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We perform analytic construction of a sphaleron-like solution in the4-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 theequation of motion, which carries the Chern-Simons number$N_\text{CS}=\frac{1}{2}$. Since we are interested in the static solution inthe low-energy effective theory, we focus on the part of the action whichcontains 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 benothing but the BPS condition for the 't Hooft-Polyakov monopole, once theextra-space component $A_y$ is identified with the adjoint scalar, needed forthe monopole solution. Thus, the sphaleron-like solution is based on the BPSmonopole embedded in the higher dimensional space-time, which may beinterpreted as a self-dual gauge field. By use of the lesson we learn in thecase of the instanton in ordinary 4D space-time, we achieve the sphaleron-likeconfiguration of $A_{i}$, which carries $N_\text{CS} = \frac{1}{2}$. As acharacteristic feature of this construction invoking higher dimensional gaugetheory, in clear contrast to the case of the ordinary BPS monopole, the VEV ofthe adjoint scalar is topologically fixed, and therefore the mass of thesphaleron-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 thesphaleron-like solution may be regarded as a saddle point of the energy in thespace of static field configurations.

Analytic construction of sphaleron-like solution invoking higher dimensional gauge theory