We study the validity of positivity bounds in the presence of a massless graviton, assuming the Regge behavior of the amplitude. Under this assumption, the problematic t-channel pole is canceled with the UV integral of the imaginary part of the amplitude in the dispersion relation, which gives rise to finite corrections to the positivity bounds. We find that low-energy effective field theories (EFT) with “wrong” sign are generically allowed. The allowed amount of the positivity violation is determined by the Regge behavior. This violation is suppressed by $$ {M}_{\mathrm{pl}}^{-2}\alpha^{\prime } $$ M pl − 2 α ′ where α′ is the scale of Reggeization. This implies that the positivity bounds can be applied only when the cutoff scale of EFT is much lower than the scale of Reggeization. We then obtain the positivity bounds on scalar-tensor EFT at one-loop level. Implications of our results on the degenerate higher-order scalar-tensor (DHOST) theory are also discussed.

In this study, we formulate a systematic way of deriving an effective equation of motion(EoM) for long wavelength modes of a massless scalar field with a general potential V(φ) on de Sitter background, and investigate whether or not the effective EoM can be described as a classical stochastic process. Our formulation gives an extension of the usual stochastic formalism to including sub-leading secular growth coming from the nonlinearity of short wavelength modes. Applying our formalism to λφ 4 theory, we explicitly derive an effective EoM which correctly recovers the nextto-leading secularly growing part at a late time, and show that this effective EoM can be seen as a classical stochastic process. Our extended stochastic formalism can describe all secularly growing terms which appear in all correlation functions with a specific operator ordering. The restriction of the operator ordering will not be a big drawback because the commutator of a light scalar field becomes negligible at large scales owing to the squeezing.

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It is known that in the theory of light scalar fields during inflation, correlation functions suffer from infrared (IR) divergences or large IR loop corrections, leading to the breakdown of perturbation theory. In order to understand the physical meaning of such IR enhancement, we investigate the stochastic properties of an effective equation of motion (EoM) for long-wavelength modes of a canonically normalized light scalar field φ with a general sufficiently flat interaction potential on de Sitter background. Firstly, we provide an alternative refined derivation of the effective action for long-wavelength modes which leads to the effective EoM that correctly reproduces all the IR correlation functions in a good approximation at a late time, by integrating out short-wavelength modes. Next, under the assumption that one can neglect non-local correlations in the influence functional exceeding the coarse-graining scale, we show that the effective EoM for IR modes of the "average field" in Schwinger-Keldysh formalism φ < c can be interpreted as a classical stochastic process in the present model.

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