Abstract
We study the threshold corrections for inclusive deep-inelastic scattering (DIS) and their all-order resummation. Using recent results for the QCD form factor, related anomalous dimensions and Mellin moments of DIS structure functions at four loops we derive the complete soft and collinear contributions to the DIS Wilson coefficients at four loops. For a general SU(nc) gauge group the results are exact in the large-nc approximation and for QCD with nc = 3 we present precise approximations. We extend the threshold resummation exponent GN in Mellin-N space to the fifth logarithmic (N4LL) order collecting the terms \( {\alpha}_{\mathrm{s}}^3 \)(αs ln N)n to all orders in the strong coupling constant αs. We study the numerical effect of the N4LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms. As a byproduct, we derive a numerical result for the complete pole structure of the QCD form factor in the parameter of dimensional regularization ε at four loops.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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 F3 , Nucl. Phys. B 813 (2009) 220 [arXiv:0812.4168] [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].
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].
A.V. Manohar, Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [INSPIRE].
J. Chay and C. Kim, Deep inelastic scattering near the endpoint in soft-collinear effective theory, Phys. Rev. D 75 (2007) 016003 [hep-ph/0511066] [INSPIRE].
A. Idilbi, X.-D. Ji and F. Yuan, Resummation of threshold logarithms in effective field theory for DIS, Drell-Yan and Higgs production, Nucl. Phys. B 753 (2006) 42 [hep-ph/0605068] [INSPIRE].
T. Becher, M. Neubert and B.D. Pecjak, Factorization and momentum-space resummation in deep-inelastic scattering, JHEP 01 (2007) 076 [hep-ph/0607228] [INSPIRE].
A. Accardi et al., A critical appraisal and evaluation of modern PDFs, Eur. Phys. J. C 76 (2016) 471 [arXiv:1603.08906] [INSPIRE].
J.C. Collins, Algorithm to compute corrections to the Sudakov form-factor, Phys. Rev. D 22 (1980) 1478 [INSPIRE].
A. Sen, Asymptotic behavior of the Sudakov form-factor in QCD, Phys. Rev. D 24 (1981) 3281 [INSPIRE].
G.P. Korchemsky, Sudakov form-factor in QCD, Phys. Lett. B 220 (1989) 629 [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].
S. Moch, J.A.M. Vermaseren and A. Vogt, The quark form-factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039] [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].
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].
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].
E. Laenen and L. Magnea, Threshold resummation for electroweak annihilation from DIS data, Phys. Lett. B 632 (2006) 270 [hep-ph/0508284] [INSPIRE].
V. Ravindran, On Sudakov and soft resummations in QCD, Nucl. Phys. B 746 (2006) 58 [hep-ph/0512249] [INSPIRE].
V. Ravindran, Higher-order threshold effects to inclusive processes in QCD, Nucl. Phys. B 752 (2006) 173 [hep-ph/0603041] [INSPIRE].
J. Henn, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and R.N. Lee, Four-loop photon quark form factor and cusp anomalous dimension in the large-Nc limit of QCD, JHEP 03 (2017) 139 [arXiv:1612.04389] [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].
B. Ruijl, T. Ueda, J.A.M. Vermaseren, J. Davies and A. Vogt, First forcer results on deep-inelastic scattering and related quantities, PoS(LL2016)071 (2016) [arXiv:1605.08408] [INSPIRE].
R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, The \( {n}_f^2 \) contributions to fermionic four-loop form factors, Phys. Rev. D 96 (2017) 014008 [arXiv:1705.06862] [INSPIRE].
A. Grozin, Four-loop cusp anomalous dimension in QED, JHEP 06 (2018) 073 [arXiv:1805.05050] [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].
R. Brüser, A. Grozin, J.M. Henn and M. Stahlhofen, Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles, JHEP 05 (2019) 186 [arXiv:1902.05076] [INSPIRE].
S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, On quartic colour factors in splitting functions and the gluon cusp anomalous dimension, Phys. Lett. B 782 (2018) 627 [arXiv:1805.09638] [INSPIRE].
R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Four-loop quark form factor with quartic fundamental colour factor, JHEP 02 (2019) 172 [arXiv:1901.02898] [INSPIRE].
A. von Manteuffel and R.M. Schabinger, Quark and gluon form factors in four loop QCD: the \( {N}_f^2 \) and Nqγ Nf contributions, Phys. Rev. D 99 (2019) 094014 [arXiv:1902.08208] [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, arXiv:1911.10174 [INSPIRE].
