Abstract
We study the perturbative structure of threshold enhanced logarithms in the coefficient functions of deep inelastic scattering (DIS) and semi-inclusive e+e− annihilation (SIA) processes and setup a framework to sum them up to all orders in perturbation theory. Threshold logarithms show up as the distributions ((1−z)−1 logi(1−z))+ from the soft plus virtual (SV) and as logarithms logi(1−z) from next to SV (NSV) contributions. We use the Sudakov differential and the renormalisation group equations along with the factorisation properties of parton level cross sections to obtain the resummed result which predicts SV as well as next to SV contributions to all orders in strong coupling constant. In Mellin N space, we resum the large logarithms of the form logi(N) keeping 1/N corrections. In particular, the towers of logarithms, each of the form \( {a}_s^n/{N}^{\alpha }{\log}^{2n-\alpha }(N),{a}_s^n/{N}^{\alpha }{\log}^{2n-1-\alpha }(N)\cdots \) etc for α = 0, 1, are summed to all orders in as.
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References
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].
T. D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].
J. A. M. Vermaseren, A. Vogt and S. Moch, The Third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B 724 (2005) 3 [hep-ph/0504242] [INSPIRE].
G. Soar, S. Moch, J. A. M. Vermaseren and A. Vogt, On Higgs-exchange DIS, physical evolution kernels and fourth-order splitting functions at large x, Nucl. Phys. B 832 (2010) 152 [arXiv:0912.0369] [INSPIRE].
J. Ablinger et al., The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function F2(x, Q2) and Transversity, Nucl. Phys. B 886 (2014) 733 [arXiv:1406.4654] [INSPIRE].
S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [Erratum ibid. 111 (2013) 199905] [arXiv:0901.0722] [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].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
A. H. Ajjath, 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].
A. H. Ajjath et al., NNLO QCD⊕QED corrections to Higgs production in bottom quark annihilation, Phys. Rev. D 100 (2019) 114016 [arXiv:1906.09028] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops, Phys. Rev. Lett. 114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].
B. Mistlberger, Higgs boson production at hadron colliders at N3 LO in QCD, JHEP 05 (2018) 028 [arXiv:1802.00833] [INSPIRE].
C. Duhr, F. Dulat and B. Mistlberger, Higgs Boson Production in Bottom-Quark Fusion to Third Order in the Strong Coupling, Phys. Rev. Lett. 125 (2020) 051804 [arXiv:1904.09990] [INSPIRE].
R. Hamberg, W. L. van Neerven and T. Matsuura, A complete calculation of the order \( \${\alpha}_s^2 \) correction to the Drell-Yan K factor, Nucl. Phys. B 359 (1991) 343 [Erratum ibid. 644 (2002) 403] [INSPIRE].
R. V. Harlander and W. B. Kilgore, Next-to-next-to-leading order Higgs production at hadron colliders, Phys. Rev. Lett. 88 (2002) 201801 [hep-ph/0201206] [INSPIRE].
C. Duhr, F. Dulat and B. Mistlberger, Drell-Yan Cross Section to Third Order in the Strong Coupling Constant, Phys. Rev. Lett. 125 (2020) 172001 [arXiv:2001.07717] [INSPIRE].
H. M. Georgi, S. L. Glashow, M. E. Machacek and D. V. Nanopoulos, Higgs Bosons from Two Gluon Annihilation in Proton Proton Collisions, Phys. Rev. Lett. 40 (1978) 692 [INSPIRE].
D. Graudenz, M. Spira and P. M. Zerwas, QCD corrections to Higgs boson production at proton proton colliders, Phys. Rev. Lett. 70 (1993) 1372 [INSPIRE].
A. Djouadi, M. Spira and P. M. Zerwas, Production of Higgs bosons in proton colliders: QCD corrections, Phys. Lett. B 264 (1991) 440 [INSPIRE].
M. Spira, A. Djouadi, D. Graudenz and P. M. Zerwas, Higgs boson production at the LHC, Nucl. Phys. B 453 (1995) 17 [hep-ph/9504378] [INSPIRE].
