Lattice Nucleon Isovector Unpolarized Parton Distribution in the Physical-Continuum Limit
MMSUHEP-20-019
Lattice Nucleon Isovector Unpolarized Parton Distribution in thePhysical-Continuum Limit
Huey-Wen Lin,
1, 2, ∗ Jiunn-Wei Chen, † and Rui Zhang
1, 2, ‡ Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824 Department of Computational Mathematics, Science and Engineering,Michigan State University, East Lansing, MI 48824 Department of Physics, Center for Theoretical Physics,and Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei, Taiwan 106
We present the first lattice-QCD calculation of the nucleon isovector unpolarized parton distri-bution functions (PDFs) at the physical-continuum limit using Large-Momentum Effective Theory(LaMET). The lattice results are calculated using ensembles with multiple sea pion masses with thelightest one around 135 MeV, 3 lattice spacings a ∈ [0 . , .
12] fm, and multiple volumes with M π L ranging 3.3 to 5.5. We perform a simultaneous chiral-continuum extrapolation to obtain RI/MOMrenormalized nucleon matrix elements with various Wilson-link displacements in the continuum limitat physical pion mass. Then, we apply one-loop perturbative matching to the quasi-PDFs to obtainthe lightcone PDFs. We find the lattice-spacing dependence to be much larger than the dependenceon pion mass and lattice volume for these LaMET matrix elements. Our physical-continuum limitunpolarized isovector nucleon PDFs are found to be consistent with global-PDF results. I. INTRODUCTION
Precision determination of parton distribution func-tions (PDFs) is not only important to probing unknownsof the Standard Model but also to advance interpretationof high-energy experiments searching for signs of physicsbeyond the Standard Model. In addition to energy-frontier experiments like the LHC, there are also manymid-energy experimental efforts around the world, suchas at Brookhaven and Jefferson Laboratory in the UnitedStates, GSI in Germany, J-PARC in Japan, or a futureelectron-ion collider (EIC). These are set to explore theless-known kinematic regions of nucleon structure andmore. The pursuit of PDFs has led to collaborationsof theorists and experimentalists working side-by-side totake advantage of all available data, evaluating differentcombinations of input theories, parameter choices andassumptions, resulting in multiple global-PDF determi-nations. Comparison of these different global-fit deter-minations of the PDFs is important to reveal hidden un-certainties in PDF data sets. Often, in kinematic regionswhere experimental data are plentiful or overconstrained,such as the mid- x region of the PDFs, there is consis-tency among different PDF sets. However, in the regionswhere data are sparse or suffer from complicated nucleareffects, such as at small- or large- x or for heavy-flavorPDFs, disagreements are seen. For more details, we re-fer readers to the non-technical review in Ref. [1]. Anonperturbative approach from first principles, such aslattice QCD (LQCD), can provide the necessary inputsto fill gaps in the experimental data or add information ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] to constrain global fits. For a long while, lattice PDFcalculations were limited to moments only, that is, wherethe x dependence of the PDF is integrated out. Precisionlattice determinations of the moments (after removing alllattice artifacts, such as discretization and finite-volumeeffects) can have significant impact on determinations ofthe PDFs [1, 2].Large-momentum effective theory (LaMET) [3, 4]enables computation of the Bjorken- x dependence ofhadron PDFs on a Euclidean lattice. LaMET relatesequal-time spatial correlators, whose Fourier transformsare called quasi-PDFs, to PDFs in the limit of infi-nite hadron momentum. For large but finite momentaaccessible on a realistic lattice, LaMET relates quasi-PDFs to physical ones through a factorization theorem,the proof of which was developed in Refs. [5–7]. SinceLaMET was proposed, a lot of progress has been