# On relationship between conformal transformations and broken chiral
symmetry

Research paper by **A. I. Machavariani**

Indexed on: **30 Nov '06**Published on: **30 Nov '06**Published in: **Mathematical Physics**

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#### 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 $0\le q^{\mu}q_{\mu}<M^2$.