Nonforward Superconvergence Relations, Higher Symmetries, and Mass Spectra

Superconvergence relations for the reactions $\mathrm{PV}\to \mathrm{PV}$, $\mathrm{PV}\to \mathrm{VV}$, and $\mathrm{VV}\to \mathrm{VV}$ between nonets of pseudoscalar and vector mesons can be saturated at zero momentum transfer with sets of particles corresponding to the representations (6, $\overline{6}$; $l$) of the rest-symmetry group $U\left(6\right)\mathrm{}\times U\left(6\right)\mathrm{}\times O\left(3\right)\mathrm{}$. Every single one of these representations saturates the relations, provided the vertices are invariant under the collinear $U\left(6\right)\mathrm{}\times O\left(2\right)\mathrm{}$ group. The infinite sequence of representations (6, $\overline{6}$; $l$), $l=0, 1, \dots$ is then used in order to saturate the nonforward superconvergence relations for the reactions $\mathrm{PV}\to \mathrm{PV}$. For mass spectra $m_l$ with accumulation points ${m_{\infty }}^2>4\mathrm{}{m_0}^2$, it is found that the resulting equations have no nontrivial solution for the coupling constants. This result remains unchanged for an oscillator-like spectrum. The possibility is discussed that mass splitting within the multiplets (6, $\overline{6}0$; $l$) and/or symmetry breaking at the vertices can in principle make a saturation possible. It is argued that an approximate saturation of the amplitudes and a few of their derivatives at $t=0\mathrm{}$ with a finite number of resonances may well be reasonable. For higher derivatives the saturation is expected to depend sensitively upon the absorptive parts at higher energies, which are more reasonably described by Regge terms than by direct-channel resonances. The formal saturation of superconvergence relations with mass-degenerate multiplets is discussed briefly.