  We perform analytic construction of a sphaleron-like solution in the
4-dimensional (4D) space-time invoking the framework of 5D SU(2) gauge theory.
By the sphaleron-like solution we mean a static finite energy solution to the
equation of motion, which carries the Chern-Simons number
$N_\text{CS}=\frac{1}{2}$. Since we are interested in the static solution in
the low-energy effective theory, we focus on the part of the action which
contains only the gauge fields in the 4D space (not space-time), $(A_{i},
A_{y}) (i = 1,2,3)$ and keep only the Kaluza-Klein zero modes of these fields.
Interestingly, the self-duality condition in this 4D space is known to be
nothing but the BPS condition for the 't Hooft-Polyakov monopole, once the
extra-space component $A_y$ is identified with the adjoint scalar, needed for
the monopole solution. Thus, the sphaleron-like solution is based on the BPS
monopole embedded in the higher dimensional space-time, which may be
interpreted as a self-dual gauge field. By use of the lesson we learn in the
case of the instanton in ordinary 4D space-time, we achieve the sphaleron-like
configuration of $A_{i}$, which carries $N_\text{CS} = \frac{1}{2}$. As a
characteristic feature of this construction invoking higher dimensional gauge
theory, in clear contrast to the case of the ordinary BPS monopole, the VEV of
the adjoint scalar is topologically fixed, and therefore the mass of the
sphaleron-like solution is determined to be $M_\text{sp} =
\frac{4\pi}{g_{4}^{2}}\frac{1}{R}$ ($g_{4}$: 4D gauge coupling constant, $R$:
the radius of the circle as the extra space). We also argue that the
sphaleron-like solution may be regarded as a saddle point of the energy in the
space of static field configurations.