madein the theoretical understanding of the formalism [6, 8–12, 12–15, 15, 16, 16–27, 27–64]. The method has beenapplied in lattice calculations of PDFs for the up anddown quark content of the nucleon [21, 27, 28, 30, 65–75, 75–78], π [79–82] and K [81] mesons, and the∆ + [83] baryon. Despite limited volumes and relativelycoarse lattice spacings, previous state-of-the-art nucleonisovector quark PDFs, determined from lattice data atthe physical pion mass, have shown reasonable agree-ment [68, 69] with phenomenological results extractedfrom the experimental data. Encouraged by this success,LaMET has also been extended to twist-three PDFs [84–86], as well as gluon [87, 88], strange and charm distri-butions [89]. It was also applied to meson distributionamplitudes [22, 90–92] and generalized parton distribu-tions (GPDs) [93–96]. Attempts have also been madeto generalize LaMET to transverse momentum depen-dent (TMD) PDFs [97–104], to calculate the nonpertur-bative Collins-Soper evolution kernel [99, 105, 106] andsoft functions [107] on the lattice. LaMET also brought a r X i v : . [ h e p - l a t ] N ov renewed interest in earlier approaches [108–114] and in-spired new ones [115–130]. For recent reviews on thesetopics, we refer readers to Refs. [1, 131–134] for moredetails.To further improve the lattice computations at phys-ical pion mass, the remaining lattice systematics mustbe treated, by extrapolation to infinite volume and thecontinuum limit. This is a critical next step to create alattice PDF calculation with fully controlled systemat-ics. Since our calculation uses the quasi-PDF method,we consider only quasi-PDF results for comparison ofsystematic uncertainty. The first study of finite-volumesystematics was done in Ref. [75] with isovector bothpolarized and unpolarized nucleon PDFs; three latticevolumes (2.88, 3.84, 4.8 fm) were studied at pion mass220-MeV and nucleon momenta 1.3 and 2.6 GeV, and nonoticeable finite-volume dependence was found. This isconsistent with a later study in chiral perturbation the-ory (ChPT) [135], which showed that momentum boostreduces the finite-volume effect, since the length contrac-tion of the hadron makes the lattice effectively bigger.ChPT also showed that for nucleon momenta greaterthan 1 GeV and the lattice size times pion mass greaterthan 3, then the finite-volume effect on the isovector nu-cleon PDF is less than 1%. This conclusion is consistentwith the numerical findings of Ref. [75] and suggests thatthe finite-volume effect is negligible at current lattice pre-cision.Continuum extrapolation is also important for LaMETdue to potential operator mixing in the nonlocal opera-tors. The nonlocal operators for the quasi-PDFs can mixwith a tower of higher-dimensional operators at O ( a ),even if all symmetries (including chiral symmetry) arerestored [16, 29, 54]. This is different from the situationfor local operators, where mixing can occur at O ( a ) ifa chiral lattice fermion action is used. To ensure thatsuch operator mixings do not contaminate the final re-sults of the lattice PDF calculations, it is important totake the continuum limit. There have been some studiesof the continuum extrapolation of the quasi-PDF methodin the pion and kaon distribution amplitudes [91] and innucleon PDFs [78]; both cases use three lattice spacingsbut a single heavy quark mass with M π >
300 MeV.Ref. [81] determines valence-quark PDFs of the pion andkaon using two lattice spacings (0.06 and 0.12 fm) and3 pion masses ( M π ∈ [220 , II. LATTICE PARAMETERS AND SETUP
In this work, we use clover lattice fermion actionfor the valence quarks on top of 2+1+1 flavors (de-generate up and down quarks plus strange and charmquarks at their physical masses in the QCD vacuum) of hypercubic (HYP)-smeared [136] highly improved stag-gered quarks (HISQ) [137, 138] in configurations gen-erated by MILC Collaboration. The lattice parametersinclude lattice spacings a ∈ [0 . , .
12] fm, pion mass M π ∈ [135 , L ∈ [2 . , .
5] fm(which make M π L ∈ [3 . , . a ≈ .
