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Properties of the $π^0$, $η$, $η'$, $σ$, $f_0(980)$ and $a_0(980)$ mesons and their relevance for the polarizabilities of the nucleon

The signs and values of the two-photon couplings $F_{Mγγ}$ of mesons $(M)$ and their couplings $g_{MNN}$ to the nucleon as entering into the $t$-channel parts of the difference of the electromagnetic polarizabilities $(α-β)$ and the backward angle spin polarizabilities $γ_π$ are determined. The excellent agreement achieved with the experimental polarizabilities of the proton makes it possible to make reliable predictions for the neutron. The results obtained are $α_n=13.4\pm 1.0$, $β_n=1.8\mp 1.0$ ($10^{-4}$ fm$^3$), and $γ^{(n)}_π=57.6\pm 1.8$ ($10^{-4}$ fm$^4$). New empirical information on the flavor wave functions of the $f_0(980)$ and the $a_0(980)$ meson is obtained.