论文标题
类似Sphaleron的分析构建，调用较高维度理论的溶液
Analytic construction of sphaleron-like solution invoking higher dimensional gauge theory
论文作者
论文摘要
我们在4维（4D）时空中进行了类似Sphaleron的解决方案的分析结构，以调用5D SU（2）仪表理论的框架。通过类似Sphaleron的解决方案，我们是指运动方程的静态有限能解决方案，该解决方案带有Chern-Simons编号$ N_ \ text {cs} = \ frac {1} {2} {2} $。由于我们对低能量有效理论中的静态解决方案感兴趣，因此我们专注于仅包含4D空间（非时空），$（a_ {i}，a_ {y}）（i = 1,2,3）的量规字段的部分（i = 1,2,3）$，并且仅保留这些字段的Kaluza-Klein零模式。有趣的是，众所周知，在这个4D空间中的自偶性条件不过是't Hooft-polyakov monopole的BPS条件，一旦将超空间组件$ a_y $识别为单极解所需的伴随标量。因此，类似SPHALERON的溶液基于嵌入在较高维时时间时的BPS单极，可以将其解释为一个自动划线仪。通过使用课程，我们在普通4D时空的情况下学习了intsanton，我们实现了$ a_ {i} $的SphaLeron样配置，该配置带有$ n_ \ text {cs} = \ frac {1} {1} {2} $。作为这种构造的特征，引用了更高的维量规理论，与普通BPS单极的情况相反，伴随标量的VEV在拓扑上是拓扑固定的，因此，Sphaleron样解决方案的质量被确定为$ m_ \ text {sp {sp} = {sp} = \ frac {4π} {g_ {4}^{2}}} \ frac {1} {r} $（$ g_ {4} $：4D量规耦合常数，$ r $：圆圈的半径作为额外的空间）。我们还认为，类似Sphaleron的解可能被视为静态场配置空间中能量的鞍点。
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 $M_\text{sp} = \frac{4π}{g_{4}^{2}}\frac{1}{R}$ ($g_{4}$: 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.
