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The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff

The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff

Eur. Phys. J. C (2018) 78:337 https://doi.org/10.1140/epjc/s10052-018-5811-3

Regular Article - Theoretical

The Weak and emergence from an ultraviolet cutoff

Ben Heidenreich1,a, Matthew Reece2,b, Tom Rudelius3,c 1 Perimeter Institute for , Waterloo, ON N2L 2Y5, Canada 2 Department of Physics, Harvard University, Cambridge, MA 02138, USA 3 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

Received: 12 March 2018 / Accepted: 16 April 2018 / Published online: 26 April 2018 © The Author(s) 2018

Abstract We study ultraviolet cutoffs associated with the ratio greater than or equal to that of a large semiclassical Weak Gravity Conjecture (WGC) and Sublattice Weak Grav- extremal black hole. ity Conjecture (sLWGC). There is a magnetic WGC cutoff at If quantum exist which violate the Weak Grav- −1/2 the scale eGN with an associated sLWGC tower of ity Conjecture (WGC), they will have unusual properties. charged . A more fundamental cutoff is the scale at In particular, large near-extremal black holes in these theo- which gravity becomes strong and field theory breaks down ries cannot completely evaporate, but instead evolve slowly entirely. By clarifying the nature of the sLWGC for non- towards extremality, resulting in a tower of stable extremal abelian gauge groups we derive a parametric upper bound on black holes. However, unlike stable black hole remnants in this strong gravity scale for arbitrary gauge theories. Intrigu- theories with global symmetries, the mass of these stable ingly, we show that in theories approximately saturating the extremal black holes increases in proportion to their charge, sLWGC, the scales at which loop corrections from the tower hence the sharpest contradictions (e.g., an infinite density of charged particles to the gauge boson and graviton prop- of states in violation of the covariant bound [4]) do agators become important are parametrically identical. This not occur, and these observations fall short of a compelling suggests a picture in which gauge fields emerge from the argument for the conjecture. scale by integrating out a tower of charged On the other hand, strong circumstantial evidence for the fields. We derive a converse statement: if a gauge the- WGC comes from the absence of counterexamples in string ory becomes strongly coupled at or below the quantum grav- theory, which provides many examples of consistent quan- ity scale, the WGC follows. We sketch some phenomenolog- tum gravities. While in some of the more involved string ical consequences of the UV cutoffs we derive. constructions it may be difficult to check the WGC explic- itly, there are many examples, such as compactifications of the perturbative heterotic string, where the conjecture is both 1 Introduction non-trivial and verified by explicit calculation. As yet there is no convincing proof of the Weak Grav- ity Conjecture (WGC) in a general setting, although recently 1.1 The Weak Gravity Conjecture there have been two interesting claims to derive a version of the WGC from entropy bounds [5,6] in specific simple grav- The Weak Gravity Conjecture [1] is an interesting proposal itational effective field theories, the latter closely related to for a universal feature of all quantum gravities, and is one of (but disagreeing with) earlier that highlighted unusual the most concrete and falsifiable observations of the swamp- features of loop corrections to black hole entropy in the pres- land program [2,3]. In its most minimal form, the conjecture ence of WGC-violating particles [7,8]. (An even earlier argu- states that in any theory of quantum gravity with a massless ment based on entropy bounds was sketched in [9]). Other gauge boson there is a charged with charge-to-mass recent work, based on detailed evidence from numerical GR, has suggested that the WGC could be a consequence of the a e-mail: [email protected] Weak Cosmic Censorship Conjecture [10Ð12]. An incom- b e-mail: [email protected] plete sampling of other recent work related to the WGC c e-mail: [email protected] includes [13Ð34]. In this paper, we will not attempt to sort 123 337 Page 2 of 33 Eur. Phys. J. C (2018) 78 :337 out the precise status of these various arguments, but merely (up to some subtleties) it follows from modular invariance note that circumstantial evidence in favor of the WGC has in tree level string theory [41] (also see [42] for a closely been steadily increasing, and it is plausible that some proof analogous AdS3/CFT2 argument). of the conjecture will eventually become generally accepted. There are some issues which arise in interpreting this con- The WGC generalizes in a straightforward manner to p- jecture. For one, what do we mean by “particle”? We have form gauge symmetries and their corresponding charged p- previously argued that multiparticle states are insufficient, branes, as well as to theories with massless scalars (see, e.g., because if the sLWGC is satisfied by multiparticle states [35]) and/or with multiple massless gauge bosons. In the it may fail to be satisfied after dimensional reduction [35]. latter case, the conjecture states that for any rational direc- (Related work on the WGC in different dimensions appeared tion in charge there is a superextremal state in the the- in [15]). Furthermore, evidence from perturbative string the- ory (possibly a multiparticle state). Here “rational direction” ory supports the sLWGC for single particles [41]. On the means any ray which intersects a site in the charge lattice and other hand, it is clear in examples that at many lattice sites the “superextremal” means that the charge-to-mass ratio of the superextremal particles required by the sLWGC are unstable, state is greater than or equal to that of a large semiclassical so the statement as formulated is only clearly defined at para-   1 extremal black hole with a parallel charge vector Q ∝ QBH. metrically weak coupling. In this case “particle” can mean This is equivalent to the graphical “convex hull condition” an unstable (though narrow) resonance. (CHC)of[36]. It is important to note that the black hole Secondly, we have purposefully excluded the four dimen- extremality bound (and hence the weak gravity bound) can sional case above because the conjecture, strictly as stated, be modified by dilatonic couplings [37Ð40], but for the pur- cannot be true in four dimensions: there are examples of poses of this paper we will ignore this possibility. four-dimensional quantum gravities with photons coupled to Since its inception, the WGC has been considered along- massless charged particles.2 In such a theory, the gauge cou- side a number of stronger variants of the conjecture. This is pling of the photon runs to zero in the deep infrared, implying in part because it is difficult to test the minimal WGC stated that parametrically large black holes can have parametrically above if we only have access to the low-energy effective field large charge-to-mass ratios, and only massless particles are theory (EFT); the superextremal charged particles which sat- superextremal; however, there cannot be massless charged isfy the conjecture could be very heavy, and not part of the particles everywhere on a sublattice because this would imply EFT. However, we must be cautious in considering stronger an infinite number of massless particles. The issue is quan- . While the WGC has no known counterexamples tum in nature: for instance, in heterotic orbifolds of this type in string theory, many stronger variants proposed in the lit- the sLWGC is naively satisfied at tree level but fails due to the erature do [41]! The earliest of these variants is the “strong one-loop running of the gauge coupling; a tower of charged WGC” of [1], proposed alongside the original WGC itself. resonances is still present, but the resonances