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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., 22 pages, 17 figures

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 scheme, RI/MOM, 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: 18 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

We outline the physics opportunities provided by the Electron Ion Collider (EIC). These include the study of the parton structure of the nucleon and nuclei, the onset of gluon saturation, the production of jets and heavy flavor, hadron spectroscopy and tests of fundamental symmetries. We review the present status and future challenges in EIC theory that have to be addressed in order to realize this ambitious and impactful physics program, including how to engage a diverse and inclusive workforce. In order to address these many-fold challenges, we propose a coordinated effort involving theory groups with differing expertise is needed. We discuss the scientific goals and scope of such an EIC Theory Alliance., Comment: 44 pages, ReVTeX, White Paper on EIC Theory Alliance

39th International Symposium on Lattice Field Theory, Lattice 2022, Bonn, Germany, 8 Aug 2022 - 13 Aug 2022; Proceedings of Science / International School for Advanced Studies (LATTICE2022), 123 (2023). doi:10.22323/1.430.0123, We present an exploratory study of the quasi-beam function using an $N_f = 2 + 1 + 1$ twisted mass lattice of size $24^3 \times 48$, with a pion mass of $350$ MeV and of lattice spacing $0.093$ fm. We show results for longitudinal momentum of $1.7$ GeV and transverse separation of up to $0.18$ fm. We also study the non-perturbative renormalization of the bare matrix element using the RI/MOM scheme., Published by SISSA, Trieste

This handbook provides a comprehensive review of transverse-momentum-dependent parton distribution functions and fragmentation functions, commonly referred to as transverse momentum distributions (TMDs). TMDs describe the distribution of partons inside the proton and other hadrons with respect to both their longitudinal and transverse momenta. They provide unique insight into the internal momentum and spin structure of hadrons, and are a key ingredient in the description of many collider physics cross sections. Understanding TMDs requires a combination of theoretical techniques from quantum field theory, nonperturbative calculations using lattice QCD, and phenomenological analysis of experimental data. The handbook covers a wide range of topics, from theoretical foundations to experimental analyses, as well as recent developments and future directions. It is intended to provide an essential reference for researchers and graduate students interested in understanding the structure of hadrons and the dynamics of partons in high energy collisions., 471 pages, many 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

We present results of the x-dependence of the unpolarized gluon PDF for the proton. We use an $N_f=2+1+1$ ensemble of maximally twisted mass fermions with clover improvement and the Iwasaki improved gluon action. The quark masses are tuned so that the pion mass is 260 MeV. We use a $32^3\times64$ lattice size with a lattice spacing $a=0.093$ fm giving a spatial extent of 3 fm. We employ the pseudo-distribution approach and obtain the light-cone Ioffe time distribution (ITD) combining data for nucleon momentum boosts up to 1.67 GeV and Wilson line lengths, $z$, up to 0.56 fm. We explore systematic effects such as the dependence on the maximum value of $z$ entering the fits to obtain the gluon PDF.

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.

38th International Symposium on Lattice Field Theory, Lattice 2021, Online, United States, 26 Jul 2021 - 30 Jul 2021; Proceedings of Science / International School for Advanced Studies (LATTICE2021), 348 (2022). doi:10.22323/1.396.0348, The gluon and quark contributions to the proton spin are computed using an ensemble of gauge configurations with twisted mass Wilson fermions with masses tuned to approximately reproduce their physical values. A non-perturbative renormalization is carried out for both quark and gluon matrix elements. The contribution of up, down, strange and charm quarks to the proton intrinsic spin, orbital angular momentum and angular momentum are determined. All scale and scheme dependent quantities are given in the $\mathrm{ \overline{MS}}$ scheme at 2~GeV., Published by SISSA, Trieste

In this document, we summarize the status and challenges of calculating parton physics in lattice QCD for the US Particle Physics Community Planning Exercise (a.k.a. "Snowmass"). While PDF-moments calculations have been very successful and been continuously improved, new methods have been developed to calculate distributions directly in $x$-space. Many recent lattice studies have been focused on calculating isovector PDFs of the pion and nucleon, learning to control systematics associated with excited-state contamination, renormalization and continuum extrapolations, pion-mass and finite-volume effects, etc. Although in some cases, the lattice results are already competitive with experimental data, to reach the level of precision in a wide range of $x$ for unpolarized nucleon PDFs impactful for future collider physics remains a challenge, and may require exascale supercomputing power. The new theoretical methods open the door for calculating other partonic observables which will be the focus of the experimental program in nuclear physics, including generalized parton distributions and transverse-momentum dependent PDFs. A fruitful interplay between experimental data and lattice-QCD calculations will usher in a new era for parton physics and hadron structure., Comment: 14 pages

19th International Conference on Hadron Spectroscopy and Structure, HADRON 2021, Mexico City, Mexico, 26 Jul 2021 - 1 Aug 2021; Suplemento de la revista mexicana de física 3(3), 0308104 (2022). doi:10.31349/SuplRevMexFis.3.0308104, 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., Published by Sociedad Mexicana de Física, Ciudad de México

