the quarks. Additionally, we perform a Fourier transform over the momentum transfer, which allows us to explore the spin structure in the impact-parameter space., Comment: 31 pages, 13 figures, version appeared in JHEP

We present the full decomposition of the momentum fraction carried by quarks and gluons in the pion and the kaon. We employ three gauge ensembles generated with $N_f=2+1+1$ Wilson twisted-mass clover-improved fermions at the physical quark masses. For both mesons we perform a continuum extrapolation directly at the physical pion mass, which allows us to determine for the first time the momentum decomposition at the physical point. We find that the total momentum fraction carried by quarks is 0.532(56) and 0.618(32) and by gluons 0.388(49) and 0.408(61) in the pion and in the kaon, respectively, in the $\overline{\mathrm{MS}}$ scheme and at the renormalization scale of 2 GeV. Having computed both the quark and gluon contributions in the continuum limit, we find that the momentum sum is 0.926(68) for the pion and 1.046(90) for the kaon, verifying the momentum sum rule., Comment: 14 pages, 10 figures

Generalized parton distributions (GPDs) are key quantities for the description of a hadron's three-dimensional structure. They are the current focus of all areas of hadronic physics -- phenomenological, experimental, and theoretical, including lattice QCD. Synergies between these areas are desirable and essential to achieve precise quantification and understanding of the structure of, particularly nucleons, as the basic ingredients of matter. In this paper, we investigate, for the first time, the numerical implementation of the pseudo-distribution approach for the extraction of zero-skewness GPDs for unpolarized quarks. Pseudo-distributions are Euclidean parton correlators computable in lattice QCD that can be perturbatively matched to the light-cone parton distributions of interest. Being closely related to the quasi-distributions and coming from the same lattice-extracted matrix elements, they are, however, subject to different systematic effects. We use the data previously utilized for quasi-GPDs and extend it with other momentum transfers and nucleon boosts, in particular a higher one ($P_3=1.67$ GeV) with eight-fold larger statistics than the largest one used for quasi-distributions ($P_3=1.25$ GeV). We renormalize the matrix elements with a ratio scheme and match the resulting Ioffe time distributions to the light cone in coordinate space. The matched distributions are then used to reconstruct the $x$-dependence with a fitting ansatz.We investigate some systematic effects related to this procedure, and we also compare the results with the ones obtained in the framework of quasi-GPDs. Our final results involve the invariant four-momentum transfer squared ($-t$) dependence of the flavor non-singlet ($u-d$) $H$ and $E$ GPDs., Comment: 35 pages, 29 figures, compatible with journal version

First lattice QCD calculations of $x$-dependent GPD have been performed in the (symmetric) Breit frame, where the momentum transfer is evenly divided between the initial and final hadron states. However, employing the asymmetric frame, we are able to obtain proton GPDs for multiple momentum transfers in a computationally efficient setup. In these proceedings, we focus on the helicity twist-2 GPD at zero skewness that gives access to the $\widetilde{H}$ GPD. We will cover the implementation of the asymmetric frame, its comparison to the Breit frame, and the dependence of the GPD on the squared four-momentum transfer, $-t$. The calculation is performed on an $N_f = 2+1+1$ ensemble of twisted mass fermions with a clover improvement. The mass of the pion for this ensemble is roughly 260 MeV., Comment: 9 pages, 4 figures, Contribution to The 40th International Symposium on Lattice Field Theory (Lattice 2023)

Understanding the intricate three-dimensional internal structure of the nucleon has been a long-standing challenge. The main quantitative tool to map out this structure are the generalized parton distributions (GPDs). In these proceedings, we present our exploratory results from applying the pseudo-distribution approach on the lattice, showing the unpolarized isovector $E$ GPD as an example. We use one ensemble of $N_f=2+1+1$ twisted mass fermions with a clover term, at a non-physical pion mass of $260 \textrm{ MeV}$ and a lattice spacing of $0.093 \textrm{ fm}$., Comment: 8 pages, 3 figures, contribution to the 40th International Symposium on Lattice Field Theory, Lattice 2023, Fermilab, Batavia, Illinois, USA

