Abstract
We evaluate the light-cone operator product expansion for unpolarized transverse momentum dependent (TMD) operator in the background-field technique up twist-3 inclusively. The next-to-leading order (NLO) matching coefficient for the Sivers function is derived. The method, as well as many details of the calculation are presented.
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References
J. Collins, Foundations of perturbative QCD, Cambridge University Press, Cambridge U.K. (2013).
M.G. Echevarria, A. Idilbi and I. Scimemi, Factorization theorem for Drell-Yan at low q T and transverse momentum distributions on-the-light-cone, JHEP 07 (2012) 002 [arXiv:1111.4996] [INSPIRE].
M.G. Echevarria, A. Idilbi and I. Scimemi, Unified treatment of the QCD evolution of all (un-)polarized transverse momentum dependent functions: Collins function as a study case, Phys. Rev. D 90 (2014) 014003 [arXiv:1402.0869] [INSPIRE].
A. Bacchetta et al., Semi-inclusive deep inelastic scattering at small transverse momentum, JHEP 02 (2007) 093 [hep-ph/0611265] [INSPIRE].
A.V. Efremov and O.V. Teryaev, On spin effects in quantum chromodynamics, Sov. J. Nucl. Phys. 36 (1982) 140 [INSPIRE].
D.W. Sivers, Single spin production asymmetries from the hard scattering of point-like constituents, Phys. Rev. D 41 (1990) 83 [INSPIRE].
HERMES collaboration, Observation of the naive-T-odd Sivers effect in deep-inelastic scattering, Phys. Rev. Lett. 103 (2009) 152002 [arXiv:0906.3918] [INSPIRE].
COMPASS collaboration, Collins and Sivers asymmetries for pions and kaons in muon-deuteron DIS, Phys. Lett. B 673 (2009) 127 [arXiv:0802.2160] [INSPIRE].
COMPASS collaboration, II — Experimental investigation of transverse spin asymmetries in μ-p SIDIS processes: Sivers asymmetries, Phys. Lett. B 717 (2012) 383 [arXiv:1205.5122] [INSPIRE].
Jefferson Lab Hall A collaboration, Single spin asymmetries in charged pion production from semi-inclusive deep inelastic scattering on a transversely polarized 3 He target, Phys. Rev. Lett. 107 (2011) 072003 [arXiv:1106.0363] [INSPIRE].
STAR collaboration, Measurement of the transverse single-spin asymmetry in p ↑ + p → W ± /Z 0 at RHIC, Phys. Rev. Lett. 116 (2016) 132301 [arXiv:1511.06003] [INSPIRE].
STAR collaboration, Measurement of transverse single spin asymmetries in π 0 production from p ↑ + p and p ↑ + A collisions at STAR, PoS(DIS2016)212 [arXiv:1805.08875] [INSPIRE].
PHENIX collaboration, J. Bok, Transverse single spin asymmetries of charged hadron from p+p and p+A Collisions at PHENIX,PoS(DIS2018)153.
A. Accardi et al., Electron ion collider: the next QCD frontier, Eur. Phys. J. A 52 (2016) 268 [arXiv:1212.1701] [INSPIRE].
M. Anselmino et al., Extracting the Sivers function from polarized SIDIS data and making predictions, Phys. Rev. D 72 (2005) 094007 [Erratum ibid. D 72 (2005) 099903] [hep-ph/0507181] [INSPIRE].
Z.-B. Kang and J.-W. Qiu, Testing the time-reversal modified universality of the Sivers function, Phys. Rev. Lett. 103 (2009) 172001 [arXiv:0903.3629] [INSPIRE].
M.G. Echevarria, A. Idilbi, Z.-B. Kang and I. Vitev, QCD evolution of the Sivers asymmetry, Phys. Rev. D 89 (2014) 074013 [arXiv:1401.5078] [INSPIRE].
M. Anselmino et al., Study of the sign change of the Sivers function from STAR Collaboration W/Z production data, JHEP 04 (2017) 046 [arXiv:1612.06413] [INSPIRE].
A. Martin, F. Bradamante and V. Barone, Direct extraction of the Sivers distributions from spin asymmetries in pion and kaon leptoproduction, Phys. Rev. D 95 (2017) 094024 [arXiv:1701.08283] [INSPIRE].