are now subex- For theories with a single photon, the strong WGC conjec- tremal due to the running. The same issue does not arise in tures that the lightest charged particle is superextremal, which higher dimensional theories as massless charged particles do implies the WGC. For multiple photons, the same statement not renormalize the gauge coupling to zero.3 does not imply the WGC, hence there have been various Nonetheless, if the sLWGC is true in higher dimen- attempts to formulate a generalization which does Ð see, e.g., sions then likely some analogous statement should hold [15,35] Ð all of which imply that the lightest charged particle in four dimensions, perhaps with the notion of superex- is superextremal. However, there are well-understood super- tremal replaced by a renormalized version. If there are no symmetric examples in string theory for which this is not the very charged particles then we expect that the sLWGC case [41], hence the strong WGC and all of its variants are should be satisfied up to order-one factors in the charge- false. to-mass ratios. Throughout the paper we will discuss four- In this paper we will be primarily concerned with a strong dimensional examples on the same footing as higher dimen- variant of the WGC for which there is substantial evidence, sional ones with this assumption in mind. Cases with very the “Sublattice WGC” (sLWGC) [41]. In more than four dimensions, the sLWGC holds that in any theory of quantum gravity with massless gauge fields, there must exist a sublat- tice of the charge lattice (of finite index) with a superextremal particle at every site. The sLWGC has been shown to hold in 2 For instance, this occurs in type II string theory compactified on a toroidal orbifolds of type II and heterotic string theory, and Calabi-Yau manifold with a conifold singularity [43]. 3 Quantum corrections may still be important in higher dimensions Ð the evidence from [41] is entirely at tree-level, which is sufficient to 1 In the case of a multiparticle state, the “mass” is simply the sum of the establish a superextremal sublattice in cases where the particles are particle masses (equal to the ADM energy in the limit of large spatial BPS but otherwise not Ð but no counterexamples to the strict statement separation between the particles). of the conjecture are known in D > 4. 123 Eur. Phys. J. C (2018) 78 :337 Page 3 of 33 337 light charged particles are interesting in their own right, but coupled. A theory that saturates an sLWGC-like bound has in this paper we consider them only very briefly in Sect. 7.3.4 the property that gravity and gauge theory become strongly Occasionally in this paper we will refer to the “LWGC” coupled at the same parametric energy scale. This is a highly [35], which is a criterion similar to the sLWGC but with suggestive property, and offers the possibility of answering superextremal states across the entire charge lattice. The some of the interpretational questions about the meaning of LWGC holds in some but not all quantum gravities [41], the sLWGC. As we approach strong coupling and the charged but Ð at least in simple examples Ð theories which violate the particles become increasingly broad, it suggests that it is the LWGC have superextremal states at an order-one fraction of density of states of different charges that must behave nicely sites in the charge lattice, hence many consequences of the in order that the evolving strengths of gravity and electromag- LWGC are robust against its violations. netism become strong at the same scale. It also suggests that we can think of the sLWGC as giving a sufficient condition 1.2 Ultraviolet cutoffs for us to be able to think of a gauge theory as emergent:the smallness of the coupling at low is a consequence It has long been appreciated that the WGC has implications of the dynamics of heavy particles in the ultraviolet. This fits for the energy scales of new physics. In particular, the “mag- very comfortably with Harlow’s proposal that the WGC is a netic version” of the WGC holds that an abelian gauge the- property of emergent gauge fields needed to enforce factoriz- ory of coupling constant e should have superextremal mag- ability of the Hilbert space in quantum gravity with multiple netic monopoles. Assuming that the mass of the magnetic asymptotic boundaries [44]. monopole is not much less than the energy /e2 stored in its The sLWGC may be thought of as saying that, in effect, magnetic field yields [1] all gauge theories in the context of quantum gravity share properties of KaluzaÐKlein theories, with associated towers   . eMPl (1.1) of charged particles. If we compactify a D + 1 dimensional Here 1/ is the radius at which the semiclassical compu- gravity theory on a circle of radius R, both the gauge the- tation of the field energy breaks down. The magnetic WGC ory coupling eKK and the gravitational coupling are obtained requires new physics at or below this scale, but the nature of by tree-level matching in terms of the higher-dimensional this new physics varies in different examples, and does not Planck scale: necessarily signal a breakdown in effective field theory in 1 = π 3 D−1 , general. The scale of quantum gravity Ð at which local quan- R M ; + (1.2) e2 Pl D 1 tum field theory breaks down entirely Ð may be much higher, KK D−2 = π D−1 , and is not directly constrained by the magnetic WGC. MPl;D 2 RMPl;D+1 (1.3) For instance, in the case of a ’t Hooft monopole,  is roughly the scale at which the abelian gauge theory com- with MPl;D the D-dimensional Planck scale. The higher- pletes to a nonabelian gauge group. Above this scale, grav- dimensional Planck scale MPl;D+1 may be interpreted as the ity remains weakly coupled, and the nonabelian gauge the- scale at which quantum gravity necessarily becomes strong,   ory description is valid. The sLWGC postulates a tower of QG MPl;D+1, and the matching ensures that this is well below the D-dimensional Planck scale. Counting Kaluza- particles arising at a scale of order eMPl, strengthening the magnetic WGC argument that this is a new physics scale. Klein modes shows that this parametrically agrees with the Nonetheless, the particles in the tower may remain weakly “species bound” [45Ð49] coupled and be treated in an effective theory. For example, a D−2  D−2, QG Nd.o.f. MPl;D (1.4) tower of KaluzaÐKlein particles can signal the breakdown of 4d effective theory but be treated within a 5d effective theory. where Nd.o.f. is the number of degrees of freedom with mass  In this paper, we will see that once we impose the sLWGC, below QG. The species bound and its gauge theory analog we can also make statements about a more fundamental cut- will play a significant role in this paper. off: the quantum gravity scale where gravity is strongly cou- For a general quantum gravity theory, such a simple tree- pled and QFT breaks down entirely. We will argue that theo- level matching argument may not apply. However, in theo- ries satisfying the sLWGC obey a nontrivial property: if we ries that satisfy the sLWGC, there will always be a tower of consider energy scales far up the tower of charged states, i.e., charged particles, and these particles affect gravitational and − / large compared to eG 1 2, loop corrections imply that both gauge interactions through loops. We will see that these loop N  gravity and the gauge theory become increasingly strongly effects generically lower the scale QG as well as the dynam- ical scale of the gauge theory, and under certain assumptions they naturally match these two scales. Furthermore, there 4 Note that for our purposes, the Standard Model electron is not “very light”, because the renormalized photon coupling near the WGC scale is a sort of converse statement: if gauge theory and gravity eMPl differs only by an order-one amount from the infrared coupling. become strongly coupled at (parametrically) the same energy 123 337 Page 4 of 33 Eur. Phys. J. C (2018) 78 :337

