Abstract
We consider extensions of the soft-gluon effective coupling that generalize the Catani-Marchesini-Webber (CMW) coupling in the context of soft-gluon resummation beyond the next-to-leading logarithmic accuracy. Starting from the probability density of correlated soft emission in d dimensions we introduce a class of soft couplings relevant for resummed QCD calculations of hard-scattering observables. We show that at the conformal point, where the d-dimensional QCD β function vanishes, all these effective couplings are equal and they are also equal to the cusp anomalous dimension. We present explicit results in d dimensions for the soft-emission probability density and the soft couplings at the second-order in the QCD coupling αS. In d = 4 dimensions we obtain the explicit relation between the soft couplings at \( \mathcal{O} \)(\( {\alpha}_{\textrm{S}}^3 \)). Finally, we study the structure of the soft coupling in the large-nF limit and we present explicit expressions to all orders in perturbation theory. We also check that, at the conformal point, our large-nF results agree with the known result of the cusp anomalous dimension.
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References
G. Luisoni and S. Marzani, QCD resummation for hadronic final states, J. Phys. G 42 (2015) 103101 [arXiv:1505.04084] [INSPIRE].
T. Becher, A. Broggio and A. Ferroglia, Introduction to soft-collinear effective theory, Springer, Cham, Switzerland (2015) [https://doi.org/10.1007/978-3-319-14848-9] [INSPIRE].
S. Catani, B.R. Webber and G. Marchesini, QCD coherent branching and semiinclusive processes at large x, Nucl. Phys. B 349 (1991) 635 [INSPIRE].
G.F. Sterman, Summation of large corrections to short distance hadronic cross-sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].
S. Catani and L. Trentadue, Resummation of the QCD perturbative series for hard processes, Nucl. Phys. B 327 (1989) 323 [INSPIRE].
S. Catani and L. Trentadue, Comment on QCD exponentiation at large x, Nucl. Phys. B 353 (1991) 183 [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].
J. Kodaira and L. Trentadue, Summing soft emission in QCD, Phys. Lett. B 112 (1982) 66 [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].
T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].
S. Catani, L. Trentadue, G. Turnock and B.R. Webber, Resummation of large logarithms in e+e− event shape distributions, Nucl. Phys. B 407 (1993) 3 [INSPIRE].
M. Dasgupta and G.P. Salam, Resummed event shape variables in DIS, JHEP 08 (2002) 032 [hep-ph/0208073] [INSPIRE].
A. Banfi, G.P. Salam and G. Zanderighi, Generalized resummation of QCD final state observables, Phys. Lett. B 584 (2004) 298 [hep-ph/0304148] [INSPIRE].
R. Bonciani, S. Catani, M.L. Mangano and P. Nason, Sudakov resummation of multiparton QCD cross-sections, Phys. Lett. B 575 (2003) 268 [hep-ph/0307035] [INSPIRE].
A. Banfi, G.P. Salam and G. Zanderighi, Principles of general final-state resummation and automated implementation, JHEP 03 (2005) 073 [hep-ph/0407286] [INSPIRE].
A. Buckley et al., General-purpose event generators for LHC physics, Phys. Rept. 504 (2011) 145 [arXiv:1101.2599] [INSPIRE].
J.M. Campbell et al., Event generators for high-energy physics experiments, in the proceedings of the Snowmass 2021, (2022) [arXiv:2203.11110] [INSPIRE].
G. Bewick, S. Ferrario Ravasio, P. Richardson and M.H. Seymour, Logarithmic accuracy of angular-ordered parton showers, JHEP 04 (2020) 019 [arXiv:1904.11866] [INSPIRE].
M. Dasgupta et al., Parton showers beyond leading logarithmic accuracy, Phys. Rev. Lett. 125 (2020) 052002 [arXiv:2002.11114] [INSPIRE].
F. Herren et al., A new approach to color-coherent parton evolution, JHEP 10 (2023) 091 [arXiv:2208.06057] [INSPIRE].
A. Banfi, B.K. El-Menoufi and P.F. Monni, The Sudakov radiator for jet observables and the soft physical coupling, JHEP 01 (2019) 083 [arXiv:1807.11487] [INSPIRE].
S. Catani, D. De Florian and M. Grazzini, Soft-gluon effective coupling and cusp anomalous dimension, Eur. Phys. J. C 79 (2019) 685 [arXiv:1904.10365] [INSPIRE].
