A dilaton-pion mass relation

Recently, Golterman and Shamir presented an effective field theory which is supposed to describe the low-energy physics of the pion and the dilaton in an $SU(N_c)$ gauge theory with $N_f$ Dirac fermions in the fundamental representation. By employing this formulation with a slight but important modification, we derive a relation between the dilaton mass squared~$m_\tau^2$, with and without the fermion mass~$m$, and the pion mass squared~$m_\pi^2$ to the leading order of the chiral logarithm. This is analogous to a similar relation obtained by Matsuzaki and~Yamawaki on the basis of a somewhat different low-energy effective field theory. Our relation reads $m_\tau^2=m_\tau^2|_{m=0}+KN_f\hat{f}_\pi^2m_\pi^2/(2\hat{f}_\tau^2)+O(m_\pi^4\ln m_\pi^2)$ with~$K=9$, where $\hat{f}_\pi$ and~$\hat{f}_\tau$ are decay constants of the pion and the dilaton, respectively. This mass relation differs from the one derived by Matsuzaki and~Yamawaki on the points that $K=(3-\gamma_m)(1+\gamma_m)$, where $\gamma_m$ is the mass anomalous dimension, and the leading chiral logarithm correction is~$O(m_\pi^2\ln m_\pi^2)$. For~$\gamma_m\sim1$, the value of the decay constant~$\hat{f}_\tau$ estimated from our mass relation becomes $\sim50\%$ larger than $\hat{f}_\tau$ estimated from the relation of Matsuzaki and~Yamawaki.