Abstract
We discuss the upper limit, kmax, of the transverse-momentum integration performed in the kt-factorization formula. Based on explicit calculations in the Yukawa theory and the study of seminal papers, we argue that kmax is equal to the factorization scale μF used to factorize the cross section into an off-shell hard coefficient and a universal factor. There is consequently a relation between kmax and the definition of unintegrated parton densities (UPDFs). The use of an inconsistent relation leads potentially to the overestimation of the cross section, which has been observed, e.g., in D-meson production [1]. One of our conclusions is that UPDFs related to collinear PDFs by an integration up to μ ∼ Q, where Q is the hard scale and μ the scale in the collinear PDFs, imply that \( {k}_{\textrm{max}}^2 \) ∼ Q2. Integrating the transverse-momentum significantly above may result in the overestimation of the cross section. On the opposite, for UPDFs related to the collinear ones by an integration of the transverse momentum up to infinity, any \( {k}_{\textrm{max}}^2 \) > Q2 is fine.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Guiot, Heavy-quark production with kt-factorization: The importance of the sea-quark distribution, Phys. Rev. D 99 (2019) 074006 [arXiv:1812.02156] [INSPIRE].
F. Aslan et al., Basics of factorization in a scalar Yukawa field theory, Phys. Rev. D 107 (2023) 074031 [arXiv:2212.00757] [INSPIRE].
S. Catani, M. Ciafaloni and F. Hautmann, Gluon contributions to small x heavy flavour production, Phys. Lett. B 242 (1990) 97 [INSPIRE].
S. Catani, M. Ciafaloni and F. Hautmann, High-energy factorization and small x heavy flavor production, Nucl. Phys. B 366 (1991) 135 [INSPIRE].
J.C. Collins and R.K. Ellis, Heavy quark production in very high-energy hadron collisions, Nucl. Phys. B 360 (1991) 3 [INSPIRE].
E.M. Levin, M.G. Ryskin, Y.M. Shabelski and A.G. Shuvaev, Heavy quark production in semihard nucleon interactions, Sov. J. Nucl. Phys. 53 (1991) 657 [INSPIRE].
V.S. Fadin, E.A. Kuraev and L.N. Lipatov, On the Pomeranchuk Singularity in Asymptotically Free Theories, Phys. Lett. B 60 (1975) 50 [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Multi-Reggeon Processes in the Yang-Mills Theory, Sov. Phys. JETP 44 (1976) 443 [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk singularity in nonabelian gauge theories, JETP 45 (1977) 199.
I.I. Balitsky and L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [INSPIRE].
M.A. Kimber, A.D. Martin and M.G. Ryskin, Unintegrated parton distributions, Phys. Rev. D 63 (2001) 114027 [hep-ph/0101348] [INSPIRE].
G. Watt, A.D. Martin and M.G. Ryskin, Unintegrated parton distributions and inclusive jet production at HERA, Eur. Phys. J. C 31 (2003) 73 [hep-ph/0306169] [INSPIRE].
K. Golec-Biernat and A.M. Staśto, On the use of the KMR unintegrated parton distribution functions, Phys. Lett. B 781 (2018) 633 [arXiv:1803.06246] [INSPIRE].
F. Hautmann, L. Keersmaekers, A. Lelek and A.M. Van Kampen, Dynamical resolution scale in transverse momentum distributions at the LHC, Nucl. Phys. B 949 (2019) 114795 [arXiv:1908.08524] [INSPIRE].
B. Guiot, Pathologies of the Kimber-Martin-Ryskin prescriptions for unintegrated PDFs: Which prescription should be preferred?, Phys. Rev. D 101 (2020) 054006 [arXiv:1910.09656] [INSPIRE].
R.K. Valeshabadi and M. Modarres, On the ambiguity between differential and integral forms of the Martin-Ryskin-Watt unintegrated parton distribution function model, Eur. Phys. J. C 82 (2022) 66 [arXiv:2111.02254] [INSPIRE].
M.A. Nefedov and V.A. Saleev, High-Energy Factorization for Drell-Yan process in pp and \( p\overline{p} \) collisions with new Unintegrated PDFs, Phys. Rev. D 102 (2020) 114018 [arXiv:2009.13188] [INSPIRE].