S. Catani, D. de Florian and M. Grazzini, Higgs production in hadron collisions: Soft and virtual QCD corrections at NNLO, JHEP 05 (2001) 025 [hep-ph/0102227] [INSPIRE].
R. V. Harlander and W. B. Kilgore, Soft and virtual corrections to proton proton → H + x at NNLO, Phys. Rev. D 64 (2001) 013015 [hep-ph/0102241] [INSPIRE].
C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
S. Catani, D. de Florian, M. Grazzini and P. Nason, Soft gluon resummation for Higgs boson production at hadron colliders, JHEP 07 (2003) 028 [hep-ph/0306211] [INSPIRE].
V. Ravindran, J. Smith and W. L. van Neerven, NNLO corrections to the total cross-section for Higgs boson production in hadron hadron collisions, Nucl. Phys. B 665 (2003) 325 [hep-ph/0302135] [INSPIRE].
S. Moch and A. Vogt, Higher-order soft corrections to lepton pair and Higgs boson production, Phys. Lett. B 631 (2005) 48 [hep-ph/0508265] [INSPIRE].
V. Ravindran, Higher-order threshold effects to inclusive processes in QCD, Nucl. Phys. B 752 (2006) 173 [hep-ph/0603041] [INSPIRE].
D. de Florian and J. Mazzitelli, A next-to-next-to-leading order calculation of soft-virtual cross sections, JHEP 12 (2012) 088 [arXiv:1209.0673] [INSPIRE].
M. Bonvini, R. D. Ball, S. Forte, S. Marzani and G. Ridolfi, Updated Higgs cross section at approximate N3 LO, J. Phys. G 41 (2014) 095002 [arXiv:1404.3204] [INSPIRE].
D. de Florian, J. Mazzitelli, S. Moch and A. Vogt, Approximate N3 LO Higgs-boson production cross section using physical-kernel constraints, JHEP 10 (2014) 176 [arXiv:1408.6277] [INSPIRE].
C. Anastasiou et al., Higgs boson gluon-fusion production at threshold in N3 LO QCD, Phys. Lett. B 737 (2014) 325 [arXiv:1403.4616] [INSPIRE].
Y. Li, A. von Manteuffel, R. M. Schabinger and H. X. Zhu, Soft-virtual corrections to Higgs production at N3 LO, Phys. Rev. D 91 (2015) 036008 [arXiv:1412.2771] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, F. Herzog and B. Mistlberger, Soft expansion of double-real-virtual corrections to Higgs production at N3 LO, JHEP 08 (2015) 051 [arXiv:1505.04110] [INSPIRE].
G. Das, S. Moch and A. Vogt, Approximate four-loop QCD corrections to the Higgs-boson production cross section, Phys. Lett. B 807 (2020) 135546 [arXiv:2004.00563] [INSPIRE].
G. Altarelli, R. Ellis and G. Martinelli, Leptoproduction and Drell-Yan Processes Beyond the Leading Approximation in Chromodynamics, Nucl. Phys. B 143 (1978) 521.
G. Altarelli, R. K. Ellis and G. Martinelli, Large Perturbative Corrections to the Drell-Yan Process in QCD, Nucl. Phys. B 157 (1979) 461 [INSPIRE].
T. Matsuura and W. L. van Neerven, Second Order Logarithmic Corrections to the Drell-Yan Cross-section, Z. Phys. C 38 (1988) 623 [INSPIRE].
T. Matsuura, S. C. van der Marck and W. L. van Neerven, The Order α− s2 Contribution to the K Factor of the Drell-Yan Process, Phys. Lett. B 211 (1988) 171 [INSPIRE].
T. Matsuura, S. C. van der Marck and W. L. van Neerven, The Calculation of the Second Order Soft and Virtual Contributions to the Drell-Yan Cross-Section, Nucl. Phys. B 319 (1989) 570 [INSPIRE].
T. Matsuura, R. Hamberg and W. L. van Neerven, The Contribution of the Gluon-gluon Subprocess to the Drell-Yan K Factor, Nucl. Phys. B 345 (1990) 331 [INSPIRE].
