- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Gluon amplitudes in first quantization

Fiorenzo Bastianelli1,4,*, Roberto Bonezzi2,†, Olindo Corradini3,4,‡, and Filippo Fecit1,4,§

  • *Contact author: fiorenzo.bastianelli@unibo.it
  • †Contact author: roberto.bonezzi@physik.hu-berlin.de
  • ‡Contact author: olindo.corradini@unimore.it
  • §Contact author: filippo.fecit2@unibo.it
PDF

Phys. Rev. D 112, 105015 – Published 21 November, 2025

DOI: https://doi.org/10.1103/krjy-csy2

Abstract

We compute tree-level gluon amplitudes as worldline correlators of vertex operators in a bosonic spinning particle model. In this framework, the particle’s position degrees of freedom are extended by complex bosonic variables that encode its spin. In the free theory, the model exhibits a first-class constraint algebra whose gauging ensures the unitarity of the quantum theory. This algebra is a contraction of the sl(2,R) algebra, which is by itself a subalgebra of the Virasoro algebra. In string theory, gauging the Virasoro algebra plays a similar role. Our model admits a consistent truncation to describe a pure spin-1 particle. Non-Abelian interactions are introduced by using Becchi-Rouet-Stora-Tyutin (BRST) techniques, which allow us to extract the vertex operators of the theory as suitable deformations of the BRST charge. BRST invariance is central to ensuring the consistency of the tree-level amplitudes analyzed in this work. We discuss connections with similar worldline constructions and comment on the potential relevance of this framework for uncovering the structures underlying the double-copy program in gauge and gravitational theories.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (56)