M. Boglione, U. D’Alesio, C. Flore and J.O. Gonzalez-Hernandez, Assessing signals of TMD physics in SIDIS azimuthal asymmetries and in the extraction of the Sivers function, JHEP 07 (2018) 148 [arXiv:1806.10645] [INSPIRE].
E.C. Aschenauer, U. D’Alesio and F. Murgia, TMDs and SSAs in hadronic interactions, Eur. Phys. J. A 52 (2016) 156 [arXiv:1512.05379] [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].
X. Ji, J.-W. Qiu, W. Vogelsang and F. Yuan, A unified picture for single transverse-spin asymmetries in hard processes, Phys. Rev. Lett. 97 (2006) 082002 [hep-ph/0602239] [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].
Y. Koike, W. Vogelsang and F. Yuan, On the relation between mechanisms for single-transverse-spin asymmetries, Phys. Lett. B 659 (2008) 878 [arXiv:0711.0636] [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].
P. Sun and F. Yuan, Transverse momentum dependent evolution: matching semi-inclusive deep inelastic scattering processes to Drell-Yan and W/Z boson production, Phys. Rev. D 88 (2013) 114012 [arXiv:1308.5003] [INSPIRE].
L.-Y. Dai, Z.-B. Kang, A. Prokudin and I. Vitev, Next-to-leading order transverse momentum-weighted Sivers asymmetry in semi-inclusive deep inelastic scattering: the role of the three-gluon correlator, Phys. Rev. D 92 (2015) 114024 [arXiv:1409.5851] [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Transverse momentum distribution in Drell-Yan pair and W and Z boson production, Nucl. Phys. B 250 (1985) 199 [INSPIRE].
I. Scimemi and A. Vladimirov, Analysis of vector boson production within TMD factorization, Eur. Phys. J. C 78 (2018) 89 [arXiv:1706.01473] [INSPIRE].
A. Bacchetta et al., Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production, JHEP 06 (2017) 081 [arXiv:1703.10157] [INSPIRE].
M.G. Echevarria, I. Scimemi and A. Vladimirov, Unpolarized transverse momentum dependent parton distribution and fragmentation functions at next-to-next-to-leading order, JHEP 09 (2016) 004 [arXiv:1604.07869] [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].
A. Bacchetta, D. Boer, M. Diehl and P.J. Mulders, Matches and mismatches in the descriptions of semi-inclusive processes at low and high transverse momentum, JHEP 08 (2008) 023 [arXiv:0803.0227] [INSPIRE].
A. Bacchetta and A. Prokudin, Evolution of the helicity and transversity Transverse-Momentum-Dependent parton distributions, Nucl. Phys. B 875 (2013) 536 [arXiv:1303.2129] [INSPIRE].
M.G. Echevarria, T. Kasemets, P.J. Mulders and C. Pisano, QCD evolution of (un)polarized gluon TMDPDFs and the Higgs q T -distribution, JHEP 07 (2015) 158 [Erratum ibid. 1705 (2017) 073] [arXiv:1502.05354] [INSPIRE].
M.G. Echevarria, I. Scimemi and A. Vladimirov, Transverse momentum dependent fragmentation function at next-to-next-to-leading order, Phys. Rev. D 93 (2016) 011502 [Erratum ibid. D 94 (2016) 099904] [arXiv:1509.06392] [INSPIRE].
D. Gutierrez-Reyes, I. Scimemi and A. Vladimirov, Transverse momentum dependent transversely polarized distributions at next-to-next-to-leading-order, JHEP 07 (2018) 172 [arXiv:1805.07243] [INSPIRE].
X.P. Chai, K.B. Chen and J.P. Ma, A note on pretzelosity TMD parton distribution, Phys. Lett. B 789 (2019) 360 [arXiv:1808.10560] [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].
I. Scimemi and A. Vladimirov, Power corrections and renormalons in Transverse Momentum Distributions, JHEP 03 (2017) 002 [arXiv:1609.06047] [INSPIRE].
I. Balitsky and A. Tarasov, Power corrections to TMD factorization for Z-boson production, JHEP 05 (2018) 150 [arXiv:1712.09389] [INSPIRE].
