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 infra- trized by a scalar field ϕ; we take the singular point to be red divergences if there is a massless particle coupling ϕ ¼ 0 without the loss of generality. We assume that the to ϕ). spectrum is dominated by an infinite tower of particles that Parametrically, the large-p behavior of the contribution become uniformly light as ϕ → 0,so to the one-loop integral from fermion n is
051601-2 PHYSICAL REVIEW LETTERS 121, 051601 (2018) μ2 D−4 1 ∂ 2 D−4 That is, for a wide variety of spectra for the tower of states jΠ ð 2 ≫ 2Þj ∼ np ∼ mn p ð Þ n p mn ; 5 that become light as ϕ → 0, the condition that the modulus Kðϕ0ÞlD Kðϕ0Þ ∂ϕ lD becomes strongly coupled at the quantum gravity scale 2 where lD is a loop factor (e.g., l4 ≃ 16π ) which we omit fixes the kinetic term of the modulus to be henceforward. To assess the strong-coupling scale, we λ ð Þ 1 define a function ϕ p that captures the parametric L ∼ MD−2 ð∂ϕÞ2: ð12Þ contribution to Πðp2Þ from the sum over only those kin Pl ϕ2 particles with mass less than p.Wehave In particular, distances in the field space grow logarithmi- D−4 X ∂ 2 cally with the value of ϕ. Equivalently, the particles in the λ ð Þ ≔ p mn ð Þ ϕ p ðϕ Þ ∂ϕ : 6 tower become exponentially light in the field-space dis- K 0 j ðϕ Þ n mn 0
energies above p0, similar to the unification of gauge and QG ðΛ Þ 8 GNN QG gravity loops discussed in Ref. [5]. Following (12), we can canonically normalize the In terms of this scale, we have φ ≃ ðD−2Þ=2 ðϕ ϕ Þ modulus as MPl log = 0 , expanding about ϕ ¼ ϕ 1 2 0. From (3), this leads to couplings of the form λϕðΛ Þ¼ hμ iΛ ; ð9Þ QG Λ2 ðϕ Þ QG QGK 0 GN m m L ⊃− n φψ¯ ψ − n φ2ψ¯ ψ − : ð14Þ ðD−2Þ=2 n n 2MD−2 n n where MPl Pl 1 X ∂ 2 We see that the modulus has gravitational-strength cou- hμ2i ≔ mn Λ ðΛÞ ∂ϕ plings to the particles in the tower; this is a well-known N j ðϕ Þ Λ n mn 0 < characteristic of moduli, here a natural consequence of our 1 X Λ2 assumptions. ≈ m2 ∼ : ð Þ ðΛÞϕ2 n ϕ2 10 Trans-Planckian distances and EFT.—Conjecture 2 N j ðϕ Þ Λ n mn 0 < implies that a single EFT cannot describe parametrically large distances in moduli space, since a parametrically large In the last line, we have made a mild but crucial assumption number of massive particles inevitably become light, and that most of the particles in the tower lie near the cutoff, as we need UV information to determine their properties. in any tower with an increasing density of states dN=dp or However, the conjecture places no restrictions on tra- with a power-law (increasing or decreasing) density of versals that are larger than MPl but not parametrically large. states. For instance, there are examples in which the density of Given this assumption, if we demand that the modulus states is essentially constant along a trajectory of super- “ ” becomes strongly coupled at the quantum gravity scale Planckian length. (Such a trajectory is not necessarily a Λ QG in (8), we constrain the form of the kinetic term: geodesic in moduli space [12].) This typically occurs when 1 the modulus in question is an axion, consistent with λϕðΛ Þ ∼ 1 ⇒ Kðϕ0Þ ∼ : ð Þ Conjecture 2 because the compact field space prevents QG ϕ2 11 GN 0 an arbitrarily large excursion. While axion excursions do
051601-3 PHYSICAL REVIEW LETTERS 121, 051601 (2018) not lead to a tower of particles becoming light, we still To illustrate these points, consider an axion θ arising expect a tower of particles with masses that change as the from compactification of a five-dimensional Uð1Þ gauge axion expectation value varies. When the axion field theory with coupling constant e5 on a circle of radius R. completes a full circuit, the spectrum must return to where Consistent with recent ideas about the WGC [23–26] and it started, perhaps with a nontrivial monodromy. the analysis of Ref. [5], we assume the 5D theory has a Motivated by axions, we reconsider the effect of a tower tower of near-extremal charged particles, which leads to a of particles with masses mnðϕÞ that depend on a modulus 4D KK spectrum of the form ϕ, dropping the assumption that the masses vanish as ϕ → 0 2 ∼ ½ 2 2 þ 2ð − θÞ2 2 ð Þ . By the same analysis as above, if the modulus mn1;n2 n1e1 e2 n2 n1 MPl: 18 becomes strongly coupled at the quantum gravity scale Λ λ ðΛ Þ ∼ 1 2 ¼ 2 ð2π Þ 2 ¼ 2 ð Þ2 QG, i.e., if ϕ QG , then the modulus kinetic term is Here e1 e5= R and e2 = RMPl are the 4D approximately gauge couplings, and θ ≅ θ þ 1 is the axion. Assuming a sufficiently large number of modes charged under each ð1Þ X ∂m 2 U lie below the cutoff, we can approximate the sums ðϕÞ ∼ Λ ðϕÞD−4 n ð Þ K QG ∂ϕ : 15 over n1 and n2 in (15) as integrals, giving njmnðϕÞ
MPl : A mode with a mass of the order of traversal almost all the modes will be recycled. Λ QG generically acquires a different mass of the order of What does this mean for EFT and axion inflation? On the Λ QG. On the other hand, (17) does imply that generically an one hand, we could integrate out all the modes with mass order-one fraction of the modes will pass through the cutoff e1MPl or above, obtaining an EFT with a lower cutoff but during this traversal. (Similar phenomena were observed with no apparent drama as the axion traverses a super- in a particular model with towers of spectator fields Planckian distance. On the other hand, if we wish to in Ref. [22].) compute the axion potential using EFT, then we cannot
