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

@misc{backus2026halfladder,
  title = {{Half-ladder partial waves for all $n,d$ with applications to string theory}},
  author = {Jeffrey V. Backus},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28617},
  eprint = {2609.28617},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28617},
}

TY  - UNPB
TI  - Half-ladder partial waves for all n,d with applications to string theory
AU  - Jeffrey V. Backus
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28617
UR  - https://arxiv.org/abs/2609.28617
AB  - 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 ≤ 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.
ER  -