Skip to main content
Download PDF

QCD resummation on single hadron transverse momentum distribution with the thrust axis

Journal of High Energy Physics volume 2020, Article number: 127 (2020) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We derive the transverse momentum dependent (TMD) factorization and resummation formula of the unpolarized transverse momentum distribution (jT) for the single hadron production with the thrust axis in an electron-positron collision. Two different kinematic regions are considered, including small transverse momentum limit jT « Q, and joint transverse momentum and threshold limit jT « Q(1 − zh) « Q, where Q and zh are the hard scattering energy and the observed hadron momentum fraction. Using effective theory methods, we resum logarithms ln(Q/jT) and ln(1 − zh) to all orders. In the end, we present the differential cross sections and Gaussian widths calculated for the inclusive charged pion production and find that our results are consistent with the measurements reported by the Belle collaboration.

References

  1. A. Accardi et al., Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all, Eur. Phys. J. A 52 (2016) 268 [arXiv:1212.1701] [INSPIRE].

    ADS  Article  Google Scholar 

  2. D. Boer et al., Gluons and the quark sea at high energies: Distributions, polarization, tomography, arXiv:1108.1713 [INSPIRE].

  3. A. Prokudin, Y. Hatta, Y. Kovchegov and C. Marquet eds., Probing Nucleons and Nuclei in High Energy Collisions, proceedings of the INT Program INT-18-3 , Seattle, WA, U.S.A., 1 October–16 November 2018, World Scientific (2020).

  4. X. Liu, F. Ringer, W. Vogelsang and F. Yuan, Lepton-jet Correlation in Deep Inelastic Scattering, Phys. Rev. D 102 (2020) 094022 [arXiv:2007.12866] [INSPIRE].

  5. Jefferson Lab Angular Momentum collaboration, Origin of single transverse-spin asymmetries in high-energy collisions, Phys. Rev. D 102 (2020) 054002 [arXiv:2002.08384] [INSPIRE].

  6. A. Bacchetta, F. Delcarro, C. Pisano and M. Radici, The three-dimensional distribution of quarks in momentum space, arXiv:2004.14278 [INSPIRE].

  7. A. Bacchetta et al., Transverse-momentum-dependent parton distributions up to N3LL from Drell-Yan data, JHEP 07 (2020) 117 [arXiv:1912.07550] [INSPIRE].

    ADS  Article  Google Scholar 

  8. I. Scimemi and A. Vladimirov, Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum, JHEP 06 (2020) 137 [arXiv:1912.06532] [INSPIRE].

    ADS  Article  Google Scholar 

  9. X. Liu, F. Ringer, W. Vogelsang and F. Yuan, Lepton-jet Correlations in Deep Inelastic Scattering at the Electron-Ion Collider, Phys. Rev. Lett. 122 (2019) 192003 [arXiv:1812.08077] [INSPIRE].

    ADS  Article  Google Scholar 

  10. Y.-T. Chien, R. Rahn, S. Schrijnder van Velzen, D.Y. Shao, W.J. Waalewijn and B. Wu, Azimuthal angle for boson-jet production in the back-to-back limit, arXiv:2005.12279 [INSPIRE].

  11. Y.-T. Chien, D.Y. Shao and B. Wu, Resummation of Boson-Jet Correlation at Hadron Colliders, JHEP 11 (2019) 025 [arXiv:1905.01335] [INSPIRE].

    ADS  Article  Google Scholar 

  12. S. Fleming, Y. Makris and T. Mehen, An effective field theory approach to quarkonium at small transverse momentum, JHEP 04 (2020) 122 [arXiv:1910.03586] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  13. M.G.A. Buffing, Z.-B. Kang, K. Lee and X. Liu, A transverse momentum dependent framework for back-to-back photon + jet production, arXiv:1812.07549 [INSPIRE].

  14. M.G. Echevarria, A. Idilbi, Z.-B. Kang and I. Vitev, QCD Evolution of the Sivers Asymmetry, Phys. Rev. D 89 (2014) 074013 [arXiv:1401.5078] [INSPIRE].

  15. D. Boer, C. Lorcé, C. Pisano and J. Zhou, The gluon Sivers distribution: status and future prospects, Adv. High Energy Phys. 2015 (2015) 371396 [arXiv:1504.04332] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  16. A. Bacchetta, M. Diehl, K. Goeke, A. Metz, P.J. Mulders and M. Schlegel, Semi-inclusive deep inelastic scattering at small transverse momentum, JHEP 02 (2007) 093 [hep-ph/0611265] [INSPIRE].