15 fm and 0.12 fm at M π ≈
135 MeV, and theseensembles are excluded from use in mixed-action calcu-lations. The other ensembles are carefully checked forthe relevant signatures, and exceptional configurationsare absent for the M π ∈ { , } MeV MILC ensem-bles [138] at 0.12 fm and finer lattice spacings, as well asfor 0.09 and 0.06 fm near the physical pion mass. Thereare no issues that we have observed for any observableon the ensembles used in this calculation.For the nucleon matrix-element measurement, we useGaussian momentum smearing [143] for the quark field ψ ( x ) → S mom ψ ( x ) =11 + 6 α (cid:16) ψ ( x ) + α (cid:88) j U j ( x ) e ik ˆ e j ψ ( x + ˆ e j ) (cid:17) , (1)where k is the momentum-smearing parameter, whichcan be tuned separately on each ensemble for optimalsignal-to-noise ratios in the matrix elements of the de-sired nucleon boost momentum. U j ( x ) are the gaugelinks in the j direction, and α is a tunable parameteras in traditional Gaussian smearing. Such a momentumsource is designed to increase the overlap with nucleonsof the desired boost momentum, and we are able to reachhigher boost momentum for the nucleon states than ourprevious work [27].On the lattice, we first calculate the time-independent,nonlocal (in space, chosen to be the z direction) correla-tors of a nucleon with finite- P z boost˜ h lat ( z, P z ) = (cid:68) (cid:126)P (cid:12)(cid:12)(cid:12) ¯ ψ ( z )Γ (cid:16) (cid:89) n U z ( n ˆ z ) (cid:17) ψ (0) (cid:12)(cid:12)(cid:12) (cid:126)P (cid:69) , (2)where U z is a discrete gauge link in the z direction and (cid:126)P = { , , P z } is the momentum of the nucleon. Γ = γ t for the unpolarized parton distribution. Note that ourprevious work on the unpolarized quark distribution usesΓ = γ z ; this operator has mixing with matrix elements Ensemble ID a (fm) N s × N t M val π (MeV) M val π L t sep /a P z N cfg N meas a12m310 0.1207(11) 24 ×
64 310(3) 4.55 { , , , } { , , } πL { , , , } a12m220S 0.1202(12) 24 ×
64 225(2) 3.29 { , , , } { , , } πL { , , , } a12m220 0.1184(10) 32 ×
64 228(2) 4.38 { , , , } { , , } πL { , , , } a12m220L 0.1189(09) 40 ×
64 228(2) 5.5 { , , , } { , , , , } πL { , , , } a09m130 0.0871(6) 64 ×
96 138(1) 3.90 { , , , } { , , } πL { , , , } a06m310 0.0582(4) 48 ×
96 320(2) 4.52 { / , , , } { , , , } πL { , , , } TABLE I: Ensemble information and parameters used in this calculation. N meas is the total number of measurements of thethree-point correlators for different values of t sep . L indicates the spatial length which is aN s (in fm). with Γ = 1 [15, 29], while the γ t case is free from suchmixing at O ( a ). In this work, we only study the isovec-tor unpolarized quark PDF.As we increase the nucleon boost momentum, we antic-ipate that excited-state contamination worsens, since thestates are relatively closer to each other; therefore, a care-ful study of the excited-state contamination is necessaryfor the LaMET (or quasi-/pseudo-PDF) approach. Tomake sure the excited-state contamination is under con-trol, we measure at least four nucleon three-point source-sink separations, and we perform a number of differentextraction and analysis schemes. We use multigrid algo-rithm [144, 145] in the Chroma software package [146] tospeed up the inversion of the quark propagator for theclover fermions. Details of our calculation parameterscan be found in Table I.Figure 1 shows an example analysis we did on the en-semble with a ≈ .
06 fm and 310-MeV pion mass. Onethis ensemble, we use multiple values of nucleon boostmomenta, P z = { , , n πL } , with n ∈ { , , } , which cor-respond to 1.7, 2.15 and 2.6 GeV nucleon momenta. Weconsider multiple analysis methods to remove excited-state systematics among the 5 source-sink separations,0.60, 0.72, 0.84, 0.96, 1.08 fm, used in this work: First,we use the “two-simRR” analysis described in Ref. [142]to obtain the ground-state nucleon matrix elements us-ing all five source-sink separations. (This analysis notonly obtains the ground-state matrix element but alsothe transition and excited-state matrix elements.) A sec-ond extraction uses the same method but only the largestfour separations. Finally, we use the “two-sim” analy-sis, which includes both the ground state and the tran-sition matrix elements but without the excited matrixelements. Figure 1 shows the real and imaginary parts ofthe matrix elements for all three momenta using variouscombinations of data and analysis strategies. There isno clear observation of excited-state contamination us-ing any of these analyses. If the excited states were notunder control, we should see these different analyses giv-ing very different ground-state signals. Similar analysishas been done in all ensembles. For the rest of this paper,we will take the middle analysis, focusing on the matrixelement using “two-simRR” with source-sink separation t sep ≤ .
