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quantum chromodynamics ( qcd ) has been successful in describing many properties of the strong interaction . in the weak - coupling regime , we can rely on perturbation theory to work out the path integral which describes physical observables of interest
. however , for long distances perturbative qcd no longer converges .
instead , we use a discretization of space and time in a finite volume to calculate these quantities from first principles numerically ; such research forms the regime of lattice qcd@xcite .    to keep the systematic error due to discretization under control , one follows symanzik improvement order by order in terms of the ultraviolet cutoff ( @xmath0 ) for both the action and operators .
however , the breaking of continuous ( euclidean ) so(4 ) symmetry allows many new degrees of freedom , leading to various lattice actions that return to the same continuum action once the symmetry is restored .
thus , there exist many gauge and fermion actions for us to choose from .
today , most gauge actions used are @xmath1-improved and leave small discretization effects ( @xmath2 ) due to gauge choices . on the other hand ,
most fermion actions are only @xmath3-improved and have systematic errors of @xmath4 that become dominant .
for this reason , lattice calculations are generally distinguished according to the fermion action used .
differences among the actions are benign once all systematics are included , and the choice of fermion action is constrained by limits of computational and human power and by the main physics focus .
the commonly used actions are : domain - wall fermions ( dwf ) , overlap fermions , wilson / clover fermions , twisted - wilson fermions and staggered fermions .
since the real world is effectively continuous and infinitely large , we will have to take limits of @xmath5 and @xmath6 to eliminate the artifacts introduced in a discretized finite box . with the most state - of - the - art supercomputer ,
we are close but yet to simulate at the physical pion mass . using calculations at multiple heavier pion masses , which are affordable for available computational resources
, we can apply chiral perturbation theory to extrapolate quantities of interest to the physical limit .
a recent work by the bmw collaboration@xcite calculating multiple lattice spacings , volumes and pion masses as light as 180  mev provided an excellent demonstration of how ground - state hadron masses with fully understood and controlled systematics are consistent with experiment .
such calculations with multiple pion masses also help to determine the low - energy constants of chiral effective theory .
a typical nucleon interpolating field used in the lattice calculation is @xmath7 u_{c } , $ ] and the nucleon two- and three - point green functions are obtained from @xmath8 where @xmath9 is the operator of interest . for the vector ( axial )
current , the operator is @xmath10 . for the structure functions , the operators are    [ cols="^,^,^ , < " , ]     where @xmath11 is the difference between forward and backward covariant derivatives .
we calculate only the `` connected '' diagrams , which means the inserted quark current is contracted with the valence quarks in the baryon interpolating fields , as in the majority of lattice three - point calculations .
however , due to isospin symmetry , isovector quantities have a cancellation that removes the unknown disconnected piece .
( disconnected diagrams are notoriously difficult to calculate directly on the lattice .
they require that expensive fermion - matrix inversion be applied to numerous source vectors .
much effort has been devoted to solving this difficulty in the near future with new techniques . ) for more details , please refer to a selection of plenary talks : refs .
@xcite and references within .
the nucleon axial charge is well measured in neutron @xmath12-decay experiments , so it is a natural candidate for demonstrating how well lattice qcd can be extrapolated to the physical pion point .
the isovector axial charges @xmath13 are defined as the zero - momentum - transfer limits of the isovector axial form factors . results from various collaborations are shown on the left - hand side of figure  [ fig : ga_all ] .
we show a small - scale  expansion fit ( gray band ) to the 2 + 1-flavor ( dwf sea quarks ) rbc data@xcite ( blue filled circles ) .
the results are consistent with the lhpc data@xcite ( dwf valence with staggered sea ) ; rbc s 2f and 0f results are consistent with qcdsf s 2f clover and 0f overlap fermion numbers@xcite respectively . the lowest  pion - mass values among rbc s 2 + 1f and 2f results may suffer from sizable finite - volume systematic errors ; larger - volume calculations should be carried out to confirm these suspicions .
results for the zeroth moment of transversity @xmath14 are given on the right - hand side of figure  [ fig : ga_all ] .
we observe rather weak dependence ( roughly linear ) on the quark mass , which remains consistent for calculations with the same number of sea - quark flavors .
the chiral extrapolation form@xcite is applied to the rbc 2 + 1f data@xcite , yielding 0.56(4 ) at the physical pion mass .
however , this is close to what has been found by lhpc with mixed action@xcite : about 0.7 .
these extrapolated values are significantly smaller than those found at the simulated pion masses , which are near 1 .
we urgently need data at the lightest pion mass to confirm the rapidly decreasing behavior predicted by the chiral effective theory .
figure  [ fig : allmoments ] shows the latest calculations of the first moments of the quark momentum fraction ( left ) and helicity ( right ) distributions . here again the 0f , 2f and 2 + 1f results are compared between different collaborations with different choices of fermion actions and are seen to be consistent among calculations with the same number of sea - quark flavors .
the chiral extrapolation@xcite is performed using rbc s 2 + 1f data , which gives 0.133(13 ) and 0.203(23 ) for the first moments of the quark momentum fraction and helicity distributions respectively , consistent with experiment .
we see strong curvature due to the chiral form ; more light - pion points should be taken to reduce extrapolation uncertainties .
these extrapolation numbers are also consistent with the lhpc s mixed - action calculation@xcite .
higher moments of the isovector distributions are calculated by lhpc ( scri , sesam ) with 0f and 2f wilson and clover fermions@xcite and by qcdsf with 0f clover fermions at multiple lattice spacings@xcite .
consistent results are seen among different groups : the second and third moments are about 25% and 10% of the first moments respectively .