38th International Symposium on Lattice Field Theory, Lattice 2021, Online, United States, 26 Jul 2021 - 30 Jul 2021; Proceedings of Science / International School for Advanced Studies (LATTICE2021), 129 (2022). doi:10.22323/1.396.0129, 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., Published by SISSA, Trieste

38th International Symposium on Lattice Field Theory , (LATTICE2021), Zoom/Gather@Massachusetts Institute of Technology, USA, 26 Jul 2021 - 30 Jul 2021; Proceedings of Science / International School for Advanced Studies (LATTICE2021), 250 (2022). doi:10.22323/1.396.0250, We present the nucleon axial and electromagnetic form factors using $N_\textrm{f}$=2+1+1 ensembles of twisted mass fermions with clover improvement and with masses tuned to their physical values. Excited state effects are studied using several sink-source time separations in the range 0.8 fm - 1.6 fm, exponentially increasing statistics with the separation such that statistical errors remain approximately constant. In addition, quark loop disconnected diagrams are included in order to extract the isoscalar axial form factors and the proton and neutron electromagnetic form factors, as well as their strange-quark contributions. The radii and moments are extracted by modelling the $Q^2$ dependence, including using the so-called $z$-expansion. A preliminary assessment of lattice cut-off effects is presented using two lattice spacings directly at the physical point., Published by SISSA, Trieste

In this article, we review recent lattice calculations on the $x$-dependence of PDFs and GPDs from lattice QCD., Proceedings of plenary talk at ISMD 2021, 12-16 July 2021

We present a calculation of the pion and kaon Mellin moment ⟨x3⟩ extracted directly in lattice QCD using a three-derivative local operator. We use one ensemble of gauge configurations with two degenerate light, a strange and a charm quark (Nf=2+1+1) of maximally twisted mass fermions with clover improvement. The ensemble reproduces a pion mass ∼260 MeV, and a kaon mass ∼530 MeV. Excited-states contamination is evaluated using four values of the source-sink time separation within the range of 1.12–1.67 fm. We use an operator that is free of mixing, and apply a multiplicative renormalization function calculated nonperturbatively. Our results are converted to the MS¯ scheme and evolved at a scale of 2 GeV, using three-loop expressions in perturbation theory. The final values are ⟨x3⟩πu+=0.024(18)stat(2)syst, ⟨x3⟩Ku+=0.035(6)stat(3)syst, and ⟨x3⟩Ks+=0.075(5)stat(1)syst, where the systematic error is the uncertainty due to excited state contamination. We combine ⟨x3⟩ with the two lower moments, namely ⟨x⟩ and ⟨x2⟩, to obtain the ratios ⟨x3⟩/⟨x⟩ and ⟨x3⟩/⟨x2⟩, as well as ⟨x3⟩Ku+/⟨x3⟩πu+ and ⟨x3⟩Ku+/⟨x3⟩πu+. In addition, we reconstruct the x-dependence of the pion and kaon PDFs via 2- and 3-parameter fits to our results. We find that the reconstruction is feasible and that our lattice data favor a large x-dependence that falls as (1−x)2 for both the pion and kaon PDFs. We integrate the reconstructed PDFs to extract the higher moments with 4≤n≤6. Finally, we compare the pion and kaon PDFs, as well as the ratios of their moments, to address the effect of SU(3) flavor symmetry breaking.

We present a calculation of the pion and kaon Mellin moment $\langle x^3 \rangle$ extracted directly in lattice QCD using a three-derivative local operator. We use one 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 clover improvement. The ensemble reproduces a pion mass $\sim260$ MeV, and a kaon mass $\sim530$ MeV. Excited-states contamination is evaluated using four values of the source-sink time separation within the range of $1.12-1.67$ fm. We use an operator that is free of mixing, and apply a multiplicative renormalization function calculated non-perturbatively. Our results are converted to the $\overline{\rm MS}$ scheme and evolved at a scale of 2 GeV, using three-loop expressions in perturbation theory. The final values are $\langle x^3 \rangle_\pi^{u^+}=0.024(18)_{\rm stat}(2)_{\rm syst}$, $\langle x^3 \rangle_K^{u^+}=0.035(6)_{\rm stat}(3)_{\rm syst}$, and $\langle x^3 \rangle_K^{s^+}=0.075(5)_{\rm stat}(1)_{\rm syst}$, where the systematic error is the uncertainty due to excited state contamination. We combine $\langle x^3 \rangle$ with the two lower moments to obtain the ratios $\langle x^3 \rangle/\langle x \rangle$ and $\langle x^3 \rangle/\langle x^2 \rangle$, as well as ratios between the pion and kaon moments. In addition, we reconstruct the $x$-dependence of the pion and kaon PDFs via 2- and 3-parameter fits to our results. We find that the reconstruction is feasible and that our lattice data favor a large $x$-dependence that falls as $(1-x)^2$ for both the pion and kaon PDFs. We integrate the reconstructed PDFs to extract the higher moments with $4\leq n\leq 6$. Finally, we compare the pion and kaon PDFs, as well as the ratios of their moments, to address the effect of SU(3) flavor symmetry breaking., Comment: 24 pages, 12 figures

