Abstract
In models with a U(1) gauge extension beyond the Standard Model, one can derive sum rules for the couplings of the theory that are a consequence of tree-level unitarity. In this paper, we provide a comprehensive list of coupling sum rules for a general \( \textrm{SU}{(2)}_L\times \textrm{U}{(1)}_Y\times \textrm{U}{(1)}_{Y^{\prime }} \) gauge theory coupled to an arbitrary set of fermion and scalar multiplets. These results are of particular interest for models of dark matter that employ an extended gauge sector mediated by a new (dark) Z′ gauge boson. For the case of a minimal extension of the Standard Model with a \( \textrm{U}{(1)}_{Y^{\prime }} \) gauge boson, we clarify the definitions of the weak mixing angle and the electroweak ρ parameter. We demonstrate the utility of a generalized ρ parameter (denoted by ρ′) whose definition naturally follows from the unitarity sum rules developed in this paper.
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Acknowledgments
We are grateful to Bruno Bento, Ricardo Florentino, and Jorge C. Romão for discussions. The work of M.P.B. is supported in part by the Portuguese Fundação para a Ciência e Tecnologia (FCT) under contract SFRH/BD/146718/2019. The work of M.P.B. and J.P.S. supported in part by FCT under Contracts CERN/FIS-PAR/0008/2019, PTDC/FIS-PAR/29436/2017, UIDB/00777/2020, and UIDP/00777/2020; these projects are partially funded through POCTI (FEDER), COMPETE, QREN, and the EU. The work of H.E.H. is supported in part by the U.S. Department of Energy Grant No. DE-SC0010107. H.E.H. is grateful for the hospitality and support during his visit to the Instituto Superior Técnico, Universidade de Lisboa.
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Bento, M.P., Haber, H.E. & Silva, J.P. Tree-level Unitarity in \( \textrm{SU}{(2)}_L\times \textrm{U}{(1)}_Y\times \textrm{U}{(1)}_{Y^{\prime }} \) Models. J. High Energ. Phys. 2023, 83 (2023). https://doi.org/10.1007/JHEP10(2023)083
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DOI: https://doi.org/10.1007/JHEP10(2023)083