We work in mostly-plus metric signature throughout, where the Mandelstam invariants are $s=-(p_1+p_2{)}^2$, $t=-(p_2+p_3{)}^2$, and $u=-(p_1+p_3{)}^2$ for cyclically labeled incoming momenta.

Note that the $t>0$ behavior is unspecified, so a finite or infinite tower of spins is a priori allowed.

Strictly speaking, we impose this property on the real component of $\alpha (t)$, which controls the magnitude of the amplitude in the Regge limit, along the real $t$ axis.

Mathematically, we assume that the amplitude is a meromorphic function in the entire complex $s$ plane, including infinity.

Recall that the famous Regge formula in Eq. (1) is an exact formula for asymptotically large $s$ everywhere in the complex $s$ plane except the positive real axis [104]. Case in point, any tree-level amplitude $A(s,t)$ is required to have simple poles on the real $s$ axis separated by regions of analyticity, which are all clearly absent from Eq. (1). In fact, Eq. (1) smears the discontinuities on the positive $s$ axis into a single branch cut, fusing the discrete spectrum of states into a continuum whose microscopic resolution is controlled by the gaps between poles.

The case of infinite $J(n)$ at finite $n$ corresponds to an infinite tower of spins residing on an isolated, narrow resonance. This describes a propagating mode of unbounded spatial extent, in violation of locality [10, 52, 76].

Here we directly integrate Eq. (1), which exhibits a branch cut along the positive real $s$ axis corresponding to all physical discontinuities. To compute Eq. (3) we transform to polar coordinates, $s=s_{*}{e}^{i\theta }$, and integrate around the branch cut in the domain $0<\theta <2\pi$.

This condition is trivially satisfied when $\alpha (t)$ is a constant integer with no $t$ dependence whatsoever. In this case, the amplitude is a rational function, so for sufficiently large $s_{*}$ there are no poles nearby, and thus any residue computed there will be zero. While our logic still applies, this scenario is obviously uninteresting, so we will assume throughout that the leading Regge trajectory $\alpha (t)$ actually depends on $t$.

Note that the hypergeometric [78] and ${}_4{F}_3$ satellite [90] amplitudes exhibit power-law Regge behavior reminiscent of quantum field theory, where $\alpha (t)$ crosses few if any integers, yielding few if any Regge zeros, as expected.

In the notation of Ref. [109], this deformed worldsheet integral corresponds to taking ${\tilde{f}}_4(u)=g/4$, a constant, and is hence the simplest possible model in that family. We can write the amplitude in closed form, $$A(s,t)=\frac{\mathrm{\Gamma }(-s)\mathrm{\Gamma }(-t)}{\mathrm{\Gamma }(-s-t)}{{}_{2}F}_2[\begin{array}{c}-s,-t\\ -\frac{s+t}{2},\frac{1-s-t}{2}\end{array};\frac{g}{4}],$$by expanding the exponential and resumming.

The original Coon amplitude [99], whose residues are consistent with unitarity only for $q<1$, exhibits an accumulation point in the spectrum and is neither dual resonant nor meromorphic, on account of a branch cut dressing factor [78, 114]. While amplitudes of this form are realized in nonperturbative string theoretic constructions [77], they depart from our assumption of tree-level dynamics, so we leave them to future work.

Here we have implicitly assumed that every zero of $\alpha (t)+r=0$ appears in $R(n,t)$ as a block. Said another way, $R(n,t)$ does not include partial branches of solutions where only a subset of the zeros of $\alpha (t)+r=0$ appear. This structure is justified because of the underlying logic of the Regge zeros: If one solution of $\alpha (t)+r=0$ appears in $R(n,t)$, then $n$ is sufficiently large to justify the arc integral argument that implies the Regge zeros, and thus the other solutions of $\alpha (t)+r=0$ should be exhibited as well.

To be precise, the result of this bootstrap procedure is the $Z$-theory amplitude [120] for colored external scalars, which is trivially related to the amplitudes of the superstring and bosonic string by a simple prefactor and shift of the Mandelstam invariants, respectively. The latter correspond to offsets in $\alpha (t)$, $J(n)$, or $\mu (n)$, which can also be used to generate so-called satellite amplitudes.

This representation of the string amplitude is obtained by transforming the Koba-Nielsen integral into one involving the $u$ variables [80, 109, 124, 125, 126, 127, 128, 129, 130], with integrand going like $u_{13}^{X_{13}}{u}_{14}^{X_{14}}{u}_{24}^{X_{24}}{u}_{25}^{X_{25}}{u}_{35}^{X_{35}}$. One then uses the $u$ equations that relate the different $u$ variables among each other to eliminate all but two of them [93].