On relationship between conformal transformations and broken chiral symmetry

Abstract

Starting with the conformal transformations in the momentum space, the nonlinear $\sigma$-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 $q_{\mu}$ ($q'_{\mu}=q_{\mu}+h_{\mu}$, $q'_{\mu}=\Lambda^{\nu}_{\mu}q_{\nu}$, $q'_{\mu}=\lambda q_{\mu}$ and $q'_{\mu}=-M^2q_{\mu}/q^2$) the equivalence rotations in the 6D space were used. The derived five dimensional Lagrangians consists of the parts defined in the two different region $q_{\mu}q^{\mu}\pm q_5^2=\pm M^2$ which are connected by the inversion $q'_{\mu}=-M^2 q_{\mu}/q^{2}$, where $M$ is a scale parameter. For the $\sigma$-model $M$ is determined by the pion mass $M^2= m_{\pi}^2/2$. 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 $q^{\mu}q_{\mu}>M^2$ and $0\le q^{\mu}q_{\mu}<M^2$.