Abstract
The transverse component of the axial-vector correlation function of quark fields is a natural starting object for lattice calculations of twist-3 nucleon parton distribution functions. In this work we derive the corresponding factorization expression in terms of twist-2 and twist-3 collinear distributions to one-loop accuracy. The results are presented both in position space, as the factorization theorem for Ioffe-time distributions, and in momentum space, for the axial-vector quasi- and pseudodistributions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Burkhardt and W.N. Cottingham, Sum rules for forward virtual Compton scattering, Annals Phys. 56 (1970) 453 [INSPIRE].
S. Wandzura and F. Wilczek, Sum Rules for Spin Dependent Electroproduction: Test of Relativistic Constituent Quarks, Phys. Lett. B 72 (1977) 195 [INSPIRE].
J. Kodaira, S. Matsuda, T. Muta, K. Sasaki and T. Uematsu, QCD Effects in Polarized Electroproduction, Phys. Rev. D 20 (1979) 627 [INSPIRE].
E.V. Shuryak and A.I. Vainshtein, Theory of Power Corrections to Deep Inelastic Scattering in Quantum Chromodynamics. 2. Q4 Effects, Polarized Target, Nucl. Phys. B 201 (1982) 141 [INSPIRE].
A.P. Bukhvostov, E.A. Kuraev and L.N. Lipatov, Deep inelastic scattering by a polarized target in quantum chromodynamics, Sov. Phys. JETP 60 (1984) 22 [INSPIRE].
P.G. Ratcliffe, Transverse Spin and Higher Twist in QCD, Nucl. Phys. B 264 (1986) 493 [INSPIRE].
I.I. Balitsky and V.M. Braun, Evolution Equations for QCD String Operators, Nucl. Phys. B 311 (1989) 541 [INSPIRE].
R.L. Jaffe and X.-D. Ji, Studies of the Transverse Spin Dependent Structure Function g2(x, Q2), Phys. Rev. D 43 (1991) 724 [INSPIRE].
X.-D. Ji and C.-h. Chou, QCD radiative corrections to the transverse spin structure function g2(x, Q2): 1. Nonsinglet operators, Phys. Rev. D 42 (1990) 3637 [INSPIRE].
A. Ali, V.M. Braun and G. Hiller, Asymptotic solutions of the evolution equation for the polarized nucleon structure function g2(x, Q2), Phys. Lett. B 266 (1991) 117 [INSPIRE].
J. Kodaira, Y. Yasui and T. Uematsu, Spin structure function g2(x, Q2) and twist - three operators in QCD, Phys. Lett. B 344 (1995) 348 [hep-ph/9408354] [INSPIRE].
J. Kodaira, Y. Yasui, K. Tanaka and T. Uematsu, QCD corrections to the nucleon’s spin structure function g2(x, Q2), Phys. Lett. B 387 (1996) 855 [hep-ph/9603377] [INSPIRE].
V.M. Braun, S.E. Derkachov and A.N. Manashov, Integrability of three particle evolution equations in QCD, Phys. Rev. Lett. 81 (1998) 2020 [hep-ph/9805225] [INSPIRE].
A.V. Belitsky, Renormalization of twist - three operators and integrable lattice models, Nucl. Phys. B 574 (2000) 407 [hep-ph/9907420] [INSPIRE].
S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Evolution equations for quark gluon distributions in multicolor QCD and open spin chains, Nucl. Phys. B 566 (2000) 203 [hep-ph/9909539] [INSPIRE].
V.M. Braun, G.P. Korchemsky and A.N. Manashov, Evolution of twist - three parton distributions in QCD beyond the large Nc limit, Phys. Lett. B 476 (2000) 455 [hep-ph/0001130] [INSPIRE].
V.M. Braun, G.P. Korchemsky and A.N. Manashov, Gluon contribution to the structure function g2(x, Q2), Nucl. Phys. B 597 (2001) 370 [hep-ph/0010128] [INSPIRE].
