Utility of a Special Second Scalar Doublet

This Brief Review deals with the recent resurgence of interest in adding a second scalar doublet (eta^+,eta^0) to the Standard Model of particle interactions. In most studies, it is taken for granted that eta^0 should have a nonzero vacuum expectation value, even if it may be very small. What if there is an exactly conserved symmetry which ensures =0? The phenomenological ramifications of this idea include dark matter, radiative neutrino mass, leptogenesis, and grand unification.Comment: 9 pages, 1 figur

Supersymmetric U(1) Gauge Realization of the Dark Scalar Doublet Model of Radiative Neutrino Mass

Adding a second scalar doublet (eta^+,eta^0) and three neutral singlet fermions N_{1,2,3} to the Standard Model of particle interactions with a new Z_2 symmetry, it has been shown that Re(eta^0) or Im(eta^0) is a good dark-matter candidate and seesaw neutrino masses are generated radiatively. A supersymmetric U(1) gauge extension of this new idea is proposed, which enforces the usual R parity of the Minimal Supersymmetric Standard Model, and allows this new Z_2 symmetry to emerge as a discrete remnant.Comment: 8 pages, 3 figure

A Remark on Soliton Equation of Mean Curvature Flow

In this short note, we consider self-similar immersions $F: \mathbb{R}^n \to \mathbb{R}^{n+k}$ of the Graphic Mean Curvature Flow of higher co-dimension. We show that the following is true: Let $F(x) = (x,f(x)), x \in \mathbb{R}^{n}$ be a graph solution to the soliton equation $$\bar{H}(x) + F^{\bot}(x) = 0.$$ Assume $\sup_{\mathbb{R}^{n}}|Df(x)| \le C_{0} < + \infty$. Then there exists a unique smooth function $f_{\infty}: \mathbb{R}^{n}\to \mathbb{R}^k$ such that $$f_{\infty}(x) = \lim_{\lambda \to \infty}f_{\lambda}(x)$$ and $$f_{\infty}(r x)=r f_{\infty}(x)$$ for any real number $r\not= 0$, where $$f_{\lambda}(x) = \lambda^{-1}f(\lambda x).$$Comment: 6 page

Osgood-Hartogs type properties of power series and smooth functions

We study the convergence of a formal power series of two variables if its restrictions on curves belonging to a certain family are convergent. Also analyticity of a given $C^\infty$ function $f$ is proved when the restriction of $f$ on analytic curves belonging to some family is analytic. Our results generalize two known statements: a theorem of P. Lelong and the Bochnak-Siciak Theorem. The questions we study fall into the category of "Osgood-Hartogs-type" problems.Comment: 13 page