Abstract
Sum rules for structure functions and their twist-2 relations have important roles in constraining their magnitudes and x dependencies and in studying higher-twist effects. The Wandzura-Wilczek (WW) relation and the Burkhardt-Cottingham (BC) sum rule are such examples for the polarized structure functions g1 and g2. Recently, new twist-3 and twist-4 parton distribution functions were proposed for spin-1 hadrons, so that it became possible to investigate spin-1 structure functions including higher-twist ones. We show in this work that an analogous twist-2 relation and a sum rule exist for the tensor-polarized parton distribution functions f1LL and fLT, where f1LL is a twist-2 function and fLT is a twist-3 one. Namely, the twist-2 part of fLT is expressed by an integral of f1LL (or b1) and the integral of the function f2LT = (2/3)fLT − f1LL over x vanishes. If the parton-model sum rule for f1LL (b1) is applied by assuming vanishing tensor-polarized antiquark distributions, another sum rule also exists for fLT itself. These relations should be valuable for studying tensor-polarized distribution functions of spin-1 hadrons and for separating twist-2 components from higher-twist terms, as the WW relation and BC sum rule have been used for investigating x dependence and higher-twist effects in g2. In deriving these relations, we indicate that four twist-3 multiparton distribution functions FLT, GLT, \( {H}_{LL}^{\perp } \), and HTT exist for tensor-polarized spin-1 hadrons. These multiparton distribution functions are also interesting to probe multiparton correlations in spin-1 hadrons. In the near future, we expect that physics of spin-1 hadrons will become a popular topic, since there are experimental projects to investigate spin structure of the spin-1 deuteron at the Jefferson Laboratory, the Fermilab, the nuclotron-based ion collider facility, the electron-ion colliders in US and China in 2020’s and 2030’s.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.L. Frankfurt and M.I. Strikman, High-momentum-transfer processes with polarized deuterons, Nucl. Phys. A 405 (1983) 557 [INSPIRE].
P. Hoodbhoy, R.L. Jaffe and A. Manohar, Novel Effects in Deep Inelastic Scattering from Spin 1 Hadrons, Nucl. Phys. B 312 (1989) 571 [INSPIRE].
F.E. Close and S. Kumano, Sum rule for the spin-dependent structure function b1(x) for spin-one hadrons, Phys. Rev. D 42 (1990) 2377 [INSPIRE].
S. Kumano, Tensor-polarized structure functions: Tensor structure of deuteron in 2020’s, J. Phys. Conf. Ser. 543 (2014) 012001 [arXiv:1407.3852] [INSPIRE].
HERMES collaboration, Measurement of the Tensor Structure Function b1 of the Deuteron, Phys. Rev. Lett. 95 (2005) 242001 [hep-ex/0506018] [INSPIRE].
S. Kumano, Tensor-polarized quark and antiquark distribution functions in a spin-one hadron, Phys. Rev. D 82 (2010) 017501 [arXiv:1005.4524] [INSPIRE].
J.-P. Chen et al., The Deuteron Tensor Structure Function b1, Proposal to Jefferson Lab PAC-38, PR12-11-110 (2011) https://www.jlab.org/exp_prog/proposals/11/PR12-11-110.pdf.
W. Cosyn, Y.-B. Dong, S. Kumano and M. Sargsian, Tensor-polarized structure function b1 in standard convolution description of deuteron, Phys. Rev. D 95 (2017) 074036 [arXiv:1702.05337] [INSPIRE].
R.L. Jaffe and A. Manohar, Nuclear gluonometry, Phys. Lett. B 223 (1989) 218 [INSPIRE].
M. Jones et al., Search for Exotic Gluonic States in the Nucleus, A Letter of Intent to Jefferson Lab PAC 44, LOI12-16-006 (2016) https://www.jlab.org/exp_prog/proposals/16/LOI12-16-006.pdf.
J.P. Ma, C. Wang and G.P. Zhang, Azimuthal Dependence of DIS with Spin-1 Target, arXiv:1306.6693 [INSPIRE].
R. Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, arXiv:2103.05419 [INSPIRE].
