NUCLEON AND PION GLUON PARTON DISTRIBUTION FUNCTION FROM LATTICE QCD CALCULATION By Zhouyou Fan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics – Doctor of Philosophy 2022 ABSTRACT NUCLEON AND PION GLUON PARTON DISTRIBUTION FUNCTION FROM LATTICE QCD CALCULATION By Zhouyou Fan Parton distribution functions (PDFs) are important to characterize the structure of the hadrons such as protons and neutrons. The contribution to the structure from quarks has been studied in detail during the past few decades. The structure in the gluon sector is also important but less studied. For high-energy hadrons, the gluon contribution dominates at small x, where x is the momentum fraction carried by a quark or gluon. At large x, the uncertainty of the gluon PDF is large, especially compared to that of the quark PDFs at large x. Gluon PDFs for nucleons and pions are mostly extracted from global analysis of experimental data using perturbation theory as a guide. Theoretically, lattice QCD provides an independent non-perturbative theoretical approach to calculate the gluon PDFs. We present the exploratory study of nucleon gluon PDFs from lattice QCD using the quasi-PDF approach. Using valence overlap fermions on the 2 + 1- flavor domain-wall fermion gauge ensemble, the quasi-PDF matrix elements we obtain agree with the Fourier transform of the global-fit PDF within statistical uncertainty. We further study the x-dependent nucleon and pion gluon distributions via the pseudo-PDF approach on lattice ensembles with 2 + 1 + 1 flavors of highly improved staggered quarks (HISQ) generated by the MILC Collaboration. Using clover fermions for the valence action, and adding momentum smearing, PDFs are found for pion boost momenta up to 2.56 GeV. Several lattice sizes and spacings (a ≈ 0.9, 0.12 and 0.15 fm) were evaluated, resulting in three pion masses, Mπ ≈ 220, 310 and 690 MeV/c2 . ACKNOWLEDGEMENTS The researches described in this dissertation used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Of- fice of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 through 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 (iCER). The projects are partially supported by the US National Science Foundation under grant PHY 1653405 “CAREER: Constraining Parton Distribution Functions for New-Physics Searches”. I am deeply grateful to my advisor Professor Huey-Wen Lin for the thorough mentor-ship during my graduate study. I am not able to graduate within 5 years without her continual support and suggestions in daily research, rigorous work attitude and time arrangement. I would like to thank Yi-Bo Yang for his help in the first exploratory study of gluon PDF and his patiently introduction of some basic ideas on LQCD and coding, Jian-Hui Zhang and Jiunn- Wei Chen for earlier discussions on the gluon quasi-PDF, Rui Zhang for useful discussions in the earlier stage of gluon pseudo-PDF study, Alejandro Salas and Santanu Mondal for their comments on this dissertation. I thank MILC Collaboration for sharing the lattices used to perform this study. The LQCD calculations were performed using the Chroma software suite [1]. I thank JAM Collaboration for providing us the pion xg(x) data with uncertainties for comparison. Moreover, I would like to say thank you to my committee members, Professor Chien-Peng Yuan, Professor Scott Pratt, Professor Alexei Bazavov and Professor Joey Huston for their guidance and support. Finally, I have my special thanks to my parents, who have encouraged me throughout my graduate career, especially when I felt depressed. iii TABLE OF CONTENTS LIST OF TABLES vi LIST OF FIGURES vii I Introduction to Parton distribution functions (PDFs) 1 I.1 unpolarized nucleon gluon PDFs 1 I.2 pion gluon PDFs 2 II Lattice QCD 4 II.1 The continuum QCD 4 II.2 The formulation of Lattice QCD 5 II.2.1 Gauge actions 6 II.2.2 Fermion action 7 II.2.2.1 Staggered Fermion 9 II.2.2.2 Wilson-like Fermions 12 II.2.2.3 Other fermion actions 13 II.2.2.4 Mixed-action 14 II.3 Correlation functions 14 II.3.1 Smearing 15 II.3.2 propagator and inversion 17 II.3.3 Two-point correlators 20 II.4 Nonperturbative renormalization 22 III Bjorken x-dependence PDF from lattice QCD 27 III.1 Large Momentum Effective Theory 28 III.1.1 Non-singlet quark quasi-PDF 29 III.1.2 Gluon quasi-PDF 30 III.2 Pseudo-PDF method 31 III.2.1 Quark pseudo-PDF 32 III.2.2 Gluon pseudo-PDF 33 III.3 Nucleon Isovector Quark PDFs 34 IV Meson gluon PDF results 36 IV.1 Ioffe-time distribution 38 IV.2 Pion gluon PDF 44 IV.3 Summary 47 V Nucleon gluon PDF 50 V.1 First Exploratory Study 50 V.2 First Pseudo-PDF Study 57 V.2.1 Results and Discussions 62 V.2.2 Summary and Outlook 67 V.3 Updated Pseudo-PDF Study 67 V.3.1 Lattice correlators and matrix elements 68 V.3.2 Results and Discussions 69 V.3.2.1 xg(x)/ < x >g Results 69 iv V.3.2.2 Renormalized gluon moments 74 V.3.2.3 xg(x) Results 80 V.3.3 Summary 81 VI Conclusion 83 APPENDIX 86 BIBLIOGRAPHY 113 v LIST OF TABLES Table 1 Lattice spacing a, valence pion mass Mπval and ηs mass Mηval s , lattice size L3 × T , number of configurations Ncfg , number of total two-point correlator 2pt measurements Nmeas , and separation times tsep used in the three-point corre- lator fits of Nf = 2 + 1 + 1 clover valence fermions on HISQ ensembles generated by MILC Collaboration and analyzed in this study. 39 Table 2 Our gluon PDF fit parameters, A and C, from Eq. V.13, and goodness of the fit, χ2 /dof, for calculations at two valence pion masses and the extrapolated physical pion mass. 64 Table 3 Predictions for the higher gluon moments from this work at pion mass for about 690 MeV, 310 MeV, and the extrapolated 135 MeV. The moments predictions are compared with the corresponding ones obtained from CT18 NNLO and NNPDF3.1 NNLO global fits. The first error in our number cor- responds to the statistical errors from the calculation and the second errors are the systematic errors. 66 Table 4 Lattice spacing a, valence pion mass Mπval and ηs mass Mηval s , lattice size L3 × T , number of configurations Ncfg , number of total two-point correlator 2pt measurements Nmeas , and separation times tsep used in the three-point corre- lator fits of Nf = 2 + 1 + 1 clover valence fermions on HISQ ensembles generated by the MILC collaboration and analyzed in this study. More de- tails on the parameters used in the calculation are included in the Table 7 in the appendix. 69 Table 5 The truncation length Lc , the number of configurations Ncfg and measure- ments Nmeas that we used for different lattice ensembles. We use 16 sources for the truncation on each configurations; thus, Nmeas is 16 times Ncfg . 76  −1 Table 6 The complete multiplicative renormalization constant ZOMS (0), the bare gluon momentum fraction hxig bare , and the renormalized gluon momentum fraction hxiMSg for four ensembles used in this calculation. We use the a12m310 NPR factors for a12m220 hxiMS g calculation, since the mass dependence is weak for the NPR factors. 79 Table 7 Lattice spacing a, valence pion mass Mπval and ηs mass Mηval s , lattice size 3 L × T , number of configurations Ncfg , number of total two-point correlator 2pt measurements Nmeas , the Gaussian smearing parameters {α, Ninteration }, the 2π momentum smearing parameters k in q(x) + α j Uj (x)ei( L )kêj q(x + êj ), P mass parameters ml and ms for light and strange quarks respectively, and separation times tsep used in the three-point correlator fits of Nf = 2 + 1 + 1 clover valence fermions on HISQ ensembles generated by the MILC collab- oration and analyzed in this study. 86 vi LIST OF FIGURES Figure 1 The CT18 unpolarized nucleon PDFs for u, u, d, d, s = s, and g at Q = 2 GeV and Q = 100 GeV. The gluon PDF g(x, Q) has been scaled down as g(x, Q)/5. This figure is taken from reference [2]. 1 Figure 2 Comparison between the pion PDFs from the determination by xFitter col- laboration [3], the JAM collaboration [4], and the GRVPI1 group [5]. This figure is taken from Fig. 3 in Ref. [3]. 3 Figure 3 Non-perturbative renormalization condition. In the left hand side is the tree amputated Green’s function, and the right hand side are the bare amputated Green’s function and renormalization factors. 24 Figure 4 The lattice calculations of isovector nucleon unpolarized (top), helicity (middle) and transversity (bottom) with quark&antiquark, left&right col- umn respectively, taken from [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. These figures are taken from reference [23]. 35 Figure 5 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from the lightest pion mass a ≈ 0.12 fm, Mπ ≈ 220 MeV for Pz = 2 × 2π/L, z = 1 (upper row) and Pz = 4 × 2π/L, z = 4 (lower row). The gray band shown on all plots is the extracted ground-state matrix element from the two-sim fit using tsep ∈ [5, 9]. From left to right, the columns are: the ratio of the three-point to two-point correlators with the reconstructed fit bands from the two-sim fit using tsep ∈ [5, 9], shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 9], the two-sim fit results using tsep ∈ [tmin sep , 9] as functions of tsep , and the two-sim fit results using tsep ∈ [5, tsep ] as functions of tmax min max sep . 40 Figure 6 Example RpITDs from the a12m220 ensemble as functions of tmin sep for Pz = 2 × 2π/L, z = 1 (top) and Pz = 4 × 2π/L, z = 4 (bottom). The two-sim fit RpITD results using tsep ∈ [tminsep , 9] are consistent with the ones final chosen tsep ∈ [5, 9]. 42 Figure 7 The ηs (top) and pion (bottom) RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV, respectively, for the a12m220, a12m310, and a15m310 ensembles. In both cases, we observe weak lattice-spacing and pion-mass dependence. 42 Figure 8 The RpITDs M with reconstructed bands from “z-expansion” fits (top) and the EpITDs G with reconstructed bands from the fits to the Eq. V.13 form (bottom) calculated on ensembles with lattice spacing a ≈ 0.12 fm, pion masses Mπ ≈ {220, 310, 690} MeV, and a ≈ 0.15 fm, Mπ ≈ 310 MeV, noticing that a ≈ 0.12, Mπ ≈ 690 MeV results are from a12m220 ensemble here. 43 vii Figure 9 The xg(x, µ)/hxig at µ2 = 4 GeV2 as function of x (bottom) calculated with lattice spacing a ≈ 0.12 fm, pion masses Mπ ≈ 220 MeV with the fitted bands of zmax ≈ 0.6 fm from the 1-, 2- and 3-parameter fits described in Eq. V.13 and the paragraph after it. 45 Figure 10 The pion gluon PDF xg(x, µ)/hxig as a function of x obtained from the fit to the lattice data on ensembles with lattice spacing a ≈ {0.12, 0.15} fm, pion masses Mπ ≈ {220, 310, 690} MeV (left plot and its inserted plot), and xg(x, µ)/hxig (x2 g(x, µ)/hxig in the inserted plot) as function of x obtained from lattices of a ≈ 0.12 fm, Mπ ≈ 220 MeV (right), compared with the NLO pion gluon PDFs from xFitter’20 and JAM’21, and the pion gluon PDF from DSE’20 at µ = 2 GeV in the MS scheme. The JAM’21 error shown is overestimated due to lack of available correlated uncertainties in its constituent components. Our PDF results are consistent with JAM [4, 24] and DSE [25] for x > 0.2, and xFitter [3] for x > 0.5. 48 Figure 11 The ratio R(tsep , t) for H̃0 (0, 0) at different tsep as a function of operator insertion time t (left panel), and the ratio R̃(tsep ) as a function of source- sink seperation tsep (right panel). Four colored points in the right panel corresponds to the R̃ at the separations plotted in the left-panel. 53 Figure 12 The bare H̃(z, Pz = 0.46 GeV) and the renormalized one H̃ Ra at 2 GeV with 1,3,5 HYP smearing steps, as functions of z. In H̃ Ra , the exponential falloff in the bare H̃ due to the linear divergence is removed by the “ratio renormalization factor” Z(µ, z) ≡ H0MS (0, 0, µ)/H̃0 (z, 0). Some data us- ing the same HYP smearing steps are shifted horizontally to enhance the legibility. 53 Ra Figure 13 The renormalized H̃i=0,1,2,3 (z, Pz ) as a functions of Pz at z=0 (top) and 3 (bottom). Some data with the same Pz are shifted horizontally to enhance the legibility. The case with Oi=3 suffers from a large contamination from higher-twist distributions, while the results with Oi=0,1,2 are consistent with each other, especially at larger Pz . 54 Figure 14 The final results of H̃0Ra (z, Pz ) at 678 MeV (top) and 340 MeV (bottom) pion mass as a functions of zPz , in comparison with the FT of the gluon PDF from the global fits CT14 [26] and PDF4LHC15 NNLO [27]. The data with Pz = 0.92 GeV are shifted horizontally to enhance the legibility. They are consistent with each other within the uncertainty. 56 Figure 15 The similar figure for the pion gluon quasi-PDF matrix elements with Mπ = 678 MeV. The shape is quite similar to the case in Fig. 14. 56 Figure 16 Nucleon effective-mass plots for Mπ ≈ 690 MeV (left) and Mπ ≈ 310 MeV (right) at z = 0, Pz = [0, 5] × 2π L on the a12m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 5 2π L is the largest momentum we used, and it is the noisiest data set. 59 viii Figure 17 Dispersion relations of the nucleon energy from the two-state fits for Mπ ≈ 690 MeV (left) and Mπ ≈ 310 MeV (right) 59 Figure 18 The three-point ratio plots for Mπ ≈ 690 MeV (top row) and Mπ ≈ 310 MeV (bottom row)nucleons z = 1 as functions of t−tsep /2, as defined in Eq. IV.5. The results for nucleon momentum Pz = 2 × 2π/L are shown. The gray bands in each panel indicate the extracted ground-state matrix elements of the operator Og . In each column, the plots show the fitted ratio and the ex- tracted ground-state matrix elements from two-simRR and two-sim fits with all 4 source-sink separations, and the two-state fits using only the smallest and largest tsep from left to right, respectively. The second column, which are the two-sim extracted ground-state matrix elements, are used in the sub- sequent analysis. The ground-state matrix elements extracted are stable and consistent among different fitting methods and three-point data input used. 61 Figure 19 The fitted bare ground-state matrix elements without normalization by kine- matic factors as functions of z obtained from the two-sim fit using different two- and three-point fit ranges for nucleon momentum Pz ∈ {0, 2, 4}×2π/L from left to right, respectively, for Mπ ≈ 690 MeV (first row) and Mπ ≈ 310 MeV (second row) nucleons. The green points, which represent the fit- 2pt range choice t3pt skip = 1, tmin = 3 are used in the following analysis, because the errors of the matrix elements of this fit range are relatively smaller than the error of the red points. The orange points, which represent the fit-range choice t3pt 2pt 2 skip = 1, tmin = 2, are not used because the χ /dof of the 2-point correlator fits with t2pt 2pt min = 2 are much larger than tmin = 3 cases. 61 Figure 20 The reduced ITDs M (ν, z 2 ) as functions of ν and their extrapolation to the physical pion mass at Pz = 1 × 2π/L (left) and Pz = 5 × 2π/L (right). The blue bands represent the fitted results of the reduced ITDs at the physical pion mass Mπ = 135 MeV. 62 Figure 21 The reduced ITDs M (ν, z 2 ) as functions of ν at pion masses Mπ = 690, 310 and extrapolated 135 MeV from left to right, respectively. The points of different colors represent the reduced ITDs M (ν, z 2 ) of different z 2 and the red band represents the z-expansion fit band. 63 Figure 22 The evolved ITDs G as functions of ν at pion masses Mπ ≈ 690, 310 and extrapolated 135 MeV from left to right, respectively. The points of different colors represent the evolved ITDs G(ν, z 2 ) of different z values. The red band represents the fitted band of evolved ITD matched from the functional form PDF using the matching formula Eq. V.12. The yellow and pink bands represent the evolved ITD matched from the CT18 NNLO and NNPDF3.1 NNLO unpolarized gluon PDF, respectively. The evolution and matching are both performed at µ = 2 GeV in the MS scheme. 63 ix Figure 23 The unpolarized gluon PDF, xg(x, µ)/hxg iµ2 (left), xg(x, µ) (middle), x2 g(x, µ) in the large-x region as a function of x (right), obtained from the fit to the lattice data at pion masses Mπ = 135 (extrapolated), 310 and 690 MeV com- pared with the CT18 NNLO (red band with dot-dashed line) and NNPDF3.1 NNLO (orange band with solid line) gluon PDFs. Our x > 0.3 PDF results are consistent with the CT18 NNLO and NNPDF3.1 NNLO unpolarized gluon PDFs at µ = 2 GeV in the MS scheme. 65 Figure 24 Left: The evolved ITDs G as functions of ν at Mπ ≈ 310 MeV with fits per- formed using different νmax cutoff in the evolved ITDs. As we can see from the tightening of the fit band, the evolved ITDs at larger ν are still useful in constraining the fit despite their larger errors. Middle: The unpolarized gluon PDF obtained from the fits to the evolved ITDs at 310-MeV pion mass with different νmax . The evolution and matching are both performed at µ = 2 GeV in the MS scheme. The larger the ν input, the more precise the PDF obtained. Right: The 2-GeV MS renormalized unpolarized gluon PDF obtained from a fit to the evolved ITDs generated from the CT18 NNLO PDF with νmax ∈ {3, 5, 6.54}, compared with the original CT18 NNLO un- polarized gluon PDFs. As ν increases, we can see the gluon PDF is better reproduced toward small x. Using this exercise, we can see that our lattice PDF is only reliable in the x > 0.25 region. By taking the moments ob- tained from CT18 with a cutoff of νmax = 6.54 compared to those from the original PDF, we can estimate the higher-moment systematics in our lattice calculation. 66 Figure 25 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from a12m220 and a09m310 ensembles light nucleon cor- relators at pion masses Mπ ≈ {220, 310} MeV. The gray band shown on all plots is the extracted ground-state matrix element from the two-sim fit we used as our final fit. From left to right, the columns are: the ratio of the three-point to two-point correlators with the reconstructed fit bands from the two-sim fit using the final tsep inputs, shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 10], the two-sim fit results using tsep ∈ [tmin max min max sep , tsep ] varying tsep and tsep . 70 Figure 26 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of z obtained from the fitted bare ground-state matrix elements for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. 70 Figure 27 The RpITDs M with the reconstructed bands from fits in Eq. V.14 on the a09m310, a12m310, a15m310 lattice ensembles for nucleon respectively. 72 Figure 28 The preliminary RpITDs M with the example reconstructed bands from fits in Eq. V.15 on the a09m310 (red), a12m220 (green) lattice ensembles in the left-hand plot and the extrapolated bands at physical pion mass and continuum limit in the right one. The lattice spacing a dependence and gluon-in-quark contribution are studied and compared in the right-hand plot. 73 x Figure 29 The preliminary RpITDs M with reconstructed bands from fits at phyis- cal pion mass and continuum limit, comparing with the RpITDs matched from global-fit PDFs, and the RpITD mean value reconstructed included the gluon-in-quark contribution. 74 Figure 30 The preliminary unpolarized gluon PDF, xg(x, µ)/hxig in the large-x region as a function of x and its zoomed in plot, obtained from the fits to the dif- ferent lattice ensembles data compared with the fit to the extrapolated data at physical pion mass and continuum limit, and the mean value fit including the gluon-in-quark term in the matching. 74 Figure 31 The preliminary multiplicative renormalization constants (ZOMS )−1 ((µ = 2 GeV)2 , p2 ) as function of p2 for a09m310, a12m310, a15m310 ensem- bles are shown in the upper-left, upper-right, lower-left plots, respectively. A comparison of different ensembles renormalization factor and their fit bands are shown in the lower-right plot. The fit band comes from the fit form in Eq. V.25. The lower limits of the fit range of the momentum are chosen to be the same as in Ref. [28]. 78 Figure 32 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from the a15m310 light nucleon correlators at pion masses Mπ ≈ 310 MeV. The gray band shown on all plots is the extracted ground- state matrix element from the two-sim fit using tsep ∈ [5, 8]. From left to right, the columns are: the ratio of the three-point to two-point correlators with the reconstructed fit bands from the two-sim fit using tsep ∈ [5, 8], shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 9], the two-sim fit results using tsep ∈ [tmin sep , 8] as min max functions of tsep , and the two-sim fit results using tsep ∈ [5, tsep ] as functions of tmax sep . 79 Figure 33 The preliminary renormalized gluon momentum fraction hxiMS g extrapola- tion in lattice spacing a and pion mass Mπ . The reconstructed fit bands at Mπ ∈ {135, 310, 690} MeV as function of a and the bands at a ∈ {0, 0.09, 0.12, 0.15} fm as function of Mπ are shown in the left and right plots, respectively. 80 Figure 34 The preliminary unpolarized gluon PDF, xg(x, µ) in the large-x region as a function of x and a zoomed plot, obtained from the fit to the different lattice ensembles data compared with the CT18 NNLO (red band with dot-dashed line) and NNPDF3.1 NNLO (orange band with solid line) gluon PDFs. Our PDF results are consistent with the CT18 NNLO and NNPDF3.1 NNLO unpolarized gluon PDFs at µ = 2 GeV in the MS scheme within errors. 81 Figure 35 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 11] for the a12m220 ensemble strange nucleon at pion masses Mπ ≈ 700 MeV, at the momentum Pz ∈ [0, 7] × 2π/L. tmin = 4 is used in the final 2-state fits for a12m220 strange nucleon 2-point correlators. 87 xi Figure 36 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 11] for the a12m310 ensemble strange nucleon at pion masses Mπ ≈ 690 MeV, at the momentum Pz ∈ [0, 5] × 2π/L. tmin = 4 is used in the final 2-state fits for a12m310 strange nucleon 2-point correlators. 88 Figure 37 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 10] for the a15m310 ensemble strange nucleon at pion masses Mπ ≈ 690 MeV, at the momentum Pz ∈ [0, 5] × 2π/L. tmin = 1 is used in the final 2-state fits for a15m310 strange nucleon 2-point correlators. 89 Figure 38 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 11] for the a12m220 ensemble light nucleon at pion masses Mπ ≈ 220 MeV, at the momentum Pz ∈ [0, 7] × 2π/L. tmin = 4 is used in the final 2-state fits for a12m220 light nucleon 2-point correlators. 90 Figure 39 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 13] for the a09m310 ensemble light nucleon at pion masses Mπ ≈ 310 MeV, at the momentum Pz ∈ [0, 5] × 2π/L. tmin = 4 is used in the final 2-state fits for a09m310 light nucleon 2-point correlators. 91 Figure 40 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 11] for the a12m310 ensemble light nucleon at pion masses Mπ ≈ 310 MeV, at the momentum Pz ∈ [0, 5] × 2π/L. tmin = 4 is used in the final 2-state fits for a12m310 light nucleon 2-point correlators. 92 Figure 41 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 10] for the a15m310 ensemble light nucleon at pion masses Mπ ≈ 310 MeV, at the momentum Pz ∈ [0, 5] × 2π/L. tmin = 1 is used in the final 2-state fits for a15m310 light nucleon 2-point correlators. 93 Figure 42 Nucleon effective-mass plots for Mπ ≈ 700 MeV, at Pz = [0, 7] × 2π L on the a12m220 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 7 2π L is the largest momentum we used, and it is the noisiest data set. 95 Figure 43 Nucleon effective-mass plots for Mπ ≈ 690 MeV, at Pz = [0, 5] × 2π L on the a12m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 5 2π L is the largest momentum we used, and it is the noisiest data set. 95 xii Figure 44 Nucleon effective-mass plots for Mπ ≈ 690 MeV, at Pz = [0, 5] × 2π L on the a15m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 5 2π L is the largest momentum we used, and it is the noisiest data set. 96 Figure 45 Nucleon effective-mass plots for Mπ ≈ 220 MeV, at Pz = [0, 7] × 2π L on the a12m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 7 2π L is the largest momentum we used, and it is the noisiest data set. 97 Figure 46 Nucleon effective-mass plots for Mπ ≈ 310 MeV, at Pz = [0, 5] × 2π L on the a09m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 5 2π L is the largest momentum we used, and it is the noisiest data set. 98 Figure 47 Nucleon effective-mass plots for Mπ ≈ 310 MeV, at Pz = [0, 5] × 2π L on the a12m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 5 2π L is the largest momentum we used, and it is the noisiest data set. 98 Figure 48 Nucleon effective-mass plots for Mπ ≈ 310 MeV, at Pz = [0, 5] × 2π L on the a15m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 5 2π L is the largest momentum we used, and it is the noisiest data set. 99 Figure 49 Dispersion relations of the nucleon energy from the two-state fits for Mπ ≈ {700, 690, 690, 690} MeV (left) on a12m220, a09m310, a12m310, a15m310 ensembles respectively. The speed of ligt c = 0.9638(24), 0.9695(48), 0.9067(47) repespectively. 100 Figure 50 Dispersion relations of the nucleon energy from the two-state fits for Mπ ≈ {220, 310, 310, 310} MeV (left) on a12m220, a09m310, a12m310, a15m310 ensembles respectively. The speed of ligt c = 0.986(12), 1.0174(89), 0.997(14), 0.931(29) repespectively. 101 Figure 51 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from the a12m220, a09m310, a12m310, a15m310 ensem- bles light nucleon correlators from top to bottom, at pion masses Mπ ≈ {700, 690, 690, 690} MeV. The gray band shown on all plots is the extracted ground-state matrix element from the two-sim fit using tsep ∈ [5, 8]. From left to right, the columns are: the ratio of the three-point to two-point corre- lators with the reconstructed fit bands from the two-sim fit using tsep ∈ [5, 8], shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 9], the two-sim fit results using tsep ∈ [tmin sep , 8] as min max functions of tsep , and the two-sim fit results using tsep ∈ [5, tsep ] as functions of tmax sep . 