  17. Z.-B. Kang, A. Prokudin, P. Sun and F. Yuan, Extraction of Quark Transversity Distribution and Collins Fragmentation Functions with QCD Evolution, Phys. Rev. D 93 (2016) 014009 [arXiv:1505.05589] [INSPIRE].

  18. M.G. Echevarria, Y. Makris and I. Scimemi, Quarkonium TMD fragmentation functions in NRQCD, JHEP 10 (2020) 164 [arXiv:2007.05547] [INSPIRE].

    ADS  Article  Google Scholar 

  19. D. Callos, Z.-B. Kang and J. Terry, Extracting the transverse momentum dependent polarizing fragmentation functions, Phys. Rev. D 102 (2020) 096007 [arXiv:2003.04828] [INSPIRE].

  20. M. Arratia, Z.-B. Kang, A. Prokudin and F. Ringer, Jet-based measurements of Sivers and Collins asymmetries at the future electron-ion collider, Phys. Rev. D 102 (2020) 074015 [arXiv:2007.07281] [INSPIRE].

  21. D. Neill, I. Scimemi and W.J. Waalewijn, Jet axes and universal transverse-momentum- dependent fragmentation, JHEP 04 (2017) 020 [arXiv:1612.04817] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  22. Z.-B. Kang, A. Prokudin, F. Ringer and F. Yuan, Collins azimuthal asymmetries of hadron production inside jets, Phys. Lett. B 774 (2017) 635 [arXiv:1707.00913] [INSPIRE].

    ADS  Article  Google Scholar 

  23. Z.-B. Kang, X. Liu, F. Ringer and H. Xing, The transverse momentum distribution of hadrons within jets, JHEP 11 (2017) 068 [arXiv:1705.08443] [INSPIRE].

    ADS  Article  Google Scholar 

  24. Z.-B. Kang, K. Lee, J. Terry and H. Xing, Jet fragmentation functions for Z-tagged jets, Phys. Lett. B 798 (2019) 134978 [arXiv:1906.07187] [INSPIRE].

    Article  Google Scholar 

  25. Z.-B. Kang, K. Lee and F. Zhao, Polarized jet fragmentation functions, Phys. Lett. B 809 (2020) 135756 [arXiv:2005.02398] [INSPIRE].

    Article  Google Scholar 

  26. D. Gutierrez-Reyes, Y. Makris, V. Vaidya, I. Scimemi and L. Zoppi, Probing Transverse- Momentum Distributions With Groomed Jets, JHEP 08 (2019) 161 [arXiv:1907.05896] [INSPIRE].

    ADS  Article  Google Scholar 

  27. A. Metz and A. Vossen, Parton Fragmentation Functions, Prog. Part. Nucl. Phys. 91 (2016) 136 [arXiv:1607.02521] [INSPIRE].

    ADS  Article  Google Scholar 

  28. J.C. Collins, D.E. Soper and G.F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, Nucl. Phys. B 250 (1985) 199 [INSPIRE].

    ADS  Google Scholar 

  29. J. Collins, Foundations of perturbative QCD, in Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology 32, Cambridge University Press, Cambridge U.K. (2011) [INSPIRE].

  30. X.-d. Ji, J.-p. Ma and F. Yuan, QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum, Phys. Rev. D 71 (2005) 034005 [hep-ph/0404183] [INSPIRE].

  31. J.C. Collins, Fragmentation of transversely polarized quarks probed in transverse momentum distributions, Nucl. Phys. B 396 (1993) 161 [hep-ph/9208213] [INSPIRE].

  32. D. Boer, P.J. Mulders and F. Pijlman, Universality of T odd effects in single spin and azimuthal asymmetries, Nucl. Phys. B 667 (2003) 201 [hep-ph/0303034] [INSPIRE].

  33. J.C. Collins and D.E. Soper, Back-To-Back Jets in QCD, Nucl. Phys. B 193 (1981) 381 [Erratum ibid. 213 (1983) 545] [INSPIRE].

  34. D. Boer, R. Jakob and P.J. Mulders, Asymmetries in polarized hadron production in e+e− annihilation up to order 1/Q, Nucl. Phys. B 504 (1997) 345 [hep-ph/9702281] [INSPIRE].

  35. D. Pitonyak, M. Schlegel and A. Metz, Polarized hadron pair production from electron-positron annihilation, Phys. Rev. D 89 (2014) 054032 [arXiv:1310.6240] [INSPIRE].

  36. P. Sun, J. Isaacson, C.P. Yuan and F. Yuan, Nonperturbative functions for SIDIS and Drell-Yan processes, Int. J. Mod. Phys. A 33 (2018) 1841006 [arXiv:1406.3073] [INSPIRE].