72 fm only. Before we can study the PDFs, we first need renormal-ize the bare matrix elements obtained on the ensembles.To do so, we calculate the RI/MOM renormalization con-stant ˜ Z nonperturbatively on the lattice by imposing thefollowing momentum-subtraction condition on the ma-trix element of the quasi-PDF in an off-shell quark state: Z ( p Rz , /a, µ R ) =Tr[ /p (cid:80) s (cid:104) ps | ¯ ψ f ( λ ˜ n ) / ˜ n t W ( λ ˜ n, ψ f (0) | ps (cid:105) ]Tr[ /p (cid:80) s (cid:104) ps | ¯ ψ f ( λ ˜ n ) / ˜ n t W ( λ ˜ n, ψ f (0) | ps (cid:105) tree ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p − µ Rpz = pRz . (3)On the lattice, (cid:104) ps | O γ t ( z ) | ps (cid:105) is calculated from the am-putated Green function of O γ t with Euclidean externalmomentum. In Fig. 2 we show the RI/MOM renormal-ization factors calculated from all ensembles as a functionof Wilson-line displacement z . We observe a strong de-pendence of the renormalization factors on lattice spac-ing; this is expected, since the renormalization factorsserve as counterterms to cancel the ultraviolet (UV) di-vergence of the bare matrix elements. On the other hand,the dependence on pion mass is negligible; the renormal-ization factors from a12m220 and a12m310 overlap oneanother.Figure 3 shows an example comparison of the realrenormalized isovector nucleon matrix elements for allensembles. We observe a small pion-mass dependence forthe a ≈ .
12 fm ensembles between the ensembles with220- and 310-MeV pions, and no sizable finite-volumeeffects. When comparing lattice-spacing dependence, wenoted a small trend of the matrix elements moving down-ward from 0.12 fm to 0.06 fm (green to blue points) butoverall within 2 standard deviations. We also comparewith the results from a single superfine lattice-spacingstudy from Ref. [63] with similar nucleon boost momen-tum, and the results are consistent as well (due to thelarger uncertainties). For the work below, we will fo-cus on a continuum extrapolation without the superfinelattice spacing, since the data is unlikely change the ex-trapolation much. z R e M E z v - - - - - z I m M E z v FIG. 1: The real (left) and imaginary (right) parts of the bare isovector nucleon matrix elements for unpolarized PDFs asfunctions of z at different momenta. Their kinematic factors are omitted to enhance visibility by separating the z = 0 matrixelements. The colors indicate the different nucleon boost momenta: blue, red and green for matrix elements from 1.7, 2.15 and2.6 GeV, respectively. At a given positive z value, the data is slightly offset to show different ground-state extraction strategies;from left to right they are: two-simRR using all t sep , two-simRR using the largest 4 t sep , two-sim using the largest 3 t sep .Different analyses are consistent within statistical errors, which suggests the excited-state contamination is well controlled. a042m310a06m310a09m130a12m310a12m220S0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.0 z ( fm ) R e [ Z - ( μ R = . G e V , p z R = ) ] a042m310a06m310a09m130a12m310a12m220S0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.0 z ( fm ) R e [ Z - ( μ R = . G e V , p z R = . G e V ) ] FIG. 2: The real component of the inverse renormalization factor from all ensembles as functions of Wilson-line displacement z with RI/MOM renormalization scales µ R = 3 . p Rz = 0 (left) and p Rz = 2 . III. RESULTS AND DISCUSSIONS
