Abstract
We provide the proper definition of all the leading-twist (un)polarized gluon transverse momentum dependent parton distribution functions (TMDPDFs), by considering the Higgs boson transverse momentum distribution in hadron-hadron collisions and deriving the factorization theorem in terms of them. We show that the evolution of all the (un)polarized gluon TMDPDFs is driven by a universal evolution kernel, which can be resummed up to next-to-next-to-leading-logarithmic accuracy. Considering the proper definition of gluon TMDPDFs, we perform an explicit next-to-leading-order calculation of the unpolarized (f g1 ), linearly polarized (h ⊥ g1 ) and helicity (g g1L ) gluon TMDPDFs, and show that, as expected, they are free from rapidity divergences. As a byproduct, we obtain the Wilson coefficients of the refactorization of these TMDPDFs at large transverse momentum. In particular, the coefficient of g g1L , which has never been calculated before, constitutes a new and necessary ingredient for a reliable phenomenological extraction of this quantity, for instance at RHIC or the future AFTER@LHC or Electron-Ion Collider. The coefficients of f g1 and h ⊥ g1 have never been calculated in the present formalism, although they could be obtained by carefully collecting and recasting previous results in the new TMD formalism. We apply these results to analyze the contribution of linearly polarized gluons at different scales, relevant, for instance, for the inclusive production of the Higgs boson and the C-even pseudoscalar bottomonium state η b . Applying our resummation scheme we finally provide predictions for the Higgs boson q T -distribution at the LHC.
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References
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].
J.P. Ralston and D.E. Soper, Production of Dimuons from High-Energy Polarized Proton Proton Collisions, Nucl. Phys. B 152 (1979) 109 [INSPIRE].
J.P. Ralston and D.E. Soper, Drell-Yan Model at Measured Q−transverse: Asymptotic Smallness of One Loop Corrections, Nucl. Phys. B 172 (1980) 445 [INSPIRE].
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].
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].
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].
J.C. Collins, Foundations of perturbative QCD, Cambridge University Press, Cambridge, U.K. (2011).
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. 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].
G. Bozzi, S. Catani, D. de Florian and M. Grazzini, The q T spectrum of the Higgs boson at the LHC in QCD perturbation theory, Phys. Lett. B 564 (2003) 65 [hep-ph/0302104] [INSPIRE].
S. Catani, D. de Florian, M. Grazzini and P. Nason, Soft gluon resummation for Higgs boson production at hadron colliders, JHEP 07 (2003) 028 [hep-ph/0306211] [INSPIRE].
G. Bozzi, S. Catani, D. de Florian and M. Grazzini, Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC, Nucl. Phys. B 737 (2006) 73 [hep-ph/0508068] [INSPIRE].
G. Bozzi, S. Catani, D. de Florian and M. Grazzini, Higgs boson production at the LHC: Transverse-momentum resummation and rapidity dependence, Nucl. Phys. B 791 (2008) 1 [arXiv:0705.3887] [INSPIRE].
P. Sun, B.-W. Xiao and F. Yuan, Gluon Distribution Functions and Higgs Boson Production at Moderate Transverse Momentum, Phys. Rev. D 84 (2011) 094005 [arXiv:1109.1354] [INSPIRE].
S. Catani, E. D’Emilio and L. Trentadue, The Gluon Form-factor to Higher Orders: Gluon Gluon Annihilation at Small Q−transverse, Phys. Lett. B 211 (1988) 335 [INSPIRE].
S. Catani, D. de Florian and M. Grazzini, Universality of nonleading logarithmic contributions in transverse momentum distributions, Nucl. Phys. B 596 (2001) 299 [hep-ph/0008184] [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].
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].
C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [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].
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].
S. Mantry and F. Petriello, Factorization and Resummation of Higgs Boson Differential Distributions in Soft-Collinear Effective Theory, Phys. Rev. D 81 (2010) 093007 [arXiv:0911.4135] [INSPIRE].
T. Becher, M. Neubert and D. Wilhelm, Higgs-Boson Production at Small Transverse Momentum, JHEP 05 (2013) 110 [arXiv:1212.2621] [INSPIRE].
D. Neill, I.Z. Rothstein and V. Vaidya, The Higgs Transverse Momentum Distribution at NNLL and its Theoretical Errors, arXiv:1503.00005 [INSPIRE].
X.-d. Ji, J.-P. Ma and F. Yuan, Transverse-momentum-dependent gluon distributions and semi-inclusive processes at hadron colliders, JHEP 07 (2005) 020 [hep-ph/0503015] [INSPIRE].
