New self-dual solutions of $SU(2)$ Yang-Mills theory in Euclidean Schwarzschild space

We present a systematic study of spherically symmetric self-dual solutions of $SU(2)$ Yang-Mills theory on Euclidean Schwarzschild space. All the previously known solutions are recovered and a new one-parameter family of instantons is obtained. The newly found solutions have continuous actions and interpolate between the classic Charap and Duff instantons. We examine the physical properties of this family and show that it consists of dyons of unit (magnetic and electric) charge.

Figure 1

Plots of ${\varphi }_{\kappa }(r)$ for evenly spaced values of $\kappa$ going from $-4$ (lowest curve) to $-1$ (uppermost curve). The thick curves correspond to the Charap and Duff solutions ($\kappa =-3$ for the lower thick curve and $\kappa =-2$ for the upper thick curve).

Figure 2

Top: Plots of ${\varphi }_{\kappa }(r)$ as a function of $r$ for evenly spaced values of $\kappa$ going from $\kappa =-3$ (lowest curve) to $\kappa =-2$ (uppermost curve). Bottom: Semilog plots of ${\varphi }_{\kappa }(s+2m)$ as a function of $s$ associated with the same values of $\kappa$ as before.

Figure 3

Action $S_{\kappa }$ (associated with the solution ${\varphi }_{\kappa }$) as a function of $\kappa$.

Figure 4

Plots of the series expressions obtained before (solid line) and after (dotted line) applying the mapping method. The dashed line corresponds to the numerical solution obtained in the previous section. Notice that the series solution given by the mapping method (truncated here at $n=30000$) is virtually indistinguishable from the numerical solution. In all the cases, $\kappa =-2.5$.

Figure 5

Plot of the coefficients $b_n$ as a function of $n$ for $\kappa =-2.5$.