A. Banfi, H. McAslan, P.F. Monni and G. Zanderighi, A general method for the resummation of event-shape distributions in e+e− annihilation, JHEP 05 (2015) 102 [arXiv:1412.2126] [INSPIRE].
G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov evolution kernels of parton distributions, Mod. Phys. Lett. A 4 (1989) 1257 [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].
A. Vogt, S. Moch and J.A.M. Vermaseren, The three-loop splitting functions in QCD: the singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].
S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [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].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [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].
J. Blümlein, P. Marquard, C. Schneider and K. Schönwald, The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements, Nucl. Phys. B 971 (2021) 115542 [arXiv:2107.06267] [INSPIRE].
J.M. Henn, G.P. Korchemsky and B. Mistlberger, The full four-loop cusp anomalous dimension in N = 4 super Yang-Mills and QCD, JHEP 04 (2020) 018 [arXiv:1911.10174] [INSPIRE].
A. von Manteuffel, E. Panzer and R.M. Schabinger, Cusp and collinear anomalous dimensions in four-loop QCD from form factors, Phys. Rev. Lett. 124 (2020) 162001 [arXiv:2002.04617] [INSPIRE].
C.G. Bollini and J.J. Giambiagi, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].
J.F. Ashmore, A method of gauge invariant regularization, Lett. Nuovo Cim. 4 (1972) 289 [INSPIRE].
G.M. Cicuta and E. Montaldi, Analytic renormalization via continuous space dimension, Lett. Nuovo Cim. 4 (1972) 329 [INSPIRE].
R. Gastmans and R. Meuldermans, Dimensional regularization of the infrared problem, Nucl. Phys. B 63 (1973) 277 [INSPIRE].
Z. Bern, V. Del Duca, W.B. Kilgore and C.R. Schmidt, The infrared behavior of one loop QCD amplitudes at next-to-next-to leading order, Phys. Rev. D 60 (1999) 116001 [hep-ph/9903516] [INSPIRE].
S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys. B 591 (2000) 435 [hep-ph/0007142] [INSPIRE].
I. Feige and M.D. Schwartz, Hard-soft-collinear factorization to all orders, Phys. Rev. D 90 (2014) 105020 [arXiv:1403.6472] [INSPIRE].
S. Catani, D. Colferai and A. Torrini, Triple (and quadruple) soft-gluon radiation in QCD hard scattering, JHEP 01 (2020) 118 [arXiv:1908.01616] [INSPIRE].
S. Catani and M. Grazzini, Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond, Nucl. Phys. B 570 (2000) 287 [hep-ph/9908523] [INSPIRE].
Y. Li and H.X. Zhu, Single soft gluon emission at two loops, JHEP 11 (2013) 080 [arXiv:1309.4391] [INSPIRE].
C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization, Phys. Lett. B 727 (2013) 452 [arXiv:1309.4393] [INSPIRE].
T. Becher and M. Neubert, Drell-Yan production at small qT, transverse parton distributions and the collinear anomaly, Eur. Phys. J. C 71 (2011) 1665 [arXiv:1007.4005] [INSPIRE].
S. Catani, S. Devoto, M. Grazzini and J. Mazzitelli, Soft-parton contributions to heavy-quark production at low transverse momentum, JHEP 04 (2023) 144 [arXiv:2301.11786] [INSPIRE].
G. Somogyi, Angular integrals in d dimensions, J. Math. Phys. 52 (2011) 083501 [arXiv:1101.3557] [INSPIRE].
M. Beneke and V.M. Braun, Power corrections and renormalons in Drell-Yan production, Nucl. Phys. B 454 (1995) 253 [hep-ph/9506452] [INSPIRE].
Acknowledgments
This work is supported in part by the Swiss National Science Foundation (SNSF) under contract 200020_188464. One of us (MG) would like to thank the CERN Theoretical Physics Department for the kind hospitality and financial support during the completion of this work.
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Catani, S., de Florian, D., Devoto, S. et al. Soft-gluon effective coupling: perturbative results and the large-nF limit to all orders. J. High Energ. Phys. 2023, 217 (2023). https://doi.org/10.1007/JHEP11(2023)217
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DOI: https://doi.org/10.1007/JHEP11(2023)217