In this article, we review recent lattice calculations on the $x$-dependence of parton distributions, with the latter providing information on hadron structure. These calculations are based on matrix elements of boosted hadrons coupled to non-local operators and can be related to the standard, light-cone distribution functions via an appropriate factorization formalism. There is significant progress in several directions, including calculations of flavor singlet parton distribution functions (PDFs), first calculations of generalized parton distributions (GPDs), as well as the implementation of some of the approaches for the transverse-momentum-dependent PDFs (TMD PDFs). This new direction of lattice calculations is particularly interesting for phenomenological fits on experimental data sets, as the lattice results can help to improve the constraints on the distribution functions., Invited article in lieu of a plenary talk at the 38th International Symposium on Lattice Field Theory (Lattice 2020). Prepared for the EPJA special issue "Lattice Field Theory during the COVID-19 pandemic". (25 pages)

We present a calculation of the pion quark momentum fraction, ⟨x⟩, and its third Mellin moment ⟨x2⟩. We also obtain directly, for the first time, ⟨x⟩ and ⟨x2⟩ for the kaon using local operators. We use an ensemble of two degenerate light, a strange, and a charm quark (Nf=2+1+1) of maximally twisted mass fermions with clover improvement. The quark masses are chosen so that they reproduce a pion mass of about 260 MeV and a kaon mass of 530 MeV. The lattice spacing of the ensemble is 0.093 fm and the lattice has a spatial extent of 3 fm. We analyze several values of the source-sink time separation within the range of 1.12–2.23 fm to study and eliminate excited-state contributions. The necessary renormalization functions are calculated nonperturbatively in the RI′ scheme and are converted to the modified minimal subtraction scheme at a scale of 2 GeV. The final values for the momentum fraction are ⟨x⟩u+π=0.261(3)stat(6)syst, ⟨x⟩u+K=0.246(2)stat(2)syst, and ⟨x⟩s+K=0.317(2)stat(1)syst. For the third Mellin moments, we find ⟨x2⟩u+π=0.082(21)stat(17)syst, ⟨x2⟩u+K=0.093(5)stat(3)syst, and ⟨x2⟩s+K=0.134(5)stat(2)syst. The reported systematic uncertainties are due to excited-state contamination. We also give the ratio ⟨x2⟩/⟨x⟩ which is an indication of how quickly the parton distribution functions lose support at large x.

We present preliminary results of a calculation of the QCD renormalization constants (RCs) for quark bilinear operators, evaluated non-perturbatively on the lattice in the RI'-MOM scheme. The calculation is performed by using dedicated ensembles with $N_f=4$ degenerate dynamical twisted mass (clover) fermions and the Iwasaki gauge action. A detailed analysis is reported, with emphasis on the control or subtraction of the hadronic contaminations occurring in the lattice estimators of RCs and a check of proper scaling with $a^2$ of the final results. Such a careful study of systematic errors is the counterpart of the high statistical precision reached by current calculations of RCs in the RI'-MOM scheme and is important in order to quote accurate results in phenomenological applications, such as the computation of quark masses., Comment: 10 pages, 6 figures, 3 tables. Talk given at the 38th International Symposium on Lattice Field Theory (LATTICE2021)

We present a calculation of the scalar, vector, and tensor form factors for the pion and kaon in lattice QCD. We use an ensemble of two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with clover improvement. The corresponding pion and kaon masses are about 265 MeV and 530 MeV, respectively. The calculation is done in both rest and boosted frames obtaining data for four-vector momentum transfer squared up to $-q^2=2.5$ GeV$^2$ for the pion and 3 GeV$^2$ for the kaon. The excited-states effects are studied by analyzing six values of the source-sink time separation for the rest frame ($1.12-2.23$ fm) and for four values for the boosted frame ($1.12-1.67$ fm). The lattice data are renormalized non-perturbatively and the results for the scheme- and scale-dependent scalar and tensor form factors are presented in the $\overline{\rm MS}$ scheme at a scale of 2 GeV. We apply different parametrizations to describe $q^2$-dependence of the form factors to extract the scalar, vector, and tensor radii, as well as the tensor anomalous magnetic moment. We compare the pion and kaon form factors to study SU(3) flavor symmetry breaking effects. By combining the data for the vector and tensor form factors we also obtain the lowest moment of the densities of transversely polarized quarks in the impact parameter space. Finally, we give an estimate for the average transverse shift in the $y$ direction for polarized quarks in the $x$ direction., Comment: 28 pages, 23 pages, 9 tables

We present, for the first time, an ab initio calculation of the individual up, down, and strange quark helicity parton distribution functions for the proton. The calculation is performed within the twisted mass clover-improved fermion formulation of lattice QCD. The analysis is performed using one ensemble of dynamical up, down, strange, and charm quarks with a pion mass of 260 MeV. The lattice matrix elements are nonperturbatively renormalized and the final results are presented in the $\overline{MS}$ scheme at a scale of 2 GeV. We give results for $��u^+(x), ��d^+(x) ��u^-(x), ��d^-(x)$, including disconnected quark loop contributions, as well as for $��s^+(x)$ and $��s^-(x)$. For the latter we achieve unprecedented precision compared to the phenomenological estimates.

