Abstract
We analyze in detail the most singular behaviour of processes involving triple-collinear splittings with massive particles in the quasi-collinear limit, and present compact expressions for the splitting amplitudes and the corresponding splitting kernels at the squared-amplitude level. Our expressions fully agree with well-known triple-collinear splittings in the massless limit, which are used as a guide to achieve the final expressions. These results are important to quantify dominant mass effects in many observables, and constitute an essential ingredient of current high-precision computational frameworks for collider phenomenology.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1989) 1 [hep-ph/0409313] [INSPIRE].
S. Catani, D. de Florian and G. Rodrigo, Space-like (versus time-like) collinear limits in QCD: Is factorization violated?, JHEP 07 (2012) 026 [arXiv:1112.4405] [INSPIRE].
J.R. Forshaw, M.H. Seymour and A. Siodmok, On the Breaking of Collinear Factorization in QCD, JHEP 11 (2012) 066 [arXiv:1206.6363] [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].
G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
G.M. Cicuta and E. Montaldi, Analytic renormalization via continuous space dimension, Lett. Nuovo Cim. 4 (1972) 329 [INSPIRE].
S. Catani and M.H. Seymour, A general algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [hep-ph/9605323] [INSPIRE].
S. Frixione, Z. Kunszt and A. Signer, Three jet cross-sections to next-to-leading order, Nucl. Phys. B 467 (1996) 399 [hep-ph/9512328] [INSPIRE].
W.J. Torres Bobadilla et al., May the four be with you: Novel IR-subtraction methods to tackle NNLO calculations, Eur. Phys. J. C 81 (2021) 250 [arXiv:2012.02567] [INSPIRE].
F. Caola et al., The path forward to N3LO, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2203.06730] [INSPIRE].
N. Agarwal, L. Magnea, C. Signorile-Signorile and A. Tripathi, The infrared structure of perturbative gauge theories, Phys. Rept. 994 (2023) 1 [arXiv:2112.07099] [INSPIRE].
S. Catani and P.K. Dhani, Collinear functions for QCD resummations, JHEP 03 (2023) 200 [arXiv:2208.05840] [INSPIRE].
P.K. Dhani, Azimuthally-correlated contributions to QCD transverse-momentum resummation at \( \mathcal{O}\left({\alpha}_{\textrm{S}}^2\right) \), PoS LL2022 (2022) 045 [arXiv:2208.07321] [INSPIRE].
S. Höche, Introduction to parton-shower event generators, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders, Boulder, U.S.A, June 2–27 (2014), p. 235–295 [https://doi.org/10.1142/9789814678766_0005] [arXiv:1411.4085] [INSPIRE].
Z. Bern, G. Chalmers, L.J. Dixon and D.A. Kosower, One loop N gluon amplitudes with maximal helicity violation via collinear limits, Phys. Rev. Lett. 72 (1994) 2134 [hep-ph/9312333] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
Z. Bern and G. Chalmers, Factorization in one loop gauge theory, Nucl. Phys. B 447 (1995) 465 [hep-ph/9503236] [INSPIRE].
D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [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, 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].
S. Catani, D. de Florian and G. Rodrigo, The triple collinear limit of one loop QCD amplitudes, Phys. Lett. B 586 (2004) 323 [hep-ph/0312067] [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].
J.M. Campbell and E.W.N. Glover, Double unresolved approximations to multiparton scattering amplitudes, Nucl. Phys. B 527 (1998) 264 [hep-ph/9710255] [INSPIRE].
S. Catani and M. Grazzini, Collinear factorization and splitting functions for next-to-next-to-leading order QCD calculations, Phys. Lett. B 446 (1999) 143 [hep-ph/9810389] [INSPIRE].
Z. Bern, V. Del Duca and C.R. Schmidt, The infrared behavior of one loop gluon amplitudes at next-to-next-to-leading order, Phys. Lett. B 445 (1998) 168 [hep-ph/9810409] [INSPIRE].
D.A. Kosower and P. Uwer, One loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515] [INSPIRE].