however , for moments with @xmath15 , divergences occur involving lower - dimension operators at finite @xmath0 , which limits the number of moments accessible to lattice qcd .     and their chiral extrapolations .
the bands in both plots are chiral extrapolations fit to the rbc 2 + 1f data.,title="fig : " ]   and their chiral extrapolations .
the bands in both plots are chiral extrapolations fit to the rbc 2 + 1f data.,title="fig : " ]    the twist-3 first moment of the polarized structure function @xmath16 is another interesting feature to consider .
it is related to the polarized structure functions @xmath17 and @xmath18 and the wandzura - wilczek relation@xcite .
the lowest moment @xmath19 from rbc s 2 + 1f data extrapolated to the physical pion mass is consistent with zero @xmath20 . combined with the small value of @xmath21 found by qcdsf s 2f calculation@xcite , we conclude that the wandzura - wilczek relation between the moments of @xmath17 and @xmath18 , which asserts vanishing @xmath16 , is at least approximately true .
studying the momentum - transfer ( @xmath22 ) dependence of the elastic electromagnetic form factors is important in understanding the structure of hadrons at different scales .
there have been many experimental studies of these form factors on the nucleon .
a recent such experiment , the jefferson lab double - polarization experiment ( with both a polarized target and longitudinally polarized beam ) revealed a non - trivial momentum dependence for the ratio @xmath23 .
this contradicts results from the rosenbluth separation method , which suggested @xmath24 .
the contradiction has been attributed to systematic errors due to two - photon exchange that contaminate the rosenbluth separation method more than the double - polarization .
( for details and further references , see the recent review articles : refs .
lattice calculations can make valuable contributions to the study of nucleon form factors , since they allow access to both the pion - mass and momentum dependence of such form factors .
recently , the limitations of the largest - available @xmath22 ( in terms of the quality of the signal - to - noise ratios ) has been overcome@xcite .
an exploratory study using clover fermions extends the range of momentum transfer to 6  gev@xmath25 , as shown in figure  [ fig : ff ] .
the @xmath22 dependence of the neutron has exceeded the range of the current existing data .
such calculations will provide interesting comparisons for data collected after the future 12-gev upgrade at jefferson lab .
the generalized parton distributions ( gpds ) have been calculated by lhpc ( 2 + 1f mixed action , @xmath26760  mev)@xcite and qcdsf ( 2f clover action , @xmath27950  mev)@xcite .
one of the main topics of physics interest derived from gpds is to study the origins of the nucleon s spin .
the quark spin ( @xmath28 ) and total quark contribution ( @xmath29 ) to the angular momentum can be connected to gpds via@xcite @xmath30 , % -\frac{1}{2}\delta \sigma^q \label{ionflux}\ ] ] where @xmath31 and @xmath32 are the polarized and unpolarized generalized form factors .
the left - hand side of figure  [ fig : spin - gems ] shows the quark spin and orbital angular momentum ( @xmath33 ) calculated by lhpc and qcdsf ; they are consistent .
one found the total @xmath34 quark angular momentum and the angular momentum from the sum of @xmath35 and @xmath34 quarks to be consistent with zero .
this is because @xmath36 and @xmath37 are of the same magnitude but have opposite signs .
a similar relation holds for @xmath38 and @xmath36 .
note that in both of the calculations only connected diagrams are included .    for the transverse structure of the proton
, qcdsf calculates the lowest two moments of the transverse spin densities of quarks in the nucleon@xcite .
an exploratory attempt has been made to calculate transverse - momentum distributions ( tmds ) using lattice link products to approximate the wilson line in light - cone frame in spatial coordinates and fourier transforming into momentum - space to map the density distributions@xcite ; this calculation used 2 + 1f mixed action ( dwf on staggered , @xmath39  mev ) .
another new development during the last year is progress on disconnected diagrams .
an indirect way of calculating the strangeness form factors via charge symmetry in 2 + 1f lattices is shown on the right - hand side of figure  [ fig : spin - gems ] .
the direct approach has also made great progress .
for example , the strange - quark distribution @xmath40 has been calculated by the @xmath41qcd collaboration on 2 + 1f lattices@xcite .
the same group has also calculated the gluon momentum fraction @xmath42 with improved signal .
this is an exciting era for the use of lattice qcd in nuclear physics : there have been huge leaps due to increasing computational resources worldwide and improved algorithms , allowing continual improvement in lighter pion masses , larger volumes and finer lattice spacings .
various groups have demonstrated universality with the consistent results coming from independent calculations using different lattice actions . by reproducing well measured experimental values ,
we solidify confidence in lattice predictions of quantities that have not or can not be measured by experiment .
there are many different aspects of hadron structure that one can do with lattice qcd ; only a few examples have been presented here .    in the near future , pion masses around 200  mev or lighter with multiple volumes and lattice spacings will become commonplace .
there will be less need to depend on chiral perturbation theory for extrapolations , and once the physical pion mass becomes accessible , we can check its correctness .
full - contraction calculations ( including disconnected diagrams ) in all matrix elements , form factors and gpds will lead to precision calculations for individual quark components or individual proton and neutron quantities .
further improvements in methodology and expanding computational resources will allow direct calculations of the gluon content of the nucleon .
the future is unlimited with lattice qcd .
_ acknowledgements _ : authored by jefferson science associates , llc under u.s .
doe contract no .
de - ac05 - 06or23177 .
government retains a non - exclusive , paid - up , irrevocable , world - wide license to publish or reproduce this manuscript for u.s .
government purposes .
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 .     
 address = jefferson laboratory , 12000 jefferson avenue , newport news , va 23606