Abstract
Transverse momentum dependent (TMD) parton distribution functions (PDFs), TMDs for short, are defined as the Fourier transform of matrix elements of non-local combinations of quark and gluon fields. The nonlocality is bridged by gauge links, which for TMDs have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. It is possible, however, to construct sets of universal TMDs of which in a given process particular combinations are needed with calculable, process-dependent, coefficients. This occurs for both T-odd and T-even TMDs, including also the unpolarized quark and gluon TMDs. This extends the by now well-known example of T-odd TMDs that appear with opposite sign in single-spin azimuthal asymmetries in semi-inclusive deep inelastic scattering or in the Drell-Yan process. In this paper we ana-lyze the cases where TMDs enter multiplied by products of two transverse momenta, which includes besides the p T -broadening observable, also instances with rank-2 structures. To experimentally demonstrate the process dependence of the latter cases requires measurements of second harmonic azimuthal asymmetries, while the p T -broadening will require measurements of processes beyond semi-inclusive deep inelastic scattering or the Drell-Yan process. Furthermore, we propose specific quantities that will allow for theoretical studies of the process dependence of TMDs using lattice QCD calculations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 (1982) 445 [INSPIRE].
J.C. Collins and D.E. Soper, Back-To-Back Jets in QCD, Nucl. Phys. B 193 (1981) 381 [Erratum ibid. B 213 (1983) 545] [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].
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].
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].
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].
C.J. Bomhof, P.J. Mulders and F. Pijlman, The construction of gauge-links in arbitrary hard processes, Eur. Phys. J. C 47 (2006) 147 [hep-ph/0601171] [INSPIRE].
C.J. Bomhof and P.J. Mulders, Non-universality of transverse momentum dependent parton distribution functions, Nucl. Phys. B 795 (2008) 409 [arXiv:0709.1390] [INSPIRE].
M.G.A. Buffing, A. Mukherjee and P.J. Mulders, Generalized universality of higher transverse moments of quark transverse momentum dependent correlators, Phys. Rev. D 86 (2012) 074030 [arXiv:1207.3221] [INSPIRE].
M.G.A. Buffing, A. Mukherjee and P.J. Mulders, Generalized universality of definite rank gluon transverse momentum dependent correlators, Phys. Rev. D 88 (2013) 054027 [arXiv:1306.5897] [INSPIRE].
R.K. Ellis, W. Furmanski and R. Petronzio, Power Corrections to the Parton Model in QCD, Nucl. Phys. B 207 (1982) 1 [INSPIRE].
R.K. Ellis, W. Furmanski and R. Petronzio, Unraveling Higher Twists, Nucl. Phys. B 212 (1983) 29 [INSPIRE].
J.P. Ralston and D.E. Soper, Production of Dimuons from High-Energy Polarized Proton Proton Collisions, Nucl. Phys. B 152 (1979) 109 [INSPIRE].
P.J. Mulders and R.D. Tangerman, The complete tree level result up to order 1/Q for polarized deep inelastic leptoproduction, Nucl. Phys. B 461 (1996) 197 [Erratum ibid. B 484 (1997) 538] [hep-ph/9510301] [INSPIRE].
D. Boer and P.J. Mulders, Time reversal odd distribution functions in leptoproduction, Phys. Rev. D 57 (1998) 5780 [hep-ph/9711485] [INSPIRE].
J. Collins, Foundations of perturbative QCD, Cambridge University Press, (2011).
M.G. Echevarría, 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].
A.V. Efremov and O.V. Teryaev, On Spin Effects in Quantum Chromodynamics, Sov. J. Nucl. Phys. 36 (1982) 140 [INSPIRE].
A.V. Efremov and O.V. Teryaev, QCD Asymmetry and Polarized Hadron Structure Functions, Phys. Lett. B 150 (1985) 383 [INSPIRE].
J.-w. Qiu and G.F. Sterman, Single transverse spin asymmetries, Phys. Rev. Lett. 67 (1991) 2264 [INSPIRE].
J.-w. Qiu and G.F. Sterman, Single transverse spin asymmetries in direct photon production, Nucl. Phys. B 378 (1992) 52 [INSPIRE].
J.-w. Qiu and G.F. Sterman, Single transverse spin asymmetries in hadronic pion production, Phys. Rev. D 59 (1999) 014004 [hep-ph/9806356] [INSPIRE].