R. Zhu, P. Sun and F. Yuan, Low Transverse Momentum Heavy Quark Pair Production to Probe Gluon Tomography, Phys. Lett. B 727 (2013) 474 [arXiv:1309.0780] [INSPIRE].
J.P. Ma, J.X. Wang and S. Zhao, Transverse momentum dependent factorization for quarkonium production at low transverse momentum, Phys. Rev. D 88 (2013) 014027 [arXiv:1211.7144] [INSPIRE].
G.-P. Zhang, Probing transverse momentum dependent gluon distribution functions from hadronic quarkonium pair production, Phys. Rev. D 90 (2014) 094011 [arXiv:1406.5476] [INSPIRE].
J.-W. Qiu, P. Sun, B.-W. Xiao and F. Yuan, Universal Suppression of Heavy Quarkonium Production in pA Collisions at Low Transverse Momentum, Phys. Rev. D 89 (2014) 034007 [arXiv:1310.2230] [INSPIRE].
H.T. Li, C.S. Li, D.Y. Shao, L.L. Yang and H.X. Zhu, Top quark pair production at small transverse momentum in hadronic collisions, Phys. Rev. D 88 (2013) 074004 [arXiv:1307.2464] [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].
M. Diehl, T. Kasemets and S. Keane, Correlations in double parton distributions: effects of evolution, JHEP 05 (2014) 118 [arXiv:1401.1233] [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 and C. Pisano, Polarized gluon studies with charmonium and bottomonium at LHCb and AFTER, Phys. Rev. D 86 (2012) 094007 [arXiv:1208.3642] [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].
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].
W.J. den Dunnen, J.-P. Lansberg, C. Pisano and M. Schlegel, Accessing the Transverse Dynamics and Polarization of Gluons inside the Proton at the LHC, Phys. Rev. Lett. 112 (2014) 212001 [arXiv:1401.7611] [INSPIRE].
D. Boer and W.J. den Dunnen, TMD evolution and the Higgs transverse momentum distribution, Nucl. Phys. B 886 (2014) 421 [arXiv:1404.6753] [INSPIRE].
J.R. Ellis, M.K. Gaillard and D.V. Nanopoulos, A Phenomenological Profile of the Higgs Boson, Nucl. Phys. B 106 (1976) 292 [INSPIRE].
M.A. Shifman, A.I. Vainshtein, M.B. Voloshin and V.I. Zakharov, Low-Energy Theorems for Higgs Boson Couplings to Photons, Sov. J. Nucl. Phys. 30 (1979) 711 [INSPIRE].
A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, Higgs Particles, Sov. Phys. Usp. 23 (1980) 429 [INSPIRE].
T. Inami, T. Kubota and Y. Okada, Effective Gauge Theory and the Effect of Heavy Quarks in Higgs Boson Decays, Z. Phys. C 18 (1983) 69 [INSPIRE].
M.B. Voloshin, Once Again About the Role of Gluonic Mechanism in Interaction of Light Higgs Boson with Hadrons, Sov. J. Nucl. Phys. 44 (1986) 478 [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].
M. Garcia-Echevarria, A. Idilbi and I. Scimemi, SCET, Light-Cone Gauge and the T-Wilson Lines, Phys. Rev. D 84 (2011) 011502 [arXiv:1104.0686] [INSPIRE].
L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [INSPIRE].
V. Ahrens, T. Becher, M. Neubert and L.L. Yang, Origin of the Large Perturbative Corrections to Higgs Production at Hadron Colliders, Phys. Rev. D 79 (2009) 033013 [arXiv:0808.3008] [INSPIRE].
A.V. Manohar and I.W. Stewart, The Zero-Bin and Mode Factorization in Quantum Field Theory, Phys. Rev. D 76 (2007) 074002 [hep-ph/0605001] [INSPIRE].
J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The Rapidity Renormalization Group, Phys. Rev. Lett. 108 (2012) 151601 [arXiv:1104.0881] [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].
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].
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 and P.J. Mulders, Color entanglement for azimuthal asymmetries in the Drell-Yan process, Phys. Rev. Lett. 112 (2014) 092002 [arXiv:1309.4681] [INSPIRE].
M.G. Echevarria, A. Idilbi, A. Schäfer and I. Scimemi, Model-Independent Evolution of Transverse Momentum Dependent Distribution Functions (TMDs) at NNLL, Eur. Phys. J. C 73 (2013) 2636 [arXiv:1208.1281] [INSPIRE].