102 xiii Figure 52 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from the a12m220, a09m310, a12m310, a15m310 ensem- bles light nucleon correlators from top to bottom, at pion masses Mπ ≈ {220, 310, 310, 310} MeV. The gray band shown on all plots is the extracted ground-state matrix element from the two-sim fit using tsep ∈ [5, 8]. From left to right, the columns are: the ratio of the three-point to two-point corre- lators with the reconstructed fit bands from the two-sim fit using tsep ∈ [5, 8], shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 9], the two-sim fit results using tsep ∈ [tmin sep , 8] as min max functions of tsep , and the two-sim fit results using tsep ∈ [5, tsep ] as functions of tmax sep . 103 Figure 53 Example ratio plots from the a15m310 light nucleon correlators at pion masses Mπ ≈ 310 MeV from momentum smearing parameter k 6= 0 listed in Tab. 7 and k = 0 from the PNDME collaboration from left to right, respectively. 104 Figure 54 The bare gluon momentum fraction hxibare g and fitted bands normaliza- tion by kinematic factors as functions of momentum Pz = 2π × Nz /L for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. 105 Figure 55 The bare gluon momentum fraction hxibare g and fitted bands normalization by kinematic factors as functions of momentum Pz = 2π × Nz /L for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. 105 Figure 56 The fitted bare ground-state matrix elements without normalization by kine- matic factors as functions of z obtained from the two-sim fit for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. 106 Figure 57 The fitted bare ground-state matrix elements without normalization by kine- matic factors as functions of z obtained from the two-sim fit for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. 107 Figure 58 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of ν obtained from the fitted bare ground-state matrix elements for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. 108 Figure 59 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of s obtained from the fitted bare ground-state matrix elements for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. 108 xiv Figure 60 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of ν obtained from the fitted bare ground-state matrix elements for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. 109 Figure 61 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of z obtained from the fitted bare ground-state matrix elements for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. 109 Figure 62 RpITD fits in Eq. V.14 with different fit range z ∈ [0, zcut ] for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. The χ2 /dof of the fits are listed in the plot legends. 110 Figure 63 RpITD fits in Eq. V.14 with different fit range z ∈ [0, zcut ] for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 en- sembles respectively. The χ2 /dof of the fits are listed in the plot legends. 111 xv I. Introduction to Parton distribution functions (PDFs) Parton Distribution Functions (PDFs) represent the probability density to find a parton car- rying a momentum fraction x at energy scale µ, where the parton is quark or gluon inside a hadron. PDFs are important inputs for the calculations of the strong and weak interactions, one of the four fundamental forces. Therefore, the accuracy of the PDFs determines the accuracy of these calculations. Gluons play an important role in binding quarks and the generation of most of the mass of light quark hadrons through the mechanism of dynamical breaking of chiral symmetry (DCSB). The gluon nucleon spin and momentum fractions are about 35−50% at 2 GeV scale [29, 30, 31]. In current PDF analyses, gluon PDF dominates at low-x region especially at large scale µ. Gluon PDF g(x) contributes to the next-to-leading order (NLO) in the deep inelastic scattering (DIS) cross section, and enters at leading order in jet production [32, 33]. To calculate the cross section for these processes in pp collisions, g(x) needs to be known precisely. In this thesis, we mainly focus on the unpolarized nucleon and pion gluon PDFs. I.1. unpolarized nucleon gluon PDFs 2.0 2.0 CT18 at 2 GeV CT18 at 100 GeV s s 1.5 g/5 1.5 g/5 u u d d x*f(x,Q) x*f(x,Q) – – d d 1.0 – 1.0 – u u c c b 0.5 0.5 0.0 -6 0.0 -6 10 10-4 10-3 10-2 10-1 0.2 0.5 0.9 10 10-4 10-3 10-2 10-1 0.2 0.5 0.9 x x Figure 1 The CT18 unpolarized nucleon PDFs for u, u, d, d, s = s, and g at Q = 2 GeV and Q = 100 GeV. The gluon PDF g(x, Q) has been scaled down as g(x, Q)/5. This figure is taken from reference [2]. With the increasing high-precision measurements from LHC, RHIC, Tevatron, Jefferson Lab and HERA, quark distributions are now well-determined with only few-percent level in 1 many cases. However, gluon PDFs still have relatively large uncertainties because of the lim- ited experiments in constraining them. In the unpolarized gluon PDF case, it is now con- strained by the inclusion of processes such as inclusive jet production [34], top-quark pair distributions [35, 36], and direct photon production [37], inclusive deep-inelastic scattering (DIS) [38], D-meson production [39, 40], and the transverse momentum of Z bosons [41]. Al- though there is experimental data, e.g. top-quark pair production, which constrains g(x) in the large-x region, and charm production, which constrains g(x) in the small-x region. g(x) is still experimentally the least known unpolarized PDF because the gluon does not couple to electromagnetic probes. The future U.S.-based Electron-Ion Collider (EIC) [42], planned to be built at Brookhaven National Lab, will further our knowledge of gluon distribution [43, 44]. The Electron-Ion Collider in China (EicC) [45], is also aim at contributing to the gluon distri- butions. Currently, the global analysis of experimental measurements is the main approach to de- termine the PDFs. There are different collaborations have released their analyses of nucleon PDFs, including the unpolarized nucleon gluon PDF which of interest in this paper. The most recent global analyses PDF sets are CT18 PDF [2], NNPDF [46], and the MMHT14, ABMP, CJ, JAM, HERAPDF sets [47, 14, 16, 48, 49]. We show the PDF set from CT18 in Fig. 1. Different flavor quark and gluon unpolarized PDFs at Q = 2 GeV or Q = 100 GeV are shown in same plot. Since the gluon PDF g(x, Q) has been scaled down as g(x, Q)/5 to fit with other quark PDFs in Fig. 1, it is easy to see that the gluon PDFs are dominated at small-x region, especially with larger energy scale. I.2. pion gluon PDFs Global analyses of pion PDFs mostly rely on Drell-Yan data. The early studies of pion PDFs were based mostly on pion-induced Drell-Yan data in conjunction with J/ψ-production data or direct photon production data to constrain the pion gluon PDF [50, 51, 52, 5, 53]. There are more recent studies, such as the work by Bourrely and Soffer [54], that extract the pion PDF based on Drell-Yan π + W data. JAM Collaboration [4, 24] uses a Monte-Carlo approach to analyze the Drell-Yan πA and leading-neutron electroproduction data from HERA to reach the lower-x region, and revealed that gluons carry a significantly higher momentum fraction (about 2 Figure 2 Comparison between the pion PDFs from the determination by xFitter collabora- tion [3], the JAM collaboration [4], and the GRVPI1 group [5]. This figure is taken from Fig. 3 in Ref. [3]. 30%) in the pion than had been inferred from Drell-Yan data alone. The xFitter group [3] ana- lyzed Drell-Yan πA and photoproduction data using their open-source QCD fit framework for PDF extraction and found that these data can constrain the valence distribution well but are not sensitive enough for the sea and gluon distributions to be precisely determined. The anal- ysis done Ref. [55] suggests that the pion-induced J/ψ-production data provides an additional constraint on pion PDFs, particularly in the pion gluon PDF in the large-x region. The pion valence- and sea-quark, gluon PDFs from xFitter collaboration [3], the JAM col- laboration [4], and the GRVPI1 group [5] are compared in Fig. 2. Similar to the nucleon PDFs case, pion gluon PDFs are dominated at small-x region. The valence-quark PDFs have the smallest relative errors among all the three plots, which indicates that they are better deter- mined. Ultimately, the pion valence-quark distributions are better constrained than the gluon distribution from the global analysis of experimental data. 3 II. Lattice QCD II.1. The continuum QCD Lattice gauge theory is the main numerical tool to study the nonperturbative properties of QCD suggested by K. G. Wilson [56], which is a non-perturbative implementation of field theory using the Feynman path integral approach by introducing a finite lattice spacing and finite lattice size. Space-time is discretized and the path integral becomes finite due to the finite size. This discretization introduces deviations from continuum QCD calculations. Such deviations vanish when the lattice spacing is taken to zero, which is referred as the continuum limit. We introduce QCD starting with its Lagrangian in the continuum Minkowski space-time, X 1 X µν LQCD = ψ̄fα (x)(D / αβ − mf δαβ )ψfβ (x) − F (x)Fµνa (x) (II.1) f 4 a a where Greek letters α, β, µ, and ν are spinor indices, a is the color index, Nc = 3 is the number of colors for QCD and mf is the quark mass with flavor f . The first and second terms in the right hand side are the fermion and gluon Lagrangians, respectively. The covariant derivative D/ and the field strength tensor Fµν are, λa a µ D/ =i(∂µ − ig A )γ (II.2) 2 µ a Fµν =∂µ Aaν − ∂ν Aaµ + gf abc Abµ Acν (II.3) where Aaµ and ψfα are the gauge and quark fields, g is the bare coupling constant, λa are the generators of SU (Nc ), and fabc are the corresponding structure constants. The Euclidean Lagrangian LE QCD is transformed from Eq. II.1 by substituting t → iτ , LE E E QCD =Lgluon + Lf ermion 1 X µν /E X β a = Fa (x)Fµν (x) + ψ̄fα (x)(D αβ + mf δαβ )ψf (x) (II.4) 4 a f E where the Euclidean covariant D / is, λa a E D/ E = γµE DµE = (∂µ + ig A )γ (II.5) 2 µ µ 4 where the γµE and λa are the Euclidean Dirac matrices and SU (N c) generators. The partition function in Euclidean space-time is, Z E Z= DAµ Dψ Dψ̄ e−S (II.6) where the Euclidean QCD action S E is, Z E S = d4 xLE QCD (II.7) The thermal expectation value of physical observables can be obtained by, R Nf Πµ DAµ Πf =1 Dψf Dψ̄f O exp(−S E ) hOi = R Nf (II.8) Πµ DAµ Πf =1 Dψf Dψ̄f exp(−S E ) In practice there are two approaches to evaluating the physical observables in QCD. One ap- proach is to use perturbative methods to do the calculation when the strong coupling constant αs is small in high energy or short distance interactions. Another approach is lattice gauge theory which regularizes on a four-dimensional discretized Euclidean space-time. The infrared and ultraviolet momentum cut-offs are introduced by the finite lattice size L3 × T and lattice spacing a, which are π/L and π/a respectively. Equation II.8 can be computed via high per- formance computer because it becomes multiple integrations. A consequence of lattice gauge theory is that the Lorentz symmetry is lost. Since all the symmetries are restored in the con- tinuum limit, to recover the real physics, we remove the discretization by taking the continuum limit a → 0, L → ∞ and T → ∞. In the following sections we will briefly review the discretized version of the fields, the correlation functions, and the nonperturbative renormalization on the lattice. II.2. The formulation of Lattice QCD On the lattice, the gluon fields are defined as the links between lattice sites and the fermion fields are defined on the lattice sites. We will discuss them in detail in Sec. II.2.1 and Sec. II.2.2. 5 II.2.1. Gauge actions The original gauge action was introduced by Wilson in Ref. [56, 57]. The Wilson gauge action can be formed by the summation of the trace of the smallest closed loops on all lattice sites, XX 1 SG [U ] = β tr(1 − RePµν ) (II.9) x µ<ν Nc where the inverse the coupling β = 2Nc /g 2 , g is the bare gauge coupling on the lattice, P is the plaquette representing the smallest closed loop, Pµ,ν = Uµ (x)Uν (x + aµ̂)Uµ† (x + aν̂)Uν† (x). (II.10) The wilson link Uµ (x) is related to the continuum gauge fields Aµ (x) through, Uµ (x) = eiagAµ (x+µ/2) . (II.11) Based on the above Wilson gauge action construction, the leading correction is O(a2 ) in the continuum limit by taking a → 0. There are many different improved actions which have less discretization errors. To improve the gauge action, we write down an effective action which describes the behav- ior of Wilson’s form of lattice QCD at finite a. Following [1, 2, 5–7] we write the effective action in the form, Z SGef f = d4 x(L0 (x) + aL1 (x) + a2 L2 (x) + ...), (II.12) where L0 is the usual QCD Lagrangian, the terms Lk , k > 0 are the additional correction terms which are built from products of quark and gluon fields with dimensions d = 4 + k. The leading correction term L1 can be written as a linear combination of the following dimension-5 operators, L11 (x) =ψ̄(x)σµν Fµν (x)ψ(x), → − → − ←− ←− L12 (x) =ψ̄(x) D µ (x) D µ (x)ψ(x) + ψ̄(x) D µ (x) D µ (x)ψ(x), L13 (x) =mtr[Fµν (x)Fµν (x)], → − ← − L14 (x) =m(ψ̄(x)γµ D µ (x)ψ(x) − ψ̄(x)γµ D µ (x)ψ(x)), 6 L15 (x) =m2 ψ̄(x)ψ(x). (II.13) This list of operators may be further reduced by using the field equation (γµ Dµ + m)ψ = 0, which gives rise to the two relations L11 −L12 + 2L15 = 0, L14 +L15 = 0. (II.14) These relations may be used to eliminate the terms L12 and L14 from the set of operators. Thus it is sufficient to work with only the terms L11 , L13 and L15 . For O(a) improvement it is sufficient to add a single term including the L1 terms to the fermion action. Relevant purely gluonic operators appear only at dimension 6, i.e., they contribute at O(a2 ). For the improvement of the gauge action we refer the reader to the original literature, where the Luscher–Weisz gauge action is presented [58, 59, 60]. The general form of the O(a) improvement of the gauge action, Z SG = d4 x(L0 (x) + c2 L12 (x) + c3 L13 (x) + c5 L15 (x)), (II.15) One-loop Symanzik improved gauge action is introduced by K. Symanzik in the book [61], where the corresponding coefficients are c2 = −1/12, c3 = 0, c5 = 1/2. The ”Iwasaki gauge action” introduced by Y.Iwasaki in 1983 [62] has the coefficient c2 = −0.331. Such gauge actions are adopted by lattice collaborations to generate lattice ensembles. The MILC collab- oration [63, 64] and the RBC collaboration [65] utilize the Symanzik improved gauge action and Iwasaki gauge action respectively. II.2.2. Fermion action To discretize the Dirac action, Wilson replaced the derivative with the symmetrized difference and included appropriate gauge links to maintain gauge invariance 1 X ψ̄Dψ = ψ̄(x) γµ [Uµ (x)ψ(x + µ̂) − [Uµ† (x − µ̂)ψ(x − µ̂)] (II.16) 2a µ It is easy to see that one recovers the Dirac action in the limit a → 0 by Taylor expanding the Uµ and ψ(x + µ̂) in powers of the lattice spacing a, keeping only the leading term in a, 1 µ̂ 0 ψ̄(x)γµ [(1 + iagAµ (x + ) + ...)(ψ(x) + aψ (x) + ...) 2a 2 7 µ̂ 0 − (1 − iagAµ (x − ) + ...)(ψ(x) − aψ (x) + ...)] 2 a2 3 =ψ̄(x)γµ (∂µ + ∂ + ...)ψ(x) 6 µ a2 1 + ig ψ̄(x)γµ [Aµ + ( ∂µ2 Aµ + (∂µ Aµ )∂µ + Aµ ∂µ2 ) + ...]ψ(x), (II.17) 2 4 which is the kinetic part of the continuum Dirac action to O(a2 ) in Euclidean space-time. Thus one arrives at the simplest, so called “naive” lattice action for fermions, X S N = mq ψ̄(x)ψ(x) x 1 X + ψ̄(x)γµ [Uµ (x)ψ(x + µ̂) − Uµ† (x − µ̂)ψ(x − µ̂)] 2a x X N = ψ̄(x)Mxy [U ]ψ(x) (II.18) x where the interaction matrix M N is N 1 X † Mi,j [U ] = mq δij + [γµ Ui,µ δi,j−µ − γµ Ui−µ,µ δi,j+µ ] (II.19) 2a µ The Euclidean γ matrices are hermitian, γµ = 㵆 , and satisfy γµ , γν = 2δµν . The naive-quark action has an exact “doubling” symmetry under the transformation: ψ(x) → ψ̃(x) ≡γ5 γρ (−1)xρ /a ψ(x) =γ5 γρ exp(ixρ π/a)ψ(x). (II.20) Thus any low energy-momentum mode ψ(x) of the theory is equivalent to another mode ψ̃(x) that has momentum pρ ≈ π/a, the maximum value allowed on the lattice. This new mode is one of the “doublers” of the naive quark action. The doubling transformation can be applied successively in two or more directions. There are 15 doublers because of the four dimensions. It is not possible to construct a lattice fermion action that is ultra local, has chiral sym- metric and the correct continuum limit, and undoubled at the same time. There are several improved fermions that solve the doubling problem. The Wilson fermion [56] solves the dou- bling problem but breaks chiral symmetry. Staggered fermion also solves the doubling problem but brakes taste symmetry. Ginsparg-Wilson fermion extends chiral symmetry. This fermion action is local but not ultra local, which means it is still universal as the Wilson fermion. 8 II.2.2.1 Staggered Fermion The staggered-quark discretization of the quark action is equivalent to the naive discretization of the quark action. Staggering is an important optimization in simulations. Consider the following local transformation of the naive-quark field: ψ(x) → Ω(x) χ(x) ψ̄(x) → χ̄(x) Ω† (x) (II.21) where Y3 Ω(x) ≡ γx ≡ (γµ )xµ , (II.22) µ=0 and we have set the lattice spacing a = 1 for convenience. (We will use lattice units, where a = 1, in this and all succeeding appendices.) Note that Ω(x) = γn for nµ = xµ mod 2; (II.23) there are only 16 different Ωs. It is easy to show that < αµ (x) ≡Ω† (x)γµ Ω(x ± µ̂) = (−1)xµ , (II.24) 1 =Ω† (x)Ω(x) (II.25) where x< µ ≡ x0 + x1 + · · · + xµ−1 . Therefore, the staggered-quark action can be formulated by rewriting the naive-quark action as ψ̄(x)(γ · ∆ + m)ψ(x) = χ̄(x)(α(x) · ∆ + m)χ(x). (II.26) Remarkably the χ action is diagonal in spinor space; each component of χ is exactly equivalent to every other component. Consequently the χ propagator is diagonal in spinor space in any background gauge field: hχ(x)χ̄(y)iχ = s(x, y) 1spinor , (II.27) where s(x, y) is the one-spinor-component staggered-quark propagator. Transforming back to the original naive-quark field we find that SF ≡ hψ(x)ψ̄(y)iψ = s(x, y) Ω(x)Ω† (y). (II.28) 9 This last result is a somewhat surprising consequence of the doubling symmetry, it illus- trates that the spinor structure of the naive-quark propagator is completely independent of the gauge field. This is certainly not the case for individual tastes of naive quark, whose spins will flip back and forth as they scatter off fluctuations in the chromomagnetic field. The six- teen tastes of the naive quark field are packaged in such a way that all gauge-field dependence vanishes in the spinor structure. We can introduce such a form factor by replacing the link operator Uµ (x) in the action with Fµ Uµ (x), where smearing operator Fµ is defined by ! (2) Y a2 δρ Fµ ≡ 1+ , (II.29) ρ6=µ 4 symm. (2) and δρ approximates a covariant second derivative when acting on link fields as 1 δρ(2) Uµ (x) ≡ 2 Uρ (x) Uµ (x + aρ̂) Uρ† (x + aµ̂) a − 2Uµ (x) + Uρ† (x − aρ̂) Uµ (x − aρ̂) Uρ (x − aρ̂ + aµ̂) .  (II.30) (2) Equation II.29 holds given that δρ ≈ −4/a2 (and Fµ vanishes) when acting on a link field that carries momentum qρ ≈ π/a [66]. Smearing the links with Fµ removes the leading O(a2 ) taste-exchange interactions, but introduces new O(a2 ) errors. These can be removed by replacing Fµ with [67] X a2 (δρ )2 Fµ ≡ Fµ − , (II.31) ρ6=µ 4 where δρ approximates a covariant first derivative: 1 δρ Uµ (x) ≡ Uρ (x) Uµ (x + aρ̂) Uρ† (x + aµ̂) a − Uρ† (x − aρ̂) Uµ (x − aρ̂) Uρ (x − aρ̂ + aµ̂) .  (II.32) The new term has no effect on taste exchange but cancels the O(a2 ) part of Fµ . Improving the a2 3 derivative by ∆µ → ∆µ − 6 ∆µ , and replacing links by a2 -accurate smeared links removes all tree-level O(a2 ) errors in the naive-quark action. The result is the widely used “ASQTAD” 10 action [67], ! a2 3 X X   ψ̄(x) γµ ∆µ (V ) − ∆µ (U ) + m0 ψ(x), (II.33) x µ 6 where in the first difference operator, Vµ (x) ≡ FµA Uµ (x). (II.34) In practice, the operator Vµ is usually “tadpole” improved [68]; however, in reality, tadpole improvement is not needed when links are smeared and re-unitarized [69, 70]. The doubly smeared operator is simplified if we rearrange it as follows ! X a2 (δρ )2 FµH ≡ Fµ − U Fµ , (II.35) ρ6=µ 2 where the entire correction for a2 errors is moved to the outermost smearing. The highly improved stag- gered quark (HISQ) discretization of the quark action is X ψ̄(x) γ · DHISQ + m ψ(x),  (II.36) x where a2 DµHISQ ≡ ∆µ (W ) − (1 + )∆3µ (X), (II.37) 6 where the difference operators are, Wµ (x) ≡ FµH Uµ (x), (II.38) and, Xµ (x) ≡ U FµH Uµ (x). (II.39) The approaches to computing radiative correction to  are discussed in Ref. [71]. One approach is to adjust  until the relativistic dispersion relation fulfilled nonperturbatively. Another ap- proach is to compute the one-loop correction to  using perturbation theory, by requiring the correct dispersion relation for a quark in 1-loop order. Staggered fermions are widely used for dynamical simulations due to the reduced number of degrees of freedom. This property ensures staggered fermions to be numerically cheaper 11 to simulate while preserving chiral symmetry. However, a problem is that the action describes four degenerate tastes of quarks, while in a realistic QCD simulation one would like to have two light mass-degenerate u and d quarks and one heavier strange quark. Although the conceptual problems are not all resolved, simulations with staggered fermions have found good agreement with experimental results, as shown in Refs. [72, 73, 74]. II.2.2.2 Wilson-like Fermions Wilson’s solution to the doubling problem was to add a dimension five operator arψ̄ψ, whereby the extra fifteen species at pµ = π get a mass proportional to r/a [56]. The Wil- son action is X S W = mq ψ̄(x)ψ(x) x 1 X + ψ̄(x)γµ [Uµ (x)ψ(x + µ̂) − Uµ† (x − µ̂)ψ(x − µ̂)] 2a x r X − ψ̄(x)[Uµ (x)ψ(x + µ̂) − 2ψ(x) + Uµ† (x − µ̂)ψ(x − µ̂)] 2a x X = ψ̄ L (x)Mxy W [U ]ψ L (x), (II.40) x where the interaction matrix M W is X † W Mi,j [U ] = δij + κ [(r − γµ )Ui,µ δi,j−µ + (r + γµ )Ui−µ,µ δi,j+µ ], (II.41) µ with the rescalling 1 p ψ κ= ψ L = mq a + 4rψ = √ . (II.42) 2mq a + 8r 2κ Even though the Wilson fermion fixes the doublers, it introduces O(a) artifacts. To remove these artifacts, the clover-improvement is introduced by adding an additional dimension-5 op- erator to the Wilson action S W . The resulting clover action is, iacsw κr S clover =S W − ψ̄(x)σµν Fµν ψ(x) 4 X =mq ψ̄(x)ψ(x) x 1 X + ψ̄(x)γµ [Uµ (x)ψ(x + µ̂) − Uµ† (x − µ̂)ψ(x − µ̂)] 2a x 12 r X − ψ̄(x)[Uµ (x)ψ(x + µ̂) − 2ψ(x) + Uµ† (x − µ̂)ψ(x − µ̂)] 2a x iacsw κr − ψ̄(x)σµν Fµν ψ(x) 4 X = ψ̄ L (x)MxyC [U ]ψ L (x), (II.43) x the fermion matrix M C is given by [59]   i M [U ]C i,j = 1 − csw κ σµν Fµν (x) δi,j 2 X −κ (1 − γµ ) Uµ (x) δi+µ̂,j + (1 + γµ ) Uµ† (x − µ̂) δi−µ̂,j , (II.44) µ where we sum over µ and ν. The anti-symmetric and anti-Hermitian tensor F C is given by 1 C Fµ,ν = Uµ (x)Uν (x + µ̂)Uµ† (x + ν̂)Uν† (x) 8 +Uν (x)Uµ† (x + ν̂ − µ̂)Uν† (x − µ̂)Uµ (x − µ̂) +Uµ† (x − µ̂)Uν† (x − ν̂ − µ̂)Uµ (x − ν̂ − µ̂)Uν (x − ν̂) +Uν† (x − ν̂)Uµ (x − ν̂)Uν (x − ν̂ + µ̂)Uµ† (x) − h.c.] . (II.45) Staggered or Wilson-like fermions have their relative advantages and disadvantages. Stag- gered fermions do better when the chiral symmetry plays an essential role and the external states are Goldstone bosons. Wilson fermions are preferred due to their correspondence with Dirac fermions in terms of spin and flavor. II.2.2.3 Other fermion actions There are other fermion actions that are also commonly used in the lattice calculations. Overlap fermions originated from the initial papers on the overlap formulation [75, 76]. Neuberger pre- sented the modern form of the overlap Dirac operator in Ref. [77] and showed in Ref. [78] that it is a solution of the Ginsparg–Wilson equation. Domain wall fermions were outlined in the seminal paper by Kaplan [79]. The ideas developed further in Ref. [80, 81, 82] gave rise to the formulation of domain wall fermions mainly used now. An example of dynamical simulations with domain wall fermions is given in Ref. [83]. Twisted mass QCD is a formulation which in its simplest form pertains to QCD with two mass-degenerate quark flavors of Wilson fermions 13 (QCD with isospin). Twisted mass QCD was first outlined in Refs. [84, 85]. An example of a lattice calculation of the nucleon parton distribution function is presented in Ref. [9]. Different fermion actions are used due to different physics goals and different computation resources. II.2.2.4 Mixed-action As described in the previous section, Wilson fermions break chiral and flavour symmetries. However, they are computationally expensive compared to improved staggered quarks. For example, in the Nf = 2+1 improved staggered case, the square root of the fermion determinant is employed to reduce the number of dynamical flavours from four to two for the up and down quarks, and the fourth root is taken to reduce the number of flavours from four to one for the strange quark [72]. Ensembles of gauge field configurations are then generated with these frac- tional power determinants as weight factors. A mixed action is defined [86] as one where the action used to generate the ensemble of gauge configurations, or sea quark action, is different from the valence quark action used to determine hadronic observables on those configurations. Since all lattice Dirac operators give the same continuum limit, the differences between the actions are O(a) and vanish at the continuum limit in the mixed-action. One advantage of using mixed-action fermions is that one can use computationally cheaper fermion actions for sea quarks, such as staggered-like