D.P. Anderle et al., Electron-ion collider in China, Front. Phys. (Beijing) 16 (2021) 64701 [arXiv:2102.09222] [INSPIRE].
S. Hino and S. Kumano, Structure functions in the polarized Drell-Yan processes with spin-1/2 and spin-1 hadrons. 1. General formalism, Phys. Rev. D 59 (1999) 094026 [hep-ph/9810425] [INSPIRE].
S. Hino and S. Kumano, Structure functions in the polarized Drell-Yan processes with spin-1/2 and spin-1 hadrons. 2. Parton model, Phys. Rev. D 60 (1999) 054018 [hep-ph/9902258] [INSPIRE].
S. Kumano and Q.-T. Song, Theoretical estimate on tensor-polarization asymmetry in proton-deuteron Drell-Yan process, Phys. Rev. D 94 (2016) 054022 [arXiv:1606.03149] [INSPIRE].
S. Kumano and Q.-T. Song, Gluon transversity in polarized proton-deuteron Drell-Yan process, Phys. Rev. D 101 (2020) 054011 [arXiv:1910.12523] [INSPIRE].
S. Kumano and Q.-T. Song, Deuteron polarizations in the proton-deuteron Drell-Yan process for finding the gluon transversity, Phys. Rev. D 101 (2020) 094013 [arXiv:2003.06623] [INSPIRE].
D. Geesaman et al., Letter of Intent for a Drell-Yan experiment with a polarized proton target, Proposal to Fermilab PAC, P-1039 (2013) https://www.fnal.gov/directorate/program_planning/June2013PACPublic/P- 1039_LOI_polarized_DY.pdf.
D. Keller, D. Crabb and D. Day, Enhanced tensor polarization in solid-state targets, Nucl. Instrum. Meth. A 981 (2020) 164504 [arXiv:2008.09515] [INSPIRE].
NICA, For the Spin Physics Detector (SPD) project at NICA, http://spd.jinr.ru/.
A. Arbuzov et al., On the physics potential to study the gluon content of proton and deuteron at NICA SPD, Prog. Nucl. Part. Phys. 119 (2021) 103858 [arXiv:2011.15005] [INSPIRE].
G.A. Miller, Pionic and Hidden-Color, Six-Quark Contributions to the Deuteron b1 Structure Function, Phys. Rev. C 89 (2014) 045203 [arXiv:1311.4561] [INSPIRE].
M. Nzar and P. Hoodbhoy, Estimation of the double helicity flip deuteron structure function, Phys. Rev. D 45 (1992) 2264 [INSPIRE].
S. Kumano and Q.-T. Song, Transverse-momentum-dependent parton distribution functions up to twist 4 for spin-1 hadrons, Phys. Rev. D 103 (2021) 014025 [arXiv:2011.08583] [INSPIRE].
A. Bacchetta and P.J. Mulders, Deep inelastic leptoproduction of spin-one hadrons, Phys. Rev. D 62 (2000) 114004 [hep-ph/0007120] [INSPIRE].
A. Bacchetta, Probing the Transverse Spin of Quarks in Deep Inelastic Scattering, Ph.D. thesis, Vrije University, Amsterdam (2002) hep-ph/0212025 [INSPIRE].
T.A.A. van Daal, Mapping the internal structure of hadrons through color and spin effects, Ph.D. thesis, Groningen Univesity (2018) arXiv:1812.07336 [INSPIRE].
S. Cotogno, Polarized partons in hadrons at high energy, Ph.D. thesis, Free University, Amsterdam (2018) [INSPIRE].
S. Wandzura and F. Wilczek, Sum Rules for Spin Dependent Electroproduction: Test of Relativistic Constituent Quarks, Phys. Lett. B 72 (1977) 195 [INSPIRE].
H. Burkhardt and W.N. Cottingham, Sum rules for forward virtual Compton scattering, Annals Phys. 56 (1970) 453 [INSPIRE].
J. Kodaira and K. Tanaka, Polarized structure functions in QCD, Prog. Theor. Phys. 101 (1999) 191 [hep-ph/9812449] [INSPIRE].
