PHYSICAL REVIEW LETTERS 121, 051601 (2018) Emergence of Weak Coupling at Large Distance in Quantum Gravity Ben Heidenreich* Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5 † Matthew Reece Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA ‡ Tom Rudelius School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA (Received 9 March 2018; published 3 August 2018) The Ooguri-Vafa swampland conjectures claim that in any consistent theory of quantum gravity, when venturing to large distances in scalar field space, a tower of particles will become light at a rate that is exponential in the field-space distance. We provide a novel viewpoint on this claim: If we assume that a tower of states becomes light near a particular point in field space, and we further demand that loop corrections drive both gravity and the scalar to strong coupling at a common energy scale, then the requirement that the particles become light exponentially fast in the field-space distance in Planck units follows automatically. Furthermore, the same assumption of a common strong-coupling scale for scalar fields and gravitons implies that, when a scalar field evolves over a super-Planckian distance, the average particle mass changes by an amount of the order of the cutoff energy. This supports earlier suggestions that significantly super-Planckian excursions in field space cannot be described within a single effective field theory. We comment on the relationship of our results to the weak gravity conjecture. DOI: 10.1103/PhysRevLett.121.051601 Introduction.—A question of fundamental interest in the the weak gravity conjecture (WGC) [7]. Other recent works study of quantum gravity is the following: Which low- on the swampland conjectures include Refs. [8–13]. energy gravitational effective field theories (EFTs) admit an Controlled string compactifications typically have many ultraviolet (UV) completion? Said differently, what are the “moduli”: light scalars with gravitational-strength cou- low-energy predictions of quantum gravity? plings. This is related to the fact that the string theory The string theory provides a very large class of quantum has no continuous parameters, so any freely adjustable gravities, with a wide variety of possible low-energy EFTs coupling must be controlled by the vacuum expectation (see, e.g., [1,2]). But, despite this enormous “landscape” of value (VEV) of a scalar field. In the simplest supersym- vacua, there is growing evidence that they all share metric examples, the moduli are exactly massless and identifiable common features, distinguishing them from parametrize a continuous moduli space of vacua. More “swampland” [3], a large class of seemingly consistent realistic examples require a moduli potential with isolated gravitational EFTs with no quantum gravity UV comple- minima—the vacua of the theory—but the only reliable tion. Thus, our original question becomes this: How do we way to generate vacua in the weak-coupling regime is distinguish the swampland from the landscape? through nonperturbative effects. Since these are exponen- There are several candidate criteria for fencing off parts of tially small at weak coupling, the moduli masses are “ the swampland. In this Letter, we focus on the swampland likewise exponentially suppressed relative to, e.g., the ” conjectures of Refs. [3,4] concerning the moduli space of Kaluza-Klein (KK) scale or the string scale. quantum gravity. We will apply ideas about the effective “Conjecture 0” of Ref. [4] generalizes these observations field theory and the infrared emergence of weak coupling to any quantum gravity. recently developed in Ref. [5] (see also [6]), where they were Conjecture 0.—Every continuous parameter in a quan- used to understand another candidate swampland criterion, tum gravity is controlled by the VEV of a scalar field. Thus, we expect moduli to be ubiquitous in the land- scape, and it is natural to investigate the properties of the Published by the American Physical Society under the terms of moduli space of quantum gravity. In particular, we focus on the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to two related conjectures from Ref. [4]. the author(s) and the published article’s title, journal citation, Conjecture 1.—The moduli space has an infinite diam- and DOI. Funded by SCOAP3. eter (despite a finite volume [3,14]). 0031-9007=18=121(5)=051601(6) 051601-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 121, 051601 (2018) ≈ μ ϕ þ Oðϕ2ÞðÞ Conjecture 2.