We present a calculation of the connected-diagram contributions to the first three non-trivial Mellin moments for the pion and kaon, extracted using local operators with up to 3 covariant derivatives. We use one 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 clover improvement. The ensemble has a pion mass $\sim$260 MeV, and a kaon mass $\sim$530 MeV. We reconstruct the $x$-dependence of the PDFs via fits to our results, and find that our lattice data favor a $(1-x)^2$-behavior in the large-$x$ region for both the pion and kaon PDFs. We integrate the reconstructed PDFs to extract the higher moments, $\langle x^n \rangle$, with $4 \leq n \leq 6$. Finally, we compare the pion and kaon PDFs, as well as the ratios of their Mellin moments, to address the effect of SU(3) flavor symmetry breaking., Comment: 11 pages, 5 figures, Proceedings of the 38th Annual International Symposium on Lattice Field Theory - LATTICE2021

We present a calculation of the pion quark momentum fraction, $\langle x \rangle$, and its third Mellin moment $\langle x^2 \rangle$. We also obtain directly, for the first time, $\langle x \rangle$ and $\langle x^2 \rangle$ for the kaon using local operators. We use an ensemble of two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with clover improvement. The quark masses are chosen so that they reproduce a pion mass of about 260 MeV, and a kaon mass of 530 MeV. The lattice spacing of the ensemble is 0.093 fm and the lattice has a spatial extent of 3 fm. We analyze several values of the source-sink time separation within the range of $1.12-2.23$ fm to study and eliminate excited-states contributions. The necessary renormalization functions are calculated non-perturbatively in the RI$'$ scheme, and are converted to the $\overline{\rm MS}$ scheme at a scale of 2 GeV. The final values for the momentum fraction are $\langle x \rangle^\pi_{u^+}=0.261(3)_{\rm stat}(6)_{\rm syst}$, $\langle x \rangle^K_{u^+}=0.246(2)_{\rm stat}(2)_{\rm syst}$, and $\langle x \rangle^K_{s^+}=0.317(2)_{\rm stat}(1)_{\rm syst}$. For the third Mellin moments we find $\langle x^2 \rangle^\pi_{u^+}=0.082(21)_{\rm stat}(17)_{\rm syst}$, $\langle x^2 \rangle^K_{u^+}=0.093(5)_{\rm stat}(3)_{\rm syst}$, and $\langle x^2 \rangle^K_{s^+}=0.134(5)_{\rm stat}(2)_{\rm syst}$. The reported systematic uncertainties are due to excited-state contamination. We also give the ratio $\langle x^2 \rangle/\langle x \rangle$ which is an indication of how quickly the PDFs lose support at large $x$., Comment: 21 pages, 10 figures

Perturbative matching relates the parton quasidistributions, 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 gT(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 gT(x) in the MS¯ and modified MS¯ schemes. We also discuss the role played by a zero-mode contribution. Our results have already been used for the extraction of gT(x) from lattice QCD calculations.

Physical review / D 103(9), 094512 (2021). doi:10.1103/PhysRevD.103.094512, 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., Published by Inst., Melville, NY

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

Physical review / D 99(11), 114504 (2019). doi:10.1103/PhysRevD.99.114504, 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 nonperturbative renormalization scheme used here and its systematics, effects of truncating the Fourier transform and different matching prescriptions. Finally, we show improved results for the PDFs and discuss future directions, challenges and prospects for evaluating precisely PDFs from lattice QCD with fully quantified uncertainties., Published by APS, College Park, MD

We present results for renormalized matrix elements related to the unpolarized quasi-distribution function of the $��^+$ 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 $��$ state significantly reducing in this way the statistical error of both two- and three-point functions., 7 pages, 4 figures, conference

