Spin-Dependent Parton Distribution Function with Large Momentum at Physical Pion Mass

Spin-Dependent Parton Distribution Function with Large Momentum at Physical Pion Mass


We present a lattice-QCD calculation of the longitudinally polarized isovector parton distribution function (PDF) using ensembles at the physical pion mass with large proton boost momenta  GeV within the framework of large-momentum effective theory (LaMET). In contrast with our previous physical-pion PDF result in 2017, we increase the statistics significantly, double the boost momentum and increase the investment in controlling excited-state contamination by using six source-sink separations in our analysis. We detail the systematics that affect spin-dependent PDF calculations, providing a guide so that future calculations can further improve the precision of lattice-determined PDFs. We find our final parton distribution to be in reasonable agreement with the current global PDF curves provided by NNPDF, JAM and DSSV.


Lattice Parton Physics Project () Collaboration

Introduction: The nucleon spin-dependent parton distribution function (PDF) or helicity distribution, describes the number density of partons (quark and gluon) with given longitudinal momentum fraction and given polarization within a nucleon polarized longitudinally with respect to its motion. Understanding the spin structure of the proton is one of the most challenging frontier problems of modern physics. Decades of polarized deep inelastic scattering (DIS) and semi-inclusive DIS (SIDIS) data at a wide range of kinematics has greatly improved our understanding of these structures. Significant progress has been made in recent years: the determination of the polarized gluon distribution at small  de Florian et al. (2014) using the data on inclusive jet and pion production in polarized collisions at the Relativistic Heavy-Ion Collider (RHIC) Adamczyk et al. (2015); Adare et al. (2014, 2016a) and double spin asymmetries from open-charm muon production at COMPASS Adolph et al. (2013); the constraints on the polarization of sea quarks and antiquarks with longitudinal single-spin asymmetries in -boson production Adamczyk et al. (2014); Adare et al. (2016b), just to name a few. In the future, the kinematic coverage for spin-dependent PDFs is expected to be greatly extended with new data on DIS and SIDIS from JLab 12-GeV Dudek et al. (2012) and a future facility for polarized high-energy collisions, the Electron-Ion Collider (EIC) Accardi et al. (2016).

There are currently a number of global-analysis efforts to combine all available experimental data to extract helicity distribution. Using the positivity of the cross section, the polarized PDF must be smaller or equal to its unpolarized counterpart; that is, . and symmetry are often imposed in the global analysis with the integration of the isovector polarized PDF giving the nucleon axial charge fixed at the ultra-cold neutron -decay value, around 1.27 (with small variance). A few of the latest well known global analyses, such as DSSV14 de Florian et al. (2014), JAM17 Ethier et al. (2017), NNPDFpol1.1 Nocera et al. (2014), are being updated with improved statistical techniques. However, a few open questions remain: symmetry violation can be large Cabibbo et al. (2003); some claim up to 30% uncertainty in  Flores-Mendieta et al. (1998). Significant uncertainties dominate the polarized PDFs in the large- () valence region, which is important to discriminate among various QCD models Nocera (2015). The fragmentation functions (FFs) needed to extract polarized PDFs from SIDIS data are less well known than the unpolarized cases. The determination of the strange polarized PDF is nontrivial in global analysis, and precision kaon SIDIS data are needed to reduce the uncertainty in and test breaking. While many issues can be resolved with future experimental data in the long term, direct input from lattice QCD in the next couple years can provide a speed-up in the spin-dependent PDFs and work hand-in-hand with global-fitting PDF analyses to provide more precise spin-dependent PDFs.

Using lattice QCD to study the Bjorken- dependent PDFs is not an easy task. On the theoretical side, PDFs are defined through the lightcone correlations of quark fields within a single-hadron state which involves strong infrared (IR) dynamics. Nonperturbative methods, such as lattice QCD, are ideal to study their characteristics at low energy scales (around 1 GeV or so). The direct calculation of PDFs based on lattice QCD is still extremely hard, as it is infeasible to perform the lattice calculation directly in the Minkowski spacetime to extract lightcone correlators. Although the lower moments of the PDFs were investigated decades ago Martinelli and Sachrajda (1987, 1989); Detmold et al. (2001); Dolgov et al. (2002) based on operator product expansion (OPE), higher-moment calculations were not feasible for a long time due to symmetry breaking. Reconstructing the dependence using only a few low moments is not sufficient to even disentangle the quark and antiquark contributions.

