A review of recent lattice calculations of nucleon structure and matrix elements of operators in nucleons is presented. It primarily covers developments in the calculation of the matrix elements of the scalar, tensor, pseudo-scalar, axial-vector and vector operators in the ground state of neutrons and protons in the isospin symmetric limit. Some preliminary calculations of the electric dipole moment, the gravitational moments and stress-energy distribution, and the magnetic polarizability are briefly described.
We present the status of our calculation of the first few moments of the nucleon structure functions. Our calculations are done using domain wall fermions in the quenched approximation with the DBW2 gauge action at 1.3GeV inverse lattice spacing.
Preliminary results are presented in our program to calculate low moments of structure functions for the proton and neutron on a $24^3\times32$ lattice at $β=6.2$. A comparison is made for a variety of smeared nucleon sources and preliminary results for the calculation of the nucleon tensor charge are presented.
We review recent progress on lattice calculations of nucleon spin structure, including the parton distribution functions, form factors, generalization parton distributions, and recent developments in lattice techniques.
We perform a high-statistics precision calculation of nucleon matrix elements using an open sink method allowing us to explore a wide range of sink-source time separations. In this way the influence of excited states of nucleon matrix elements can be studied. As particular examples we present results for the nucleon axial charge $g_A$ and for the first moment of the isovector unpolarized parton distribution $<x>_{u-d}$. In addition, we report on preliminary results using the generalized eigenvalue method for nucleon matrix elements. All calculations are performed using $N_f=2+1+1$ maximally tw…
This talk presents recent calculations in full QCD of the lowest three moments of generalized parton distributions and the insight they provide into the behavior of nucleon electromagnetic form factors, the origin of the nucleon spin, and the transverse structure of the nucleon. In addition, new exploratory calculations in the chiral regime of full QCD are discussed.
In this paper we review recent progress in hadron structure using lattice QCD simulations, with main focus in the evaluation of nucleon matrix elements. We highlight developments that may guide new Physics searches, such as the scalar and tensor charges, as well as, the neutron electric dipole moment.
We report on the nucleon decay matrix elements with domain-wall fermions in quenched approximation. Results from direct and indirect method are compared with a focus on the process of a proton decaying to a pion and a lepton. We discuss the renormalization necessary for the matching to the continuum theory. Preliminary results for the renormalized chiral lagrangian parameters are presented.
We employ leading order covariant chiral perturbation theory to compute the nucleon-pion-state contribution to the 3-point correlation functions one typically measures in lattice QCD to extract the isovector nucleon charges $g_A,g_T$ and $g_S$. We estimate the impact of the nucleon-pion-state contribution on both the plateau and the summation method for lattice simulations with physical pion masses. The nucleon-pion-state contribution results in an overestimation of all charges with both methods. The overestimation is roughly equal for the axial and the tensor charge, and about fifty percent l…
The contributions of strange quarks to the nucleon mass and the nucleon spin are investigated in a mixed action scheme employing domain wall valence quarks and quark loops on MILC asqtad dynamical fermion ensembles. Results are presented for pion masses 495 MeV and 356 MeV.
We present a calculation of the nucleon sigma term on two-flavor QCD configurations with dynamical overlap fermions. We analyse the lattice data for the nucleon mass using the baryon chiral perturbation theory. Using partially quenched data sets, we extract the connected and disconnected contributions to the nucleon sigma term separately. Chiral symmetry on the lattice simplifies the determination of the disconnected contribution. We find that the strange quark content, which determines the neutralino dark matter reaction rate with nucleon through the Higgs boson exchange, is much smaller than…
The nucleon-pion-state contribution to QCD two- and three-point functions used in the calculation of the axial form factors of the nucleon are studied in chiral perturbation theory. For physically small quark masses the nucleon-pion states are expected to dominante the excited-state contamination at large euclidean time separations. To leading order in chiral perturbation theory the results depend on two experimentally well-known low-energy constants only and the nucleon-pion-state contribution can be reliably estimated. The nucleon-pion-state contribution to the axial form factor $G_{\rm A}(Q…
We report on preliminary results of a high statistics quenched lattice QCD calculation of nucleon matrix elements within the Symanzik improvement programme. Using the recently determined renormalisation constants from the Alpha Collaboration we present a fully non-pertubative calculation of the forward nucleon axial matrix element with $O(a)$ lattice artifacts completely removed. Runs are made at $β=6.0$ and $β=6.2$, in an attempt to check scaling and $O(a^2)$ effects. We shall also briefly describe results for $<x>$, the matrix element of a higher derivative operator.