1 1  b  scale, there must be a particle satisfying the WGC (up to order = − i q2 log UV , (2.1) e2 e2 8π 2 i m one factors). UV i i Our paper is organized as follows. In Sect. 2, we will exam- ine the familiar case of a U(1) theory in 4d, showing that an with mi and qi the mass and charge of the particles in the LWGC-saturating tower of particles leads to a coincidence of tower and bi a beta function coefficient. For gravity, the UV cutoff can be understood in terms of the “species bound,” the U(1) Landau pole scale U(1) and the species bound scale  . In Sect. 3, we discuss the form of loop corrections to the which can be thought of as a result of divergent quantum QG  photon and graviton propagators in a general D-dimensional corrections to the Einstein-Hilbert term cut off at QG [45Ð theory and generalize the argument to D dimensions; some 49] details are left to appendix A. In Sect. 4, we revisit our argu- 2 2 M  Nd.o.f. . (2.2) ments for the sLWGC from [41] to show that for nonabelian Pl QG gauge theories the WGC-obeying particles at each sublat- Beyond this perturbative argument, there are various other tice site should be taken to be the highest-weight state in motivations of the species bound, for instance based on −1 their representation, so that different sublattice sites corre- demanding that semiclassical black holes of radius QG not spond to different representations of the nonabelian group. evaporate too quickly [49, ¤3.1]. In Sect. 5, we apply this newfound understanding of the non- Suppose now that we have a tower of particles with masses abelian sLWGC to give a general argument for the UV cutoff approximately saturating the LWGC bound; that is, there is a scale QG demanded by the nonabelian sLWGC. We also particle of every charge q with m ∼ eqMPl. Then the number show that the coincidence of strong coupling scales for gauge of particles below a mass scale  is N() ≥ /(eMPl), theory and gravity persists for arbitrary gauge groups. which implies the species bound In Sect. 6, we consider converse statements. In particular,  2  ( )2 ≥ QG 2 , assuming that a gauge theory becomes strong below the quan- MPl N QG QG QG (2.3) tum gravity scale is sufficient to derive the original WGC. eMPl Assuming that the parametric fractional size of loop correc- and hence tions to the gauge boson and graviton propagators are the   1/3 . same over a range of energies allows a stronger sLWGC-like QG e MPl (2.4) statement to be derived. In this section we also consider the Compare this to the gauge theory Landau pole U(1):ifwe case of Higgsed gauge theories, clarifying some arguments treat the logarithms and numerical prefactors as parametri- from earlier literature. In Sect. 7 we consider some examples 5 cally order one, and ask for the scale at which eUV → 0 of quantum gravity theories that do not fit in the paradigm we according to (2.1), we find have discussed elsewhere in this paper. For instance, string  Q theories with gs 1 have a Hagedorn density of states that 1  invalidates some of our arguments. In these cases our use of ∼ q2 ∼ Q3, (2.5) e2 simple EFT loop calculations is no longer valid, so modified q=1 arguments may carry over. In Sect. 8 we briefly discuss some where Q is the largest charge in the tower, Q ∼ U(1)/ possible phenomenological applications of our UV cutoffs to (eMPl). Again, this leads to the conclusion nonabelian theories with very small gauge couplings. Finally, 1/3 in Sect. 9 we summarize our conclusions and discuss some U(1) ∼ e MPl. (2.6) open questions. Thus we see that a tower of charged particles implies UV cutoffs on both gauge theory and gravity. If the spectrum consists solely of a tower of near-extremal particles, then 2 Warmup: Landau pole and species bound for a 4d parametrically both the gauge theory and gravity cutoffs are 1/3 LWGC spectrum at the scale e MPl. We can think of this, loosely speaking, as a form of “gauge-gravity unification.” We do not mean In theories with a tower of charged particles, both gauge inter- that gravity and gauge theory are unified in the same way actions and gravity become strongly coupled in the ultravi- that different gauge groups are unified in GUTs, but simply olet. Let us begin with the familiar case of four dimensions, that we can think of the weakness of the two as having where it is well-known that charged particles lead to a Lan- emerged in the infrared from integrating out a tower of states  dau pole for abelian gauge theories due to the running of starting at a common scale QG in which all kinetic terms have their naive, order-one size (in appropriate units). the coupling. At one loop, the gauge coupling eUV at a scale UV is related to the low-energy gauge coupling e according to: 5 See Sect. 7.3 for a discussion of large logarithms. 123 Eur. Phys. J. C (2018) 78 :337 Page 5 of 33 337