W. L. van Neerven and E. B. Zijlstra, The \( O\left({\alpha}_s^2\right) \) corrected Drell-Yan K factor in the DIS and MS scheme, Nucl. Phys. B 382 (1992) 11 [Erratum ibid. 680 (2004) 513] [INSPIRE].
T. Ahmed, M. Mahakhud, N. Rana and V. Ravindran, Drell-Yan Production at Threshold to Third Order in QCD, Phys. Rev. Lett. 113 (2014) 112002 [arXiv:1404.0366] [INSPIRE].
S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Threshold resummation at N3 LL accuracy and soft-virtual cross sections at N3 LO, Nucl. Phys. B 888 (2014) 75 [arXiv:1405.4827] [INSPIRE].
V. Ravindran, On Sudakov and soft resummations in QCD, Nucl. Phys. B 746 (2006) 58 [hep-ph/0512249] [INSPIRE].
T. Ahmed, N. Rana and V. Ravindran, Higgs boson production through \( b\overline{b} \) annihilation at threshold in N3 LO QCD, JHEP 10 (2014) 139 [arXiv:1408.0787] [INSPIRE].
M. C. Kumar, M. K. Mandal and V. Ravindran, Associated production of Higgs boson with vector boson at threshold N3 LO in QCD, JHEP 03 (2015) 037 [arXiv:1412.3357] [INSPIRE].
Y. Li, A. von Manteuffel, R. M. Schabinger and H. X. Zhu, N3 LO Higgs boson and Drell-Yan production at threshold: The one-loop two-emission contribution, Phys. Rev. D 90 (2014) 053006 [arXiv:1404.5839] [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, M. L. Mangano, P. Nason and L. Trentadue, The Resummation of soft gluons in hadronic collisions, Nucl. Phys. B 478 (1996) 273 [hep-ph/9604351] [INSPIRE].
S. Moch, J. A. M. Vermaseren and A. Vogt, Higher-order corrections in threshold resummation, Nucl. Phys. B 726 (2005) 317 [hep-ph/0506288] [INSPIRE].
M. Bonvini, S. Forte and G. Ridolfi, The Threshold region for Higgs production in gluon fusion, Phys. Rev. Lett. 109 (2012) 102002 [arXiv:1204.5473] [INSPIRE].
M. Bonvini and S. Marzani, Resummed Higgs cross section at N3 LL, JHEP 09 (2014) 007 [arXiv:1405.3654] [INSPIRE].
M. Bonvini and L. Rottoli, Three loop soft function for N3 LL′ gluon fusion Higgs production in soft-collinear effective theory, Phys. Rev. D 91 (2015) 051301 [arXiv:1412.3791] [INSPIRE].
M. Bonvini, S. Marzani, C. Muselli and L. Rottoli, On the Higgs cross section at N3 LO+N3 LL and its uncertainty, JHEP 08 (2016) 105 [arXiv:1603.08000] [INSPIRE].
M. Bonvini, A. S. Papanastasiou and F. J. Tackmann, Matched predictions for the \( b\overline{b}H \) cross section at the 13 TeV LHC, JHEP 10 (2016) 053 [arXiv:1605.01733] [INSPIRE].
A. H. Ajjath, A. Chakraborty, G. Das, P. Mukherjee and V. Ravindran, Resummed prediction for Higgs boson production through \( b\overline{b} \) annihilation at N3 LL, JHEP 11 (2019) 006 [arXiv:1905.03771] [INSPIRE].
M. Bonvini, Threshold resummation for Drell-Yan production: Theory and phenomenology, PoS DIS2010 (2010) 100 [arXiv:1006.5918] [INSPIRE].
M. Bonvini, Resummation of soft and hard gluon radiation in perturbative QCD, Ph.D. thesis, Genoa U., 2012. arXiv:1212.0480 [INSPIRE].