fermion actions. There are disadvan- tages to using mixed-action fermions. One possible problem for the Wilson-type fermions is that they have “exceptional” configurations in the simulation because they do not preserve the chiral symmetry. The use of chiral fermions like the HISQ can help with this problem due to the condition number of the fermion matrix going like a single inverse power of the quark mass. One disadvantage of the mixed-action fermion is that one cannot match the valence and sea quarks to restore unitarity at finite lattice spacing, since the valence and sea quarks have different discretization effects. For example, one can utilize clover valence fermion on MILC collaboration generated 2 + 1 + 1- flavor HISQ ensembles at physical pion mass with multiple fine lattice spacings [87]. II.3. Correlation functions The evaluation of lattice correlation functions is a standard procedure in most lattice calcula- tions . We compute quark propagators for each gauge configuration, combine them to construct 14 the hadron propagators, and average over all gauge configurations. We need to first identify the hadron interpolators and from these we can obtain the Euclidean correlator. The two-point and three-point correlators are discussed in detail in the following subsections. II.3.1. Smearing Link smearing The correlation functions signal can be improved by smoothing or smearing the gauge field over time slices or over both space and time. Such smearing can be use because we are in- terested in long distance correlation, and it is typical for gauge theories to have violent gauge field short distance fluctuations. Smearing is done typically by replacing the link variables with local averages over short paths connecting the endpoints of the replaced link. One does not have to fix the gauge for smearing because it is a gauge covariant procedure. We can then obtain the operators and propagators constructed on the smeared configura- tions. The smearing operator combines a fixed number of links which ensures the smearing to be local. Thus, it should have a negligible effect on the operators’ long distance correlation signals in the continuum limit. There are several algorithms to smear the gauge fields. Here, we will briefly introduce three of them. APE smearing is the first proposed smearing transformation used for operator improve- ment [88]. APE smearing averages over the original link Uµ and the six perpendicular staples Cµν connecting its endpoints, 0 αX Uµ (x) =(1 − α)Uµ (x) + Cµν (x) (II.46) 6 ν6=µ Cµν (x) =Uν (x)Uµ (x + ν̂)Uν (x + µ̂)† +Uν (x − ν̂)† Uµ (x − ν̂)Uν (x − ν̂ + µ̂) (II.47) where the α parameter can be adjusted depending on the gauge coupling. The APE smearing reduces scaling violations, improves chiral symmetry, and reduces taste breaking. However, APE smearing has some disadvantages, it smears out short scale physical properties if repeated too many times, its projected link is not differentiable and it cannot be used in dynamical simulations. The HYP smearing procedure is similar to the APE smearing, it contains 3 sets of APE 15 smearing that stays within a hypercube [89, 90]. α1 X V̄µ,νσ (x) =(1−α1 )Uµ (x) + Uρ (x) Uµ (x+ ρ̂) Uρ (x+ µ̂)† 2 ±ρ6=(µ,ν,σ) α2 X Ṽµ,ν (x) =(1−α2 )Uµ (x) + V̄σ,µν (x) V̄µ,νσ (x+ σ̂) V̄σ,µν (x+ µ̂)† 4 ±σ6=(µ,ν) 0 α3 X Uµ (x) =(1−α3 )Uµ (x) + Ṽν,µ (x) Ṽµ,ν (x+ ν̂) Ṽν,µ (x+ µ̂)† (II.48) 6 ±ν6=(µ) where the α1 , α2 and α3 are tunable parameters in the HYP smearing procedure. The HYP smearing is more compact and effective than APE smearing, but still not differentiable. The stout smearing method [91] uses a particular way of projection by defining the new link after a smearing step as 0 Uµ (x) = eiQµ (x) Uµ (x), (II.49) Qµ (x) is a traceless hermitian matrix constructed from staples, i 1 Qµ (x) = (Ωµ (x)† − Ωµ (x) − tr[Ωµ (x)† − Ωµ (x)]), 2 3 X Ωµ (x) =( ρµν Cµν (x))Uµ (x)† (II.50) ν6=µ where the staples Cµν are given in Eq. II.47 and the factors ρµν are tunable parameters. The new links have gauge transformation properties like the original ones. The advantage of stout 0 smearing is that Uµ (n) is differentiable with respect to the link variables, which is beneficial in the applications like the hybrid Monte Carlo method for dynamical fermions. Smeared action simulations are usually faster than unsmeared ones. One can even have multiple iterations of all these smearing steps. There are longer distance links getting involved with larger iteration steps, and the asymptotic behavior of the operators and propagators will be affected more with larger iteration steps. Therefore, one should carefully examine them to avoid the potential problems. Quark smearing Quark smearing within hadronic sources or sinks is used to increase the overlap with the de- sired physical state. The gauge link smearing was generalized by allowing iterative smearing of quark fields in interpolators that create hadronic states, in particular gauge covariant Wuppertal 16 smearing [92, 93, 94], hydrogen-like smearing [94], Jacobi smearing [95, 96], APE smearing was employed for the spatial gauge transporters within the quark smearing in Refs. [93, 94], Gaussian smearing [97, 98], “free form smearing” [99], Gauge fixed sources have been uti- lized in parallel to gauge covariant iterative smearing functions, wall sources for zero [100], non-zero momentum [101], box [102] sources, Gaussian “shell sources” [103], and sources with nodes [104]. Newer attempts have been made to introduce an anisotropy into Wuppertal smearing [105, 106]. In this thesis, Gaussian momentum smearing is used for the quark field which introduced in, Ref. [107], 1 X Uj (x)eikêj Ψ(x + êj ) ,  Smom Ψ(x) = Ψ(x) + α (II.51) 1 + 6α j where k is the momentum-smearing parameter and α is the Gaussian smearing parameter. Gaussian momentum smearing is helpful to getting us a better signal at a higher boost nucleon momentum for the correlators. II.3.2. propagator and inversion The quark propagator governs the behavior of n-point functions and is important to analyze it. For free fermions this analysis is best done with the momentum space propagator D̃(p)−1 . For the case of massless fermions m = 0, the fixed p propagator has the correct naive continuum limit as a → 0, −ia−1 µ γµ sin(pµ a) a→0 −i µ γµ pµ P P D̃(p)−1 |m=0 = P → (II.52) a−2 µ sin(pµ a)2 p2 The general quark propagator Gij i j αβ (x, y) = hqα (x)q̄β (y)i (II.53) satisfies Mαγ ik (x, z)Gkjγβ (z, y) = δij δαβ δxy , (II.54) where M is the lattice Dirac operator. The anti-quark propagator is related through G(y, x) = γ5 G(x, y)† γ5 , (II.55) 17 which is an evident property from the lattice Dirac equation. The sources to all propagators can be obtained by finding the solutions xi to the Matrix equation, Axi = bi (II.56) where A is the Dirac matrix, bi are the constructed our quark sources. The system of lin- ear equations can be solved by iterative methods such as Conjugate Gradient (CG) [108] for symmetric positive definite matrices, the MINRES-method [109] for symmetric non-definite matrices or some sort of Bi-ConjugateGradient (BiCG) method [110] for non-symmetric ma- trices. We use multigrid-ConjugateGradient (MG-CG) algorithm [111, 112] in the Chroma software package [1]. The procedure of MG-CG is as following. To solve Ax = b (II.57) and (A + σ)xσ = b (II.58) using the method of CG. The (i + 1)th iterate xi+1 to the solution can be obtained from xi , xi+1 = xi + αi pi (II.59) where x1 = 0 is the starting point, and pi are the search directions which obey the recursion relation, p1 = g1 ; pi+1 = gi+1 + βi pi ∀i ≥ 1 (II.60) where gi is the residue equal to Axi − b and also obeys a recursion relation, g1 = −b; gi+1 = gi + αi Api ∀i ≥ 1 (II.61) Then, the coefficients αi and βi in Eqs. II.60 and II.61 are obtained as, (gi , gi ) αi = − (II.62) (pi , Api ) (gi+1 , gi+1 ) βi = (II.63) (gi , gi ) Using the Lanczos connection, we write down the recursion relation for xσi and pσi with that 18 the normalized Lanczos vectors for A and A + σ are the same, giσ = cσi gi (II.64) Then assuming that xσ1 = 0, we have cσ1 = 1. Because giσ obeys the following recursion relation: σ h αiσ βi−1 σ i σ σ σ αiσ βi−1 σ σ gi+1 = 1+ σ gi + αi (A + σ)gi − σ gi−1 . (II.65) αi−1 αi−1 Then we substitute Eq. II.64 into Eq. II.65, cσi+1 αiσ = αi (II.66) cσi cσi+1 2 βiσ = βi [ ] (II.67) cσi cσi βi−1  cσi  = 1 − α i σ + α i 1 − (II.68) cσi+1 αi−1 cσi−1 Note that cσ1 = 1 and 1 cσ2 = . (II.69) 1 − α1 σ The recursion relations for pσi and xσi are pσ1 = g1 ; pσi+1 = cσi+1 gi+1 + βiσ pσi ∀i ≥ 1, (II.70) xσi+1 = xσi + αiσ pσi . (II.71) The advantage of this method is that giσ is trivially modified and the Eq. II.61 that involves the multiplication of matrix with a vector remains the same independent of σ. MG-CG Algorithm [113] Starting with g1 = −b, p1 = −b, cσ1 = 1, pσ1 = −b and xσ1 = 0. The iteration proceeds as follows: i. Compute Api and (pi , Api ). ii. Compute (gi , gi ) and save this number, and compute αi . iii. Use Eq. II.61 to compute gi+1 then overwrite gi . iv. Compute (gi+1 , gi+1 ) and βi using the gi+1 in step iii. v. Compute pi+1 using Eq. II.60 and overwrite pi . 19 vi. Compute cσi+1 using Eq. II.69 in the first iteration and using Eq. II.62 from the second iteration onwards. vii. Compute αiσ and βiσ using Eq. II.66 and Eq. II.67 respectively. viii. Compute xσi+1 using Eq. II.71 and overwrite xσi . ix. Compute pσi+1 using Eq. II.70 and overwrite pσi . Once the solutions to the propagator equation are computed, we can obtain the correlation functions. We will expand the discussion of the correlation functions in the section.II.3.3. II.3.3. Two-point correlators The two-point correlation functions can be used to determine the hadron mass and are important in the calculation of the matrix elements discussed in the next subsection. Given an operator with pion quantum numbers, such as χΦ (x) = q̄1 (x)γ5 q2 (x) (II.72) where the q̄1 (y) and q2 (y) are the anti-quark and quark operators, the dimensionless two-point correlator from Euclidean time ti to Euclidean time tf with momentum p~ is X D E Φ CAB (ti , tf , p~) =a6 e−i(~xf −~xi )·~p 0 χΦ,B (xf )χ†Φ,A (xi ) 0 ~ xf XX D E =a 6 e −i(~xf −~ xi )·~ p ~ 0 |χΦ,B (xf )| n(k) × n,~k ~ xf 1 D E n(~k) χ†Φ,A (xi ) 0 2V En(~k) XX D E =a6 e−i(~xf −~xi )·~p 0 χΦ,B (xi )ei(xf −xi )·k n(~k) × n,~k ~ xf 1 D ~ † E n(k) χΦ,A (xi ) 0 2V En(~k) XXD E =a6 0 |χΦ,B (xi )| n(~k) × n,~k ~ xf e−(tf −ti )En(~k) D ~ E ~ n(k) χ†Φ,A (xi ) 0 ei(~xf −~xi )·(k−~p) 2V En(~k) XD E =a 3 0 |χΦ,B (xi )| n(k) × ~ n,~k 20 e−(tf −ti )En(~k) D ~ E ~ n(k) χ†Φ,A (xi ) 0 δ~k,~p e−i~xi ·(k−~p) (3) 2En(~k) X D E e−(tf −ti )En(~p) 3 † =a h0 |χΦ,B (xi )| n(~p)i n(~p) χΦ,B (xi ) 0 n 2En(~p) For tf  ti , the ground state meson i.e. the pion dominates and the result becomes D E e−(tf −ti )Eπ(~p) 3 † Γππ (t AB i f , t , p ~ ) → a h0 |χ Φ,B (x i )| π(~p )i π(~ p) χ (x Φ,A i ) 0 (II.73) 2Eπ(~p) For operator with nucleon quantum numbers, T χN (y) = lmn [q1 (y)l iγ4 γ2 γ5 q2m (y)]q3n (y), (II.74) where {l, m, n} are color indices, q1 (y), q2 (y) and q3 (y) are the quark operators, the nucleon like two-point correlator is X D E CABN (ti , tf , p~) = a6 Γe−i(~xf −~xi )·~p 0 χN,B (xf )χ†N,A (xi ) 0 , (II.75) ~xf where the projection operator is Γ = 21 (1 + γ4 ), let |0+ (~p) > to be the positive parity ground + state i.e. the nucleon state with energy e−(tf −ti )En(~p) X D E N CAB (ti , tf , p~) =a 6 h0 |χΦ,B (xi )| n(~p)i n(~p) χ†Φ,B (xi ) 0 × n + + + −(tf −ti )En(~ En(~ p) + m e p) + (II.76) 2En(~ p) The dimensionless correlator from Euclidean time ti to Euclidean time tf with momentum p~ is X CABN (ti , tf , p~ ; T ) =a9 e−i(~xf −~xi )·~p Tαβ 0 χN N βB (xf )χ̄αA (xi ) 0 ~ xf XX D E =a 9 e −i(~xf −~ xi )·~ p Tαβ 0 χN ~ n(k, s) × βB (xf ) n,~k,s ~xf mn D ~ E n(k, s) χ̄N αA i (x ) 0 V En(~k) XX D E =a9 e−i(~xf −~xi )·~p Tαβ 0 χN (x i )ei(xf −xi )·k n(~k, s) × βB n,~k,s ~xf mn D ~ N E n(k, s) χ̄αA (xi ) 0 V En(~k) XX D E =a9 Tαβ 0 χN (x i ) n(~k, s) × βB n,~k,s ~xf 21 mn e−(tf −ti )En(~k) D ~ E ~ n(k, s) χ̄N (x αA i ) 0 ei(~xf −~xi )·(k−~p) V En(~k) X D E =a6 Tαβ 0 χN (x i ) n(~k, s) × βB n,~k,s mn e−(tf −ti )En(~k) D ~ E (3) ~ n(k, s) χ̄αA (xi ) 0 δ~k,~p e−i~xi ·(k−~p) N En(~k) X =a6 Tαβ 0 χN βB (xi ) n(~ p, s) n(~p, s) χ̄N αA (xi ) 0 × n,s mn −(tf −ti )En(~p) e (II.77) En(~p) where Tαβ is some generic 4 × 4 matrix in Dirac spin space, and α, β are Dirac indices. For tf  ti , the ground nucleon state dominates and the result becomes N CAB (ti , tf , p~ ; T ) (II.78) X mN −(tf −ti )EN (~p) → a6 Tαβ 0 χN βB (xi ) N (~ p, s) N (~p, s) χ̄NαA (xi ) 0 e s EN (~p) II.4. Nonperturbative renormalization Renormalization of lattice operators is a necessary ingredient to obtain many physical results from numerical simulations. There are approaches to do the nonperturbative renormalization, such as the chiral Ward identities to determine the renormalization constants [114, 115, 115], fix non-perturbative renormalization conditions directly on hadronic matrix elements [116], and calculate the quantities perturbatively in one-loop both in the continuum and on the lattice in the Landau gauge [117]. A regularization independent momentum subtraction (RI/MOM) scheme method which avoids completely the use of lattice perturbation theory and allows a non-perturbative deter- mination of the renormalization constants of any composite operator is proposed in Ref. [118]. In the following subsection, I will introduce this method through a simple fermion operator case. Fermion Operator Renormalization: For simplicity, we consider the forward amputated Green function ΓO (pa) of a two-fermion bare lattice operator Oq (a) = ψ̄(x)Γψ(x), computed between off-shell quark states with four-momentum p, with p2 = µ2 in the Landau gauge. We 22 define the renormalized operator Oq (µ),   Oq (µ) = ZORI-MOM µa, g(a) Oq (a). (II.79) where the ZORI-MOM renormalization constant. By imposing the renormalization condition, as shown in Fig. 3,     −1 ZORI-MOM µa, g(a) Zψ µa, g(a) ΓO (pa)|p2 =µ2 = 1, (II.80) 1/2 where Zψ is the field renormalization constant, to be defined below. This procedure defines a renormalized operator O(µ) which is independent of the regularization scheme [119, 120]. It depends, however, on the external states and on the gauge. This does not affect the final results, which, combined with the continuum calculation of the renormalization conditions, at any given order of perturbation theory, will be gauge invariant and independent of the external states. Let us specify the different quantities entering Eq. II.80. ΓO is defined in terms of the expectation value of the non-amputated Green function GO (pa), and of the quark propagator S(pa) 1   ΓO (pa) = tr ΛO (pa)P̂O , (II.81) 12 where ΛO (pa) = S(pa)−1 GO (pa)S(pa)−1 . (II.82) P̂O is a suitable projector on the tree-level operator: P̂O = 1̂ (P̂O = γ5 ) for the scalar (pseudoscalar) density. The factor 1/12 ensures the correct overall normalization of the trace (colour×spin=12). Projectors are very convenient when defining Green functions, particularly in the non-perturbative case. They have been extensively used in refs. [120]. Of course one can also use other definitions of ZO . 1/2 Zψ is the renormalization constant of the fermion field. It can be defined in different ways, some of which are equivalently perturbative. Beyond perturbation theory, the most natural definition of Zψ is obtained from the amputated Green function of the conserve vector current V C . Indeed, one knows that for V C the renormalization constant is equal to one: 1   ZV−1C = ΓV C × Zψ−1 = tr ΛVµC (pa)γµ |p2 =µ2 × Zψ−1 = 1, (II.83) 48 23 Figure 3 Non-perturbative renormalization condition. In the left hand side is the tree ampu- tated Green’s function, and the right hand side are the bare amputated Green’s function and renormalization factors. which implies 1   Zψ = tr ΛVµC (pa)γµ |p2 =µ2 . (II.84) 48 The complete multiplicative renormalization constant in the MS scheme needs a perturba- tive matching factor which converts from the RI-MOM renormalization at scale p2 to the MS scheme, in addition to the RI-MOM factors ZORI-MOM (µ2R ), Z MS (µ) = RMS (µ, p2 ) ZORI-MOM (p2 ). (II.85) The conversion ratio RMS (µ, p2 ) is derived up to 3-loop in Ref. [121],    MS 2 517 5 αs 2 R (µ, p ) = 1 + − + 12ζ3 + nf 18 3 4π   1287283 14197 79 1165 18014 368 1102 2 + − + ζ3 + ζ4 − ζ5 + nf − ζ3 nf − n 648 12 4 3 81 9 243 f  α 3 s × + O(αs4 ). (II.86) 4π where nf is the number of flavors and ζn is the Riemann zeta function evaluated at n. The strong coupling constant αs (µ) is evaluated in the MS scheme by using its perturbative running to four loops [122]. The beta functions in the MS scheme to 4-loops can be found in Ref. [123]. The renormalization condition needs to satisfy ΛQCD  µ  1/a, (II.87) to keep under control both non-perturbative and discretization effects. If such a window for µ does not exist, in current lattice simulations, an accurate matching of lattice operators to continuum ones becomes impossible in all methods. Gluon Operator Renormalization: The common gluon operators definitions are, X 1 Oµν (z) ≡ (O(F µα , Fαν ; z) + O(F να , Fαµ ; z)) α=0,1,2,3 2 24 1 X X − O(F αβ , Fβα ; z), (II.88) 4 α=0,1,2,3 β=0,1,2,3 with the operator O(Fνµ , Fβα ; ) defined by Fνµ (z)U (z, 0)Fβα (0). The field tensor Fµν needed in the definition of the quasi-PDF operators is defined by i Fµν = (P[µ,ν] + P[ν,−µ] + P[−µ,−ν] + P[−ν,µ] ) (II.89) 8a2 g where the plaquette Pµ,ν = Uµ (x)Uν (x+aµ̂)Uµ† (x+aν̂)Uν† (x) and P[µ,ν] = Pµ,ν −Pν,µ . For the common gluon operators, the RI-MOM renormalization factor ZO (µ2R ) can be obtained with the non-perturbative renormalization condition, −1 Zg (p2 )ZORI-MOM (p2 )Λbare tree O (p)(ΛO (p)) |p2 =µ2R = 1, (II.90) where Zg (p2 ) is the gluon field renormalization and ΛO (p) is the amputated Green’s function for the operator O in the Landau gauge-fixed gluon state. The tree level amputated Green’s function is [124], Λtree O (p) = hOµν T r[Aσ (p)Aτ (−p)]i tree Nc2 − 1 = (2pµ pν gστ − pτ pν gσµ − pτ pµ gσν − pσ pν gτ µ 2 − pσ pµ gτ ν + pσ pτ gµν − p2 gστ gµν + p2 gσµ gτ ν + p2 gσν gτ µ ) (II.91) The gluon field renormalization Zg (p2 ) is obtained through the gluon two-point function which is the trace of the gluon propagator, Nc2 − 1 pµ pν Dµν (p) = hT r[Aµ (p)Aν (−p)]i = Zg (p2 ) (gµν − ) (II.92) 2p2 p2 The gluon fields are calculated from the Landau gauge-fixed wilson link Uµ (x), 1 1 Aµ (x + aêµ /2) = [(Uµ (x) − Uµ† (x)) − Tr(Uµ (x) − Uµ† (x))] (II.93) 2ig0 Nc The momentum space lattice gluon fieds can be obtained with the Fourier transformation, X ˙ Aµ (p) = e−ip(x+aêµ /2) Aµ (x + aêµ /2) (II.94) x 2πnµ where pµ = aLµ , nµ = 0, ..., Lµ − 1. Therefore, the RI-MOM renormalization factor ZORI-MOM (µ2R ) 25 can be obtained from the wilson link Uµ (x). Similar to the quark case, the complete multiplicative renormalization constant in the MS scheme is, Z MS (µ) = RMS (µ, p2 ) ZORI-MOM (p2 ). (II.95) In this work, the 1-loop expression for this matching, derived in Ref. [125], is used: g 2 Nf g 2 Nc ξ2     2 2 10 4 R MS (µ , µ2R ) =1− log(µ2 /µ2R ) + − − 2ξ + . (II.96) 16π 2 3 9 16π 2 3 4 where Nf and Nc are the numbers of flavors and colors respectively, ξ = 0 in the Landau gauge, and g 2 is defined by 1/α(µ) [126, 127, 128]. The renormalization constant Z MS can be used to calculate the renormalized gluon moments and gluon gravitational form factors of the nucleon and pion, where the renormalized gluon moments are calculated and used in the calculation of gluon PDF in the following pseudo-PDF sections. 26 III. Bjorken x-dependence PDF from lattice QCD For decades, Bjorken x-dependence parton distributions cannot be directly determined in Eu- clidean lattice QCD, because their field-theoretic definition involves fields at light-like separa- tions. One way to obtain the PDFs in lattice QCD is to calculate the Mellin moments. PDFs can be reconstructed from the inverse Mellin transform with a sufficient number of Mellin moments. However, the calculation is limited to the lowest three moments, because power- divergent mixing occurs between twist-two operators on the lattice. With this limitation, the lowest three moments are insufficient to fully reconstruct the momentum dependence PDFs without significant model dependence [129]. Moving beyond the lowest three moments re- quires overcoming the challenge of power-divergent mixing for lattice-QCD twist-two opera- tors. One novel approach to this problem [130] builds upon the physical intuition that as long as the scale associated with the operator is taken to be much larger than the hadronic scale but much smaller than the inverse lattice spacing. Other approaches that avoid power-divergent mixing have also been suggested, including “auxiliary heavy/light quark” [131, 132, 133, 134], “operator product expansion (OPE) without OPE” [135, 136, 137, 138, 139, 140, 141, 142], and “hadronic tensor” [143, 144, 145, 146, 147, 148]. Following the proposal of Large-Momentum Effective Theory (LaMET) [149, 150, 151], many approaches are proposed to determining the x- dependence of PDFs directly from lattice QCD. The LaMET method calculates lattice quasi-distribution functions, defined in terms of matrix elements of equal-time and spatially separated operators, and then takes the infinite- momentum limit to extract the lightcone distribution. The quasi-PDF can be related to the Pz - independent lightcone PDF through a factorization theorem that factors from it a perturbative matching coefficient with corrections suppressed by the hadron momentum [150]. The factor- ization can be calculated exactly in perturbation theory [152, 153]. Many lattice works have been done on nucleon and meson PDFs, and generalized parton distributions (GPDs) based on the quasi-PDF approach [154, 155, 156, 157, 158, 159, 160, 161, 9, 6, 162, 10, 7, 163, 8, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181]. Alterna- tive approaches to lightcone PDFs in lattice QCD are “good lattice cross sections” [152, 182, 183, 184, 185] and the pseudo-PDF approach [186, 187, 188, 189, 190, 191, 192, 193, 194, 13, 12, 195, 196, 197]. In this chapter, we will mainly focus on the introduction of LaMET and 27 pseudo-PDF approaches in the Sec. III.1 and Sec. III.2 respectively. III.1. Large Momentum Effective Theory Feynman defined parton density in an infinite momentum frame (IMF). One can view a hadron as a beam of non-interacting partons (quarks and gluons) with the parton momentum density q(x) and g(x), where Bjorken x = kz /Pz is the fraction of longitudinal momentum of the parton, kz is the parton momentum and Pz is the hadron momentum which goes to infinity to approach the light-cone property. Later, people found that the most convenient formulation of parton density is in the formalism of light-cone correlation. The unpolarized quark and gluon distribution in the nucleon in the light-cone coordinates [198], dξ − −ixξ− P + Z R ξ− − − 2 q(x, µ ) = e hP | ψ̄(ξ − )γ+ eig 0 A+ (ξ )dξ ψ(0) |P i , 4π dξ − −ixξ− P + Z R − + − −ig 0ξ dη − A+ (η − ) µ+ xg(x, µ2 ) = e hP |F µ (ξ )Pe F (0)|P i 2πP + where the nucleon momentum P µ is along the z-direction, P µ = (P 0 , 0, 0, P z ), and |P i is the hadron state with momentum P , µ2 is the renormalization scale, A+ is a gluon potential matrix in the fundamental representation, Fµν = T a Gaµν = T a (∂µ Aaν − ∂ν Aaµ − gf abc Abµ Acν ) is the gluon field tensor, and 1 ξ ± = (ξ 0 + ξ 3 ) (III.1) 2 ξ µ (µ = 0, 1, 2, 3) is the space-time coordinates. However, one cannot directly calculate time-dependent correlations in the framework of non-perturbative QCD on a Euclidean lattice. Large momentum effective theory is proposed to overcome this difficulty. In LaMET, one can fist construct a Euclidean quasi operator O which becomes the light-cone operator o in the infinite momentum frame (IMF) limit. There could be many operators O become to the same light-cone operator in the IMF limit. Any two operators leading to the same light-cone operator could linear combine into operators that become same light-cone operator in the IMF limit. Because O is an Euclidean operator, one can compute the matrix element of O on the lattice. O(Pz , a) depends on the momentum of the hadron Pz and the lattice spacing a (providing UV cutoff). The light-front operator o(µ) can be extracted from 28 O(Pz , a) at a large Pz through an effective theory, O(Pz , a) = Z(Pz , µ)o(µ) + O(1/(Pz )2 ) + ... (III.2) Z contains all the lattice artifact, but only depends on the UV physics, can be calculated in perturbation theory. Parton distribution can be extracted by accurately calculating the matching factor Z and higher-order corrections. III.1.1. Non-singlet quark quasi-PDF In the early days, the quasi-PDF studies are mostly limited to isovector quark distributions in nucleon and valence-quark distribution in meson. The non-singlet quark quasi distribution is defined as[149], Z 2 dz −ixzPz q̃(x, µ , Pz ) = e hP | OΓ (z) |P i , (III.3) 4π where the OΓ (z) is u - d isovector qPDF operator: OΓ (z) ≡ u(z)ΓU (z, 0)u(0) − d(z)ΓU (z, 0)d(0), (III.4) where the Dirac Γ used will determine the quantum numbers of the quark PDF — Γ = γt , γz γ5 , iσzj (with j 6= z), for the unpolarized, longitudinally polarized, transversity case respectively. The Wilson link U is defined along the z direction Z z2 U (z2 , z1 ) = Pexp(−ig dηAz (η)), (III.5) z1 As we discuss in Chapter. II, the renormalization factor for the local operators can be cal- culated in the RI/MOM scheme. For the LaMET (non-local) operators, the quark NPR factors were done in Refs. [161, 160]. For the quark PDF, we use the RI-MOM renormalization con- stant defined via Tr[PΛtree (p, z, γt )] ZΓmp (z, pR −1 z , a , µR ) = . (III.6) Tr[PΛ(p, z, γt )] p2 =µ2R , pz =pR z For the unpolarized case, Λtree (p, z, γt ) = γt