A breakthrough in direct calculation of dependence in hadron structure using lattice QCD emerged in recent years through large-momentum effective theory (LaMET) Ji et al. (2013); Ji (2013); Hatta et al. (2014); Ji (2014); Ji et al. (2015a). On the lattice, one can calculate the matrix element of certain static operators in a boosted hadron state, and then further process them to extract the PDFs with the LaMET matching and factorization theorem. To study spin-dependent PDFs, one starts with calculating the helicity quark “quasi-PDF”


where , with the Wilson line . With large , and proper renormalization, it can be factorized into a perturbative matching kernel and the corresponding light-cone helicity quark PDF, up to power corrections suppressed by .

There has been many progress made since the first LaMET paper Ji et al. (2013) providing further theoretical supports in its completeness and supporting calculations in connecting Lattice QCD quasi-distribution to the true lightcone distribution. On the theoretical side, works has been done in computing the matching kernel connecting the quasi-PDFs to the PDFs at one-loop order Xiong et al. (2014); Ji and Zhang (2015); Ji et al. (2015b); Xiong and Zhang (2015); Ji et al. (2015c, 2018a); Stewart and Zhao (2018); Constantinou and Panagopoulos (2017); Green et al. (2018); Izubuchi et al. (2018); Wang et al. (2018); Wang and Zhao (2017), correcting the nucleon-mass effects Chen et al. (2016), and studying the renormalization properties of the quasi-PDF operator Ji and Zhang (2015); Ishikawa et al. (2016); Chen et al. (2017a); Xiong et al. (2017); Constantinou and Panagopoulos (2017); Ji et al. (2018b); Ishikawa et al. (2017); Green et al. (2018); Chen et al. (2018a). On the lattice-calculation side, progress has been made in the isovector quark PDF of the nucleon Lin et al. (2015); Alexandrou et al. (2015); Chen et al. (2016); Alexandrou et al. (2017a); Lin et al. (2017); Alexandrou et al. (2018) the meson distribution amplitudes (DAs) Zhang et al. (2017); Chen et al. (2017b), and the nonperturbative renormalization (NPR) of the nonlocal quark bilinear operator in the regularization-independent momentum-subtraction (RI/MOM) scheme Chen et al. (2018a); Alexandrou et al. (2017b). Certain technical issues regarding the NPR were raised and resolved in Refs. Constantinou and Panagopoulos (2017); Alexandrou et al. (2017b); Green et al. (2018); Chen et al. (2018a, 2017c); Lin et al. (2017); Chen et al. (2017d). In parallel, there are other proposals to calculate the PDFs from lattice QCD Ma and Qiu (2014, 2018); Radyushkin (2017); Orginos et al. (2017); Liu and Dong (1994); Liang et al. (2018); Detmold and Lin (2006); Braun and Mueller (2008); Bali et al. (2018); Chambers et al. (2017) complementary to LaMET approach.

In addition, progress in the lattice QCD has been made in spin-dependent PDFs. Our earlier reference Lin et al. (2015) reports a clear sign for the total polarized sea asymmetry , which was later confirmed in updated measurements by STAR Adamczyk et al. (2014) and PHENIX Adare et al. (2016b) collaborations. The lattice calculation was done with a heavier than physical pion mass  MeV, which reduced the cost of the exploratory calculation and enhanced the signal-to-noise ratio. Three different nucleon momenta () were used to study the suppressed power correction with largest momentum around 1.3 GeV. The full matching calculations and mass corrections were reported in a later paper Chen et al. (2016). A followup work done by ETMC Alexandrou et al. (2017a) at slightly heavier pion mass and similar nucleon momenta shows similar results. The first spin-dependent PDFs with fully RI/MOM NPR were reported by  Lin et al. (2017), and recently with high statistics by ETMC  Alexandrou et al. (2018) with similar boost momenta used.

In this paper, we report calculations at physical pion mass that are improved over our previous work Lin et al. (2017) on the helicity isovector quark distribution. High statistics and large boost nucleon momentum (up to 3.0 GeV) are presented and shown necessary to access antiquark results. The nucleon matrix elements are renormalized using RI/MOM scheme, and a new matching calculation is done to connect the RI/MOM quasi-PDF to the renormalized lightcone PDF. We increase the number of source-sink separations in combination with multiple-state analysis to remove excited-state contamination from the wanted nucleon ground-state. We found that in the moderate to large- region, our result shows a significant improvement compared to previous lattice studies, and are consistent with the global PDF analyses by NNPDF, JAM and DSSV groups. We also see evidence of , but further improvements in the future work are needed to get more precise distribution in these regions.

