0,\ \ \lambda_3>-\sqrt{\lambda_1(\lambda_2+\lambda_4/3)};$ \item $\lambda_4<0,\ \ \lambda_1>0,\ \ \lambda_2>-\lambda_4, \ \ \lambda_3>-\sqrt{\lambda_1(\lambda_2+\lambda_4)}; $ \item $\lambda_2<0,\ \ \lambda_1>0,\ \ \lambda_4>-3\lambda_2, \ \ \lambda_3>-\sqrt{\lambda_1(\lambda_2+\lambda_4/3)}. $ \end{itemize} One of these conditions has to hold \emph{for all scales} between the electroweak and the Planck scales to avoid the problem of vacuum instability, and thus to overcome one of the main open problems of the SM in its current form. More concretely, we will require them to hold for the running couplings $\lambda_i(\mu)$ over this whole range when these are evolved with the $\beta$-functions~(\ref{beta}). Assuming the following values of the mass parameters \beq m_1^2=-2\lambda_{1}v_{H}^{2}-6\lambda_{3}v_{\phi}^{2},\qquad m_2^2=-2\lambda_{3}v_{H}^{2}-\left(6\lambda_{2}+2{\lambda_{4}}\right)v_{\phi}^{2}, \eeq (and thus parametrising them directly in terms of the positive parameters $v_H$ and $v_\phi$) it is straightforward to show that the global minimum of the potential takes the form\footnote{Thus, $v_H$ and $v_\phi$ are the expectation values of the complex fields (we here drop the customary factor~$1/\sqrt{2}$).} \beq\label{vevs} \langle {H} \rangle = \left(\begin{array}{c} 0 \\[1mm] v_H \end{array}\right), \qquad \langle \phi \rangle = \mathcal{U}_0\left(\begin{array}{ccc} v_\phi&0&0 \\[1mm] 0&v_\phi&0 \\[1mm] 0&0&v_\phi \end{array}\right)\mathcal{U}_0^{T}, \qquad \mathcal{U}_0\in {\rm U}(3), \eeq provided that (in addition to the above positivity conditions) the following inequalities are also satisfied \eq{\nn \lambda_1 \left\{{\lambda}_{2}+\frac{{\lambda}_{4}}{3}\right\}-\lambda^2_3 >0, \qquad \lambda_4>0. } At the classical level the U(3) matrix $\cU_0$ remains undetermined. The explicit breaking of SU(3)$_N$ symmetry to be discussed below will, however, lift this degeneracy and produce a `vacuum alignment' with $\cU_0\neq \mathbbm{1}$ according to~\cite{DASHEN}, and also introduce small corrections that will lift the degeneracy of eigenvalues in $\langle\phi_{ij}\rangle$. A second motivation for the replacement of a single complex scalar by a sextet is the following. Because the SU(3)$_N$ invariance is assumed to be broken both spontaneously and explicitly (by the Yukawa interaction coupling right-chiral neutrinos to the lepton doublets via the matrix $Y^\nu_{ij}$, see~(\ref{L'}) below) there exist various light particles, i.e.\ \hbox{(pseudo-)Goldstone} bosons. It is a general result that the manifold of Goldstone bosons $\cal{M}$ is the quotient of the symmetry group by the symmetry of the vacuum. For~(\ref{vevs}) the residual symmetry is SO(3), and therefore \beq \label{MGoldstone} {\cal M} = {\rm U}(3)\big/ {\rm SO}(3) \, \equiv \, {\rm U}(1)_{B-L} \,\times\, {\rm SU}(3)_N\big/{\rm SO}(3), \eeq whence there are altogether six (pseudo-)Goldstone bosons in our model. One of them is the \emph{genuine} Goldstone boson associated with the exact U(1)$_{B - L}$ symmetry (so we can take out the U(1) factor). After the symmetry breaking we have as usual the real Higgs field $H_0$ \beq H(x)= \left(\begin{array}{c} 0 \\[1mm] v_H +\frac{1}{\sqrt{2}}H_0(x)\end{array}\right), \eeq {(in the unitary gauge)} while a convenient parametrisation of the coset space $\cal M$ is given by \beq\label{ReDef1} \phi(x)=\mathcal{U}_0\,e^{\ri \tilde{A} (x) }\,(v_\phi+\tilde{R}(x))\, e^{\ri \tilde{A}(x)}\, \mathcal{U}_0^{\rm T}, \eeq where $\tilde{A}_{ij}$ and $\tilde{R}_{ij}$ are real symmetric matrices. The trace part of \beq\label{calG} \mathcal{G}(x)\equiv\mathcal{U}_0\tilde{A}(x)\mathcal{U}_0^{\dagger}, \eeq is the $(B - L)$ Goldstone boson $\Ca(x)$ that remains a Goldstone boson even when the U(3) symmetry is broken, while the traceless part of $\mathcal{G}(x)$ yields the five Goldstone bosons that will be converted