Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imp
rimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.
The left-right twin Higgs model predicts one neutral Higgs boson $phi_{0}$ and it acquires mass $m_{phi_{0}}sim mu_{r}$ with the $mu$ term, which can be lighter than half the SM-like Higgs boson mass in a portion of parameter space. Thus, the SM-like
Higgs boson $h$ can dominantly decay into a pair of light neutral Higgs bosons especially when $m_{h}$ is below the $WW$ threshold. First, we examine the branching ratios of the SM-like Higgs boson decays and find that the new decay mode $hrightarrow phi_{0}phi_{0}$ is dominant for the case of $m_{h}>2m_{phi_{0}}$. Then we study the production via gluon fusion followed by the decay into two photons or two weak gauge bosons and found that the production rate can be significantly suppressed for some part of parameter space. Finally, we comparatively study the process $gammagammarightarrow h rightarrow bbar{b}$ at ILC in the cases of $m_{h}>2m_{phi_{0}}$ and $m_{h}<2m_{phi_{0}}$, respectively. We find that these predictions can significantly deviated from the SM predictions, e.g., the gluon-gluon fusion channel, in the cases of $m_{h}>2m_{phi_{0}}$ and $m_{h}<2m_{phi_{0}}$, can be suppressed by about 80% and 45%, respectively. Therefor, it is possible to probe the left-right twin Higgs model via these Higgs boson production processes at the LHC experiment or in the future ILC experiment.
The left-right twin Higgs(LRTH) model predicts the existence of three additional Higgs bosons: one neutral Higgs $phi^{0}$ and a pair of charged Higgs bosons $phi^{pm}$. In this paper, we studied the production of a pair of charged and neutral Higgs
bosons associated with standard model gauge boson $Z$ at the ILC, i.e., $e^{+}e^{-}rightarrow Zphi^{+}phi^{-}$ and $e^{+}e^{-}rightarrow Zphi^{0}phi^{0}$. We calculate the production rate and present the distributions of the various observables, such as, the distributions of the energy and the transverse momenta of final $Z$-boson and charged Higgs boson $phi^{-}$, the differential cross section of the invariant mass of charged Higgs bosons pair, the distribution of the angle between charged Higgs bosons pair and the production angle distributions of $Z$-boson and charged Higgs boson $phi^{-}$. Our numerical results show that, for the process $e^{+}e^{-}rightarrow Zphi^{+}phi^{-}$, the production rates are at the level of $10^{-1} fb$ with reasonable parameter values. For the process of $e^{+}e^{-}rightarrow Zphi^{0}phi^{0}$, we find that the production cross section are smaller than $6times 10^{-3} fb$ in most of parameter space. However, the resonance production cross section can be significantly enhanced.