Recently, we made significant advancements in improving the computational efficiency of lattice QCD calculations for Generalized Parton Distributions (GPDs). This progress was achieved by adopting calculations of matrix elements in asymmetric frames, deviating from the computationally-expensive symmetric frame typically used, and allowing freedom in the choice for the distribution of the momentum transfer between the initial and final states. A crucial aspect of this approach involves the adoption of a Lorentz covariant parameterization for the matrix elements, introducing Lorentz-invariant amplitudes. This approach also allows us to propose an alternative definition of quasi-GPDs, ensuring frame independence and potentially reduce power corrections in matching to light-cone GPDs. In our previous work, we presented lattice QCD results for twist-2 unpolarized GPDs ($H$ and $E$) of quarks obtained from calculations performed in asymmetric frames at zero skewness. Building upon this work, we now introduce a novel Lorentz covariant parameterization for the axial-vector matrix elements. We employ this parameterization to compute the axial-vector GPD $\widetilde{H}$ at zero skewness, using an $N_f=2+1+1$ ensemble of twisted mass fermions with clover improvement. The light-quark masses employed in our calculations correspond to a pion mass of approximately 260 MeV., Comment: 32 pages, 20 figures. Version accepted for publication in Physical Review D

This work presents the first lattice-QCD calculation of the twist-3 axial quark GPDs for the proton using the large-momentum effective theory approach. We calculate matrix elements with momentum-boosted proton states and a non-local axial operator. The calculation is performed using one ensemble of two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with a clover term. The ensemble has a volume $32^3\times64$, lattice spacing 0.0934 fm, and corresponds to a pion mass of 260 MeV. The matrix elements are calculated for three values of the proton momentum, namely 0.83, 1.25, and 1.67 GeV. The light-cone GPDs are defined in the symmetric frame, which we implement here with a (negative) 4-momentum transfer squared of 0.69, 1.38, and 2.76 GeV$^2$, all at zero skewness. We also conduct several consistency checks, including assessing the local limit of the twist-3 GPDs and examining the Burkhardt-Cottingham-type as well as Efremov-Teryaev-Leader-type sum rules., Comment: 22 pages, 17 figures. Version accepted for publication in Physical Review D

Staple-shaped Wilson line operators are necessary for the study of transverse momentum-dependent parton distribution functions (TMDPDFs) in lattice QCD and beyond. In this work, we study the renormalization of such operators in the general case of an asymmetric staple. We analyze the mixing pattern of these operators using their symmetry properties, where we find that the possible mixing is restricted within groups of four operators. We then present numerical results using the regularization independent momentum subtraction (RI/MOM) scheme to study the importance of mixing using one operator in particular, the $\gamma_0$ operator. Based on these results, we consider the short distance ratio (SDR) scheme, which is desirable in the absence of mixing. Finally, we investigate a variant of the RI/MOM scheme, where the renormalization factors are computed at short distances., Comment: 20 pages, 14 figures

For the first time, we present a lattice QCD determination of Mellin moments of unpolarized generalized parton distributions (GPDs) of the proton from an analysis of the quasi-GPD matrix elements within the short-distance factorization framework. We perform our calculation on an $N_f$=2+1+1 twisted mass fermions ensemble with a clover improvement at lattice spacing $a=0.093$ fm and a pion mass of $m_\pi=260$ MeV. Focusing on the zero-skewness case, the iso-vector and iso-scalar quasi-GPDs are calculated from the $\gamma_0$ definition, as well as a recently proposed Lorentz-invariant definition. We utilize data on both symmetric and asymmetric kinematic frames, which allows us to obtain the Mellin moments for several values of the momentum transfer, $-t$, in the range 0.17 to $2.77~\rm{GeV}^2$. We use the ratio scheme for GPDs, i.e. renormalization group invariant ratios with leading-twist factorization formula and perturbatively calculated matching coefficients up to the next-next-to-leading order (NNLO) to extract Mellin moments of GPDs, which are consistent with renormalization-group improved results. We compare our determination from quasi-GPDs with the results extracted using standard calculations of Mellin moments of local operators, specifically those related to the electromagnetic and gravitational form factors. We estimated the moments of GPDs up to the fifth ones for the first time. By extrapolating the Mellin moments to $-t=0$, we obtained the quark charges, momentum fraction, as well as the angular momentum contributions to the proton spin. The impact parameter space interpretation of the GPD moments is discussed, which provides insights into the spatial distribution of unpolarized quarks and their correlations in the transverse plane of an unpolarized or transversely polarized proton., Comment: 28 pages, 22 figures

In recent years, there has been a breakthrough in lattice calculations of $x$-dependent partonic distributions. This encompasses also distributions describing the 3D structure of the nucleon, such as generalized parton distributions (GPDs). We report a new method of accessing GPDs in asymmetric frames of reference, relying on a novel Lorentz-covariant parametrization of the accessed off-forward matrix elements in boosted nucleon states. The approach offers the possibility of computationally more efficient determination of the full parameter dependence of GPDs and as such, it can contribute to better understanding of nucleon's structure., Comment: Proceedings of XXIX Cracow Epiphany Conference on Physics at the EIC and Future Facilities, 16-19 January 2023; 15 pages, 7 figures

