关于保角变换与手征对称性破缺的关系
On relationship between conformal transformations and broken chiral symmetry
摘要：
从动量空间的共形变换出发，再现了非线性$σ$模型和$SU(2)\times U(1)$对称性自发破缺的标准模型。相应的手征拉格朗日量以五维形式给出，因为对于四动量$q_μ$（$q'_μ=q_μ+h_μ$、$q'_μ=Λ^ν_μq_ν$、$q'_μ=λq_μ$和$q'_μ=-M^2q_μ/q^2$）的保角变换，使用了6D空间中的等效旋转。导出的五维拉格朗日量由定义在两个不同区域$q_μq^μ\pm q_5^2=\pm M^2$中的部分组成，这两个区域由反演$q'_μ=-M^2 q_μ/q^{2}$连接，其中$M$是尺度参数。对于$σ$ -模型，$M$由π质量$M^2= m_π^2/2$确定。对于$SU(2)\times U(1)$对称性自发破缺的5维拉格朗日量，尺度参数$M^2$由Higgs粒子质量$8m^2_{Higgs}=9M^2$定义.与通常的四维公式不同，本方法中的手征对称性破缺项是由保角变换得到的，并且证明了拉格朗日函数中相应的相互作用部分在$q^μq_μ>M^2$和$0\le q^μq_μ<M^2$区域中具有相反的符号.
英文摘要：
Starting with the conformal transformations in the momentum space, the nonlinear $σ$-model and the standard model with the spontaneous broken $SU(2)\times U(1)$ symmetry are reproduced. The corresponding chiral Lagrangians are given in the five dimensional form because for the conformal transformations of the four-momentum $q_μ$ ($q'_μ=q_μ+h_μ$, $q'_μ=Λ^ν_μq_ν$, $q'_μ=λq_μ$ and $q'_μ=-M^2q_μ/q^2$) the equivalence rotations in the 6D space were used. The derived five dimensional Lagrangians consists of the parts defined in the two different region $q_μq^μ\pm q_5^2=\pm M^2$ which are connected by the inversion $q'_μ=-M^2 q_μ/q^{2}$, where $M$ is a scale parameter. For the $σ$-model $M$ is determined by the pion mass $M^2= m_π^2/2$. For the 5D Lagrangian with the spontaneous broken $SU(2)\times U(1)$ symmetry the scale parameter $M^2$ is defined by the Higgs particle mass $8m^2_{Higgs}=9M^2$. Unlike to the usual four-dimensional formulation in the present approach the chiral symmetry breaking terms are obtained from the conformal transformations and it is demonstrated, that the corresponding interaction parts of Lagrangians have the opposite sign in the regions $q^μq_μ>M^2$ and $0\le q^μq_μ<M^2$.
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