V.M. Braun, G.P. Korchemsky and A.N. Manashov, Evolution equation for the structure function g2(x, Q2), Nucl. Phys. B 603 (2001) 69 [hep-ph/0102313] [INSPIRE].
J. Blumlein and H. Bottcher, QCD analysis of polarized deep inelastic data and parton distributions, Nucl. Phys. B 636 (2002) 225 [hep-ph/0203155] [INSPIRE].
E155 collaboration, Measurement of the proton and deuteron spin structure functions g2 and asymmetry A2, Phys. Lett. B 458 (1999) 529 [hep-ex/9901006] [INSPIRE].
E155 collaboration, Precision measurement of the proton and deuteron spin structure functions g2 and asymmetries A2, Phys. Lett. B 553 (2003) 18 [hep-ex/0204028] [INSPIRE].
Resonance Spin Structure collaboration, Probing Quark-Gluon Interactions with Transverse Polarized Scattering, Phys. Rev. Lett. 105 (2010) 101601 [arXiv:0812.0031] [INSPIRE].
HERMES collaboration, Measurement of the virtual-photon asymmetry A2 and the spin-structure function g2 of the proton, Eur. Phys. J. C 72 (2012) 1921 [arXiv:1112.5584] [INSPIRE].
Jefferson Lab Hall A collaboration, Measurements of \( {d}_2^n \) and \( {A}_1^n \): Probing the neutron spin structure, Phys. Rev. D 94 (2016) 052003 [arXiv:1603.03612] [INSPIRE].
SANE collaboration, Revealing Color Forces with Transverse Polarized Electron Scattering, Phys. Rev. Lett. 122 (2019) 022002 [arXiv:1805.08835] [INSPIRE].
Jefferson Lab Angular Momentum collaboration, Iterative Monte Carlo analysis of spin-dependent parton distributions, Phys. Rev. D 93 (2016) 074005 [arXiv:1601.07782] [INSPIRE].
I.I. Balitsky, V.M. Braun and A.V. Kolesnichenko, Power corrections 1/Q2 to parton sum rules for deep inelastic scattering from polarized targets, Phys. Lett. B 242 (1990) 245 [Erratum ibid. 318 (1993) 648] [hep-ph/9310316] [INSPIRE].
E. Stein, P. Gornicki, L. Mankiewicz, A. Schafer and W. Greiner, QCD sum rule calculation of twist - three contributions to polarized nucleon structure functions, Phys. Lett. B 343 (1995) 369 [hep-ph/9409212] [INSPIRE].
J. Balla, M.V. Polyakov and C. Weiss, Nucleon matrix elements of higher twist operators from the instanton vacuum, Nucl. Phys. B 510 (1998) 327 [hep-ph/9707515] [INSPIRE].
V.M. Braun, T. Lautenschlager, A.N. Manashov and B. Pirnay, Higher twist parton distributions from light-cone wave functions, Phys. Rev. D 83 (2011) 094023 [arXiv:1103.1269] [INSPIRE].
M. Gockeler et al., Investigation of the second moment of the nucleon’s g1 and g2 structure functions in two-flavor lattice QCD, Phys. Rev. D 72 (2005) 054507 [hep-lat/0506017] [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].
R. Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, arXiv:2103.05419 [INSPIRE].
D. Boer, P.J. Mulders and F. Pijlman, Universality of T odd effects in single spin and azimuthal asymmetries, Nucl. Phys. B 667 (2003) 201 [hep-ph/0303034] [INSPIRE].
K. Kanazawa, Y. Koike, A. Metz, D. Pitonyak and M. Schlegel, Operator Constraints for Twist-3 Functions and Lorentz Invariance Properties of Twist-3 Observables, Phys. Rev. D 93 (2016) 054024 [arXiv:1512.07233] [INSPIRE].
