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Momentum shift and on-shell recursion relation for electroweak theory

Yohei Ema1,2,*, Ting Gao1,†, Wenqi Ke1,2,‡, Zhen Liu1,§, Kun-Feng Lyu1,∥, and Ishmam Mahbub1,¶

  • 1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
  • 2William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

  • *Contact author: ema00001@umn.edu
  • †Contact author: gao00212@umn.edu
  • ‡Contact author: wke@umn.edu
  • §Contact author: zliuphys@umn.edu
  • ∥Contact author: lyu00145@umn.edu
  • Contact author: mahbu008@umn.edu
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Phys. Rev. D 110, 105002 – Published 7 November, 2024

DOI: https://doi.org/10.1103/PhysRevD.110.105002

Abstract

We study the all-line transverse (ALT) shift which we developed for on-shell recursion of amplitudes for particles of any mass. We discuss the validity of the shift for general theories of spin ≤1, and illustrate the connection between Ward identity and constructibility for massive spin-1 amplitude under the ALT shift. We apply the shift to the electroweak theory, and various four-point scattering amplitudes among electroweak gauge bosons and fermions are constructed. We show explicitly that the four-point gauge boson contact terms in massive electroweak theory automatically arise after recursive construction, independent of UV completion, and they automatically cancel the terms growing as (energy)4 at high energy. We explore UV completion of the electroweak theory that cancels the remaining (energy)2 terms and impose unitarity requirements to constrain additional couplings. The ALT shift framework allows consistent treatment in dealing with contact term ambiguities for renormalizable massive and massless theories, which we show can be useful in studying real-world amplitudes with massive spinors.

Physics Subject Headings (PhySH)

See Also

Momentum shift and on-shell constructible massive amplitudes

Yohei Ema, Ting Gao, Wenqi Ke, Zhen Liu, Kun-Feng Lyu, and Ishmam Mahbub
Phys. Rev. D 110, 105003 (2024)

Article Text

References (56)

