Recent discovery of the SM-like Higgs boson with
Article funded by SCOAP3
0$ and $q_{Y}<0$ so that $X$ is tachyonic whereas $Y$ is not, at the origin of field space.
Then the PQ symmetry is broken as $X$ takes the VEV and it induces non-zero $Y$ VEV:
\dis{\langle |X| \rangle &\simeq \frac{1}{(n+2)^{1/2(n+1)}|y|^{1/(n+1)}} \sqrt[n+1]{ |m_{X}| M_*^n},
\\
\langle Y \rangle &\simeq -\frac{1}{(n+2)^{(n+2)/2(n+1)}y^{1/(n+1)}}\frac{A^*|m_{X}|}{(n+2)|m_{X}|^2+|m_{Y}|^2} \sqrt[n+1]{ |m_{X}| M_*^n} .\label{Eq:XYVEV}}
Note that the linear dependence of superpotential on $Y$ results in the $ Y $ VEV proportional to the $A$-term which is suppressed by $\epsilon_2$, so there appears a hierarchy between $X$ and $Y$ VEVs,
\diseq{\bigg|\frac{Y}{X}\bigg|=\frac{|A||m_{X}|}{\sqrt{n+2}[(n+2)|m_{X}|^2+m_{Y}^2]}\sim\frac{|A|}{\sqrt{D_A}}\sim \epsilon_2.}
We identify the higher scale $X$ VEV as the PQ scale $v_{\rm PQ}$.
As $|m_{X, Y}|\sim m_{3/2} \sim \sqrt{D_A}$, for $M_* =\mplanck$, we have
\diseq{v_{\rm PQ}\equiv \langle X \rangle \sim |y|^{-1/(n+1)} \sqrt[n+1]{ m_{3/2} M_*^n}= \left\{ \begin{array}{ll}
|y|^{-1/2}10^{10}\gev& n=1\\
|y|^{-1/3}10^{12-13}\gev&n=2 \\
|y|^{-1/4}10^{14}\gev & n=3
\end{array}\right. .}
On the other hand, for $M_*=M_{\rm GUT}$,
\diseq{v_{\rm PQ} \sim \left\{ \begin{array}{ll}
|y|^{-1/2}10^{9}\gev& n=1\\
|y|^{-1/3}10^{11-12}\gev&n=2 \\
|y|^{-1/4}10^{13}\gev & n=3
\end{array}\right. .}
Therefore, regarding a bound $10^9\gev
0 $ from the PQ charges in table~\ref{Table:model1}. Hence $ S_1 $ gets non-zero VEV with its quartic potential, while $ S_2 $ does so from its tadpole term after $ \langle S_1 \rangle $ becomes non-zero. Assuming $\mu_2'^2 (\sim \kappa_2^2 D_A) \lesssim m_{S_{1, 2}}^2 (\sim D_A)$ with $ \kappa_2 \lesssim {\cal O}(1) $, $S_1$ and $S_2$ get their VEVs as \dis{ \label{Eq:signletvevs} |\langle S_1 \rangle| &\simeq\sqrt{\frac{-2m_{S_1}^2}{\kappa_1^2}}\sim \frac{|m_{S_1}|}{\kappa_1}, \\ |\langle S_2 \rangle| &=\frac12 \frac{\kappa_1 \mu_2' |S_1| ^2}{m_{S_2}^2+\mu_2'^2+\kappa_1^2 |S_1|^2}\sim \frac{\mu_2'}{\kappa_1}.} The parametric relations between the doublet Higgs sector and singlet sector are very different depending on which singlet field couples to $H_u H_d$. For the model~(\ref{eq:model1}) in which $ S_H = S_1 $, $\mu_\textrm{eff} =\lambda \langle S_H \rangle $ and $ (B\mu)_\textrm{eff} \simeq \lambda \langle \partial_{S_H} f \rangle$ are given by\footnote{The $A$-term contributions to $(B\mu)_\textrm{eff}$ can be neglected since they are suppressed by $\epsilon$ as discussed in section~\ref{Sec:Model}.} \dis{ \label{model1mu} \mu_{\rm eff} &= \lambda\langle S_1 \rangle \sim \frac{m_{S_1}}{\kappa_1} \sim \frac{\sqrt{D_A}}{\kappa_1}, \\ (B\mu)_{\rm eff}&\simeq\lambda \kappa_1 \langle S_1S_2 \rangle \sim \frac{m_{S_1}\mu_2'}{\kappa_1} \sim \frac{\kappa_2}{\kappa_1}D_A.