The decay $J/\psi\to\gamma\phi\phi$: spin dependence of amplitude and angular distributions of photons with linear polarizations

Based on the effective invariant amplitudes of the transitions $J/\psi\to\gamma X(J^P)$ and $X(J^P)\to\phi\phi$, the spin dependence of the $J/\psi\to\gamma X(J^P)\to\gamma\phi\phi$ decay amplitude is obtained in the case of the intermediate resonances $X$ with $J^P=0^\pm,1^\pm,2^\pm$. Angular distributions of photons with the definite linear polarizations relative to the plane spanned by the momentum of initial electron and the total momentum of the $\phi\phi$ pair in the reaction $e^+e^-\to J/\psi\to\gamma X(J^P)\to\gamma\phi\phi$ are calculated. It is shown that the sign of the asymmetry of the distributions of the photons polarized in the above plane and orthogonal to it correlates with the signature $P_X(-1)^{J_X}$ of the $X$ resonance with given spin-parity $J_X^{P_X}$ so it may help to establish this quantum number in a way which does not depend on the specific model of the $X(J^P)\to\phi\phi$ amplitude.