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  • Open Access

Weak bosons as partons below 10 TeV partonic center of momentum

Innes Bigaran1,2,3,* and Richard Ruiz4,†

  • *Contact author: ibigaran@fnal.gov
  • †Contact author: rruiz@ifj.edu.pl
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Phys. Rev. D 113, 113006 – Published 15 June, 2026

DOI: https://doi.org/10.1103/2pqd-jg7c

Abstract

We investigate the modeling of weak boson number densities for leptons and hadrons in practical calculations in the Standard Model. In the framework of the Effective W Approximation (EWA) and in the Rξ and axial gauges, we derive the unrenormalized, tree-level parton number densities for weak bosons from massless fermions at next-to-leading power in the collinear expansion. Corrections exhibit various pathologies and properties, including those conjectured but not proven, and parallel heavy quark factorization. We suppress pathologies through a new set of kinematical consistency conditions. When satisfied, shapes and normalizations of full matrix elements for many-leg processes can be well approximated by the EWA and fragmentation contributions at leading power, suggesting the onset of tree-level factorization. Findings also suggest that the EWA is testable at the LHC with L=450  fb−1 of same-sign WW scattering data at s=13.6  TeV.

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References (76)

  1. E. Witten, Heavy quark contributions to deep inelastic scattering, Nucl. Phys. B104, 445 (1976).
  2. M. A. G. Aivazis, F. I. Olness, and W.-K. Tung, Leptoproduction of heavy quarks. I. General formalism and kinematics of charged current and neutral current production processes, Phys. Rev. D 50, 3085 (1994).
  3. M. A. G. Aivazis, J. C. Collins, F. I. Olness, and W.-K. Tung, Leptoproduction of heavy quarks. II. A unified QCD formulation of charged and neutral current processes from fixed target to collider energies, Phys. Rev. D 50, 3102 (1994).
  4. T. Han, J. Sayre, and S. Westhoff, Top-quark initiated processes at high-energy hadron colliders, J. High Energy Phys. 04 (2015) 145.
  5. S. Dawson, A. Ismail, and I. Low, Redux on “When is the top quark a parton?,” Phys. Rev. D 90, 014005 (2014).
  6. A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, Parton distributions incorporating QED contributions, Eur. Phys. J. C 39, 155 (2005).
  7. A. Manohar, P. Nason, G. P. Salam, and G. Zanderighi, How bright is the proton? A precise determination of the photon parton distribution function, Phys. Rev. Lett. 117, 242002 (2016).
  8. A. V. Manohar, P. Nason, G. P. Salam, and G. Zanderighi, The photon content of the proton, J. High Energy Phys. 12 (2017) 046.
  9. K. Xie, B. Zhou, and T. J. Hobbs (CTEQ-TEA Collaboration), The photon content of the neutron, J. High Energy Phys. 04 (2024) 022.
  10. S. Dawson, The effective W approximation, Nucl. Phys. B249, 42 (1985).
  11. G. L. Kane, W. W. Repko, and W. B. Rolnick, The effective W±, Z0 approximation for high-energy collisions, Phys. Lett. 148B, 367 (1984).
  12. Z. Kunszt and D. E. Soper, On the validity of the effective W approximation, Nucl. Phys. B296, 253 (1988).
  13. M. Ciafaloni, P. Ciafaloni, and D. Comelli, Electroweak Bloch-Nordsieck violation at the TeV scale: ’Strong’ weak interactions?, Nucl. Phys. B589, 359 (2000).
  14. M. Ciafaloni, P. Ciafaloni, and D. Comelli, Bloch-Nordsieck violating electroweak corrections to inclusive TeV scale hard processes, Phys. Rev. Lett. 84, 4810 (2000).
  15. M. Ciafaloni, P. Ciafaloni, and D. Comelli, Bloch-Nordsieck violation in spontaneously broken Abelian theories, Phys. Rev. Lett. 87, 211802 (2001).
  16. P. Ciafaloni and D. Comelli, Electroweak evolution equations, J. High Energy Phys. 11 (2005) 022.
  17. J.-y. Chiu, F. Golf, R. Kelley, and A. V. Manohar, Electroweak Sudakov corrections using effective field theory, Phys. Rev. Lett. 100, 021802 (2008).
  18. J.-y. Chiu, A. Fuhrer, R. Kelley, and A. V. Manohar, Factorization structure of gauge theory amplitudes and application to hard scattering processes at the LHC, Phys. Rev. D 80, 094013 (2009).
  19. A. Manohar, B. Shotwell, C. Bauer, and S. Turczyk, Non-cancellation of electroweak logarithms in high-energy scattering, Phys. Lett. B 740, 179 (2015).
  20. J. Chen, T. Han, and B. Tweedie, Electroweak splitting functions and high energy showering, J. High Energy Phys. 11 (2017) 093.
  21. C. W. Bauer, N. Ferland, and B. R. Webber, Standard model parton distributions at very high energies, J. High Energy Phys. 08 (2017) 036.
  22. C. W. Bauer and B. R. Webber, Polarization effects in standard model parton distributions at very high energies, J. High Energy Phys. 03 (2019) 013.
