Higgs Singlet Extension of the Standard Model

The Higgs Singlet Extension of the Standard Model (HSESM) is a simple extended Higgs sector with one additional Higgs singlet field with hypercharge \(Y_S=0\). Our conventions differs from the literature [SW05], [PW06], [BCW07], [PR13]. Translation rules are given in Appendix A in [DDL18].

The most general CP-conserving \(Z_2\)-symmetric renormalizable scalar potential reads

(1)\[V_{\mathrm{HSESM}} = m_1^2 \Phi^\dagger \Phi + m_2^2 S^2 +\frac{\lambda_1}{2} \left(\Phi^\dagger \Phi\right)^2 +\frac{\lambda_2}{2} S^4 +\lambda_3 \Phi^\dagger \Phi S^2,\]

with \(\Phi\) being the SM Higgs doublet and \(S\) being a singlet field, and all parameters are real. We choose the following set of physical parameters:

Basis HSESM potential Gauge part
before SSB \(m_1\), \(m_2\), \(m_{12}\), \(\lambda_1\), \(\lambda_2\), \(\lambda_3\) \(g\), \(g^\prime\)
Recola2 input \(M_{\mathrm{H}_l}\), \(M_{\mathrm{H}_h}\), \(s_{\alpha}\), \(t_\beta\), \(M_\mathrm{W}\) \(\alpha_\mathrm{em}\), \(M_\mathrm{Z}\)

The angle \(\boldsymbol \alpha\) (sas = \(\sin(\alpha)\)) is defined in the same way as in the THDM and \(\boldsymbol{t_\beta} = \tan(\beta)\) (=tb) is defined as \(t_\beta = \mathrm{vev}_s/\mathrm{vev}\), i.e. the ratio of the vevs associated to \(S\) and \(\Phi\) [DLU17].

For comparison we list key couplings of type VVS (\(g^{\mu \nu}\) omitted) and SSS:

\[\begin{split}\mathrm{i}\lambda_{\mathrm{Z} \mathrm{Z} \mathrm{H}_l} &= -\mathrm{i} s_\alpha \frac{e M_\mathrm{Z} }{c_\mathrm{w} s_\mathrm{w}}\\ \mathrm{i}\lambda_{\mathrm{Z} \mathrm{Z} \mathrm{H}_l} &= +\mathrm{i} c_\alpha \frac{e M_\mathrm{Z} }{c_\mathrm{w} s_\mathrm{w}}\\ \mathrm{i}\lambda_{\mathrm{H}_l \mathrm{H}_l \mathrm{H}_h} &= -\mathrm{i} c_\alpha s_\alpha \frac{ e}{2 M_\mathrm{W} s_\mathrm{w} t_\beta} \left( M_{H_\mathrm{h}}^2 + 2 M_{H_\mathrm{l}}^2\right) \left( c_\alpha + s_\alpha t_\beta\right)\\ \mathrm{i}\lambda_{\mathrm{H}_l \mathrm{H}_h \mathrm{H}_h} &= -\mathrm{i} c_\alpha s_\alpha \frac{e}{2 M_\mathrm{W} s_\mathrm{w} t_\beta} \left( M_{H_\mathrm{l}}^2 + 2 M_{H_\mathrm{h}}^2\right) \left( s_\alpha - c_\alpha t_\beta\right)\end{split}\]

The fields extend the ones in the SM by

Fields Recola identifier
\(H_\mathrm{l}\) 'Hl'
\(H_\mathrm{h}\) 'Hh'

where \(H_\mathrm{l}\) is the lighter Higgs-boson which typically takes the role of the SM one.

HS interface

The HS comes with special functions which can be accessed Recola2:

set_sa_rcl(sa) Sets the value for \(\sin(\alpha)\) to sa.
set_tb_rcl(tb) Sets the value for \(\tan(\beta)\) to tb.
set_pole_mass_hl_hh_rcl(ml,gl,mh,gh) Sets the pole masses, widths of the light and heavy Higgs bosons to ml, gl and mh, gh, respectively.

The standard renormalization schemes are accessed by the following special functions:

use_mixing_alpha_msbar_scheme_rcl(s) Sets the renormalization scheme for the mixing angle \(\alpha\) to an \(\overline{\mathrm{MS}}\) scheme.
use_tb_msbar_scheme_rcl(s) Sets the renormalization scheme for \(t_\beta\) to an \(\overline{\mathrm{MS}}\) scheme.
use_mixing_alpha_onshell_scheme_rcl(s) Sets the renormalization scheme for the mixing angle \(\alpha\) to an on-shell or BFM scheme.
use_tb_onshell_scheme_rcl(s) Sets the renormalization scheme for \(t_\beta\) to an on-shell or BFM scheme.

For details on the schems we refer to [DDL18]. On request we can provide other renormalization schemes.



[BCW07]Matthew Bowen, Yanou Cui, and James D. Wells. Narrow trans-TeV Higgs bosons and H -> hh decays: Two LHC search paths for a hidden sector Higgs boson. JHEP, 03:036, 2007. arXiv:hep-ph/0701035, doi:10.1088/1126-6708/2007/03/036.
[DDL18](1, 2) Ansgar Denner, Stefan Dittmaier, and Jean-Nicolas Lang. Renormalization of mixing angles. PREPRINT, 2018. arXiv:1808.03466.
[DLU17]Ansgar Denner, Jean-Nicolas Lang, and Sandro Uccirati. NLO electroweak corrections in extended Higgs Sectors with RECOLA2. JHEP, 07:087, 2017. arXiv:1705.06053, doi:10.1007/JHEP07(2017)087.
[PW06]Brian Patt and Frank Wilczek. Higgs-field portal into hidden sectors. 2006. arXiv:hep-ph/0605188.
[PR13]Giovanni Marco Pruna and Tania Robens. Higgs singlet extension parameter space in the light of the LHC discovery. Phys. Rev., D88(11):115012, 2013. arXiv:1303.1150, doi:10.1103/PhysRevD.88.115012.
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