} On the other hand, for the model~(\ref{eq:model2}) in which $ S_H = S_2 $, we have \dis{ \label{model2mu} \mu_{\rm eff} &= \lambda \langle S_2 \rangle \sim \frac{\mu_2'}{\kappa_1}\sim \frac{\kappa_2}{\kappa_1}\sqrt{D_A}, \\ (B\mu)_{\rm eff} &\simeq \lambda\langle\frac12\kappa_1 S_1^2 +\mu_2' S_2 \rangle\sim \frac{m_{S_1}^2}{\kappa_1}\sim \frac{D_A}{\kappa_1}.} From the above relations, one can find that the dimensionless coefficients $ \kappa_1, \kappa_2 $ of the model~(\ref{eq:model1}) should be of $ {\cal O}(1) $ with $ \sqrt{D_A} \sim \msusy $ to satisfy the conditions $ \mu_\textrm{eff} \lesssim \Omsusy$ and $(B\mu)_\textrm{eff} \sim \Omsusysq $. On the contrary, for the model~(\ref{eq:model2}), we observe that $ \sqrt{D_A} $ smaller than $ \msusy$ is allowed even satisfying $ \mu_\textrm{eff} \lesssim \Omsusy$ and $(B\mu)_\textrm{eff} \sim \Omsusysq $ if $ \kappa_1, \kappa_2 $ are smaller than order unity.\footnote{Smaller $ \sqrt{D_A}$ than $m_{3/2}$ can be obtained by assuming larger sequestering $\epsilon_1 \sim g^2/8\pi^2 < 1/8\pi^2$ between the $\U(1)_A $ sector and SUSY breaking modulus in eq.~(\ref{Eq:Dtermval}). In this case, if $\epsilon_2$ is also of similar order with $\epsilon_1$, which is shown to be quite plausible in appendix~\ref{Sec:appA}, $m_\textrm{GM} $ in eq.~(\ref{Eq:mSUSY}) is still around $m_{3/2}$.} Since the singlet masses are governed by the scale of $ \sqrt{D_A} $, this means that the singlet sector of the model~(\ref{eq:model1}) must be around $ m_{3/2} \sim 1$\,TeV similarly with the other SUSY sectors, while the singlet sector of the model~(\ref{eq:model2}) can be parametrically lighter. A caveat must be placed, however, for the possibility of the relatively light singlet sector of the model~(\ref{eq:model2}). Smaller $ \sqrt{D_A} $ than $ m_\textrm{GM} \sim m_{3/2} $ means that SUSY breaking is dominantly mediated by the gauge mediation. If we take the \emph{minimal} gauge mediation for the simplest case, it is known that $ \mu_\textrm{eff} $ cannot be smaller than the scale of $ m_{H_u}(m_{\tilde t}) \sim m_{\tilde t} \sim m_{3/2} $ for the EWSB to occur (see ref.~\cite{Agashe-ml-1999ct}, for instance). This can be problematic because of the mixing between the singlet scalar $ S_H $ and the SM-like Higgs boson $ h $: \diseq{ \label{mixing-hS_H} m_{hS_H}^2 = \lambda v \left(2\mu_\textrm{eff} - (A_\lambda + \partial_{S_H}^2 f(S_1, S_2)) \sin 2\beta \right), } where $ f(S_1, S_2) $ is the singlet sector superpotential. For either $ S_H = S_1 $ or $ S_2$ , we find that $ \partial_{S_H}^2 f(S_1, S_2) \sim \mu_2' $ from~(\ref{Eq:signletvevs}). Thus it will be around $ \lambda v \mu_\textrm{eff} $ unless there occurs some fine cancellation between $ \mu_\textrm{eff} $ and $ \mu_2' $. To ensure the stability of the electroweak vacuum, the diagonal elements of the mass matrix must satisfy $ m_{hh}^2 m_{S_H S_H}^2 > m_{h S_H}^4 $ so that \diseq{ \label{m_SHSH} m_{S_H S_H}^2 \sim D_A \gtrsim \mu_\textrm{eff}^2.