G.F.R. Sborlini, D. de Florian and G. Rodrigo, Double collinear splitting amplitudes at next-to-leading order, JHEP 01 (2014) 018 [arXiv:1310.6841] [INSPIRE].
V. Del Duca, A. Frizzo and F. Maltoni, Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit, Nucl. Phys. B 568 (2000) 211 [hep-ph/9909464] [INSPIRE].
T.G. Birthwright, E.W.N. Glover, V.V. Khoze and P. Marquard, Multi-gluon collinear limits from MHV diagrams, JHEP 05 (2005) 013 [hep-ph/0503063] [INSPIRE].
T.G. Birthwright, E.W.N. Glover, V.V. Khoze and P. Marquard, Collinear limits in QCD from MHV rules, JHEP 07 (2005) 068 [hep-ph/0505219] [INSPIRE].
M. Delto and K. Melnikov, Integrated triple-collinear counter-terms for the nested soft-collinear subtraction scheme, JHEP 05 (2019) 148 [arXiv:1901.05213] [INSPIRE].
V. Del Duca et al., Tree-level splitting amplitudes for a gluon into four collinear partons, JHEP 10 (2020) 093 [arXiv:2007.05345] [INSPIRE].
G.F.R. Sborlini, D. de Florian and G. Rodrigo, Triple collinear splitting functions at NLO for scattering processes with photons, JHEP 10 (2014) 161 [arXiv:1408.4821] [INSPIRE].
G.F.R. Sborlini, D. de Florian and G. Rodrigo, Polarized triple-collinear splitting functions at NLO for processes with photons, JHEP 03 (2015) 021 [arXiv:1409.6137] [INSPIRE].
S. Badger, F. Buciuni and T. Peraro, One-loop triple collinear splitting amplitudes in QCD, JHEP 09 (2015) 188 [arXiv:1507.05070] [INSPIRE].
M. Czakon and S. Sapeta, Complete collection of one-loop triple-collinear splitting operators for dimensionally-regulated QCD, JHEP 07 (2022) 052 [arXiv:2204.11801] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Two-loop g → gg splitting amplitudes in QCD, JHEP 08 (2004) 012 [hep-ph/0404293] [INSPIRE].
S.D. Badger and E.W.N. Glover, Two loop splitting functions in QCD, JHEP 07 (2004) 040 [hep-ph/0405236] [INSPIRE].
L.J. Dixon, E. Herrmann, K. Yan and H.X. Zhu, Soft gluon emission at two loops in full color, JHEP 05 (2020) 135 [arXiv:1912.09370] [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].
A. Vogt, S. Moch and J. Vermaseren, Photon-parton splitting functions at the next-to-next-to-leading order of QCD, Acta Phys. Polon. B 37 (2006) 683 [hep-ph/0511112] [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].
S. Höche and S. Prestel, Triple collinear emissions in parton showers, Phys. Rev. D 96 (2017) 074017 [arXiv:1705.00742] [INSPIRE].
A.M. Snigirev, Triple parton scattering in collinear approximation of perturbative QCD, Phys. Rev. D 94 (2016) 034026 [arXiv:1603.08187] [INSPIRE].
Y.L. Dokshitzer, V.A. Khoze and S.I. Troian, On specific QCD properties of heavy quark fragmentation (’dead cone’), J. Phys. G 17 (1991) 1602 [INSPIRE].
S. Catani, S. Dittmaier and Z. Trocsanyi, One loop singular behavior of QCD and SUSY QCD amplitudes with massive partons, Phys. Lett. B 500 (2001) 149 [hep-ph/0011222] [INSPIRE].
S. Keller and E. Laenen, Next-to-leading order cross-sections for tagged reactions, Phys. Rev. D 59 (1999) 114004 [hep-ph/9812415] [INSPIRE].
S. Dittmaier, A general approach to photon radiation off fermions, Nucl. Phys. B 565 (2000) 69 [hep-ph/9904440] [INSPIRE].
