Abstract
We compute the next-to-leading order (NLO) QCD corrections to the 1 → 2 splitting amplitudes in different dimensional regularization (DREG) schemes. Besides recovering previously known results, we explore new DREG schemes and analyze their consistency by comparing the divergent structure with the expected behavior predicted by Catani’s formula. Through the introduction of scalar-gluons, we show the relation among splittings matrices computed using different schemes. Also, we extended this analysis to cover the double collinear limit of scattering amplitudes in the context of QCD + QED.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Bollini and J. Giambiagi, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].
G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].
S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop anomalous dimension matrix for soft gluon exchange, Phys. Rev. Lett. 97 (2006) 072001 [hep-ph/0606254] [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, L. Magnea and G.F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP 08 (2008) 022 [arXiv:0805.3515] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722] [INSPIRE].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [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].
L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [arXiv:0910.3653] [INSPIRE].
S. Catani and M. 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].
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of hard processes in QCD, in Perturbative quantum chromodynamics, A.H. Mueller ed., Adv. Ser. Direct. High Energy Phys. 5 (1988) 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].
S. Catani, D. de Florian and G. Rodrigo, Factorization violation in the multiparton collinear limit, PoS (LL2012) 035 [arXiv:1211.7274] [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].
F.A. Berends and W. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [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].
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].
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 and P. Uwer, One loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515] [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. Badger and E.N. Glover, Two loop splitting functions in QCD, JHEP 07 (2004) 040 [hep-ph/0405236] [INSPIRE].
D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [INSPIRE].
J.M. Campbell and E.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].
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. Birthwright, E.N. Glover, V. Khoze and P. Marquard, Multi-gluon collinear limits from MHV diagrams, JHEP 05 (2005) 013 [hep-ph/0503063] [INSPIRE].
T. Birthwright, E.N. Glover, V. Khoze and P. Marquard, Collinear limits in QCD from MHV rules, JHEP 07 (2005) 068 [hep-ph/0505219] [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].
S. Catani, M. Seymour and Z. Trócsányi, Regularization scheme independence and unitarity in QCD cross-sections, Phys. Rev. D 55 (1997) 6819 [hep-ph/9610553] [INSPIRE].
Z. Kunszt, A. Signer and Z. Trócsányi, One loop helicity amplitudes for all 2 → 2 processes in QCD and N = 1 supersymmetric Yang-Mills theory, Nucl. Phys. B 411 (1994) 397 [hep-ph/9305239] [INSPIRE].
R. Harlander, P. Kant, L. Mihaila and M. Steinhauser, Dimensional reduction applied to QCD at three loops, JHEP 09 (2006) 053 [hep-ph/0607240] [INSPIRE].
W.B. Kilgore, Regularization schemes and higher order corrections, Phys. Rev. D 83 (2011) 114005 [arXiv:1102.5353] [INSPIRE].
A. Signer and D. Stöckinger, Factorization and regularization by dimensional reduction, Phys. Lett. B 626 (2005) 127 [hep-ph/0508203] [INSPIRE].
A. Signer and D. Stöckinger, Using dimensional reduction for hadronic collisions, Nucl. Phys. B 808 (2009) 88 [arXiv:0807.4424] [INSPIRE].
R. Gastmans and R. Meuldermans, Dimensional regularization of the infrared problem, Nucl. Phys. B 63 (1973) 277 [INSPIRE].
W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. B 84 (1979) 193 [INSPIRE].
D. Capper, D. Jones and P. van Nieuwenhuizen, Regularization by dimensional reduction of supersymmetric and nonsupersymmetric gauge theories, Nucl. Phys. B 167 (1980) 479 [INSPIRE].
R. Harlander, P. Kant, L. Mihaila and M. Steinhauser, Dimensional reduction applied to QCD at higher orders, arXiv:0706.2982 [INSPIRE].
R. Mertig, M. Böhm and A. Denner, FEYN CALC: computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
K. Chetyrkin and F. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
A. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
A. Smirnov and V. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184 (2013) 2820 [arXiv:1302.5885] [INSPIRE].
S. Catani and M. Seymour, The dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order, Phys. Lett. B 378 (1996) 287 [hep-ph/9602277] [INSPIRE].
D. Pritchard and W.J. Stirling, QCD calculations in the light cone gauge. 1, Nucl. Phys. B 165 (1980) 237 [INSPIRE].
S. Catani, P. Draggiotis and G. Rodrigo, Recursion relations for the multiparton collinear limit and splitting functions, PoS (LL2012) 054 [arXiv:1210.0698] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1310.6841
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Sborlini, G.F.R., de Florian, D. & Rodrigo, G. Double collinear splitting amplitudes at next-to-leading order. J. High Energ. Phys. 2014, 18 (2014). https://doi.org/10.1007/JHEP01(2014)018
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2014)018