Abstract
A detailed analysis of the systematic uncertainties in the calculation of the isovector momentum fraction, 〈x〉u − d, helicity moment, 〈x〉Δu − Δd, and the transversity moment, 〈x〉δu − δd, of the nucleon is presented using high-statistics data on seven ensembles of gauge configurations generated by the JLab/W&M/LANL/MIT collaborations using 2 + 1-flavors of dynamical Wilson-clover quarks. The much higher statistics have facilitated better control over all systematics compared to previous lattice calculations. The least understood systematic — excited-state contamination — is quantified by studying the variation of the results as a function of different estimates of the mass gap of the first excited state, obtained from two- and three-point correlation functions, and as a function of the pion mass Mπ. The final results are obtained using a simultaneous fit in the lattice spacing a, pion mass Mπ and the finite volume parameter MπL keeping leading order corrections. The data show no significant dependence on the lattice spacing and some evidence for finite-volume corrections. Our final results, in the \( \overline{\mathrm{MS}} \) scheme at 2 GeV, are 〈x〉u − d = 0.155(17)(20), 〈x〉Δu − Δd = 0.183(14)(20) and 〈x〉δu − δd = 0.220(18)(20), where the first error is the overall analysis uncertainty assuming excited-state contributions have been removed, and the second is an additional systematic uncertainty due to possible residual excited-state contributions. These results are consistent with phenomenological global fit values.
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STAR collaboration, Gluon polarization and jet production at STAR, Nuovo Cim. C 036 (2013) 35 [arXiv:1303.0543] [INSPIRE].
PHENIX collaboration, Inclusive double-helicity asymmetries in neutral-pion and η-meson production in \( \overrightarrow{p}+\overrightarrow{p} \) collisions at \( \sqrt{s} \) = 200 GeV, Phys. Rev. D 90 (2014) 012007 [arXiv:1402.6296] [INSPIRE].
J. Dudek et al., Physics Opportunities with the 12 GeV upgrade at Jefferson Lab, Eur. Phys. J. A 48 (2012) 187 [arXiv:1208.1244] [INSPIRE].
A. Accardi et al., Electron ion collider: the next QCD frontier: understanding the glue that binds us all, Eur. Phys. J. A 52 (2016) 268 [arXiv:1212.1701] [INSPIRE].
CTEQ collaboration, Handbook of perturbative QCD: version 1.0, Rev. Mod. Phys. 67 (1995) 157 [INSPIRE].
X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang and Y. Zhao, Large-momentum effective theory, arXiv:2004.03543 [INSPIRE].
B. Yoon et al., Nucleon transverse momentum-dependent parton distributions in lattice QCD: renormalization patterns and discretization effects, Phys. Rev. D 96 (2017) 094508 [arXiv:1706.03406] [INSPIRE].
M. Diehl, Generalized parton distributions, Phys. Rept. 388 (2003) 41 [hep-ph/0307382] [INSPIRE].
K. Cichy and M. Constantinou, A guide to light-cone PDFs from Lattice QCD: an overview of approaches, techniques and results, Adv. High Energy Phys. 2019 (2019) 3036904 [arXiv:1811.07248] [INSPIRE].
N. Karthik, Lattice computations of PDF: challenges and progress, talk given at 37th International Conference on Lattice Field Theory (LATTICE 2019), June 16–22, Wuhan, China (2019).
H.-W. Lin et al., Parton distributions and lattice QCD calculations: a community white paper, Prog. Part. Nucl. Phys. 100 (2018) 107 [arXiv:1711.07916] [INSPIRE].
M. Constantinou et al., Parton distributions and lattice QCD calculations: toward 3D structure, arXiv:2006.08636 [INSPIRE].
Flavour Lattice Averaging Group collaboration, FLAG review 2019: Flavour Lattice Averaging Group (FLAG), Eur. Phys. J. C 80 (2020) 113 [arXiv:1902.08191] [INSPIRE].
R. Gupta et al., Isovector charges of the nucleon from 2 + 1 + 1-flavor lattice QCD, Phys. Rev. D 98 (2018) 034503 [arXiv:1806.09006] [INSPIRE].
R. Edwards, U.S. 2 + 1 flavor clover lattice generation program, unpublished (2016).
S. Mondal, R. Gupta, S. Park, B. Yoon, T. Bhattacharya and H.-W. Lin, Moments of nucleon isovector structure functions in 2 + 1 + 1-flavor QCD, Phys. Rev. D 102 (2020) 054512 [arXiv:2005.13779] [INSPIRE].
C. Morningstar and M.J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D 69 (2004) 054501 [hep-lat/0311018] [INSPIRE].
B. Sheikholeslami and R. Wohlert, Improved continuum limit lattice action for QCD with Wilson fermions, Nucl. Phys. B 259 (1985) 572 [INSPIRE].
