Status of Chiral-Scale Perturbation Theory

Abstract

Chiral-scale perturbation theory $\chi$PT${}_{\sigma }$ has been proposed as an alternative to chiral $SU(3{)}_L\times SU(3{)}_R$ perturbation theory which explains the $\Delta I=1/2$ rule for kaon decays. It is based on a low-energy expansion about an infrared fixed point in three-flavor QCD. In $\chi$PT${}_{\sigma }$, quark condensation $⟨\bar{q}q{⟩}_{\mathrm{vac}}\ne 0$ induces nine Nambu-Goldstone bosons: $\pi ,K,\eta$ and a QCD dilaton $\sigma$ which we identify with the $f_0(500)$ resonance. Partial conservation of the dilatation and chiral currents constrains low-energy constants which enter the effective Lagrangian of $\chi$PT${}_{\sigma }$. These constraints allow us to obtain new phenomenological bounds on the dilaton decay constant via the coupling of $\sigma /f_0$ to pions, whose value is known precisely from dispersive analyses of $\pi \pi$ scattering. Improved predictions for $\sigma \to \gamma \gamma$ and the $\sigma NN$ coupling are also noted. To test $\chi$PT${}_{\sigma }$ for kaon decays, we revive a 1985 proposal for lattice methods to be applied to $K\to \pi$ on-shell.

arXiv comment10 pages, 1 figure. Presented at the 8th International Workshop on Chiral Dynamics, 29 June 2015 - 03 July 2015, Pisa, Italy. Revision: references and comment added

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