I. Scimemi and A. Vladimirov, Matching of transverse momentum dependent distributions at twist-3, Eur. Phys. J. C 78 (2018) 802 [arXiv:1804.08148] [INSPIRE].
V. Moos and A. Vladimirov, Calculation of transverse momentum dependent distributions beyond the leading power, JHEP 12 (2020) 145 [arXiv:2008.01744] [INSPIRE].
D.W. Sivers, Single Spin Production Asymmetries from the Hard Scattering of Point-Like Constituents, Phys. Rev. D 41 (1990) 83 [INSPIRE].
J.-w. Qiu and G.F. Sterman, Single transverse spin asymmetries, Phys. Rev. Lett. 67 (1991) 2264 [INSPIRE].
Z.-B. Kang, B.-W. Xiao and F. Yuan, QCD Resummation for Single Spin Asymmetries, Phys. Rev. Lett. 107 (2011) 152002 [arXiv:1106.0266] [INSPIRE].
V.M. Braun, A.N. Manashov and B. Pirnay, Scale dependence of twist-three contributions to single spin asymmetries, Phys. Rev. D 80 (2009) 114002 [Erratum ibid. 86 (2012) 119902] [arXiv:0909.3410] [INSPIRE].
I. Scimemi, A. Tarasov and A. Vladimirov, Collinear matching for Sivers function at next-to-leading order, JHEP 05 (2019) 125 [arXiv:1901.04519] [INSPIRE].
M. Bury, A. Prokudin and A. Vladimirov, N3LO extraction of the Sivers function from SIDIS, Drell-Yan, and W±/Z data, Phys. Rev. Lett. 126 (2021) 112002 [arXiv:2012.05135] [INSPIRE].
M. Bury, A. Prokudin and A. Vladimirov, Extraction of the Sivers function from SIDIS, Drell-Yan, and W±/Z boson production data with TMD evolution, arXiv:2103.03270 [INSPIRE].
M. Constantinou et al., Parton distributions and lattice QCD calculations: toward 3D structure, arXiv:2006.08636 [INSPIRE].
X. Ji, Y. Liu, Y.-S. Liu, J.-H. Zhang and Y. Zhao, Large-Momentum Effective Theory, arXiv:2004.03543 [INSPIRE].
T. Izubuchi, X. Ji, L. Jin, I.W. Stewart and Y. Zhao, Factorization Theorem Relating Euclidean and Light-Cone Parton Distributions, Phys. Rev. D 98 (2018) 056004 [arXiv:1801.03917] [INSPIRE].
L. Del Debbio, T. Giani, J. Karpie, K. Orginos, A. Radyushkin and S. Zafeiropoulos, Neural-network analysis of Parton Distribution Functions from Ioffe-time pseudodistributions, JHEP 02 (2021) 138 [arXiv:2010.03996] [INSPIRE].
X. Ji, Parton Physics on a Euclidean Lattice, Phys. Rev. Lett. 110 (2013) 262002 [arXiv:1305.1539] [INSPIRE].
A.V. Radyushkin, Quasi-parton distribution functions, momentum distributions, and pseudo-parton distribution functions, Phys. Rev. D 96 (2017) 034025 [arXiv:1705.01488] [INSPIRE].
S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato and F. Steffens, Insights on proton structure from lattice QCD: The twist-3 parton distribution function gT(x), Phys. Rev. D 102 (2020) 111501 [arXiv:2004.04130] [INSPIRE].
S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato and F. Steffens, One-loop matching for the twist-3 parton distribution gT(x), Phys. Rev. D 102 (2020) 034005 [arXiv:2005.10939] [INSPIRE].
R.L. Jaffe, Spin, twist and hadron structure in deep inelastic processes, in Ettore Majorana International School of Nucleon Structure: 1st Course: The Spin Structure of the Nucleon, (1996) [hep-ph/9602236] [INSPIRE].