Within the theory of Quantum Chromodynamics (QCD), the rich structure of hadrons can be quantitatively characterized, among others, using a basis of universal nonperturbative functions: parton distribution functions (PDFs), generalized parton distributions (GPDs), transverse momentum dependent parton distributions (TMDs), and distribution amplitudes (DAs). For more than half a century, there has been a joint experimental and theoretical effort to obtain these partonic functions. However, the complexity of the strong interactions has placed severe limitations, and first-principle information on these distributions was extracted mostly from their moments computed in Lattice QCD. Recently, breakthrough ideas changed the landscape and several approaches were proposed to access the distributions themselves on the lattice. In this paper, we review in considerable detail approaches directly related to partonic distributions. We highlight a recent idea proposed by X. Ji on extracting quasidistributions that spawned renewed interest in the whole field and sparked the largest amount of numerical studies within Lattice QCD. We discuss theoretical and practical developments, including challenges that had to be overcome, with some yet to be handled. We also review numerical results, including a discussion based on evolving understanding of the underlying concepts and the theoretical and practical progress. Particular attention is given to important aspects that validated the quasidistribution approach, such as renormalization, matching to light-cone distributions, and lattice techniques. In addition to a thorough discussion of quasidistributions, we consider other approaches: hadronic tensor, auxiliary quark methods, pseudodistributions, OPE without OPE, and good lattice cross-sections. In the last part of the paper, we provide a summary and prospects of the field, with emphasis on the necessary conditions to obtain results with controlled uncertainties.

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., 6 pages, 6 figures, 1 table

Physical review letters 121(11), 112001 (2018). doi:10.1103/PhysRevLett.121.112001, We extract parton distribution functions (PDFs) of the nucleon from lattice QCD using an ensemble of gauge field configurations simulated with light quark masses fixed to their physical values. Theoretical and algorithmic improvements that allow such a calculation include momentum smearing to reach large nucleon boosts with reduced statistical errors, nonperturbative renormalization, target mass corrections, and a novel modified matching of lattice QCD results to connect to what is extracted from experimental measurements. We give results on the unpolarized and helicity PDFs in the modified minimal subtraction scheme at a scale of 2 GeV and reproduce the main features of the experimentally determined quantities, showing an overlap for a range of Bjorken-x values. This first direct nonperturbative evaluation opens a most promising path to compute PDFs in an ab initio way on the lattice and provides a framework for investigating also a wider class of similar quantities, which require the evaluation of hadronic matrix elements of nonlocal operators., Published by APS, College Park, Md.

Physical review / D 97(7), 074503 (2018). doi:10.1103/PhysRevD.97.074503, We compare lattice QCD determinations of topological susceptibility using a gluonic definition from the gradient flow and a fermionic definition from the spectral projector method. We use ensembles with dynamical light, strange and charm flavors of maximally twisted mass fermions. For both definitions of the susceptibility we employ ensembles at three values of the lattice spacing and several quark masses at each spacing. The data are fitted to chiral perturbation theory predictions with a discretization term to determine the continuum chiral condensate in the massless limit and estimate the overall discretization errors. We find that both approaches lead to compatible results in the continuum limit, but the gluonic ones are much more affected by cut-off effects. This finally yields a much smaller total error in the spectral projector results. We show that there exists, in principle, a value of the spectral cutoff which would completely eliminate discretization effects in the topological susceptibility., Published by Inst., Woodbury, NY

In this paper we present the perturbative evaluation of the difference between the renormalization functions of flavor singlet and nonsinglet bilinear quark operators on the lattice. The computation is performed to two loops and to lowest order in the lattice spacing, for a class of improved lattice actions, including Wilson, tree-level (TL) Symanzik and Iwasaki gluons, twisted mass and SLiNC Wilson fermions, as well as staggered fermions with twice stout-smeared links. In the staggered formalism, the stout smearing procedure is also applied to the definition of bilinear operators., Comment: 20 pages, 11 figures. The only difference between v2 and v1 is the inclusion of an Addendum in which conversion factors for different renormalization schemes are individually listed

In this paper we present results for the renormalization of gauge invariant nonlocal fermion operators which contain a Wilson line, to one-loop level in lattice perturbation theory. Our calculations have been performed for Wilson/clover fermions and a wide class of Symanzik improved gluon actions. The extended nature of such long-link operators results in a nontrivial renormalization, including contributions which diverge linearly as well as logarithmically with the lattice spacing, along with additional finite factors. On the lattice there is also mixing among certain subsets of these nonlocal operators; we calculate the corresponding finite mixing coefficients, which are necessary in order to disentangle individual matrix elements for each operator from lattice simulation data. Finally, extending our perturbative setup, we present non-perturbative prescriptions to extract the linearly divergent contributions., 24 pages, 4 figures , 2 tables. Version accepted for publication in Physical Review D. A typo was corrected in Eq.(38). A misprint appearing in Eq.(46), as well as three lines above it, has been corrected

We present results on the nucleon axial and induced pseudo-scalar form factors using an ensemble of two degenerate twisted mass clover-improved fermions generated at the physical value of the pion mass. We evaluate the isovector and the isoscalar, as well as, the strange and the charm axial form factors. The disconnected contributions are evaluated using recently developed methods that include deflation of the lower eigenstates, allowing us to extract the isoscalar, strange and charm axial form factors. We find that the disconnected quark loop contributions are non-zero and particularly large for the induced pseudo-scalar form factor., Replaced with the published version