e−izpz is the tree level matrix element in the mo- mentum space, P = γt − (pt /px )γx is the projection operator corresponding to the so-called minimal projection, where only the term with the Dirac structure proportional to γt is kept [199, 200]. For the polarized case, Λtree (z, pz , γz γ5 ) = γz γ5 e−ipz z , and the projection operator 29 P is chosen to be P = γ5 γz /4. Λtree (z, pz , γz γ5 ) = γz γ5 e−ipz z . For the transversity case, Λtree (z, pz , γz γ5 ) = γz γ5 e−ipz z , and the projection operator P is chosen to be P = γ5 γz /4. Λtree (z, pz , γz γ5 ) = γz γ5 e−ipz z . The renormalization constant ZΓ (z, pR −1 z , a , µR ) depends on the lattice spacing a, as well as the other two scales pR z and µR which are unphysical scales in- troduced in the RI/MOM scheme [199, 161]. The non-singlet quark quasi-PDF can be related to the P z -independent light-front PDF through, 1 M 2 Λ2QCD Z     dy x µ Λ z q̃(x, Λ, pz ) = Z , , q(y, p , µ) + O , 2 (III.7) −1 |y| y ypz pz p2z pz where Λ indicates the approximate non-perturbative scale of QCD, µ is the renormalization scale, Z is a matching kernel and M is the nucleon mass. Here the O (M 2 /p2z ) terms are  target-mass corrections and the O Λ2QCD /p2z terms are higher-twist effects, both of which are suppressed at large nucleon momentum. III.1.2. Gluon quasi-PDF Similar to the the quark quasi distribution, the gluon quasi distribution is defined as, Z 2 dz ixzPz g̃(x, µ , Pz ) = e hP |Og (z)|P i (III.8) 2πxPz We use the gluon operators defined in Ref. [201], which are multiplicatively renormalizable, X Og,1 (z) ≡ O(F ti , F zi ; z), i6=z,t X Og,2 (z) ≡ O(F ti , F ti ; z), i6=z,t X Og,3 (z) ≡ O(F zi , F zi ; z), i6=z,t X Og,4 (z) ≡ O(F zµ , F zµ ; z), µ=0,1,2,3 (III.9) where O(F µν , F αβ ; z) = Fνµ (z)U (z, 0)Fβα (0). The renormalization factor of the gluon quasi- PDF in the RI/MOM scheme is provide in Refs. [201, 164], [eδm|z| ZOg,i Zg ]−1 (zn, pRz , 1/a, µR ) = 30 Pijab h0|T [Aa,i (p)Og,i (z, 0)Ab,j (−p)]|0i|amp , (III.10) Pijab h0|T [Aa,i (p)Og,i (z, 0)Ab,j (−p)]|0iamp,tree p2 = −µ2R pz = pR z in the case of Og,1 , where δm ∼ O(1/a) is the mass counterterm. Here Aa,i (p) denotes the momentum space gluon field with momentum p. Zg is the gluon field renormalization constant. Pijab is a projection operator with color indices a, b and Lorentz indices i, j. A simple example of the projection is Pijab = δ ab g⊥,ij . However, calculating the gluon renormalization nonperturbatively suffers worse signal-to-noise than the corresponding nucleon calculation, making it harder to apply this strategies. The factorization for the renormalized singlet quark and gluon quasi-PDFs after the non- perturbative renormalization is provided in Ref. [201], {q̃i , g̃}(x, Pz , µMS , µRI , pRI z ) Z 1  2 ! dy x µRI yPz yPz = C{qi ,g},qj , , MS , RI qj (y, µMS ) 0 |y| y pz RI pz µ Z 1 2 ! x µRI   dy yPz yPz + C{qi ,g},g , , MS , RI g(y, µMS ) 0 |y| y pz RI pz µ  2 ΛQCD Λ2QCD m2N  +O , , (III.11) x2 Pz2 (1 − x)2 Pz2 Pz2 where pRIz and µ RI are the momentum of the off-shell quark and the renormalization scale in the RI/MOM-scheme nonperturbative renormalization (NPR), the summation of j is over all quarks/anti-quarks, the coefficients Cgg , Cqg , Cqi qj and Cgq are derived in Ref. [164]. The  Λ2 Λ2QCD  m2N residual terms, O x2QCD , P 2 (1−x)2 P 2 P 2 , , come from the nucleon-mass correction and higher- z z z twist effects, suppressed by the nucleon momentum. Beginning from the quark/gluon quasi-PDF operators, then implementing the RI/MOM scheme for renormalization and the matching conditions for quark/gluon, the theoretical basis for directly extracting the unpolarized quark and gluon PDFs from lattice simulations is well established through quasi-PDF method. III.2. Pseudo-PDF method The pseudo-PDF method was first introduced in Refs. [202, 203] for quark PDFs. The unpo- larized gluon PDF case using pseudo-PDF approach was proposed in Ref. [204] and polarized 31 case was later presented in 2022 in Ref. [205] after this thesis began. The general dependence of the matrix element M (z, pz ) on the hadron momentum p and the displacement of the quark and antiquark fields z can be expressed as a function of the Lorentz invariants ν = z · p (Ioffe time [206, 207]) and z 2 , where z and p are general 4-vectors. We can thus introduce the Ioffe time pseudo-distribution (pITD), M(ν, z 2 ) ≡ M (z, pz ) . (III.12) To eliminate the ultraviolet divergences in the pITD, we construct the reduced pseudo-ITD (RpITD) by taking the ratio of the pITD to the corresponding z-dependent matrix element at Pz = 0, and further normalize the ratio by the matrix element at z 2 = 0 as done in the first quark pseudo-PDF calculation [186], M(zPz , z 2 )M(0 · Pz , 0) M (ν, z 2 ) = . (III.13) M(z · 0, z 2 )/M(0 · 0, 0) By construction, the RpITD double ratios employed here are normalized to one at z = 0. It is directly related to the PDFs as Mq,g (ν, z 2 ) = Iq,g (ν, µ2 ) + O(z 2 ) , (III.14) with µ2 = 1/z 2 . Here I(ν, µ2 ) is the Ioffe time PDF [206, 207], III.2.1. Quark pseudo-PDF The quark Ioffe time PDF Iq (ν, µ2 ) is the Fourier transform of the quark PDF, Z dν −ixν 2 q(x, µ ) = e Iq (ν, µ2 ), (III.15) 2π the quark RpITDs are connected to through the matching below while ignoring the O(z 2 ) terms, 1 q(x, µ2 ) Z Mq (ν, z ) = 2 dx Rqq (xν, z 2 µ2 ), (III.16) 0 hxiq R1 where µ is the renormalization scale in MS scheme and hxig = 0 dx xg(x, µ2 ) is the gluon momentum fraction of the nucleon. αs (µ) Rqq (y, z 2 , µ2 ) = cos y − Nc × 2π 32 2γE +1      2 2e ln z µ + 2 Rq,B (y) + Rq,L (y) , (III.17) 4 where αs is the strong coupling at scale µ, Nc = 3 is the number of colors, and γE = 0.5772 is the Euler-Mascheroni constant. For the term Rqq (y, z 2 , µ2 ), z was chosen to be 2e−γE −1/2 /µ so that the logarithmic term vanishes, which suppresses the residuals containing higher order logarithmic terms, following the previous publication regarding the one-loop evolution of the pseudo-PDF [192]. Equation III.16 and the terms Rq,B (y), Rq,L (y) are defined in Eqs. 16 and 24 in Ref. [192]. III.2.2. Gluon pseudo-PDF All the quark and gluon operators in Eq. III.4 and III.9 can be also used in pseudo-PDF method. Another unpolarized gluon operator is introduced in Ref. [204], X X O(z) ≡ O(F ti , F ti ; z) − O(F ij , F ij ; z), (III.18) i6=z,t i,j6=z,t which can directly used in the pseudo-PDF matching Eq. V.12. The gluon RpITDs are con- nected to through the matching below while ignoring the O(z 2 ) terms, 1 xg(x, µ2 ) Z Mg (ν, z ) = 2 dx Rgg (xν, z 2 µ2 ) 0 hxig Z 1 Pz xqS (x, µ2 ) + dx Rgq (xν, z 2 µ2 ), (III.19) P0 0 hxig R1 where µ is the renormalization scale in MS scheme and hxig = 0 dx xg(x, µ2 ) is the gluon momentum fraction of the nucleon. αs (µ) Rgg (y, z 2 , µ2 ) = cos y − Nc × 2π 2γE +1      2 2e ln z µ + 2 Rg,B (y) + Rg,L (y) + Rg,C (y) , (III.20) 4 Equation V.12 and the terms Rg,B (y), Rg,L (y), Rg,C (y) are defined in Eqs. 7.21–23 and in the paragraph below Eq. 7.23 in Ref. [204]. Note that the lattice-calculated RpITDs are also con- nected to the singlet quark-PDF qs via the quark-gluon matching kernel Rgq with an additional non-singlet quark term added to Eq. V.12. Quasi-PDF (LaMET) and the pseudo-PDF approaches are faced with different technical problems of inferring the PDF from a Fourier transform due to the data in a limited range of z 33 or ν. Such effects have been discussed in [157, 208]. In particular, because x is the Fourier dual of ν, accessing a limited range of ν (or z) has the effect of introducing uncontrolled systematic errors at small x (roughly x . 0.15 for existing lattice calculations). These systematic errors can be controlled using increasingly large values of ν, although this requires an increased com- putational power. Therefore, improved computational methods are required to reliably extract PDFs at small x. III.3. Nucleon Isovector Quark PDFs With the well established quasi-PDF and pseudo-PDF approaches for quark PDF, lattice com- munity significantly improves quantitative results on extracting x-dependent PDFs from lattice QCD. In the beginning of this chapter, we introduced many of these quark quasi-PDF and pseudo-PDF calculations. In this section, we will present more recent calculations of nucleon isovector quark distributions done at the physical pion mass. The first unpolarized PDFs at the physical pion mass [209, 157] using the quasi-PDFs ap- proach were determined using small momentum for unpolarized, helicity and transversity quark and antiquark distributions [210, 155, 211, 158, 156]. In the more recent studies, unpolarized, helicity and transversity quark PDFs are determined using much larger momentum at phyiscal pion mass. From top to bottom, each row of Fig. 4 shows the newer PDF results on ensem- √ bles with momenta above 2 GeV, and then renormalized at µ = {3, 3, 2} GeV, for isovector quark unpolarized [7], helicity [7], and transversity [8] PDFs, respectively. The lattice results agree with global fit unpolarized PDFs CT14 [26], NNPDF3.1 [15], CJ15 [16], helicity PDFs NNPDFpol1.1 [18], JAM17 [19], and transversity PDFs JAM17 [19], LMPSS17 [20], respec- tively. In the small-x region, larger zPz data are needed for lattice calculations to have better control. The lattice QCD calculations can now obtain the non-singlet quark PDFs at physi- cal pion mass at higher and higher momentum. The lattice-QCD calculations is not far away from providing an great impact on global analyzed PDFs with a better control on the current uncertainties. Most x-dependence PDF lattice QCD calculations, however, are done using the popular quasi-PDF and pseudo-PDF techniques and mostly limited to isovector quark distributions in the nucleon and the valence-quark distribution in the pion and kaon. Gluon distributions in the 34 5 LP3'18, Pmax   3.0 GeV 1.5 LP3'18, Pmax   3.0 GeV ETMC'20, Pmax   1.4 GeV ETMC'20, Pmax   1.4 GeV 4 ETMC'18, pmax   1.4 GeV 1.0 ETMC'18, Pmax   1.4 GeV JLab/W&M'20, Pmax   3.3 GeV NNPDF3.1 3 NNPDF 3.1 0.5 ABP16 u-d ABP16 u-d CJ15 2 CJ15 0 1 -0.5 0 -1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 x x 2.0 LP3'18, Pmax   3.0 GeV LP3'18, Pmax   3.0 GeV 4 1.5 ETMC'18, Pmax   1.4 GeV ETMC'18, Pmax   1.4 GeV NNPDFpol1.1 NNPDFpol1.1 3 JAM'17 1.0 JAM'17 Δu - Δd Δu - Δd DSSV'08 DSSV'08 2 0.5 0 1 -0.5 0 -1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 x x 2.0 LP3'18, Pmax   3.0 GeV LP3'18, Pmax   3.0 GeV 4 1.5 ETMC'18, Pmax   1.4 GeV ETMC'18, Pmax   1.4 GeV MEX'19 3 PV'18 1.0 δu - δd JAM'17 δu - δd 2 LMPSS'17 0.5 0 1 -0.5 0 -1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 x x Figure 4 The lattice calculations of isovector nucleon unpolarized (top), helicity (mid- dle) and transversity (bottom) with quark&antiquark, left&right column respectively, taken from [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. These figures are taken from reference [23]. nucleon, pion and kaon are not studied on the lattice until this thesis started [163, 195, 212, 213, 214]. In this thesis, we present the first study of x-dependence gluon nucleon and pion distributions in the following Chapter. V and IV. 35 IV. Meson gluon PDF results The lightest bound state in quantum chromodynamics (QCD), the pion, plays a funda- mental role, since it is the Nambu-Goldstone boson of dynamical chiral symmetry breaking (DCSB). Studies of pion and kaon structure reveal the physics of DCSB, help to reveal the relative impact of DCSB versus the chiral symmetry breaking by the quark masses, and are important to understand nonperturbative QCD. Studying the pion parton distribution functions (PDFs) is important to characterize the structure of the pion and further understand DCSB and nonperturbative QCD [215, 43, 44]. Currently, we know less about the pion PDFs than the nu- cleon PDFs, because there are fewer experimental data sets for the global analysis of the pion PDFs, especially for the sea-quark and gluon distributions. The future U.S.-based Electron-Ion Collider (EIC) [42], planned to be built at Brookhaven National Lab, will further our knowl- edge of pion structure [43, 44]. In China, a similar machine, the Electron-Ion Collider in China (EicC) [45], is also planned to make impacts on the pion gluon and sea-quark distributions. In Europe, the Drell-Yan and J/ψ-production experiments from COMPASS++/AMBER [216] will aim at improving our knowledge of both the pion gluon and quark PDFs. Global analyses of pion PDFs mostly rely on Drell-Yan data. The early studies of pion PDFs were based mostly on pion-induced Drell-Yan data and used J/ψ-production data or direct photon production to constrain the pion gluon PDF [50, 51, 52, 5, 53]. There are more recent studies, such as the work by Bourrely and Soffer [54], that extract the pion PDF based on Drell-Yan π + W data. JAM Collaboration [4, 24] uses a Monte-Carlo approach to analyze the Drell-Yan πA and leading-neutron electroproduction data from HERA to reach the lower-x region, and revealed that gluons carry a significantly higher momentum fraction (about 30%) in the pion than had been inferred from Drell-Yan data alone. The xFitter group [3] analyzed Drell-Yan πA and photoproduction data using their open-source QCD fit framework for PDF extraction and found that these data can constrain the valence distribution well but are not sensitive enough for the sea and gluon distributions to be precisely determined. The analysis done Ref. [55] suggests that the pion-induced J/ψ-production data has additional constraint on pion PDFs, particularly in the pion gluon PDF in the large-x region. All in all, the pion valence- quark distributions are better constrained than the gluon distribution from the global analysis of 36 experimental data. While waiting for more experimental data sets, the study of the pion gluon distribution from theoretical side can provide useful information for the experiments. The pion gluon PDF is rarely studied using continuum-QCD phenomenological models or through lattice-QCD (LQCD) simulations. Most model studies only predict the pion valence- quark distribution [217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231], but the gluon and sea PDFs are predicted by the Dyson-Schwinger equation (DSE) continuum approach [25, 232]. The prediction of the pion gluon PDF in DSE, based on an implemen- tation of rainbow-ladder truncation of DSE, is consistent with the JAM pion gluon PDF re- sult [4, 24] within two sigma. LQCD provides an first-principles calculations to improve our knowledge of nonperturbative pion gluon structure; however, there have been only two efforts to determine the first moment of pion gluon PDF [233, 234]. An early calculation in 2000 using quenched QCD predicted hxig = 0.37(8)(12), using Wilson fermion action with a lat- tice spacing a = 0.093 fm, lattice size L3 × T = 244 , a large 890-MeV pion mass and 3,066 configurations at µ2 = 4 GeV2 [233]. A more recent study in 2018 using Nf = 2 + 1 clover fermion action with a lattice spacing a = 0.1167(16) fm, larger lattice size 323 × 96, 450-MeV pion mass, and 572,663 measurements, gave a larger first-moment result, hxig = 0.61(9) at µ2 = 4 GeV2 [234]. In principle, a series of moments can be used to reconstruct the PDF. Although there are calculations of the first moment of the pion gluon PDF, there is little chance that sufficient higher moments of the pion gluon PDF can be obtained to perform such a recon- struction. A direct lattice calculation of the x-dependence of the pion gluon PDF is needed. Only recently have lattice calculations of the gluon PDF become possible, when the neces- sary one-loop matching relations of the gluon PDF were computed for the pseudo-PDF [204] and quasi-PDF [201, 164] approaches. Both approaches make direct calculation of the x depen- dence of the pion gluon PDF feasible. In this work, we apply the pseudo-PDF method by using the ratio renormalization scheme to avoid the difficulty of calculating the gluon renormalization factors. We follow a developed procedure for using the pseudo-PDF method to obtain light- cone PDFs from Ioffe-time distributions (ITDs) by matching through two steps, evolution and scheme conversion [192, 186, 190, 12]. Another commonly used procedure is direct matching to the lightcone ITDs [191, 13]. Using the pseudo-PDF method also allows us to use lattice correlators at all boost momenta at small Ioffe-time. There have been a number of successful 37 pseudo-PDF calculations of nucleon isovector PDFs [186, 190, 13, 12] and pion valence-quark PDFs [191]. The earliest calculation was done on a quenched lattice [186], then the pion masses were set closer to the physical pion mass [190, 191, 13], and the calculation at physical pion mass was done recently [12]. The lattice-calculated PDFs in Refs. [190, 191, 13, 12] show good agreement with the global-analysis PDFs. In this work, we present the first calculation of the full x-dependent pion gluon distribu- tion using the pseudo-PDF method from two lattice spacings, 0.12 and 0.15 fm, and three pion masses: 690, 310 and 220 MeV. The rest of the paper is organized as follows. We present the procedure to obtain the lightcone gluon PDF from the reduced pseudo-ITDs, the numerical setup of lattice simulation, and how we extracted the reduced pseudo-ITDs from lattice calcu- lated correlators, the final determination of the pion gluon PDF from our lattice calculations is compared with the NLO xFitter [3] and JAM pion gluon PDFs [4, 24], and the systematics induced by different steps are studied, and the lattice-spacing and pion-mass dependence are investigated. IV.1. Ioffe-time distribution In this thesis, we use the gluon pseudo-PDF matching introduced in Sec. III.2 and the unpolar- ized gluon operator defined in Eq. V.26. We neglect the pion quark PDF, since the total quark PDF is found to be much smaller than the gluon PDF in global fits [4, 3]. We will later estimate the systematic uncertainty introduced by this assumption. The gluon evolved pITD (EpITD), G is obtained by using the evolution term R1 (y, z 2 µ2 ), Z 1 G(ν, µ) = M (ν, z ) + 2 dxR1 (x, z 2 µ2 )M (xν, z 2 ). (IV.1) 0 The z dependence of the EpITDs should be compensated by the ln z 2 term in the evolution formula. In principle, the EpITD G is free of z dependence and is connected to the lightcone gluon PDF g(x, µ2 ) through the scheme-conversion term R2 (y), 1 xg(x, µ2 ) Z G(ν, µ) = dx R2 (xν), (IV.2) 0 hxig so the gluon PDF g(x, µ2 ) can be extracted by inverting this equation. On the lattice, we use clover valence fermions on three ensembles with Nf = 2 + 1 + 1 38 highly improved staggered quarks (HISQ) [71] generated by the MILC Collaboration [63, 64, 235, 236] with two different lattice spacings (a ≈ 0.12 and 0.15 fm) and three pion masses (220, 310, 690 MeV). The masses of the clover quarks are tuned to reproduce the lightest light and strange sea pseudoscalar meson masses used by PNDME Collaboration [237, 238, 239, 240]. We use five HYP-smearing [89] steps on the gluon loops to reduce the statistical uncertainties, as studied in Ref. [163]. We use Gaussian momentum smearing for the quark fields [107] 2π q(x)+α j Uj (x)ei( L )kêj q(x+êj ), to reach higher meson boost momenta with the momentum- P smearing parameter k listed in Table 4. Table 4 gives the lattice spacing a, valence pion mass Mπval and ηs mass Mηval s , lattice size L3 × T , number of configurations Ncfg , number of total two- 2pt point correlator measurements Nmeas , and separation time tsep used in the three-point correlator fits for the three ensembles. This allows us to reach the continuum limit and physical pion mass through extrapolation. The total amount of measurements vary in 105 –106 for different ensembles. ensemble a12m220 a12m310 a15m310 a (fm) 0.1184(10) 0.1207(11) 0.1510(20) val Mπ (MeV) 226.6(3) 311.1(6) 319.1(31) Mηval s (MeV) 696.9(2) 684.1(6) 687.3(13) 3 L ×T 323 × 64 243 × 64 163 × 48 Pz (GeV) [0, 2.29] [0, 2.14] [0, 2.05] k 3.9 2.9 2.3 Ncfg 957 1013 900 2pt Nmeas 731,200 324,160 21,600 tsep {5,6,7,8,9} {5,6,7,8,9} {4,5,6,7} Table 1 Lattice spacing a, valence pion mass Mπval and ηs mass Mηval s , lattice size L3 × T , 2pt number of configurations Ncfg , number of total two-point correlator measurements Nmeas , and separation times tsep used in the three-point correlator fits of Nf = 2 + 1 + 1 clover valence fermions on HISQ ensembles generated by MILC Collaboration and analyzed in this study. The two-point correlator for a meson Φ is Z CΦ2pt (Pz ; t) = dy 3 e−iy·Pz hχΦ (~y , t)|χΦ (~0, 0)i = |AΦ,0 |2 e−EΦ,0 t + |AΦ,1 |2 e−EΦ,1 t + ..., (IV.3) where Pz is the meson momentum in the z-direction, χΦ = q̄1 γ5 q2 is the pseudoscalar-meson interpolation operator, t is the Euclidean time, and |AΦ,i |2 and EΦ,i are the amplitude and energy for the ground-state (i = 0) and the first excited state (i = 1), respectively. 39 Figure 5 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from the lightest pion mass a ≈ 0.12 fm, Mπ ≈ 220 MeV for Pz = 2 × 2π/L, z = 1 (upper row) and Pz = 4 × 2π/L, z = 4 (lower row). The gray band shown on all plots is the extracted ground-state matrix element from the two-sim fit using tsep ∈ [5, 9]. From left to right, the columns are: the ratio of the three-point to two-point correlators with the reconstructed fit bands from the two-sim fit using tsep ∈ [5, 9], shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 9], the two-sim fit results min max using tsep ∈ [tminsep , 9] as functions of tsep , and the two-sim fit results using tsep ∈ [5, tsep ] as functions of tmax sep . The three-point gluon correlators are obtained by combining the gluon loop with pion two- point correlators. The matrix elements of the gluon operators can be obtained by fitting the three-point correlators to the energy-eigenstate expansion, CΦ3pt (z, Pz ; tsep , t) Z = d3 y e−iy·Pz hχΦ (~y , tsep )|O(z, t)|χΦ (~0, 0)i = |AΦ,0 |2 h0|O|0ie−EΦ,0 tsep + |AΦ,0 ||AΦ,1 |h0|O|1ie−EΦ,1 (tsep −t) e−EΦ,0 t + |AΦ,0 ||AΦ,1 |h1|O|0ie−EΦ,0 (tsep −t) e−EΦ,1 t + |AΦ,1 |2 h1|O|1ie−EΦ,1 tsep + ..., (IV.4) where tsep is the source-sink time separation, and t is the gluon-operator insertion time. The amplitudes and energies, AΦ,0 , AΦ,1 , EΦ,0 and EΦ,1 , are obtained from the two-state fits of the two-point correlators. h0|O|0i, h0|O|1i (h1|O|0i), and h1|O|1i are the ground-state ma- trix element, the ground–excited-state matrix element, and the excited-state matrix element, respectively. We extract the ground-state matrix element h0|O|0i from the two-state fit of the three-point correlators, or a two-state simultaneous “two-sim” fit on multiple separation times 40 with the h0|O|0i, h0|O|1i and h1|O|0i terms. To verify that our fitted matrix elements are reliably extracted, we compare to ratios of the three-point to the two-point correlator C 3pt (z, Pz ; tsep , t) Rratio (z, Pz ; tsep , t) = ; (IV.5) C 2pt (Pz ; tsep ) if there were no excited states, the ratio would be the ground-state matrix element. The left- hand side of Fig. 25 shows example ratios for the gluon matrix elements from the lightest pion ensemble, a12m220, at selected momenta Pz and Wilson-line length z. We see the ratios increase with increasing source-sink separation going from 0.60 to 1.08 fm. At large separation, the ratios begin to converge, indicating the neglect of excited states becomes less problematic. The gray bands indicate the ground-state matrix elements extracted using the two-sim fit to three-point correlators at five tsep . The convergence of the fits that neglect excited states can also be seen in second column of Fig. 25, where we compare one-state fits from each source- sink separations: the one-state fit results increase as tsep increases, starting to converge at large tsep to the two-sim fit results. The third and fourth columns of Fig. 25 show two-sim fits using tsep ∈ [tmin sep , 9] and tsep ∈ [5, tmax sep ] to study how the two-sim ground-state matrix elements depend on the source-sink sep- arations input into fit. We observe that the matrix elements are consistent with each other within one standard deviation, showing consistent extraction of the ground-state matrix ele- ment, though the statistical errors are larger than those of the one-state fits. We observe larger fluctuations in the matrix element extractions when small tmin max sep = 3 and 4, or small tsep = 6 and 7, are used. The ground state matrix element extracted from two-sim fits becomes very stable when tmin max sep > 4 and tsep > 7. Figure 6 shows the RpITD of the same examples Pz = 2×2π/L, z = 1 and Pz = 4×2π/L, z = 4 from two-sim fit results using tsep ∈ [tmin sep , 9]. The RpITD results, which are constructed to suppress lattice fluctuations, are very stable over the range of different fits considered. For a12m310 and a15m310 ensembles, the tsep dependence of RpITDs is milder than those from a12m220 ensemble due to the heavier pion mass. Overall, our ground-state RpITDs from the two-sim fit are stable, and we use them to extract the gluon PDF. Using the RpITDs extracted in the previous section, we examine the pion-mass and lattice- 41 1.10 1.2 1.05 1.0 M(z=1, Pz =2) M(z=4, Pz =4) 1.00 0.8 0.95 0.6 0.90 0.4 0.85 0.2 0.80 0.0 3 4 5 6 7 8 3 4 5 6 7 8 tmin sep tmin sep Figure 6 Example RpITDs from the a12m220 ensemble as functions of tmin sep for Pz = 2 × 2π/L, z = 1 (top) and Pz = 4 × 2π/L, z = 4 (bottom). The two-sim fit RpITD results using tsep ∈ [tmin sep , 9] are consistent with the ones final chosen tsep ∈ [5, 9]. spacing dependence. The top of Fig. 26 shows the ηs RpITDs at boost momentum around 2 GeV as functions of the Wilson-line length z for the a12m220, a12m310, and a15m310 ensembles. We see no noticeable lattice-spacing dependence. The bottom of Fig. 26 shows the pion RpITDs with boost momentum around 1.3 GeV for the same ensembles. Again, there is no visible lattice-spacing or pion-mass dependence. 