—At large distances in moduli space, an mn n 2 infinite tower of resonances becomes light exponentially quickly with an increasing distance. after an appropriate redefinition of ϕ. In making this ansatz, Here distances in moduli space are defined by geodesic we are no longer free to assume that ϕ has a canonical kinetic term. In fact, we will soon show that the kinetic term distances with respect to the moduli space metric gijðϕÞ: for ϕ blows up as ϕ → 0, and the geodesic distance to ð Þ 1 ϕ ¼ 0 L mod ¼ g ðϕÞ∂ϕi · ∂ϕj: ð1Þ diverges logarithmically in accordance with kin 2 ij Conjecture 2. The ansatz (2) is quite general. For instance, suppose Implicit in these conjectures is the notion that parametri- there are multiple towers with masses trending to zero at cally distant points in moduli space correspond to weak- different rates. Then, by moving far enough in the moduli coupling limits. space, we can focus on the tower that becomes light fastest In this Letter, we explore the connection between weak and parametrize the modulus ϕ so that this tower becomes coupling, a tower of light resonances, and large distances in light in a linear fashion. Thus, aside from being justified by moduli space. Central to our arguments is the assumption, specific examples, we believe that (2) captures the most developed in Ref. [5], that weak coupling in quantum general intended meaning of “tower” in Conjecture 2. gravities is a long-distance phenomenon, with all physics For concreteness, we take the particles in the tower to be becoming strongly coupled in the ultraviolet at a common Dirac fermions, though similar results apply for other spins. “quantum gravity scale.” This occurs in many concrete The Lagrangian is then quantum gravities, such as large volume compactifications of M theory. While there are possible counterexamples as 1 X L ¼ ðϕÞð∂ϕÞ2 − ðϕÞþ ½ψ¯ ð ∂ − μ ϕÞψ ð Þ well (see [5]), a suitable generalization of this notion may 2 K V n i= n n : 3 address these, and it is worthwhile to understand the n consequences regardless. ϕ This assumption clarifies the connection between light Fermion loops will correct the propagator as well as ϕ resonances and weak coupling. For example, light, charged S-matrix elements for scattering of particles. At some Λ resonances screen gauge forces, leading to weak gauge energy scale , these loop corrections become as large as couplings in the infrared: This is essentially the mechanism the tree-level contribution, and the EFT breaks down. We Λ of Ref. [5]. follow the approach of Ref. [5] to ascertain the scale , ΛðϕÞ We now explain the connection between light resonances with the new ingredient that depends on the value of and parametrically large distances in moduli space, apply- the modulus. (We work in the Einstein frame, holding the ’ ϕ ing the same assumption. We find that, if a tower of fields D-dimensional Newton s constant fixed as varies. Λ becomes light, the requirement of a common quantum Alternatively, one could hold the cutoff fixed and view gravity scale implies they do so exponentially quickly, so the strength of gravity as varying across the moduli space, ϕ → 0 the quantitative portion of Conjecture 2 can be derived from which may be more natural when corresponds to a our qualitative assumptions. Applying similar reasoning, decompactification limit.) ϕ we further consider the effect of moving a large but We now compute the loop correction to the propagator bounded distance in moduli space, with particular attention from the tower of fermions, perturbing around a fixed hϕi¼ϕ ðϕÞ to the case of an axion with a trans-Planckian decay expectation value 0. [For general V and general ϕ constant. In that case, our arguments imply that many 0, there is a tadpole, and the field will begin to roll. modes have masses that change by an amount large Implicit in the notion of a modulus is that the potential is compared to the quantum gravity cutoff energy. This casts relatively flat; hence, there is a finite timescale over hϕi¼ϕ some doubt on the utility of the effective field theory for the which our computation of scattering with fixed 0 treatment of trans-Planckian field ranges in quantum makes sense. We comment further on the flatness of the gravity. [While this paper was in preparation (following potentialpffiffiffiffiffiffiffiffiffiffiffiffiffi below.] The canonically normalized fluctuation ¯ a strategy sketched in the conclusions of Ref. [5]), we ϕ ≔ Kðϕ0Þðϕ − ϕ0Þ has the propagator became aware of independent work [15] with some over- 1 1 lapping results.] hϕ¯˜ ð Þϕ¯˜ ð− Þi ∼ ð Þ p p 2 2 2 : 4 Light resonances and large distances.—Motivated by p − mϕ þ iε 1 þ Πðp Þ Conjecture 2, we consider a trajectory in moduli space approaching a singular point where an infinite tower of By adding appropriate counterterms, we can choose the resonances becomes massless. This trajectory is parame- renormalization condition Πð0Þ¼0 (up to possible