  1. R. Britto, F. Cachazo, and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B715, 499 (2005).
  2. R. Britto, F. Cachazo, B. Feng, and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94, 181602 (2005).
  3. J. Bedford, A. Brandhuber, B. J. Spence, and G. Travaglini, A recursion relation for gravity amplitudes, Nucl. Phys. B721, 98 (2005).
  4. F. Cachazo and P. Svrcek, Tree level recursion relations in general relativity, arXiv:hep-th/0502160.
  5. N. Arkani-Hamed, F. Cachazo, and J. Kaplan, What is the simplest quantum field theory?, J. High Energy Phys. 09 (2010) 016.
  6. T. Cohen, H. Elvang, and M. Kiermaier, On-shell constructibility of tree amplitudes in general field theories, J. High Energy Phys. 04 (2011) 053.
  7. H. Elvang and Y.-t. Huang, Scattering amplitudes, arXiv:1308.1697.
  8. J. M. Henn and J. C. Plefka, Scattering Amplitudes in Gauge Theories (Springer, Berlin, 2014), Vol. 883.
  9. C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, and J. Trnka, On-shell recursion relations for effective field theories, Phys. Rev. Lett. 116, 041601 (2016).
  10. C. Cheung, TASI lectures on scattering amplitudes, in Anticipating the Next Discoveries in Particle Physics (World Scientific, Singapore, 2018), pp. 571–623.
  11. G. Travaglini et al., The SAGEX review on scattering amplitudes, J. Phys. A 55, 443001 (2022).
  12. S. D. Badger, E. W. N. Glover, V. V. Khoze, and P. Svrcek, Recursion relations for gauge theory amplitudes with massive particles, J. High Energy Phys. 07 (2005) 025.
  13. S. D. Badger, E. W. N. Glover, and V. V. Khoze, Recursion relations for gauge theory amplitudes with massive vector bosons and fermions, J. High Energy Phys. 01 (2006) 066.
  14. K. J. Ozeren and W. J. Stirling, Scattering amplitudes with massive fermions using BCFW recursion, Eur. Phys. J. C 48, 159 (2006).
  15. R. Boels and C. Schwinn, CSW rules for a massive scalar, Phys. Lett. B 662, 80 (2008).
  16. S. Ballav and A. Manna, Recursion relations for scattering amplitudes with massive particles, J. High Energy Phys. 03 (2021) 295.
  17. N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang, Scattering amplitudes for all masses and spins, J. High Energy Phys. 11 (2021) 070.
  18. A. Herderschee, S. Koren, and T. Trott, Constructing N=4 Coulomb branch superamplitudes, J. High Energy Phys. 08 (2019) 107.
  19. A. Herderschee, S. Koren, and T. Trott, Massive on-shell supersymmetric scattering amplitudes, J. High Energy Phys. 10 (2019) 092.
  20. R. Franken and C. Schwinn, On-shell constructibility of born amplitudes in spontaneously broken gauge theories, J. High Energy Phys. 02 (2020) 073.
  21. C. Wu and S.-H. Zhu, Massive on-shell recursion relations for n-point amplitudes, J. High Energy Phys. 06 (2022) 117.
  22. Y. Ema, T. Gao, W. Ke, Z. Liu, K.-F. Lyu, and I. Mahbub, following paper, Momentum shift and on-shell constructible massive amplitudes, Phys. Rev. D 110, 105003 (2024).
  23. B. Bachu and A. Yelleshpur, On-shell electroweak sector and the Higgs mechanism, J. High Energy Phys. 08 (2020) 039.
  24. D. Liu and Z. Yin, Gauge invariance from on-shell massive amplitudes and tree-level unitarity, Phys. Rev. D 106, 076003 (2022).
  25. N. Christensen, perturbative unitarity and the 4-point vertices in the constructive Standard Model, Phys. Rev. D 109, 116014 (2024).
  26. N. Christensen, A complete set of 4-point amplitudes in the constructive standard model, Eur. Phys. J. C 84, 620 (2024).
  27. N. Christensen, H. Diaz-Quiroz, B. Field, J. Hayward, and J. Miles, Challenges with internal photons in constructive QED, Nucl. Phys. B993, 116278 (2023).
  28. H.-Y. Lai, D. Liu, and J. Terning, The constructive method for massive particles in QED, J. High Energy Phys. 06 (2024) 086.
  29. C. Cheung, C.-H. Shen, and J. Trnka, Simple recursion relations for general field theories, J. High Energy Phys. 06 (2015) 118.
  30. Y. Shadmi and Y. Weiss, Effective field theory amplitudes the on-shell way: Scalar and vector couplings to gluons, J. High Energy Phys. 02 (2019) 165.
  31. R. Aoude and C. S. Machado, The rise of SMEFT on-shell amplitudes, J. High Energy Phys. 12 (2019) 058.
  32. G. Durieux, T. Kitahara, Y. Shadmi, and Y. Weiss, The electroweak effective field theory from on-shell amplitudes, J. High Energy Phys. 01 (2020) 119.
  33. G. Durieux, T. Kitahara, C. S. Machado, Y. Shadmi, and Y. Weiss, Constructing massive on-shell contact terms, J. High Energy Phys. 12 (2020) 175.
  34. R. Balkin, G. Durieux, T. Kitahara, Y. Shadmi, and Y. Weiss, On-shell Higgsing for EFTs, J. High Energy Phys. 03 (2022) 129.
  35. H. Liu, T. Ma, Y. Shadmi, and M. Waterbury, An EFT Hunter’s guide to two-to-two scattering: HEFT and SMEFT on-shell amplitudes, J. High Energy Phys. 05 (2023) 241.
  36. C. H. Llewellyn Smith, High-energy behavior and gauge symmetry, Phys. Lett. 46B, 233 (1973).
  37. B. W. Lee, C. Quigg, and H. B. Thacker, The strength of weak interactions at very high-energies and the Higgs boson mass, Phys. Rev. Lett. 38, 883 (1977).
  38. B. W. Lee, C. Quigg, and H. B. Thacker, Weak interactions at very high-energies: The role of the Higgs boson mass, Phys. Rev. D 16, 1519 (1977).
  39. J. Bonifacio and K. Hinterbichler, Bounds on amplitudes in effective theories with massive spinning particles, Phys. Rev. D 98, 045003 (2018).
  40. C. Csaki, C. Grojean, H. Murayama, L. Pilo, and J. Terning, Gauge theories on an interval: Unitarity without a Higgs, Phys. Rev. D 69, 055006 (2004).
  41. C. Csaki, C. Grojean, L. Pilo, and J. Terning, Towards a realistic model of Higgsless electroweak symmetry breaking, Phys. Rev. Lett. 92, 101802 (2004).
  42. F. Nardi, L. Ricci, and A. Wulzer, Low-virtuality splitting in the standard model, arXiv:2405.08220.
  43. M.-Z. Chung, Y.-T. Huang, J.-W. Kim, and S. Lee, The simplest massive S-matrix: From minimal coupling to black holes, J. High Energy Phys. 04 (2019) 156.
  44. W.-M. Chen, M.-Z. Chung, Y.-t. Huang, and J.-W. Kim, The 2PM Hamiltonian for binary Kerr to quartic in spin, J. High Energy Phys. 08 (2022) 148.
  45. M. Chiodaroli, H. Johansson, and P. Pichini, Compton black-hole scattering for S ≤5/2, J. High Energy Phys. 02 (2022) 156.
  46. K. Haddad, Recursion in the classical limit and the neutron-star Compton amplitude, J. High Energy Phys. 05 (2023) 177.
  47. A. Pokraka, S. Rajan, L. Ren, A. Volovich, and W. W. Zhao, Five-dimensional spinor helicity for all masses and spins, arXiv:2405.09533.
  48. H. Kawai, D. C. Lewellen, and S. H. H. Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B269, 1 (1986).
  49. Z. Bern, J. J. M. Carrasco, and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78, 085011 (2008).
  50. S. H. Henry Tye and Y. Zhang, Dual identities inside the gluon and the graviton scattering amplitudes, J. High Energy Phys. 06 (2010) 071; 04 (2011) 114(E).
  51. B. Feng, R. Huang, and Y. Jia, Gauge amplitude identities by on-shell recursion relation in S-matrix program, Phys. Lett. B 695, 350 (2011).
  52. Z. Bern, T. Dennen, Y.-t. Huang, and M. Kiermaier, Gravity as the square of gauge theory, Phys. Rev. D 82, 065003 (2010).
  53. N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard, and P. Vanhove, The momentum kernel of gauge and gravity theories, J. High Energy Phys. 01 (2011) 001.
  54. C. R. Mafra, O. Schlotterer, and S. Stieberger, Explicit BCJ numerators from pure spinors, J. High Energy Phys. 07 (2011) 092.
  55. C.-H. Fu, Y.-J. Du, and B. Feng, An algebraic approach to BCJ numerators, J. High Energy Phys. 03 (2013) 050.
  56. J. Ellis, tikz-feynman: Feynman diagrams with tikz, Comput. Phys. Commun. 210, 103 (2017).

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Information

Phys. Rev. D 110, 105002– Published 7 November, 2024
  • Received 6 August 2024
  • Accepted 2 October 2024
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