2.0 2.0 a12m220, Pz =1.31 GeV a12m220, Pz =1.96 GeV 1.5 1.5 a12m310, Pz =1.28 GeV a12m310, Pz =2.14 GeV a15m310, Pz =1.54 GeV a15m310, Pz =2.05 GeV 1.0 1.0 M(ν,z2) M(ν,z2) 0.5 0.5 0.0 0.0 -0.5 -0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z (fm) z (fm) Figure 7 The ηs (top) and pion (bottom) RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV, respectively, for the a12m220, a12m310, and a15m310 ensembles. In both cases, we observe weak lattice-spacing and pion-mass dependence. To extract gluon PDFs, we follow the steps in Sec. V.3.1 between Eq. V.10 and Eq. IV.1 by first obtaining EpITDs and using Eq. IV.2 to extract g(x). To obtain EpITDs, we need the RpITD M (ν, z 2 ) to be a continuous function of ν to evaluate the x ∈ [0, 1] integral in Eq. IV.1. We achieve this by using a “z-expansion”1 fit [241, 242] following previous quark pseudo-PDF 1 Note that the z in the “z-expansion” is not related to the Wilson link length z we use elsewhere. 42 calculations [191]. The following form is used [191]: kmax X M (ν, z , a, Mπ ) = 2 λk τ k , (IV.6) k=0 √ √ ν +ν− νcut where τ = √ cut √ νcut +ν+ νcut . Then, we use the fitted M (ν, z 2 ) in the integral in Eq. IV.1. The fits are performed by minimizing the χ2 function, X (M (ν, z 2 ) − M (ν, z 2 , a, Mπ )2 χ2M (a, Mπ ) = 2 . (IV.7) ν,z σM (ν, z 2 , a, Mπ ) The z-dependence in the M (uν, z 2 ) term of the evolution function comes from the one-loop matching term, which is a higher-order correction compared to the tree-level term; thus, the z-dependence can be neglected in M (ν, z 2 ) in the integral in Eq. IV.1. We adopt as the best value νcut = 1, as used in Ref. [191], but we also vary νcut in the range [0.5, 2], and the results are consistent. We fix λ0 = 1 to enforce the RpITD M (ν, z 2 ) in Eq. V.11. The expansion order kmax = 3 is used, because we can fit all the data points of Pz ∈ [1, 5]×2π/L (Pz ∈ [1, 7]×2π/L for a12m220 ensemble) and z up to 0.6 fm with χ2 /dof < 1 using a 4-term z-expansion for each ensemble. The reconstructed bands from “z-expansion” on RpITDs are shown in the upper plot in Fig. 8. They describe the RpITD data points well for all ensembles. 2.0 2.0 ■ a=0.12 fm, Mπ ∼220 MeV ● a=0.12 fm, Mπ ∼690 MeV ■ a=0.12 fm, Mπ ∼220 MeV ● a=0.12 fm, Mπ ∼690 MeV ▲ a=0.12 fm, Mπ ∼310 MeV ▼ a=0.15 fm, Mπ ∼310 MeV ▲ a=0.12 fm, Mπ ∼310 MeV ▼ a=0.15 fm, Mπ ∼310 MeV 1.5 1.5 1.0●▼▲■ ●■▲ ●▼■■ ▲●■ ●▼▲■ ●■▲ ● ■ ▲● 1.0●▼▲■ ●■▲●▼■■ ▲●■ ●▼▲■ ●■▲▲●■▼■ ▲●■ ●▼■ ●■ ● ■ ● ▼ ■● ■●■ ▲ ■ ▼ M(ν,z2 ) ▲▼ ■ ●● ●● ■● ▼ ▼ ■● ■ ▲ ▲ ● ■ ■ ▲● ● ■▲ ■ ▲ ■ ● ● ■ ●▲ ▲ ● ▼ ▲ ● G(ν,μ) ■ ▲ ■ ■▲ ■ ▲ ■▲ ●■ ▲● ● ▼ ■ ▼ ▲ ■ ■ ▼ ● ● ▼ ▼ ▲ ■ ■ ● ■ ▼ ■▲ ■▲ ▲ ▼ ▲■ ▲ ■ ▲ ▼ ▲ ▲ ▲ ■● ■● ▲ ■ ■ ▼▲ ■ ▼ ▼ ▲ ■ ■ ▲● ● ● ▼ ● ▼▲ ▼ ▼ ▼ ■ ▼ ▲ ■ ▲ ● ● ▼ ▼ 0.5 ▼ ▼ ▲ ■ ■ ■ ■ ▲ ▲ ● 0.5 ▼ ■ ● ▼ ▼ ▲ ■ ▼ 0.0 ▼ ■ 0.0 ▼ xFitterPI_NLO -0.5 -0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ν ν Figure 8 The RpITDs M with reconstructed bands from “z-expansion” fits (top) and the EpITDs G with reconstructed bands from the fits to the Eq. V.13 form (bottom) calculated on ensembles with lattice spacing a ≈ 0.12 fm, pion masses Mπ ≈ {220, 310, 690} MeV, and a ≈ 0.15 fm, Mπ ≈ 310 MeV, noticing that a ≈ 0.12, Mπ ≈ 690 MeV results are from a12m220 ensemble here. After we have the continuous-ν fitted RpITDs, we obtain the EpITDs through Eq. IV.1. The RpITDs M and EpITDs G as functions of ν on all ensembles studied in this work are shown in Fig. 8. At some ν values, there are multiple z and Pz combinations for a fixed ν value. 43 Therefore, there are points in the same color and symbol overlapping at the same ν from the same lattice spacing and pion mass. To match with the lightcone gluon PDF through Eq. IV.2, the EpITDs G(ν, µ) should be free of z 2 dependence. However, the EpITDs obtained from Eq. IV.1 have z 2 dependence from neglecting the gluon-in-quark contribution and higher-order terms in the matching. The EpITDs also depend on lattice-spacing a and pion-mass Mπ . Recall that the RpITDs show weak dependence on lattice spacing a and pion mass Mπ . We see that the effects of a and Mπ dependence on the EpITDs are also not large; the EpITD results from different a, Mπ are mostly consistent with each other, as shown in the second row of Fig. 8. We also observe a weak dependence on z 2 for the RpITDs and EpITDs in Fig. 8. IV.2. Pion gluon PDF The gluon PDF g(x, µ2 ) can now be extracted from the EpITDs using Eq. IV.2. We assume a functional form, also used by JAM [4, 24], for the lightcone PDF to fit the EpITD, xg(x, µ) xA (1 − x)C fg (x, µ) = = , (IV.8) hxig (µ) B(A + 1, C + 1) R1 for x ∈ [0, 1] and zero elsewhere. The beta function B(A + 1, C + 1) = 0 dx xA (1 − x)C is used to normalize the area to unity. Then, we apply the matching formula to obtain the EpITD G from the functional form PDF using Eq. IV.2. We fit the EpITDs G(ν, µ) obtained from the parametrization to the EpITDs G(ν, z 2 , µ, a, Mπ ) from the lattice calculation. The fits are performed by minimizing the χ2 function, X (G(ν, µ) − G(ν, µ, a, Mπ ))2 χ2G (µ, a, Mπ ) = 2 . (IV.9) ν σG (ν, µ, a, Mπ ) We investigate the systematic uncertainty introduced by the different parametrization forms which are commonly used for fg (x, µ) in PDF global analysis and some lattice calculations. The first one is the 2-parameter form in Eq. V.13. Second, we consider the 1-parameter form N1 (1 − x)C used in xFitter’s analysis [3] (also used in Ref. [51, 52]), which is equivalent to Eq. V.13 with A = 0. Third, we consider a 3-parameter form, √ xA (1 − x)C (1 + D x) fg,3 (x, µ) = , (IV.10) B(A + 1, C + 1) + DB(A + 1 + 1/2, C + 1) We fit the three different forms to the EpITDs of lattice data with zmax ≈ 0.6 fm by applying 44 3.0 2.5 1-parameter 2-parameter xg(x,μ=2 GeV)/〈x〉g 2.0 3-parameter 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x Figure 9 The xg(x, µ)/hxig at µ2 = 4 GeV2 as function of x (bottom) calculated with lattice spacing a ≈ 0.12 fm, pion masses Mπ ≈ 220 MeV with the fitted bands of zmax ≈ 0.6 fm from the 1-, 2- and 3-parameter fits described in Eq. V.13 and the paragraph after it. the scheme conversion Eq. IV.2 to the 1-, 2- and 3-parameter PDF forms. Here, we focus on the result from the lightest pion mass Mπ ≈ 220 MeV at lattice spacing a ≈ 0.12 fm. The χ2 /dof of the fits decreases as 1.47(72), 1.08(68), to 1.04(41), shows slightly better fit quality for 2- and 3-parameter fits. As shown in Fig. 9, there is a big discrepancy between the fg (x, µ) fit bands from the 1-parameter fit and the 2-parameter fit in the x < 0.4 region, but the discrepancy between the 2- and 3-parameter fits is much smaller. Therefore, we conclude that 1-parameter fit on lattice data here is not quite reliable, and the fit results converge at the 2- and 3-parameter fits. The same conclusions hold for all other ensembles and pion masses. Therefore, using the 2-parameter form defined in Eq. V.13 (same parametrization as JAM) for our final results is very reasonable. Another source of systematic uncertainty comes from neglecting the contribution of the quark term in the matching based on the assumption (motivated by global fits) that the pion qS (x) is smaller than the gluon PDF. Currently, there are no qS (x) results from lattice simulation since only the valence distribution of the pion has been done. Thus, we estimate the systematic due to omitting the qS (x) contribution by using the pion quark PDFs from xFitter [3] at NLO. Using these, we obtain revised RpITDs and EpITDs including the gluon-in-quark Rgq term focusing on example from the a ≈ 0.12 fm, pion mass Mπ ≈ 220 MeV lattice, repeating the same procedure from Eq. IV.6 and fitting the EpITDs with Eq. V.13. On the left-hand side of Fig. 10, we show the mean value of xg(x, µ)/hxig with both gluon-in-gluon (gg) and gluon-in- quark (gq) contributions (the blue solid line) compared to the a12m220 results using the gluon- in-gluon contribution only (the green solid line). There are 5 to 10% differences in the mean 45 value including the gluon-in-gluon contribution for x < 0.9, which indicates that the gluon- in-quark contribution is relatively small at µ2 = 4 GeV2 compared to the current statistical errors in the small-x region. In the x > 0.9 region, the gluon-in-quark contribution becomes more significant, but it remains smaller than the statistical error. Once studies are available with sufficiently reduced statistical uncertainty in the large-x region, the quark contribution will need to be included. From the above analyses of the choice of fit form and the contribution of the quark term, we conclude that these systematics are negligible relative to the current statistics. Finite-volume systematics have not been taken into account in this work. However, the results of the finite- volume study on the nucleon isovector PDFs on the a12m220 ensemble with multiple lattice volumes (2.88, 3.84, 4.8 fm) suggest that the finite-volume effect is negligible at the current lattice precision [165]. This is consistent with a later study using chiral perturbation theory (ChPT), [243], also showing that momentum boost reduces the finite-volume effect, since the length contraction of the hadron makes the lattice effectively bigger. We expect the finite- volume error to be much smaller than the statistical ones. Therefore, we adopt the zmax ≈ 0.6 fm (zmax ≈ 0.75 fm for a15m310 ensembles) fits to the EpITDs, neglect the quark contribution term in the matching, and use the Eq. IV.2 fit form for our final results on all lattice ensembles. The xg(x, µ)/hxig reconstructed fit bands of these ensembles are shown in the left plot in Fig. 10, comparing results from different lattice spacings and pion masses. The reconstructed fit bands with different pion mass Mπ ≈ {220, 310, 690} MeV are consistent at the same lattice spacing a ≈ 0.12 fm, indicating mild gluon PDF dependence on pion mass. Similarly, when comparing lattice-spacing dependence of pion PDFs using data around pion mass Mπ ≈ 310 MeV (Mπ ≈ 690 MeV in the inserted plot), we find that fitted PDF is slightly smaller in the x > 0.1 region for the 0.12-fm lattice, but still within one sigma, which indicates the lattice-spacing dependence is also mild. We also note that the bands from different ensembles show a differing speed of fall-off as x → 1 in the large-x region. We study this fall-off behavior in more depth below. The behavior of the gluon PDF fall-off in the large-x region is widely studied in both theory and global analyses. Perturbative QCD studies [244, 245] and DSE calculations [230, 25, 232] suggest that the gluon distribution g(x, µ2 ) ∼ (1 − x)C with C ≈ 3 in the limit x → 1. The 46 prediction from perturbative QCD [245] is based on the idea that the gluon PDF should be suppressed at large x relative to the quark PDF, because the quarks are the sources of large- x gluons; that is, g(x, µ2 )/qv (x, µ2 ) → 0 as x → 1. Early fits of experimental data gave C ≈ 2 [51, 52] or C < 2 [5, 53], but the more recent global analysis from JAM collaboration yielded C > 3 [4, 24] and xFitter collaboration found C ≈ 3 [3]. Our fitted parameter C is 3.6(1.5), 3.3(2.0), 4.7(2.8) for Mπ ≈ {690, 310, 220} MeV, respectively, at lattice spacing a ≈ 0.12 fm. These C results are consistent with each other and show a slightly increasing trend as the pion mass approaches the physical pion mass. For lattice spacings a ≈ {0.15, 0.12} fm, C = {2.2(1.5), 3.3(2.0)}, respectively, at Mπ ≈ 310 MeV, which suggests that C will increase toward the continuum limit. We also investigate the effect of the gluon-in-quark contribution on the C value, and it makes about 0.1 difference, which we neglect. Given that both the pion-mass and lattice-spacing extrapolations seem to show increasing C, it seems reasonable to conclude from this lattice-QCD study that C > 3. We compare our reconstructed gluon PDF to those from global fits on the right-hand side of Fig. 10. It shows the xg(x, µ)/hxig reconstructed fit band of a ≈ 0.12 fm, Mπ ≈ 220 MeV lattice, from DSE calculation [25], and NLO pion gluon PDFs from xFitter [3] and JAM [4, 24] at µ2 = 4 GeV2 . The JAM band appears somewhat wider than expected, because we reconstruct it by dividing xg(x, µ) by the mean value of hxig ; the correlated values needed for a correct error estimation were not available. Note that xFitter uses the fit form of Eq. V.13 with A = 0. Our fitted pion gluon PDF is consistent with JAM and DSE for x > 0.2, and with xFitter for x > 0.5 within one sigma. We also show x2 g(x, µ)/hxig for x > 0.5 region in the inserted plot on the right-hand side of Fig. 10. We see in this comparison that our results are of similar error size as the global-fit analysis and are useful to provide constraints from theoretical calculation in addition to the experimental data. IV.3. Summary In this work, we presented the first calculation of the pion gluon PDF from lattice QCD and studied its pion-mass and lattice-spacing dependence using the pseudo-PDF approach. We employed clover valence fermions on ensembles with Nf = 2+1+1 highly improved staggered quarks (HISQ) at two lattice spacings (a ≈ 0.12 and 0.15 fm) and three pion masses (220, 310 47 3.0 3.0 a∼0.12 fm, Mπ ∼220 MeV, gg+gq MSULat'21 2.5 a∼0.12 fm, Mπ ∼220 MeV 2.5 xFitter'20 a∼0.12 fm, Mπ ∼310 MeV xg(x,μ=2 GeV)/〈x〉 xg(x,μ=2 GeV)/〈x〉 2.0 2.0 JAM'21 a∼0.12 fm, Mπ ∼690 MeV DSE'20 1.5 a∼0.15 fm, Mπ ∼310 MeV 1.5 1.0 1.0 0.5 0.5 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x Figure 10 The pion gluon PDF xg(x, µ)/hxig as a function of x obtained from the fit to the lattice data on ensembles with lattice spacing a ≈ {0.12, 0.15} fm, pion masses Mπ ≈ {220, 310, 690} MeV (left plot and its inserted plot), and xg(x, µ)/hxig (x2 g(x, µ)/hxig in the inserted plot) as function of x obtained from lattices of a ≈ 0.12 fm, Mπ ≈ 220 MeV (right), compared with the NLO pion gluon PDFs from xFitter’20 and JAM’21, and the pion gluon PDF from DSE’20 at µ = 2 GeV in the MS scheme. The JAM’21 error shown is overestimated due to lack of available correlated uncertainties in its constituent components. Our PDF results are consistent with JAM [4, 24] and DSE [25] for x > 0.2, and xFitter [3] for x > 0.5. and 690 MeV). These ensembles allowed us to probe the dependence of the pion gluon PDF on pion mass and lattice spacing. In both cases, the dependence appears to be weak compared to the current statistical uncertainty. We investigated the systematics associated with the functional form used in the reconstruc- tion fits as well as the systematics caused by neglecting the quark contribution in the matching. The effect of the assumed gluon PDF fit form was investigated by using various forms, which are all commonly used or proposed in other PDF works. We observe large effects changing the fit to xg(x, µ)/hxig from 1- to 2-parameter form but convergence at 3 parameters. This implies the 2-parameter fits are sufficient for our calculation, and our finial pion gluon PDF results are presented using the 2-parameter fit results. We used the pion quark PDF from xFitter to make an estimation of the quark contribution to the pion gluon RpITD. We found the systematic errors it contributed are smaller than 10% of the statistical errors. Our pion gluon PDF for the lightest pion mass is consistent with JAM’21 and DSE’20 for x > 0.2, and with xFitter’20 for x > 0.5 within uncertainty, as shown in our final comparison plots of the pion gluon PDF. We also studied the asymptotic behavior of the pion gluon PDF in the large-x region in terms of (1 − x)C . C > 3 is implied from our study at two lattice spacings and three pion masses. The future study of the pion gluon PDF from the lattice QCD with improved precision and systematic control when combined in global-fit analyses with the 48 results of anticipated experiments [216, 43, 45, 216] will provide best determination of the gluon content within the pion. 49 V. Nucleon gluon PDF The unpolarized gluon parton distribution functions (PDFs) g(x) and quark PDFs q(x) are important inputs to many theory predictions for hadron colliders [26, 47, 15, 14, 16, 246, 247, 248]. For example, both g(x) and q(x) contribute to the deep inelastic scattering (DIS) cross section, and g(x) enters at leading order in jet production [32, 33]. To calculate the cross section for these processes in pp collisions, g(x) needs to be known precisely. Although there are experimental data like top-quark pair production, which constrains g(x) in the large-x region, and charm production, which constrains g(x) in the small-x region, g(x) is still experimentally the least known unpolarized PDF because the gluon does not couple to electromagnetic probes. The Electron-Ion Collider (EIC), which aims to understand the role of gluons in binding quarks and gluons into nucleons and nuclei, is at least in part intended to address this gap in our experimental knowledge [42]. In addition to experimental studies, the theoretical approaches to determining gluon structure by calculation are continually improving. The recent calculations on nucleon PDFs based on quasi-PDF, pseudo-PDF, ”good lattice cross sections” approaches are listed in the beginning of Chap. III V.1. First Exploratory Study In our first calculation of gluon quasi-PDF, we defined the gluon quasi-PDF matrix element and operator different with the LaMET operators we discussed in Sec. III.1, H̃0 (z, Pz ) = hP |O0 (z)|P i, (V.1) O(F tµ , F µt ; z) − 14 g tt O(F µν , F νµ ; z)  P0 O0 ≡ 3 2 , P + 41 Pz2 4 0 renormalized at the scale µ with O(F ρµ , F µτ ; z) = F ρµ (z)U (z, 0)F µτ (0). When z = 0, H̃0 (0, Pz ) is a local operator and equals to hxig . In the large momentum limit, only the leading twist contribution in g̃(x) survives, and then g̃(x) can be factorized into the the gluon PDF g(y) and a perturbative calculable kernel C(x, y), up to mixing with the quark PDF and the higher- twist corrections O(1/Pz2 ). This operator is later proved not multiplicatively renormalizable in Ref. [201] and not used in our following calculations. Since the Lattice calculation of H̃0 (z, Pz ) is under the lattice regularization, a non-perturbative renormalization (NPR) of the glue operators O0 (z) is required to convert H̃0 (z, Pz ) into that 50 under the MS scheme with the perturbative matching in the continuum. This can be achieved following the glue NPR strategy introduced in Ref. [30] just recently for hxig . As shown in Refs. [201, 249], the O(F zµ , F µz ; z) and O(F µν , F νµ ; z) (µ, ν 6= z) struc- tures in O0 should be renormalized separately before combined together, but its linear diver- gence [250, 251] is an overall multiplicative factor depending on the Wilson-link length z. For the linear divergence introduced by the Wilson link, an empirical observation in the quark un- polarized quasi-PDF case is that, the non-perturbative RI/MOM renormalization constant with pR z = 0 can be approximated by the nucleon iso-vector matrix element with Pz = 0 in the z < 0.5 fm region, with ∼10% deviation, while the systematic uncertainties due to the hadron IR structure is hard to estimate [200]. If the gluon case is similar, the linear divergence of the gluon quasi-PDF matrix element can be removed by defining the “ratio renormalization” (similar to the reduce Ioffe-time distribution considered in the quark case [203, 186, 208]) H̃0MS (0, 0, µ) H̃0Ra (z, Pz , µ) = H̃0 (z, Pz ) (V.2) H̃0 (z, 0) as an approximation of the RI/MOM renormalized one, with H̃0Ra (z, Pz , µ) = hxiMS g (µ). After the renormalization, both the quark and gluon PDF contribute to the factorization of the gluon qausi-PDF [250], and the case with the gluon quasi-PDF operator defined here will be investigated in a future study. In this work, we will calculate the gluon quasi-PDF matrix element and apply the “ratio renormalization” to have a glimpse on the range of z and Pz one can reach on the lattice, and compare it with the FT of the gluon PDF. Numerical setup: The lattice calculation is carried out with valence overlap fermions on 203 configurations of the 2 + 1-flavor domain-wall fermion gauge ensemble “24I” [252] with L3 × T = 243 × 64, a = 0.1105(3) fm, and Mπsea =330 MeV. For the nucleon two-point function, we calculate with the overlap fermion and loop over all timeslices with a 2-2-2 Z3 grid source and low-mode substitution [253, 254], and set the valence-quark mass to be roughly the same as the sea and strange-quark masses (the corresponding pion masses are 340 and 678 MeV, respectively). Counting independent smeared-point sources, the statistics of the two- point functions are 203 × 64 × 8 × 2 = 207, 872, where the last factor of 2 coming from the averaging between the forward and backward nucleon propagators. 51 On the lattice, O0 is defined by P0 OE (Ftµ , Fµt , z) − 14 OE (Fµν , Fνµ ; z)  O0 = − (V.3) P + 14 Pz2 3 2 4 0   where OE (Fρµ , Fµτ , z) = 2Tr Fρµ (z)U (z, 0)Fµτ (0)U (0, z) is defined in the Euclidean space with the gauge link U (z, 0) in the fundamental representation, and the clover definition of the field tensor Fµν is the same as that used in our previous calculation of the glue momentum fraction [30]. The choice for the quasi-PDF operator is not unique. Any operator that approaches the lightcone one in the large-momentum limit is a candidate, such as the other choices inspired by Eq. (??) 1 O1 (z) ≡ O(Ftµ , Fzµ ; z), Pz P0 O(Fzµ , Fµz ; z) − 14 g zz O(Fµν , Fνµ ; z)  O2 (z) ≡ 1 2 , (V.4) P + 34 Pz2 4 0 as well as 1 O3 (z) ≡ O(Fzµ , Fµz ; z) (V.5) P0 proposed in Ref. [149]. These alternative operators O1,2,3 can be defined on the lattice similarly. As we will address in the latter part of this work, the quasi-PDF using O1,2,3 has larger higher- twist corrections and/or statistical uncertainty compared to that from using O0 . The bare glue matrix element H̃0 (z, Pz ) with the Wilson link length z and nucleon momen- tum {0, 0, Pz } can be obtained from the derivative of the summed ratio following the recent high-precision calculation of nucleon matrix elements [255, 256], X X R̃(z, Pz ; tsep ) = R(z, Pz ; tsep , t) − R(z, Pz ; tsep − 1, t) 0 6 and a constant fit can provide the same result as what can be obtained from the two-state fit of R with larger tsep . In the tsep  t  0 limit, both R̃ and R saturate to the same H̃0 (0, 0) = hxibare g = 0.55(8) as in the figure, while such a limit can be reached with smaller tsep in the R̃ case. Using the renormalization constant of hxig in MS at 2 GeV 53 with 5 steps of the HYP smearing calculated in Ref. [30] of 0.90(10) and ignoring mixing from Ra the quark momentum fraction, the MS renormalized hxiMS g (2 GeV) = H̃0 (0, 0, 2 GeV) = 0.50(7)(5) agrees with the phenomenological determination 0.42(2) [26] within uncertainties. 1.5 ● O0 HYP5 ■ O1 HYP5 1.5 ● O0 HYP5 ■ O1 HYP5 ◆ O2 HYP5 ▲ O3 HYP5 ◆ O2 HYP5 ▲ O3 HYP5 1.0 1.0 (z=0,Pz ) (z=3,Pz ) ■ 0.5 ● ◆ ●■◆ ● ◆ 0.5 ● ● ●■◆ ■ ◆ ◆ 0.0 0.0 ∼ Ra ∼ Ra Hi Hi ▲ -0.5 -0.5 ▲ ▲ ▲ ▲ -1.0 ▲ -1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Pz (GeV) Pz (GeV) Ra Figure 13 The renormalized H̃i=0,1,2,3 (z, Pz ) as a functions of Pz at z=0 (top) and 3 (bottom). Some data with the same Pz are shifted horizontally to enhance the legibility. The case with Oi=3 suffers from a large contamination from higher-twist distributions, while the results with Oi=0,1,2 are consistent with each other, especially at larger Pz . Due to its linear divergence [251], the bare H̃0 (z, Pz ) decays exponentially as |z| increases. Fig. 12 shows the z dependence of H̃0 (z, Pz ) with Pz = 0.46 GeV and 1, 3 and 5 HYP smearing steps. It is obvious to see that the decay rates decreases when more steps of smearing are applied, since the corresponding linear divergence becomes smaller. Note that H̃0 (z, Pz ) is purely real and symmetric with respect to z; thus, we just plot the real part in the positive-z region. The “ratio renormalized” matrix elements H̃0Ra (z, Pz ) with different HYP smearing steps are consistent with each other, as shown in Fig. 12, while more HYP smearing can reduce the statistical uncertainties significantly. Ra H̃0MS (0,0,µ) Then, we plot the “ratio renormalized” H̃i=0,1,2,3 (z = 0, Pz ) using Z(µ, z) ≡ H̃0 (0,0,µ) for the glue operator Oi with 5 HYP smearing steps and Pz = 0.0, 0.46, 0.92 GeV in the top panel of Fig. 13. All the cases with Oi=0,1,2 provide consistent results, except O3 which suffers from large mixing with the higher-twist operator O(Fνµ , Fµν ; z). With larger Pz , the value of H̃3Ra (0, Pz ) becomes less negative as higher-twist contamination becomes smaller. Ra The lower panel of Fig. 13 shows H̃i=0,1,2,3 (z = 3, Pz ) with different operators and Pz = 0.0, 0.46, 0.92 GeV. The O3 case also suffers from large higher-twist contamination like the z = 0 case; the results with Oi=0,1,2 seem to be slightly different from each other at Pz = 0.46 GeV, while the consistency at Pz = 0.92 GeV is much better. Since the operators O0,1,2 can provide consistent results but the uncertainty using O0 is slightly smaller than the other two 54 cases, we will concentrate on this case in the following discussion. Finally, the coordinate-space gluon quasi-PDF matrix element ratios H̃0Ra (z, Pz ) are plotted in Fig. 14, compared with the corresponding FT of the gluon PDF, H(z, µ=2 GeV), based on the global fits from CT14 [26] and PDF4LHC15 NNLO [27]. Since the uncertainties increases exponentially at larger z, our present lattice data with good signals are limited to the range zPz <2 or so, and the values at different zPz are consistent with each other. At the same time, H(z, 2 GeV) doesn’t changes much either in this region as in Fig. 14, as investigated in Ref. [208]. Up to perturbative matching and power correction at O(1/Pz2 ), they should be the same, and our simulation results are within the statistical uncertainty at large z. The results at the lighter pion mass (at the unitary point) of 340 MeV is also shown in Fig. 14, which is consistent with those from the strange quark mass case but with larger uncertainties. We also study the pion gluon quasi-PDF (see Fig. 15) and similar features are observed. In a recent work [257] involving part of the present authors, the glue momentum fraction hxiMS (corresponds to H̃ Ra (0) here) is calculated on configurations with different lattice spac- ing, valence and sea quark masses. The value of hxiMS tend to be slightly larger with smaller quark mass, but the dependence is weak. Thus it hints that the entire gluon distribution may be also insensitive to either the valence or sea quark mass given the current statistical errors, up to ∼ 400 MeV pion mass or so. The quark case is similar; thus we don’t expect the gluon quasi-PDF and the mixing with the quark PDF through the factorization to be very sensitive to the quark mass unless the statistical uncertainty can be reduced significantly. If H̃0Ra (z, Pz ) keeps flat outside the region where we have good signal, the gluon quasi-PDF g̃(x) will be a delta function at x = 0 through FT. On the other hand, the width of g̃(x) will be ∼ 0.5 in x if we suppose H̃0Ra (z, Pz ) = 0 for all the zPz >3. We conclude the FT of our present results of H̃0Ra (z, Pz ) cannot provide any meaningful constraint on the gluon PDF g(x). Summary and outlook: In summary, we present the first gluon quasi-PDF result for the nucleon and pion with multiple hadron boost momenta Pz and explore different choices of the operators. With proper renormalization, the quasi-PDF matrix elements we obtain agree with the FT of the global-fit PDF within statistical uncertainty, up to mixing from the quark PDF, perturbative matching and higher-twist correction O(1/Pz2 ). Since global fitting results shows that most of the contribution of g(x) comes from the 55 0.7 0.7 ▲ 0.6 0.6 ▲ ▲ ▲ ▲ 0.5◆▲ ◆ ◆▲ ◆ ◆ 0.5 ▲ ◆▲ ◆ ◆▲ ◆ Ho (z,Pz ) Ho (z,Pz ) 0.4 0.4 ∼ Ra 0.3 ∼ Ra 0.3 ● PDF4LHC15 NLO ◆ ● PDF4LHC15 NLO 0.2 ■ CT14 NNLO 0.2 ■ CT14 NNLO ◆ Pz =0.46 GeV 0.1 0.1 ◆ Pz =0.46 GeV ▲ Pz =0.92 GeV ▲ Pz =0.92 GeV 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 zPz zPz Figure 14 The final results of H̃0Ra (z, Pz ) at 678 MeV (top) and 340 MeV (bottom) pion mass as a functions of zPz , in comparison with the FT of the gluon PDF from the global fits CT14 [26] and PDF4LHC15 NNLO [27]. The data with Pz = 0.92 GeV are shifted horizontally to en- hance the legibility. They are consistent with each other within the uncertainty. 