Y. Kanazawa and Y. Koike, Chiral odd contribution to single transverse spin asymmetry in hadronic pion production, Phys. Lett. B 478 (2000) 121 [hep-ph/0001021] [INSPIRE].
C.J. Bomhof, P.J. Mulders and F. Pijlman, Gauge link structure in quark-quark correlators in hard processes, Phys. Lett. B 596 (2004) 277 [hep-ph/0406099] [INSPIRE].
A. Bacchetta, C.J. Bomhof, P.J. Mulders and F. Pijlman, Single spin asymmetries in hadron-hadron collisions, Phys. Rev. D 72 (2005) 034030 [hep-ph/0505268] [INSPIRE].
C.J. Bomhof and P.J. Mulders, Gluonic pole cross-sections and single spin asymmetries in hadron-hadron scattering, JHEP 02 (2007) 029 [hep-ph/0609206] [INSPIRE].
F. Dominguez, C. Marquet, B.-W. Xiao and F. Yuan, Universality of Unintegrated Gluon Distributions at small x, Phys. Rev. D 83 (2011) 105005 [arXiv:1101.0715] [INSPIRE].
A. Schäfer and J. Zhou, Process dependent nuclear k ⊥ broadening effect, Phys. Rev. D 88 (2013) 074012 [arXiv:1305.5042] [INSPIRE].
B.-W. Xiao and F. Yuan, Non-Universality of Transverse Momentum Dependent Parton Distributions at Small-x, Phys. Rev. Lett. 105 (2010) 062001 [arXiv:1003.0482] [INSPIRE].
B.-W. Xiao and F. Yuan, Initial and Final State Interaction Effects in Small-x Quark Distributions, Phys. Rev. D 82 (2010) 114009 [arXiv:1008.4432] [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, Average transverse momentum quantities approaching the lightfront, arXiv:1409.8317.
Fermilab E706 collaboration, L. Apanasevich et al., Evidence for parton k T effects in high p T particle production, Phys. Rev. Lett. 81 (1998) 2642 [hep-ex/9711017] [INSPIRE].
L. Apanasevich et al., k T effects in direct photon production, Phys. Rev. D 59 (1999) 074007 [hep-ph/9808467] [INSPIRE].
M. Aghasyan et al., Studies of Transverse Momentum Dependent Parton Distributions and Bessel Weighting, JHEP 03 (2015) 039 [arXiv:1409.0487] [INSPIRE].
D. Boer, P.J. Mulders and C. Pisano, Dijet imbalance in hadronic collisions, Phys. Rev. D 80 (2009) 094017 [arXiv:0909.4652] [INSPIRE].
C. Pisano, D. Boer, S.J. Brodsky, M.G.A. Buffing and P.J. Mulders, Linear polarization of gluons and photons in unpolarized collider experiments, JHEP 10 (2013) 024 [arXiv:1307.3417] [INSPIRE].
T.C. Rogers and P.J. Mulders, No Generalized TMD-Factorization in Hadro-Production of High Transverse Momentum Hadrons, Phys. Rev. D 81 (2010) 094006 [arXiv:1001.2977] [INSPIRE].
A. Bacchetta, M. Diehl, K. Goeke, A. Metz, P.J. Mulders and M. Schlegel, Semi-inclusive deep inelastic scattering at small transverse momentum, JHEP 02 (2007) 093 [hep-ph/0611265] [INSPIRE].
P.J. Mulders and J. Rodrigues, Transverse momentum dependence in gluon distribution and fragmentation functions, Phys. Rev. D 63 (2001) 094021 [hep-ph/0009343] [INSPIRE].
J.C. Collins, What exactly is a parton density?, Acta Phys. Polon. B 34 (2003) 3103 [hep-ph/0304122] [INSPIRE].
J. Collins, New definition of TMD parton densities, Int. J. Mod. Phys. Conf. Ser. 4 (2011) 85 [arXiv:1107.4123] [INSPIRE].
S. Catani and M. Grazzini, QCD transverse-momentum resummation in gluon fusion processes, Nucl. Phys. B 845 (2011) 297 [arXiv:1011.3918] [INSPIRE].
S. Catani and M. Grazzini, Higgs Boson Production at Hadron Colliders: Hard-Collinear Coefficients at the NNLO, Eur. Phys. J. C 72 (2012) 2013 [Erratum ibid. C 72 (2012) 2132] [arXiv:1106.4652] [INSPIRE].
