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We study the origin of neutrino mass from $SO(10)$ arising from $M$ Theory compactified on a $G_2$-manifold. This is linked to the problem of the breaking of the extra $U(1)$ gauge group, in the $SU(5)\times U(1)$ subgroup of $SO(10)$, which we show can achieved via a (generalised) Kolda-Martin mechanism. The resulting neutrino masses arise from a combination of the seesaw mechanism and induced R-parity breaking contributions. The rather complicated neutrino mass matrix is analysed for one neutrino family and it is shown how phenomenologically acceptable neutrino masses can emerge., 32 pages, 12 figures

We consider grand unified theories based on $SO(10)$ which originate from $\text{string}/M$ theory on ${G}_{2}$ manifolds or Calabi-Yau spaces with discrete symmetries. In this framework we are naturally led to a novel solution of the doublet-triplet splitting problem previously considered by Dvali which involves an extra vectorlike Standard Model family and light but weakly coupled color triplets. These additional states are predicted to be accessible at the LHC and also induce R-parity violation. Gauge coupling unification occurs with a larger GUT coupling.

We investigate the prospects for discovering charginos and neutralinos at a future $pp$ collider with $\sqrt{s} = 100$ TeV. We focus on models where squarks and sleptons are decoupled -- as motivated by the LHC data. Our initial study is based on models where Higgsinos form the main component of the LSP and $W$-inos compose the heavier chargino states ($M_2 > ��$), though it is straightforward to consider the reverse situation also. We show that in such scenarios $W$-inos decay into $W^\pm$, $Z$ and $h$ plus neutralinos almost universally. In the $W Z$ channel we compare signal and background in various kinematical distributions. We design simple but effective signal regions for the trilepton channel and evaluate discovery reach and exclusion limits. Assuming 3000 fb$^{-1}$ of integrated luminosity, $W$-inos could be discovered (excluded) up to 1.1 (1.8) TeV if the spectrum is not compressed., 19 pages, 9 figures

We propose a new method for Monte Carlo solution of non-linear integral equations by combining the Newton-Kantorovich method for solving non-linear equations with the Markov Chain Monte Carlo (MCMC) method for solving linear equations. The Newton-Kantorovich method allows to express the non-linear equation as a system of the linear equations which then can be treated by the MCMC (random walk) algorithm. We apply this method to the Balitsky-Kovchegov (BK) equation describing evolution of gluon density at low x. Results of numerical computations show that the MCMC method is both precise and efficient. The presented algorithm may be particularly suited for solving more complicated and higher-dimensional non-linear integral equation, for which traditional methods become unfeasible., 10 pages, 2 figures, typos corrected, accepted by JHEP