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Can \( \overline{\mathrm{MS}} \) parton distributions be negative?

Journal of High Energy Physics volume 2020, Article number: 129 (2020) Cite this article

A preprint version of the article is available at arXiv.

Abstract

It is common lore that Parton Distribution Functions (PDFs) in the \( \overline{\mathrm{MS}} \) factorization scheme can become negative beyond leading order due to the collinear subtraction which is needed in order to define partonic cross sections. We show that this is in fact not the case and next-to-leading order (NLO) \( \overline{\mathrm{MS}} \) PDFs are actually positive in the perturbative regime. In order to prove this, we modify the subtraction prescription, and perform the collinear subtraction in such a way that partonic cross sections remain positive. This defines a factorization scheme in which PDFs are positive. We then show that positivity of the PDFs is preserved when transforming from this scheme to \( \overline{\mathrm{MS}} \), provided only the strong coupling is in the perturbative regime, such that the NLO scheme change is smaller than the LO term.

References

  1. [1]

    G. Altarelli, S. Forte and G. Ridolfi, On positivity of parton distributions, Nucl. Phys. B 534 (1998) 277 [hep-ph/9806345] [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    S. Forte, G. Altarelli and G. Ridolfi, Are parton distributions positive?, Nucl. Phys. B Proc. Suppl. 74 (1999) 138 [hep-ph/9808462] [INSPIRE].

    ADS  Article  Google Scholar 

  3. [3]

    NNPDF collaboration, A determination of parton distributions with faithful uncertainty estimation, Nucl. Phys. B 809 (2009) 1 [Erratum ibid. 816 (2009) 293] [arXiv:0808.1231] [INSPIRE].

  4. [4]

    F. Faura, S. Iranipour, E.R. Nocera, J. Rojo and M. Ubiali, The Strangest Proton?, arXiv:2009.00014 [INSPIRE].

  5. [5]

    R.D. Ball et al., A first unbiased global NLO determination of parton distributions and their uncertainties, Nucl. Phys. B 838 (2010) 136 [arXiv:1002.4407] [INSPIRE].

    ADS  Article  Google Scholar 

  6. [6]

    NNPDF collaboration, Parton distributions for the LHC Run II, JHEP 04 (2015) 040 [arXiv:1410.8849] [INSPIRE].

  7. [7]

    R.K. Ellis, W.J. Stirling and B.R. Webber, QCD and collider physics, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 8 (1996) 1.

    Google Scholar 

  8. [8]

    J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 (1982) 445 [INSPIRE].

    ADS  Article  Google Scholar 

  9. [9]

    G. Curci, W. Furmanski and R. Petronzio, Evolution of Parton Densities Beyond Leading Order: The Nonsinglet Case, Nucl. Phys. B 175 (1980) 27 [INSPIRE].

    ADS  Article  Google Scholar 

  10. [10]

    J. Collins, Foundations of perturbative QCD, vol. 32, Cambridge University Press (2013).

  11. [11]

    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].

    ADS  Article  Google Scholar 

  12. [12]

    S. Catani and L. Trentadue, Resummation of the QCD Perturbative Series for Hard Processes, Nucl. Phys. B 327 (1989) 323 [INSPIRE].

    ADS  Article  Google Scholar 

  13. [13]

    G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].

    ADS  Article  Google Scholar 

  14. [14]

    S. Forte and G. Ridolfi, Renormalization group approach to soft gluon resummation, Nucl. Phys. B 650 (2003) 229 [hep-ph/0209154] [INSPIRE].

    ADS  Article  Google Scholar 

  15. [15]

    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].

    ADS  Article  Google Scholar 

  16. [16]

    S. Jadach, W. Płaczek, S. Sapeta, A. Siodmok and M. Skrzypek, Parton distribution functions in Monte Carlo factorisation scheme, Eur. Phys. J. C 76 (2016) 649 [arXiv:1606.00355] [INSPIRE].

    ADS  Article  Google Scholar 

  17. [17]

    F. Maltoni, Basics of QCD for the LHC: pp → H + X as a case study, CERN Yellow Rep. School Proc. 2 (2018) 41.

    Google Scholar 

  18. [18]

    S. Catani, Comment on quarks and gluons at small x and the SDIS factorization scheme, Z. Phys. C 70 (1996) 263 [hep-ph/9506357] [INSPIRE].

    Article  Google Scholar 

  19. [19]

    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].

    ADS  Article  Google Scholar 

  20. [20]

    S. Jadach, private communications, (2020).

  21. [21]

    S. Forte, E. Laenen, P. Nason and J. Rojo, Heavy quarks in deep-inelastic scattering, Nucl. Phys. B 834 (2010) 116 [arXiv:1001.2312] [INSPIRE].

    ADS  Article  Google Scholar 

  22. [22]

    M.A.G. Aivazis, J.C. Collins, F.I. Olness and W.-K. Tung, Leptoproduction of heavy quarks. 2. A unified QCD formulation of charged and neutral current processes from fixed target to collider energies, Phys. Rev. D 50 (1994) 3102 [hep-ph/9312319] [INSPIRE].

    ADS  Article  Google Scholar 

  23. [23]

    M. Buza, Y. Matiounine, J. Smith and W.L. van Neerven, Charm electroproduction viewed in the variable flavor number scheme versus fixed order perturbation theory, Eur. Phys. J. C 1 (1998) 301 [hep-ph/9612398] [INSPIRE].

    ADS  Article  Google Scholar 

  24. [24]

    R.D. Ball, M. Bonvini and L. Rottoli, Charm in Deep-Inelastic Scattering, JHEP 11 (2015) 122 [arXiv:1510.02491] [INSPIRE].

    ADS  Article  Google Scholar 

  25. [25]

    NNPDF collaboration, A Determination of the Charm Content of the Proton, Eur. Phys. J. C 76 (2016) 647 [arXiv:1605.06515] [INSPIRE].

  26. [26]

    NNPDF collaboration, Parton distributions from high-precision collider data, Eur. Phys. J. C 77 (2017) 663 [arXiv:1706.00428] [INSPIRE].

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Affiliations

  1. Tif Lab, Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano, Via Celoria 16, I-20133, Milano, Italy

    Alessandro Candido, Stefano Forte & Felix Hekhorn

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  1. Alessandro Candido
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  2. Stefano Forte
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  3. Felix Hekhorn
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Correspondence to Stefano Forte.

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ArXiv ePrint: 2006.07377

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Candido, A., Forte, S. & Hekhorn, F. Can \( \overline{\mathrm{MS}} \) parton distributions be negative?. J. High Energ. Phys. 2020, 129 (2020). https://doi.org/10.1007/JHEP11(2020)129

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Keywords

  • Deep Inelastic Scattering (Phenomenology)
  • QCD Phenomenology