Abstract
We discuss the structure of rapidity divergences that are presented in the soft factors of transverse momentum dependent (TMD) factorization theorems. To provide the discussion on the most general level we consider soft factors for multi-parton scattering. We show that the rapidity divergences are result of the gluon exchanges with the distant transverse plane, and are structurally equivalent to the ultraviolet divergences. It allows to formulate and to prove the renormalization theorem for rapidity divergences. The proof is made with the help the conformal transformation which maps rapidity divergences to ultraviolet divergences. The theorem is the systematic form of the factorization of rapidity divergences, which is required for the definition of TMD parton distributions. In particular, the definition of multi parton distributions is presented. The equivalence of ultraviolet and rapidity divergences leads to the exact relation between soft and rapidity anomalous dimensions. Using this relation we derive the rapidity anomalous dimension at the three-loop order.
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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].
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].
J. Collins, Foundations of perturbative QCD. Cambridge University Press, (2013).
J. Collins, New definition of TMD parton densities, Int. J. Mod. Phys. Conf. Ser. 4 (2011) 85 [arXiv:1107.4123] [INSPIRE].
M.G. Echevarría, 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].
J.C. Collins and F.V. Tkachov, Breakdown of dimensional regularization in the Sudakov problem, Phys. Lett. B 294 (1992) 403 [hep-ph/9208209] [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].
V.A. Smirnov, Asymptotic expansions of two loop Feynman diagrams in the Sudakov limit, Phys. Lett. B 404 (1997) 101 [hep-ph/9703357] [INSPIRE].
T. Becher and G. Bell, Analytic Regularization in Soft-Collinear Effective Theory, Phys. Lett. B 713 (2012) 41 [arXiv:1112.3907] [INSPIRE].
T. Becher, M. Neubert and D. Wilhelm, Higgs-Boson Production at Small Transverse Momentum, JHEP 05 (2013) 110 [arXiv:1212.2621] [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].
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].
M.G. Echevarría, I. Scimemi and A. Vladimirov, Universal transverse momentum dependent soft function at NNLO, Phys. Rev. D 93 (2016) 054004 [arXiv:1511.05590] [INSPIRE].
M.G. Echevarría, 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].
T. Gehrmann, T. Lübbert and L.L. Yang, Transverse parton distribution functions at next-to-next-to-leading order: the quark-to-quark case, Phys. Rev. Lett. 109 (2012) 242003 [arXiv:1209.0682] [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].
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].
Y. Li, D. Neill and H.X. Zhu, An Exponential Regulator for Rapidity Divergences, [arXiv:1604.00392] [INSPIRE].
M. Diehl, D. Ostermeier and A. Schäfer, Elements of a theory for multiparton interactions in QCD, JHEP 03 (2012) 089 [Erratum ibid. 03 (2016) 001] [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].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, N-Jettiness: An Inclusive Event Shape to Veto Jets, Phys. Rev. Lett. 105 (2010) 092002 [arXiv:1004.2489] [INSPIRE].
T.T. Jouttenus, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The Soft Function for Exclusive N-Jet Production at Hadron Colliders, Phys. Rev. D 83 (2011) 114030 [arXiv:1102.4344] [INSPIRE].
A.A. Vladimirov, Correspondence between Soft and Rapidity Anomalous Dimensions, Phys. Rev. Lett. 118 (2017) 062001 [arXiv:1610.05791] [INSPIRE].
S. Catani and M.H. Seymour, A general algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. B 510 (1998) 503] [hep-ph/9605323] [INSPIRE].
M. Beneke, P. Falgari and C. Schwinn, Soft radiation in heavy-particle pair production: All-order colour structure and two-loop anomalous dimension, Nucl. Phys. B 828 (2010) 69 [arXiv:0907.1443] [INSPIRE].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [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].
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].
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].
I. Scimemi and A. Vladimirov, Power corrections and renormalons in Transverse Momentum Distributions, JHEP 03 (2017) 002 [arXiv:1609.06047] [INSPIRE].
A. Vladimirov, Soft factors for double parton scattering at NNLO, JHEP 12 (2016) 038 [arXiv:1608.04920] [INSPIRE].
A.A. Vladimirov, Exponentiation for products of Wilson lines within the generating function approach, JHEP 06 (2015) 120 [arXiv:1501.03316] [INSPIRE].
A.A. Vladimirov, Generating function for web diagrams, Phys. Rev. D 90 (2014) 066007 [arXiv:1406.6253] [INSPIRE].
O. Erdoğan and G. Sterman, Gauge Theory Webs and Surfaces, Phys. Rev. D 91 (2015) 016003 [arXiv:1112.4564] [INSPIRE].
V.S. Dotsenko and S.N. Vergeles, Renormalizability of Phase Factors in the Nonabelian Gauge Theory, Nucl. Phys. B 169 (1980) 527 [INSPIRE].
R.A. Brandt, F. Neri and M.-a. Sato, Renormalization of Loop Functions for All Loops, Phys. Rev. D 24 (1981) 879 [INSPIRE].
Y. Hatta, Relating e + e − annihilation to high energy scattering at weak and strong coupling, JHEP 11 (2008) 057 [arXiv:0810.0889] [INSPIRE].
V.M. Braun and A.N. Manashov, Two-loop evolution equations for light-ray operators, Phys. Lett. B 734 (2014) 137 [arXiv:1404.0863] [INSPIRE].
T. Banks and A. Zaks, On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions, Nucl. Phys. B 196 (1982) 189 [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [INSPIRE].
J.M. Drummond, G.P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].
M.G. Echevarría, 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].
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 Dimensions for Transverse-Momentum Resummation, Phys. Rev. Lett. 118 (2017) 022004 [arXiv:1604.01404] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The three loop splitting functions in QCD: The nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
O. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multileg scattering, Phys. Rev. Lett. 117 (2016) 172002 [arXiv:1507.00047] [INSPIRE].
S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].
L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [arXiv:0910.3653] [INSPIRE].
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Vladimirov, A. Structure of rapidity divergences in multi-parton scattering soft factors. J. High Energ. Phys. 2018, 45 (2018). https://doi.org/10.1007/JHEP04(2018)045
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DOI: https://doi.org/10.1007/JHEP04(2018)045