Spontaneous Breakdown of Conformal and Chiral Invariance

We discuss the implications of a theory in which scale and chiral invariance are spontaneously broken, and the dilaton appears as a mixture of the two isoscalar members of the scalar nonet. The usual assumptions for the conformal properties of the axial-vector current constrain the low-energy behavior of the spin-2 form factor, $F_1\mathrm{}\left(t\right)\mathrm{}$, of the pionic matrix element of the stress-energy tensor. In the limit of scale invariance, we find ${F_1}^{\prime }\mathrm{}\left(0\right)=\frac{{F_{\sigma \pi }}^{\prime }\mathrm{}\left(0\right)\mathrm{}}{F_{\sigma \pi }\mathrm{}\left(0\right)\mathrm{}}$, where $F_{\sigma \pi }\mathrm{}\left(t\right)\mathrm{}$ is the axial-vector form factor obtained from the coupling of the dilaton to a pion via the axial-vector current, and the prime denotes differentiation. This relation connects the assumptions of $f$ dominance of $F_1\mathrm{}\left(t\right)\mathrm{}$ and $A_1$ dominance of $F_{\sigma \pi }\mathrm{}\left(t\right)\mathrm{}$. Using the method of collinear dispersion relations, we estimate the effects of violation of scale invariance. A result previously obtained in the limit of scale invariance becomes $F_{\sigma }{F}_{\sigma \pi }\left({m_{\sigma }}^2\right)f_{\pi }\simeq \frac{1}{2}$, where $f_{\pi }$ is the decay constant of the pion, and $F_{\sigma }$ couples the dilation to the vacuum via the stress-energy tensor. Similar corrections to the scale-invariant prediction for ${F_1}^{\prime }\mathrm{}\left(0\right)\mathrm{}$ are calculated. The magnitudes of the corrections are controlled by the $A_1\sigma \pi$ coupling constant. According to the usual estimates of this constant, the predicted width of the dilaton is compatible with the Adler-Weisberger sum rule for $\pi \pi$ scattering and phenomenological estimates of the $\sigma \mathrm{NN}$ coupling constant. While the relation for ${F_1}^{\prime }\mathrm{}\left(0\right)\mathrm{}$ obtained in the limit of scale invariance is compatible with the assumption of $f$ dominance, the effects of symmetry breaking are large. In the real world, we find that $f$ dominance is a poor approximation, a conclusion which is supported by recent estimates of the $\mathrm{fNN}$ coupling constants. We discuss the relation of our work to the magnitude of parameters measuring symmetry violation in the energy density. Our interpretation of a recent result of Cheng and Dashen is that scale invariance is spontaneously broken, and chiral $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ is a much better symmetry than $\mathrm{SU}\left(3\right)\mathrm{}$.