J. Blümlein and N. Kochelev, On the twist -2 and twist - three contributions to the spin dependent electroweak structure functions, Nucl. Phys. B 498 (1997) 285 [hep-ph/9612318] [INSPIRE].
J. Blümlein and A. Tkabladze, Target mass corrections for polarized structure functions and new sum rules, Nucl. Phys. B 553 (1999) 427 [hep-ph/9812478] [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].
M. Anselmino, A. Efremov and E. Leader, The Theory and phenomenology of polarized deep inelastic scattering, Phys. Rept. 261 (1995) 1 [Erratum ibid. 281 (1997) 399] [hep-ph/9501369] [INSPIRE].
B. Lampe and E. Reya, Spin physics and polarized structure functions, Phys. Rept. 332 (2000) 1 [hep-ph/9810270] [INSPIRE].
E. Leader, Spin in Particle Physics, Cambridge University Press (2001) [DOI].
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].
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), pp. 42–129 [hep-ph/9602236] [INSPIRE].
C. Itzkyson and J.-B. Zuber, Quantum Field Theory McGraw-Hill Inc. (1980).
R.L. Jaffe and X.-D. Ji, Chiral-odd parton distributions and polarized Drell-Yan process, Phys. Rev. Lett. 67 (1991) 552 [INSPIRE].
R.L. Jaffe and X.-D. Ji, Chiral odd parton distributions and Drell-Yan processes, Nucl. Phys. B 375 (1992) 527 [INSPIRE].
A.V. Belitsky and D. Mueller, Scale dependence of the chiral-odd twist-3 distributions hL (x) and e(x), Nucl. Phys. B 503 (1997) 279 [hep-ph/9702354] [INSPIRE].
A.V. Belitsky, Leading order analysis of twist-3 space- and time-like cut vertices in QCD, Int. J. Mod. Phys. A 32 (2017) 1730018 [hep-ph/9703432] [INSPIRE].
V. Barone and R.G. Ratcliffe, Transverse Spin Physics, World Scientific, Singapore (2003) [DOI].
A. Accardi, A. Bacchetta, W. Melnitchouk and M. Schlegel, What can break the Wandzura-Wilczek relation?, JHEP 11 (2009) 093 [arXiv:0907.2942] [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].
A. Deur, S.J. Brodsky and G. F de Teramond, The spin structure of the nucleon, Rept. Prog. Phys. 82 (2019) 076201 [arXiv:1801.09154] [INSPIRE].
I.I. Balitsky and V.M. Braun, Evolution Equations for QCD String Operators, Nucl. Phys. B 311 (1989) 541 [INSPIRE].
T. Rogers, Transverse moments of TMD parton densities and ultraviolet divergences, Mod. Phys. Lett. A 35 (2020) 37. [arXiv:2008.05351] [INSPIRE]
P.J. Mulders, Lectures at the 17th Taiwan Nuclear Physics School, Institute of Physics, Academia Sinica, Taipei, August 25–28, 2014, http://www.nat.vu.nl/∼mulders/.
R.D. Tangerman, Higher-twist correlations in polarized hadrons, Ph.D. thesis, Free University Amsterdam (1996), https://inis.iaea.org/collection/NCLCollectionStore/_Public/28/015/28015381.pdf.
P. Ball and V.M. Braun, Higher twist distribution amplitudes of vector mesons in QCD: twist-4 distributions and meson mass corrections, Nucl. Phys. B 543 (1999) 201 [hep-ph/9810475] [INSPIRE].
S.J. Brodsky, H.-C. Pauli and S.S. Pinsky, Quantum chromodynamics and other field theories on the light cone, Phys. Rept. 301 (1998) 299 [hep-ph/9705477] [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].
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: 2106.15849
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
Kumano, S., Song, QT. Twist-2 relation and sum rule for tensor-polarized parton distribution functions of spin-1 hadrons. J. High Energ. Phys. 2021, 141 (2021). https://doi.org/10.1007/JHEP09(2021)141
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2021)141