A dilaton-pion mass relation

Abstract

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 $m_{\tau }^2=m_{\tau }^2{|}_{m=0}+K{N}_f{\hat{f}}_{\pi }^2{m}_{\pi }^2/(2{\hat{f}}_{\tau }^2)+O(m_{\pi }^4\ln m_{\pi }^2)$ 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~${\gamma }_m\sim 1$, the value of the decay constant~${\hat{f}}_{\tau }$ estimated from our mass relation becomes $\sim 50\%$ larger than ${\hat{f}}_{\tau }$ estimated from the relation of Matsuzaki and~Yamawaki.

Comments: 13 pages