  23. T. Han, Y. Ma, and K. Xie, High energy leptonic collisions and electroweak parton distribution functions, Phys. Rev. D 103, L031301 (2021).
  24. S. Plätzer and M. Sjodahl, Amplitude factorization in the electroweak standard model, Phys. Rev. D 110, 056023 (2024).
  25. S. Frixione and G. Stagnitto, The muon parton distribution functions, J. High Energy Phys. 12 (2023) 170.
  26. J. Lindfors, Distribution functions for heavy vector bosons inside colliding particle beams, Z. Phys. C 28, 427 (1985).
  27. R. Kleiss and W. J. Stirling, Anomalous high-energy behavior in boson fusion, Phys. Lett. B 182, 75 (1986).
  28. P. W. Johnson, F. I. Olness, and W.-K. Tung, The effective vector boson method for high-energy collisions, Phys. Rev. D 36, 291 (1987).
  29. A. Abbasabadi, W. W. Repko, D. A. Dicus, and R. Vega, Comparison of exact and effective gauge boson calculations for gauge boson fusion processes, Phys. Rev. D 38, 2770 (1988).
  30. I. Kuss and H. Spiesberger, Luminosities for vector boson—vector boson scattering at high-energy colliders, Phys. Rev. D 53, 6078 (1996).
  31. I. Kuss, Improved effective vector boson approximation for hadron hadron collisions, Phys. Rev. D 55, 7165 (1997).
  32. E. Accomando, A. Ballestrero, A. Belhouari, and E. Maina, Isolating vector boson scattering at the LHC: Gauge cancellations and the equivalent vector boson approximation vs complete calculations, Phys. Rev. D 74, 073010 (2006).
  33. J. Brehmer, J. Jaeckel, and T. Plehn, Polarized WW scattering on the Higgs pole, Phys. Rev. D 90, 054023 (2014).
  34. P. Borel, R. Franceschini, R. Rattazzi, and A. Wulzer, Probing the scattering of equivalent electroweak bosons, J. High Energy Phys. 06 (2012) 122.
  35. A. Costantini, F. De Lillo, F. Maltoni, L. Mantani, O. Mattelaer, R. Ruiz, and X. Zhao, Vector boson fusion at multi-TeV muon colliders, J. High Energy Phys. 09 (2020) 080.
  36. B. Fuks, J. Neundorf, K. Peters, R. Ruiz, and M. Saimpert, Majorana neutrinos in same-sign W±W± scattering at the LHC: Breaking the TeV barrier, Phys. Rev. D 103, 055005 (2021).
  37. R. Ruiz, A. Costantini, F. Maltoni, and O. Mattelaer, The effective vector boson approximation in high-energy muon collisions, J. High Energy Phys. 06 (2022) 114.
  38. A. M. Sirunyan et al. (CMS Collaboration), Observation of electroweak production of same-sign W boson pairs in the two jet and two same-sign lepton final state in proton-proton collisions at s=13  TeV, Phys. Rev. Lett. 120, 081801 (2018).
  39. M. Aaboud et al. (ATLAS Collaboration), Observation of electroweak production of a same-sign W boson pair in association with two jets in pp collisions at s=13  TeV with the ATLAS detector, Phys. Rev. Lett. 123, 161801 (2019).
  40. G. Aad et al. (ATLAS Collaboration), Measurement and interpretation of same-sign W boson pair production in association with two jets in pp collisions at s=13  TeV with the ATLAS detector, J. High Energy Phys. 04 (2024) 026.
  41. A. M. Sirunyan et al. (CMS Collaboration), Measurements of production cross sections of polarized same-sign W boson pairs in association with two jets in proton-proton collisions at s=13  TeV, Phys. Lett. B 812, 136018 (2021).
  42. A. Ballestrero et al., Precise predictions for same-sign W-boson scattering at the LHC, Eur. Phys. J. C 78, 671 (2018).
  43. D. Buarque Franzosi et al., Vector boson scattering processes: Status and prospects, Rev. Phys. 8, 100071 (2022).
  44. S. Dittmaier, P. Maierhöfer, C. Schwan, and R. Winterhalder, Like-sign W-boson scattering at the LHC—approximations and full next-to-leading-order predictions, J. High Energy Phys. 11 (2023) 022.
  45. B. Henning, D. Lombardo, M. Riembau, and F. Riva, Measuring Higgs couplings without Higgs bosons, Phys. Rev. Lett. 123, 181801 (2019).
  46. R. Bellan et al., A sensitivity study of VBS and diboson WW to dimension-6 EFT operators at the LHC, J. High Energy Phys. 05 (2022) 039.
  47. D. A. Dicus and S. Willenbrock, Higgs boson production from heavy quark fusion, Phys. Rev. D 39, 751 (1989).
  48. F. Maltoni, Z. Sullivan, and S. Willenbrock, Higgs-boson production via bottom-quark fusion, Phys. Rev. D 67, 093005 (2003).
  49. P. Huang, A. J. Long, and L.-T. Wang, Probing the electroweak phase transition with Higgs factories and gravitational waves, Phys. Rev. D 94, 075008 (2016).
  50. M. J. Ramsey-Musolf, The electroweak phase transition: A collider target, J. High Energy Phys. 09 (2019) 179.
  51. M. O. Olea Romacho, Higgs physics as a window to the electroweak epoch, Ph.D. thesis, Hamburg University, 2022.