Figure 1: The real (left) and imaginary (right) parts of the bare isovector nucleon matrix elements for spin-dependent PDFs as functions of at all three momenta (2.2, and 2.6 and 3.0 GeV indicated by red, blue and green, respectively). Their kinematic factors are omitted to enhance visibility by separating the small- matrix elements. At a given positive value, the data are slightly offset to show different ground-state extraction strategies; from left to right they are: two-simRR using all , two-simRR using the largest 5 , two-sim using the largest 4 , and two-sim using the largest 3 . Different analyses are consistent within statistical errors, which suggests the excited-state contamination is well controlled.

Lattice-QCD Calculation Setup: In this paper, we report the results of a lattice-QCD calculation using clover valence fermions on an ensemble of gauge configurations with lattice spacing  fm, box size  fm and pion mass  MeV with (degenerate up/down, strange and charm) flavors of highly improved staggered quarks (HISQ) Follana et al. (2007) generated by MILC Collaboration Bazavov et al. (2013). The gauge links are hypercubic (HYP)-smeared Hasenfratz and Knechtli (2001) and then the clover parameters are tuned to recover the lowest pion mass of the staggered quarks in the sea Gupta et al. (2017); Bhattacharya et al. (2015a, b); Bhattacharya et al. (2014). Only one step of HYP smearing is used to improve the discretization effects; too much smearing may alter the ultraviolet results for the PDF. We use Gaussian momentum smearing Bali et al. (2016) for the quark field , where is the input momentum parameter, are the gauge links in the direction, and is a tunable parameter as in traditional Gaussian smearing. Such a momentum source is designed to increase the overlap with nucleons of the desired boost momentum, and allows us reach higher boost momentum for the nucleon states than our previous work at physical pion mass Lin et al. (2017). We use multigrid algorithm Babich et al. (2010); Osborn et al. (2010) in Chroma software package Edwards and Joo (2005) to speed up the physical pion mass clover fermion inversion on the quark propagator, allowing us to reach high-statistical calculations. We use multiple values of nucleon boost momenta, , with , which correspond to 2.2, 2.6 and 3.0 GeV nucleon momenta.

We carefully investigate the excited-state contamination in the nucleon matrix elements. Excited-state contamination is notorious for contaminating the nucleon axial charge in many past lattice-QCD calculations. As we increase the nucleon boost momentum, we anticipate that excited-state contamination worsens, since the states are relatively closer to each other; therefore, a careful study in the excited-state contamination is necessary for the LaMET (or quasi-/pseudo-PDF) approach. First, we measure the nucleon matrix elements with six source-sink separations,  fm with the number of measurements K respectively. We use multiple analysis methods to remove excited-state systematics among these source-sink separations: Fit #1 uses the “two-sim” analysis described in Ref. Bhattacharya et al. (2014) to obtain the ground-state nucleon matrix elements using largest four source-sink separations. Fit #2 uses the same strategy as in Fit #1 but with only the largest three source-sink separations. Fit #3 uses the “two-simRR”analysis Bhattacharya et al. (2014), which includes an additional matrix element related to excited states; to counter the increase of degrees of freedom, we use the largest five separations and all five separations as Fit #4. Figure 1 shows the bare matrix elements at a range of positive ranges for all three-momenta. All four fitting strategies yield consistent results. The two-simRR analysis uses the as small as 0.54 fm, gives consistent results from the two-sim analysis using of 0.81 fm with roughly the same statistical errors after removing the excited-state contamination. Similar results are given by Fit #1 and Fit #3, which have about the same uncertainties but with much larger error than the other 2 fits when using fewer three-point nucleon correlators. We dropped the smallest from our final analysis.

A nonperturbative renormalization on the lattice is required to obtain the continuum limit of the quasi-PDF matrix element, which is subject to UV power divergences. In this work, we follow the RI/MOM scheme elaborated in Refs. Stewart and Zhao (2018); Chen et al. (2018a), and match the result to the PDF with the one-loop matching coefficient calculated with the method developed in Ref. Stewart and Zhao (2018). We apply RI/MOM scheme nonperturbative renormalization (NPR) as described in our earlier work Chen et al. (2018a, b) but switch the to . The renormalization constant depends on the lattice spacing as well as the other two scales and , defined in Eq. 2 in Ref. Chen et al. (2018b)). It is used to renormalize the bare nucleon matrix element of the quasi-PDF in coordinate space,


Note that the continuum limit of is well defined and should be taken before the matching to the PDF; continuum extrapolation is, however, beyond the scope of this work.