B. Guiot and A. van Hameren, D and B-meson production using kt-factorization calculations in a variable-flavor-number scheme, Phys. Rev. D 104 (2021) 094038 [arXiv:2108.06419] [INSPIRE].
B. Guiot, Normalization of unintegrated parton densities, Phys. Rev. D 107 (2023) 014015 [arXiv:2205.02873] [INSPIRE].
A. van Hameren, KaTie: For parton-level event generation with kT-dependent initial states, Comput. Phys. Commun. 224 (2018) 371 [arXiv:1611.00680] [INSPIRE].
F. Hautmann et al., Soft-gluon resolution scale in QCD evolution equations, Phys. Lett. B 772 (2017) 446 [arXiv:1704.01757] [INSPIRE].
F. Hautmann et al., Collinear and TMD Quark and Gluon Densities from Parton Branching Solution of QCD Evolution Equations, JHEP 01 (2018) 070 [arXiv:1708.03279] [INSPIRE].
A. Bermudez Martinez et al., Collinear and TMD parton densities from fits to precision DIS measurements in the parton branching method, Phys. Rev. D 99 (2019) 074008 [arXiv:1804.11152] [INSPIRE].
S. Catani, M. Ciafaloni and F. Hautmann, High-energy factorization in QCD and minimal subtraction scheme, Phys. Lett. B 307 (1993) 147 [INSPIRE].
M. Ciafaloni, Coherence Effects in Initial Jets at Small Q2/s, Nucl. Phys. B 296 (1988) 49 [INSPIRE].
S. Catani, F. Fiorani and G. Marchesini, Small x Behavior of Initial State Radiation in Perturbative QCD, Nucl. Phys. B 336 (1990) 18 [INSPIRE].
A. Gawron and J. Kwieciński, Resummation effects in Higgs boson transverse momentum distribution within the framework of unintegrated parton distributions, Phys. Rev. D 70 (2004) 014003 [hep-ph/0309303] [INSPIRE].
G.P. Salam, Soft emissions and the equivalence of BFKL and CCFM final states, JHEP 03 (1999) 009 [hep-ph/9902324] [INSPIRE].
E. Avsar and E. Iancu, CCFM Evolution with Unitarity Corrections, Nucl. Phys. A 829 (2009) 31 [arXiv:0906.2683] [INSPIRE].
A. Gawron, J. Kwieciński and W. Broniowski, Unintegrated parton distributions of pions and nucleons from the CCFM equations in the single loop approximation, Phys. Rev. D 68 (2003) 054001 [hep-ph/0305219] [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1989) 1 [hep-ph/0409313] [INSPIRE].
R.D. Ball and R.K. Ellis, Heavy quark production at high-energy, JHEP 05 (2001) 053 [hep-ph/0101199] [INSPIRE].
J.-P. Lansberg, M. Nefedov and M.A. Ozcelik, Matching next-to-leading-order and high-energy-resummed calculations of heavy-quarkonium-hadroproduction cross sections, JHEP 05 (2022) 083 [arXiv:2112.06789] [INSPIRE].
R. Maciuła, A. Szczurek and A. Cisek, J/ψ-meson production within improved color evaporation model with the kT-factorization approach for \( c\overline{c} \) production, Phys. Rev. D 99 (2019) 054014 [arXiv:1810.08063] [INSPIRE].
B. Guiot, A. Radic, I. Schmidt and K. Werner, J/ψ production at NLO with a scale-dependent color-evaporation model, Phys. Rev. D 108 (2023) 114003 [arXiv:2306.11032] [INSPIRE].
Acknowledgments
BG acknowledge support from ANID PIA/APOYO AFB230003. AvH is supported by grant no. 2019/35/B/ST2/03531 of the Polish National Science Centre. We are grateful to Francesco Hautmann for his comments on the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2401.06888
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
Guiot, B., van Hameren, A. Examination of kt-factorization in a Yukawa theory. J. High Energ. Phys. 2024, 85 (2024). https://doi.org/10.1007/JHEP04(2024)085
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2024)085