It is often taken for granted that Generalized Parton Distributions (GPDs) are defined in the "symmetric" frame, where the transferred momentum is symmetrically distributed between the incoming/outgoing hadrons. However, such frames pose computational challenges for the lattice QCD practitioners. In these proceedings, we lay the foundation for lattice QCD calculations of GPDs in "asymmetric" frames, where the transferred momentum is not symmetrically distributed between the incoming/outgoing hadrons. The novelty of our work relies on the parameterization of the matrix elements in terms of Lorentz-invariant amplitudes, which not only helps in establishing relations between the said frames but also helps in isolating higher-twist contaminations. As an example, we focus on the unpolarized GPDs for spin-1/2 particles., Comment: 10 pages, 1 figure, contribution to the 39th International Symposium on Lattice Field Theory, Lattice 2022, Bonn, Germany

Pseudoscalar-pole diagrams are an important component of estimates of the hadronic light-by-light (HLbL) contribution to the muon $g-2$. We report on our computation of the transition form factors $\mathcal{F}_{P \rightarrow \gamma^* \gamma^*}$ for the neutral pseudoscalar mesons $P=\pi^0$ and $\eta$. The calculation is performed using twisted-mass lattice QCD with physical quark masses. On the lattice, we have access to a broad range of (space-like) photon four-momenta and therefore produce form factor data complementary to the experimentally accessible single-virtual direction, which directly leads to an estimate of the pion- and $\eta$-pole components of the muon $g-2$. For the pion, our result for the $g-2$ contribution in the continuum is comparable with previous lattice and data-driven determinations, with combined relative uncertainties below $10\%$. For the $\eta$ meson, we report on a preliminary determination from a single lattice spacing.

In this work, we present a numerical investigation of a novel Lorentz-covariant parametrization to extract $x$-dependent GPDs using off-forward matrix elements of momentum-boosted hadrons coupled to non-local operators. The novelty of the method is the implementation of an asymmetric frame for the momentum transfer between the initial and final hadron state and the parametrization of the matrix elements into Lorentz-invariant amplitudes. The amplitudes can then be related to the standard light-cone GPDs. GPDs are defined in the symmetric frame, which requires a separate calculation for each value of the momentum transfer, increasing the computational cost significantly. The proposed method is powerful, as one can extract the GPDs at multiple values of the momentum transfer at the computational cost of a single value. For this proof-of-concept calculation, we use one ensemble of $N_f=2+1+1$ twisted mass fermions and a clover improvement with a pion mass of 260 MeV to calculate the proton unpolarized GPDs., Comment: 10 pages, 5 figures, contribution to the 39th International Symposium on Lattice Field Theory, Lattice 2022, Bonn, Germany

Traditionally, lattice QCD computations of generalized parton distributions (GPDs) have been carried out in a symmetric frame, where the transferred momentum is symmetrically distributed between the incoming and outgoing hadrons. However, such frames are inconvenient since they require a separate calculation for each value of the momentum transfer, increasing significantly the computational cost. In this work, by focusing on the quasi-distribution approach, we lay the foundation for faster and more effective lattice QCD calculations of GPDs exploiting asymmetric frames, with freedom in the transferred momentum distribution. An important ingredient of our approach is the Lorentz covariant parameterization of the matrix elements in terms of Lorentz-invariant amplitudes, which allows one to relate matrix elements in different frames. We also use this amplitude approach to propose a new definition of quasi-GPDs that is frame-independent and, more importantly, may lead to smaller power corrections in the matching relations to the light-cone GPDs. We demonstrate the efficacy of the formalism through numerical calculations using one ensemble of $N_f$=2+1+1 twisted mass fermions with a clover improvement. The value of the light-quark masses lead to a pion mass of about 260 MeV. Concentrating on the proton, and limiting ourselves to a vanishing longitudinal momentum transfer to the target, we extract the invariant amplitudes from matrix element calculations in both the symmetric and asymmetric frame, and obtain results for the twist-2 light-cone GPDs for unpolarized quarks, that is, $H$ and $E$., Comment: 39 pages, 19 figures, matches with published version

