Deformation of instanton in external color fields

We study instanton properties in a color electro-magnetic field. In the presence of external color fields, the 't Hooft ansatz is no longer a solution of the Yang-Mills field equation, and therefore we numerically investigate the instanton solution which minimizes the total Yang-Mills action in the nontrivial topological sector. For simplicity, the external color field is set to be a constant abelian magnetic (electric) one. Starting from the total gauge field by superposing the 't Hooft ansatz in the singular gauge and the external color field, we perform the numerical minimization of the total Yang-Mills action under the global color orientation, the anisotropic scale transformation and the local transformation of the instanton configuration. We obtain the instanton solution which provides the minimal total action in the external color field. In this solution, the 4-dimensionally rotational symmetry O(4) is reduced into two 2-dimensionally rotational symmetries O(2) x O(2), due to the homogeneous external field. In the space R3 at a fixed t, we find a quadrapole deformation of this instanton solution. In the magnetic field H vector, the prolate deformation occurs along H vector. In the electric field E vector, the oblate deformation occurs along E vector. We further discuss the correlation between the instanton and the monopole in the external field. The external field plays the important role on the determination of the monopole trajectory formed by the instanton. In fact, the monopole and anti-monopole pair appears around the instanton in the external field and such a monopole loop seems to screen the external field in part. (author)