See Refs. [14, 15, 16, 17] for related discussions on positivity bounds in gravitational theories. Also, see Refs. [10, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26] for earlier applications of positivity bounds to the swampland program.

A similar cutoff $\mathrm{\Lambda }\sim \sqrt{m_e{M}_{\mathrm{Pl}}/e}$ is implied from positivity in the presence of a hidden sector that is coupled to photon only through gravity [26].

These properties are indeed satisfied in known amplitudes in perturbative string theory. Furthermore, the analyticity and the mild behavior are inferred from causality and locality at least for a gapped system, so in the literature these properties are sometimes called “axioms”.

The same order-estimate was performed in [10], but the recent work [11] refined the bound more explicitly in terms of $f(t)$, ${\alpha }^{\prime }$, and ${\alpha }^{\prime \prime }$. Also, more recently, the paper [25] considers the gravitational positivity bounds without assuming the mild high-energy behavior in the Regge limit, and argues that the negativity of the rhs may be allowed from the EFT perspective. In this case, however, the order of magnitude of the rhs cannot be determined.

The one-loop diagrams are calculated by using the Mathematica packages feynarts [33] and feyncalc [34], and the loop integrals are evaluated by package-x [35]. As a consistency check, we confirm the desired crossing symmetries, the relation (8), and the agreement with two different gauge choices, the Feynman-’t Hooft gauge and the unitary gauge.

The inclusion of a charged spin-0 particle does not change the situation as well. See Ref. [25] for the analysis in scalar QED.

We use the same references cited by [37]. First, the transition constants are $C_{\gamma \to \rho }^2=(\alpha /2.54)$, $C_{\gamma \to \omega }^2=(\alpha /20.5)$, $C_{\gamma \to \varphi }^2=(\alpha /11.7)$, taken from [39]. Adopting [40], the Pomeron exchange contribution grows as $s^{1.08}$ and the overall factor is chosen such that $\mathrm{Im}{\mathcal{A}}_{VV\to VV}=8.56\mathrm{mb}s(s/s_0{)}^{0.08}$ [41].

Also note that we have assumed a weakly coupled UV completion where the graviton is Reggeized below the Planck scale. Our precise statement is if the SM coupled to GR is UV completed at a scale below the Planck scale, the scale should be less than ${10}^{16}\mathrm{GeV}$. It would be interesting to generalize our argument to strongly coupled UV completion of gravity. For this, one would need to first carefully reconsider the standard assumptions of positivity bounds such as locality and unitarity because super-Planckian physics such as black hole creation cannot be ignored.