The momentum source is used to generate NPR propagators with a number of quark momenta ranging . We study the momentum dependence using ranging from 2.3 to 3.7 GeV and quark momenta . Figure 2 shows the dependence of at fixed  GeV. We find in the small- region, there is a notable change in renormalization constants, while at large , it seems to reach a plateau and become stable. Similar behavior is also observed in the  GeV case. Therefore, we pick = as our central value for the renormalization constant.

Figure 2: The values of (the inverse of the renormalization constant) with  GeV at (top,  fm) and (bottom,  fm) as functions of . Note that becomes stable at large .
Figure 3: The real (top) and imaginary (bottom) parts of the renormalized nucleon matrix elements as functions of displacement. Different colors here (red, green, blue) indicate different boost nucleon momenta 2.2, 2.6 and 3.0 GeV, respectively. We normalize all the matrix elements with .

Parton Distribution Function Results and Discussion: We use the “derivative” method proposed in our earlier work Lin et al. (2017) to improve the truncation error due to the Fourier transformation into space:


Note that is consistent with zero for , and we vary to estimate the systematics due to this parameter choice.

Following the framework described in Refs. Stewart and Zhao (2018); Izubuchi et al. (2018), the matching between the quasi-PDF and the PDF at scale we obtain is


where , and the antiquark distribution falls in the region . The matching coefficient definitions at one-loop level can be found in the Ref. Liu et al. (2018). We take a small- expansion with the matching condition that is to be convolved with to obtain the PDF , and predict the PDF using the quasi-PDF obtained by lattice calculation. We include the systematic uncertainty introduced by this assumption in the final error budget.

Using the renormalized quasi-PDF and applying the matching, we show in Fig. 4 one of the three nucleon boost momenta,  GeV, before and after applying the matching formula of Eq. Spin-Dependent Parton Distribution Function with Large Momentum at Physical Pion Mass and mass correction. The matching raises the antiquark (negative- region) distribution to the same asymmetry for , and lowers the positive mid- to large quark distribution, compared with our exploratory study and heavier-pion PDF. After matching, we study the dependence on the nucleon boost momentum, shown in Fig. 5. Within the statistical errors, the distribution seems to converge across the three momenta. However, the central values shift noticeably from 2.2 to 3.0 GeV, moving the antiquark distribution toward the asymmetry measured in experiment: .

Figure 4: The spin-dependent quasi-PDF, matched PDF, and matched PDF with mass correction at nucleon boost momentum 3.0 GeV. The matching process lowers the quasi-PDF at large positive and enhances the small- region’s quark asymmetry. It significantly changes the antiquark asymmetry at this nucleon momentum.
Figure 5: Nucleon boost momentum dependence of the matched polarized isovector PDFs. For quark asymmetry, the shape is consistent throughout most regions. However, in the antiquark region, there is a significant change in distribution as momentum increases.

Our final result, focusing at the largest boost momentum of 3 GeV, is shown in Fig. 6. It significantly improves on our previous results at physical pion mass Lin et al. (2017). We increase the nucleon momenta used in the calculation from 0.4, 0.8, 1.3 GeV to 2.2, 2.6 and 3.0 GeV. We increase at least a factor of 10 in statistics and provide a more complete analysis in removing excited-state contamination. We use the derivative method to improve the truncation systematics in the quasi-PDF Fourier transformation. In addition, we apply the complete one-loop matching needed from the RI/MOM NPR for the quasi-PDF and the all-order mass correction to obtain the lightcone PDF. In comparing the results from ETMC Alexandrou et al. (2018), which use similar lattice spacing, our result uses larger boost nucleon momenta , up to 3 GeV, while theirs are only up to 1.3 GeV. Instead of performing a two-step matching like ETMC, we choose to directly match to the PDF, and the matching coefficient for the polarized case has already been calculated at one-loop order in perturbation theory Stewart and Zhao (2018). Although in principle it is equivalent to the two-step procedure described above, the direct matching can possibly save us from additional systematic uncertainties when we implement them numerically on the lattice data. Their result focused on large using a one-state analysis, whereas we simultaneously fit multiple with two or three nucleon matrix elements involving excited states. In this way, the ground state can be obtained more reliably before noise comes to dominate at large . As a result of smaller , the smaller in Fourier transformation means there will be a smaller reliable -region in their distribution. This is possibly due to the oscillation induced by truncation of the Fourier transform, which is not treated in their work; they see oscillation that continues into the regions.