To obtain the physical-continuum matrix elements, weextrapolate the lattice spacing to zero and the pion massto its physical value through the following ansatz: h R ( zP z , a, M π ) = h R ( zP z ) (cid:0) c a,i ( zP z ) a i + c M,j ( zP z ) M jπ (cid:1) (4)To allow for flexibility in the extrapolation form, wevary the order of dependence on lattice spacing ( i ) andpion-mass ( j ) between linear and quadratic. The finite-volume effects are small in Fig. 3, consistent with theChPT study [135], suggesting that finite-volume effectsare negligible for current lattice precision. In this work,we attempt to use a common set of momenta across asmany ensembles as possible and keep any interpolationclose to an existing data point. For this reason, we use P z ≈ . P z ≈ . n z = 5 in lattice momentumunits on a06m310, a12m310, and a12m220S, n z = 10 on a09m130, and n z ≈ . z -expansion [147, 148] tothe matrix elements at five momenta n z = 4 , , , ,
10 onthe a12m220L lattice: h R ( z , a, M π , L ) = (cid:80) i =0 c z,i z ( P z ) i ,then evaluate the polynomial at P z = 2 . n z = 8, we need not worry much about the possibility ofoverfitting.We extrapolate the renormalized matrix elements tothe physical limit with four combinations of i and j in Eq. (4), and obtained 4 different physical-continuummatrix elements, as shown in solid (central value) anddashed (error band) lines in Fig. 4. We find that thefour different fits in real matrix elements are in goodagreement, and more fluctuations are seen in the imag-inary matrix elements. The fluctuation is mainly domi-nated by the lattice-spacing extrapolation. Using a lin-ear lattice-spacing extrapolation form results in slightlyhigher continuum-limit matrix elements than those ob-tained from a quadratic form. a12m310a12m220Sa12m220Ma12m220L0 2 4 6 8 10 120.00.20.40.60.81.0 zP z h R ( z P z , μ R = . G e V , p z R = ) a06m310a09m130a12m310a042m3100 5 10 15 - z h R ( z P z , μ R = . G e V , p z R = ) FIG. 3: Example of the real renormalized matrix elements with µ R = 3 . p Rz = 0 GeV comparison of selected a ≈ .
12 fm lattices (left) and lattice-spacing dependent 310 MeV results (right) along with physical pion mass ensemble. Thea42m310 data is taken from Ref. [63], which uses a similar mixed action but much finer lattice spacing. The lattice ensemblesare distinguished by color while different symbols indicate different boost momenta within each ensemble. ( a , M π )( a , M π )( a , M π )( a , M π ) AIC AVG a06m310a09m130a12m310a12m220Sa12m220L0 2 4 6 8 10 12 - z h R ( z P z , P z = . G e V , μ R = . G e V , p z R = ) ( a , M π )( a , M π )( a , M π )( a , M π ) AIC AVG a06m310a09m130a12m310a12m220Sa12m220L0 2 4 6 8 10 12 - - - - - - z h R ( z P z , P z = . G e V , μ R = . G e V , p z R = ) FIG. 4: Example of the physical-continuum extrapolation of the real (left) and imaginary (right) matrix elements fromthe ensembles with nucleon boost momentum around 2.2 GeV. Various ans¨atze with linear/quadratic extrapolation in latticespacing and pion mass are shown as solid lines for the central values and dashed lines for uncertainties. The filled band showsthe AIC-averaged physical-continuum matrix elements.
