Analytic Boundaries of Infinite-Spin-Tower Amplitudes from Hidden Zero

We study the general form of meromorphic amplitudes that are compatible with unitarity, analyticity, crossing symmetry, polynomial boundedness and the hidden-zero and corresponding splitting conditions. These amplitudes are infinite-spin-tower (IST) amplitudes and are characterized by the distribution of poles. With infinitely many evenly separated poles, the IST amplitudes reduce to Veneziano amplitudes. We construct such unitary amplitudes in a primal way (rule in) and find the bounds analytically. The allowed region we derive is smaller than, but close to, the region allowed by the positivity bounds (rule out). We argue that, in the absence of an accumulation point in the energy levels, the analytic boundary we derive is the largest possible boundary in the primal construction of meromorphic amplitudes. This type of IST amplitude can also be extended to the fully crossing-symmetric case related to the Virasoro-Shapiro amplitude. We found from the IST amplitudes that graviton pole imposes unitarity constraints on UV spectrum.