2 2 It is straightforward to generalize the above argument to For p mi , however, the result is that the size of the the case where the LWGC is violated but the sLWGC is sat- loop correction grows with momentum Ð and faster than log- isfied on a sublattice of index k > 1. In this case, we obtain arithmically, when D > 4. Although this may sometimes 1/3 be referred to as a “power-law running” of the coupling,  , ( )  (ke) M , (2.7) QG U 1 Pl there is no straightforward sense in which the momentum with the two scales again parametrically the same when the dependence of loops can be absorbed in a running cou- spectrum is dominated by near-extremal particles. In string pling in a process-independent manner in a general non- theory examples k cannot be parametrically large, thus Ð at renormalizable field theory [50]. least in these cases Ð the consequences of the sLWGC are Nonetheless, the lack of a well-defined renormalized cou- similar to those of the LWGC. pling does not prevent us from estimating the energy scale at If the spectrum differs greatly from our assumptions Ð for which loop amplitudes become large and perturbation theory instance, if there are many more neutral particles that enter in breaks down. Consider, for example, the two-point function the species bound but do not affect the running of the gauge of the photon. The sum of iterated 1PI loop corrections to the theory Ð then the sLWGC does not necessarily imply gauge- photon propagator, with the leading one-loop 1PI graph, has gravity unification. However, the sLWGC always implies a the form cutoff on the quantum gravity scale that goes to zero as e →   μν μ ν 2 ˜ ˜ η − p p /p 1 0. Aμ(p)Aν(−p) = . (3.2) p2 + i 1 + (p2) 3 Loops and UV cutoffs for gauge theory and gravity in The function (p2) can be read off from the standard one- D dimensions loop QED vacuum polarization calculation. For example, for a set of charged scalars of charge qi and mass mi , we compute The discussion in the previous section focused on the familiar  2e2 1 case of four dimensions, where the Landau pole and species  (p2) = (2 − D/2) dx x(2x − 1) unreg (4π)D/2 bound arguments for UV cutoffs are familiar. Similar results  0  D/2−2 hold in a general D-dimensional theory, where both gauge × 2 2 − 2 ( − ) . qi mi p x 1 x (3.3) theory and gravity are generically non-renormalizable. If i many particles run in loops, the loop expansion can break For D odd this expression is finite as written, while for D down at prematurely low scales. To explain this point it is even we can use dimensional regularization D → D −  useful to adopt a somewhat different language that is suit- to see that it contains additional logarithmic dependence on able for both gauge theory and gravity. p2. The divergent piece in even dimensions can be absorbed by counterterms (including certain higher-derivative opera- 3.1 Growth of amplitudes with energy tors in D > 4). By rescaling the photon field we impose the renormalization condition (0) = 0.6 If all the charged From the kinetic terms for gauge theory and gravity,    scalars are light, m2  p2, then we can estimate √ i D 1 1 μν  S = d x −g R − Fμν F , − 2 (3.1) |( 2)|∼ 2 D 4 2, 16πG N 4e p e p qi (3.4) i 2 we read off that G N has mass dimension 2 − D and e has mass dimension 4 − D. We customarily introduce a reduced up to order-one factors and logarithms. D−2 = /( π ) The lesson from this is that loop amplitudes grow with Planck mass MPl 1 8 G N , and we could likewise introduce a mass scale associated with the gauge theory, momentum p at a rate that depends on both the space- − M D 4 = 1/e2. We might guess that in D > 4, where both dimension and the spectrum of charged particles with U(1)  gravity and gauge theory are nonrenormalizable, effective m p. In the above example, strong coupling arises when |(p2)|∼1. For a tower of approximately LWGC- field theory breaks down at the scale MPl or MU(1).How- saturating particles, that is a tower for which m2 ∼ ever, in a theory with a large number of degrees of freedom − e2q2 M D 2 p N we know that this naive dimensional analysis can be mod- Pl , if we sum up to energy we reach a maximum ∼ /( (D−2)/2) ified by powers of N. charge Q p eMPl and find In four dimensions we discussed the gauge theory cut- off in terms of a logarithmically running coupling constant. 6 With this renormalization condition, the gauge coupling which In higher dimensions, we should be more cautious. For appears in (3.3) is the infrared gauge coupling. This condition cannot = p2  m2 we can integrate out a heavy particle i, expand- be imposed in D 4 with massless charged particles due to infrared i divergences, which corresponds to the fact that the gauge coupling 2 / 2 inginpowersofp to obtain a threshold correction to 1 e flows to zero in the deep infrared; this special case is discussed fur- together with an infinite sum of higher-derivative operators. ther in Sect. 7.3. 123 337 Page 6 of 33 Eur. Phys. J. C (2018) 78 :337