We present physics results from simulations of QCD using Nf=2 dynamical Wilson twisted mass fermions at the physical value of the pion mass. These simulations are enabled by the addition of the clover term to the twisted mass quark action. We show evidence that compared to previous simulations without this term, the pion mass splitting due to isospin breaking is almost completely eliminated. Using this new action, we compute the masses and decay constants of pseudoscalar mesons involving the dynamical up and down as well as valence strange and charm quarks at one value of the lattice spacing, a≈0.09 fm. Further, we determine renormalized quark masses as well as their scale-independent ratios, in excellent agreement with other lattice determinations in the continuum limit. In the baryon sector, we show that the nucleon mass is compatible with its physical value and that the masses of the Δ baryons do not show any sign of isospin breaking. Finally, we compute the electron, muon and tau lepton anomalous magnetic moments and show the results to be consistent with extrapolations of older ETMC data to the continuum and physical pion mass limits. We mostly find remarkably good agreement with phenomenology, even though we cannot take the continuum and thermodynamic limits. © 2017 American Physical Society. 95 9 Cited By :4

Physical review / D 96(1), 014513(2017). doi:10.1103/PhysRevD.96.014513, We provide a high statistics analysis of the $x$-dependence of the bare unpolarized, helicity and transversity iso-vector parton distribution functions (PDFs) from lattice calculations employing (maximally) twisted mass fermions. The $x$-dependence of the calculated PDFs resembles those of the phenomenological parameterizations, a feature that makes this approach promising despite the lack of a full renormalization program for them. Furthermore, we apply momentum smearing for the relevant matrix elements to compute the lattice PDFs and find a large improvement factor when compared to conventional Gaussian smearing. This allows us to extend the lattice computation of the distributions to higher values of the nucleon momentum., Published by APS, Woodbury, NY

We compare lattice QCD determinations of topological susceptibility using a gluonic definition from the gradient flow and a fermionic definition from the spectral projector method. We use ensembles with dynamical light, strange and charm flavors of maximally twisted mass fermions. For both definitions of the susceptibility we employ ensembles at three values of the lattice spacing and several quark masses at each spacing. The data are fitted to chiral perturbation theory predictions with a discretization term to determine the continuum chiral condensate in the massless limit and estimate the overall discretization errors. We find that both approaches lead to compatible results in the continuum limit, but the gluonic ones are much more affected by cut-off effects. This finally yields a much smaller total error in the spectral projector results. We show that there exists, in principle, a value of the spectral cutoff which would completely eliminate discretization effects in the topological susceptibility.

We present results for the nucleon electromagnetic and axial form factors using an N$_f$=2 twisted mass fermion ensemble with pion mass of about 131 MeV. We use multiple sink-source separations to identify excited state contamination. Dipole masses for the momentum dependence of the form factors are extracted and compared to experiment, as is the nucleon magnetic moment and charge and magnetic radii., Comment: 7 pages, 5 figures. Talk presented at the 34th annual International Symposium on Lattice Field Theory, 24-30 July 2016, University of Southampton, UK

We present results on the renormalization functions of the quark field and fermion bilinears with up to one covariant derivative. For the fermion part of the action, we employ the twisted mass formulation with Nf=2 and Nf=4 degenerate dynamical quarks, while in the gluon sector, we use the Iwasaki-improved action. The simulations for Nf=4 have been performed for pion masses in the range of 390-760 MeV and at three values of the lattice spacing, a, corresponding to β=1.90, 1.95, 2.10. The Nf=2 action includes a clover term with csw=1.57551 at β=2.10, and three ensembles at different values of mπ. The evaluation of the renormalization functions is carried out in the RI′ scheme using a momentum source. The nonperturbative evaluation is complemented with a perturbative computation, which is carried out at one-loop level and to all orders in the lattice spacing, a. For each renormalization function computed nonperturbatively, we subtract the corresponding lattice artifacts to all orders in a, so that a large part of the cutoff effects is eliminated. The renormalization functions are converted to the MS scheme at a reference energy scale of μ=2 GeV after taking the chiral limit. © 2017 American Physical Society. 95 3 Cited By :2

In these proceedings we discuss recent progress in nucleon structure using lattice QCD simulations at or near the physical value of the pion mass. Main focus will be given in observables such as the nucleon axial charge and the first moments of parton distributions, for both the valence and sea quark contributions, and discuss their implications on the spin content of the nucleon. We will will also report developments on the evaluation of the gluon momentum fraction, which contributes significantly to the nucleon spin., Comment: 6 pages, 11 Figures, Invited talk at the "Mini Symposium on Nucleon Spin Structure and Lattice QCD" at 22nd International Spin Symposium, Urbana-Champaign, Sep. 25-30, 2016

