Abstract
The electron, positron, and photon Parton Distribution Functions (PDFs) of the unpolarised electron have recently been computed at the next-to-leading logarithmic accuracy in QED, by adopting the \( \overline{\mathrm{MS}} \) factorisation scheme. We present here analogous results, obtained by working in a different framework that is inspired by the so-called DIS scheme. We derive analytical solutions relevant to the large-z region, where we show that the behaviour of the PDFs depends in a dramatic way on whether running-α effects are included to all orders, as opposed to being truncated to some fixed order. By means of suitable initial and evolution conditions, next-to-leading logarithmic accurate PDFs are obtained whose large-z functional forms are identical to those of their leading logarithmic counterparts.
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28 December 2021
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP12(2021)196
References
D. R. Yennie, S. C. Frautschi and H. Suura, The infrared divergence phenomena and high-energy processes, Annals Phys. 13 (1961) 379 [INSPIRE].
S. Jadach, B. F. L. Ward and Z. Was, Coherent exclusive exponentiation for precision Monte Carlo calculations, Phys. Rev. D 63 (2001) 113009 [hep-ph/0006359] [INSPIRE].
H. Anlauf, H. D. Dahmen, P. Manakos, T. Mannel and T. Ohl, KRONOS: A Monte Carlo event generator for higher order electromagnetic radiative corrections to deep inelastic scattering at HERA, Comput. Phys. Commun. 70 (1992) 97 [INSPIRE].
J. Fujimoto, Y. Shimizu and T. Munehisa, Monte Carlo approach to radiative processes in e+ e− annihilation, Prog. Theor. Phys. 90 (1993) 177 [INSPIRE].
T. Munehisa, J. Fujimoto, Y. Kurihara and Y. Shimizu, Improved QEDPS for radiative corrections in e+ e− annihilation, Prog. Theor. Phys. 95 (1996) 375 [hep-ph/9603322] [INSPIRE].
C. M. Carloni Calame, C. Lunardini, G. Montagna, O. Nicrosini and F. Piccinini, Large angle Bhabha scattering and luminosity at flavor factories, Nucl. Phys. B 584 (2000) 459 [hep-ph/0003268] [INSPIRE].
E. A. Kuraev and V. S. Fadin, On Radiative Corrections to e+ e− Single Photon Annihilation at High-Energy, Sov. J. Nucl. Phys. 41 (1985) 466 [INSPIRE].
J. R. Ellis and R. Peccei, eds., Physics AT LEP. 1, CERN, Geneva, Switzerland (1986) [INSPIRE].
M. Skrzypek and S. Jadach, Exact and approximate solutions for the electron nonsinglet structure function in QED, Z. Phys. C 49 (1991) 577 [INSPIRE].
M. Skrzypek, Leading logarithmic calculations of QED corrections at LEP, Acta Phys. Polon. B 23 (1992) 135 [INSPIRE].
M. Cacciari, A. Deandrea, G. Montagna and O. Nicrosini, QED structure functions: A Systematic approach, Europhys. Lett. 17 (1992) 123 [INSPIRE].
S. Frixione, Initial conditions for electron and photon structure and fragmentation functions, JHEP 11 (2019) 158 [arXiv:1909.03886] [INSPIRE].
V. Bertone, M. Cacciari, S. Frixione and G. Stagnitto, The partonic structure of the electron at the next-to-leading logarithmic accuracy in QED, JHEP 03 (2020) 135 [arXiv:1911.12040] [INSPIRE].
J. Blumlein and H. Kawamura, Universal higher order singlet QED corrections to unpolarized lepton scattering, Eur. Phys. J. C 51 (2007) 317 [hep-ph/0701019] [INSPIRE].
J. Blumlein, A. De Freitas and W. van Neerven, Two-loop QED Operator Matrix Elements with Massive External Fermion Lines, Nucl. Phys. B 855 (2012) 508 [arXiv:1107.4638] [INSPIRE].
G. Altarelli, R. K. Ellis and G. Martinelli, Leptoproduction and Drell-Yan Processes Beyond the Leading Approximation in Chromodynamics, Nucl. Phys. B 143 (1978) 521 [Erratum ibid. B 146 (1978) 544].
V. Bertone, M. Cacciari, S. Frixione, G. Stagnitto, M. Zaro and X. Zhao, Studies of e+ e− cross sections at the next-to-leading logarithmic accuracy, in preparation.
V. N. Gribov and L. N. Lipatov, Deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438 [INSPIRE].
L. N. Lipatov, The parton model and perturbation theory, Yad. Fiz. 20 (1974) 181 [Sov. J. Nucl. Phys. 20 (1975) 94] [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].
Y. L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e− Annihilation by Perturbation Theory in Quantum Chromodynamics, Sov. Phys. JETP 46 (1977) 641 [INSPIRE].
M. Diemoz, F. Ferroni, E. Longo and G. Martinelli, Parton Densities from Deep Inelastic Scattering to Hadronic Processes at Super Collider Energies, Z. Phys. C 39 (1988) 21 [INSPIRE].
W. Furmanski and R. Petronzio, Lepton - Hadron Processes Beyond Leading Order in Quantum Chromodynamics, Z. Phys. C 11 (1982) 293 [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].
S. Frixione, A General approach to jet cross-sections in QCD, Nucl. Phys. B 507 (1997) 295 [hep-ph/9706545] [INSPIRE].
R. Frederix, S. Frixione, V. Hirschi, D. Pagani, H. S. Shao and M. Zaro, The automation of next-to-leading order electroweak calculations, JHEP 07 (2018) 185 [arXiv:1804.10017] [INSPIRE].
M. Cacciari, M. Greco and P. Nason, The pT spectrum in heavy flavor hadroproduction, JHEP 05 (1998) 007 [hep-ph/9803400] [INSPIRE].
B. Mele and P. Nason, The Fragmentation function for heavy quarks in QCD, Nucl. Phys. B 361 (1991) 626 [Erratum ibid. 921 (2017) 841] [INSPIRE].
W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7 (1954) 649.
S. Blanes, F. Casas, J. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rept. 470 (2009) 151 [arXiv:0810.5488].
J. Blumlein and S. Kurth, Harmonic sums and Mellin transforms up to two loop order, Phys. Rev. D 60 (1999) 014018 [hep-ph/9810241] [INSPIRE].
M. Bonvini, Resummation of soft and hard gluon radiation in perturbative QCD, Ph.D. thesis, Genoa University (2012) arXiv:1212.0480 [INSPIRE].
A. A. Almasy, N. A. Lo Presti and A. Vogt, Generalized threshold resummation in inclusive DIS and semi-inclusive electron-positron annihilation, JHEP 01 (2016) 028 [arXiv:1511.08612] [INSPIRE].
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Frixione, S. On factorisation schemes for the electron parton distribution functions in QED. J. High Energ. Phys. 2021, 180 (2021). https://doi.org/10.1007/JHEP07(2021)180
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DOI: https://doi.org/10.1007/JHEP07(2021)180