M. Lüscher, S. Sint, R. Sommer, P. Weisz and U. Wolff, Nonperturbative O(a) improvement of lattice QCD, Nucl. Phys. B 491 (1997) 323 [hep-lat/9609035] [INSPIRE].
S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195 (1987) 216 [INSPIRE].
B. Yoon et al., Isovector charges of the nucleon from 2 + 1-flavor QCD with clover fermions, Phys. Rev. D 95 (2017) 074508 [arXiv:1611.07452] [INSPIRE].
G.S. Bali, S. Collins and A. Schafer, Effective noise reduction techniques for disconnected loops in Lattice QCD, Comput. Phys. Commun. 181 (2010) 1570 [arXiv:0910.3970] [INSPIRE].
T. Blum, T. Izubuchi and E. Shintani, New class of variance-reduction techniques using lattice symmetries, Phys. Rev. D 88 (2013) 094503 [arXiv:1208.4349] [INSPIRE].
S. Gusken, K. Schilling, R. Sommer, K.H. Mutter and A. Patel, Mass splittings in the baryon octet and the nucleon σ term in lattice QCD, Phys. Lett. B 212 (1988) 216 [INSPIRE].
B. Yoon et al., Controlling excited-state contamination in nucleon matrix elements, Phys. Rev. D 93 (2016) 114506 [arXiv:1602.07737] [INSPIRE].
R. Babich et al., Adaptive multigrid algorithm for the lattice Wilson-Dirac operator, Phys. Rev. Lett. 105 (2010) 201602 [arXiv:1005.3043] [INSPIRE].
M.A. Clark, R. Babich, K. Barros, R.C. Brower and C. Rebbi, Solving lattice QCD systems of equations using mixed precision solvers on GPUs, Comput. Phys. Commun. 181 (2010) 1517 [arXiv:0911.3191] [INSPIRE].
M.A. Clark et al., Accelerating lattice QCD multigrid on GPUs using fine-grained parallelization, in the proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (SC16), November 13–18, Salt Lake City, U.S.A. (2016).
S. Gusken, U. Low, K.H. Mutter, R. Sommer, A. Patel and K. Schilling, Nonsinglet axial vector couplings of the baryon octet in lattice QCD, Phys. Lett. B 227 (1989) 266 [INSPIRE].
SciDAC, LHPC, UKQCD collaboration, The Chroma software system for lattice QCD, Nucl. Phys. B Proc. Suppl. 140 (2005) 832 [hep-lat/0409003] [INSPIRE].
M. Gockeler et al., Polarized and unpolarized nucleon structure functions from lattice QCD, Phys. Rev. D 53 (1996) 2317 [hep-lat/9508004] [INSPIRE].
T. Harris et al., Nucleon isovector charges and twist-2 matrix elements with Nf = 2 + 1 dynamical Wilson quarks, Phys. Rev. D 100 (2019) 034513 [arXiv:1905.01291] [INSPIRE].
T. Bhattacharya, S.D. Cohen, R. Gupta, A. Joseph, H.-W. Lin and B. Yoon, Nucleon charges and electromagnetic form factors from 2 + 1 + 1-flavor lattice QCD, Phys. Rev. D 89 (2014) 094502 [arXiv:1306.5435] [INSPIRE].
G.S. Bali et al., The moment 〈x〉u − d of the nucleon from Nf = 2 lattice QCD down to nearly physical quark masses, Phys. Rev. D 90 (2014) 074510 [arXiv:1408.6850] [INSPIRE].
G.S. Bali et al., 〈x〉u − d from lattice QCD at nearly physical quark masses, Phys. Rev. D 86 (2012) 054504 [arXiv:1207.1110] [INSPIRE].
Y.-C. Jang, R. Gupta, B. Yoon and T. Bhattacharya, Axial vector form factors from lattice QCD that satisfy the PCAC relation, Phys. Rev. Lett. 124 (2020) 072002 [arXiv:1905.06470] [INSPIRE].
H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Contr. 19 (1974) 716.
C. Alexandrou et al., Complete flavor decomposition of the spin and momentum fraction of the proton using lattice QCD simulations at physical pion mass, Phys. Rev. D 101 (2020) 094513 [arXiv:2003.08486] [INSPIRE].
C. Alexandrou et al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys. Rev. D 101 (2020) 034519 [arXiv:1908.10706] [INSPIRE].
Y.B. Yang et al., Proton mass decomposition from the QCD energy momentum tensor, Phys. Rev. Lett. 121 (2018) 212001 [arXiv:1808.08677] [INSPIRE].
G.S. Bali, S. Collins, M. Göckeler, R. Rödl, A. Schäfer and A. Sternbeck, Nucleon generalized form factors from two-flavor lattice QCD, Phys. Rev. D 100 (2019) 014507 [arXiv:1812.08256] [INSPIRE].