Nuclear physics / B 923, 394-415 (2017). doi:10.1016/j.nuclphysb.2017.08.012, In this work we present, for the first time, the non-perturbative renormalization for the unpolarized, helicity and transversity quasi-PDFs, in an RI′ scheme. The proposed prescription addresses simultaneously all aspects of renormalization: logarithmic divergences, finite renormalization as well as the linear divergence which is present in the matrix elements of fermion operators with Wilson lines. Furthermore, for the case of the unpolarized quasi-PDF, we describe how to eliminate the unwanted mixing with the twist-3 scalar operator. We utilize perturbation theory for the one-loop conversion factor that brings the renormalization functions to the $\overline{MS}$-scheme at a scale of 2 GeV. We also explain how to improve the estimates on the renormalization functions by eliminating lattice artifacts. The latter can be computed in one-loop perturbation theory and to all orders in the lattice spacing. We apply the methodology for the renormalization to an ensemble of twisted mass fermions with $N_f = 2 + 1 + 1$ dynamical quarks, and a pion mass of around 375 MeV., Published by North-Holland Publ. Co., Amsterdam

Physical review / D 95(11), 114514(2017). doi:10.1103/PhysRevD.95.114514, We present results on the light, strange and charm nucleon scalar and tensor charges from lattice QCD, using simulations with $N_f$ = 2 flavors of twisted mass clover-improved fermions with a physical value of the pion mass. Both connected and disconnected contributions are included, enabling us to extract the isoscalar, strange and charm charges for the first time directly at the physical point. Furthermore, the renormalization is computed nonperturbatively for both isovector and isoscalar quantities. We investigate excited state effects by analyzing several sink-source time separations and by employing a set of methods to probe ground state dominance. Our final results for the scalar charges are $g^{u}_{S} =$ 5.20(42)(15)(12), $g^{d}_{S} =$ 4.27(26)(15)(12), $g^{s}_{S} =$ 0.33(7)(1)(4), and $g^{c}_{S} =$ 0.062(13)(3)(5) and for the tensor charges $g^{c}_{T} =$ 0.794(16)(2)(13), $g^{d}_{T} =$ −0.210(10)(2)(13), $g^{s}_{T} =$ 0.00032(24)(0), and $g^{c}_{T} =$ 0.00062(85)(0) in the $\overline{MS}$ scheme at 2 GeV. The first error is statistical, the second is the systematic error due to the renormalization and the third the systematic arising from estimating the contamination due to the excited states, when our data are precise enough to probe the first excited state., Published by APS, Woodbury, NY

The renormalization factors of local quark-bilinear operators are computed nonperturbatively for Nf=3 flavors of stout link nonperturbative clover (SLiNC) fermions, with emphasis on the various procedures for the chiral and continuum extrapolations. The simulations are performed at a lattice spacing a=0.074fm, and for five values of the pion mass in the range of 290-465 MeV, allowing a safe and stable chiral extrapolation. Emphasis is given in the subtraction of the well-known pion pole which affects the renormalization factor of the pseudoscalar current. We also compute the inverse propagator and the Green's functions of the local bilinears to one loop in perturbation theory. We investigate lattice artifacts by computing them perturbatively to second order as well as to all orders in the lattice spacing. The renormalization conditions are defined in the RI′-MOM scheme, for both the perturbative and nonperturbative results. The renormalization factors, obtained at different values of the renormalization scale, are translated to the MS¯ scheme and are evolved perturbatively to 2 GeV. Any residual dependence on the initial renormalization scale is eliminated by an extrapolation to the continuum limit. We also study the various sources of systematic errors. Particular care is taken in correcting the nonperturbative estimates by subtracting lattice artifacts computed to one-loop perturbation theory using the same action. We test two different methods, by subtracting either the O(g2a2) contributions, or the complete (all orders in a) one-loop lattice artifacts. © 2015 American Physical Society. 91 1 Cited By :12

We present our study of the renormalization of the chromomagnetic operator, ${\mathcal{O}}_{\mathrm{CM}}$, which appears in the effective Hamiltonian describing $\mathrm{\ensuremath{\Delta}}S=1$ transitions in and beyond the Standard Model. We have computed, perturbatively to one loop, the relevant Green's functions with two (quark-quark) and three (quark-quark-gluon) external fields, at nonzero quark masses, using both the lattice and dimensional regularizations. The perturbative computation on the lattice is carried out using the maximally twisted-mass action for the fermions, while for the gluons we employed the Symanzik improved gauge action for different sets of values of the Symanzik coefficients. We have identified all the operators which can possibly mix with ${\mathcal{O}}_{\mathrm{CM}}$, including lower-dimensional and nongauge invariant operators, and we have calculated those elements of the mixing matrix which are relevant for the renormalization of ${\mathcal{O}}_{\mathrm{CM}}$. We have also performed numerical lattice calculations to determine nonperturbatively the mixings of the chromomagnetic operator with lower-dimensional operators, through proper renormalization conditions. For the first time, the $1/{a}^{2}$-divergent mixing of the chromomagnetic operator with the scalar density has been determined nonperturbatively with high precision. Moreover, the $1/a$-divergent mixing with the pseudoscalar density, due to the breaking of parity within the twisted-mass regularization of QCD, has been calculated nonperturbatively and found to be smaller than its one-loop perturbative estimate. The QCD simulations have been carried out using the gauge configurations produced by the European Twisted Mass Collaboration with ${N}_{f}=2+1+1$ dynamical quarks, which include in the sea, besides two light mass degenerate quarks, also the strange and charm quarks with masses close to their physical values.

