Fixed-angle behavior in 4-point dual models. I. Absolutely convergent series of beta functions

The large-s fixed-angle asymptotic behavior of scattering amplitudes of the form T(s,t,u)=A(s,t)+A(s,u)+A(u,t), with $A(s,t)=\Sigma {h=0}^{\infty }{C}_{h}B(h-\alpha (s),h-\alpha (t\left)\right)$, is studied. We find that the asymptotic behavior is determined by the function $F\left(z\right)=\Sigma {h=0}^{\infty }{C}_h{z}^h$. If F(z) is entire the Veneziano fixed-angle behavior is preserved. If F(z) is singular at z=λ, λ real and larger than one, a different, but still exponentially damped, fixed-angle behavior is produced around the forward and backward regions. If λ=1, we find a behavior which depends critically on the growth of Im(S) as Re(S)→∞ [S≡α(s)]. We find, in particular, that the models of Mandelstam and Frampton have this last type of fixed-angle behavior. The same is true of the model of Gervais and Neveu, unless their choice α₀=-1 is made. The model of Neveu and Schwarz, however, does have the usual Veneziano fixed-angle behavior. We finally relate our work to that of Ellis and Freund, and find some discrepancy, which we comment on.