Q 3 neglecting heavy particles will never change our estimate of |( 2)|∼ 2 D−4 2 ∼ 2 p D−4. p e p q e ( − )/ p where the loop expansion breaks down. eM D 2 2 q=1 Pl A second objection is that we have only considered the (3.5) photon two-point function at one-loop, which is moreover an off-shell quantity (meaning that it may not be well- We denote the scale p at which this becomes order one by defined outside of the effective field theory description). In  ( ), given by: U 1 “Appendix A.2” we briefly discuss higher loop diagrams and /( − ) 3(D−2)/2 1 D 1 on-shell S-matrix elements, arguing that the loop expansion  ( ) ∼ eM . (3.6) U 1 Pl breaks down at parametrically the same scale as above. This is the D-dimensional analogue of the 4d Landau pole To capture the heuristics discussed above, we find it con- bound (2.6). Notice that we are not saying that the gauge venient to define  coupling becomes strong at this scale, since the meaning of 2 D−4 λgauge(E) := e E I (i), (3.7) a running gauge coupling is unclear away from four dimen- i: mi 4, in which case the remaining finite contribution is what we theory. Below we will systematically neglect this and other consider here. order-one factors. 123 Eur. Phys. J. C (2018) 78 :337 Page 7 of 33 337

To begin, we consider the effect of these particles on the representation of the group rather than simply with each Car- strong coupling scale of gravity. There is a tower of particles tan charge. For instance, in an SU(2) gauge theory there is a up to charge knmax with masses below the cutoff QG, with lattice site with Cartan charge 1, but states of this charge exist  in all representations with positive integer spin. We would n ∼ QG . (3.11) postulate that the correct nonabelian sLWGC cannot be sat- max (D−2)/2 ekMPl isfied at this lattice site with representations of higher spin, but requires a spin 1 representation with superextremal par- Below the scale QG, there are therefore Nd.o.f. ∼ nmax ticles of charge 1, as shown in Fig. 1. This stronger statement such particles with mass below QG. Plugging into (3.9) is satisfied, for instance, in the SO(32) and E8 × E8 heterotic and solving for QG, we find string theories. 3(D−2) 1 2(D−1) To make a precise conjecture, we first review some basic U(1) :   (ek) D−1 M , (3.12) QG Pl facts about a compact nonabelian Lie group G.Let denote as we previously derived in [41]. For the case D = 4, this the set of roots of G, each designated by a weight vector /  ∈ reads   (ek)1 3 M . Q , i.e., a set of Cartan charges. We choose a set of QG Pl +   ∈ + Next, consider the Landau pole of the U(1) gauge theory. positive roots such that for any root Q, Q iff −  ∈/ +  ∈ +  +  + In this case, we have Q and for Q1,2 , Q1 Q2 is in if it is a root. Simple roots are positive roots which are not the n max sum of two other positive roots. The simple roots are linearly λ (E) ∼ e2 E D−4 (kn)2 gauge independent and span the space of roots, hence the number n=1 of simple roots equals the rank of G minus the rank of its 3 ( ) ∼ 2 2 D−4 E . center Z G . e k E ( − )/ (3.13) ekM D 2 2 Given a set of positive roots, there is a partial ordering Pl     on weights with Q1 ≥ Q2 if Q1 − Q2 is a non-negative In the second step we used Eq. (3.11). We see then that the linear combination of positive (equivalently, simple) roots. λ () ∼   condition gauge 1 leads to parametrically matching The highest weight Q R of representation R (if it it exists) is    the scale QG in (3.12). Notice that in this analysis we have the unique weight which satisfies Q R ≥ Q for all weights     ignored various constant factors as well as logarithms. We Q in R. A weight Q is dominant if Q · Qα ≥ 0 for all  have also assumed a tower of approximately equally-spaced simple roots Qα.8 An irreducible representation (irrep) R has  states. We have also assumed that the density of states of a highest weight Q R which is dominant. Moreover, for any  9 a given mass is dominated by these charged states; if this dominant Q in the weight lattice G of G there is a unique assumption is violated, e.g., if there is a large number of light irrep R of G and all finite dimensional representations of G uncharged states, then the quantum gravity scale QG could are direct sums of these. be significantly below the scale of the Landau pole. Under The hyperplanes orthogonal to the roots divide the space of these assumptions, and as a parametric statement (rather weights into Weyl chambers, which are permuted by the Weyl than a precise quantitative one), this suggests that theories group (acting freely and transitively on them). The dominant   that approximately saturate the sLWGC bounds will tend weights lie in a Weyl chamber Q · Qα ≥ 0, known as the to exhibit the phenomenon of gauge-gravity unification as fundamental Weyl chamber. A choice of positive roots is defined in Sect. 2. equivalent to a choice of fundamental Weyl chamber.