By following an exact analytical procedure we study the transmission properties of a network of quantum wires arranged in a so-called T3 configuration and in the presence of a perpendicular magnetic field. It is shown that proper use of a single unit cell is sufficient to demonstrate a recently discovered localization phenomenon for all wavelengths and for special values of the external field. This Aharonov-Bohm localization is shown to be independent of a vertex coupling that appears as a free parameter in the probability flux conservation. © 2002 Elsevier Science B.V. All rights reserved. 302 1 39 47 Cited By :1

We study matrix elements of the "chromomagnetic" operator on the lattice. This operator is contained in the strangeness-changing effective Hamiltonian which describes electroweak effects in the Standard Model and beyond. Having dimension 5, the chromomagnetic operator is characterized by a rich pattern of mixing with other operators of equal and lower dimensionality, including also non gauge invariant quantities it is thus quite a challenge to extract from lattice simulations a clear signal for the hadronic matrix elements of this operator. We compute all relevant mixing coefficients to one loop in lattice perturbation theory this necessitates calculating both 2-point (quark-antiquark) and 3-point (gluon-quark-antiquark) Green's functions at nonzero quark masses. We use the twisted mass lattice formulation, with Symanzik improved gluon action. We also provide a nonperturbative method to compute mixing with lower dimensional operators we use this method in numerical simulations, and extract the mixing with the 3-dimensional scalar density, finding good agreement with one-loop results. © Published under licence by IOP Publishing Ltd. 562 1 Sponsors: Canakkale Onsekiz Mart University HadronPhysics2 Consortium Middle East Technical University Ozyegin University Turkish Atomic Energy Authority Conference code: 113030 Cited By :1

This study presents the preparation and characterization of double networks (DN) based on a first amphiphilic polymethacrylate conetwork (APCN) and a second polyacrylamide network. The APCN first network comprised interconnected "in-out" star copolymers of 2-(dimethylamino)ethyl methacrylate (DMAEMA, hydrophilic ionizable monomer) and 2-ethylhexyl methacrylate (EHMA, hydrophobic comonomer) or lauryl methacrylate (LauMA, second hydrophobic comonomer), synthesized using group transfer polymerization, following one-pot, sequential, monomer, and hydrophobic cross-linker (ethylene glycol dimethacrylate, EGDMA) additions. The second network was prepared by the aqueous photopolymerization of acrylamide (AAm) at two different concentrations, 2 and 5 M, and N,N′-methylenebis(acrylamide) cross-linker in the presence of the fully ionized (via HCl addition) APCN. After synthesis, all DNs and single (first and second) (co)networks, equilibrium-swollen in water, were characterized in terms of their mechanical properties in compression. The DNs exhibited improved mechanical properties (stress and strain at break, and elastic modulus) compared to the corresponding single networks. Better reinforcement was achieved in the DNs whose APCN first networks bore a lower hydrophobic content and whose hydrophobic monomer was EHMA rather than LauMA. The best DN exhibited stress at break above 8 MPa and strain at break nearly 80%, close to the values of the best DNs in the literature. Nanoindentation studies were also performed on selected DNs which proved again the enhanced mechanical properties of the present DNs, manifested as high resistance to penetration and low creep displacement. Small-angle X-ray scattering (SAXS) indicated a broad correlation peak for all APCN first networks, suggestive of microphase separation with short-range order, arising from the presence of the hydrophobic segments. The single correlation peak was preserved in the SAXS profiles of the DNs, which was, however, shifted to lower q-values, consistent with further network swelling. Despite the SAXS evidence for only weak phase separation on the nanoscale in the DNs, half of the water-swollen DNs (the ones with a 5 M AAm concentration in the second network) exhibited strong birefringence which probably arose from the stretching of the charged DMAEMA segments rather than the presence of anisotropic nanophases. © 2016 American Chemical Society. 49 5 1731 1742 Cited By :12

We evaluate the light, strange and charm scalar content of the nucleon using one lattice QCD ensemble generated with two degenerate light quarks with mass fixed to their physical value. We use improved techniques to evaluate the disconnected quark loops to sufficient accuracy to determine the strange and charm nucleon $\sigma$-terms. in addition to the light quark content $\sigma_{\pi N}$. We find $\sigma_{\pi N}$=$37.2(2.6)(^{4.7}_{2.9})$ MeV $\sigma_{s}$=$41.1(8.2)(^{7.8}_{5.8})$ MeV and $\sigma_c$=$79(21)(^{12}_{\phantom{1}8})$ MeV, where the first error is statistical and the second is the systematic error due to the determination of the lattice spacing, the assessment of finite volume and residual excited state effects., Comment: Version accepted for publication in Phys. Rev. Lett,, 6 pages, 4 figures