Momentum and spatial distributions of quarks and gluons inside hadrons are typically encoded in the so-called generalized parton distributions (GPDs). GPDs are multi-dimensional quantities that are very challenging to extract, both experimentally and within lattice QCD. We present the first lattice results on the $x$-dependence of isovector unpolarized, helicity and transversity GPDs of the proton, obtained from lattice QCD using an ensemble of $N_f=2+1+1$ maximally twisted mass fermions, with pion mass $M_\pi=260$ MeV and lattice spacing $a\simeq 0.093$ fm. Our calculations use the quasi-distribution formalism and the final distributions are presented in the MS-bar scheme at a renormalization scale of 2 GeV., Comment: Invited talk at the Hadron 2021 conference. 7 pages, 4 figures

We present first results on selected twist-3 quark GPDs using the quasi-distribution method. This approach relates lattice QCD data and light-cone distribution functions using Large Momentum Effective Theory (LaMET). We calculate quark-antiquark correlators of boosted nucleons coupled to non-local operators with vector and axial Dirac structure, which is transverse to the momentum boost. We use three values of the momentum boost, namely 0.83, 1.25, 1.67 GeV. The GPDs are defined in the symmetric (Breit) frame, which we implement here with 4-vector momentum transfer squared of 0, 0.69 and 1.39 GeV 2 , all at zero skewness. The calculation is performed using one ensemble of two degenerate light, a strange and a charm quark ($N_f$ = 2 + 1 + 1) of maximally twisted mass fermions with a clover term, corresponding to a pion mass of 260 MeV.

We report on our computation of the pion transition form factor ${\cal F}_{P\rightarrow \gamma^*\gamma^*}$ from twisted mass lattice QCD in order to determine the numerically dominant light pseudoscalar pole contribution in the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon $a_\mu =(g-2)_\mu$. The pion transition form factor is computed directly at the physical point. We present first results for our estimate of the pion-pole contribution with kinematic setup for the pion at rest., Comment: 9 pages, 7 figures; Proceedings of the 38th International Symposium on Lattice Field Theory - Lattice 2021, Zoom/Gather@Massachusetts Institute of Technology

Generalized parton distributions (GPDs) are among the most fundamental quantities for describing the internal structure of hadrons. In this work, we present results on isovector GPDs of the proton obtained within lattice Quantum Chromodynamics. We use the quasi-distribution formalism and perform the calculations on an ensemble of $N_f = 2 + 1 + 1$ twisted mass fermions, with pion mass $M_\pi=260$~MeV and lattice spacing $a\simeq 0.093$~fm. Results are presented for unpolarized, helicity, and transversity GPDs at zero and nonzero skewness with controlled statistical uncertainties. Comparisons with their forward limit show qualitative features anticipated from model calculations., Comment: Talk at the 38th International Symposium on Lattice Field Theory (LATTICE 2021), 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

In these proceedings, we summarize the main results from the first-ever calculations of the chiral-even and chiral-odd twist-3 parton distributions, $g_T(x)$ and $h_L(x)$, of the proton from lattice QCD. We use an $N_f=2+1+1$ ensemble of maximally twisted mass fermions with a clover improvement. The lattice has a spatial extent of 3~fm, the lattice spacing is 0.093~fm, and the pion mass is $260$~MeV. The matrix elements are obtained with a source-sink time separation of 1.12~fm to control contamination from excited states. The calculation is based on the quasi-PDF approach and employs three values for the proton momentum: 0.83~GeV, 1.25~GeV, and 1.67~GeV. The lattice data are renormalized non-perturbatively using the RI$'$ scheme, and the final results are presented in the $\overline{\rm MS}$ scheme at the scale of 2~GeV. Furthermore, we compute in the same setup the helicity, $g_1(x)$, and transversity, $h_1(x)$, distributions, which are used to compare $g_T(x)$ and $h_L(x)$ to their Wandzura-Wilczek approximations. For $h_L(x)$, we combine results for the isovector and isoscalar flavor combinations to disentangle the individual up- and down-quark contributions., Comment: Talk at the 38th International Symposium on Lattice Field Theory (LATTICE 2021), 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We perform the first full decomposition of the pion momentum into its gluon and quark contributions. We employ an ensemble generated by the Extended Twisted Mass Collaboration with $N_f=2 + 1 +1$ Wilson twisted mass clover fermions at maximal twist tuned to reproduce the physical pion mass. We present our results in the $\overline{\mathrm{MS}}$ scheme at $2\gev$. We find $\avgx_{u+d}=0.601(28)$, $\avgx_s=0.059(13)$, $\avgx_c=0.019(05)$, and $\avgx_g=0.52(11)$ for the separate contributions, respectively, whose sum saturates the momentum sum rule.

