Interacting gauge fields and the zero-energy eigenstates in two dimensions

Gauge fields are formulated in terms of the zero-energy eigenstates of 2-dimensional Schr$\ddot{\mathrm{o}}$dinger equations with central potentials $V_a(\rho )=-a^2{g}_a{\rho }^{2(a-1)}$ ($a≠0$, $g_a>0$ and $\rho =\sqrt{x^2+y^2}$). It is shown that the zero-energy states can naturally be interpreted as a kind of interacting gauge fields of which effects are solved as the factors $e^{i{g}_c{\chi }_A}$, where ${\chi }_A$ are complex gauge functions written by the zero-energy eigenfunctions. We see that the gauge fields for $a=1$ are nothing but tachyons that have negative squared-mass $m^2=-g_1$. We also find out U(1)-type gauge fields for $a=1\mathrm{/}2$ and SU(3)-type gauge fields for $a=3\mathrm{/}2$. Massive particles with internal structures described by the zero-energy states are also studied.