M.G.A. Buffing and P.J. Mulders, Gauge links for transverse momentum dependent correlators at tree-level, JHEP 07 (2011) 065 [arXiv:1105.4804] [INSPIRE].
B.U. Musch, P. Hägler, M. Engelhardt, J.W. Negele and A. Schäfer, Sivers and Boer-Mulders observables from lattice QCD, Phys. Rev. D 85 (2012) 094510 [arXiv:1111.4249] [INSPIRE].
M. Engelhardt, B. Musch, P. Hägler, J. Negele and A. Schäfer, Lattice study of the Boer-Mulders transverse momentum distribution in the pion, PoS(LATTICE 2013)284 [arXiv:1310.8335] [INSPIRE].
D. Boer, W.J. den Dunnen, C. Pisano and M. Schlegel, Determining the Higgs spin and parity in the diphoton decay channel, Phys. Rev. Lett. 111 (2013) 032002 [arXiv:1304.2654] [INSPIRE].
M.G. Echevarría, 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 [arXiv:1502.05354] [INSPIRE].
D. Boer and C. Pisano, Impact of gluon polarization on Higgs boson plus jet production at the LHC, Phys. Rev. D 91 (2015) 074024 [arXiv:1412.5556] [INSPIRE].
D. Boer, W.J. den Dunnen, C. Pisano, M. Schlegel and W. Vogelsang, Linearly Polarized Gluons and the Higgs Transverse Momentum Distribution, Phys. Rev. Lett. 108 (2012) 032002 [arXiv:1109.1444] [INSPIRE].
D. Boer, S.J. Brodsky, P.J. Mulders and C. Pisano, Direct Probes of Linearly Polarized Gluons inside Unpolarized Hadrons, Phys. Rev. Lett. 106 (2011) 132001 [arXiv:1011.4225] [INSPIRE].
A. Metz and J. Zhou, Distribution of linearly polarized gluons inside a large nucleus, Phys. Rev. D 84 (2011) 051503 [arXiv:1105.1991] [INSPIRE].
M. Luo, J.-w. Qiu and G.F. Sterman, Nuclear dependence at large transverse momentum, Phys. Lett. B 279 (1992) 377 [INSPIRE].
M. Luo, J.-w. Qiu and G.F. Sterman, Twist four nuclear parton distributions from photoproduction, Phys. Rev. D 49 (1994) 4493 [INSPIRE].
M. Luo, J.-w. Qiu and G.F. Sterman, Anomalous nuclear enhancement in deeply inelastic scattering and photoproduction, Phys. Rev. D 50 (1994) 1951 [INSPIRE].
X.-f. Guo, Jet broadening in deeply inelastic scattering, Phys. Rev. D 58 (1998) 114033 [hep-ph/9804234] [INSPIRE].
Z.-B. Kang and J.-W. Qiu, Transverse momentum broadening of vector boson production in high energy nuclear collisions, Phys. Rev. D 77 (2008) 114027 [arXiv:0802.2904] [INSPIRE].
Z.-B. Kang and J.-W. Qiu, Nuclear modification of vector boson production in proton-lead collisions at the LHC, Phys. Lett. B 721 (2013) 277 [arXiv:1212.6541] [INSPIRE].
Z.-B. Kang, I. Vitev and H. Xing, Dihadron momentum imbalance and correlations in d+Au collisions, Phys. Rev. D 85 (2012) 054024 [arXiv:1112.6021] [INSPIRE].
H. Xing, Z.-B. Kang, I. Vitev and E. Wang, Transverse momentum imbalance of back-to-back particle production in p+A and e+A collisions, Phys. Rev. D 86 (2012) 094010 [arXiv:1206.1826] [INSPIRE].
Z.-B. Kang, I. Vitev and H. Xing, Multiple scattering effects on inclusive particle production in the large-x regime, Phys. Rev. D 88 (2013) 054010 [arXiv:1307.3557] [INSPIRE].
R.L. Jaffe, Parton Distribution Functions for Twist Four, Nucl. Phys. B 229 (1983) 205 [INSPIRE].
J.-W. Qiu, Twist Four Contributions to the Parton Structure Functions, Phys. Rev. D 42 (1990) 30 [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1503.03760
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Boer, D., Buffing, M.G.A. & Mulders, P.J. Operator analysis of p T -widths of TMDs. J. High Energ. Phys. 2015, 53 (2015). https://doi.org/10.1007/JHEP08(2015)053
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2015)053