General collapsed-limit connected trispectrum template

Construct a general template for the connected four-point function of vector fields in the collapsed limit k\to 0 and for arbitrary spins, in order to determine its contribution to the infrared tail of parity-violating gravitational waves.

Background

The paper decomposes the source four-point correlation function into Gaussian products of two-point functions and a connected trispectrum. The Gaussian contributions can be analyzed explicitly, but the infrared behavior of the connected part depends on the trispectrum’s momentum and spin structure.

The collapsed limit, in which the external momentum carried by each source pair tends to zero, controls the low-frequency gravitational-wave tail. A general template valid in this limit and for arbitrary spins would therefore extend the paper’s analysis beyond the specific Gaussian sources and scalar trispectrum templates considered later.

References

Unfortunately, we are not aware of a general template for the connected four-point function in the $k\to 0$ limit and for general spins.

— Unpolarized Low-Frequency Tail of Causal Chiral Gravitational Waves  (2609.28828 - Chabour et al., 23 Sep 2026) in Section 2, immediately following Eq. (\ref{eq:four-point-a})