I. Balitsky and A. Tarasov, Rapidity evolution of gluon TMD from low to moderate x, JHEP 10 (2015) 017 [arXiv:1505.02151] [INSPIRE].
I. Scimemi and A. Vladimirov, Systematic analysis of double-scale evolution, JHEP 08 (2018) 003 [arXiv:1803.11089] [INSPIRE].
B.S. DeWitt, Quantum theory of gravity. 2. The manifestly covariant theory, Phys. Rev. 162 (1967) 1195 [INSPIRE].
I.I. Balitsky and V.M. Braun, Evolution equations for QCD string operators, Nucl. Phys. B 311 (1989) 541 [INSPIRE].
I. Balitsky and A. Tarasov, Gluon TMD in particle production from low to moderate x, JHEP 06 (2016) 164 [arXiv:1603.06548] [INSPIRE].
A. Vladimirov, Structure of rapidity divergences in multi-parton scattering soft factors, JHEP 04 (2018) 045 [arXiv:1707.07606] [INSPIRE].
J.C. Collins, Leading twist single transverse-spin asymmetries: Drell-Yan and deep inelastic scattering, Phys. Lett. B 536 (2002) 43 [hep-ph/0204004] [INSPIRE].
W. Vogelsang and F. Yuan, Single-transverse spin asymmetries: From DIS to hadronic collisions, Phys. Rev. D 72 (2005) 054028 [hep-ph/0507266] [INSPIRE].
D. Boer, L. Gamberg, B. Musch and A. Prokudin, Bessel-weighted asymmetries in semi inclusive deep inelastic scattering, JHEP 10 (2011) 021 [arXiv:1107.5294] [INSPIRE].
D. Boer, Investigating the origins of transverse spin asymmetries at RHIC, Phys. Rev. D 60 (1999) 014012 [hep-ph/9902255] [INSPIRE].
M. Anselmino, U. D’Alesio and F. Murgia, Transverse single spin asymmetries in Drell-Yan processes, Phys. Rev. D 67 (2003) 074010 [hep-ph/0210371] [INSPIRE].
A.V. Efremov et al., Sivers effect in semi-inclusive DIS and in the Drell-Yan process, Phys. Lett. B 612 (2005) 233 [hep-ph/0412353] [INSPIRE].
S. Arnold, A. Metz and M. Schlegel, Dilepton production from polarized hadron hadron collisions, Phys. Rev. D 79 (2009) 034005 [arXiv:0809.2262] [INSPIRE].
S.J. Brodsky, D.S. Hwang and I. Schmidt, Final state interactions and single spin asymmetries in semiinclusive deep inelastic scattering, Phys. Lett. B 530 (2002) 99 [hep-ph/0201296] [INSPIRE].
S.J. Brodsky, D.S. Hwang and I. Schmidt, Initial state interactions and single spin asymmetries in Drell-Yan processes, Nucl. Phys. B 642 (2002) 344 [hep-ph/0206259] [INSPIRE].
A.A. Vladimirov, Correspondence between soft and rapidity anomalous dimensions, Phys. Rev. Lett. 118 (2017) 062001 [arXiv:1610.05791] [INSPIRE].
V. Bertone, I. Scimemi and A. Vladimirov, Exraction of unpolarized TMDPDF and rapidity anomalous dimension from the global fit of Drell-Yan data, in preparation.
T. Gehrmann et al., Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [arXiv:1004.3653] [INSPIRE].
R.D. Tangerman and P.J. Mulders, Intrinsic transverse momentum and the polarized Drell-Yan process, Phys. Rev. D 51 (1995) 3357 [hep-ph/9403227] [INSPIRE].
K. Goeke, A. Metz and M. Schlegel, Parameterization of the quark-quark correlator of a spin-1/2 hadron, Phys. Lett. B 618 (2005) 90 [hep-ph/0504130] [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. D 86 (2012) 119902] [arXiv:0909.3410] [INSPIRE].
M.G. Echevarr´ıa, A. Idilbi and I. Scimemi, Soft and collinear factorization and transverse momentum dependent parton distribution functions, Phys. Lett. B 726 (2013) 795 [arXiv:1211.1947] [INSPIRE].
J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A formalism for the systematic treatment of rapidity logarithms in quantum field theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].
