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 (see [DDL18]) differs from the literature [SW05], [PW06], [BCW07], [PR13].

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

(1)\[V_{\mathrm{HSESM}} = -m_1^2 \Phi^\dagger \Phi - \frac{m_2^2}{2} S^2 +\frac{\lambda_1}{8} S^4 +\frac{\lambda_2}{4} \left(\Phi^\dagger \Phi\right)^2 +\frac{\lambda_3}{2} \Phi^\dagger \Phi S^2,\]

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 \(M_{\mathrm{H}_l}\), \(M_{\mathrm{H}_h}\), \(s_{\alpha}\), \(\lambda_3\), \(M_\mathrm{W}\) \(\alpha_\mathrm{em}\), \(M_\mathrm{Z}\)

The angle \(\boldsymbol \alpha\) (sas = \(\sin(\alpha)\)) is defined in the same way as in the THDM.

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

\[\begin{split}\mathrm{i}\lambda_{\mathrm{Z} \mathrm{Z} \mathrm{H}_l} &= +\mathrm{i} c_\alpha \frac{e M_\mathrm{Z} }{c_\mathrm{w} s_\mathrm{w}}\\ \mathrm{i}\lambda_{\mathrm{Z} \mathrm{Z} \mathrm{H}_l} &= +\mathrm{i} s_\alpha \frac{e M_\mathrm{Z} }{c_\mathrm{w} s_\mathrm{w}}\end{split}\]

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_l3_rcl(l3) Sets the value for \(\lambda_3\) to l3.
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_l3_msbar_scheme_rcl(s) Sets the renormalization scheme for \(\lambda_3\) 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_l3_onshell_scheme_rcl(s) Sets the renormalization scheme for \(\lambda_3\) 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. JHEP, 11:104, 2018. arXiv:1808.03466, doi:10.1007/JHEP11(2018)104.
[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.
[SW05]Robert M. Schabinger and James D. Wells. A Minimal spontaneously broken hidden sector and its impact on Higgs boson physics at the large hadron collider. Phys. Rev., D72:093007, 2005. arXiv:hep-ph/0509209, doi:10.1103/PhysRevD.72.093007.