Couplings of light I=0 scalar mesons to simple operators in the complex plane

Abstract

The flavour and glue structure of the light scalar mesons in QCD are probed by studying the couplings of the I=0 mesons $\sigma (600)$ and $f_0(980)$ to the operators $\bar{q}q$, ${\alpha }_s{G}^2$ and to two photons. The Roy dispersive representation for the $\pi \pi$ amplitude $t_0^0(s)$ is used to determine the pole positions as well as the residues in the complex plane. On the real axis, $t_0^0$ is constrained to solve the Roy equation together with elastic unitarity up to the $K\Kbar$ threshold leading to an improved description of the $f_0(980)$. The problem of using a two-particle threshold as a matching point is discussed. A simple relation is established between the coupling of a scalar meson to an operator $j_S$ and the value of the related pion form-factor computed at the resonance pole. Pion scalar form-factors as well as two-photon partial-wave amplitudes are expressed as coupled-channel Omnès dispersive representations. Subtraction constants are constrained by chiral symmetry and experimental data. Comparison of our results for the $\bar{q}q$ couplings with earlier determinations of the analogous couplings of the lightest I=1 and $I=1/2$ scalar mesons are compatible with an assignment of the $\sigma$, $\kappa$, $a_0(980)$, $f_0(980)$ into a nonet. Concerning the gluonic operator ${\alpha }_s{G}^2$ we find a significant coupling to both the $\sigma$ and the $f_0(980)$.

31 pages, 5 figures

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