We then average the results the above fits using theAkaike information criterion (AIC): h AIC = (cid:80) i,j h i,j e − (2 k i,j + χ i,j ) / (cid:80) i,j e − (2 k i,j + χ i,j ) / , (5)where k i,j is the number of free parameters to fit, and χ i,j represents the fit quality, which is shown as thegray band in Fig. 4. The AIC-average results are withintwo standard deviations of each of the individual fit-ted matrix elements. When the nucleon momentum P z (cid:29) { M N , Λ QCD } , the quasi-PDF can be matched tothe PDF through the factorization theorem [3, 4, 6],˜ q ( x, P z , p Rz , µ R ) = (cid:90) − dy | y | C (cid:18) xy , r, yP z µ , yP z p Rz (cid:19) q ( y, µ ) , (6)where r = µ R /p Rz and C is a perturbative matchingkernel, which has been used in previous works [70, 72,79, 93]. The flavor indices of q , ˜ q , and C are implied.There are two main sources of residual P z depen-dence in removing the frame dependence from the light- cone PDFs: the target-mass correction and twist-4 ef-fects. For the former, the nucleon mass ( M N ) correc-tions can be corrected to all orders in M N /P z [21]. Thetwist-4 effect is O (Λ /P z ) from dimensional analysis;however, Ref. [50] suggested the effect could be up to O (Λ /x P z ) in order to cancel the renormalon ambi-guity in the kernel. However, a recent study of bubble-chain diagrams in Ref. [64] did not find slow convergenceof the kernel at three-loop order, indicating that therenormalon effect could be mild to this order in quasi-PDFs.A related issue is that Ref. [43] asserted that the twist-4operator is set by the lattice spacing a ; hence, its suppres-sion factor compared with twist-2 is O (1 / ( P z a ) ) insteadof O (Λ /P z ) [43]. However, the twist-4 contributionthat needs to be subtracted from the quasi-distributionoperator can be written as an equal-time correlator withtwo more mass dimensions than the original [21]. Hence,they should not cause power-divergent mixings that needto be subtracted before applying RI/MOM renormaliza-tion. Furthermore, the RI/MOM renormalization factor Z is well fitted by e ( m − /a − m ) | z | | z | d /c z (cid:29) a [64].If a power divergence appeared, Z would have extra pow-ers of 1 /a dependence, which are not observed.Recently, it was argued that in addition to the typical O (1 /P z ) power corrections, nonperturbative renormal-ization could introduce an even more important O (1 /P z )infrared contribution that cannot be removed by thematching kernel [64]. Hence, we will use this more con-servative estimate for our error.We estimate the systematic associated with the trans-formation of the lightcone distribution through the fol-lowing procedure. First, we take the nucleon isovectorPDF from CT18 global fit, and create set of mock matrixelements as functions of zP z using the same parametersused in the lattice calculation. We then run these mockmatrix elements through the same analysis used to cal-culate the PDFs; this should yield the same PDFs thatwere originally used to create the data, but they will dif-fer due to the inverse problem in the transformation. Asimilar analysis has been done in Ref. [30]. The differ-ence between the input and reconstructed PDFs providesa measure of the size of the transformation systematicuncertainty. As expected, the reconstructed PDFs havemuch larger uncertainties in the small- x and negative- x regions. We neglect the small- x and antiquark results dueto the large uncertainty associated with nucleon boostedmomenta less than 2.6 GeV. We added the difference as asystematic error in quadrature with the twist-4 errors, es-timated to be O (Λ QCD /P z ) by using Λ QCD ≈ . .These errors are shown as the outer uncertainty band inFig. 5.We focus on comparing our results with previous lat-tice quasi-PDF calculations done at the physical pionmass (but with a single lattice spacing) and with a se-lection of global-fit PDFs. The first generation of un-polarized PDFs at the physical pion mass [30, 69] usingthe quasi-PDFs approach were determined using smallmomentum with P max ≈ . a ≈ .
09 fm). This, in addition to the chal-lenges in reconstructing the x dependence, were shownto have led to the wrong sign of sea-flavor asymme-try [30]. Later calculations at physical pion mass pushedthe nucleon boost momentum 3 GeV [70]. However,the lattice discretization systematics were not taken intoaccount, and the twist-4 effects were assumed to be O (Λ /P z ). The latter estimated systematic is negli-gible, since these few-percent effects at this large momen-tum are much smaller than the statistical and other sys-tematics. Since we account for all these neglected system- This is because in the e ( m − /a − m ) | z | structure of the renor-malization factor, m − does not depend on the matrix elementused for its extraction, but m does. It was then shown thatthis uncertainty induced an O (1 /P z ) uncertainty in the quasi-PDFs [64]. The error coming from O (Λ QCD /P z ) is only a few percent, toosmall to be seen at the scale of the results, so we ignore it hereand focus on the larger sources of uncertainty. atics in this work, the total uncertainty appears largerthan those of previous quasi-PDF works, even thoughthe statistical error remains comparable. When com-paring our continuum-physical nucleon isovector PDFswith those obtained from global fits, CT18NNLO [149],NNPDF3.1NNLO [151], ABP16 [150], and CJ15 [152], wefound our results, even with only the errors considered byinner statistical bands, have nice agreement. The errorsincrease toward the smaller- x region for both lattice andglobal fitted PDFs, but overall, they agree within twostandard deviations. IV. SUMMARY AND OUTLOOK
In this work, we presented the first determination inthe physical-continuum limit of the nucleon isovector par-ton distribution, using six lattice ensembles, including 3lattice spacings, multiple volumes and a physical pionmass. We found small a small pion-mass dependenceand no sizable finite-volume effects, but a noticeabletrend of the matrix elements changing from 0.12 fm to0.06 fm. The resulting continuum-physical matrix ele-ments are dominated by the lattice-spacing extrapola-tion. Our analysis results in PDFs consistent with var-ious global PDF fits with excellent agreements for mid-to large- x regions, and compatible within 2 standard de-viations for x < .
