Abstract
We present analytic all-order results for the highest three threshold logarithms of the space-like and time-like off-diagonal splitting functions and the corresponding coefficient functions for inclusive deep-inelastic scattering (DIS) and semi-inclusive e + e − annihilation. All these results, obtained through an order-by-order analysis of the structure of the corresponding unfactorized quantities in dimensional regularization, can be expressed in terms of the Bernoulli functions introduced by one of us and leading-logarithmic soft-gluon exponentials. The resulting numerical corrections are small for the splitting functions but large for the coefficient functions. In both cases more terms in the threshold expansion need to be determined in order to arrive at quantitatively reliable results.
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Particle Data Group collaboration, K.A. Olive et al., Review of Particle Physics, Chin. Phys. C 38 (2014) 090001 [INSPIRE].
W.L. van Neerven and E.B. Zijlstra, Order α 2 S contributions to the deep inelastic Wilson coefficient, Phys. Lett. B 272 (1991) 127 [INSPIRE].
E.B. Zijlstra and W.L. van Neerven, Contribution of the second order gluonic Wilson coefficient to the deep inelastic structure function, Phys. Lett. B 273 (1991) 476 [INSPIRE].
E.B. Zijlstra and W.L. van Neerven, Order α 2 S correction to the structure function F 3(x, Q 2) in deep inelastic neutrino-hadron scattering, Phys. Lett. B 297 (1992) 377 [INSPIRE].
S. Moch and J.A.M. Vermaseren, Deep inelastic structure functions at two loops, Nucl. Phys. B 573 (2000) 853 [hep-ph/9912355] [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].
S. Moch, J.A.M. Vermaseren and A. Vogt, Third-order QCD corrections to the charged-current structure function F 3, Nucl. Phys. B 813 (2009) 220 [arXiv:0812.4168].
P.J. Rijken and W.L. van Neerven, O(α 2 S ) contributions to the longitudinal fragmentation function in e + e − annihilation, Phys. Lett. B 386 (1996) 422 [hep-ph/9604436] [INSPIRE].
P.J. Rijken and W.L. van Neerven, Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation, Nucl. Phys. B 487 (1997) 233 [hep-ph/9609377] [INSPIRE].
P.J. Rijken and W.L. van Neerven, O(α 2 S ) contributions to the asymmetric fragmentation function in e + e − annihilation, Phys. Lett. B 392 (1997) 207 [hep-ph/9609379] [INSPIRE].
A. Mitov and S.-O. Moch, QCD Corrections to Semi-Inclusive Hadron Production in Electron-Positron Annihilation at Two Loops, Nucl. Phys. B 751 (2006) 18 [hep-ph/0604160] [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. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [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].
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].
C. Anzai et al., Exact N 3 LO results for qq′ → H + X, JHEP 07 (2015) 140 [arXiv:1506.02674] [INSPIRE].
G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions, Mod. Phys. Lett. A 4 (1989) 1257 [INSPIRE].
Yu. L. Dokshitzer, G. Marchesini and G.P. Salam, Revisiting parton evolution and the large-x limit, Phys. Lett. B 634 (2006) 504 [hep-ph/0511302] [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. Mitov, S. Moch and A. Vogt, Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions, Phys. Lett. B 638 (2006) 61 [hep-ph/0604053] [INSPIRE].
S. Moch and A. Vogt, On third-order timelike splitting functions and top-mediated Higgs decay into hadrons, Phys. Lett. B 659 (2008) 290 [arXiv:0709.3899] [INSPIRE].
A.A. Almasy, S. Moch and A. Vogt, On the Next-to-Next-to-Leading Order Evolution of Flavour-Singlet Fragmentation Functions, Nucl. Phys. B 854 (2012) 133 [arXiv:1107.2263] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case, Nucl. Phys. B 889 (2014) 351 [arXiv:1409.5131] [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].
L. Magnea, All Order Summation and Two Loop Results for the Drell-Yan Cross-section, Nucl. Phys. B 349 (1991) 703 [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].
H. Contopanagos, E. Laenen and G.F. Sterman, Sudakov factorization and resummation, Nucl. Phys. B 484 (1997) 303 [hep-ph/9604313] [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].
S. Moch and A. Vogt, Higher-order threshold resummation for semi-inclusive e + e − annihilation, Phys. Lett. B 680 (2009) 239 [arXiv:0908.2746] [INSPIRE].
