Banana diagrams as functions of geodesic distance

We extend the study of banana diagrams in coordinate representation to the case of curved space-times. If the space is harmonic, the Green functions continue to depend on a single variable – the geodesic distance. But now this dependence can be somewhat non-trivial. We demonstrate that, like in the flat case, the coordinate differential equations for powers of Green functions can still be expressed as determinants of certain operators. Therefore, not-surprisingly, the coordinate equations remain straightforward – while their reformulation in terms of momentum integrals and Picard-Fuchs equations can seem problematic. However we show that the Feynman parameter representation can also be generalized, at least for banana diagrams in simple harmonic spaces, so that the Picard-Fuchs equations retain their Euclidean form with just a minor modification. A separate story is the transfer to the case when the Green function essentially depends on several rather than a single argument. In this case, we provide just one example, that the equations are still there, but conceptual issues in the more general case will be discussed elsewhere.

💡 Research Summary

**
The paper extends the coordinate‑space analysis of “banana” (or sunrise) Feynman diagrams from flat Minkowski space to curved space‑times, focusing on a class of curved manifolds called harmonic spaces. A harmonic space is defined by the property that the Laplacian of the geodesic distance σ depends only on σ itself, i.e. Δσ = p(σ). This condition is equivalent to the space being Einstein (R_{\muν}=κ g_{\muν}) and guarantees that any Green function that depends solely on σ can be treated as a function of a single variable even in a curved background.

The authors first review several important examples of harmonic spaces:

1. Simple harmonic spaces (SH) – spaces where Δσ takes the same form as in flat Euclidean space, Δσ = (d‑1)/σ. An explicit four‑dimensional Ricci‑flat but curvature‑non‑trivial metric is given as an illustration.
2. Maximally symmetric spaces – the d‑sphere, hyperbolic space and their analytic continuations, embedded via z²+η_{\muν}x^{\mu}x^{\nu}=r². Here σ is the angle between two points and Δσ = (d‑1) cot(σ)/r.
3. Complex projective space CPⁿ – equipped with the Fubini‑Study metric, a Kähler manifold. The geodesic distance is the Hermitian angle between two complex lines, and Δσ = (dim CPⁿ‑1) cot σ − tan σ.
4. Grassmannians GR(k,n) – generalizations of CPⁿ, also Kähler, with the same functional form for Δσ.

Having established that Δσ is a function of σ in these settings, the paper rewrites the massive, non‑minimally coupled Klein‑Gordon equation for the Green function G(σ): \