The fomalism is developed to express nucleon matrix elements of the electromagnetic current in terms of form factors consistent with the translational, rotational, and parity symmetries of a cubic lattice. We calculate the number of these form factors and show how appropriate linear combinations approach the continuum limit.
We employ chiral perturbation theory to calculate the nucleon-pion-state contribution to the 3-point correlation functions measured in lattice QCD to compute various moments of parton distribution functions (quark momentum fraction, helicity and transversity moment). We estimate the impact of the nucleon-pion-state contribution on the plateau method for lattice simulations with a physical pion mass. The nucleon-pion-state contribution results in an overestimation of all three moments. The overestimation is at the 5-20% level for source-sink separations of about 1.5 fm.
We investigate scalar matrix elements of the nucleon using $N_f=2+1+1$ flavors of maximally twisted mass fermions at a fixed value of the lattice spacing of $a\approx 0.078{\rm fm}$. We compute disconnected contributions to the relevant three-point functions using an efficient noise reduction technique. Using these methods together with an only multiplicative renormalization applicable for twisted mass fermions, allows us to obtain accurate results in the light and strange sector.
The current status of calculations of simple nucleon structure observables is reviewed, with a focus on the axial charge. A major challenge is the combination of an exponentially decaying signal-to-noise ratio and the need for large source-sink separations to eliminate excited-state contributions; efforts to understand and deal with this problem are the focus of the largest section of this review. Finite-volume effects and chiral extrapolation are also briefly discussed.
The extraction of hadron matrix elements in lattice QCD using the standard two- and three-point correlator functions demands careful attention to systematic uncertainties. One of the most commonly studied sources of systematic error is contamination from excited states. We apply the variational method to calculate the axial vector current $g_{A}$, the scalar current $g_{S}$ and the quark momentum fraction $\left<x\right>$ of the nucleon and we compare the results to the more commonly used summation and two-exponential fit methods. The results demonstrate that the variational approach offers a …
The finite size effects on nucleon masses are calculated in relativistic chiral perturbation theory. Results are compared with two-flavor lattice results.
We investigate a systematic error coming from higher excited state contributions in the energy shift of light nucleus in the two-nucleon channel by comparing two different source calculations with the exponential and wall sources. Since it is hard to obtain a clear signal of the wall source correlation function in a plateau region, we employ a large quark mass as the pion mass is 0.8 GeV in quenched QCD. We discuss the systematic error in the spin-triplet channel of the two-nucleon system, and the volume dependence of the energy shift.
Protons and neutrons have a rich structure in terms of their constituents, the quarks and gluons. Understanding this structure requires solving Quantum Chromodynamics (QCD). However QCD is extremely complicated, so we must numerically solve the equations of QCD using a method known as lattice QCD. Here we describe a typical lattice QCD calculation by examining our recent computation of the nucleon axial charge.
Excited state contributions represent a formidable challenge for hadron structure calculations in lattice QCD. For physical systems that exhibit an exponential signal-to-noise problem they often hinder the extraction of ground state matrix elements, introducing a major source of systematic error in lattice calculations of such quantities. The development of methods to treat the contribution of excited states and the current status of related lattice studies are reviewed with focus on nucleon structure calculations that are notoriously affected by excited state contamination.
We report the status of our nucleon structure calculations with 2+1 flavors of domain wall fermions on the RBC-UKQCD $32^3\times64$ gauge ensembles with the Iwasaki+DSDR action. These ensembles have a fixed lattice scale of 1/a = 1.37 GeV, and two pion masses of about 170 and 250 MeV. Preliminary results for the isovector electromagnectic form factors and their corresponding root-mean-squared (r.m.s.) radii will be presented.