T. Huber, A. von Manteuffel, E. Panzer, R.M. Schabinger and G. Yang, The four-loop cusp anomalous dimension from the N = 4 Sudakov form factor, arXiv:1912.13459 [INSPIRE].
G. Das, S.-O. Moch and A. Vogt, Soft corrections to inclusive DIS at four loops and beyond, PoS(DIS2019)010 (2019) [arXiv:1908.03071] [INSPIRE].
S. Catani, M.L. Mangano and P. Nason, Sudakov resummation for prompt photon production in hadron collisions, JHEP 07 (1998) 024 [hep-ph/9806484] [INSPIRE].
S. Forte and G. Ridolfi, Renormalization group approach to soft gluon resummation, Nucl. Phys. B 650 (2003) 229 [hep-ph/0209154] [INSPIRE].
E. Gardi and R.G. Roberts, The interplay between Sudakov resummation, renormalons and higher twist in deep inelastic scattering, Nucl. Phys. B 653 (2003) 227 [hep-ph/0210429] [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].
A. Vogt, Next-to-next-to-leading logarithmic threshold resummation for deep inelastic scattering and the Drell-Yan process, Phys. Lett. B 497 (2001) 228 [hep-ph/0010146] [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].
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].
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].
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].
F. Herzog, S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, Five-loop contributions to low-N non-singlet anomalous dimensions in QCD, Phys. Lett. B 790 (2019) 436 [arXiv:1812.11818] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, Onγ5 in higher-order QCD calculations and the NNLO evolution of the polarized valence distribution, Phys. Lett. B 748 (2015) 432 [arXiv:1506.04517] [INSPIRE].
G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov evolution kernels of parton distributions, Mod. Phys. Lett. A 4 (1989) 1257 [INSPIRE].
S. Albino and R.D. Ball, Soft resummation of quark anomalous dimensions and coefficient functions in MS-bar factorization, Phys. Lett. B 513 (2001) 93 [hep-ph/0011133] [INSPIRE].
J. Davies, A. Vogt, B. Ruijl, T. Ueda and J.A.M. Vermaseren, Large-Nf contributions to the four-loop splitting functions in QCD, Nucl. Phys. B 915 (2017) 335 [arXiv:1610.07477] [INSPIRE].
J.A. Gracey, Anomalous dimension of nonsinglet Wilson operators at O(1/Nf) in deep inelastic scattering, Phys. Lett. B 322 (1994) 141 [hep-ph/9401214] [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].
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, 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(∈2), JHEP 11 (2010) 102 [arXiv:1010.4478] [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].
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].
G. Falcioni, E. Gardi and C. Milloy, Relating amplitude and PDF factorisation through Wilson-line geometries, JHEP 11 (2019) 100 [arXiv:1909.00697] [INSPIRE].
A. Vogt, Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS, Comput. Phys. Commun. 170 (2005) 65 [hep-ph/0408244] [INSPIRE].
A. Vogt, On soft gluon effects in deep inelastic structure functions, Phys. Lett. B 471 (1999) 97 [hep-ph/9910545] [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].
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].
H. Kawamura, N.A. Lo Presti, S. Moch and A. Vogt, On the next-to-next-to-leading order QCD corrections to heavy-quark production in deep-inelastic scattering, Nucl. Phys. B 864 (2012) 399 [arXiv:1205.5727] [INSPIRE].
A.H. Hoang, P. Pietrulewicz and D. Samitz, Variable flavor number scheme for final state jets in DIS, Phys. Rev. D 93 (2016) 034034 [arXiv:1508.04323] [INSPIRE].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2, arXiv:1707.06453 [INSPIRE].
A. von Manteuffel, E. Panzer and R.M. Schabinger, Analytic four-loop anomalous dimensions in massless QCD from form factors, arXiv:2002.04617 [INSPIRE].
W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Deep inelastic scattering beyond the leading order in asymptotically free gauge theories, Phys. Rev. D 18 (1978) 3998 [INSPIRE].
W.L. van Neerven and E.B. Zijlstra, Order \( {\alpha}_s^2 \) contributions to the deep inelastic Wilson coefficient, Phys. Lett. B 272 (1991) 127 [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].
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: 1912.12920
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
Das, G., Moch, SO. & Vogt, A. Soft corrections to inclusive deep-inelastic scattering at four loops and beyond. J. High Energ. Phys. 2020, 116 (2020). https://doi.org/10.1007/JHEP03(2020)116
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2020)116