Broken Chiral and Conformal Symmetry in an Effective-Lagrangian Formalism

The simultaneous breaking of conformal and chiral symmetry is investigated within the framework of nonlinear realizations and effective Lagrangians. The explicit introduction of a massless dilaton field, $\chi$ enables conformal invariance to be preserved in Lagrangians for massive matter fields. It is shown that the equation of Callan, Coleman, and Jackiw, ${\partial }_{\mu }{D}_{\mu }={\theta }_{\mu \mu }$, remains valid notwithstanding the introduction of this particle, and also that it is possible to construct Lagrangians which are simultaneously invariant under the chiral and conformal groups. If we introduce a term which explicitly violates both symmetries, then the dilaton acquires a definite (bare) mass which can be expressed in terms of the masses of the chiral bosons —the pion in the case of chiral $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$, the pion and kaon in the case of chiral $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$. The precise form of this mass relation depends upon the type of symmetry-breaking term adopted.