Excited-state contamination is one of the dominant uncertainties in lattice calculations of the nucleon axial-charge, $g_A$. Recently published results in leading-order chiral perturbation theory (ChPT) predict the excited-state contamination to be independent of the nucleon interpolator and positive. However, empirical results from numerical lattice calculations show negative contamination (downward curvature), indicating that present-day calculations are not in the regime where the leading-order ChPT predictions apply. In this paper we show that, under plausible assumptions, one can reproduc…
The nucleon-pion-state contribution to QCD two-point and three-point functions relevant for lattice calculations of the nucleon electromagnetic form factors are studied in chiral perturbation theory. To leading order the results depend on a few experimentally known low-energy constants only, and the nucleon-pion-state contribution to the form factors can be estimated. The nucleon-pion-state contribution to the electric form factor $G_{\rm E}(Q^2)$ is at the +5 percent level for a source-sink separation of 2 fm, and it increases with increasing momentum transfer $Q^2$. For the magnetic form fac…
QCD matrix elements of axial and vector currents between nucleons are required for the Monte Carlo reconstruction of the energy of neutrinos that are detected in long baseline oscillation experiments in the quasi-elastic regime. The cleanest approach for determining the axial matrix elements is lattice QCD. However, the extraction of these from the corresponding correlation functions is complicated by very large excited state contributions, that are related to transitions from the nucleon to a nucleon-pion pair. In this pilot study with a pion mass $m_π= 429~ \mathrm{MeV}$, we demonstrate for …
Precision measurements on nucleons provide constraints on the Standard Model and can also discern the signatures predicted for particles beyond the Standard Model. Knowing the Standard Model inputs to nucleon matrix elements will be necessary to constrain the couplings of dark matter candidates such as the neutralino, to relate the neutron electric dipole moment to the CP-violating theta parameter, or to search for new TeV-scale particles though non-$V-A$ interactions in neutron beta decay. However, these matrix elements derive from the properties of quantum chromodynamics (QCD) at low energie…
We present preliminary results of a new lattice computation of hadronic matrix elements of baryon number violating operators which appear in the low-energy effective Lagrangian of (SUSY-)Grand Unified Theories. The contribution of irrelevant form factor which has caused an underestimate of the matrix elements in previous studies is subtracted in this calculation. Our results are 2$\sim$4 times larger than the most conservative values often employed in phenomenological analyses of nucleon decay with specific GUT models.
We calculate the nucleon strange quark content directly from disconnected three-point functions. Numerical simulations are carried out in two-flavor QCD using the overlap quark action with up and down quark masses down to a fifth of the physical strange quark mass. To improve the statistical accuracy, we calculate the nucleon two-point functions with the low-mode averaging technique, whereas the all-to-all quark propagator is used for the disconnected quark loop. We obtain the y parameter, which is the ratio of the strange and light quark contents, y = 0.024(45) at the physical point. This is …
We present a preliminary domain-wall fermion lattice-QCD calculation of isovector vector and axial charges, g_V and g_A, of the nucleon. Since the lattice renormalizations, Z_V and Z_A, of the currents are identical with DWF, the lattice ratio (g_A/g_V)^{lattice} directly yields the continuum value. Indeed Z_V determined from the matrix element of the vector current agrees closely with Z_A from a non-perturbative renormalization study of quark bilinears. We also obtain spin related quantities Delta-q/g_V and delta-q/g_V.
In this work, we report a novel technique in lattice QCD for studying the high momentum-transfer region of nucleon form factors. These calculations could give important theoretical input to experiments, such as those of JLab's 12-GeV program and studies of nucleon deformation. There is an extensive history of form-factor calculations on the lattice, primarily with ground states for both the initial and final state. However, determining form factors at large momentum transfer ($Q^2$) has been difficult due to large statistical and systematic errors in this regime. We study the nucleon form fact…
We present the first lattice calculation of nucleon matrix elements using four dynamical flavors. We use the $N_f=2+1+1$ maximally twisted mass formulation. The renormalization is performed non-perturbatively in the RI$^\prime$-MOM scheme and results are given for the vector and axial vector operators with up to one-derivative. Our calculation of the average momentum of the unpolarized non-singlet parton distribution is presented and compared to our previous results obtained from the $N_f=2$ case.
