I am not clear with the difference of heavy and light mesons; 1. what is their difference? 2. how can we treat their decay properties? 3.for instance many people describe them using the Bethe-Salpeter equation at 4-dimension or at 3-dimension. which one is the better method? To help me, thank you very much!!
Heavy mesons have at least one heavy quark. This question is way to broad to give an answer more specific than "with quantum field theory".
One pretty neat difference is that in the case of heavy mesons (which, as mfb said, contain at least one heavy quark) we can use an effective field theory called Heavy Quark Effective Theory (HQET). The idea is that in this case we can expand our physical quantities in a series in powers of [itex]1/m_H[/itex], where [itex]m_H[/itex] is the mass of the heavy quark. As you can see, if the quark is reasonably heavy this approach can simplify your work and allow a fairly extended perturbative treatment. On the other hand if the quarks are all light ones then you can't do anything useful with that and most of the time you need some non-perturbative methods to compute, for example decay amplitudes. One good example are the [itex]D[/itex] (containing a charm quark) and the [itex]B[/itex] (containing a bottom) mesons. While the bottom is pretty heavy with respect to the scale of QCD (more or less 500 MeV), the charm quark is a little bit in between the two regimes. That's why (I you read some literature) we have a much better theoretical understanding of, for example, neutral B meson oscillation than we have for the neutral D meson.