# 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.

## Releases¶

References

 [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.