Y. Ji and A.V. Belitsky, Renormalization of twist-four operators in light-cone gauge, Nucl. Phys. B 894 (2015) 161 [arXiv:1405.2828] [INSPIRE].
K.G. Chetyrkin and A.G. Grozin, Three loop anomalous dimension of the heavy light quark current in HQET, Nucl. Phys. B 666 (2003) 289 [hep-ph/0303113] [INSPIRE].
V.M. Braun, K.G. Chetyrkin and B.A. Kniehl, Renormalization of parton quasi-distributions beyond the leading order: spacelike vs. timelike, JHEP 07 (2020) 161 [arXiv:2004.01043] [INSPIRE].
V.S. Dotsenko and S.N. Vergeles, Renormalizability of Phase Factors in the Nonabelian Gauge Theory, Nucl. Phys. B 169 (1980) 527 [INSPIRE].
K. Orginos, A. Radyushkin, J. Karpie and S. Zafeiropoulos, Lattice QCD exploration of parton pseudo-distribution functions, Phys. Rev. D 96 (2017) 094503 [arXiv:1706.05373] [INSPIRE].
V.M. Braun, A. Vladimirov and J.-H. Zhang, Power corrections and renormalons in parton quasidistributions, Phys. Rev. D 99 (2019) 014013 [arXiv:1810.00048] [INSPIRE].
B.L. Ioffe, Space-time picture of photon and neutrino scattering and electroproduction cross-section asymptotics, Phys. Lett. B 30 (1969) 123 [INSPIRE].
V. Braun, P. Gornicki and L. Mankiewicz, Ioffe - time distributions instead of parton momentum distributions in description of deep inelastic scattering, Phys. Rev. D 51 (1995) 6036 [hep-ph/9410318] [INSPIRE].
L.-B. Chen, W. Wang and R. Zhu, Quasi parton distribution functions at NNLO: flavor non-diagonal quark contributions, Phys. Rev. D 102 (2020) 011503 [arXiv:2005.13757] [INSPIRE].
L.-B. Chen, W. Wang and R. Zhu, Next-to-Next-to-Leading Order Calculation of Quasiparton Distribution Functions, Phys. Rev. Lett. 126 (2021) 072002 [arXiv:2006.14825] [INSPIRE].
Z.-Y. Li, Y.-Q. Ma and J.-W. Qiu, Extraction of Next-to-Next-to-Leading-Order Parton Distribution Functions from Lattice QCD Calculations, Phys. Rev. Lett. 126 (2021) 072001 [arXiv:2006.12370] [INSPIRE].
S.A. Anikin and O.I. Zavyalov, Short Distance and Light Cone Expansions for Products of Currents, Annals Phys. 116 (1978) 135 [INSPIRE].
S.A. Anikin, O.I. Zavyalov and N.I. Karchev, Nonlocal light-cone expansion of the product of currents and its renormalization group analysis, Theor. Math. Phys. 38 (1979) 193 [INSPIRE].
D. Müller, D. Robaschik, B. Geyer, F.M. Dittes and J. Hořejši, Wave functions, evolution equations and evolution kernels from light ray operators of QCD, Fortsch. Phys. 42 (1994) 101 [hep-ph/9812448] [INSPIRE].
I.I. Balitsky and V.M. Braun, The Nonlocal operator expansion for inclusive particle production in e+e− annihilation, Nucl. Phys. B 361 (1991) 93 [INSPIRE].
B. Geyer, M. Lazar and D. Robaschik, Decomposition of nonlocal light cone operators into harmonic operators of definite twist, Nucl. Phys. B 559 (1999) 339 [hep-th/9901090] [INSPIRE].
V.M. Braun and A.N. Manashov, Operator product expansion in QCD in off-forward kinematics: Separation of kinematic and dynamical contributions, JHEP 01 (2012) 085 [arXiv:1111.6765] [INSPIRE].