    ADS  Article  Google Scholar 

  37. M. Boglione, J. Collins, L. Gamberg, J.O. Gonzalez-Hernandez, T.C. Rogers and N. Sato, Kinematics of Current Region Fragmentation in Semi-Inclusive Deeply Inelastic Scattering, Phys. Lett. B 766 (2017) 245 [arXiv:1611.10329] [INSPIRE].

    ADS  Article  Google Scholar 

  38. F. Hautmann, I. Scimemi and A. Vladimirov, Non-perturbative contributions to vector-boson transverse momentum spectra in hadronic collisions, Phys. Lett. B 806 (2020) 135478 [arXiv:2002.12810] [INSPIRE].

    Article  Google Scholar 

  39. J.C. Collins and A. Metz, Universality of soft and collinear factors in hard-scattering factorization, Phys. Rev. Lett. 93 (2004) 252001 [hep-ph/0408249] [INSPIRE].

  40. A. Bacchetta, F. Delcarro, C. Pisano, M. Radici and A. Signori, Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production, JHEP 06 (2017) 081 [Erratum JHEP 06 (2019) 051] [arXiv:1703.10157] [INSPIRE].

  41. C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].

  42. C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].

  43. C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].

  44. M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].

  45. T. Becher and M. Neubert, Drell-Yan Production at Small qT, Transverse Parton Distributions and the Collinear Anomaly, Eur. Phys. J. C 71 (2011) 1665 [arXiv:1007.4005] [INSPIRE].

    ADS  Google Scholar 

  46. J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The Rapidity Renormalization Group, Phys. Rev. Lett. 108 (2012) 151601 [arXiv:1104.0881] [INSPIRE].

    ADS  Article  Google Scholar 

  47. M.G. Echevarria, A. Idilbi and I. Scimemi, Factorization Theorem For Drell-Yan At Low qT And Transverse Momentum Distributions On-The-Light-Cone, JHEP 07 (2012) 002 [arXiv:1111.4996] [INSPIRE].

    ADS  Article  Google Scholar 

  48. Belle collaboration, Transverse momentum dependent production cross sections of charged pions, kaons and protons produced in inclusive e+e− annihilation at \( \sqrt{s} \) = 10.58 GeV, Phys. Rev. D 99 (2019) 112006 [arXiv:1902.01552] [INSPIRE].

  49. M. Boglione, J.O. Gonzalez-Hernandez and R. Taghavi, Transverse parton momenta in single inclusive hadron production in e+e− annihilation processes, Phys. Lett. B 772 (2017) 78 [arXiv:1704.08882] [INSPIRE].

    ADS  Article  Google Scholar 

  50. M. Soleymaninia and H. Khanpour, Transverse momentum dependent of charged pion, kaon, and proton/antiproton fragmentation functions from e+e− annihilation process, Phys. Rev. D 100 (2019) 094033 [arXiv:1907.12294] [INSPIRE].

  51. M. Dasgupta and G.P. Salam, Resummation of nonglobal QCD observables, Phys. Lett. B 512 (2001) 323 [hep-ph/0104277] [INSPIRE].

  52. G.F. Sterman, Resummations, power corrections and interjet radiation, Acta Phys. Polon. B 36 (2005) 389 [hep-ph/0410014] [INSPIRE].

  53. A. Banfi, G. Marchesini and G. Smye, Away from jet energy flow, JHEP 08 (2002) 006 [hep-ph/0206076] [INSPIRE].

  54. T. Becher, R. Rahn and D.Y. Shao, Non-global and rapidity logarithms in narrow jet broadening, JHEP 10 (2017) 030 [arXiv:1708.04516] [INSPIRE].

    ADS  Article  Google Scholar 

  55. S. Brandt, C. Peyrou, R. Sosnowski and A. Wroblewski, The Principal axis of jets. An Attempt to analyze high-energy collisions as two-body processes, Phys. Lett. 12 (1964) 57 [INSPIRE].

  56. A. Jain, M. Procura and W.J. Waalewijn, Fully-Unintegrated Parton Distribution and Fragmentation Functions at Perturbative kT, JHEP 04 (2012) 132 [arXiv:1110.0839] [INSPIRE].

    ADS  Article  Google Scholar 

  57. M. Dasgupta and G.P. Salam, Accounting for coherence in interjet Et flow: A Case study, JHEP 03 (2002) 017 [hep-ph/0203009] [INSPIRE].

  58. T. Becher, B.D. Pecjak and D.Y. Shao, Factorization for the light-jet mass and hemisphere soft function, JHEP 12 (2016) 018 [arXiv:1610.01608] [INSPIRE].

    ADS  Article  Google Scholar 

  59. T. Becher, M. Neubert, L. Rothen and D.Y. Shao, Factorization and Resummation for Jet Processes, JHEP 11 (2016) 019 [Erratum JHEP 05 (2017) 154] [arXiv:1605.02737] [INSPIRE].

