Veneziano Representation and the Boson Spectrum

The Veneziano representation for the scattering of two pseudoscalar mesons is constructed under the assumptions of (1) linear trajectories with identical slopes, (2) Adler conditions for soft $K$ as well as for soft $\pi$, (3) no $I\ge \frac{3}{2}$ or $S\ge 2$ direct-channel resonances, and (4) no nonleading terms in the Veneziano representation. It is found that a nonet of vector-tensor trajectories is consistent with the above assumptions, while a pure octet is not. The $\omega -\varphi$ and the $f^0-f^{\prime }$ mixing angles are predicted to be given by $sin\theta =\frac{1}{\sqrt{3}}$. The Adler conditions lead to the requirements that the $\omega$, $\rho$, $f^0$, and $A_2$ mesons lie on a degenerate trajectory ½ unit above the $\pi$, that the $K^{*}-K_n$ trajectory lie ½ unit above the $K$, and that an $A_2^{}{}_{}{}^{\prime }$ trajectory with only even-$J$ members appear ½ unit above the $\eta$. The Adler conditions also give the Okubo mass formula for the $\varphi$ and $f^{\prime }$ mesons, and the Gell-Mann-Okubo formula for the $\eta$. The possibility of the splitting of $f^{\prime }$ into two peaks is also discussed. In this model, the residues of leading poles satisfy factorization, while the daughter poles manifestly violate factorization. Since unitarity implies factorization, it is argued that the daughters must be significantly modified by any unitarization procedure, but the leading poles can be interpreted as resonances in the usual way. The lack of simple algebraic relations among the residues of daughter poles also suggests that the daughters need not belong to ${\mathrm{SU}}_3$ multiplets. The $J=1\mathrm{}$ recurrences of the pseudoscalar mesons are identified as the axial vectors $A$, $K_A$, and $D\left(1285\right)\mathrm{}$. The $B$ meson, however, seems to fit well as a daughter of the $\rho$, and hence need not belong to an ${\mathrm{SU}}_3$ multiplet.