Gravitational positivity bounds on scalar potentials

We derive constraints on scalar field theories coupled to gravity by using recently developed positivity bounds in the presence of gravity. It is found that a canonically normalized real scalar cannot have an arbitrarily flat potential unless some new physics enters well below the Planck scale. An upper bound on the scale of new physics is determined by loop corrections to the self-energy. Our result provides a swampland condition for scalar potentials.

Figure 1

Analytic structure of $\tilde{\mathcal{M}}(s,t)$ on the complex $s$-plane and the integration contour to derive the relation (2). The wavy line is a brunch cut and the point $s=s_{*}$ is the reference point. We choose $s_{*}=2m_{\text{ph}}^2-(t/2)+i\mu$ ($\mu >0$).

Figure 2

Diagrams (i): Relevant 1PI diagrams for an effective vertex ${\varphi }^{2}h$ in the present analysis. Solid lines and double wavy lines denote the propagators of $\varphi$ and $h_{\mu \nu }$, respectively. Diagrams (ii): The $t$-channel diagrams which give negative contributions to $c_{2,\text{impr}}(0)$, expressed in terms of the 1PI vertices shown in the diagrams (i).

Figure 3

Gravitational positivity bound on $\lambda {\varphi }^4$ theory. Lines saturating our bound (18) and the applicability condition (16) in $\lambda {\varphi }^4$ theory are shown in the solid red line and dashes blue line, respectively. We substitute $M_{\mathrm{s}}=10^{16}\mathrm{GeV}$ and $\lambda =10^{-2}$ to draw the dashed lines. The shaded region is excluded by the bound (18) under the condition (20).

Figure 4

Diagrams relevant for the self-energy of $\varphi$ in the present analysis. The first and the second diagrams are one-loop diagrams, and the third diagram is a two-loop diagram. The second diagram is independent of the external momentum and so the leading corrections from the quartic coupling $\lambda$ to the momentum-dependence arise at the two-loop level.

Figure 5

1PI diagrams for nongravitational four-point scattering up to $\mathcal{O}({\lambda }^2,\lambda g^2,g^4)$. All the possible assignments of external momenta should be considered.

Figure 6

The process $\varphi \varphi \to \varphi \varphi$ expressed by the non-1PI diagrams up to one-loop. All the possible assignments of external momenta should be considered. Diagrams (d) represent the self-energy corrections to the $\varphi \varphi \to \varphi \varphi$ process. The diagram (e) is the contribution from the one-loop corrections to ${\varphi }^3$ vertex.

Figure 7

One-loop 1PI diagrams for the ${\varphi }^{2}h$-vertex. Counterterm diagrams are also shown which are necessary for renormalization. The diagram with a field renormalization $\delta Z_{\varphi }$ is not included since it is not necessary at one-loop level in the present model. External in-going momenta for scalar lines are $k_1$ and $k_2$. In-going momentum for external graviton is $q=-(k_1+k_2)$. All the possible assignment of external momenta should be considered.

Figure 8

Two-loop 1PI diagrams for the ${\varphi }^{2}h$-vertex from the quartic coupling $\lambda$ which are relevant for the computation of $Q(k_1,k_2)$ and $R(k_1,k_2)$. Other diagrams are relevant only for the computation of $T(k_1,k_2)$, the trace component of $V^{\mu \nu }$. External in-going momenta for scalar lines are $k_1$ and $k_2$ and that for the graviton is referred to as $q=-(k_1+k_2)$.

Figure 9

Classification of diagrams with graviton up to $\mathcal{O}(M_{\text{pl}}^{-2})$. A solid thick line and a thin double wavy line denote a loop-corrected propagator of $\varphi$ and a free propagator of graviton, respectively. Diagrams which contribute to $c_{\text{grav},\text{others}}$ are underlined: (1) $s$, $u$-channel graviton exchange diagrams, (2) diagrams with a graviton-scalar conversion, and (3) those with a graviton propagator inside the loop. The $t$-channel graviton exchange diagrams contribute to $c_{\text{grav},t\text{−}\text{ch}}$.

Figure 10

One-loop graviton exchange diagrams relevant for $c_{\text{grav}}$ at order $M_{\text{pl}}^{-2}$. Each type has both $s$- and $u$-channel versions. There are other one-loop diagrams for $s$, $u$-channel graviton exchange, but they do not contribute to $c_{\text{grav}}$.

Figure 11

The one-loop diagram with a graviton-scalar conversion that are relevant for $c_{\text{grav}}$ at order $M_{\text{pl}}^{-2}$. For this type, the $s,u$-channel diagrams have a nonzero contribution to $c_{\text{grav}}$, but the $t$-channel diagram does not. Also, there are other one-loop diagrams with a graviton-scalar conversion, but they do not contribute to $c_{\text{grav}}$.

Figure 12

Diagrams with a graviton propagator inside loops which are of $\mathcal{O}(M_{\text{pl}}^{-2})$. Again, all the possible assignment of external momenta have to be considered. The diagrams (A), (B), and (C) can be understood as the contributions from the $\mathcal{O}(M_{\text{pl}}^{-2})$ corrections to the 1PI self-energy, the effective ${\varphi }^3$ vertex, and the effective ${\varphi }^4$ vertex, respectively.