4 The nonabelian sLWGC 8 Here the Cartan generators are normalized so that within each simple subalgebra Tr Hi H j ∝ δij in any representation. This doesn’t com- plete fix the inner product, which encodes additional in the The sLWGC can be applied to commuting generators within worldsheet argument to follow. a nonabelian gauge group. In [35,41,51] we have essentially 9 The weight lattice is the set of all possible weights in finite dimen- ignored the nonabelian nature of the group and discussed the sional unitary representations of G, which form a lattice since G is  (s)LWGC as a statement about the Cartan generators. This compact. While all weights Q must be algebraically integral, i.e.,  · /  2 ∈ Z, can be motivated, for example, by compactifying the theory 2Qα Q Qα only for G simply connected does this completely fix the weight lattice. Otherwise there are further conditions; for instance, on a circle with a Wilson line that breaks the gauge group to a there are no spin-1/2 representations of SO(3), only of its simply con- product of U(1)s. In this way, much of the evidence we have nected double cover SU(2). Moreover, when Z(G) has non-zero rank found for the sLWGC in string theory applies to nonabelian the abelian charges are quantized as well. The weight lattice need not groups. factor between the abelian and semi-simple components of the Lie alge- bra, as demonstrated by, e.g., G = U(N) = (SU(N) × U(1))/ZN ,for However, we would like to postulate a slightly stronger which the U(1) charge mod N is fixed to be equal to the charge under statement: there should exist superextremal particles in each the ZN center of SU(N). 123 337 Page 8 of 33 Eur. Phys. J. C (2018) 78 :337

M M

Q3 Q3

Fig. 1 The nonabelian sLWGC (left) and abelian sLWGC (right) for below a given mass scale than the abelian sLWGC does, as the latter can an SU(2) gauge group. For a sublattice of fixed index, the nonabelian be satisfied by particles charged under a sparse set of representations, sLWGC requires many more particles charged under the U(1) Cartan provided they are sufficiently light

We can now state the nonabelian version of the sublattice weight lattice need not factor between left and right movers),   Weak Gravity Conjecture: hence simple roots have either QL = 0orQ R = 0. The nonabelian sLWGC For any quantum gravity in D ≥ 5 We introduce a chemical potential for the Cartan, as in the with zero cosmological constant and unbroken gauge group abelian case:  G, there is a finite-index Weyl-invariant sublattice 0 of the Q Q  Z = Tr(q L q¯ R y L y¯ R ). (4.2) weight lattice G such that for every dominant weight Q R ∈ 0 there is a superextremal resonance transforming in the G By the same arguments as before, the spectrum is invariant  → + ρ ρ ∈ ∗ irrep R with highest weight Q R. under Q Q for Q with TL,R, defined by Here “Weyl-invariant” means that 0 is invariant under the ( ) = + 1 2 , action of the Weyl group W G , which ensures that the con- L,R TL,R QL,R (4.3) jecture is independent of the choice of fundamental Weyl 2 2 = 2 − 2 chamber. “Superextremal” means the same as in the abelian held fixed. The norm Q QL Q R is invariant under  ∗ case: the charge-to-mass ratio of the resonance is greater than the Weyl group, as is the weight lattice Q, hence so is Q. or equal to that of a large extremal black hole with a paral- Starting with the graviton state L = R = 0 and QL,R = = 1 2 ∈ ∗ lel weight vector. Note that in the special case where G is 0, we produce a state with L,R 2 QL,R for every Q Q. abelian this reduces to the abelian sLWGC of [41]; other- To show that this state is the highest weight in its G wise the nonabelian sLWGC is strictly stronger. irrep, we proceed by contradiction. If not, there is at least  We will make two arguments in favor of this conjecture. one simple root Q+ (a left-mover for definiteness) such that    First, we show that it holds in the NSNS sector of tree-level there is a state with charge (QL + Q+, Q R) and L = 1 2 = 1 (  +  )2 −  ·  − 1  2 = 1 2 string theory (with caveats similar to those for the abelian 2 QL 2 QL Q+ QL Q+ 2 Q+ and R 2 Q R.  sLWGC). Second, we show that it is preserved upon com- Thus, there is a corresponding state with charge (Q+, 0) and   pactification of a higher dimensional theory which satisfies L =−QL · Q+ and R = 0, where L is a non-positive ∗ the conjecture on a Ricci flat manifold. Based on these argu- integer because Q is dominant and lies in Q. Since the ments, we conclude that the evidence for the nonabelian graviton has spin 2, this state also has spin 2, but then either sLWGC is similar to that for the abelian sLWGC of [41]. (1) L = 0 and there are additional (charged) massless spin NSNS sector gauge bosons correspond to worldsheet con- 2 particles, or (2) L < 0, and by turning on left-moving served currents, with OPEs oscillators in the non-compact directions we obtain massless particles of spin greater than 2. In either case, the low energy limit is not Einstein gravity.10 kab icab J a(z)J b( ) ∼ + c J c( ) +··· , The rest of the argument for the nonabelian sLWGC from 0 2 0 z z modular invariance goes the same way as in the abelian case, ˜ ˜a˜b˜ a˜b ˜ ˜ k ic˜ ˜ ˜ with the same caveats as in [41]. J a(z¯)J b(0) ∼ + c J c(0) +··· , (4.1) z¯2 z¯ We now consider compactification. As in the abelian case, the non-trivial ingredients that can lift some of the KK modes

