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Convergence target of the recursive action series in the self-interaction program

Determine what the overall series of action terms ∑ S_i[γ,h], specified only by the recursive sequence rule and initial terms in the self-interaction construction, converges to, without presupposing the Einstein–Hilbert action.

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Background

Derivation attempts often build a sequence of action terms S_i by expanding around a background and defining successive terms via a recursive functional relation. Although specific constructions reproduce the first few terms of the Einstein–Hilbert expansion, the general limit of the series is not fixed solely by the recursive prescription.

The authors stress that, absent independent appeal to general relativity, it is unclear what the entire series converges to, highlighting a gap in the claimed derivations from spin-2 self-interaction.

References

Even if it were for instance discovered that the ``sequence within the series'' $(S_i)_{i \in \mathbb{N}$ is explicitly given by $S_i[\gamma, h] = 2/i! \left[ \int d4 x h{\mu \nu} \frac{\delta}{\gamma{\mu \nu}\right]{i-2} S_2[\gamma, h]$, it is not at all clear from the mere sequence rule what the overall series would converge to.

— GR as a classical spin-2 theory? (2403.08637 - Linnemann et al., 13 Mar 2024) in Section 2.1 (Ambiguities in derivation)