Orbit Portraits of Unicritical Anti-polynomials

Orbit portraits were introduced by Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical anti-polynomials and give an explicit description of the orbit portraits that can occur for such maps in terms of their characteristic angles, which turns out to be rather restricted when compared with the holomorphic case. Finally, we prove a realization theorem for these combinatorial objects. The results obtained in this paper serve as a combinatorial foundation for a detailed understanding of the combinatorics and topology of the parameter spaces of unicritical anti-polynomials and their connectedness loci, known as the multicorns

Lepton flavour violating decay of 125 GeV Higgs boson to μτ\mu\tau channel and excess in ttˉHt\bar t H

A recent search for the lepton flavor violating (LFV) decays of the Higgs boson, performed by CMS collaboration, reports an interesting deviation from the standard model (SM). The search conducted in the channel $H\rightarrow \mu\tau_e$ and $H\rightarrow \mu\tau_{\textrm{had}}$ shows an excess of $2.4\sigma$ signal events with 19.7 fb$^{-1}$ data at a center-of-mass energy $\sqrt s=8$ TeV. On the other hand, a search performed by CMS collaboration for the SM Higgs boson produced in association with a top quark pair ($t\bar t H$) also showed an excess in the same-sign di-muon final state. In this work we try to find out if these two seemingly uncorrelated excesses are related or not. Our analysis reveals that a lepton flavour violating Higgs decay ($H\rightarrow\mu\tau$) can partially explain the excess in the same sign di-muon final state in the $t\bar t H$ search, infact brings down the excess well within 2$\sigma$ error of the SM expectation. Probing such non-standard Higgs boson decay is of interest and might contain hints of new physics at the electroweak scale.Comment: 10 pages, 2 figures and 3 table

Rational Parameter Rays of The Multibrot Sets

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof is inspired by previous works of Schleicher and Milnor on the combinatorics of the Mandelbrot set; in particular, we make essential use of combinatorial tools such as orbit portraits and kneading sequences. However, we avoid the standard global counting arguments in our proof and replace them by local analytic arguments to show that the parabolic and the Misiurewicz parameters are landing points of rational parameter rays