to pseudo-Goldstone bosons. In accordance with the decomposition ${\bf 6} \ra {\bf 1} \oplus {\bf 5}$ under the residual SO(3) symmetry we can thus write \beq\label{A} \tilde{A}_{ij} (x) = \frac{1}{2\sqrt{6}v_\phi}\Ca(x) \delta_{ij} \, + \, A_{ij}(x) \;\;, \qquad \Tr A(x) = 0, \eeq with \beq\label{G} A_{ij} \equiv \frac1{v_\phi} G(x) \equiv \frac{1}{4v_\phi} {\sum_a}' G_a\la^a_{ij}, \eeq where the restricted sum is only over the five \emph{symmetric} Gell-Mann matrices (with the standard normalization $\Tr(\la^a \la^b)=2\delta^{ab}$) and where the real fields $\Ca(x)$ and $G_a(x)$ are canonically normalized. The matrix $\tilde{R}_{ij}(x)$ likewise can be split into a trace and a traceless part, \emph{viz.} \beq\label{R} \tilde{R}_{ij}(x) = \frac{1}{\sqrt{6}}r(x) \delta_{ij} \, + \frac{1}{2}\, {\sum_a}' R_a\la^a_{ij} (x). \eeq Because the new scalars are thus only very weakly coupled to the remaining SM fields, the main observable effects are due to the mixing between the standard Higgs boson and the new scalars. In fact, the five modes $R_a$ are already the proper mass eigenstates with eigenvalues \beq M_R^2=4\lambda_4 v_\phi^2, \eeq The field $r$ can mix with $H_0$ and the combined mass matrix for the fields $(H_0,r)$ reads \beq \mathcal{M}^2 =\left(\begin{array}{cc} 4\lambda_1 v_H^2 & 4\sqrt{3} \lambda_3 v_H v_\phi \\[2mm] \ 4\sqrt{3} \lambda_3 v_H v_\phi\ &\ 4(3\lambda_2+\lambda_4) v_\phi^2\ \end{array}\right), \eeq and determines the mass eigenstates $h_0$ and $h'$ \beq\label{hh'} h_0 \,=\, \cos\beta\, H_0 \, + \, \sin\beta \, r \;\; , \quad h' \, = \, - \sin\beta \, H_0 \, + \, \cos\beta \, r, \eeq with the mixing angle $\beta$. We identify the lighter of the two mass eigenstates $h_0$ with the observed Higgs boson, with $M_{h_0}\approx 125\GeV$. The mixing will lead to a second resonance associated with $h'$, which is one of the main predictions of the present model. This resonance should be rather narrow because of the factor $\sin^2\beta$~\cite{MN325}. It will have the same decay channels to the SM particles as the standard Higgs boson (hence look like a `shadow Higgs'), but depending on the actual mass of $h'$, there may also be other decay channels which could broaden the resonance. We will return to this point below. The possibility of further extension of the Higgs sector in the framework of a U(3) symmetric scalar sector is considered in the appendix. ]]>

V_{\rm eff}^{\rm max}$ we have thermal equilibrium, and only for $T < V_{\rm eff}^{\rm max} $ can the pseudo-Goldstone particles start to be produced non-thermally by coherent oscillations. Therefore setting $T_* = (V_{\rm eff}^{\rm max})^{1/4}$ seems a reasonable choice; this~gives \beq \Omega \;\sim \; \rho_{crit}^{-1} V_{\rm eff}^{\rm max} \left(\frac{T_0}{T_*}\right)^3 \; \sim \; \rho_{crit}^{-1} \left( V_{\rm eff}^{\rm max}\right)^{1/4} T_0^3 \;\sim\; \rho_{crit}^{-1} \sqrt{m_\nu M} \, T_0^3. \eeq This is indeed an estimate that also gives about the right order of magnitude for standard axions, with $V_{\rm eff}^{\rm max} = \Lambda_{QCD}^4$. In our case, the result comes out too small by two or three orders of magnitude. However, the above estimate is fraught with several uncertainties, apart from the precise details of the production mechanism, which may give rise to all kinds of `fudge factors'. In particular, since there is a `collective' of scalar fields involved in this process it is not clear whether there cannot exist new enhancement effects, similar to the resonant effects giving rise to leptogenesis as in~\cite{PU,PU2}. Furthermore, a modification of the scalar sector along the lines of section~2.2 might change the value of $V_{\rm eff}^{\rm max}$, for instance replacing $m_\nu^2 M^2$ by $m_\nu M^3$ in~(\ref{Vmax}) which would give the desired number. So this issue clearly requires further and more detailed study. ]]>