4. The nucleon isovector moments (cid:104) x n (cid:105) are around 0.2, 0.06, and 0.04 for n = 1 , ,
3, re-spectively. Currently, we use a conservative systematicerror estimate, mainly dominated by twist-4 systematicson the order of O (Λ QCD /P z ). The small- x and antiquarkPDFs are not reliably extracted in this work; future workwill focus on reducing the twist-4 systematics and push-ing toward improving the lattice determination of small- x and antiquark PDFs. Acknowledgments
We thank the MILC Collaboration for sharing thelattices used to perform this study. The LQCD cal-culations were performed using the Chroma softwaresuite [146] with the multigrid solver algorithm [144, 145].This research used resources of the National Energy Re-search Scientific Computing Center, a DOE Office ofScience User Facility supported by the Office of Sci-ence of the U.S. Department of Energy under ContractNo. DE-AC02-05CH11231 through ERCAP and ALCC;the Extreme Science and Engineering Discovery Environ-ment (XSEDE), which is supported by National ScienceFoundation grant number ACI-1548562; facilities of theUSQCD Collaboration, which are funded by the Office ofScience of the U.S. Department of Energy, and supportedin part by Michigan State University through compu-tational resources provided by the Institute for Cyber-Enabled Research (iCER). The work of HL is also partlysupported by the Research Corporation for Science Ad-
MSULat'20, P z a → P z a → P z a ≈ P z a ≈ x u - d MSULat'20, P z a → P z a → x u - d FIG. 5: The nucleon isovector unpolarized PDFs from our lattice calculation in the physical-continuum limit, compared withpast lattice quasi-PDF results from LP and ETMC (left) [69, 70], and global fits from Refs. [149–152] (right). Note thatthe previous lattice work by LP3’18 and ETMC’18 were done using a single lattice spacing at physical pion mass and did nottake into account the systematics due to twist-4 effects, while our work (MSULat’20) includes this systematic as well as thereconstruction errors. vancement through the Cottrell Scholar Award. HL andRZ are supported by the US National Science Foundationunder grant PHY 1653405 “CAREER: Constraining Par-ton Distribution Functions for New-Physics Searches”. JWC is partly supported by the Ministry of Science andTechnology, Taiwan, under Grant No. 108- 2112-M-002-003-MY3 and the Kenda Foundation. [1] H.-W. Lin et al., Prog. Part. Nucl. Phys. , 107(2018), 1711.07916.[2] H.-W. Lin, W. Melnitchouk, A. Prokudin, N. Sato,and H. Shows, Phys. Rev. Lett. , 152502 (2018),1710.09858.[3] X. Ji, Phys. Rev. Lett. , 262002 (2013), 1305.1539.[4] X. Ji, Sci. China Phys. Mech. Astron. , 1407 (2014),1404.6680.[5] Y.-Q. Ma and J.-W. Qiu, Phys. Rev. Lett. , 022003(2018), 1709.03018.[6] T. Izubuchi, X. Ji, L. Jin, I. W. Stewart, and Y. Zhao,Phys. Rev. D98 , 056004 (2018), 1801.03917.[7] Y.-S. Liu, W. Wang, J. Xu, Q.-A. Zhang, J.-H. Zhang,S. Zhao, and Y. Zhao, Phys. Rev. D , 034006 (2019),1902.00307.[8] X. Xiong, X. Ji, J.-H. Zhang, and Y. Zhao, Phys. Rev.
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