We present the first calculation of the $x$-dependence of the isovector transversity generalized parton distributions (GPDs) for the proton within lattice QCD. We compute the matrix elements with non-local operators containing a Wilson line. The calculation implements the Breit symmetric frame. The proton momenta are chosen as $0.83,\,1.25,\,1.67$ GeV, and the values of the momentum transfer squared are $0.69,\,1.02$ GeV$^2$. These combinations include cases with zero and nonzero skewness. The calculation is performed using one ensemble of two degenerate-mass light, a strange and a charm quark of maximally twisted mass fermions with a clover term. The lattice results are renormalized non-perturbatively and finally matched to the light-cone GPDs using one-loop perturbation theory within the framework of large momentum effective theory. The final GPDs are given in the $\overline{\rm MS}$ scheme at a scale of 2 GeV. In addition to the individual GPDs, we form the combination of the transversity GPDs that is related to the transverse spin structure of the proton. Finally, we extract the lowest two moments of GPDs and draw a number of important qualitative conclusions., Comment: 24 pages, 11 figures, version accepted in Phys. Rev. D

Twist-3 partonic distributions contain important information that characterizes nucleon's structure. In this work, we show our lattice exploration of the twist-3 PDFs $g_T(x)$, and $h_L(x)$. We also present our preliminary results on the twist-3 GPD $\tilde{G}_2(x)$. We use the quasi-distribution approach to connect the lattice-extracted matrix elements, renormalized in the RI/MOM scheme, to light-cone distributions, applying the matching procedure that we developed in parallel. We also calculate the twist-2 counterparts of $g_T(x)$ and $h_L(x)$, i.e. $g_1(x)$, and $h_1(x)$, and test the Wandzura-Wilczek approximation., Comment: 6 pages, 4 figures, submission to SciPost Proceedings (DIS 2021)

We report the first-ever calculation of the isovector flavor combination of the chiral-odd twist-3 parton distribution $h_L(x)$ for the proton from lattice QCD. We employ gauge configurations with two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with a clover improvement. The lattice has a spatial extent of 3 fm and lattice spacing of 0.093 fm. The values of the quark masses lead to a pion mass of $260$ MeV. We use a source-sink time separation of 1.12 fm to control contamination from excited states. Our calculation is based on the quasi-distribution approach, with three values for the proton momentum: 0.83 GeV, 1.25 GeV, and 1.67 GeV. The lattice data are renormalized non-perturbatively using the RI$'$ scheme, and the final result for $h_L(x)$ is presented in the $\overline{\rm MS}$ scheme at the scale of 2 GeV. Furthermore, we compute in the same setup the transversity distribution, $h_1(x)$, which allows us, in particular, to compare $h_L(x)$ to its Wandzura-Wilczek approximation. We also combine results for the isovector and isoscalar flavor combinations to disentangle the individual quark contributions for $h_1(x)$ and $h_L(x)$, and address the Wandzura-Wilczek approximation in that case as well., Comment: 18 pages, 13 figures

In this work, we perform a lattice QCD study of the intrinsic, rapidity-independent soft function within the framework of large momentum effective theory. The computation is carried out using a gauge ensemble of $N_f=2+1+1$ clover-improved twisted mass fermion. After applying an appropriate renormalization procedure and the removal of significant higher-twist contamination, we obtain the intrinsic soft function that is comparable to the one-loop perturbative result at large external momentum. The determination of the nonperturbative soft function from first principles is crucial to sharpen our understanding of the processes with small transverse momentum such as the Drell-Yan production and the semi-inclusive deep inelastic scattering. Additionally, we calculate the Collins-Soper evolution kernel using the quasi-transverse-momentum-dependent wave function as input., Comment: 7 pages, 4 figures

The quasi-PDF approach provides a path to computing parton distribution functions (PDFs) using lattice QCD. This approach requires matrix elements of a power-divergent operator in a nucleon at high momentum and one generically expects discretization effects starting at first order in the lattice spacing $a$. Therefore, it is important to demonstrate that the continuum limit can be reliably taken and to understand the size and shape of lattice artifacts. In this work, we report a calculation of isovector unpolarized and helicity PDFs using lattice ensembles with $N_f=2+1+1$ Wilson twisted mass fermions, a pion mass of approximately 370 MeV, and three different lattice spacings. Our results show a significant dependence on $a$, and the continuum extrapolation produces a better agreement with phenomenology. The latter is particularly true for the antiquark distribution at small momentum fraction $x$, where the extrapolation changes its sign., Comment: 24 pages, 21 figures. v2: additional references, discussion of systematics, and position-space comparison with phenomenology. v3: applied workaround for problem affecting Greek symbols in figures with some PDF viewers