} Therefore, relatively small $ \sqrt{D_A} $ requires also small $ \mu_\textrm{eff} $, which is impossible in the minimal gauge mediation. It means that, in the simplest case, the singlet sector is expected to be as heavy as the other SUSY sectors around $ \msusy \sim 1 $\,TeV for both models. Still, the gauge mediation can be realized more generally as in refs.~\cite{GGM, Meade-ml-2008wd, Carpenter-ml-2008wi} with small $ \mu_\textrm{eff} < \msusy$. A non-minimal gauge mediation, however, needs another SUSY breaking term comparable to $F_Y/Y \sim 16\pi^2 \msusy$ in section~\ref{subsec:PQbreaking}. It can arise from another copy of the spontaneous PQ breaking sector with $ X', Y'$, for example. Thus, the singlet sector in the model~(\ref{eq:model2}) can be light but implies some complication of the model. Let us more specifically describe the mass spectrum of the light singlet sector of the model~(\ref{eq:model2}). From~(\ref{model2mu}), we find that $ \kappa_2 \lesssim \kappa_1 $ to satisfy $ \mu_\textrm{eff} \lesssim \Omsusy$ and $(B\mu)_\textrm{eff} \sim \Omsusysq $ as well as the condition~(\ref{m_SHSH}). This gives \dis{ \sqrt{D_A} &\sim \sqrt{\kappa_1} \msusy,~~ \mu_2' \sim \kappa_2\sqrt{\kappa_1}\msusy \lesssim \kappa_1^{3/2} \msusy, \\ \mu_\textrm{eff} &\sim \frac{\kappa_2}{\sqrt{\kappa_1}}\msusy \lesssim \sqrt{\kappa_1}\msusy. } Hence, for the coupling constants $ \kappa_2 \lesssim \kappa_1 < {\cal O}(1) $, there appear hierarchical mass scales $ \mu_2' < \mu_\textrm{eff} \lesssim \sqrt{D_A} < \msusy $. For instance, $ \kappa_1 \sim 0.01 $ gives $ \sqrt{D_A} \sim 0.1 \msusy \sim {\cal O}(100) $\,GeV and $ \mu_2' \lesssim {\cal O}(1) $\,GeV. Notice that the limit $ \mu_2' \rightarrow 0 $ corresponds to the PQ symmetric limit in which one pseudoscalar becomes massless, and we find that the pseudoscalar mass is $ \sim \mu_2'^2 $ from the last term in the scalar potential~(\ref{singlet_scalar_potential}). Therefore, through the small coupling constants, we obtain a relatively light singlet scalar sector of their masses given by $ \sqrt{D_A} \sim \sqrt{\kappa_1} \msusy$ with an even lighter singlet pseudoscalar of mass $ \mu_2' \lesssim \kappa_1^{3/2} \msusy $. The singlet scalars have mixing with the doublet Higgs bosons. Assuming no cancellation between $ \mu_\textrm{eff} $ and $ \partial_{S_H}^2 f(S_1, S_2) \sim \mu_2' $ in~(\ref{mixing-hS_H}), the mixing angle between the SM-like Higgs boson $ h $ and $ S_H $ is \diseq{ \theta_{hS_H} \simeq \frac{m^2_{hS_H}}{m^2_{S_HS_H} - m^2_{hh}} \sim {\cal O}\left(\frac{\kappa_2}{\kappa_1^{3/2}}\frac{\lambda v}{m_{3/2}} \right) \lesssim {\cal O}\left(\frac{0.1}{\sqrt{\kappa_1}}\right), } where $ \kappa_2 \lesssim \kappa_1 $ for the model~(\ref{eq:model2}), while both $ \kappa_1 $ and $\kappa_2 $ should be of $ {\cal O}(1) $ for the model~(\ref{eq:model1}). Similarly, the mixing angle between the SM-like Higgs boson $h$ and another scalar $S_j$ other than $ S_H $ is estimated to be \dis{ \theta_{hS_j} \simeq \frac{m^2_{hS_j}}{m^2_{S_jS_j} - m^2_{hh}} &= \frac{-\lambda v\partial_{S_j} \partial_{S_H} f(S_1, S_2) \sin 2\beta}{m^2_{S_jS_j} - m^2_{hh}}\\ &\sim {\cal O}\left(\frac{1}{\sqrt{\kappa_1}}\frac{\lambda v}{m_{3/2}} \right) \sim {\cal O}\left(\frac{0.1}{\sqrt{\kappa_1}}\right), } where $ \partial_{S_j} \partial_{S_H} f(S_1, S_2) = \partial_{S_1} \partial_{S_2} f(S_1, S_2) = \kappa_1 S_1 \sim m_{S_1} $ for either $ S_H = S_1 $ or $ S_2 $. Therefore, the mixing angles can be quite sizable for the model~(\ref{eq:model2}) with small $ \kappa_1 $ which results in departure from the SM Higgs boson properties with detectable signatures, while they are always as small as $ {\cal O}(0.1) $ for the model~(\ref{eq:model1}). We briefly describe the neutralino sector. The singlino mass matrix in the basis of $(\tilde{S_1}, \tilde{S_2})$ is given by \diseq{M_{\rm singlino}=\left( \begin{array}{cc} \kappa_1 \langle S_2 \rangle &\kappa_1 \langle S_1 \rangle\\ \kappa_1 \langle S_1 \rangle & \mu_2' \end{array}\right) \sim \left( \begin{array}{cc} \mu_2' & m_{S_1}\\ m_{S_1} & \mu_2' \end{array}\right),} \looseness=-1 and they mix with the doublet Higgsinos through the superpotential $ \lambda S_{H} H_u H_d $ with the off-diagonal elements of $ {\cal O}(\lambda v) $. Thus the mixing angles between the doublet Higgsinos and singlinos are around ${\cal O}(\lambda v/ \sqrt{\kappa_1}\msusy) \lesssim {\cal O}(0.1/\sqrt{\kappa_1})$. When we assume the \emph{minimal} gauge mediation, we have argued that all mass parameters $ m_{S_1}, \mu_2' $ and $ \mu_\textrm{eff} $ must be around $ \msusy \sim 1$\,TeV with $ \kappa_1$, $ \kappa_2 $ of $ {\cal O}(1)$. In this case, the Higgsinos and singlinos do not mix with each other so much, and they are all heavy around $ \msusy $ unless there occurs fine cancellation between $ \mu_2' $ and $ m_{S_1} $ in the singlino mass matrix, while the bino and winos are lighter than the Higgsinos and singlinos if the gluino mass is not far above the current lower bound at the LHC around 1.3\,TeV~\cite{Chatrchyan-ml-2013wxa, Aad-ml-2013wta}. However, for the model~(\ref{eq:model2}) of small $ \kappa_1 $ with general gauge mediation, $ m_{S_1}, \mu_2' $ and $ \mu_\textrm{eff} $ can be much smaller than the typical SUSY scale $\msusy$ with $ \mu_2' < \mu_\textrm{eff} \lesssim \sqrt{D_A} < \msusy $ as discussed before. In this case, there can be large mixing between the Higgsinos and singlinos, and they can be lighter than the gauginos. Also notice that the singlinos are almost Dirac-like when $\kappa_1$ is small, because $ \mu_2' \ll m_{S_1} $. ]]>