051601-4 PHYSICAL REVIEW LETTERS 121, 051601 (2018) take this approach, since in this example the axion potential Conclusions.—The sublattice WGC [23–25] and the is generated by the Casimir energy of charged particles in recently proposed tower WGC [26]—strengthened versions – the 5D parent theory [27 29], whose masses start at e1MPl. of the weak gravity conjecture motivated by dimensional If we raise the cutoff to include some of these charged reduction and evidence from the string theory—require an particles in the EFT, then we once again face the twin issues infinite tower of light particles, closely linking them to of modes emerging from the cutoff and becoming light and the swampland conjectures. In some cases, the connection a large fraction of the modes passing through the cutoff is direct: The gauge coupling g is related to the VEV of a during the axion traversal. (In the language of the 5D EFT, scalar modulus, and the g → 0 limit brings down a these issues are related to the difficulty of imposing a single tower of light, charged particles, satisfying both gauge-invariant cutoff on loops of charged particles.) Both conjectures. issues suggest that an EFT incorporating these modes is not In other cases, the swampland conjectures strengthen the well controlled for a super-Planckian field excursion in the sublattice WGC by demanding that a tower of particles can absence of additional UV input. be accessed within the EFT. For instance, in Ref. [5],we Thus, there are potential subtleties in using an EFT (even observed that in some examples the WGC tower is Λ a string-theory-derived EFT) to compute the potential of an predicted to lie above QG. A concrete case is an approx- axion over a very super-Planckian field range. Our argu- imately isotropic 4D compactification of the type IIB string ments suggest that these subtleties extend beyond the theory with gauge fields on D7 branes. The gauge coupling ∼ 1 ð Þ2 extranatural context explored in Ref. [18] and to other g = MsR , so the WGC tower is at gMPl; the string 3=2 moduli besides axions. It remains to be seen whether these scale is lower, at g MPl, but the Kaluza-Klein tower is 2 subtleties are of critical importance in candidate large-field lower still, at g MPl. The WGC tower is outside the low- models. energy EFT, but the KK tower is not. It accounts for the Modulus potential.—Above, we assumed a light modu- infinite distance in moduli space as g → 0 and the gen- lus that can be described within the EFT. We should test this eration of strongly coupled gravity via the species bound. assumption: Supersymmetric theories can have exactly This phenomenon, with multiple towers of particles becom- massless moduli, but, more generally, do loops tend to ing light at different rates as one moves to infinity in moduli give moduli a large mass? If we sum up the loop corrections space, can arise in a variety of examples. to moduli masses from a tower of fermions, we find power- We have argued that the assumption of a universal Λ divergent diagrams. Cutting these off at QG, we have strong-coupling scale for fields in quantum gravity can serve as a more fundamental replacement for some of the X m2 NðΛ ÞΛD swampland conjectures. It is still important to put the most δ 2 ∼ n ΛD−2 ∼ QG QG ∼ Λ2 ð Þ mmod D−2 QG D−2 QG: 22 basic aspects of these conjectures on a firmer footing: Can j Λ MPl MPl n mn< QG we prove rigorously that moduli exist and that large-moduli limits always send infinite towers of particles to zero mass? However, in the presence of supersymmetry (SUSY), These basic assumptions have a similar flavor to the contributions from scalars and their fermionic partners statement that quantum gravities have no global sym- approximately cancel to leave a remainder of the order metries and deserve more attention. of the average SUSY-breaking splitting within the tower of states: We thank the referees for helpful comments that have improved the presentation. The research of B. H. was δ 2 j ∼ h 2 − 2 i ð Þ mmod SUSY mboson mfermion : 23 supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Generically, without supersymmetry, loop corrections drive Government of Canada through the Department of Λ the modulus mass to the cutoff QG. Specific theories may Innovation, Science and Economic Development and by differ; e.g., a nonsupersymmetric theory compactified on a the Province of Ontario through the Ministry of Research, circle has a radion modulus with a controlled (Casimir) Innovation and Science. M. R. is supported in part by potential. The resolution of this puzzle is that tuning away Department of Energy Grant No. DE-SC0013607 and the the cosmological constant in the parent theory in turn fine- National Aeronautics and Space Administration ATP Grant tunes the modulus potential of the lower-dimensional No. NNX16AI12G. T. R. is supported by the Carl P. theory; a c.c. of the naive size in the parent theory becomes Feinberg Founders’ Circle Membership and National Λ2 a modulus potential with curvature QG in the daughter Science Foundation Grant No. PHY-1314311. theory. As another example, theories with no-scale struc- ture can have a volume modulus lighter than the overall – SUSY-breaking scale [30 32]. Higher-dimensional sym- *[email protected] † metries (e.g., 10D type IIB) can cause cancellations that are [email protected] ‡ nonobvious in 4D. We leave a complete discussion for [email protected] future work. [1] M. R. Douglas, C. R. Phys. 5, 965 (2004).
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