T. Becher and M. Neubert, Drell-Yan production at small q T , transverse parton distributions and the collinear anomaly, Eur. Phys. J. C 71 (2011) 1665 [arXiv:1007.4005] [INSPIRE].
M.G. Echevarria, I. Scimemi and A. Vladimirov, Universal transverse momentum dependent soft function at NNLO, Phys. Rev. D 93 (2016) 054004 [arXiv:1511.05590] [INSPIRE].
A.V. Belitsky, X. Ji and F. Yuan, Final state interactions and gauge invariant parton distributions, Nucl. Phys. B 656 (2003) 165 [hep-ph/0208038] [INSPIRE].
A. Idilbi and I. Scimemi, Singular and regular gauges in soft collinear effective theory: the introduction of the new Wilson line T, Phys. Lett. B 695 (2011) 463 [arXiv:1009.2776] [INSPIRE].
R.L. Jaffe, Parton distribution functions for twist four, Nucl. Phys. B 229 (1983) 205 [INSPIRE].
G.A. Chirilli, Y.V. Kovchegov and D.E. Wertepny, Regularization of the light-cone gauge gluon propagator singularities using sub-gauge conditions, JHEP 12 (2015) 138 [arXiv:1508.07962] [INSPIRE].
A.P. Chen, J.P. Ma and G.P. Zhang, One-loop corrections of single spin asymmetries in semi-inclusive DIS, Phys. Rev. D 97 (2018) 054003 [arXiv:1708.09091] [INSPIRE].
M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].
C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An effective field theory for collinear and soft gluons: heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].
C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
L.F. Abbott, The background field method beyond one loop, Nucl. Phys. B 185 (1981) 189 [INSPIRE].
L.F. Abbott, Introduction to the background field method, Acta Phys. Polon. B 13 (1982) 33 [INSPIRE].
A.A. Vladimirov, TMD PDFs in the Laguerre polynomial basis, JHEP 08 (2014) 089 [arXiv:1402.3182] [INSPIRE].
S.M. Aybat and T.C. Rogers, TMD parton distribution and fragmentation functions with QCD evolution, Phys. Rev. D 83 (2011) 114042 [arXiv:1101.5057] [INSPIRE].
Z.-B. Kang and J.-W. Qiu, Evolution of twist-3 multi-parton correlation functions relevant to single transverse-spin asymmetry, Phys. Rev. D 79 (2009) 016003 [arXiv:0811.3101] [INSPIRE].
H. Beppu, Y. Koike, K. Tanaka and S. Yoshida, Contribution of twist-3 multi-gluon correlation functions to single spin asymmetry in semi-inclusive deep inelastic scattering, Phys. Rev. D 82 (2010) 054005 [arXiv:1007.2034] [INSPIRE].
A.P. Chen, J.P. Ma and G.P. Zhang, One-loop corrections to single spin asymmetries at twist-3 in Drell-Yan processes, Phys. Rev. D 95 (2017) 074005 [arXiv:1607.08676] [INSPIRE].
A.V. Efremov and O.V. Teryaev, The transversal polarization in quantum chromodynamics, Sov. J. Nucl. Phys. 39 (1984) 962 [INSPIRE].
J.-w. Qiu and G.F. Sterman, Single transverse spin asymmetries, Phys. Rev. Lett. 67 (1991) 2264 [INSPIRE].
Z.-B. Kang and J.-W. Qiu, QCD evolution of naive-time-reversal-odd parton distribution functions, Phys. Lett. B 713 (2012) 273 [arXiv:1205.1019] [INSPIRE].
A. Schafer and J. Zhou, A note on the scale evolution of the ETQS function T F (x, x), Phys. Rev. D 85 (2012) 117501 [arXiv:1203.5293] [INSPIRE].
P. Cvitanovic, Group theory: Birdtracks, Lie’s and exceptional groups, Princeton University Press, Princeton U.S.A. (2008).
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Scimemi, I., Tarasov, A. & Vladimirov, A. Collinear matching for Sivers function at next-to-leading order. J. High Energ. Phys. 2019, 125 (2019). https://doi.org/10.1007/JHEP05(2019)125
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DOI: https://doi.org/10.1007/JHEP05(2019)125