We estimated our systematic uncertainty by varying the input of the scales in the NPR for and , and one-loop inversion matching to obtain lightcone PDFs, which is shown in the gray band in Fig. 6. We find the target-mass correction from Ref. Chen et al. (2016) to be extremely small for all three nucleon boost momenta (less than 1%). When we reconstruct a known PDF, there is significant difficulty in reproducing the small- region . Note that other lattice PDF calculations use smaller , yielding smaller reliable regions in -space. By extending to larger momentum at finer lattice spacings in future work, this can be improved straightforwardly. The error should be studied in the future as well.

We have presented new lattice-QCD results for the isovector spin-dependent PDF (that is, the up-down quark asymmetry in the proton), which has much potential impact on current PDF estimates in the near future: better determination at large- region of isovector spin-dependent PDF can be used as constraints in combining with global PDF analysis for better determined spin-dependent PDFs. A recent community whitepaper produced in collaboration between lattice and global-analysis practitioners Lin et al. (2018) predicted that a calculation of the large- isovector with 10% final error can improve on the current PDF, especially in the antiquark regions where experimental inputs are even scarcer. Currently, we are able to reproduce the global spin-dependent PDF results; the next step will be to plan improved calculations with total uncertainty less than 10%.

Summary and Outlook: In this work, we report the state-of-the-art isovector spin-dependent quark distribution using lattice QCD directly at physical pion mass with nucleon boosted momenta as large as 3 GeV. With high statistics, we combined multi-state analysis and multiple source-sink separations to carefully remove excited-state contamination from our analysis; its error is reflected in our statistical uncertainty. We renormalize our nucleon matrix element using the nonperturbative RI/MOM renormalization, and perform the LaMET one-loop finite-momentum matching and conversion to scheme to connect the lattice quasi-distribution to the lightcone distribution. An estimate of the systematic uncertainty introduced by the choice of scale in the nonperturbative RI/MOM renormalization and one-loop inversion matching is included in the final analysis. Our final spin-dependent PDFs are consistent with the global analyses done by NNPDF, JAM and DSSV. Future directions will be investigating finer lattice-spacing ensembles to reach even higher boost momenta, so that we can push toward smaller in advance of upcoming experiments such as at the EIC.

Figure 6: Our final isovector spin-dependent PDF at the largest boosted nucleon momentum at 3 GeV renormalized at 3 GeV. The pink distribution shows our statistical errors (which fold in the excited-state uncertainty in our analysis), and the outer gray bands indicate the estimation of the total uncertainty including systematics. Our result is consistent with the ones from NNPDFpol1.1 Nocera et al. (2014) DSSV de Florian et al. (2014), JAM Ethier et al. (2017) within .


We thank the MILC Collaboration for sharing the lattices used to perform this study. The LQCD calculations were performed using the Chroma software suite Edwards and Joo (2005). This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 through ALCC and ERCAP; facilities of the USQCD Collaboration, which are funded by the Office of Science of the U.S. Department of Energy, and supported in part by Michigan State University through computational resources provided by the Institute for Cyber-Enabled Research. HL and YY are supported by the US National Science Foundation under grant PHY 1653405 “CAREER: Constraining Parton Distribution Functions for New-Physics Searches”. JC is partly supported by the Ministry of Science and Technology, Taiwan, under Grant No. 105-2112-M-002-017-MY3 and the Kenda Foundation. LJ is supported by the Department of Energy, Laboratory Directed Research and Development (LDRD) funding of BNL, under contract DE-EC0012704. YL is supported by Science and Technology Commission of Shanghai Municipality (Grant No.16DZ2260200) and National Natural Science Foundation of China (Grant No.11655002). JZ is supported by the SFB/TRR-55 grant “Hadron Physics from Lattice QCD”, and a grant from National Science Foundation of China (No. 11405104). YZ is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, from DE-SC0011090 and within the framework of the TMD Topical Collaboration.



  1. preprint: MSUHEP-18-013,MIT-CTP/5032


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