C. Alexandrou et al., Nucleon spin and momentum decomposition using lattice QCD simulations, Phys. Rev. Lett. 119 (2017) 142002 [arXiv:1706.02973] [INSPIRE].
A. Abdel-Rehim et al., Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass, Phys. Rev. D 92 (2015) 114513 [Erratum ibid. 93 (2016) 039904] [arXiv:1507.04936] [INSPIRE].
J.R. Green et al., Nucleon structure from lattice QCD using a nearly physical pion mass, Phys. Lett. B 734 (2014) 290 [arXiv:1209.1687] [INSPIRE].
Y. Aoki et al., Nucleon isovector structure functions in (2 + 1)-flavor QCD with domain wall fermions, Phys. Rev. D 82 (2010) 014501 [arXiv:1003.3387] [INSPIRE].
LHPC collaboration, Nucleon structure from mixed action calculations using 2 + 1 flavors of asqtad sea and domain wall valence fermions, Phys. Rev. D 82 (2010) 094502 [arXiv:1001.3620] [INSPIRE].
T.-J. Hou et al., New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC, Phys. Rev. D 103 (2021) 014013 [arXiv:1912.10053] [INSPIRE].
J.J. Ethier, N. Sato and W. Melnitchouk, First simultaneous extraction of spin-dependent parton distributions and fragmentation functions from a global QCD analysis, Phys. Rev. Lett. 119 (2017) 132001 [arXiv:1705.05889] [INSPIRE].
NNPDF collaboration, Parton distributions from high-precision collider data, Eur. Phys. J. C 77 (2017) 663 [arXiv:1706.00428] [INSPIRE].
S. Alekhin, J. Blümlein, S. Moch and R. Placakyte, Parton distribution functions, αs, and heavy-quark masses for LHC Run II, Phys. Rev. D 96 (2017) 014011 [arXiv:1701.05838] [INSPIRE].
A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens and N. Sato, Constraints on large-x parton distributions from new weak boson production and deep-inelastic scattering data, Phys. Rev. D 93 (2016) 114017 [arXiv:1602.03154] [INSPIRE].
H1 and ZEUS collaborations, Combination of measurements of inclusive deep inelastic e±p scattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C 75 (2015) 580 [arXiv:1506.06042] [INSPIRE].
S. Dulat et al., New parton distribution functions from a global analysis of quantum chromodynamics, Phys. Rev. D 93 (2016) 033006 [arXiv:1506.07443] [INSPIRE].
L.A. Harland-Lang, A.D. Martin, P. Motylinski and R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs, Eur. Phys. J. C 75 (2015) 204 [arXiv:1412.3989] [INSPIRE].
NNPDF collaboration, A first unbiased global determination of polarized PDFs and their uncertainties, Nucl. Phys. B 887 (2014) 276 [arXiv:1406.5539] [INSPIRE].
D. de Florian, R. Sassot, M. Stratmann and W. Vogelsang, Extraction of spin-dependent parton densities and their uncertainties, Phys. Rev. D 80 (2009) 034030 [arXiv:0904.3821] [INSPIRE].
D. de Florian, R. Sassot, M. Stratmann and W. Vogelsang, Global analysis of helicity parton densities and their uncertainties, Phys. Rev. Lett. 101 (2008) 072001 [arXiv:0804.0422] [INSPIRE].
M. Gockeler et al., Perturbative and nonperturbative renormalization in lattice QCD, Phys. Rev. D 82 (2010) 114511 [Erratum ibid. 86 (2012) 099903] [arXiv:1003.5756] [INSPIRE].
M. Constantinou et al., Perturbatively improving regularization-invariant momentum scheme renormalization constants, Phys. Rev. D 87 (2013) 096019 [arXiv:1303.6776] [INSPIRE].
J.A. Gracey, Three loop anomalous dimension of the second moment of the transversity operator in the MS-bar and RI-prime schemes, Nucl. Phys. B 667 (2003) 242 [hep-ph/0306163] [INSPIRE].
M. Constantinou et al., Renormalization of local quark-bilinear operators for Nf = 3 flavors of stout link nonperturbative clover fermions, Phys. Rev. D 91 (2015) 014502 [arXiv:1408.6047] [INSPIRE].
T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, H.-W. Lin and B. Yoon, Axial, scalar and tensor charges of the nucleon from 2 + 1 + 1-flavor lattice QCD, Phys. Rev. D 94 (2016) 054508 [arXiv:1606.07049] [INSPIRE].
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Nucleon Matrix Elements (NME) collaboration., Mondal, S., Gupta, R. et al. Nucleon momentum fraction, helicity and transversity from 2+1-flavor lattice QCD. J. High Energ. Phys. 2021, 44 (2021). https://doi.org/10.1007/JHEP04(2021)044
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DOI: https://doi.org/10.1007/JHEP04(2021)044