Daughter Sequences in Unequal-Mass Vector-Meson-Scalar-Meson Scattering

We study the requirements of Mandelstam analyticity at zero energy in unequal-mass vector-meson-scalar-meson elastic scattering. Daughter sequences of Regge poles are required, and we study their properties. By factorization we determine how the sequences couple to an equal-mass vector-scalar channel, and establish that the sequences correspond to $M=0\mathrm{}$ and $M=1\mathrm{}$ Toller poles. We find how analyticity relates the slopes and second derivatives of the trajectories at $t=0\mathrm{}$ for both $M=0\mathrm{}$ and $M=1\mathrm{}$ [breaking of $O\left(4\right)\mathrm{}$ symmetry by nonzero energy]. We study the first two leading terms of the daughter residues at $t=0\mathrm{}$ [breaking of $O\left(4\right)\mathrm{}$ symmetry by unequal external masses and nonzero energy]. The roles of factorization and the equal-mass conspiracy relation are central in our work.