We present a dedicated analysis of the influence of excited states on the calculation of nucleon matrix elements. This calculation is performed at a fixed value of the lattice spacing, volume and pion mass that are typical of contemporary lattice computations. We focus on the nucleon axial charge, g_A, for which we use about 7,500 measurements, and on the average momentum of the unpolarized isovector parton distribution, <x>_{u-d}, for which we use about 23,000 measurements. All computations are done employing N_f=2+1+1 maximally-twisted-mass Wilson fermions and using non-perturbatively calcul…
Among the sources of systematic error in nucleon structure calculations is contamination from unwanted excited states. In order to measure this systematic error, we vary the operator insertion time and source-sink separation independently. We compute observables for three source-sink separations between 0.93 fm and 1.39 fm using clover-improved Wilson fermions and pion masses as low as 150 MeV. We explore the use of a two-state model fit to subtract off the contribution from excited states.
The RQCD collaboration proposed a projection method to remove the excited state contamination in lattice OCD calculations of nuclear form factors. The effectiveness of this method in removing the two-particle nucleon-pion-state contamination is examined using chiral perturbation theory. It is shown that the projection method has practically no impact in the calculation of the axial and induced pseudoscalar form factors. In the pseudoscalar form factor the projection method strongly enhances the nucleon-pion-state contamination. The generalized Goldberger-Treiman relation is satisfied even thou…
We study the strangeness electromagnetic form factors of the nucleon from the N_f=2+1 clover fermion lattice QCD calculation. The disconnected insertions are evaluated using the Z(4) stochastic method, along with unbiased subtractions from the hopping parameter expansion. In addition to increasing the number of Z(4) noises, we find that increasing the number of nucleon sources for each configuration improves the signal significantly. We obtain G_M^s(0) = -0.017(25)(07), where the first error is statistical, and the second is the uncertainties in Q^2 and chiral extrapolations. This is consisten…
We present initial results of computing nucleon electric dipole moment induced by quark chromo-EDM, CP-violating quark-gluon coupling. Using chirally-symmetric domain wall and Möbius fermions with pion mass $m_π=172\text{ MeV}$, we calculate the connected part of the electric dipole form factor $F_3(Q^2)$. In addition, we perform an exploratory study of the method to calculate EDM using uniform background electric field on a lattice introduced without breaking the periodicity in the time direction.
The nucleon-pion-state contribution in the QCD 3-point function of the pseudoscalar density is calculated to leading order in chiral perturbation theory. It predicts a nucleon-pion-state contamination in lattice estimates for the pseudoscalar form factor $G_{\rm P}(Q^2)$ determined with the plateau method. Depending on the momentum transfer $Q^2$ the contamination varies between -20% and +50% for a source-sink separation of 2 fm. The nucleon-pion-state contamination also causes violations in the generalized Goldberger-Treiman relation among the pseudoscalar and the axial nucleon form factors, …
The nucleon-pion-state contribution to QCD two-point and three-point functions used in lattice calculations of the nucleon axial form factors are studied in chiral perturbation theory. For small quark masses this contribution is expected to be the dominant excited-state contamination at large time separations. To leading order in chiral perturbation theory the results depend on only two experimentally known low-energy constants and the nucleon-pion-state contribution to the form factors can be estimated. The nucleon-pion-state contribution to the axial form factor $G_{\rm A}(Q^2)$ is at the 5 …
We present the current status of our analysis of nucleon structure observables including isovector charges and twist-2 matrix elements as well as the nucleon mass. Results are computed on a large set of CLS $N_f=2+1$ gauge ensembles with $M_π\approx0.130\mathrm{MeV}\ldots 350\mathrm{MeV}$, four values of the lattice spacing $a=0.05\mathrm{fm}\ldots 0.09\mathrm{fm}$ and covering a large range of physical volumes. Compared to the results presented at last year's conference we have added data on a very fine and large box at small light quark mass ($T\times L^3=192 \times 96^3$, $M_π=172\mathrm{Me…
We report on our on-going project to calculate the nucleon decay matrix elements with domain-wall fermions. Operator mixing is discussed employing a non-perturbative renormalization. Bare matrix elements of all the possible decay modes induced by the dimension-six operators are calculated with the direct method, which are compared with the indirect calculation using chiral perturbation theory.
In order to advance lattice calculations of moments of unpolarized, helicity, and transversity distributions, electromagnetic form factors, and generalized form factors of the nucleon to a new level of precision, this work investigates several key aspects of precision lattice calculations. We calculate the number of configurations required for constant statistical errors as a function of pion mass, describe the coherent sink method to help achieve these statistics, examine the statistical correlations between separate measurements, study correlations in the behavior of form factors at differen…
Results for the disconnected contributions to matrix elements of the vector current and scalar density have been obtained for the nucleon from the Wilson action at beta=6 using a stochastic estimator technique and 2000 quenched configurations. Various methods for analysis are employed and chiral extrapolations are discussed.
