Abstract
We show at NNLO that the soft factors for double parton scattering (DPS) for both integrated and unintegrated kinematics, can be presented entirely in the terms of the soft factor for single Drell-Yan process, i.e. the transverse momentum dependent (TMD) soft factor. Using the linearity of the logarithm of TMD soft factor in rapidity divergences, we decompose the DPS soft factor matrices into a product of matrices with rapidity divergences in given sectors, and thus, define individual double parton distributions at NNLO. The rapidity anomalous dimension matrices for double parton distributions are presented in the terms of TMD rapidity anomalous dimension. The analysis is done using the generating function approach to web diagrams. Significant part of the result is obtained from the symmetry properties of web diagrams without referring to explicit expressions or a particular rapidity regularization scheme. Additionally, we present NNLO expression for the web diagram generating function for Wilson lines with two light-like directions.
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References
CDF collaboration, F. Abe et al., Double parton scattering in \( \overline{p}p \) collisions at \( \sqrt{s}=1.8 \) TeV, Phys. Rev. D 56 (1997) 3811 [INSPIRE].
UA2 collaboration, J. Alitti et al., A Study of multi-jet events at the CERN \( \overline{p}p \) collider and a search for double parton scattering, Phys. Lett. B 268 (1991) 145 [INSPIRE].
D0 collaboration, V.M. Abazov et al., Double parton interactions in γ + 3 jet events in pp − bar collisions \( \sqrt{s}=1.96 \) TeV., Phys. Rev. D 81 (2010) 052012 [arXiv:0912.5104] [INSPIRE].
A. Del Fabbro and D. Treleani, A Double parton scattering background to Higgs boson production at the LHC, Phys. Rev. D 61 (2000) 077502 [hep-ph/9911358] [INSPIRE].
D. Bandurin, G. Golovanov and N. Skachkov, Double parton interactions as a background to associated HW production at the Tevatron, JHEP 04 (2011) 054 [arXiv:1011.2186] [INSPIRE].
J.R. Gaunt, C.-H. Kom, A. Kulesza and W.J. Stirling, Same-sign W pair production as a probe of double parton scattering at the LHC, Eur. Phys. J. C 69 (2010) 53 [arXiv:1003.3953] [INSPIRE].
M. Diehl, D. Ostermeier and A. Schafer, Elements of a theory for multiparton interactions in QCD, JHEP 03 (2012) 089 [arXiv:1111.0910] [INSPIRE].
A.V. Manohar and W.J. Waalewijn, A QCD Analysis of Double Parton Scattering: Color Correlations, Interference Effects and Evolution, Phys. Rev. D 85 (2012) 114009 [arXiv:1202.3794] [INSPIRE].
M. Diehl, J.R. Gaunt, D. Ostermeier, P. Plößl and A. Schäfer, Cancellation of Glauber gluon exchange in the double Drell-Yan process, JHEP 01 (2016) 076 [arXiv:1510.08696] [INSPIRE].
J.C. Collins, Foundations of perturbative QCD, Cambridge University Press, Cambridge U.K. (2011).
A.M. Snigirev, Double parton distributions in the leading logarithm approximation of perturbative QCD, Phys. Rev. D 68 (2003) 114012 [hep-ph/0304172] [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.A. Vladimirov, Generating function for web diagrams, Phys. Rev. D 90 (2014) 066007 [arXiv:1406.6253] [INSPIRE].
A.A. Vladimirov, Exponentiation for products of Wilson lines within the generating function approach, JHEP 06 (2015) 120 [arXiv:1501.03316] [INSPIRE].
M. Diehl and A. Schafer, Theoretical considerations on multiparton interactions in QCD, Phys. Lett. B 698 (2011) 389 [arXiv:1102.3081] [INSPIRE].
A.V. Manohar and W.J. Waalewijn, What is Double Parton Scattering?, Phys. Lett. B 713 (2012) 196 [arXiv:1202.5034] [INSPIRE].
T. Becher, A. Broggio and A. Ferroglia, Introduction to Soft-Collinear Effective Theory, Lect. Notes Phys. 896 (2015) 1 [arXiv:1410.1892] [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, 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].
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. Lee and G.F. Sterman, Momentum Flow Correlations from Event Shapes: Factorized Soft Gluons and Soft-Collinear Effective Theory, Phys. Rev. D 75 (2007) 014022 [hep-ph/0611061] [INSPIRE].
J.R. Gaunt, Single Perturbative Splitting Diagrams in Double Parton Scattering, JHEP 01 (2013) 042 [arXiv:1207.0480] [INSPIRE].
M. Diehl and J.R. Gaunt, Double parton scattering in the ultraviolet: addressing the double counting problem, arXiv:1603.05468 [INSPIRE].
T. Kasemets and M. Diehl, Angular correlations in the double Drell-Yan process, JHEP 01 (2013) 121 [arXiv:1210.5434] [INSPIRE].
J.G.M. Gatheral, Exponentiation of Eikonal Cross-sections in Nonabelian Gauge Theories, Phys. Lett. B 133 (1983) 90 [INSPIRE].
J. Frenkel and J.C. Taylor, Nonabelian Eikonal Exponentiation, Nucl. Phys. B 246 (1984) 231 [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].
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 [arXiv:1509.06392] [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, arXiv:1604.07869 [INSPIRE].
T. Lübbert, J. Oredsson and M. Stahlhofen, Rapidity renormalized TMD soft and beam functions at two loops, JHEP 03 (2016) 168 [arXiv:1602.01829] [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].
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].
T. Gehrmann, T. Luebbert and L.L. Yang, Calculation of the transverse parton distribution functions at next-to-next-to-leading order, JHEP 06 (2014) 155 [arXiv:1403.6451] [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].
Y. Li and H.X. Zhu, Bootstrapping rapidity anomalous dimension for transverse-momentum resummation, Submitted to: Phys. Rev. Lett. (2016) [arXiv:1604.01404] [INSPIRE].
A.A. Vladimirov, Soft-/rapidity-anomalous dimensions correspondence, arXiv:1610.05791 [INSPIRE].
J.C. Collins and A. Metz, Universality of soft and collinear factors in hard-scattering factorization, Phys. Rev. Lett. 93 (2004) 252001 [hep-ph/0408249] [INSPIRE].
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Vladimirov, A. Soft factors for double parton scattering at NNLO. J. High Energ. Phys. 2016, 38 (2016). https://doi.org/10.1007/JHEP12(2016)038
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DOI: https://doi.org/10.1007/JHEP12(2016)038