We organize several research projects in the author's Ph.D. career which are distinct in nature into this thesis, in the view of symmetry fulfillments and breakings.

Some broken gauge symmetry may give a massive neutral gauge boson Z', and this Z' may be the mediator between the SM and the dark matter sector, forming the dark portal. We consider the scenario of a leptophobic light Z' vector boson as the mediator, and study the prospect of searching for it at the 8 TeV Large Hadron Collider (LHC). To improve the reach in the low mass region, we perform a detailed study of the processes that the Z' is produced in association with jet, photon, W^± and Z ⁰. We show that in the region where the mass of Z' is between 80 and 400 GeV, the constraint from associated production can be comparable or even stronger than the known monojet and dijet constraints.

We study an extension of the Minimal Supersymmetric Standard Model with a gauge group SU(2)₁ ⊗ SU(2) ₂ breaking to SU(2)_L. The extra wino has an enhanced gauge coupling to the SM-like Higgs boson and, if light, has a relevant impact on the weak scale phenomenology. Compared with the MSSM case, the low energy Higgs quartic coupling is modified both by extra D-term corrections and by a modification of its renormalization group evolution from high energies. At low values of tan β, the latter effect may be dominant. This leads to interesting regions of parameter space in which the model can accommodate a 125 GeV Higgs with relatively light third generation squarks and an increased h → γγ decay branching ratio, while still satisfying the constraints from electroweak precision data and Higgs vacuum stability.

We also study some toy model towards electroweak baryogenesis, which in the wino-higgsino case can be fulfilled as the above gauge extension of the MSSM model. The fermionic component have a mixing through vector like mass terms, through which the Higgs diphoton decay branching ratio can be tuned, and dominant contribution from the bosonic component triggers a strongly first order electroweak phase transition. We find a large parameter region that can simultaneously accommodate diphoton excess and electroweak baryogenesis, without electroweak precision measurement problem.

Supersymmetric SO(10) Grand Unified Theories should have Yukawa unification at high scale, but it is made difficult by large threshold corrections to the bottom mass. Here we present another possibility, in which the top and bottom GUT scale Yukawa couplings are equal to a component of the charged lepton Yukawa matrix at the GUT scale in a basis where this matrix is not diagonal. Diagonalizing the resulting Yukawa matrix introduces mixings in the neutrino sector. Specifically we find that for a large region of parameter space with relatively light sparticles, and which has not been ruled out by current LHC or other data, the mixing induced in the neutrino sector is such that sin² 2&thetas;₂₃ ≈ 1, in agreement with data.

Supersymmetric leptogenesis may raise the gravitino overproduction problem, one way out is to make them late decaying into even lighter particles. We revisited the decaying dark matter (DDM) model, in which one collisionless particle (gravitino) decays into two collisionless particles (axion and axino for example), that are potentially dark matter particles today. The effect of DDM will be manifested in the cosmic microwave background (CMB) and structure formation. We numerically calculated this effect, find a constraint based on data available at that time: the free streaming length λ_FS [special characters omitted] 0.5 Mpc for nonrelativistic decay, and Δg_ν [special characters omitted] 0.45 for relativistic decay.