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Erratum to: Vacuum stability of a general scalar potential of a few fields

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Erratum
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1 Erratum to: Eur. Phys. J. C (2016) 76:324  https://doi.org/10.1140/epjc/s10052-016-4160-3

In the original, it was erroneously assumed in the derivation of the vacuum stability conditions for the two Higgs doublet model (2HDM) with real couplings that in the case of \(\rho = 1\), it is sufficient to consider only \(\cos \phi = \pm 1\). In fact, the solution with \(\rho = 1\), \(\cos \phi \ne \pm 1\) may exist, yielding an extra condition.

The minimisation equations for \(\phi \), \(h_{1}\), \(h_{2}\) and \(\lambda \) in the case of \(\rho = 1\) are
$$\begin{aligned} 0= & {} h_{1} h_{2} \left( 2 \lambda _{5} h_{1} h_{2} \cos \phi + \lambda _{6} h_{1}^{2} + \lambda _{7} h_{2}^{2} \right) \sin \phi , \!\! \end{aligned}$$
(1)
$$\begin{aligned} \lambda h_{1}= & {} 4 \lambda _{1} h_{1}^{3} +2 (\lambda _{3} + \lambda _{4} + \lambda _{5} \cos 2 \phi ) \, h_{1} h_{2}^{2}\nonumber \\&+ \,6 \lambda _{6} \cos \phi \, h_{1}^{2} h_{2} + 2 \lambda _{7} \cos \phi \, h_{2}^{3},\end{aligned}$$
(2)
$$\begin{aligned} \lambda h_{2}= & {} 4 \lambda _{2} h_{2}^{3} + 2 (\lambda _{3} + \lambda _{4} + \lambda _{5} \cos 2 \phi ) h_{1}^{2} h_{2}\nonumber \\&+ \,2 \lambda _{6} \cos \phi \, h_{1}^{3} + 6 \lambda _{7} \cos \phi \, h_{1} h_{2}^{2},\end{aligned}$$
(3)
$$\begin{aligned} 1= & {} h_{1}^{2} + h_{2}^{2}. \end{aligned}$$
(4)
Their solutions with \(\cos \phi \ne \pm 1\) are given by
$$\begin{aligned} \cos \phi _{\rho = 1}= & {} -\frac{\lambda _{6} h_{1}^2 + \lambda _{7} h_{2}^2}{2 \lambda _{5} h_{1} h_{2}},\end{aligned}$$
(5)
$$\begin{aligned} h_{1, \rho = 1}^{2}= & {} [\lambda _{5}(2 \lambda _{2} - \lambda _{3} - \lambda _{4} + \lambda _{5}) + \lambda _{7} (\lambda _{6} - \lambda _{7})] \nonumber \\&/[2 \lambda _{5} (\lambda _{1} + \lambda _{2} - \lambda _{3} - \lambda _{4} + \lambda _{5}) \nonumber \\&- (\lambda _{6} - \lambda _{7})^2],\end{aligned}$$
(6)
$$\begin{aligned} h_{2, \rho = 1}^{2}= & {} [\lambda _{5} (2 \lambda _{1} - \lambda _{3} - \lambda _{4} + \lambda _{5})- \lambda _{6} (\lambda _{6} - \lambda _{7})]\nonumber \\&/[2 \lambda _{5} (\lambda _{1} + \lambda _{2} - \lambda _{3} - \lambda _{4} + \lambda _{5})\nonumber \\&- (\lambda _{6} - \lambda _{7})^2],\end{aligned}$$
(7)
$$\begin{aligned} V_{\text {min}, \rho = 1}= & {} \frac{1}{2} [4 \lambda _{1} \lambda _{2} \lambda _{5} - 2 \lambda _{2} \lambda _{6}^2 - 2 \lambda _{1} \lambda _{7}^2\nonumber \\&- (\lambda _{3} + \lambda _{4} - \lambda _{5})[\lambda _{5} (\lambda _{3} + \lambda _{4} - \lambda _{5})\nonumber \\&- 2 \lambda _{6} \lambda _{7}] ]/[2 \lambda _{5} (\lambda _{1} + \lambda _{2} - \lambda _{3} - \lambda _{4}+ \lambda _{5})\nonumber \\&- (\lambda _{6} - \lambda _{7})^2]. \end{aligned}$$
(8)
Altogether, the conditions for the 2HDM potential with real couplings to be bounded from below are
$$\begin{aligned} V_{\rho = 0}> & {} 0 \wedge D_{\cos \phi = \pm 1,\, \rho = 1}\nonumber \\&\wedge&(Q_{\cos \phi = \pm 1,\, \rho = 1}> 0 \vee R_{\cos \phi = \pm 1,\, \rho = 1}> 0) \nonumber \\&\wedge&( 0< h_{1, \rho = 1}^{2}< 1 \wedge 0< h_{2, \rho = 1}^{2}< 1 \nonumber \\&\wedge&0< \cos ^{2} \phi _{\rho = 1}< 1\implies V_{\text {min}, \rho = 1}> 0 )\nonumber \\&\wedge&( 0< h_{1}^{2}< 1 \wedge 0< h_{2}^{2}< 1\nonumber \\&\wedge&0< \rho ^{2} < 1 \implies V_{\text {min}} > 0 ), \end{aligned}$$
(9)
which includes the new condition involving \(\cos \phi _{\rho = 1}\), \(h_{1, \rho = 1}^{2}\), \(h_{2, \rho = 1}^{2}\) and \(V_{\text {min}, \rho = 1}\).

Figures 3 and 4 that present examples of the allowed parameter space for the 2HDM remain unaffected by the change.

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© The Author(s) 2018

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Funded by SCOAP3

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Authors and Affiliations

  1. 1.NICPBTallinnEstonia

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