1.0 ● 0.8 0.6 Ho (z,Pz ) ● ■ ● ■ ■ ●■ ● ●■ ∼ Ra 0.4 ● Pz =0.46 GeV 0.2 ■ Pz =0.92 GeV 0.0 ■ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 zPz Figure 15 The similar figure for the pion gluon quasi-PDF matrix elements with Mπ = 678 MeV. The shape is quite similar to the case in Fig. 14. 56 x < 0.1 region, the width of its FT, H(zPz ), is pretty large as the H(zPz ) becomes half of of its maximum value (at zPz =0) at zPz ∼ 7. At the same time, the signal of the lattice simulation and also the validity of the factorization limit us to the small z region. Thus to discern the width of gluon PDF, the lattice simulation with much larger nucleon momentum Pz , such as 2-3 GeV, is needed. To archive a good signal with such a large Pz , the momentum smearing [107] and cluster decomposition error reduction [258] should be helpful. In the theoretical side, the gluon quasi-PDF operator can be renormalized non-perturbatively in the RI/MOM scheme (the O(F zµ , F µz ; z) and O(F µν , F νµ ; z) (µ, ν 6= z) structures in O0 and O2 should be renormalized separately before combined together, while O1 is multiplicative renormalizable [201, 249]) based on the NPR strategy introduced in Ref. [30], and the match- ing to the gluon PDF can be calculated perturbatively following the framework used in the quark case [199]. V.2. First Pseudo-PDF Study We later presented the first lattice-QCD results that access the x-dependence of the gluon un- polarized PDF of the nucleon via pseudo-PDF approach. This calculation is carried out using the Nf = 2 + 1 + 1 highly improved staggered quarks (HISQ) [71] lattices generated by the MILC collaboration [64] with spacetime dimensions L3 × T = 243 × 64, lattice spacing a = 0.1207(11) fm, and Mπsea ≈ 310 MeV. We apply 1 step of hypercubic (HYP) smearing [89] to reduce short-distance noise. The Wilson-clover fermions are used in the valence sector where the valence-quark masses is tuned to reproduce the lightest light and strange sea pseudoscalar meson masses (which correspond to pion masses 310 and 690 MeV, respectively), as done by PNDME collaboration [237, 238, 239, 240]. As demonstrated by PNDME and through our own calculation, we do not observe any exceptional configurations in our calculations caused by the mixed-action setup. Since our strange and light pion masses are tuned to match the cor- responding sea values, we do not anticipate lattice artifacts other than potential O(a) effects. Since this is at the same level as typical corrections to LaMET-type operators [259], it requires no special treatment. Such effects will be studied in future work. 57 We use Gaussian momentum smearing [107] is used for the quark field, 1 X Uj (x)eikêj Ψ(x + êj ) ,  Smom Ψ(x) = Ψ(x) + α (V.7) 1 + 6α j where k is the momentum-smearing parameter and α is the Gaussian smearing parameter. In our calculation, we choose k = 2.9, α = 3 with 60 iterations to help us getting a better signal at a higher boost nucleon momentum. These parameters are chosen after carefully scanning a wide parameter space to best overlap with our desired boost momenta. We use 898 lattices in total and calculate 32 sources per configuration for a total 28,735 measurements. In the previous gluon-PDF work [163], the nucleon two-point function was calculated with overlap fermions using all timeslices with a 2-2-2 Z 3 grid source and low-mode substitution [253, 254], which has 8 times more statistics and best signal at zero nucleon momentum. Even though the number of measurements in this work is smaller than the previous work, we see significant improvement in the signal-to-noise at large boost momenta with our momentum smearing, which allow us to extend our calculation to momenta as high as 2.16 GeV. We studied the (ap)n discretization effects on the nucleon two-point correlators using ensembles of different lattice spacing a ≈ 0.6, 0.9, 0.12 fm, and the results indicate that these effects are not significant on the two-point correlators. We anticipate the discretization effects to be small in our calculation, based on the observation in the two-point correlators; a study using multiple lattice spacings for the gluon three-point correlators will be needed for future precision calculations. The nucleons two-point correlators are then fitted to a two-state ansatz same as what we did in the pion gluon PDF Chapter IV. In this work, we use Ns to denote a nucleon composed of quarks such that Mπ ≈ 690 MeV and Nl to denote a nucleon composed of quarks such that Mπ ≈ 310 MeV. Figure 16 shows the effective-mass plots for the nucleon two-point functions with Pz = [0, 5] 2π L for both masses. The bands show the corresponding reconstructed fits using Eq. 1 with fit range [3, 13]. The bands are consistent with the data except where Pz and t are both large. The error of the effective masses at large Pz and t region is too large to fit. However, our reconstructed effective mass bands still match the the data points for the smaller t values even for the largest Pz = 5 × 2π/L. We check the dispersion-relation E 2 = E02 + c2 Pz2 of the nucleon energy as a function of the momentum, as shown in Fig. 17, and the speed of light c for the light quark is consistent with 1 within the statistical errors. 58 1.6 1.6 1.5 1.4 1.4 ENa12m310 ENa12m310 1.2 (P) (P) 1.3 1.2 1.0 l s 1.1 0.8 1.0 0.9 0.6 0 5 10 15 0 5 10 15 t t Figure 16 Nucleon effective-mass plots for Mπ ≈ 690 MeV (left) and Mπ ≈ 310 MeV (right) at z = 0, Pz = [0, 5] × 2π L on the a12m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point correlators. The momentum Pz = 5 2π L is the largest momentum we used, and it is the noisiest data set. 1.6 1.6 1.4 1.4 1.2 aEa12m310 aEa12m310 Nl Ns 1.2 1.0 1.0 c=0.9661(46) c=1.003(11) 0.8 0.8 0.6 0 5 10 15 20 25 0 5 10 15 20 25 Nz2 Nz2 Figure 17 Dispersion relations of the nucleon energy from the two-state fits for Mπ ≈ 690 MeV (left) and Mπ ≈ 310 MeV (right) 59 We use the unpolarized gluon operator defined in Eq. V.26. We find the bare matrix ele- ments to be consistent with up to 5 HYP-smearing steps, and the signal-to-noise ratios do not improve much with more steps. For the gluon operator used in this paper, we use 4 HYP smear- ing steps to reduce the statistical uncertainties, as studied in Ref. [163]. The matrix elements of gluon operators can be obtained by fitting the three-point function to its energy-eigenstate expansion same as we introduced in Chap. IV. Figure 18 shows example correlator plots from the ratio RN (Pz , t, tsep ) as a function of the t − tsep /2 for multiple source-sink separations for at Pz = 2 × 2π/L and tsep = {6, 7, 8, 9} × a. The reconstructed ratio plot, using the fitted parameters obtained from Eqs. (V.9) and (1) are plotted for each tsep , and the gray band indi- cates the reconstructed ground-state matrix elements h0|Og |0i. The left-two plots in Fig. 18 show the two-simRR fits and two-sim fits using the tsep = {6, 7, 8, 9}a, while the remain- ing two plots show individual two-state fits to the smallest and largest source-sink separations (tsep = {6, 9}a). The plots of pion mass Mπ ≈ 690 MeV and Mπ ≈ 310 MeV are shown in the first row and second row respectively. The reconstructed ground state matrix elements (gray bands) for Og are consistent for the fits with individual tsep = {6, 9}, the two-sim fit results and the two-simRR fit within one sigma error. Therefore, the two-sim fits describe data from tsep = {6, 7, 8, 9} well for operator Og . Thus, we use the two-sim fits to extract the ground-state matrix element h0|Og |0i of different z, Pz for the rest of this paper. Our extracted bare ground-state matrix elements are stable across various fit ranges. Fig- ure 19 shows example results from Mπ ≈ 690 MeV and Mπ ≈ 310 MeV nucleons with nucleon momentum Pz ∈ [1, 5] × 2π/L as the fit ranges for two- and three-point varies. In this case, the two-point correlator fit ranges are [tmin , 13] and the three-point correlators fit ranges are [tskip , tsep − tskip ]. All the matrix elements from different fit ranges are consistent with each other in one-sigma error. The fit range choice t3pt 2pt skip = 1, tmin = 2 are not used, because the χ2 /dof of the 2-point correlator fits with t2pt 2pt min = 2 are much larger than the tmin = 3 cases. For the rest of this paper, we use the fitted matrix elements obtained from the fit-range choice t3pt 2pt skip = 1, tmin = 3. The extracted bare matrix elements are fitted for Pz ∈ [0, 5] × 2π/L and z ∈ [0, 5] × a to obtain the Ioffe-time distributions in pseudo-PDF calculation. 60 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 tsep  6 tsep  6 0.2 tsep  7 0.2 tsep  7 0.2 0.2 tsep  6 tsep  9 tsep  8 tsep  8 0.0 tsep  9 0.0 tsep  9 0.0 0.0 -4 -2 0 2 4 -4 -2 0 2 4 -3 -2 -1 0 1 2 3 -4 -2 0 2 4 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 tsep  6 tsep  6 0.2 0.2 0.2 0.2 tsep  7 tsep  7 0.0 0.0 0.0 tsep  6 0.0 tsep  9 tsep  8 tsep  8 -0.2 tsep  9 -0.2 tsep  9 -0.2 -0.2 -4 -2 0 2 4 -4 -2 0 2 4 -3 -2 -1 0 1 2 3 -4 -2 0 2 4 Figure 18 The three-point ratio plots for Mπ ≈ 690 MeV (top row) and Mπ ≈ 310 MeV (bottom row)nucleons z = 1 as functions of t − tsep /2, as defined in Eq. IV.5. The results for nucleon momentum Pz = 2 × 2π/L are shown. The gray bands in each panel indicate the extracted ground-state matrix elements of the operator Og . In each column, the plots show the fitted ratio and the extracted ground-state matrix elements from two-simRR and two-sim fits with all 4 source-sink separations, and the two-state fits using only the smallest and largest tsep from left to right, respectively. The second column, which are the two-sim extracted ground- state matrix elements, are used in the subsequent analysis. The ground-state matrix elements extracted are stable and consistent among different fitting methods and three-point data input used. 1.0 1.2 2.0 3 pt 2 pt 3 pt 2 pt two-sim(tskip =2; tmin =3) two-sim(tskip =2; tmin =3) 3 pt 2 pt two-sim(tskip =2; tmin =3) 1.0 0.8 3 pt 2 pt 3 pt 2 pt two-sim(tskip =1; tmin =2) 3 pt 2 pt two-sim(tskip =1; tmin =2) 1.5 two-sim(tskip =1; tmin =2) ha12m310-bare ha12m310-bare ha12m310-bare 3 pt 2 pt (Pz = 0) (Pz = 2) (Pz = 4) 3 pt 2 pt two-sim(tskip =1; tmin =3) 0.8 two-sim(tskip =1; tmin =3) 3 pt 2 pt two-sim(tskip =1; tmin =3) 0.6 0.6 1.0 Ns Ns Ns 0.4 0.4 0.5 0.2 0.2 0.0 0.0 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 z z z 1.0 1.2 3.0 3 pt 2 pt 3 pt 2 pt 3 pt 2 pt two-sim(tskip =2; tmin =3) two-sim(tskip =2; tmin =3) two-sim(tskip =2; tmin =3) 0.8 1.0 2.5 3 pt 2 pt 3 pt 2 pt 3 pt 2 pt two-sim(tskip =1; tmin =2) two-sim(tskip =1; tmin =2) two-sim(tskip =1; tmin =2) ha12m310-bare ha12m310-bare ha12m310-bare 3 pt 2 pt 2.0 3 pt 2 pt (Pz = 0) (Pz = 2) (Pz = 4) 3 pt 2 pt 0.6 two-sim(tskip =1; tmin =3) 0.8 two-sim(tskip =1; tmin =3) two-sim(tskip =1; tmin =3) 1.5 0.6 0.4 Nl Nl Nl 1.0 0.4 0.2 0.5 0.2 0.0 0.0 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 z z z Figure 19 The fitted bare ground-state matrix elements without normalization by kinematic factors as functions of z obtained from the two-sim fit using different two- and three-point fit ranges for nucleon momentum Pz ∈ {0, 2, 4} × 2π/L from left to right, respectively, for Mπ ≈ 690 MeV (first row) and Mπ ≈ 310 MeV (second row) nucleons. The green points, which represent the fit-range choice t3pt 2pt skip = 1, tmin = 3 are used in the following analysis, because the errors of the matrix elements of this fit range are relatively smaller than the error of the red points. The orange points, which represent the fit-range choice t3pt 2pt skip = 1, tmin = 2, are not used because the χ2 /dof of the 2-point correlator fits with t2pt min = 2 are much larger than 2pt tmin = 3 cases. 61 V.2.1. Results and Discussions We fit the reduced ITDs for each jackknife sample at each Pz and z value. The slope K is about −0.05 GeV−2 in our fit. Then, the jackknife samples of the reduced ITDs at physical pion mass are reconstructed from the fit parameters from each jackknife sample fit. Figure 20 shows the extrapolation results for the reduced ITDs at Pz ∈ {1, 5} × 2π/L. 1.5 2.0 1.0 Mπ =690 MeV Mπ =310 MeV 1.5 0.5 Mπ →135 MeV M(ν,z2) M(ν,z2) 0.0 1.0 Mπ =690 MeV -0.5 Mπ =310 MeV -1.0 Mπ →135 MeV 0.5 -1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 1 2 3 4 5 6 ν ν Figure 20 The reduced ITDs M (ν, z 2 ) as functions of ν and their extrapolation to the physical pion mass at Pz = 1 × 2π/L (left) and Pz = 5 × 2π/L (right). The blue bands represent the fitted results of the reduced ITDs at the physical pion mass Mπ = 135 MeV. As shown in Fig. 21, the reduced ITDs of different z 2 from our lattice calculation show very little z dependence, because the z dependence cancels out when dividing out the ITD at P = 0 in the ratio defining the reduced ITD. Our fitted bands from the z-expansion fit match the reduced ITDs at different pion masses within the error bands. In Fig. 21, we can see that the fitted bands are mostly controlled by the small-z reduced ITDs, because the error grows significantly with increasing z. The reduced ITDs at physical pion mass are extrapolated from the pion masses at Mπ = 690 and 310 MeV and are closer to the smaller pion mass at Mπ = 310 MeV. As ν grows, the reduced ITDs decrease from M(0, z 2 ) = 1. The decrease becomes faster when we go to smaller pion masses, but this trend is slight because the pion- mass dependence is weak in our case, as seen in Fig. 21, where the data and the fitted bands from 3 different pion masses are consistent within one sigma error. The evolved ITDs at Mπ = 690, 310 and extrapolated 135 MeV are obtained from Eq. IV.1. In the evolution, we choose µ = 2 GeV and αs (2 GeV) = 0.304. The z dependence of the evolved ITDs should be compensated by the ln z 2 term in the evolution formula, which is confirmed in our evolution results. The evolved ITDs from different z ∈ [1, 5] × a are shown in Fig. 22 as points with different colors and are consistent with each other within one sigma error. Similar to the reduced ITDs, the evolved ITDs show small pion-mass dependence, because the 62 1.4 1.4 1.4 1.2 1.2 1.2 1.0 ■ ● ■ ■ ▼◆ ▲ ● ■ ■ 1.0 ■ ● ■ ■ ▼◆ ▲ ● ■ ■ 1.0 ■ ● ■ ■ ▼◆ ▲ ● ■ ■ ● ▲ ● ▼ ▲◆ ● ● ▲ ● ▼ ▲◆ ● ▲ ● ▼ ▲◆ M(ν,z2 ) M(ν,z2 ) M(ν,z2 ) ● ● 0.8 ▲ ▼ ▲ ◆ 0.8 ▲ ▼ ▲ ◆ 0.8 ▲ ▼ ▲ ◆ ▼ ▼ ▼ ■ z=1 ◆ ▼ ■ z=1 ◆ ▼ ■ z=1 ◆ ▼ 0.6 ◆ 0.6 ◆ 0.6 ◆ ● z=2 ● z=2 ● z=2 0.4 ▲ z=3 0.4 ▲ z=3 0.4 ▲ z=3 0.2 ▼ z=4 0.2 ▼ z=4 0.2 ▼ z=4 ◆ z=5 ◆ z=5 ◆ z=5 0.0 0.0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ν ν ν Figure 21 The reduced ITDs M (ν, z 2 ) as functions of ν at pion masses Mπ = 690, 310 and extrapolated 135 MeV from left to right, respectively. The points of different colors represent the reduced ITDs M (ν, z 2 ) of different z 2 and the red band represents the z-expansion fit band. data points from 3 different pion mass are consistent within one sigma error. According to the evolution function in Eq. IV.1, we can obtain the evolved ITD G by adding the reduced ITD M and an integral term related to M . Due to the cancellation between the two terms, this can reduce the error in the evolved ITDs. This phenomenon is also seen in other pseudo-PDF calculations [190, 12]. 1.4 1.4 1.4 1.2 1.2 1.2 ◆ ◆ ◆ ◆ ◆ ◆ ◆▼ ◆ ◆ 1.0● ◆ ▼ ▲ ■ ■ ● ▼ ▲ ● ■ ■ ◆ 1.0● ◆ ▼ ▲ ■ ■ ● ■ ■ ▼ ▲ ● ▼ ▲ ▼ 1.0● ◆ ▼ ▲ ■ ■ ● ■ ■ ▼ ▲ ● ▼ ▲ ■ ■ ● ▲ ▼ ▲ ● ● ◆ ◆ ▲ ■ ■ ● ● ▲ ■ ■ ● ● ▼ ▼ ▲ ▲ ▼ ▼ ◆ ▲ ● ▼ ▲ ● G(ν,μ) G(ν,μ) G(ν,μ) 0.8 0.8 ▲ 0.8 ▲ 0.6 ■ z=1 0.6 ■ z=1 ▼ 0.6 ■ z=1 ▼ ● z=2 ● z=2 ● z=2 0.4 ▲ z=3 MSULat, Mπ =690 MeV 0.4 ▲ z=3 MSULat, Mπ =310 MeV 0.4 ▲ z=3 MSULat, Mπ =135 MeV ◆ 0.2 ▼ z=4 CT18 NNLO 0.2 ▼ z=4 CT18 NNLO 0.2 ▼ z=4 CT18 NNLO ◆ ◆ z=5 NNPDF3.1 NNLO ◆ z=5 NNPDF3.1 NNLO ◆ z=5 NNPDF3.1 NNLO 0.0 0.0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ν ν ν Figure 22 The evolved ITDs G as functions of ν at pion masses Mπ ≈ 690, 310 and extrap- olated 135 MeV from left to right, respectively. The points of different colors represent the evolved ITDs G(ν, z 2 ) of different z values. The red band represents the fitted band of evolved ITD matched from the functional form PDF using the matching formula Eq. V.12. The yel- low and pink bands represent the evolved ITD matched from the CT18 NNLO and NNPDF3.1 NNLO unpolarized gluon PDF, respectively. The evolution and matching are both performed at µ = 2 GeV in the MS scheme. The fit is performed on the evolved ITDs for Mπ = 690, 310 and extrapolated 135 MeV separately. The fitted evolved ITD represented by the red band shows a decreasing trend as ν increases. The fit results for three pion masses are consistent with each other, as well as the evolved ITD from CT18 NNLO and NNPDF3.1 NNLO gluon unpolarized PDF, within one sigma error. However, the rate at which it decreases for smaller pion mass is slightly faster. The fit parameters and the goodness of the fit, χ2 /dof, are summarized in Table 2. From the functional form, it is obvious that parameter A constrains the small-x behaviour and parameter C constrains the large-x behaviour. However, the small-x results obtained from the lattice 63 calculation are not reliable. This is because the Fourier transform of the Ioffe time ν is related to the region around the inverse of the x and the large-ν results of evolved ITDs as shown in Fig. 22 have large error, which leads to poor constraint on the small-x behaviour of xg(x, µ). In contrast, the large-x behaviour of xg(x, µ) is constrained well because of the small error in the evolved ITDs in the small-ν region. Therefore, we have a plot that specifically shows the large-x region of x2 g(x, µ) in Fig. 23. Mπ (MeV) A C χ2 /dof 690 −0.622(14) 2.5(13) 0.35(45) 310 −0.611(8) 2.3(23) 0.19(36) 135 (extrapolated) −0.611(9) 2.2(24) 0.19(38) Table 2 Our gluon PDF fit parameters, A and C, from Eq. V.13, and goodness of the fit, χ2 /dof, for calculations at two valence pion masses and the extrapolated physical pion mass. A comparison of our unpolarized gluon PDF with CT18 NNLO and NNPDF3.1 NNLO at µ = 2 GeV in the MS scheme is shown in Fig. 23. We compare our xg(x, µ)/hxg iµ2 with the phenomenological curves in the left panel. The middle panel shows the same comparison for xg(x, µ). Our xg(x, µ) extrapolated to the physical pion mass Mπ = 135 MeV is close to the 310-MeV results and there is only mild pion-mass dependence compared with the 690- MeV results. We found that our gluon PDF is consistent with the one from CT18 NNLO and NNPDF3.1 NNLO within one sigma in the x > 0.3 region. However, in the small-x region (x < 0.3), there is a strong deviation between our lattice results and the global fits. This is likely due to the fact that the largest ν used in this calculation is less than 7, and the errors in large-ν data increase quickly as ν increases. To better see the large-x behavior, we multiply an additional x factor into the fitted xg(x, µ) and zoom into the range x ∈ [0.5, 1] in the rightmost plot of Fig. 23. Our large-x results are consistent with global fits over x ∈ [0.5, 1] though with larger errorbars, except for x ∈ [0.9, 1] where our error is smaller than NNPDF, likely due to using fewer parameters in the fit. With improved calculation and systematics in the future, lattice gluon PDFs can show promising results. To demonstrate the influence of the large-ν data on the fit results, we perform fits to the evolved ITDs with νmax of 3 and 4, comparing with the original fits with νmax = 6.54. The fits with the νmax cutoff are implemented on the lattice-calculated evolved ITDs and the evolved ITDs created by matching the CT18 NNLO gluon PDF. We show the evolved ITDs from the 64 6 2.0 CT18 NNLO CT18 NNLO 0.15 CT18 NNLO 5 NNPDF3.1 NNLO NNPDF3.1 NNLO NNPDF3.1 NNLO MSULat, Mπ =690 MeV 1.5 MSULat, Mπ =690 MeV xg(x,μ=2 GeV)/〈xg〉μ2 0.10 MSULat, Mπ =690 MeV 4 MSULat, Mπ =310 MeV xg(x,μ=2 GeV) MSULat, Mπ =310 MeV x2 g(x,μ=2 GeV) MSULat, Mπ =310 MeV MSULat, Mπ =135 MeV MSULat, Mπ =135 MeV 3 1.0 MSULat, Mπ =135 MeV 0.05 2 0.5 0.00 1 0.0 0 -0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 1.0 x x x Figure 23 The unpolarized gluon PDF, xg(x, µ)/hxg iµ2 (left), xg(x, µ) (middle), x2 g(x, µ) in the large-x region as a function of x (right), obtained from the fit to the lattice data at pion masses Mπ = 135 (extrapolated), 310 and 690 MeV compared with the CT18 NNLO (red band with dot-dashed line) and NNPDF3.1 NNLO (orange band with solid line) gluon PDFs. Our x > 0.3 PDF results are consistent with the CT18 NNLO and NNPDF3.1 NNLO unpolarized gluon PDFs at µ = 2 GeV in the MS scheme. Mπ = 310 MeV lattice data and the fitted bands on the left-hand side of Fig. 24. The errors of the fit bands become smaller as larger-νmax data are included even though the errors in the input points increases. As a result, we can see in the middle of Fig. 24 that the lattice gluon PDF errors shrink when the large-ν data help to constrain the fit. Since our ability to accurately determine the PDFs in the small-x region is limited by the νmax calculated on the lattice, we study the effect of the ν cutoff on our obtained x-dependent gluon PDF. To do so, we took the CT18 NNLO gluon PDF to construct a set of evolved ITDs using the same cutoffs νmax = {3, 4, 6.54} used on the 310-MeV PDF. The right-hand side of Fig. 24 shows that when νmax increases, the region the reconstructed PDF can recover extends to smaller x. Based on this observation, we estimate that with νmax = 6.54, the smallest x at which our lattice PDF can be trusted is around 0.25. We use the difference between the original CT18 input and the one reconstructed with a ν cutoff to estimate the systematic due to this cutoff effect on the higher moments. We summarize our predictions for the second and third moments hx2g iµ2 and hx3g iµ2 at µ = 2 GeV with their statistical and systematic errors in Table 3, together with the ones from CT18 NNLO and NNPDF3.1 NNLO results. The first error on our number corresponds to the statistical errors from the calculation, while the second error comes from combining in quadra- ture the systematic errors from four different sources: 1) The normalization of the global-PDF determination of the moment used in our calculation; 2) The finite-ν cutoff in the evolved ITDs, as discussed above. 