Random conformal snowflakes

In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structure. This makes it much harder to describe extremals and to attack such problems. Many of these problems are related to the multifractal analysis of harmonic measure. We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals random conformal snowflakes and investigate its properties developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behaviour of harmonic measure, showing how to constuct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value

Fermionic decays of scalar leptoquarks and scalar gluons in the minimal four color symmetry model

Fermionic decays of the scalar leptoquarks $S=S_1^{(+)}, S_1^{(-)}, S_m$ and of the scalar gluons $F=F_1, F_2$ predicted by the four color symmetry model with the Higgs mechanism of the quark-lepton mass splitting are investigated. Widths and branching ratios of these decays are calculated and analysed in dependence on coupling constants and on masses of the decaying particles. It is shown that the decays $S_1^{(+)}\to tl^+_j, S_1^{(-)}\to \nu_i\tilde b, S_m\to t\tilde \nu_j, F_1\to t\tilde b, F_2\to t\tilde t$ are dominant with the widths of order of a few GeV for $m_S, m_F<1$ TeV and with the total branching ratios close to 1. In the case of $m_S < m_t$ the dominant scalar leptoquark decays are S_1^{(+)}\to cl_j^+, S_1^{(-)}\to \nu_i\tilde b, S_m\to b\l_j^+, S_m\to c\tilde \nu_j with the total branching ratios $Br(S_1^{(+)}\to cl^+) \approx$ $Br(S_1^{(-)}\to \nu\tilde b) \approx 1$, $Br(S_m\to bl^+) \approx 0.9$ and $Br(S_m\to c\tilde \nu) \approx 0.1.$ A search for such decays at the LHC and Tevatron may be of interest.Comment: 11 pages, 1 figure, 1 table, to be published in Modern Physics Letters

Discrete symmetries and model-independent patterns of lepton mixing

In the context of discrete flavor symmetries, we elaborate a method that allows one to obtain relations between the mixing parameters in a model-independent way. Under very general conditions, we show that flavor groups of the von Dyck type, that are not necessarily finite, determine the absolute values of the entries of one column of the mixing matrix. We apply our formalism to finite subgroups of the infinite von Dyck groups, such as the modular groups, and find cases that yield an excellent agreement with the best fit values for the mixing angles. We explore the Klein group as the residual symmetry of the neutrino sector and explain the permutation property that appears between the elements of the mixing matrix in this case.Comment: 22 pages, 12 figure

On mass limit for chiral color symmetry G′G'-boson from Tevatron data on ttˉt \bar{t} production

The contributions of $G'$-boson predicted by the chiral color symmetry of quarks to the cross section $\sigma_{t\bar{t}}$ and to the forward-backward asymmetry $A_{\rm FB}^{p \bar p}$ of $t\bar{t}$ production at the Tevatron are calculated with account of the difference of the strengths of the $\bar q q G$ and $\bar q q G'$ interactions. The results are analysed in dependence on two free parameters of the model, the mixing angle $\theta_G$ and $G'$ mass $m_{G'}$. The $G'$-boson contributions to $\sigma_{t\bar{t}}$ and $A_{\rm FB}^{p \bar p}$ are shown to be consistent with the Tevatron data on $\sigma_{t\bar{t}}$ and $A_{\rm FB}^{p \bar p}$, the allowed region in the $m_{G'} - \theta_G$ plane is discussed and around $m_{G'}=1.2 \, TeV, \; \theta_G=14^\circ$ the region of $1 \sigma$ consistency is found.Comment: 9 pages, 1 figure, misprints in formula (14) are corrected, all the other results are vali