A. H. Ajjath, G. Das, M. C. Kumar, P. Mukherjee, V. Ravindran and K. Samanta, Resummed Drell-Yan cross-section at N3 LL, JHEP 10 (2020) 153 [arXiv:2001.11377] [INSPIRE].
M. Cacciari and S. Catani, Soft gluon resummation for the fragmentation of light and heavy quarks at large x, Nucl. Phys. B 617 (2001) 253 [hep-ph/0107138] [INSPIRE].
S. Moch and A. Vogt, On non-singlet physical evolution kernels and large-x coefficient functions in perturbative QCD, JHEP 11 (2009) 099 [arXiv:0909.2124] [INSPIRE].
G. Grunberg and V. Ravindran, On threshold resummation beyond leading 1-x order, JHEP 10 (2009) 055 [arXiv:0902.2702] [INSPIRE].
E. Laenen, L. Magnea and G. Stavenga, On next-to-eikonal corrections to threshold resummation for the Drell-Yan and DIS cross sections, Phys. Lett. B 669 (2008) 173 [arXiv:0807.4412] [INSPIRE].
E. Laenen, L. Magnea, G. Stavenga and C. D. White, On Next-to-Eikonal Exponentiation, Nucl. Phys. B Proc. Suppl. 205-206 (2010) 260 [arXiv:1007.0624] [INSPIRE].
E. Laenen, L. Magnea, G. Stavenga and C. D. White, Next-to-Eikonal Corrections to Soft Gluon Radiation: A Diagrammatic Approach, JHEP 01 (2011) 141 [arXiv:1010.1860] [INSPIRE].
D. Bonocore, E. Laenen, L. Magnea, L. Vernazza and C. D. White, The method of regions and next-to-soft corrections in Drell-Yan production, Phys. Lett. B 742 (2015) 375 [arXiv:1410.6406] [INSPIRE].
D. Bonocore, E. Laenen, L. Magnea, S. Melville, L. Vernazza and C. D. White, A factorization approach to next-to-leading-power threshold logarithms, JHEP 06 (2015) 008 [arXiv:1503.05156] [INSPIRE].
M. Beneke, A. Broggio, S. Jaskiewicz and L. Vernazza, Threshold factorization of the Drell-Yan process at next-to-leading power, JHEP 07 (2020) 078 [arXiv:1912.01585] [INSPIRE].
M. Beneke, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza and J. Wang, Leading-logarithmic threshold resummation of Higgs production in gluon fusion at next-to-leading power, JHEP 01 (2020) 094 [arXiv:1910.12685] [INSPIRE].
D. Bonocore, E. Laenen, L. Magnea, L. Vernazza and C. D. White, Non-abelian factorisation for next-to-leading-power threshold logarithms, JHEP 12 (2016) 121 [arXiv:1610.06842] [INSPIRE].
V. Del Duca, E. Laenen, L. Magnea, L. Vernazza and C. D. White, Universality of next-to-leading power threshold effects for colourless final states in hadronic collisions, JHEP 11 (2017) 057 [arXiv:1706.04018] [INSPIRE].
A. H. Ajjath, P. Mukherjee and V. Ravindran, On next to soft corrections to Drell-Yan and Higgs Boson productions, arXiv:2006.06726 [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [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].
A. Gonzalez-Arroyo, C. Lopez and F. J. Yndurain, Second Order Contributions to the Structure Functions in Deep Inelastic Scattering. 1. Theoretical Calculations, Nucl. Phys. B 153 (1979) 161 [INSPIRE].
G. Curci, W. Furmanski and R. Petronzio, Evolution of Parton Densities Beyond Leading Order: The Nonsinglet Case, Nucl. Phys. B 175 (1980) 27 [INSPIRE].
W. Furmanski and R. Petronzio, Singlet Parton Densities Beyond Leading Order, Phys. Lett. B 97 (1980) 437 [INSPIRE].
R. Hamberg and W. L. van Neerven, The Correct renormalization of the gluon operator in a covariant gauge, Nucl. Phys. B 379 (1992) 143 [INSPIRE].