3) The choice of strong coupling constant. To estimate this error, we vary αs by 10%. Like previous pseudo-PDF studies [191], we find that the changes are no more 65 2.0 2.0 1.4 CT18 NNLO MSULat, νmax =3 CT18 refit, νcut =3 1.2 ◆ ◆ ◆ 1.5 1.5 CT18 refit, νcut =5 ◆ MSULat, νmax =4 1.0● ◆ ▼ ▲ ■ ■ ● ■ ■ ▼ ▲ ● ▼ ▲ ▼ ▲ ■ ■ ● ● xg(x,μ=2 GeV) CT18 refit, νcut =6.54 xg(x,μ=2 GeV) ▲ ● ▼ MSULat, νmax =6.54 G(ν,μ) 0.8 ▲ 1.0 1.0 ■ z=1 ▼ 0.6 ● z=2 0.4 ▲ z=3 νmax =3 0.5 0.5 ◆ 0.2 ▼ z=4 νmax =4 ◆ z=5 νmax =6.54 0.0 0.0 0.0 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ν x x Figure 24 Left: The evolved ITDs G as functions of ν at Mπ ≈ 310 MeV with fits performed using different νmax cutoff in the evolved ITDs. As we can see from the tightening of the fit band, the evolved ITDs at larger ν are still useful in constraining the fit despite their larger errors. Middle: The unpolarized gluon PDF obtained from the fits to the evolved ITDs at 310-MeV pion mass with different νmax . The evolution and matching are both performed at µ = 2 GeV in the MS scheme. The larger the ν input, the more precise the PDF obtained. Right: The 2-GeV MS renormalized unpolarized gluon PDF obtained from a fit to the evolved ITDs generated from the CT18 NNLO PDF with νmax ∈ {3, 5, 6.54}, compared with the original CT18 NNLO unpolarized gluon PDFs. As ν increases, we can see the gluon PDF is better reproduced toward small x. Using this exercise, we can see that our lattice PDF is only reliable in the x > 0.25 region. By taking the moments obtained from CT18 with a cutoff of νmax = 6.54 compared to those from the original PDF, we can estimate the higher-moment systematics in our lattice calculation. than 5%; 4) The mixing with the quark singlet sector. We implement the gluon pseudo-PDF full matching kernel including the quark mixing term on CT18 NNLO unpolarized gluon PDF. The contribution of quark is about 4%, which is smaller than systematic errors from other sources. A more precise study of the effects of quark mixing on the unpolarized gluon PDF can be done when we have better control of statistical errors and other systematic errors. Overall, our moments are in agreement with the global-fit results. Future work including lighter pion masses and finer lattice-spacing ensembles will further help us reduce the systematics in the calculation. moment MSULat MSULat MSULat CT18 NNPDF3.1 (690 MeV) (310 MeV) (135 MeV) hx2g iµ2 0.040(15)(3) 0.043(26)(4) 0.045(30)(4) 0.0552(76) 0.048(13) hx3g iµ2 0.011(6)(2) 0.013(14)(3) 0.014(17)(3) 0.0154(37) 0.011(9) Table 3 Predictions for the higher gluon moments from this work at pion mass for about 690 MeV, 310 MeV, and the extrapolated 135 MeV. The moments predictions are compared with the corresponding ones obtained from CT18 NNLO and NNPDF3.1 NNLO global fits. The first error in our number corresponds to the statistical errors from the calculation and the second errors are the systematic errors. 66 V.2.2. Summary and Outlook In this paper, we present the first lattice calculation of the gluon parton distribution function us- ing the pseudo-PDF method. The current calculation is only done on one ensemble with lattice spacing of 0.12 fm and two valence-quark masses, corresponding to pion masses around 310 and 690 MeV. In contrast to the prior lattice gluon calculation [163], we now use an improved gluon operator that is proved to be multiplicatively renormalizable. The gluon nucleon matrix elements were obtained using two-state fits. The use of the improved sources in the nucleon two-point correlators allowed us to reach higher nucleon boost momentum. As a result, we were able to attempt to extract the gluon PDF as a function of Bjorken-x for the first time. There are systematics yet to be studied in this work. Future work is planned to study additional ensembles at different lattice spacings so that we can include the lattice-discretization system- atics. Lighter quark masses should be used to control the chiral extrapolation to obtain more reliable results at physical pion mass. V.3. Updated Pseudo-PDF Study We present the x-dependent nucleon distribution from lattice QCD using the pseudo-PDF ap- proach, on lattice ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration. We use clover fermions for the valence action and mo- mentum smearing to achieve pion boost momentum up to 2.56 GeV on three lattice spacings a ≈ 0.9, 0.12 and 0.15 fm and three pion masses Mπ ≈ 220, 310 and 690 MeV. We calcu- late the gluon momentum fraction hxig and combine with the xg(x)/hxig calculated from the pseudo-PDF approach to nucleon gluon unpolarized PDF xg(x) for the first time through lat- tice QCD simulation. We extract our results to physical pion mass and continuum limit, and compare with the determination by global fits. In Sec. V.3.1, we present the pseudo-PDF procedure to obtain the lightcone gluon PDF and how we extracted the reduced pseudo Ioffe-time distribution (pITDs) from lattice calculated correlators. In Sec. V.3.2.2, we present our calculation of the gluon nonperturbative renormal- ization factor and obtain the renormalized gluon momentum fraction hxig . In Sec. V.3.2.1, the final determination of the nucleon unpolarized gluon PDF xg(x) is obtained through the xg(x)/hxig and hxig calculation results, and compared with the phenomenology global fit PDF 67 results. A discussion of the systematics and the outlook for the nucleon gluon PDFs are in- cluded in the last Sec. V.3.3. V.3.1. Lattice correlators and matrix elements In this work, we follow the same procedure used to calculate the pion gluon PDF in Sec. II of Ref. [195, 212], following the pseudo-PDF procedure as in Refs. [186, 204]. The gluon operator we used is also the same one as in Eq. 1 in Ref. [212]. X 1 X O(z) ≡ O(F ti , F ti ; z) − O(F ij , F ij ; z), (V.8) i6=z,t 4 i,j6=z,t where the operator O(F µν , F αβ ; z) = Fνµ (z)U (z, 0)Fβα (0), z is the Wilson link length. To extract the ground-state matrix element to construct the reduced pITD defined in Eq. 4, we use a 2-state fit on the two-point correlators and a two-sim fit on the three-point correlators in Eqs. 11 and 12 in Ref. [212]. We present our calculation of the nucleon gluon PDFs on clover valence fermions on four ensembles with Nf = 2 + 1 + 1 highly improved staggered quarks (HISQ) [71] gen- erated by the MILC Collaboration [64] with three different lattice spacings (a ≈ 0.9, 0.12 and 0.15 fm) and three pion masses (220, 310, 690 MeV), as shown in Table. 4. Following the study in Ref. [163], five HYP-smearing [89] steps are used on the gluon loops to reduce the statistical uncertainties. We use Gaussian momentum smearing for the quark fields [107] 2π q(x)+α j Uj (x)ei( L )kêj q(x+êj ), to reach higher meson boost momenta with the momentum- P smearing parameter k listed in Table 7. The measurements vary 106 –107 for different ensem- bles. More measurements and various lattice spacings are studied comparing to our previous nucleon gluon PDF calculation on one a12m310 ensemble with 105 measurements [195]. To study the reliability of our fitted matrix-element extraction, we compare to ratios of the three-point to the two-point correlator RRatio , Ratio CΦ3pt (z, Pz ; tsep , t) RΦ (z, Pz ; tsep , t) = (V.9) CΦ2pt (Pz ; t) where the three-point and two-point correlators are defined in Eqs. 11 and 12 in Ref. [212]. The left-hand side of Fig. 25 shows example ratios for the gluon matrix elements from a12m220 and a09m310 ensembles light nucleon correlators at pion masses Mπ ≈ {220, 310} MeV at 68 ensemble a09m310 a12m220 a12m310 a15m310 a (fm) 0.0888(8) 0.1184(10) 0.1207(11) 0.1510(20) L3 × T 323 × 96 323 × 64 243 × 64 163 × 48 Mπval (MeV) 313.1(13) 226.6(3) 309.0(11) 319.1(31) Mηvals (MeV) 698.0(7) 696.9(2) 684.1(6) 687.3(13) Pz (GeV) [0,2.18] [0, 2.29] [0, 2.14] [0, 2.56] Nmeas 387,456 1,466,944 324,160 129,600 tsep {6,7,8,9} {6,7,8,9} {6,7,8,9} {5,6,7,8} Table 4 Lattice spacing a, valence pion mass Mπval and ηs mass Mηval s , lattice size L3 ×T , number 2pt of configurations Ncfg , number of total two-point correlator measurements Nmeas , and separa- tion times tsep used in the three-point correlator fits of Nf = 2 + 1 + 1 clover valence fermions on HISQ ensembles generated by the MILC collaboration and analyzed in this study. More details on the parameters used in the calculation are included in the Table 7 in the appendix. selected momenta Pz and Wilson-line length z. The ratios increase with increasing source-sink separation tsep and continuously to approach the ground-state matrix elements as going to larger tsep . The gray bands represent the ground-state matrix elements extracted using the two-sim fit to three-point correlators at five tsep , where the energies are from the two-state fits of the two- point correlators. The one-state fit results increase as tsep increases, which behaves similar to the ratios in the left-hand side plot. The third and fourth columns of Fig. 25 show two-sim fits max using tsep ∈ [tmin sep , 9] and tsep ∈ [5, tsep ] to study how the two-sim ground-state matrix elements depend on the source-sink separations input into fit. We observe that the matrix elements are consistent with each other within one standard deviation, and they begin to converge at large tmin sep and tmax sep , showing consistent extraction of the ground-state matrix element. Taking a09m310 ensemble as an example, we observe larger fluctuations in the matrix element extractions when max small tminsep = 3 and 4, or small tsep = 6 and 7, are used. The ground state matrix element extracted from two-sim fits becomes very stable when tmin max sep > 5 and tsep > 8. V.3.2. Results and Discussions V.3.2.1 xg(x)/ < x >g Results The Ioffe-time pseudo-distribution (pITD) [203, 186] is: M(ν, z 2 ) = h0(Pz )|O(z)|0(Pz )i, (V.10) 69 a12m220 one-state fit two-sim fit two-sim fit 1.0 RRatio (z  1, Pz= 2) 0.5 0.0 tsep=6 tsep=8 tsep=7 tsep=9 two-sim two-sim two-sim two-sim -0.5 4 5 6 7 -4 -2 0 2 4 4 5 6 7 8 9 10 7 8 9 10 11 t - tsep/2 tsep tmin sep tmax sep a09m310 one-state fit two-sim fit two-sim fit 0.8 RRatio (z  2, Pz= 1) 0.6 0.4 0.2 0.0 tsep=5 tsep=7 tsep=6 tsep=8 two-sim two-sim two-sim two-sim -0.2 -4 -2 0 2 4 3 4 5 6 7 8 9 3 4 5 6 6 7 8 9 10 t - tsep/2 tsep tmin sep tmax sep Figure 25 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from a12m220 and a09m310 ensembles light nucleon correlators at pion masses Mπ ≈ {220, 310} MeV. The gray band shown on all plots is the extracted ground-state ma- trix element from the two-sim fit we used as our final fit. From left to right, the columns are: the ratio of the three-point to two-point correlators with the reconstructed fit bands from the two-sim fit using the final tsep inputs, shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 10], the two-sim fit results using tsep ∈ [tmin max sep , tsep ] varying tmin max sep and tsep . The reduced pITD (RpITD) [186, 201, 249] was constructed to remove the ultraviolet diver- gences in the pITD by taking a double-ratio of the pITD, M(zPz , z 2 )/M(0 · Pz , 0) M (ν, z 2 ) = . (V.11) M(z · 0, z 2 )/M(0 · 0, 0) The renormalization of O(z) and kinematic factors are cancelled in the RpITDs. By construc- tion, the RpITD double ratios employed here are normalized to one at z = 0. 2.0 2.0 ▼ a12m220, Pz=1.31 GeV ● a12m310, Pz=1.28 GeV ▼ a12m220, Pz=1.96 GeV ● a12m310, Pz=2.14 GeV ■ a09m310, Pz=1.31 GeV ▲ a15m310, Pz=1.54 GeV ■ a09m310, Pz=1.96 GeV ▲ a15m310, Pz=2.05 GeV 1.5 1.5 M(ν,z2) M(ν,z2) 1.0●▲▼■ ■▼ ●▲■ ▼ ● ▲ ▼ ▼ ▼ 1.0●▲▼■ ■▼ ●▲ ▼ ▼ ▼ ▼ ● ● ■ ● ▲ ■ ■▲ ● ● ■ ■ ● ▲ ● ■ ■ 0.5 ▲ 0.5 ▲ ▲ ▲ 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z z Figure 26 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of z obtained from the fitted bare ground-state matrix elements for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. We can then extract the gluon PDF distribution through the pseudo-PDF matching condi- 70 tion [204] that connects the RpITD M to the lightcone gluon PDF g(x, µ2 ) through 1 xg(x, µ2 ) Z M (ν, z ) = 2 dx Rgg (xν, z 2 µ2 ), (V.12) 0 hxig R1 where µ is the renormalization scale in the MS scheme and hxig = 0 dx xg(x, µ2 ) is the gluon momentum fraction of the nucleon. Rgg is the gluon-in-gluon matching kernel we used in Ref. [195, 212, 214], which originally derived in Ref. [204]. We ignore the quark PDF contributes to the RpITDs in this calculation based on our findings in the past pion gluon PDF study [212]. One can obtain the gluon PDF g(x, µ2 ) by fitting the RpITD through the matching condition in Eq. V.12; a similar procedure has also been used by HadStruc Collaboration [191, 13, 213]. We examine the pion-mass and lattice-spacing dependence on the RpITDs extracted in the previous section. The left panel of Fig. 26 shows the RpITDs at boost momentum around 1.3 GeV as functions of the Wilson-line length z for the a12m220, a09m310, a12m310, and a15m310 ensembles. We see no noticeable lattice-spacing dependence. The bottom of Fig. 26 shows the pion RpITDs with boost momentum around 12 GeV for the same ensembles. Again, there is no visible lattice-spacing or pion-mass dependence. To obtain the gluon PDF g(x, µ2 ) on the right-hand side of Eq. V.12, we adopt the phe- nomenologically motivated form xg(x, µ) xA (1 − x)C fg (x, µ) = = , (V.13) hxig (µ) B(A + 1, C + 1) R1 for x ∈ [0, 1] and zero elsewhere. The beta function B(A + 1, C + 1) = 0 dx xA (1 − x)C is used to normalize the area to unity. Such a form is also used in global fits to obtain the nucleon gluon PDF by CT18 [2] and the pion gluon PDF by JAM [4, 24]. We fit the lattice RpITDs M lat (ν, z 2 , a, Mπ ) obtained in Eq. V.11 to the parametrization form M fit (ν, µ, z 2 , a, Mπ ) in Eq. V.12 by minimizing the χ2 function, χ2 (µ, a, Mπ ) = X (M fit (ν, µ, z 2 , a, Mπ ) − M lat (ν, z 2 , a, Mπ ))2 (V.14) 2 . ν,z σM (ν, z 2 , a, Mπ ) The reconstructed fit bands of the kaon RpITDs for the a12m220, a09m310, a12m310 and a15m310 ensembles, and nucleon RpITDs at each z 2 , compared with the lattice calculation 71 points, are shown in Fig. 27 from left to right. We see almost no z 2 -dependence (labeled in different colors) in the reconstructed bands in both a12m310 ensembles, but slightly more de- pendence in the a15m310 case. We found the a12m220, a12m310 fit to the nucleon gluon PDF to have very stable quality with χ2 /dof around 1 with consistent output of fg (x, µ), regard- less of the choice of the maximum value of the Wilson-line displacement z. However, for the a09m310, a15m310 ensemble, χ2 /dof can go as large as 4.2(1.3) and 6.0(2.0) respectively. We suspect that higher-twist effects are enhanced at this coarse lattice spacing such that the fit fails to accurately describe the lattice data. Possible future work including NNLO matching may help to improve the fit on this ensemble. 1.4 1.4 1.4 1.2 1.2 1.2 1.0 ■■ ● ■■ ▲● ■■ 1.0 ■■ ● ■■ ▲● ■■ 1.0 ■■ ● ■■ ▲● ■■ ▼◆●▲ ● ◆ ▼◆●▲ ● ◆ ▼◆●▲ ● ◆ M(ν,z2 ) M(ν,z2 ) M(ν,z2 ) ● ◆ ● ◆ ● ◆ 0.8 ▼▲ ●▲ 0.8 ▼▲ ●▲ 0.8 ▼▲ ●▲ ▼ ▲ ◆ ▼ ▲ ◆ ▼ ▲ ◆ ■ z=1 ▼ ◆ ■ z=1 ▼ ◆ ■ z=1 ▼ ◆ 0.6 ▲ 0.6 ▲ 0.6 ▲ ● z=2 ▼ ◆ ● z=2 ▼ ◆ ● z=2 ▼ ◆ 0.4 ▲ z=3 0.4 ▲ z=3 0.4 ▲ z=3 0.2 ▼ z=4 0.2 ▼ z=4 0.2 ▼ z=4 ◆ z=5 ▼ ◆ z=5 ▼ ◆ z=5 ▼ 0.0 0.0 0.0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 ν ν ν Figure 27 The RpITDs M with the reconstructed bands from fits in Eq. V.14 on the a09m310, a12m310, a15m310 lattice ensembles for nucleon respectively. We also attempt to obtain the nucleon gluon PDF at physical pion mass and continuum limit based on the RpITDs for a12m220, a09m310, a12m310 and a15m310 ensembles. Similar to the calculations we performed previous based on RpITDs, we use “z-expansion” fit [241, 242] (also adopted by past pseudo-PDF calculations [191]) and include the pion mass Mπ and lattice spacing a dependence terms into the fit as well, kmax X M (ν, z , a, Mπ ) = 2 λk τ k + l1 a2 + l2 (Mπ2 − (Mπphys )2 ) + l3 z 2 , (V.15) k=0 √ √ ν +ν− νcut where τ = √ cut √ . We vary νcut between [0.5,2] and the results are consistent with each νcut +ν+ νcut other and we choose νcut = 1 in our fits. We fix the λ0 = 1 because of the normalization we have for the reduced ITD M (ν, z 2 ) in Eq. V.11, and the maximum term kmax = 3 is used. The reconstructed fitted bands of a09m310 and a12m220 light nucleons are shown in the left plot of Fig. 28. The χ2 /dof of the fit is 0.72(64) with kmax = 3. We also analysis the a dependence in first order linear term, which are compared in the right plot of Fig. 28, where the a-term RpITD band is smaller than a2 -term RpITD band. The a2 -term RpITD at physical 72 pion mass and continuum limit can be also used in the parametrization form fit in Eq. V.14. A fit range of ν ∈ [0, 10] and z ≈ 0.1 fm are used in the a2 -term RpITD fit. The extracted RpITD can be used in Eq. V.14 to obtain the gluon PDF directly. We choose to discretize the continuum fitted a2 -term RpITD by using z = 0.1 fm and aL = 0.2 × π fm to implement the same fit procedure as we did for the other RpITDs from different ensembles. The fit results are shown in Fig. 29, compared with the with the RpITDs matched from global-fit gluon PDFs from CT18 and NNPDF3.1 at NNLO. The RpITD bands are consistent with each other within the one-sigma error range. The reconstructed gluon PDF, xg(x, µ)/hxig obtained from the fits to the different lattice ensembles data for the light nucleon, and the fit to the extrapolated data at physical pion mass and continuum limit are shown in Fig. 34, where the lattice spacing and pion mass dependence are shown to be weak. 3.0 3.0 ■ a=0.09 fm, Mπ ∼310 MeV ■ a=0.09 fm, Mπ ∼310 ▼ MeV ■ a=0.09 fm, Mπ ∼310 MeV ■ a=0.09 fm, Mπ ∼310 MeV ▼ 2.5 2.5 ● a=0.12 fm, Mπ ∼220 MeV ▲ a=0.12 fm, Mπ ∼690 MeV ● a=0.12 fm, Mπ ∼220 MeV ▲ a=0.12 fm, Mπ ∼690 MeV 2.0 ▲ a=0.12 fm, Mπ ∼310 MeV ● a=0.12 fm, Mπ ∼700 MeV 2.0 ▲ a=0.12 fm, Mπ ∼310 MeV ● a=0.12 fm, Mπ ∼700 MeV ▼ a=0.15 fm, Mπ ∼310 MeV ▼ a=0.15 fm, Mπ ∼690 MeV ▼ a=0.15 fm, Mπ ∼310 MeV ▼ a=0.15 fm, Mπ ∼690 MeV 1.5 M(ν,z2) ▼ 1.5 M(ν,z2) ▼ ▼ ▼ ▼ ▼ ● ●●▼ ●●● ● ● ▼ ▼ ●▼● ▼ ●●● ● ▼ ▼ 1.0●▼▲■■■■■■ ■■■ ▲▼ ●■● ● ▲■● ▼ ■ ● ■●▼▲■● ▲ ▼ ●▼ ● ● ● ● ● ▼ ● ▼ 1.0●▼▲■ ■●▲▼■●▲■■ ● ▲■ ● ● ▲ ● ▲ ■■ ■● ▲▼ ■▲ ▼ ■ ▲ ■ ■●▼ ▼● ■▲●▼ ■▲ ▼ ■ ●■ ■● ▲ ■▼ ▲ ● ● ▼ ■ ■■ ▲ ■● ■ ■● ▲▼ ■▲ ● ▼ ■ ■●▼ ▼● ■▲●▼ ▼ ●▼● ● ●■ ■ ● ● ● ▼ ▼ ● ▼ ■■■ ■▲ ▼ ▲ ▲ ▼ ■ ▲ ● ▲● ● ▼ ■ ▼ ▼ ▼ ● ● ●● ● ▲ ■ ▼ ■■■ ■▲ ▼ ■ ■▲ ▼ ▲ ▲ ■▼ ▲● ▲ ▼ ● ■ ▼ ■ ▲ ▼ ● ▼ ● ●● ● ● ▼ ■ ■▲ ▼ ■ ■ ■ ▲ ■ ■▲ ▲ ▼■ ▲ ▲ ▲ ▲ ● ● ▼ ■■ ■▲ ▼ ■ ▲ ■▲ ■ ▲ ▼■ ▲ ●● ● ■ ▲ ▼ ● ■ ■ ▲ ▼ ▲ ▲ ● ● ■ ● ■ ■ ▲ ▲ ▼ ● 0.5 ▼ ▼ ■ ■ ▼▲ 0.5 ▼ ▼ ■ ▼ ▲ ■ ■ ▼ ■ ■ ■ ▼ ▼ ▼ ▼ ▼ ▼ 0.0 0.0 ▼ a a2 gg+gq -0.5 -0.5 0 2 4 6 8 10 0 2 4 6 8 10 ν ν Figure 28 The preliminary RpITDs M with the example reconstructed bands from fits in Eq. V.15 on the a09m310 (red), a12m220 (green) lattice ensembles in the left-hand plot and the extrapolated bands at physical pion mass and continuum limit in the right one. The lat- tice spacing a dependence and gluon-in-quark contribution are studied and compared in the right-hand plot. We investigate the systematic uncertainty comes from neglecting the contribution of the quark term based on the assumption (motivated by global fits) that the pion qS (x) is smaller than the gluon PDF. Currently, there are no qS (x) results from lattice simulation since only the isovector distribution of the nucleon has been done. Thus, we estimate the systematic due to omitting the qS (x) contribution by using the nucleon quark PDFs from CT18 at NNLO [2]. Using these, we obtain revised RpITDs including the gluon-in-quark Rgq term as shown in the right-hand side of Fig. 28, we show the mean value of xg(x)/hxig with both gluon-in-gluon (gg) and gluon-in-quark (gq) contributions (the black solid line) compared to the results using 73 the gluon-in-gluon contribution only (the black dashed line) in Fig. 34. There are smaller than 5% differences in the mean value including the gluon-in-gluon contribution for x < 0.9, which indicates that the gluon-in-quark contribution is relatively small at µ2 = 4 GeV2 compared to the current statistical errors in the small-x region. In the x > 0.9 region, the gluon-in-quark contribution becomes more significant, but it remains smaller than the statistical error. 1.4 1.2 1.0■ ■ ■ ■ M(ν,z2 ) ■ 0.8 ■ ■ ■ ■ 0.6 MSULat22 CT18 NNLO 0.4 NNPDF3.1 NNLO 0.2 gg+gq 0.0 0 2 4 6 8 ν Figure 29 The preliminary RpITDs M with reconstructed bands from fits at phyiscal pion mass and continuum limit, comparing with the RpITDs matched from global-fit PDFs, and the RpITD mean value reconstructed included the gluon-in-quark contribution. 1.0 a12m220 0.14 a12m220 0.8 a09m310 a09m310 0.12 a12m310 a12m310 a15m310 0.10 0.6 a15m310 xg(x)/〈x〉 xg(x)/〈x〉 MSULat22 0.08 MSULat22 0.4 gg+gq gg+gq 0.06 0.04 0.2 0.02 0.0 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 1.0 x x Figure 30 The preliminary unpolarized gluon PDF, xg(x, µ)/hxig in the large-x region as a function of x and its zoomed in plot, obtained from the fits to the different lattice ensembles data compared with the fit to the extrapolated data at physical pion mass and continuum limit, and the mean value fit including the gluon-in-quark term in the matching. V.3.2.2 Renormalized gluon moments The gluon momentum faction hxig is important in understanding the nucleon momentum, mass and spin [260, 261]. Thus, a lattice-QCD calculation of the hxig itself is also of fundamen- tal interest. Available calculations of hxig have significantly improved recently [262, 234]. For the bare gluon matrix element, it was found that hypercubic (HYP) smearing of the gluon 74 operators changes the bare matrix element significantly [263, 29]. Therefore, a nonperturba- tive renormalization (NPR) [118] of the the gluon momentum fraction is needed. The gluon momentum fraction operator we use is X Og,µν (x) = (F µα (x)F µα (x) − F να (x)F να (x)), (V.16) α=0,1,2,3 where the field tensor i Fµν = (P[µ,ν] + P[ν,−µ] + P[−µ,−ν] + P[−ν,µ] ), (V.17) 8a2 g where the plaquette Pµ,ν = Uµ (x)Uν (x + aµ̂)Uµ† (x + aν̂)Uν† (x) and P[µ,ν] = Pµ,ν − Pν,µ . After we obtain the gluon bare matrix element from lattice calculation, we renormalize using a nonperturbative RI-MOM scheme and then implement a perturbative matching to get the renormalized operators in the MS scheme: Og = RMS (µ2 , µ2R )ZOg (µ2R )Ogbare . (V.18) The matching factor RMS (µ2 , µ2R ) is calculated via perturbation theory in Ref. [125]. The RI- MOM renormalization factor ZOg (µ2R ) can be obtained with the nonperturbative renormaliza- tion condition, −1 Zg (p2 )ZOg (p2 )Λbare tree Og (p)(ΛOg (p)) |p2 =µ2R = 1, (V.19) where Zg (p2 ) is the gluon-field renormalization and ΛOg (p) is the amputated Green function for the operator Og in the Landau gauge-fixed gluon state. The NPR factor ZOg (p2 ) of the operator in Eq. V.16 is derived in Ref. [234, 30], p2 hOg,µν Tr[Aτ (p)Aτ (−p)]i ZO−1g (µ2R ) = . (V.20) 2(p2µ − p2ν )hTr[Aτ (p)Aτ (−p)]i p2 =µ2R ,τ 6=µ6=ν,pτ =0 Therefore, the gluon propagator Dg (p) and gluon amputated Green’s function ΛOg (p) need to be calculated for the further calculation of the NPR factor, Dµν (p) = hTr[Aτ (p)Aτ (−p)]i ΛOg (p) = hOg,µν Tr[Aτ (p)Aτ (−p)]i. (V.21) In Ref. [258, 30], a technique called cluster-decomposition error reduction (CDER) is used 75 to increase the signal to error ratio of NPR factor. The reason for such error reduction is that, for the disconnected insertions, the vacuum insertion dominates the variance, so that the relevant operators fluctuate independently and are independent of the time separation. This explains why the signal falls off exponentially, while the error remains constant in the disconnected in- sertions. The gluon operator inserted into the propagator in Eq. V.20 is a disconnected insertion on which the CDER technique can be used. Reference [30] introduces two cutoffs, r1 between the glue operator and one of the gauge fields, and r2 between the gauge fields in the gluon propagator to gluon amputated Green function ΛOg (p), DZ Z Z E 4 4 0 4 ip·r0 0 ΛOg ≡ dr dr d xe Og,µν (x + r) Tr[Aρ (x)Aρ (x + r )] . (V.22) |r| 0.2, and with xFitter’20 for x > 0.5 within uncertainty. We also studied the asymptotic behavior of the pion gluon PDF in the large-x region in terms of (1 − x)C . C > 3 is implied from our study at two lattice spacings and three pion masses. In Chap. V, we presented an exploratory study of the nucleon gluon PDF using the quasi- PDF approach, two studies of the nucleon gluon PDF using pseudo-PDF approach on one ensemble based on 2-step fit on EpITDs and on four ensembles based on 1-step fit on RpITDs, respectively. In the exploratory study using quasi-PDF, the renormalized quasi-PDF matrix el- ements are compared with the FT of the global-fit PDF and they are consistent within statistical 83 uncertainty. In the study using pseudo-PDF approach via 2-step fit, the xg(x)/hxig extrapo- lated to the physical pion mass Mπ = 135 MeV is obtained, which is consistent with the one from CT18 NNLO and NNPDF3.1 NNLO within one sigma in the x > 0.3 region. In the study using pseudo-PDF approach via 1-step fit, we calculate gluon momentum fraction langlexig , i.e. the first moment of gluon PDF under proper renomalization. The xg(x) is then calculated on four ensembles with three lattice spacings and three pion masses. 