The nucleon decay matrix elements of three-quark operators are calculated with domain-wall fermions. Operators are renormalized non-perturbatively to match the MS bar (NDR) scheme at NLO. Quenched simulation studies involve both direct measurement of the matrix elements and the chiral Lagrangian parameters, alpha and beta. We also report on the dynamical quark effects on these parameters.
The nucleon axial form factor is a dominant contribution to errors in neutrino oscillation studies. Lattice QCD calculations can help control theory errors by providing first-principles information on nucleon form factors. In these proceedings, we present preliminary results on a blinded calculation of $g_A$ and the axial form factor using HISQ staggered baryons with 2+1+1 flavors of sea quarks. Calculations are done using physical light quark masses and are absolutely normalized. We discuss fitting form factor data with the model-independent $z$ expansion parametrization.
We derive the axion-nucleon interaction Lagrangian in heavy baryon chiral perturbation theory up to next-to-next-to-leading order. The effective axion-nucleon coupling is calculated to a few percent accuracy.
As an example of an application of lattice QCD we describe a computation of four-quark operators in the nucleon. The results are interpreted in a diquark language.
Electric dipole moments (EDMs), which are sought as evidence of CP violation, require lattice calculations to connect constraints from experiments to limits on the strong CP violation within QCD or CP violation in new physics beyond the standard model. Nucleon EDM calculations on a lattice are notoriously hard due to large statistical noise, chiral symmetry violating effects, and potential mixing of the EDM and the anomalous magnetic moment of the nucleon. In this report, details of ongoing lattice calculations of proton and neutron EDMs induced by the QCD $θ$-term and the quark chromo-EDM, th…
We study the three nucleon force in the triton channel using dynamical clover fermion lattice QCD. The Nambu-Bethe-Salpeter wave function is utilized to obtain the potentials among three nucleons. Since the straightforward calculation is prohibitively expensive, two different frameworks are developed to meet the challenge. In the first method, we study the effective two nucleon potentials in the three nucleon system, where the differences between the effective two nucleon potentials and the genuine two nucleon potentials correspond to the three nucleon system effect, part of which is originate…
Previous Lattice QCD calculations of nucleon transverse momentum-dependent parton distributions (TMDs) focused on the case of transversely polarized nucleons, and thus did not encompass two leading-twist TMDs associated with longitudinal polarization, namely, the helicity TMD and the worm-gear TMD corresponding to transversely polarized quarks in a longitudinally polarized nucleon. Based on a definition of TMDs via hadronic matrix elements of quark bilocal operators containing staple-shaped gauge connections, TMD observables characterizing the aforementioned two TMDs are evaluated, utilizing a…
We introduce a stochastic sandwich method with low-mode substitution to evaluate the connected three-point functions. The isovector matrix elements of the nucleon for the axial-vector coupling $g_A^3$, scalar couplings $g_S^3$ and the quark momentum fraction $\langle x\rangle_{u -d}$ are calculated with overlap fermion on 2+1 flavor domain-wall configurations on a $24^3 \times 64$ lattice at $m_π = 330$ MeV with lattice spacing $a = 0.114$ fm.
Electric dipole moments (EDMs) of nucleons and nuclei, which are sought as evidence of CP violation, require lattice calculations to connect constraints from experiments to limits on the strong CP violation within QCD or CP violation introduced by new physics from beyond the standard model. Nucleon EDM calculations on a lattice are notoriously hard due to large statistical noise, chiral symmetry violating effects, and potential mixing of the EDM and the anomalous magnetic moment of the nucleon. In this report, details of ongoing lattice calculations of proton and neutron EDMs induced by the QC…
Lattice QCD calculations of resonant meson-meson scattering amplitudes have improved significantly due to algorithmic and computational advances. However, progress in meson-nucleon scattering has been slower due to difficulties in computing the necessary correlation functions, the exponential signal-to-noise problem, and the finite-volume treatment of scattering with fermions. Nonetheless, first benchmark calculations have now been performed. The status of lattice QCD calculations of meson-nucleon scattering amplitudes is reviewed together with comments on future prospects.