  1. Roberto Bonezzi, Yang-Mills theory from the worldline, Phys. Rev. D 110, 065022 (2024).
  2. Peng Dai, Yu-tin Huang, and Warren Siegel, Worldgraph approach to Yang-Mills amplitudes from N=2 spinning particle, J. High Energy Phys. 10 (2008) 027.
  3. F. A. Berezin and M. S. Marinov, Particle spin dynamics as the Grassmann variant of classical mechanics, Ann. Phys. (N.Y.) 104, 336 (1977).
  4. A. Barducci, R. Casalbuoni, and L. Lusanna, Supersymmetries and the pseudoclassical relativistic electron, Nuovo Cimento Soc. Ital. Fis. 35A, 377 (1976).
  5. L. Brink, Stanley Deser, B. Zumino, P. Di Vecchia, and Paul S. Howe, Local supersymmetry for spinning particles, Phys. Lett. B 64, 435 (1976); 68, 488(E) (1977).
  6. V. D. Gershun and V. I. Tkach, Classical and quantum dynamics of particles with arbitrary spin, JETP Lett. 29, 288 (1979), http://jetpletters.ru/ps/1451/article_22085.shtml.
  7. Paul S. Howe, Silvia Penati, Mario Pernici, and Paul K. Townsend, Wave equations for arbitrary spin from quantization of the extended supersymmetric spinning particle, Phys. Lett. B 215, 555 (1988).
  8. Anders K. H. Bengtsson, A unified action for higher spin gauge bosons from covariant string theory, Phys. Lett. B 182, 321 (1986).
  9. M. Henneaux and C. Teitelboim, First and second quantized point particles of any spin, in 2nd Meeting on Quantum Mechanics of Fundamental Systems (CECS) (1987).
  10. N. Bouatta, G. Compere, and A. Sagnotti, An introduction to free higher-spin fields, in 1st Solvay Workshop on Higher Spin Gauge Theories (2004), pp. 79–99.
  11. K. Hallowell and A. Waldron, Supersymmetric quantum mechanics and super-Lichnerowicz algebras, Commun. Math. Phys. 278, 775 (2008).
  12. F. Bastianelli, O. Corradini, and A. Waldron, Detours and paths: BRST complexes and worldline formalism, J. High Energy Phys. 05 (2009) 017.
  13. Fiorenzo Bastianelli, Filippo Fecit, and Alessandro Miccichè, Pair production of massive charged vector bosons from the worldline, J. High Energy Phys. 09 (2025) 201.
  14. Kays Haddad, Gustav Uhre Jakobsen, Gustav Mogull, and Jan Plefka, Spinning bodies in general relativity from bosonic worldline oscillators, J. High Energy Phys. 02 (2025) 019.
  15. Gustav Mogull, Jan Plefka, and Kathrin Stoldt, Radiated angular momentum from spinning black hole scattering trajectories, arXiv:2506.20643.
  16. Paul S. Howe, Silvia Penati, Mario Pernici, and Paul K. Townsend, A particle mechanics description of antisymmetric tensor fields, Classical Quantum Gravity 6, 1125 (1989).
  17. Fiorenzo Bastianelli, Paolo Benincasa, and Simone Giombi, Worldline approach to vector and antisymmetric tensor fields, J. High Energy Phys. 04 (2005) 010.
  18. Fiorenzo Bastianelli, Paolo Benincasa, and Simone Giombi, Worldline approach to vector and antisymmetric tensor fields. II. J. High Energy Phys. 10 (2005) 114.
  19. Eric Laenen, Gerben Stavenga, and Chris D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, J. High Energy Phys. 03 (2009) 054.
  20. Domenico Bonocore, Asymptotic dynamics on the worldline for spinning particles, J. High Energy Phys. 02 (2021) 007.
  21. Gustav Mogull, Jan Plefka, and Jan Steinhoff, Classical black hole scattering from a worldline quantum field theory, J. High Energy Phys. 02 (2021) 048.
  22. Roberto Bonezzi and Maria Foteini Kallimani, Worldline geometries for scattering amplitudes, J. High Energy Phys. 06 (2025) 167.
  23. Zvi Bern and David A. Kosower, Efficient calculation of one loop QCD amplitudes, Phys. Rev. Lett. 66, 1669 (1991).
  24. Zvi Bern and David A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B362, 389 (1991).
  25. Zvi Bern and David A. Kosower, The Computation of loop amplitudes in gauge theories, Nucl. Phys. B379, 451 (1992).
  26. Zvi Bern and David C. Dunbar, A mapping between Feynman and string motivated one loop rules in gauge theories, Nucl. Phys. B379, 562 (1992).
  27. Christian Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rep. 355, 73 (2001).
  28. James P. Edwards and Christian Schubert, Quantum mechanical path integrals in the first quantised approach to quantum field theory, arXiv:1912.10004 [Int. J. Theor. Phys. (to be published)].
  29. Frits A. Berends and W. T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B306, 759 (1988).
  30. Yuchen Du and Diana Vaman, Tree-level graviton scattering in the worldline formalism, arXiv:2308.11326.