We present the first calculation of the $x$-dependence of the proton generalized parton distributions (GPDs) within lattice QCD. Results are obtained for the isovector unpolarized and helicity GPDs. We compute the appropriate matrix elements of fast-moving protons coupled to non-local operators containing a Wilson line. We present results for proton momenta $0.83,\,1.25,\,1.67$ GeV, and momentum transfer squared $0.69,\,1.38$ GeV$^2$. These combinations include cases with zero and nonzero skewness. The calculation is performed using one ensemble of two degenerate mass light, a strange and a charm quark of maximally twisted mass fermions with a clover term. The lattice results are matched to the light-cone GPDs using one-loop perturbation theory within the framework of large momentum effective theory. The final GPDs are given in the $\overline{\rm MS}$ scheme at a scale of 2 GeV., Comment: Main document: 5 pages, Supplementary document (Appendix): 10 pages. Version accepted for publication in Physical Review Letters

The perturbative procedure of matching was proposed to connect parton quasi-distributions that are calculable in lattice QCD to the corresponding light-cone distributions which enter physical processes. Such a matching procedure has so far been limited to the twist-2 distributions. Recently, we addressed the matching for the twist-3 PDF $g_T(x)$. In this work, we extend our perturbative calculations to the remaining twist-3 PDFs, $e(x)$ and $h_{L}(x)$. In particular, we discuss the non-trivialities involved in the calculation of the singular zero-mode contributions for the quasi-PDFs., Comment: 20 pages, 2 figures, results for zero-mode contributions partly revised, associated discussion modified

Perturbative matching relates the parton quasi-distributions, defined by Euclidean correlators at finite hadron momenta, to the light-cone distributions which are accessible in experiments. Previous matching calculations have exclusively focused on twist-2 distributions. In this work, we address, for the first time, the one-loop matching for the twist-3 parton distribution function $g_T(x)$. The results have been obtained using three different infrared regulators, while dimensional regularization has been adopted to deal with the ultraviolet divergences. We present the renormalized expressions of the matching coefficient for $g_{T}(x)$ in the $\overline{\rm MS}$ and modified $\overline{\rm MS}$ schemes. We also discuss the role played by a zero-mode contribution. Our results have already been used for the extraction of $g_T(x)$ from lattice QCD calculations., Comment: 16 pages, 2 figures; V2: minor changes, PRD version; V3: results for zero-mode contribution partly revised

In this work, we present the first-ever calculation of the isovector flavor combination of the twist-3 parton distribution function $g_T(x)$ for the proton from lattice QCD. We use an ensemble of gauge configurations with two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with a clover improvement. The lattice has a spatial extent of 3~fm, lattice spacing of 0.093~fm, and reproduces a pion mass of $260$ MeV. We use the quasi-distribution approach and employ three values of the proton momentum boost, 0.83 GeV, 1.25 GeV, and 1.67 GeV. We use a source-sink separation of 1.12~fm to suppress excited-states contamination. The lattice data are renormalized non-perturbatively. We calculate the matching equation within Large Momentum Effective Theory, which is applied to the lattice data in order to obtain $g_T$. The final distribution is presented in the $\overline{\rm MS}$ scheme at a scale of 2 GeV. We also calculate the helicity distribution $g_1$ to test the Wandzura-Wilczek approximation for $g_T$. We find that the approximation works well for a broad range of $x$. This work demonstrates the feasibility of accessing twist-3 parton distribution functions from novel methods within lattice QCD and can provide essential insights into the structure of hadrons., Comment: 10 pages, 6 figures, version accepted for publication

We perform a first calculation for the unpolarized parton distribution function of the $\Delta^+$ baryon using lattice QCD simulations within the framework of Large Momentum Effective Theory. Two ensembles of $N_f=2+1+1$ twisted mass fermions are utilized with a pion mass of 270 MeV and 360 MeV, respectively. The baryon, which is treated as a stable single-particle state, is boosted with momentum $P_3$ with values $\{0.42,0.83,1.25\}$ GeV, and we utilize momentum smearing to improve the signal. The unpolarized parton distribution function of $\Delta^+$ is obtained using a non-perturbative renormalization and a one-loop formula for the matching, with encouraging precision. In particular, we compute the $\overline{d}(x)-\overline{u}(x)$ asymmetry and compare it with the same quantity in the nucleon, in a first attempt towards resolving the physical mechanism responsible for generating such asymmetry.

Nonlocal quark bilinear operators connected by link paths are used for studying parton distribution functions (PDFs) and transverse momentum-dependent PDFs of hadrons using lattice QCD. The nonlocality makes it difficult to understand the renormalization and improvement of these operators using standard methods. In previous work, we showed that by introducing an auxiliary field on the lattice, one can understand an on-axis Wilson-line operator as the product of two local operators in an extended theory. In this paper, we provide details about the calculation in perturbation theory of the factor for conversion from our lattice-suitable renormalization scheme to the MS-bar scheme. Extending our work, we study Symanzik improvement of the extended theory to understand the pattern of discretization effects linear in the lattice spacing, $a$, which are present even if the lattice fermion action exactly preserves chiral symmetry. This provides a prospect for an eventual $O(a)$ improvement of lattice calculations of PDFs. We also generalize our approach to apply to Wilson lines along lattice diagonals and to piecewise-straight link paths., Comment: 15 pages, 4 figures. v2: minor errors corrected

We discuss the recent progress in extracting partonic functions from the quasi-distribution approach, using twisted mass fermions. This concerns, among others, the investigation of several sources of systematic effects. Their careful analysis is a prerequisite to obtain precise determinations of PDFs from the lattice with realistic estimates of all uncertainties. In these proceedings, we shortly discuss systematic effects in the matching procedure. Moreover, we present preliminary results from our new simulations at the physical point. They involve, additionally, the dynamical strange and charm quarks, as well as a larger volume and a smaller lattice spacing than in our previous computations. In addition, we show first results from computations of generalized parton distributions (GPDs) in the quasi-distribution framework., Comment: 7 pages, 4 figures, proceedings of the 37th International Symposium on Lattice Field Theory - Lattice2019, Wuhan; v2 - references updated

We present results for renormalized matrix elements related to the unpolarized quasi-distribution function of the $\Delta^+$ baryon making use of the large momentum effective theory. Two ensembles of $N_f=2+1+1$ twisted mass fermions with a clover term and pion masses of 250 MeV and 330 MeV are analyzed. We employ momentum smearing to improve the overlap with the boosted $\Delta$ state significantly reducing in this way the statistical error of both two- and three-point functions., Comment: 7 pages, 4 figures, conference

We present a detailed study of the helicity-dependent and helicity-independent collinear parton distribution functions (PDFs) of the nucleon, using the quasi-PDF approach. The lattice QCD computation is performed employing twisted mass fermions with a physical value of the light quark mass. We give a systematic and in-depth account of the salient features entering in the evaluation of quasi-PDFs and their relation to the light-cone PDFs. In particular, we give details for the computation of the matrix elements, including the study of the various sources of systematic uncertainties, such as excited states contamination. In addition, we discuss the non-perturbative renormalization scheme used here and its systematics, effects of truncating the Fourier transform and different matching prescriptions., Comment: 47 pages, 33 figures, 6 tables

We present results for the flavor non-singlet u-d parton distribution functions within the nucleon using the quasi-PDF approach. The lattice calculation is performed by employing the twisted mass formulation and two gauge ensembles, having $N_f = 2$ and $N_f = 2 + 1 + 1$ dynamical fermions with masses tuned to their physical value. For the $N_f = 2$ physical point ensemble, the unpolarized, helicity and transversity distributions are computed for three values of the nucleon momentum, namely [6, 8, 10]$\pi/L$ corresponding to [0.83,1.11,1.38] GeV. Upon renormalization, we find that, as the nucleon momentum increases, the lattice results approach the phenomenological distributions resulting from analyses of deep inelastic scattering data, opening a promising path for a direct evaluation of parton distributions from the QCD Lagrangian. For the $N_f = 2 + 1 + 1$ physical point ensemble, we present preliminary results for the unpolarized distribution extracted from a nucleon boosted by $8\pi/L$ or 0.97 GeV., Comment: 7 pages, 6 figures, The 36th International Symposium on Lattice Field Theory (Lattice 2018)