J. Collins and T. Rogers, Understanding the large-distance behavior of transverse-momentum-dependent parton densities and the Collins-Soper evolution kernel, Phys. Rev. D 91 (2015) 074020 [arXiv:1412.3820] [INSPIRE].
J.P. Ma and Q. Wang, Scale Dependence of Twist-3 quark-gluon Operators for Single Spin Asymmetries, Phys. Lett. B 715 (2012) 157 [arXiv:1205.0611] [INSPIRE].
U. D’Alesio, M.G. Echevarria, S. Melis and I. Scimemi, Non-perturbative QCD effects in q T spectra of Drell-Yan and Z-boson production, JHEP 11 (2014) 098 [arXiv:1407.3311] [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. Diehl and T. Kasemets, Positivity bounds on double parton distributions, JHEP 05 (2013) 150 [arXiv:1303.0842] [INSPIRE].
D. de Florian, R. Sassot, M. Stratmann and W. Vogelsang, Evidence for polarization of gluons in the proton, Phys. Rev. Lett. 113 (2014) 012001 [arXiv:1404.4293] [INSPIRE].
A.D. Martin, W.J. Stirling, R.S. Thorne and G. Watt, Parton distributions for the LHC, Eur. Phys. J. C 63 (2009) 189 [arXiv:0901.0002] [INSPIRE].
J.P. Ma, J.X. Wang and S. Zhao, Breakdown of QCD Factorization for P-Wave Quarkonium Production at Low Transverse Momentum, Phys. Lett. B 737 (2014) 103 [arXiv:1405.3373] [INSPIRE].
A.V. Konychev and P.M. Nadolsky, Universality of the Collins-Soper-Sterman nonperturbative function in gauge boson production, Phys. Lett. B 633 (2006) 710 [hep-ph/0506225] [INSPIRE].
J.-w. Qiu and X.-f. Zhang, Role of the nonperturbative input in QCD resummed Drell-Yan Q T distributions, Phys. Rev. D 63 (2001) 114011 [hep-ph/0012348] [INSPIRE].
D. Binosi and L. Theussl, JaxoDraw: A graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun. 161 (2004) 76 [hep-ph/0309015] [INSPIRE].
R. Mertig, M. Böhm and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
T. Becher, M. Neubert and G. Xu, Dynamical Threshold Enhancement and Resummation in Drell-Yan Production, JHEP 07 (2008) 030 [arXiv:0710.0680] [INSPIRE].
A.V. Belitsky, Leading order analysis of the twist-three space-like and time-like cut vertices in QCD, hep-ph/9703432.
Y. Schröder and M. Steinhauser, Four-loop decoupling relations for the strong coupling, JHEP 01 (2006) 051 [hep-ph/0512058] [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and C. Sturm, QCD decoupling at four loops, Nucl. Phys. B 744 (2006) 121 [hep-ph/0512060] [INSPIRE].
M. Krämer, E. Laenen and M. Spira, Soft gluon radiation in Higgs boson production at the LHC, Nucl. Phys. B 511 (1998) 523 [hep-ph/9611272] [INSPIRE].
K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser, Hadronic Higgs decay to order α 4 S , Phys. Rev. Lett. 79 (1997) 353 [hep-ph/9705240] [INSPIRE].
R.V. Harlander, Virtual corrections to gg → H to two loops in the heavy top limit, Phys. Lett. B 492 (2000) 74 [hep-ph/0007289] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [INSPIRE].
R.N. Lee, A.V. Smirnov and V.A. Smirnov, Analytic Results for Massless Three-Loop Form Factors, JHEP 04 (2010) 020 [arXiv:1001.2887] [INSPIRE].
T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, The quark and gluon form factors to three loops in QCD through to O(ϵ2), JHEP 11 (2010) 102 [arXiv:1010.4478] [INSPIRE].
A. Idilbi, X.-d. Ji, J.-P. Ma and F. Yuan, Threshold resummation for Higgs production in effective field theory, Phys. Rev. D 73 (2006) 077501 [hep-ph/0509294] [INSPIRE].
A. Idilbi, X.-d. Ji and F. Yuan, Transverse momentum distribution through soft-gluon resummation in effective field theory, Phys. Lett. B 625 (2005) 253 [hep-ph/0507196] [INSPIRE].
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Echevarria, M.G., Kasemets, T., Mulders, P.J. et al. QCD evolution of (un)polarized gluon TMDPDFs and the Higgs q T -distribution. J. High Energ. Phys. 2015, 158 (2015). https://doi.org/10.1007/JHEP07(2015)158
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DOI: https://doi.org/10.1007/JHEP07(2015)158