R. K. Ellis and W. Vogelsang, The Evolution of parton distributions beyond leading order: The Singlet case, hep-ph/9602356 [INSPIRE].
J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel and C. Schneider, The three-loop splitting functions \( {P}_{qg}^{(2)} \) and \( {P}_{gg}^{\left(2,{N}_F\right)} \), Nucl. Phys. B 922 (2017) 1 [arXiv:1705.01508] [INSPIRE].
S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond, JHEP 10 (2017) 041 [arXiv:1707.08315] [INSPIRE].
K. G. Chetyrkin, J. H. Kuhn and C. Sturm, QCD decoupling at four loops, Nucl. Phys. B 744 (2006) 121 [hep-ph/0512060] [INSPIRE].
W. L. van Neerven, Dimensional Regularization of Mass and Infrared Singularities in Two Loop On-shell Vertex Functions, Nucl. Phys. B 268 (1986) 453 [INSPIRE].
R. V. Harlander, Virtual corrections to gg → H to two loops in the heavy top limit, Phys. Lett. B 492 (2000) 74 [hep-ph/0007289] [INSPIRE].
V. Ravindran, J. Smith and W. L. van Neerven, Two-loop corrections to Higgs boson production, Nucl. Phys. B 704 (2005) 332 [hep-ph/0408315] [INSPIRE].
S. Moch, J. A. M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, Phys. Lett. B 625 (2005) 245 [hep-ph/0508055] [INSPIRE].
T. Gehrmann, T. Huber and D. Maître, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B 622 (2005) 295 [hep-ph/0507061] [INSPIRE].
P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [INSPIRE].
T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [arXiv:1004.3653] [INSPIRE].
T. Gehrmann and D. Kara, The \( Hb\overline{b} \) form factor to three loops in QCD, JHEP 09 (2014) 174 [arXiv:1407.8114] [INSPIRE].
A. von Manteuffel and R. M. Schabinger, Quark and gluon form factors to four-loop order in QCD: the \( {N}_f^3 \) contributions, Phys. Rev. D 95 (2017) 034030 [arXiv:1611.00795] [INSPIRE].
J. M. Henn, A. V. Smirnov, V. A. Smirnov and M. Steinhauser, A planar four-loop form factor and cusp anomalous dimension in QCD, JHEP 05 (2016) 066 [arXiv:1604.03126] [INSPIRE].
J. M. Henn, T. Peraro, M. Stahlhofen and P. Wasser, Matter dependence of the four-loop cusp anomalous dimension, Phys. Rev. Lett. 122 (2019) 201602 [arXiv:1901.03693] [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].
T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli and C. Studerus, The quark and gluon form factors to three loops in QCD through to O(eps2), JHEP 11 (2010) 102 [arXiv:1010.4478] [INSPIRE].
V. V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics, Sov. Phys. JETP 3 (1956) 65 [INSPIRE].
A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys. Rev. D 24 (1981) 3281 [INSPIRE].
J. C. Collins, Sudakov form-factors, Adv. Ser. Direct. High Energy Phys. 5 (1989) 573 [hep-ph/0312336] [INSPIRE].
L. Magnea and G. F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [INSPIRE].
L. Magnea, Analytic resummation for the quark form-factor in QCD, Nucl. Phys. B 593 (2001) 269 [hep-ph/0006255] [INSPIRE].
G. F. Sterman and M. E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].
S. Moch, J. A. M. Vermaseren and A. Vogt, The Quark form-factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039] [INSPIRE].
C. W. Bauer and A. V. Manohar, Shape function effects in B → X(s) gamma and B → X(u) l anti-nu decays, Phys. Rev. D 70 (2004) 034024 [hep-ph/0312109] [INSPIRE].
S. W. Bosch, B. O. Lange, M. Neubert and G. Paz, Factorization and shape function effects in inclusive B meson decays, Nucl. Phys. B 699 (2004) 335 [hep-ph/0402094] [INSPIRE].
T. Becher and M. D. Schwartz, Direct photon production with effective field theory, JHEP 02 (2010) 040 [arXiv:0911.0681] [INSPIRE].