Using the approach proposed a few years ago by X. Ji, it has become feasible to extract parton distribution functions (PDFs) from lattice QCD, a task thought to be extremely difficult before Ji's proposal. In this talk, we discuss this approach, in particular different systematic effects that need to be controlled to ultimately have precise determinations of PDFs. Special attention is paid to the analysis of excited states. We emphasize that it is crucial to control excited states contamination and we show an analysis thereof for our lattice data, used to calculate quasi-PDFs and finally light-cone PDFs in the second part of this proceeding (C. Alexandrou et al., Quasi-PDFs from Twisted mass fermions at the physical point)., Comment: Proceedings of the The 36th Annual International Symposium on Lattice Field Theory LATTICE 2018. 7 pages, 1 figure

The discovery of a sizeable asymmetry in the $\bar u$ and $\bar d$ distributions in the proton was one of the more consequential experimental findings in hadron physics last century. Although widely believed to be related to the fundamental role of chiral symmetry in QCD, a definitive verification of this hypothesis has remained elusive. We propose a novel test of the role of chiral symmetry in generating the sea flavor asymmetry by comparing the $\bar d-\bar u$ content in the proton with that in the $\Delta^+$ baryon, where a significant enhancement is expected around the opening of the $N \pi$ decay channel. Recent developments in lattice QCD suggest a promising way to test this prediction in the near future., Comment: 5 pages, 5 figures

We present the first direct calculation of the transversity parton distribution function within the nucleon from lattice QCD. The calculation is performed using simulations with the light quark mass fixed to its physical value and at one value of the lattice spacing. Novel elements of the calculations are non-perturbative renormalization and extraction of a formula for the matching to light-cone PDFs. Final results are presented in the $\overline{\rm MS}$ scheme at a scale of $\sqrt{2}$ GeV., Comment: 6 pages, 3 figures, 1 table

We present the unpolarized and helicity parton distribution functions calculated within lattice QCD simulations using physical values of the light quark mass. Non-perturbative renormalization is employed and the lattice data are converted to the MSbar-scheme at a scale of 2 GeV. A matching process is applied together with target mass corrections leading to the reconstruction of light-cone parton distribution functions. For both cases we find a similar behavior between the lattice and phenomenological data, and for the polarized PDF a nice overlap for a range of Bjorken-x values. This presents a major success for the emerging field of direct calculations of quark distributions using lattice QCD., Comment: 6 pages, 6 figures, 1 table

We show the first results for parton distribution functions within the proton at the physical pion mass, employing the method of quasi-distributions. In particular, we present the matrix elements for the iso-vector combination of the unpolarized, helicity and transversity quasi-distributions, obtained with $N_f=2$ twisted mass clover-improved fermions and a proton boosted with momentum $|\vec{p}|=0.83\mbox{ GeV}$. The momentum smearing technique has been applied to improve the overlap with the proton boosted state. Moreover, we present the renormalized helicity matrix elements in the RI$'$ scheme, following the non-perturbative renormalization prescription recently developed by our group., Comment: 8 pages, 6 figures; Proceedings of the 35th International Symposium on Lattice Field Theory, Granada, Spain

We discuss the current developments by the European Twisted Mass Collaboration in extracting parton distribution functions from the quasi-PDF approach. We concentrate on the non-perturbative renormalization prescription recently developed by us, using the RI$'$ scheme. We show results for the renormalization functions of matrix elements needed for the computation of quasi-PDFs, including the conversion to the $\overline{\rm MS}$ scheme, and for renormalized matrix elements. We discuss the systematic effects present in the $Z$-factors and the possible ways of addressing them in the future., Comment: 8 pages, 3 figures; Proceedings of the 35th International Symposium on Lattice Field Theory, Granada, Spain

Quasi-PDFs provide a path toward an ab initio calculation of parton distribution functions (PDFs) using lattice QCD. One of the problems faced in calculations of quasi-PDFs is the renormalization of a nonlocal operator. By introducing an auxiliary field, we can replace the nonlocal operator with a pair of local operators in an extended theory. On the lattice, this is closely related to the static quark theory. In this approach, we show how to understand the pattern of mixing that is allowed by chiral symmetry breaking, and obtain a master formula for renormalizing the nonlocal operator that depends on three parameters. We present an approach for nonperturbatively determining these parameters and use perturbation theory to convert to the MS-bar scheme. Renormalization parameters are obtained for two lattice spacings using Wilson twisted mass fermions and for different discretizations of the Wilson line in the nonlocal operator. Using these parameters we show the effect of renormalization on nucleon matrix elements with pion mass approximately 370 MeV, and compare renormalized results for the two lattice spacings. The renormalized matrix elements are consistent among the different Wilson line discretizations and lattice spacings., Comment: 6 pages, 6 figures. v2: added perturbative matching to MS-bar and additional references