ab ab corresponding to the Kac-Moody algebra. We fix k = δ 10 A similar argument can be made for a unitary CFT2, relevant for ˜a˜b˜ = δa˜b˜ ab ˜a˜b˜ and k by normalizing the currents. The c c and c c˜ the AdS3 WGC of [42]. In this case, there is a unique operator (the =− / =− / are then structure constants, with normalizations depending identity) with L cL 24 and R cR 24. All other operators > − / > − / (  , ) on the level for the nonabelian current algebra of each simple have L cL 24 or R cR 24. If the charge Q+ 0 state constructed above has L =−cL /24 it contradicts the uniqueness of factor of G. Note in particular that each simple factor of G is the identity operator (which is uncharged), whereas if it has L < either purely left-moving or purely right-moving (though the −cL /24 it violates unitarity. 123 Eur. Phys. J. C (2018) 78 :337 Page 9 of 33 337 are Wilson lines on torsion cycles.11 These can be viewed as Weyl invariant. Thus, the minimal sLWGC-satisfying spec- a coming from a quotient trum consists of a spin kn/2 multiplet for each n ∈ Z>0.The dimension of the spin j representation is 2 j + 1, so the total G × Mˆ , (4.4) number of states up to level nmax is G0 nmax ( + ) ˆ nmax nmax 1 where M = M/G0 is a Ricci-flat compact manifold, G0 N = (kn + 1) = k + nmax. (5.1) ˆ 2 acts freely and transitively on M, and G0 is a finite subgroup n=1 ⊆ ( ) of G.ToleaveG unbroken, we need G0 Z G (other- Applying the species bound and the counting (3.11) we obtain wise, replace G with its unbroken subgroup in the following ( − )  ⊆  1 2 2 D 2 argument). If 0 G is an extremal sublattice before com- ( ) :   D D D , SU 2 QG k g MPl (5.2) pactification then the intersection of 1 := 0×KK with the  G0-invariant sublattice 2 ⊆ G × KK is an extremal sub- for SU(2) gauge coupling g 1. In particular, in the four-   1/4 1/2 lattice after compactification. Note that 1,2 are full dimen- dimensional case we have QG k g MPl. sional sublattices, hence so is 1 ∩2 (each has finite coarse- The tower of SU(2) charged states makes the gauge theory ness). 1 is Weyl-invariant by assumption, whereas since strongly coupled in the ultraviolet. As before, we compare ⊂ ( )   :=  ∩ the quantum gravity scale with the strong coupling scale of G0 Z G is Weyl-invariant, 2 is also. Thus 0 1 2 satisfies the nonabelian sLWGC in the lower-dimensional the gauge theory SU(2), which generalizes the Landau pole theory. in four dimensions. As explained in Sect. 3.1, this is the scale λ ∼ Note that the condition that 0 is a Weyl-invariant sublat- at which gauge 1. We find tice of the weight lattice G turns out to be rather restrictive. For instance, when G is simple  must be a multiple of one nmax 0 λ ( ) = 2 D−4 ( ) ∼ 2 D−4 3 4 of a finite list of “primitive” Weyl-invariant sublattices, see gauge E g E I kn g E k nmax = “Appendix B”. n 1 E D ∼ , (5.3) 2 2(D−2) g kMPl 5 UV cutoffs for general gauge groups where we use the Dynkin index I ( j) of the spin- j represen- In this section we show that the phenomenon that we have tation, ( ) seen in Sect. 3.2 for a U 1 gauge theory coupled to gravity 2 I ( j) = j( j + 1)(2 j + 1) ∼ j3, (5.4) holds for a general gauge group at weak coupling: the species 3 bound on  and the generalized Landau pole bound from QG λ ( ) ∼ loop corrections to gauge couplings coincide. We will first as well as (3.11). Solving gauge SU(2) 1, we find para- work out the case of an SU(2) gauge theory in detail, then gen- metric agreement with Eq. (5.2). That is, the parametric scal- eralize to arbitrary gauge groups including product groups. ing with g and k of the quantum gravity cutoff and the gauge We then give a very general argument for why this coinci- theory cutoff are the same, and we again find the phenomenon dence of scales occurs. we refer to as gauge-gravity unification. In the above, we have assumed that resonances which approximately saturate the sLWGC bound dominate the 5.1 SU(2) gauge theory spectrum. If there are many more subextremal resonances, or if the tower of resonances is parametrically superextremal, We now consider an SU(2) gauge theory coupled to grav- this would affect the coincidence of scales, but the upper ity, applying the nonabelian sLWGC from Sect. 4.SU(2) bound (5.2) still applies. Notice also that the SU(2) bound is invariance implies that states of large charge come in large less sensitive to the sublattice spacing k than the U(1) bound: representations, which leads to an interesting new result com- √ the former depends on the combination g k while the latter pared with the U(1) gauge theories we have considered so far depends on the combination ek. (which have the same Cartan subalgebra). The SU(2) weight lattice is Z/2 in conventions where the roots have charge ±1. An arbitrary finite-index sublattice 5.2 Larger groups takes the form kZ/2fork a positive integer. The Weyl group We generalize to larger groups, beginning with SU(3).SU(3) is Z2, generated by charge conjugation, hence all of these are has two Cartan generators, and thus irreps are labeled by