T. Becher and M. Neubert, Toward a NNLO calculation of the \( \overline{B} \) → X (s) gamma decay rate with a cut on photon energy. II. Two-loop result for the jet function, Phys. Lett. B 637 (2006) 251 [hep-ph/0603140] [INSPIRE].
T. Becher and G. Bell, The gluon jet function at two-loop order, Phys. Lett. B 695 (2011) 252 [arXiv:1008.1936] [INSPIRE].
R. Brüser, Z. L. Liu and M. Stahlhofen, Three-Loop Quark Jet Function, Phys. Rev. Lett. 121 (2018) 072003 [arXiv:1804.09722] [INSPIRE].
P. Banerjee, P. K. Dhani and V. Ravindran, Gluon jet function at three loops in QCD, Phys. Rev. D 98 (2018) 094016 [arXiv:1805.02637] [INSPIRE].
C. W. Bauer, S. Fleming and M. E. Luke, Summing Sudakov logarithms in B → X(s gamma) in effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [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 and I. W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [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. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein and I. W. Stewart, Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].
M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].
J. Blumlein and V. Ravindran, QCD threshold corrections to Higgs decay and to hadroproduction in ℓ+ ℓ− annihilation, Phys. Lett. B 640 (2006) 40 [hep-ph/0605011] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, Phys. Lett. B 667 (2008) 1 [INSPIRE].
M. Beneke et al., Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power, JHEP 03 (2019) 043 [arXiv:1809.10631] [INSPIRE].
N. Bahjat-Abbas et al., Diagrammatic resummation of leading-logarithmic threshold effects at next-to-leading power, JHEP 11 (2019) 002 [arXiv:1905.13710] [INSPIRE].
K. G. Chetyrkin, G. Falcioni, F. Herzog and J. A. M. Vermaseren, Five-loop renormalisation of QCD in covariant gauges, JHEP 10 (2017) 179 [Addendum ibid. 12 (2017) 006] [arXiv:1709.08541] [INSPIRE].
T. Luthe, A. Maier, P. Marquard and Y. Schröder, The five-loop β-function for a general gauge group and anomalous dimensions beyond Feynman gauge, JHEP 10 (2017) 166 [arXiv:1709.07718] [INSPIRE].
F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, The five-loop β-function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [arXiv:1701.01404] [INSPIRE].
P. A. Baikov, K. G. Chetyrkin and J. H. Kühn, Five-Loop Running of the QCD coupling constant, Phys. Rev. Lett. 118 (2017) 082002 [arXiv:1606.08659] [INSPIRE].
G. Das, S.-O. Moch and A. Vogt, Soft corrections to inclusive deep-inelastic scattering at four loops and beyond, JHEP 03 (2020) 116 [arXiv:1912.12920] [INSPIRE].
J. Blumlein, V. Ravindran and W. L. van Neerven, On the Drell-Levy-Yan relation to \( O\left({\alpha}_s^2\right) \), Nucl. Phys. B 586 (2000) 349 [hep-ph/0004172] [INSPIRE].
E. G. Floratos, R. Lacaze and C. Kounnas, Space and Timelike Cut Vertices in QCD Beyond the Leading Order. 2. The Singlet Sector, Phys. Lett. B 98 (1981) 285 [INSPIRE].
S. D. Drell, D. J. Levy and T.-M. Yan, A Theory of Deep Inelastic Lepton-Nucleon Scattering and Lepton Pair Annihilation Processes. 1., Phys. Rev. 187 (1969) 2159 [INSPIRE].
G. Grunberg, Large-x structure of physical evolution kernels in Deep Inelastic Scattering, Phys. Lett. B 687 (2010) 405 [arXiv:0911.4471] [INSPIRE].
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Ajjath, A.H., Mukherjee, P., Ravindran, V. et al. On next to soft threshold corrections to DIS and SIA processes. J. High Energ. Phys. 2021, 131 (2021). https://doi.org/10.1007/JHEP04(2021)131
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DOI: https://doi.org/10.1007/JHEP04(2021)131