  60. E. Moffat, T.C. Rogers, N. Sato and A. Signori, Collinear factorization in wide-angle hadron pair production in e+e− annihilation, Phys. Rev. D 100 (2019) 094014 [arXiv:1909.02951] [INSPIRE].

  61. J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  62. M.A. Ebert, I.W. Stewart and Y. Zhao, Towards Quasi-Transverse Momentum Dependent PDFs Computable on the Lattice, JHEP 09 (2019) 037 [arXiv:1901.03685] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  63. M.G. Echevarria, I. Scimemi and A. Vladimirov, Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order, JHEP 09 (2016) 004 [arXiv:1604.07869] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  64. M.-X. Luo, X. Wang, X. Xu, L.L. Yang, T.-Z. Yang and H.X. Zhu, Transverse Parton Distribution and Fragmentation Functions at NNLO: the Quark Case, JHEP 10 (2019) 083 [arXiv:1908.03831] [INSPIRE].

    ADS  Article  Google Scholar 

  65. Z.-B. Kang, A. Prokudin, N. Sato and J. Terry, Efficient Fourier Transforms for Transverse Momentum Dependent Distributions, Comput. Phys. Commun. 258 (2021) 107611 [arXiv:1906.05949] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  66. C.W. Bauer, F.J. Tackmann, J.R. Walsh and S. Zuberi, Factorization and Resummation for Dijet Invariant Mass Spectra, Phys. Rev. D 85 (2012) 074006 [arXiv:1106.6047] [INSPIRE].

  67. M. Procura, W.J. Waalewijn and L. Zeune, Resummation of Double-Differential Cross Sections and Fully-Unintegrated Parton Distribution Functions, JHEP 02 (2015) 117 [arXiv:1410.6483] [INSPIRE].

    ADS  Article  Google Scholar 

  68. Y. Li, D. Neill and H.X. Zhu, An exponential regulator for rapidity divergences, Nucl. Phys. B 960 (2020) 115193 [arXiv:1604.00392] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  69. G. Lustermans, W.J. Waalewijn and L. Zeune, Joint transverse momentum and threshold resummation beyond NLL, Phys. Lett. B 762 (2016) 447 [arXiv:1605.02740] [INSPIRE].

    ADS  Article  Google Scholar 

  70. G. Sterman and M. Zeng, Quantifying Comparisons of Threshold Resummations, JHEP 05 (2014) 132 [arXiv:1312.5397] [INSPIRE].

    ADS  Article  Google Scholar 

  71. T. Becher and M. Neubert, Threshold resummation in momentum space from effective field theory, Phys. Rev. Lett. 97 (2006) 082001 [hep-ph/0605050] [INSPIRE].

  72. M. Boglione and A. Simonelli, Universality-breaking effects in e+e− hadronic production processes, arXiv:2007.13674 [INSPIRE].

  73. M. Balsiger, T. Becher and D.Y. Shao, Non-global logarithms in jet and isolation cone cross sections, JHEP 08 (2018) 104 [arXiv:1803.07045] [INSPIRE].

    ADS  Article  Google Scholar 

  74. D. de Florian, R. Sassot, M. Epele, R.J. Hernández-Pinto and M. Stratmann, Parton-to-Pion Fragmentation Reloaded, Phys. Rev. D 91 (2015) 014035 [arXiv:1410.6027] [INSPIRE].

  75. M. Balsiger, T. Becher and D.Y. Shao, NLL′ resummation of jet mass, JHEP 04 (2019) 020 [arXiv:1901.09038] [INSPIRE].

    ADS  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, University of California, Los Angeles, CA, 90095, USA

    Zhong-Bo Kang, Ding Yu Shao & Fanyi Zhao

  2. Mani L. Bhaumik Institute for Theoretical Physics, University of California, Los Angeles, CA, 90095, USA

    Zhong-Bo Kang, Ding Yu Shao & Fanyi Zhao

  3. Center for Frontiers in Nuclear Science, Stony Brook University, Stony Brook, NY, 11794, USA

    Zhong-Bo Kang & Ding Yu Shao

Authors
  1. Zhong-Bo Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ding Yu Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fanyi Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ding Yu Shao.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2007.14425

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kang, ZB., Shao, D.Y. & Zhao, F. QCD resummation on single hadron transverse momentum distribution with the thrust axis. J. High Energ. Phys. 2020, 127 (2020). https://doi.org/10.1007/JHEP12(2020)127

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP12(2020)127

Keywords

  • Perturbative QCD
  • Resummation
  • Effective Field Theories