84 APPENDIX 85 APPENDIX In this section, we present the parameters that were not listed in Tab. 4. The plots for the 4 ensembles are shown as following order, the 2-point energy fit, effective mass, dispersion relation plots, the matrix element ratio plots, bare matrix element as function of z plots, RpITD as function of z/ν plots, and zmax fits RpITDs and xg(x) plots. ensemble a09m310 a12m220 a12m310 a15m310 a (fm) 0.0888(8) 0.1184(10) 0.1207(11) 0.1510(20) L3 × T 323 × 96 323 × 64 243 × 64 163 × 48 Mπval (MeV) 313.1(13) 226.6(3) 309.0(11) 319.1(31) Mηval s (MeV) 698.0(7) 696.9(2) 684.1(6) 687.3(13) Pz (GeV) [0,2.18] [0, 2.29] [0, 2.14] [0, 2.56] Nmeas 387,456 1,466,944 324,160 129,600 {α, Ninteration } {3,60} {3,60} {3,60} {3,60} k 3.9 3.5 2.9 2.3 ml -0.075 -0.05138 -0.0695 -0.0893 ms -0.019938 -0.017 -0.0194 -0.021 tsep {6,7,8,9} {5,6,7,8} {5,6,7,8} {6,7,8,9} Table 7 Lattice spacing a, valence pion mass Mπval and ηs mass Mηval s , lattice size L3 × T , 2pt number of configurations Ncfg , number of total two-point correlator measurements Nmeas , the Gaussian smearing parameters {α, Ninteration }, the momentum smearing parameters k in 2π q(x) + α j Uj (x)ei( L )kêj q(x + êj ), mass parameters ml and ms for light and strange quarks P respectively, and separation times tsep used in the three-point correlator fits of Nf = 2 + 1 + 1 clover valence fermions on HISQ ensembles generated by the MILC collaboration and ana- lyzed in this study. .0.1. Two-point Energy Fit The nucleons two-point correlators are then fitted to a two-state ansatz CN2pt (Pz , t) = |AN,0 |2 e−EN,0 t + |AN,1 |2 e−EN,1 t + ..., (1) where the |AN,i |2 and EN,i are the ground-state (i = 0) and first excited state (i = 1) amplitude and energy, respectively. In this work, we use Ns to denote a nucleon composed of quarks such that Mπ ≈ 690 MeV and Nl to denote a nucleon composed of quarks such that Mπ ≈ 310 MeV. We perform an analysis of two exponential fits on 2-point correlators to obtain more reliable results for the excited state energies. We used E0 as a prior to performed more stable two-exponential fits. The E0 results as function of the fit range [tmin , 11] from the two-state 86 exponential fits at Pz ∈ [0, 5] × 2π/L for a09m310, a12m310, and a15m310 ensembles, Pz ∈ [0, 7] × 2π/L for a12m220 ensemble are shown in Figs. 35, 36, 37, 38, 39, 40, 41. Taking a12m310 light nucleon at pion masses Mπ ≈ 310 MeV as an example, the E0 results reach a plateau at tmin , therefore, tmin = 4 is used in the final 2-state fits for a12m310 light nucleon 2-point correlators. .0.1.1 Strange Nucleon 0.905 p=0 0.925 p=1 0.985 p=2 Ea12m220 Ea12m220 Ea12m220 0.900 0.920 0.980 0,Ns 0,Ns 0,Ns 0.895 0.915 0.975 0.890 0.910 0.970 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 1.080 1.320 p=3 1.190 p=4 p=5 1.075 1.315 1.185 Ea12m220 Ea12m220 Ea12m220 1.310 1.070 1.180 0,Ns 0,Ns 0,Ns 1.305 1.065 1.300 1.175 1.060 1.170 1.295 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 5 1.455 1.59 p=0 1.450 p=6 p=7 4 1.58 p=1 1.445 Ea12m220 p=2 Ea12m220 3 a12m220 1.440 1.57 p=3 0,Ns 0,Ns Ns 1.435 dof 2 p=4 1.56 χ2 1.430 p=5 1 1.425 1.55 1.420 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 1 2 3 4 5 6 7 2 pt tmin tmin 2 pt tmin Figure 35 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2- point correlator fit range [tmin , 11] for the a12m220 ensemble strange nucleon at pion masses Mπ ≈ 700 MeV, at the momentum Pz ∈ [0, 7] × 2π/L. tmin = 4 is used in the final 2-state fits for a12m220 strange nucleon 2-point correlators. 87 0.920 0.955 1.205 p=0 p=1 p=3 0.915 0.950 1.200 Ea12m310 Ea12m310 Ea12m310 1.195 0,Ns 0,Ns 0,Ns 0.910 0.945 1.190 0.905 1.185 0.940 1.180 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 1.39 1.205 p=4 1.58 p=5 p=3 1.38 1.200 1.56 Ea12m310 Ea12m310 Ea12m310 1.37 1.195 1.54 0,Ns 0,Ns 1.36 0,Ns 1.190 1.52 1.35 1.185 1.50 1.180 1.34 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 5 p=0 4 p=1 p=2 3 a12m310 p=3 Ns 2 p=4 χ2 dof p=5 1 0 0 1 2 3 4 5 6 7 2 pt tmin Figure 36 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2- point correlator fit range [tmin , 11] for the a12m310 ensemble strange nucleon at pion masses Mπ ≈ 690 MeV, at the momentum Pz ∈ [0, 5] × 2π/L. tmin = 4 is used in the final 2-state fits for a12m310 strange nucleon 2-point correlators. 88 1.195 1.590 1.135 p=0 p=1 p=3 1.585 1.190 Ea15m310 Ea15m310 Ea15m310 1.130 1.580 0,Ns 1.125 0,Ns 1.185 0,Ns 1.575 1.120 1.180 1.570 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 2 pt 2 pt 2 pt tmin tmin tmin 1.590 1.835 2.06 p=5 p=3 1.830 p=4 1.585 1.825 2.04 Ea15m310 Ea15m310 Ea15m310 1.580 1.820 0,Ns 2.02 0,Ns 1.575 0,Ns 1.815 1.810 2.00 1.570 1.805 1.98 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 2 pt 2 pt 2 pt tmin tmin tmin 5 p=0 4 p=1 p=2 3 a15m310 p=3 Ns 2 p=4 χ2 dof p=5 1 0 0 1 2 3 4 2 pt tmin Figure 37 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2- point correlator fit range [tmin , 10] for the a15m310 ensemble strange nucleon at pion masses Mπ ≈ 690 MeV, at the momentum Pz ∈ [0, 5] × 2π/L. tmin = 1 is used in the final 2-state fits for a15m310 strange nucleon 2-point correlators. 89 .0.1.2 Light Nucleon 0.64 0.670 0.755 p=0 0.665 p=1 0.750 p=2 0.63 0.660 0.745 Ea12m220 Ea12m220 Ea12m220 0.655 0.740 0,Nl 0.62 0,Nl 0.650 0,Nl 0.735 0.645 0.730 0.61 0.725 0.640 0.720 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 1.20 0.88 p=3 1.02 p=4 1.18 p=5 0.87 1.16 Ea12m220 Ea12m220 Ea12m220 1.00 0.86 1.14 0,Nl 0,Nl 0,Nl 0.85 0.98 1.12 0.84 0.96 1.10 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 5 1.36 1.55 p=0 p=6 1.50 p=7 4 1.34 p=1 1.45 Ea12m220 Ea12m220 3 p=2 1.32 1.40 a12m220 p=3 0,Nl 0,Nl Nl 2 p=4 1.35 χ2 dof 1.30 p=5 1.30 1 1.28 1.25 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 2 pt 2 pt tmin tmin 2 pt tmin Figure 38 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 11] for the a12m220 ensemble light nucleon at pion masses Mπ ≈ 220 MeV, at the momentum Pz ∈ [0, 7]×2π/L. tmin = 4 is used in the final 2-state fits for a12m220 light nucleon 2-point correlators. 90 0.560 0.665 0.520 p=0 p=1 p=2 0.555 0.660 Ea09m310 Ea09m310 Ea09m310 0.515 0.655 0,Nl 0,Nl 0.550 0,Nl 0.510 0.650 0.505 0.545 0.645 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 1.15 0.810 0.97 p=4 1.14 p=5 0.805 p=3 0.96 1.13 Ea09m310 Ea09m310 Ea09m310 0.800 1.12 0,Nl 0,Nl 0.795 0,Nl 0.95 1.11 0.790 0.94 1.10 0.785 0.93 1.09 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 5 p=0 4 p=1 3 p=2 a09m310 p=3 Nl 2 p=4 χ2 dof p=5 1 0 1 2 3 4 5 6 7 2 pt tmin Figure 39 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 13] for the a09m310 ensemble light nucleon at pion masses Mπ ≈ 310 MeV, at the momentum Pz ∈ [0, 5]×2π/L. tmin = 4 is used in the final 2-state fits for a09m310 light nucleon 2-point correlators. 91 0.88 0.740 0.690 p=0 p=1 p=2 0.87 0.685 0.735 Ea12m310 Ea12m310 Ea12m310 0.680 0.730 0.86 0,Nl 0,Nl 0,Nl 0.675 0.725 0.85 0.670 0.720 0.665 0.84 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 1.28 1.6 p=5 1.06 p=3 p=4 1.26 1.5 Ea12m310 Ea12m310 Ea12m310 1.05 1.24 1.4 0,Nl 0,Nl 0,Nl 1.04 1.22 1.3 1.03 1.20 1.2 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 pt 2 pt 2 pt tmin tmin tmin 5 p=0 4 p=1 p=2 3 a12m310 p=3 Nl 2 p=4 χ2 dof p=5 1 0 0 1 2 3 4 5 6 7 2 pt tmin Figure 40 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 11] for the a12m310 ensemble light nucleon at pion masses Mπ ≈ 310 MeV, at the momentum Pz ∈ [0, 5]×2π/L. tmin = 4 is used in the final 2-state fits for a12m310 light nucleon 2-point correlators. 92 1.0 p=0 1.00 1.20 p=1 p=2 0.9 0.95 Ea15m310 Ea15m310 Ea15m310 1.15 0.8 0.90 0,Nl 0,Nl 0,Nl 0.7 1.10 0.85 0.6 0.80 1.05 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 2 pt 2 pt 2 pt tmin tmin tmin 1.46 1.44 p=3 p=4 2.0 p=5 1.70 1.42 Ea15m310 Ea15m310 Ea15m310 1.8 1.40 1.65 0,Nl 0,Nl 0,Nl 1.38 1.6 1.36 1.60 1.4 1.34 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 2 pt 2 pt 2 pt tmin tmin tmin 5 p=0 4 p=1 p=2 3 a15m310 p=3 Nl 2 p=4 χ2 dof p=5 1 0 0 1 2 3 4 2 pt tmin Figure 41 The fitted ground state energy and the χ2 /dof of 2-state fit as function of the 2-point correlator fit range [tmin , 10] for the a15m310 ensemble light nucleon at pion masses Mπ ≈ 310 MeV, at the momentum Pz ∈ [0, 5]×2π/L. tmin = 1 is used in the final 2-state fits for a15m310 light nucleon 2-point correlators. 93 .0.2. Effective mass plot and fits Figures 42, 43, 44, 45, 46, 47 and 48 shows the effective-mass plots for the nucleon two- point functions with at Pz ∈ [0, 5] × 2π/L for a09m310, a12m310, and a15m310 ensembles, Pz ∈ [0, 7]×2π/L for a12m220 ensemble. The bands show the corresponding reconstructed fits using Eq. 1 with fit range [4, 13] for a09m310 ensemble, [4, 11] for a12m310 and a12m220 en- sembles, [1, 10] for a15m310 ensemble. The bands are consistent with the data except where Pz and t are both large. The error of the effective masses at large Pz and t region is too large to fit. However, our reconstructed effective mass bands still match the the data points for the smaller t values even for the largest Pz = 5 × 2π/L. We check the dispersion-relation E 2 = E02 + c2 Pz2 of the nucleon energy as a function of the momentum, as shown in Fig. 49, 50. The speed of light c for the light quark is consistent with 1 within two sigma errors for a09m310, a12m220, a12m310 ensembles, however, deviated from 1 for the a15m310 ensemble light quark and all ensembles strange quark. .0.2.1 Strange Nucleon 94 0.95 p=1 1.03 0.97 p=2 0.94 0.96 1.02 Ea12m220 (P=0) Ea12m220 (P=1) Ea12m220 (P=2) 0.93 0.95 1.01 Ns Ns Ns 0.92 0.94 1.00 0.91 0.93 0.99 0.90 0.92 0.98 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t t 1.24 1.36 1.12 1.23 1.35 1.11 p=3 Ea12m220 Ea12m220 Ea12m220 1.22 1.34 (P=3) (P=4) (P=5) p=4 1.10 1.21 p=5 Ns Ns Ns 1.33 1.09 1.20 1.32 1.08 1.19 1.07 1.31 1.18 1.30 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t t 1.49 1.62 1.48 1.61 Ea12m220 Ea12m220 1.47 (P=6) (P=7) 1.60 Ns Ns 1.46 1.59 p=6 1.45 1.58 p=7 1.44 1.57 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t Figure 42 Nucleon effective-mass plots for Mπ ≈ 700 MeV, at Pz = [0, 7]× 2π L on the a12m220 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point cor- relators. The momentum Pz = 7 2πL is the largest momentum we used, and it is the noisiest data set. 1.2 1.2 p=1 1.3 1.1 p=2 1.1 1.2 Ea12m310 (P=0) Ea12m310 (P=1) Ea12m310 (P=2) 1.0 1.0 1.1 Ns Ns Ns 0.9 0.9 1.0 0.8 0.8 0.9 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t t 1.8 1.6 1.4 1.7 p=3 1.5 Ea12m310 Ea12m310 Ea12m310 1.3 p=4 (P=3) (P=4) (P=5) 1.6 1.4 p=5 Ns Ns Ns 1.2 1.5 1.3 1.1 1.4 1.2 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t t Figure 43 Nucleon effective-mass plots for Mπ ≈ 690 MeV, at Pz = [0, 5]× 2π L on the a12m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point cor- relators. The momentum Pz = 5 2πL is the largest momentum we used, and it is the noisiest data set. 95 1.6 1.50 p=1 1.45 1.55 1.5 p=2 Ea15m310 Ea15m310 Ea15m310 1.40 1.50 (P=0) (P=1) (P=2) 1.4 1.35 1.45 Ns Ns Ns 1.3 1.30 1.25 1.40 1.2 1.20 1.35 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 t t t 1.75 2.20 1.95 2.15 p=3 Ea15m310 Ea15m310 Ea15m310 1.70 1.90 p=4 2.10 (P=3) (P=4) (P=5) p=5 1.65 2.05 Ns Ns Ns 1.85 2.00 1.60 1.80 1.95 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 t t t Figure 44 Nucleon effective-mass plots for Mπ ≈ 690 MeV, at Pz = [0, 5]× 2π L on the a15m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point cor- relators. The momentum Pz = 5 2πL is the largest momentum we used, and it is the noisiest data set. 96 .0.2.2 Light Nucleon 0.70 0.82 p=1 0.72 0.68 0.80 p=2 Ea12m220 Ea12m220 Ea12m220 0.70 (P=0) (P=1) (P=2) 0.66 0.78 Nl Nl Nl 0.68 0.64 0.76 0.66 0.62 0.74 0.64 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t t 0.94 1.08 1.22 0.92 1.06 p=3 Ea12m220 Ea12m220 Ea12m220 1.20 (P=3) (P=4) (P=5) 0.90 1.04 p=4 p=5 Nl Nl Nl 1.02 1.18 0.88 1.00 1.16 0.86 0.98 1.14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t t 1.59 1.35 1.58 1.30 Ea12m220 Ea12m220 1.57 (P=6) (P=7) 1.25 Nl Nl 1.56 1.20 p=6 1.55 p=7 1.15 1.54 1.10 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t Figure 45 Nucleon effective-mass plots for Mπ ≈ 220 MeV, at Pz = [0, 7]× 2π L on the a12m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point cor- relators. The momentum Pz = 7 2πL is the largest momentum we used, and it is the noisiest data set. 97 0.80 0.64 0.68 p=1 0.78 0.62 p=2 0.66 0.76 Ea09m310 Ea09m310 Ea09m310 0.60 0.64 0.74 (P=0) (P=1) (P=2) 0.58 0.62 0.72 Nl 0.56 Nl 0.60 Nl 0.70 0.54 0.58 0.68 0.52 0.56 0.66 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 t t t 1.20 0.90 1.05 p=3 Ea09m310 Ea09m310 Ea09m310 1.15 (P=3) (P=4) (P=5) p=4 1.00 p=5 Nl Nl Nl 0.85 1.10 0.95 1.05 0.80 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 t t t Figure 46 Nucleon effective-mass plots for Mπ ≈ 310 MeV, at Pz = [0, 5]× 2π L on the a09m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point cor- relators. The momentum Pz = 5 2πL is the largest momentum we used, and it is the noisiest data set. 1.0 p=1 1.1 0.9 p=2 0.9 Ea12m310 Ea12m310 Ea12m310 1.0 0.8 (P=0) (P=1) (P=2) 0.8 0.9 Nl Nl Nl 0.7 0.7 0.6 0.8 0.6 0.5 0.7 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t t 1.3 1.5 1.7 1.2 p=3 1.4 1.6 Ea12m310 (P=3) Ea12m310 (P=4) Ea12m310 (P=5) p=4 1.1 1.3 1.5 p=5 Nl Nl Nl 1.0 1.2 1.4 0.9 1.1 1.3 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t t Figure 47 Nucleon effective-mass plots for Mπ ≈ 310 MeV, at Pz = [0, 5]× 2π L on the a12m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point cor- relators. The momentum Pz = 5 2πL is the largest momentum we used, and it is the noisiest data set. 98 1.1 p=1 1.1 1.3 p=2 1.0 Ea15m310 (P=0) Ea15m310 (P=1) Ea15m310 (P=2) 1.0 1.2 0.9 Nl Nl Nl 0.9 1.1 0.8 0.8 1.0 0.7 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 t t t 2.2 1.9 1.6 2.1 p=3 1.8 Ea15m310 Ea15m310 Ea15m310 1.5 (P=3) (P=4) (P=5) p=4 2.0 1.7 p=5 Nl Nl Nl 1.4 1.9 1.6 1.3 1.8 1.5 1.7 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 t t t Figure 48 Nucleon effective-mass plots for Mπ ≈ 310 MeV, at Pz = [0, 5]× 2π L on the a15m310 ensemble. The bands are reconstructed from the two-state fitted parameters of two-point cor- relators. The momentum Pz = 5 2πL is the largest momentum we used, and it is the noisiest data set. 99 .0.3. Dispersion Plots .0.3.1 Strange Nucleon 1.2 c=1.016(74) 1.5 c=0.9683(24) 1.1 1.4 1.0 1.3 aEN aEN 1.2 0.9 1.1 0.8 1.0 0.9 0.7 0 10 20 30 40 50 0 5 10 15 20 25 Nz2 Nz2 2.0 1.5 c=0.9695(48) c=0.9067(47) 1.4 1.8 1.3 aEN aEN 1.6 1.2 1.1 1.4 1.0 1.2 0.9 0 5 10 15 20 25 0 5 10 15 20 25 NZ2 Nz2 Figure 49 Dispersion relations of the nucleon energy from the two-state fits for Mπ ≈ {700, 690, 690, 690} MeV (left) on a12m220, a09m310, a12m310, a15m310 ensembles re- spectively. The speed of ligt c = 0.9638(24), 0.9695(48), 0.9067(47) repespectively. 100 .0.3.2 Light Nucleon 1.1 c=0.986(12) c=1.016(74) 1.4 1.0 1.2 0.9 aEN aEN 0.8 1.0 0.7 0.8 0.6 0.5 0 10 20 30 40 50 0 5 10 15 20 25 Nz2 Nz2 1.6 2.0 c=0.997(14) c=0.931(29) 1.8 1.4 1.6 1.2 aEN aEN 1.4 1.0 1.2 1.0 0.8 0.8 0 5 10 15 20 25 0 5 10 15 20 25 NZ2 Nz2 Figure 50 Dispersion relations of the nucleon energy from the two-state fits for Mπ ≈ {220, 310, 310, 310} MeV (left) on a12m220, a09m310, a12m310, a15m310 ensembles re- spectively. The speed of ligt c = 0.986(12), 1.0174(89), 0.997(14), 0.931(29) repespectively. 101 .0.4. Excited-State Check (ratio plot) for Matrix Elements .0.4.1 Strange Nucleon a12m220 one-state fit two-sim fit two-sim fit 0.6 RRatio (Pz= 2) 0.4 0.2 tsep=6 tsep=8 tsep=10 0.0 tsep=7 tsep=9 two-sim two-sim two-sim two-sim -4 -2 0 2 4 4 5 6 7 8 9 10 4 5 6 7 4 5 6 7 t - tsep/2 tsep tmin sep tmin sep 0.8 a09m310 one-state fit two-sim fit two-sim fit 0.6 RRatio (Pz= 2) 0.4 0.2 0.0 tsep=6 tsep=8 tsep=10 tsep=7 tsep=9 two-sim two-sim two-sim two-sim -0.2 -4 -2 0 2 4 4 5 6 7 8 9 10 4 5 6 7 7 8 9 10 11 max t - tsep/2 tsep tmin sep tsep 1.0 a12m310 one-state fit two-sim fit two-sim fit 0.8 0.6 RRatio (Pz= 2) 0.4 0.2 0.0 -0.2 tsep=7 tsep=9 tsep=11 -0.4 tsep=8 tsep=10 two-sim two-sim two-sim two-sim -4 -2 0 2 4 5 6 7 8 9 10 11 5 6 7 8 7 8 9 10 11 max t - tsep/2 tsep tmin sep tsep a15m310 one-state fit two-sim fit two-sim fit 0.6 RRatio (Pz= 2) 0.4 0.2 0.0 tsep=5 tsep=7 tsep=9 tsep=6 tsep=8 two-sim two-sim two-sim two-sim -4 -2 0 2 4 3 4 5 6 7 8 9 4 5 6 7 7 8 9 10 11 min max t - tsep/2 tsep tsep tsep Figure 51 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from the a12m220, a09m310, a12m310, a15m310 ensembles light nucleon correla- tors from top to bottom, at pion masses Mπ ≈ {700, 690, 690, 690} MeV. The gray band shown on all plots is the extracted ground-state matrix element from the two-sim fit using tsep ∈ [5, 8]. From left to right, the columns are: the ratio of the three-point to two-point correlators with the reconstructed fit bands from the two-sim fit using tsep ∈ [5, 8], shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 9], the two-sim fit results using tsep ∈ [tmin min max sep , 8] as functions of tsep , and the two-sim fit results using tsep ∈ [5, tsep ] as max functions of tsep . 102 .0.4.2 Light Nucleon 1.0 one-state fit two-sim fit two-sim fit a12m220 0.8 0.6 RRatio (Pz= 2) 0.4 0.2 0.0 tsep=7 tsep=9 tsep=11 -0.2 tsep=8 tsep=10 two-sim two-sim two-sim two-sim -4 -2 0 2 4 5 6 7 8 9 10 11 5 6 7 8 7 8 9 10 11 t - tsep/2 tsep tmin sep tmax sep 0.8 a09m310 one-state fit two-sim fit two-sim fit 0.6 RRatio (Pz= 2) 0.4 0.2 0.0 tsep=6 tsep=8 tsep=10 tsep=7 tsep=9 two-sim two-sim two-sim two-sim -0.2 -4 -2 0 2 4 4 5 6 7 8 9 10 4 5 6 7 7 8 9 10 11 t - tsep/2 tsep tmin sep tmax sep 1.0 a12m310 one-state fit two-sim fit two-sim fit 0.8 0.6 RRatio (Pz= 2) 0.4 0.2 0.0 -0.2 tsep=7 tsep=9 tsep=11 -0.4 tsep=8 tsep=10 two-sim two-sim two-sim two-sim -4 -2 0 2 4 5 6 7 8 9 10 11 5 6 7 8 7 8 9 10 11 t - tsep/2 tsep tmin sep tmax sep a15m310 one-state fit two-sim fit two-sim fit 0.8 0.6 RRatio (Pz= 2) 0.4 0.2 0.0 -0.2 tsep=5 tsep=7 tsep=9 tsep=6 tsep=8 two-sim two-sim two-sim two-sim -0.4 -4 -2 0 2 4 3 4 5 6 7 8 9 4 5 6 7 7 8 9 10 11 t - tsep/2 tsep min tsep tmax sep Figure 52 Example ratio plots (left), one-state fits (second column) and two-sim fits (last 2 columns) from the a12m220, a09m310, a12m310, a15m310 ensembles light nucleon correla- tors from top to bottom, at pion masses Mπ ≈ {220, 310, 310, 310} MeV. The gray band shown on all plots is the extracted ground-state matrix element from the two-sim fit using tsep ∈ [5, 8]. From left to right, the columns are: the ratio of the three-point to two-point correlators with the reconstructed fit bands from the two-sim fit using tsep ∈ [5, 8], shown as functions of t − tsep /2, the one-state fit results for the three-point correlators at each tsep ∈ [3, 9], the two-sim fit results using tsep ∈ [tmin min max sep , 8] as functions of tsep , and the two-sim fit results using tsep ∈ [5, tsep ] as max functions of tsep . 103 a12m220 a12m220 0.5 0.5 RRatio (Pz= 0) RRatio (Pz= 0) 0.0 0.0 -0.5 tsep=6 tsep=8 tsep=10 -0.5 tsep=6 tsep=8 tsep=10 tsep=7 tsep=9 two-sim tsep=7 tsep=9 two-sim -4 -2 0 2 4 -4 -2 0 2 4 t - tsep/2 t - tsep/2 a15m310 a15m310 1.0 1.0 0.5 0.5 RRatio (Pz= 0) 0.0 RRatio (Pz= 0) 0.0 -0.5 tsep=5 tsep=7 tsep=9 -0.5 tsep=6 tsep=8 tsep=10 tsep=6 tsep=8 two-sim tsep=7 tsep=9 two-sim -1.0 -1.0 -4 -2 0 2 4 -4 -2 0 2 4 t - tsep/2 t - tsep/2 Figure 53 Example ratio plots from the a15m310 light nucleon correlators at pion masses Mπ ≈ 310 MeV from momentum smearing parameter k 6= 0 listed in Tab. 7 and k = 0 from the PNDME collaboration from left to right, respectively. 104 .0.5. Bare gluon momentum fraction .0.5.1 Strange Nucleon a12m220 a09m310 a12m310 a15m310 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 〈x〉bare g 〈x〉bare g 〈x〉bare g 〈x〉gbare 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Nz Nz Nz Nz Figure 54 The bare gluon momentum fraction hxibare g and fitted bands normalization by kine- matic factors as functions of momentum Pz = 2π × Nz /L for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. .0.5.2 Light Nucleon a12m220 a09m310 a12m310 a15m310 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 〈x〉bare g 〈x〉bare g 〈x〉bare g 〈x〉gbare 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Nz Nz Nz Nz Figure 55 The bare gluon momentum fraction hxibare g and fitted bands normalization by kine- matic factors as functions of momentum Pz = 2π × Nz /L for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. 105 .0.6. Bare Matrix Elements .0.6.1 Strange Nucleon Figures. 56 show the fitted bare ground-state matrix elements without normalization by kine- matic factors as functions of z obtained from the two-sim fit for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. MEBare MEBare 1.4 1.5 ■ Pz=0 1.2 ■ Pz=0 ● Pz=1 ● Pz=1 1.0 ▲ Pz=2 ▲ Pz=2 a12m220-bare a09m310-bare ●▲ 1.0 ◆■ Pz=3 0.8 Pz=3 hN hN ●▲ ▼ ▼ ▲ ▼ ◆■ s ,O0IB ▼ s ,O0IB ■● ▲ ■● ◆ Pz=4 0.6 ◆■ ◆ Pz=4 ▼◆ ■●▲ ▼ ●▲ ◆■ ▲ ▼ ●▲ ■●▲ ■ Pz=5 ■● ■●▲ ◆■ ■ Pz=5 ▼ ● ■●▲ ▲ 0.5 ■●▲ ▼◆■●▲ 0.4 ▼◆■● ● Pz=6 ■●▲ ▲ ● Pz=6 ■●▲▼◆■●▲ ■●▲▼◆■●▲ ■●▲▼◆■●▲ ▲ Pz=7 0.2 ■●▲▼◆■ ▲ Pz=7 ●▲ 0.0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 z z MEBare MEBare 1.4 1.4 ■ 1.2 ■ Pz=0 ■ Pz=0 ◆ ■ 1.2 ■ ● Pz=1 ● Pz=1 1.0 ▼ ◆ ■ ▲ ▼ ▲ Pz=2 1.0 ▲ Pz=2 a12m310-bare a15m310-bare ● ▲ ◆ 0.8 ■ ■ Pz=3 Pz=3 hN hN ▼ ■ ▼ ■● ◆ 0.8 ▼ ◆ ▼ s ,O0IB s ,O0IB ▲ ▲ ▼ 0.6 ■● ◆ Pz=4 ■● ▲ ◆ Pz=4 ■ 0.6 ■● ◆ ■ ■ Pz=5 ■ Pz=5 ▼◆ ▲▼ ■ ■●▲ ■● 0.4 0.4 ◆ ■ ■●▲▼◆■ ■●▲▼ ◆ 0.2 ■●▲▼◆ 0.2 ■●▲▼ ◆ ■ ■●▲▼ 0.0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 z z Figure 56 The fitted bare ground-state matrix elements without normalization by kinematic factors as functions of z obtained from the two-sim fit for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. .0.6.2 Light Nucleon Figures. 57 show the fitted bare ground-state matrix elements without normalization by kine- matic factors as functions of z obtained from the two-sim fit for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. respectively. 106 MEBare MEBare 1.0 ▲ 1.5 ■ Pz=0 ■ Pz=0 ▲ 0.8 ● Pz=1 ● Pz=1 ■● ▲ Pz=2 ▲ Pz=2 ha12m220-bare ha09m310-bare ▲ 0.6 ◆ ■● ▼◆■●▲ 1.0 ◆ ▼ Pz=3 ▲ ▼◆■●▲ ▼ Pz=3 ▲ Nl ,O0IB ▼ Nl ,O0IB ■● ■●▲ ▼ ■● ◆ Pz=4 ■● ▼◆ ▲ ▲ ■● ◆ Pz=4 ■●▲ ◆ ▲ 0.4 ■● ▼ ■ Pz=5 ▼ ■ Pz=5 ■●▲ ■●▲ ◆■ ◆■● ▲ 0.5 ▼ ● Pz=6 ● ■●▲▼ ● Pz=6 ■●▲ ▲ ◆ ▼◆■ ● 0.2 ■ ■●▲▼◆ ■●▲ ▲ Pz=7 ■● ▲ Pz=7 ▼◆■●▲ ●▲ ■●▲ ▲ 0.0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 z z MEBare MEBare 1.4 1.4 1.2 ■ Pz=0 ■ Pz=0 1.2 ■ ● Pz=1 ■ ● Pz=1 1.0 ◆ ■ ▲ Pz=2 1.0 ◆ ■ ▲ Pz=2 ha12m310-bare ha15m310-bare ▼ ◆ ▼ ◆ 0.8 ▼ ▼ Pz=3 ▼ Pz=3 ■● ▲ ▲ ■ 0.8 ▲ ▼ Nl ,O0IB Nl ,O0IB ■● ● ▲ 0.6 ▼◆ ◆ Pz=4 ◆ Pz=4 ● ◆■ ■●▲ 0.6 ▼ ■ ■ Pz=5 ■ ● ▲ ■ Pz=5 ▼◆ ■ 0.4 ■●▲ 0.4 ■ ■ ●▲▼◆■ ■●▲▼◆ ■ 0.2 0.2 ● ■ ▲▼◆ ■●▲▼◆■ ■ ■● ▲ ▼◆ 0.0 0.0 ■ 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 z z Figure 57 The fitted bare ground-state matrix elements without normalization by kinematic factors as functions of z obtained from the two-sim fit for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. 107 .0.7. Lattice Spacing Dependence on RpITD .0.7.1 Strange Nucleon 2.0 2.0 ▼ a12m700, Pz=1.31 GeV ● a12m690, Pz=1.28 GeV ▼ a12m700, Pz=1.31 GeV ● a12m690, Pz=2.14 GeV ■ a09m690, Pz=1.31 GeV ▲ a15m690, Pz=1.54 GeV ■ a09m690, Pz=1.96 GeV ▲ a15m690, Pz=2.05 GeV 1.5 1.5 M(ν,z2) M(ν,z2) 1.0●▲▼■ ■ ▼▲ ● ■ ● ▼ ▲ ▼ ▲ 1.0●▲▼■ ▼ ● ■ ▲ ▼ ▼ ▲ ■ ▲ ▲ ● ■ ● ▲ ■ ▼ ▲ ▼ ● ▼ ▼ ■ ● ● ■ ● ■ ■ 0.5 0.5 ● 0.0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 z z Figure 58 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of ν obtained from the fitted bare ground-state matrix elements for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. 2.0 2.0 ▼ a12m700, Pz=1.31 GeV ● a12m690, Pz=1.28 GeV ▼ a12m700, Pz=1.96 GeV ● a12m690, Pz=2.14 GeV ■ a09m690, Pz=1.31 GeV ▲ a15m690, Pz=1.54 GeV ■ a09m690, Pz=1.96 GeV ▲ a15m690, Pz=2.05 GeV 1.5 1.5 ▲ M(ν,z2) M(ν,z2) 1.0●▲▼■ ■▼ ●▲■ ▼ ● ▲ ▼ ▲ 1.0●▲▼■ ■▼ ●▲ ▼ ▲ ■ ▲ ▲ ■ ● ■▲ ▼● ▼▲ ● ▼ ▼ ■ ● ■ ● ▼ ■ ■ ● 0.5 0.5 ● 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z z Figure 59 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of s obtained from the fitted bare ground-state matrix elements for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. 108 .0.7.2 Light Nucleon 2.0 2.0 ▼ a12m220, Pz=1.31 GeV ● a12m310, Pz=1.28 GeV ▼ a12m220, Pz=1.96 GeV ● a12m310, Pz=2.14 GeV ■ a09m310, Pz=1.31 GeV ▲ a15m310, Pz=1.54 GeV ■ a09m310, Pz=1.96 GeV ▲ a15m310, Pz=2.05 GeV 1.5 1.5 M(ν,z2) M(ν,z2) 1.0●▲▼■ ■ ● ▼▲ ■ ● ▼ ▲ ▼ ▼ ▼ 1.0●▲▼■ ■ ● ▼ ▲ ▼ ▼ ▼ ● ● ■ ● ▲ ■ ■● ▲ ● ■ ■ ● ▲ ● ■ ■ 0.5 ▲ 0.5 ▲ ▲ 0.0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ν ν Figure 60 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of ν obtained from the fitted bare ground-state matrix elements for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. 2.0 2.0 ▼ a12m220, Pz=1.31 GeV ● a12m310, Pz=1.28 GeV ▼ a12m220, Pz=1.96 GeV ● a12m310, Pz=2.14 GeV ■ a09m310, Pz=1.31 GeV ▲ a15m310, Pz=1.54 GeV ■ a09m310, Pz=1.96 GeV ▲ a15m310, Pz=2.05 GeV 1.5 1.5 M(ν,z2) M(ν,z2) 1.0●▲▼■ ■▼ ●▲■ ▼ ● ▲ ▼ ▼ ▼ 1.0●▲▼■ ■▼ ●▲ ▼ ▼ ▼ ▼ ● ● ■ ● ▲ ■ ■▲ ● ● ■ ■ ● ▲ ● ■ ■ 0.5 ▲ 0.5 ▲ ▲ ▲ 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z z Figure 61 The RpITDs at boost momenta Pz ≈ 2 GeV and 1.3 GeV as functions of z obtained from the fitted bare ground-state matrix elements for Mπ ≈ {220, 310, 310, 310} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. 109 .0.8. zcut -dependent fits Figures. 62 and 63 show the RpITD fit in Eq. V.14 with different fit range z ∈ [0, zcut ], with the χ2 /dof of each fit listed in the plot legends. a12m220 a09m310 0.10 0.20 zcut =1, χ2 /dof=1.9(0.7) zcut =1, χ2 /dof=0.9(0.5) 2 0.08 zcut =2, χ /dof=2.7(1.2) zcut =2, χ2 /dof=0.9(0.5) 2 zcut =3, χ /dof=2.9(1.3) 0.15 zcut =3, χ2 /dof=0.9(0.6) xg(x,μ=2 GeV)/〈x〉 xg(x,μ=2 GeV)/〈x〉 0.06 zcut =4, χ2 /dof=2.8(1.3) zcut =4, χ2 /dof=1.(0.6) zcut =5, χ2 /dof=2.6(1.2) 0.10 zcut =5, χ2 /dof=1.1(0.7) 0.04 0.05 0.02 0.00 0.00 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 x x a12m310 a15m310 0.20 0.20 zcut =1, χ2 /dof=4.2(0.9) zcut =1, χ2 /dof=1.2(0.7) 2 zcut =2, χ /dof=5.1(1.2) zcut =2, χ2 /dof=5.5(15.1) 0.15 2 zcut =3, χ /dof=4.9(1.2) 0.15 zcut =3, χ2 /dof=7.5(21.8) xg(x,μ=2 GeV)/〈x〉 xg(x,μ=2 GeV)/〈x〉 zcut =4, χ2 /dof=4.4(1.) zcut =4, χ2 /dof=8.9(26.3) 0.10 zcut =5, χ2 /dof=4.(0.9) 0.10 zcut =5, χ2 /dof=10.1(30.3) 0.05 0.05 0.00 0.00 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 x x Figure 62 RpITD fits in Eq. V.14 with different fit range z ∈ [0, zcut ] for Mπ ≈ {700, 690, 690, 690} MeV on a12m220, a09m310, a12m310, a15m310 ensembles respectively. The χ2 /dof of the fits are listed in the plot legends. 110 a12m220 a09m310 0.10 0.20 2 zcut =1, χ /dof=1.1(0.5) zcut =1, χ2 /dof=1.(0.6) 0.08 2 zcut =2, χ2 /dof=1.6(0.9) zcut =2, χ /dof=4.3(1.5) 0.15 zcut =3, χ2 /dof=2.(1.1) 0.06 zcut =3, χ2 /dof=6.(2.2) xg(x,μ=2 GeV)/〈x〉 xg(x,μ=2 GeV)/〈x〉 2 zcut =4, χ /dof=6.8(2.8) zcut =4, χ2 /dof=2.3(1.2) 0.04 2 zcut =5, χ /dof=6.9(3.1) 0.10 zcut =5, χ2 /dof=3.4(1.2) 0.02 0.00 0.05 -0.02 -0.04 0.00 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 x x a12m310 a15m310 0.20 0.20 zcut =1, χ2 /dof=3.7(1.) zcut =1, χ2 /dof=0.4(67.3) zcut =2, χ2 /dof=3.2(0.8) zcut =2, χ2 /dof=0.4(24.6) 0.15 2 zcut =3, χ /dof=3.(0.7) 0.15 zcut =3, χ2 /dof=0.3(14.9) xg(x,μ=2 GeV)/〈x〉 xg(x,μ=2 GeV)/〈x〉 2 zcut =4, χ /dof=2.8(0.6) zcut =4, χ2 /dof=0.8(8.9) 0.10 zcut =5, χ2 /dof=3.(0.6) 0.10 zcut =5, χ2 /dof=2.5(1.7) 0.05 0.05 0.00 0.00 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 x x Figure 63 RpITD fits in Eq. 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