A.A. Almasy, G. Soar and A. Vogt, Generalized double-logarithmic large-x resummation in inclusive deep-inelastic scattering, JHEP 03 (2011) 030 [arXiv:1012.3352] [INSPIRE].
N.A. Lo Presti, A. Vogt and A.A. Almasy, Generalized Threshold Resummation for Semi-Inclusive e + e − Annihilation, arXiv:1202.5224 [INSPIRE].
A. Vogt et al., Progress on double-logarithmic large-x and small-x resummations for (semi-)inclusive hard processes, arXiv:1212.2932 [INSPIRE].
N.A. Lo Presti, A.A. Almasy and A. Vogt, Leading large-x logarithms of the quark-gluon contributions to inclusive Higgs-boson and lepton-pair production, Phys. Lett. B 737 (2014) 120 [arXiv:1407.1553] [INSPIRE].
S. Moch and A. Vogt, Threshold Resummation of the Structure Function F(L), JHEP 04 (2009) 081 [arXiv:0902.2342] [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. 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].
A. Vogt, G. Soar, S. Moch and J.A.M. Vermaseren, On higher-order flavour-singlet splitting and coefficient functions at large x, PoS DIS2010 (2010) 139 [arXiv:1008.0952] [INSPIRE].
D. de Florian, J. Mazzitelli, S. Moch and A. Vogt, Approximate N 3 LO Higgs-boson production cross section using physical-kernel constraints, JHEP 10 (2014) 176 [arXiv:1408.6277] [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, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [INSPIRE].
E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP 11 (2010) 155 [arXiv:1008.0098] [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].
C.D. White, Diagrammatic insights into next-to-soft corrections, Phys. Lett. B 737 (2014) 216 [arXiv:1406.7184] [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].
G. Grunberg and V. Ravindran, On threshold resummation beyond leading 1-x order, JHEP 10 (2009) 055 [arXiv:0902.2702] [INSPIRE].
G. Grunberg, Large-x structure of physical evolution kernels in Deep Inelastic Scattering, Phys. Lett. B 687 (2010) 405 [arXiv:0911.4471] [INSPIRE].
G. Grunberg, On threshold resummation of singlet structure and fragmentation functions, Nucl. Phys. B 851 (2011) 30 [arXiv:1101.5377] [INSPIRE].
A. Vogt, Leading logarithmic large-x resummation of off-diagonal splitting functions and coefficient functions, Phys. Lett. B 691 (2010) 77 [arXiv:1005.1606] [INSPIRE].
M. Abramowitz and I.A. Stegun eds., Handbook of Mathematical Functions, Dover, New York U.S.A. (1965).
A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, JHEP 10 (2011) 025 [arXiv:1108.2993] [INSPIRE].
C.H. Kom, A. Vogt and K. Yeats, Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD, JHEP 10 (2012) 033 [arXiv:1207.5631] [INSPIRE].
P.A. Baikov and K.G. Chetyrkin, New four loop results in QCD, Nucl. Phys. Proc. Suppl. 160 (2006) 76 [INSPIRE].
V.N. Velizhanin, Four loop anomalous dimension of the second moment of the non-singlet twist-2 operator in QCD, Nucl. Phys. B 860 (2012) 288 [arXiv:1112.3954] [INSPIRE].
V.N. Velizhanin, Four loop anomalous dimension of the third and fourth moments of the non-singlet twist-2 operator in QCD, arXiv:1411.1331 [INSPIRE].
P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Massless Propagators, R(s) and Multiloop QCD, Nucl. Part. Phys. Proc. 261-262 (2015) 3 [arXiv:1501.06739] [INSPIRE].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
M. Tentyukov and J.A.M. Vermaseren, The Multithreaded version of FORM, Comput. Phys. Commun. 181 (2010) 1419 [hep-ph/0702279] [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].
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ArXiv ePrint: 1511.08612
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Almasy, A.A., Lo Presti, N.A. & Vogt, A. Generalized threshold resummation in inclusive DIS and semi-inclusive electron-positron annihilation. J. High Energ. Phys. 2016, 28 (2016). https://doi.org/10.1007/JHEP01(2016)028
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DOI: https://doi.org/10.1007/JHEP01(2016)028