R. Jaffe, Parton Distribution Functions for Twist Four, Nucl. Phys. B 229 (1983) 205.
D. Mueller, Calculation of higher twist evolution kernels for polarized deep inelastic scattering, Phys. Lett. B 407 (1997) 314 [hep-ph/9701338] [INSPIRE].
X. Ji, J.-w. Qiu, W. Vogelsang and F. Yuan, Single Transverse-Spin Asymmetry in Drell-Yan Production at Large and Moderate Transverse Momentum, Phys. Rev. D 73 (2006) 094017 [hep-ph/0604023] [INSPIRE].
J.S. Schwinger, The Theory of quantized fields. 1, Phys. Rev. 82 (1951) 914 [INSPIRE].
L.F. Abbott, The Background Field Method Beyond One Loop, Nucl. Phys. B 185 (1981) 189 [INSPIRE].
L. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 (1982) 33.
E.V. Shuryak and A.I. Vainshtein, Theory of Power Corrections to Deep Inelastic Scattering in Quantum Chromodynamics. 1. Q2 Effects, Nucl. Phys. B 199 (1982) 451 [INSPIRE].
V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Calculations in External Fields in Quantum Chromodynamics. Technical Review, Fortsch. Phys. 32 (1984) 585 [INSPIRE].
I. Balitsky, Factorization and high-energy effective action, Phys. Rev. D 60 (1999) 014020 [hep-ph/9812311] [INSPIRE].
P. Pascual and R. Tarrach, QCD: Renormalization for the Practitioner, vol. 194 of Lect. Notes Phys. Springer (1984)[DOI].
M.A. Shifman and M.B. Voloshin, On Production of d and D* Mesons in B Meson Decays, Sov. J. Nucl. Phys. 47 (1988) 511 [INSPIRE].
Y. Matiounine, J. Smith and W.L. van Neerven, Two loop operator matrix elements calculated up to finite terms for polarized deep inelastic lepton - hadron scattering, Phys. Rev. D 58 (1998) 076002 [hep-ph/9803439] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case, Nucl. Phys. B 889 (2014) 351 [arXiv:1409.5131] [INSPIRE].
D. Gutiérrez-Reyes, I. Scimemi and A.A. Vladimirov, Twist-2 matching of transverse momentum dependent distributions, Phys. Lett. B 769 (2017) 84 [arXiv:1702.06558] [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
P. Breitenlohner and D. Maison, Dimensional Renormalization and the Action Principle, Commun. Math. Phys. 52 (1977) 11 [INSPIRE].
S.A. Larin, The Renormalization of the axial anomaly in dimensional regularization, Phys. Lett. B 303 (1993) 113 [hep-ph/9302240] [INSPIRE].
A.V. Radyushkin, Quark pseudodistributions at short distances, Phys. Lett. B 781 (2018) 433 [arXiv:1710.08813] [INSPIRE].
A.V. Belitsky, V.M. Braun, A.S. Gorsky and G.P. Korchemsky, Integrability in QCD and beyond, Int. J. Mod. Phys. A 19 (2004) 4715 [hep-th/0407232] [INSPIRE].
A. Radyushkin, Nonperturbative Evolution of Parton Quasi-Distributions, Phys. Lett. B 767 (2017) 314 [arXiv:1612.05170] [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].
J. Blumlein and H. Bottcher, QCD Analysis of Polarized Deep Inelastic Scattering Data, Nucl. Phys. B 841 (2010) 205 [arXiv:1005.3113] [INSPIRE].
S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato and F. Steffens, The role of zero-mode contributions in the matching for the twist-3 PDFs e(x) and hL(x), Phys. Rev. D 102 (2020) 114025 [arXiv:2006.12347] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2103.12105
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Braun, V.M., Ji, Y. & Vladimirov, A. QCD factorization for twist-three axial-vector parton quasidistributions. J. High Energ. Phys. 2021, 86 (2021). https://doi.org/10.1007/JHEP05(2021)086
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)086