0 $ in our convention. Now we require that at least one non-tachyonic state gets a relatively small VEV proportional to the small $ A_3 $ to generate a sizable gauge mediation as discussed in section~\ref{subsec:PQbreaking}. To this end, one can find that $ n_2 $ should be 1 so that the $A_3$-term is linear in $Y$. Also we observe that $ X, Z $ must get their non-zero VEVs when $Y=0$ in order to make $A_3$-proportional tadpole term for $Y$. Then $Y$ can get its small VEV proportional to $A_3$. With $ Y=0 $, we examine the potential in arbitrary field directions in $X$-$Z$ plane by parametrizing the fields as $ |X|=|\varphi| \cos \alpha, |Z|=|\varphi| \sin \alpha $ with $ 0 \leq \alpha \leq \pi/2$. In the field direction $\varphi$ with a constant value of $ \alpha $, the potential becomes \diseq{ V_{\rm PQ} = (m_{X}^2 \cos^2 \alpha + m_{Z}^2 \sin^2 \alpha)|\varphi|^2 + (\cos\alpha)^{2n_1} (\sin\alpha)^{2n_3} \frac{|\varphi|^{2(n+2)}}{M_*^{2n}}, } where $ m_{X}^2 < 0 $. If $ (m_{X}^2 \cos^2 \alpha + m_{Z}^2 \sin^2 \alpha) < 0 $, the potential will be minimized along $\varphi$ direction at \dis{ |\varphi| &= \frac{1}{(\cos\alpha)^{n_1/(n+1)} (\sin\alpha)^{n_3/(n+1)}} \left( M_*^n \sqrt{\left| m_{X}^2 \cos^2 \alpha + m_{Z}^2 \sin^2 \alpha \right|} \right)^{1/(n+1)}\\ &\sim \frac{v_{PQ}}{(\cos\alpha)^{n_1/(n+1)} (\sin\alpha)^{n_3/(n+1)}}. } At this field value, the potential turns out that $V_{\rm PQ} \sim -(\cos \alpha)^{-2n_1/(n+1)} (\sin \alpha)^{-2n_3/(n+1)}$ and $\partial^2V_{\rm PQ}/\partial \alpha^2 < 0 $. Thus this point actually corresponds to a saddle point in the $X$-$Z$ field space. Also the potential is unbounded from below in the field direction $ Z=0~(\alpha=0)$ unless $ n_3=0 $. Therefore, $ \varphi $ is destabilized in the $ X$-$Z $ plane. It means that generically one cannot make the required pattern of PQ symmetry breaking with three fields. We are thus led to consider even more than three fields. The simplest possibility with four fields will be \diseq{ \label{4PQbreaking} W_{\rm PQ} = y_1\frac{X^{n+2} Y}{M_*^n} + y_2\frac{X'^{n+2} Y'}{M_*^n}, } \looseness=-1 where $ (X', Y') $ fields have some different PQ charges from the $(X, Y)$ fields so that any interaction betweem them is suppressed. Then each term will realize the correct PQ symmetry breaking pattern as in the two fields case, and we can use the two kinds of fields of different PQ charges to generate two parameters among ($ \xi $, $ C\xi $, or $\mu' $) with the PQ charge assignments in table~\ref{Table:1singlet}. In this way, one can realize the low fine-tuned EWSB with one singlet field extended Higgs sector, but it requires such a complication of the PQ breaking sector that there must be more than three fields and a non-trivial PQ charge relation among them. ]]>