M. Roth, Precise predictions for four fermion production in electron positron annihilation, M.Sc. thesis, Departement Physik (D-PHYS), Eidgenössische Technische Hochschule (ETH) Zürich, CH-8093 Zürich, Switzerland (1999) [hep-ph/0008033] [INSPIRE].
L. Phaf and S. Weinzierl, Dipole formalism with heavy fermions, JHEP 04 (2001) 006 [hep-ph/0102207] [INSPIRE].
S. Catani, S. Dittmaier, M.H. Seymour and Z. Trocsanyi, The dipole formalism for next-to-leading order QCD calculations with massive partons, Nucl. Phys. B 627 (2002) 189 [hep-ph/0201036] [INSPIRE].
P. Nogueira, Automatic Feynman Graph Generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].
J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016) 432 [arXiv:1601.01167] [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, FeynCalc 9.3: New features and improvements, Comput. Phys. Commun. 256 (2020) 107478 [arXiv:2001.04407] [INSPIRE].
T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
P.K. Dhani, G. Rodrigo and G. Sborlini, Ancillary files for “Triple-collinear splittings with massive particles”, 2023 [https://doi.org/10.5281/zenodo.10246009].
D. de Florian, G.F.R. Sborlini and G. Rodrigo, QED corrections to the Altarelli-Parisi splitting functions, Eur. Phys. J. C 76 (2016) 282 [arXiv:1512.00612] [INSPIRE].
D. de Florian, G.F.R. Sborlini and G. Rodrigo, Two-loop QED corrections to the Altarelli-Parisi splitting functions, JHEP 10 (2016) 056 [arXiv:1606.02887] [INSPIRE].
D. de Florian, M. Der and I. Fabre, QCD⊕QED NNLO corrections to Drell Yan production, Phys. Rev. D 98 (2018) 094008 [arXiv:1805.12214] [INSPIRE].
A. A H et al., NNLO QCD⊕QED corrections to Higgs production in bottom quark annihilation, Phys. Rev. D 100 (2019) 114016 [arXiv:1906.09028] [INSPIRE].
A. A H, P. Mukherjee and V. Ravindran, Infrared structure of SU(N) × U(1) gauge theory to three loops, JHEP 08 (2020) 156 [arXiv:1912.13386] [INSPIRE].
R. Maciula and A. Szczurek, Far-forward production of charm mesons and neutrinos at forward physics facilities at the LHC and the intrinsic charm in the proton, Phys. Rev. D 107 (2023) 034002 [arXiv:2210.08890] [INSPIRE].
E. Craft, M. Gonzalez, K. Lee, B. Mecaj and I. Moult, The 1 → 3 Massive Splitting Functions from QCD Factorization and SCET, [arXiv:2310.06736] [INSPIRE].
Acknowledgments
We warmly thank the Galileo Galilei Institute (Firenze), where this work was completed, and S. Catani, for hospitality and discussing during the Workshop Theory Challenges in the Precision Era of the Large Hadron Collider, A. Szczurek and M. Wiesemann for their fruitful comments when presenting this work at EPS-HEP 2023, and the authors of ref. [64] for a careful comparison of the quark-initiated massive triple-collinear splitting kernels. This work is supported by the Spanish Government (Agencia Estatal de Investigación MCIN/AEI/10.13039/501100011033) Grants No. PID2020-114473GB-I00, PID2022-141910NB-I00 and Generalitat Valenciana Grant No. PROMETEO/2021/071. The work of PKD is supported by European Commission MSCA Action COLLINEAR-FRACTURE, Grant Agreement No. 101108573. The work of GS is partially supported by EU Horizon 2020 research and innovation programme STRONG-2020 project under Grant Agreement No. 824093 and H2020-MSCA-COFUND-2020 USAL4EXCELLENCE-PROOPI-391 project under Grant Agreement No 101034371.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2310.05803
Rights and permissions
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.
About this article
Cite this article
Dhani, P.K., Rodrigo, G. & Sborlini, G.F.R. Triple-collinear splittings with massive particles. J. High Energ. Phys. 2023, 188 (2023). https://doi.org/10.1007/JHEP12(2023)188
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2023)188