  31. Yuchen Du, Siddarth Ajith, Ravisankar Rajagopal, and Diana Vaman, Worldline proof of eikonal exponentiation, J. High Energy Phys. 09 (2025) 161.
  32. Siddarth Ajith, Yuchen Du, Ravisankar Rajagopal, and Diana Vaman, Worldline formalism, eikonal expansion and the classical limit of scattering amplitudes, arXiv:2409.17866.
  33. Andrea Nützi and Michael Reiterer, Amplitudes in YM and GR as a minimal model and recursive characterization, Commun. Math. Phys. 392, 427 (2022).
  34. Alex S. Arvanitakis, The L∞-algebra of the S-matrix, J. High Energy Phys. 07 (2019) 115.
  35. Tommaso Macrelli, Christian Sämann, and Martin Wolf, Scattering amplitude recursion relations in Batalin-Vilkovisky-quantizable theories, Phys. Rev. D 100, 045017 (2019).
  36. Cristhiam Lopez-Arcos and Alexander Quintero Vélez, L∞-algebras and the perturbiner expansion, J. High Energy Phys. 11 (2019) 010.
  37. Roberto Bonezzi, Christoph Chiaffrino, Felipe Diaz-Jaramillo, and Olaf Hohm, Tree-level scattering amplitudes via homotopy transfer, arXiv:2312.09306.
  38. Roberto Bonezzi, Christoph Chiaffrino, and Olaf Hohm, Vertex operators for the kinematic algebra of Yang-Mills theory, Phys. Rev. D 111, 065002 (2025).
  39. F. Bastianelli and P. van Nieuwenhuizen, Path Integrals and Anomalies in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2006.
  40. Anton M. Zeitlin, Conformal field theory and algebraic structure of gauge theory, J. High Energy Phys. 03 (2010) 056.
  41. Roberto Bonezzi, Felipe Diaz-Jaramillo, and Olaf Hohm, The gauge structure of double field theory follows from Yang-Mills theory, Phys. Rev. D 106, 026004 (2022).
  42. K. G. Selivanov, SD perturbiner in Yang-Mills + gravity, Phys. Lett. B 420, 274 (1998).
  43. Sebastian Mizera and Barbara Skrzypek, Perturbiner methods for effective field theories and the double copy, J. High Energy Phys. 10 (2018) 018.
  44. K. Daikouji, M. Shino, and Y. Sumino, Bern-Kosower rule for scalar QED, Phys. Rev. D 53, 4598 (1996).
  45. Zvi Bern, String based perturbative methods for gauge theories, in Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles (World Scientific, Singapore, 1992), pp. 0471–536.
  46. Z. Bern, J. J. M. Carrasco, and Henrik Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78, 085011 (2008).
  47. Zvi Bern, John Joseph Carrasco, Marco Chiodaroli, Henrik Johansson, and Radu Roiban, The duality between color and kinematics and its applications, J. Phys. A 57, 333002 (2024).
  48. Zvi Bern, John Joseph M. Carrasco, and Henrik Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105, 061602 (2010).
  49. Zvi Bern, John Joseph Carrasco, Marco Chiodaroli, Henrik Johansson, and Radu Roiban, The SAGEX review on scattering amplitudes Chapter 2: An invitation to color-kinematics duality and the double copy, J. Phys. A 55, 443003 (2022).
  50. Tim Adamo, John Joseph M. Carrasco, Mariana Carrillo-González, Marco Chiodaroli, Henriette Elvang, Henrik Johansson, Donal O’Connell, Radu Roiban, and Oliver Schlotterer, Snowmass White Paper: The Double Copy and its Applications, in Snowmass 2021 (2022), https://www.slac.stanford.edu/econf/C210711/.
  51. Roberto Bonezzi, Christoph Chiaffrino, Felipe Diaz-Jaramillo, and Olaf Hohm, Gauge invariant double copy of Yang-Mills theory: The quartic theory, Phys. Rev. D 107, 126015 (2023).
  52. Naser Ahmadiniaz, Filippo Maria Balli, Cristhiam Lopez-Arcos, Alexander Quintero Velez, and Christian Schubert, Color-kinematics duality from the Bern-Kosower formalism, Phys. Rev. D 104, L041702 (2021).
  53. Naser Ahmadiniaz, Filippo Maria Balli, Olindo Corradini, Cristhiam Lopez-Arcos, Alexander Quintero Velez, and Christian Schubert, Manifest colour-kinematics duality and double-copy in the string-based formalism, Nucl. Phys. B975, 115690 (2022).
  54. Canxin Shi and Jan Plefka, Classical double copy of worldline quantum field theory, Phys. Rev. D 105, 026007 (2022).
  55. Fiorenzo Bastianelli, Francesco Comberiati, and Leonardo de la Cruz, Worldline description of a bi-adjoint scalar and the zeroth copy, J. High Energy Phys. 12 (2021) 023.
  56. Naser Ahmadiniaz, Fiorenzo Bastianelli, and Olindo Corradini, Dressed scalar propagator in a non-Abelian background from the worldline formalism, Phys. Rev. D 93, 025035 (2016); 93, 049904(A) (2016).

Outline

Information

Phys. Rev. D 112, 105015– Published 21 November, 2025
  • Received 20 August 2025
  • Accepted 4 November 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation