Half-ladder partial waves for all $n,d$ with applications to string theory

Abstract: Given any four-point scattering amplitude, the partial wave expansion resolves its residues into contributions from individual spin-$s$ internal particles, where each coefficient in the expansion must be non-negative due to unitarity. In this paper, we introduce a partial wave expansion tailored to "half-ladder" Feynman diagrams at generic multiplicity $n$ and spacetime dimension $d$, with scalars on the external lines and massive bosons on the internal lines. Derived by enforcing that the half-ladder residue is an eigenvector of each internal particle's angular momentum, each basis element is written as a sum over products of special functions related to the Gegenbauer polynomials, and we give a set of simple graphical rules for explicitly building any of them at generic $n$ and $d$. To put the formalism into action, we apply our technology to deformed "string theory" amplitudes, defined a priori in any dimension and with arbitrary Regge intercept $α_0$. By constructing multi-particle positivity tests for the half-ladder and "twisted" half-ladder diagrams, we show that the only amplitudes consistent with unitarity are those belonging to open bosonic string theory ($α_0 = -1$) below the critical dimension $d \leq 26$. In particular, we confirm the expectation that $Z$-theory ($α_0 = 0$) fails to define a unitary string theory at finite $α'$. Finally, for the open bosonic string, we show how knowledge of the partial wave expansion allows us to resolve the contributions to the half-ladder diagram from individual degenerate states at a given level and spin in a purely on-shell analysis. As an ancillary file, we include a Python script combwaves.py which implements the partial wave expansion given a generic (twisted) half-ladder residue.