The weak-field expansion of the charged fermion propagator under a uniform magnetic field is studied. Starting from Schwinger's proper-time representation, we express the charged fermion propagator as an infinite series corresponding to different Landau levels. This infinite series is then reorganized according to the powers of the external field strength B. For illustration, we apply this expansion to γ → νν and ν → νγ decays, which involve charged fermions in the internal loop. The leading and subleading magnetic-field effects to the above processes are computed.

To generate small neutrino masses radiatively, the Zee model introduces two Higgs doublets and one weak-singlet charged Higgs boson to its Higgs sector. From analyzing the renormalization group equations, we determine the possible range of the lightest CP-even Higgs boson ͑h͒ mass and the Higgs boson self-couplings as a function of the cutoff scale beyond which either some of the coupling constants are strong enough to invalidate the perturbative analysis or the stability of the electroweak vacuum is no longer guaranteed. Using the results obtained from the above analysis, we find that the singlet charged Higgs boson can significantly modify the partial decay width of h→␥␥ via radiative corrections, and its collider phenomenology can also be drastically different from that of the charged Higgs bosons in the usual two-Higgs-doublet models.

We present a semi-analytic calculation of the tau-lepton flux emerging from the Earth, induced by the incident high energy neutrinos interacting inside the Earth for 10 5 ≤ E ν /GeV ≤ 10 10 . We obtain results for the energy dependence of the tau-lepton flux coming from the Earth-skimming neutrinos, because of the neutrino-nucleon charged-current scattering as well as the resonantν e e − scattering. We illustrate our results for several anticipated high energy astrophysical neutrino sources such as the AGNs, the GRBs, and the GZK neutrino fluxes. The tau lepton fluxes resulting from rock-skimming and ocean-skimming neutrinos are compared. Such comparisons can render useful information for the spectral indices of incident neutrino fluxes.

Real world markets display power-law features in variables such as price fluctuations in stocks. To further understand market behavior, we have conducted a series of market experiments on our web-based prediction market platform which allows us to reconstruct transaction networks among traders. From these networks, we are able to record the degree of a trader, the size of a community of traders, the transaction time interval among traders and other variables that are of interest. The distributions of all these variables show power-law behavior. On the other hand, agent-based models have been proposed to study the properties of real financial markets. We here study the statistical properties of these agent-based models and compare them with the results from our web-based market experiments. In this work, three agentbased models are studied, namely, zero-intelligence (ZI), zero-intelligenceplus (ZIP) and Gjerstad-Dickhaut (GD). Computer simulations of variables based on these three agent-based models were carried out. We found that although being the most naive agent-based model, ZI indeed best describes the properties observed in real markets. Our study suggests that the basic ingredient to produce the observed properties from real world markets could in fact be the result of a continuously evolving dynamical system with basic features similar to the ZI model.