  52. Note that this decoupling of Goldstone bosons, at least at tree level, can possibly soften Bloch-Nordsieck violations [15], i.e., uncancelled infrared EW logarithms.

  53. W. Bernreuther and L. Chen, Improved effective vector boson approximation revisited, Phys. Rev. D 93, 053018 (2016).
  54. T. Basu and R. Ruiz, The four polarizations of the W at high energies, arXiv:2512.10015.
  55. G. Altarelli, B. Mele, and F. Pitolli, Heavy Higgs production at future colliders, Nucl. Phys. B287, 205 (1987).
  56. M. E. Peskin and D. V. Schroeder, An introduction to Quantum Field Theory (Addison-Wesley, Reading, USA, 1995).
  57. R. P. Kauffman, Production of top quarks via vector boson fusion in e+e− collisions, Phys. Rev. D 41, 3343 (1990).
  58. S. Frixione, M. L. Mangano, P. Nason, and G. Ridolfi, Improving the Weizsacker-Williams approximation in electron—proton collisions, Phys. Lett. B 319, 339 (1993).
  59. C. Dams and R. Kleiss, The electroweak standard model in the axial gauge, Eur. Phys. J. C 34, 419 (2004).
  60. K. M. Black et al., Muon collider forum report, J. Instrum. 19, T02015 (2024).
  61. S. Asai et al. (P5 Collaboration), Exploring the quantum universe: Pathways to innovation and discovery in particle physics, arXiv:2407.19176.
  62. C. Accettura et al., Towards a muon collider, Eur. Phys. J. C 83, 864 (2023); 84, 36(E) (2024).
  63. D. Buarque Franzosi, O. Mattelaer, R. Ruiz, and S. Shil, Automated predictions from polarized matrix elements, J. High Energy Phys. 04 (2020) 082.
  64. Z. Was, Gauge invariance, infrared / collinear singularities and tree level matrix element for e+e−→νeνe¯γγ, Eur. Phys. J. C 44, 489 (2005).
  65. T. Stelzer and W. F. Long, Automatic generation of tree level helicity amplitudes, Comput. Phys. Commun. 81, 357 (1994).
  66. J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, J. High Energy Phys. 07 (2014) 079.
  67. 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. B250, 199 (1985).
  68. C. W. Bauer and N. Ferland, Resummation of electroweak Sudakov logarithms for real radiation, J. High Energy Phys. 09 (2016) 025.
  69. H. Contopanagos, E. Laenen, and G. F. Sterman, Sudakov factorization and resummation, Nucl. Phys. B484, 303 (1997).
  70. H. Brooks, P. Skands, and R. Verheyen, Interleaved resonance decays and electroweak radiation in the Vincia parton shower, SciPost Phys. 12, 101 (2022).
  71. P. M. Bredt, W. Kilian, J. Reuter, and P. Stienemeier, NLO electroweak corrections to multi-boson processes at a muon collider, J. High Energy Phys. 12 (2022) 138.
  72. A. Denner and S. Rode, Automated resummation of electroweak sudakov logarithms in diboson production at future colliders, Eur. Phys. J. C 84, 542 (2024).
  73. Y. Ma, D. Pagani, and M. Zaro, EW corrections and heavy boson radiation at a high-energy muon collider, Phys. Rev. D 111, 053002 (2025).
  74. V. Ahrens, T. Becher, M. Neubert, and L. L. Yang, Renormalization-group improved prediction for Higgs production at hadron colliders, Eur. Phys. J. C 62, 333 (2009).
  75. T. Becher, A. Broggio, and A. Ferroglia, Introduction to soft-collinear effective theory, Lect. Notes Phys. 896, 1 (2015).
  76. I. Bigaran, EWA_MG5, https://github.com/innesbigaran/EWA_MG5 (2026), gitHub repository.

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Phys. Rev. D 113, 113006– Published 15 June, 2026
  • Received 28 February 2025
  • Accepted 24 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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