11 two non-negative integers p, q. Each irrep has dimension Wilson lines on non-torsion cycles are moduli whose values affect ( + )( + )( + + )/ the masses of KK modes, but do not make them subextremal, since they p 1 q 1 p q 2 2 and a highest-weight state with affect the masses of extremal black holes in exactly the same way. Q2 = (2/3)(p2 + pq+q2) in conventions where Q2 = 2for 123 337 Page 10 of 33 Eur. Phys. J. C (2018) 78 :337 the roots. By the results of appendix B, the Weyl-invariant where we use an integral approximation at large Qmax as sublattices are k the weight lattice and k times the root before. Thus, for a superextremal tower lattice. We focus on the former case for definiteness, the latter D−2 7 2 being similar. Thus, p, q ∈ kZ≥0 for irreps whose highest- E/gM 2 D−4 Pl weight states fall on this sublattice. λgauge(E)  g E , (5.10) k2 We now estimate the species bound on the quantum grav- ity scale. The total number of states in irreps whose high- which gives parametrically the same bound as (5.7). est weights lie on k times the weight lattice with charge As before, if the spectrum is dominated by a tower of 2 ≤ 2 near extremal resonances then gauge and gravitational loops Q Qmax is become large at parametrically the same scale. In other cases ( / ) 2( 2+ + 2)≤ 2 this coincidence of scales may not occur, but (5.7) still applies 2 3 k p pq q Qmax N = if the sLWGC holds. It is straightforward to generalize these arguments to an p,q∈Z≥0 (kp + 1)(kq + 1)(kp + kq + 2) arbitrary simple gauge group G, as follows. The dimension × . (5.5) of an arbitrary irrep R with highest weight Q is determined 2 R by the Weyl dimension formula Approximating the sum with an integral, we find N ∼  √  · (  +  ) 5 /( 2) Q∈ + Q Q R Q0 Qmax 5 6k asymptotically at large Qmax. Roughly dim(R) =  , Q ( )  ·  speaking, the fifth power of appears here because SU 3 Q∈ + Q Q0 has three negative roots (half the total number of roots, which   1  equals the dimension of the group minus its rank). The neg- Q0 := Q, (5.11) 2  ative roots act as lowering operators on the highest-weight Q∈ + state, and lead to representations of size ∼ Q3. Summing where + denotes the set of positive roots, as in Sect. 4. over the Cartan then gives ∼ Q5 total states. (We generalize Asymptotically for large Q we find this argument below). R 2  2 2 ˆ  G Thus, for an superextremal sLWGC tower m g Q dim(R) ∼ f (Q R)|Q R| , D−2  M we estimate  ˆ Pl  Q · Q ˆ Q∈ + R f (Q R) :=  , (5.12) −   D 2 5  Q · Q0 − / 2 Q∈ + E D 2 E gMPl λgrav(E)  , (5.6)  := | |=| |/ D−2 k2 where G + 2 is the number of positive roots MPl ˆ and f (Q R) is an order-one function which depends only  which gives on the direction of Q R within the fundamental Weyl cham- ber. This makes precise the intuition from above that more 7(D−2) 2 5 2(D−3) raising/lowering operators (positive/negative roots) leads to ( ) :   k D+3 g D+3 M . SU 3 QG Pl (5.7) larger representations. We take k times the weight lattice as a representative exam-   2 5 In four dimensions this is QG k 7 g 7 MPl. ple of the Weyl-invariant sLWGC sublattice.12 Thus, there We now consider the ultraviolet behavior of the gauge are superextremal irreps for all theory. The Dynkin index for the (p, q) irrep of SU(3) is    Q R(n) = k ni Qi , ni ∈ Z≥0, (5.13) (p + 1)(q + 1)(p + q + 2)(p2 + pq + q2 + 3(p + q)) i I (p, q) = . 24 2  2  2 D−2  (5.8) with mass m g Q R MPl , where Qi are the fundamental weights. The number of states below some scale E is then

The total index for irreps with highest weights on k times the Q (n)2≤Q2 + R  max QrG G 2 ≤ 2 ˆ  G max weight lattice and Q Qmax is then N(E)  f (Q R(n)) |Q R(n)| ∼ , rG ,..., ≥ k n1 nrG 0 ( / ) 2( 2+ + 2)≤ 2 2 3 k p pq q Qmax (5.14) Itot = p,q∈Z≥0 ( + )( + )( + + )( 2( 2 + + 2) + ( + )) 12 As shown in appendix B, since G is simple any Weyl-invariant sublat- × kp 1 kq 1 kp kq 2 k p pq q 3k p q 24 tice is a multiple of one of a finite number of “primitive” Weyl-invariant 7 sublattices. The parametric dependence on k will be the same regardless Qmax ∼ √ , (5.9) of which primitive Weyl-invariant sublattice we start with. 56 6k2 123 Eur. Phys. J. C (2018) 78 :337 Page 11 of 33 337

2  2/( 2 D−2) where Qmax E g MPl and rG is the rank of G. 5.3 Product groups Here we again use an integral approximation but drop all ˆ numerical factors including the angular integral over f (Q R) Next, we consider the case of product groups. For a product and the volume of the fundamental domain of the weight group we can no longer describe the Weyl-invariant sublattice lattice. Putting this into the species bound, we obtain 0 ⊆ G satisfying the sLWGC by a single integer (plus a finite number of choices) as above. While it is possible to pro- n +2 D−2 rG nG G + − + − n +D−2 2 ceed carefully and catalog the possibilities, we will assume :   nG D 2 nG D 2 G , simple G QG k g MPl the full LWGC in this section for simplicity. Unless 0 is (5.15) very sparse within G , the effect of the sublattice index is competitive with other numerical factors that we consistently up to numerical factors, where n := r +  is the rank G G G ignore. plus half the number of roots, equal to (r + d )/2 where G G As a simple example, we consider an SU(2)×U(1) gauge dG is the dimension of G. In four dimensions, this takes the r/(n+2) n/(n+2) group with small gauge couplings g and e respectively. The simpler form QG  k g MPl. irreps are labeled by ( j, q) for j ∈ Z≥ /2 and q ∈ Z.The For instance, for SU(N), d = N 2 − 1 and r = N − 1, 0 G G LWGC requires a particle in each irrep with mass at most implying that 2 2 2 2 2 D−2 m( , )  g j + e q M . (5.20) N 2 + N − 2 j q Pl n ( ) = . (5.16) SU N 2 Since the dimension of each irrep is 2 j + 1, the number of states below some mass scale E is at least Notice that as N →∞, the bound (5.15) asymptotically 2 2+ 2 2≤ 2/ D−2 brings the quantum gravity cutoff close to the “magnetic g j e qE MPl 3 ( )  ( + ) ∼ 1 E , WGC cutoff” of (1.1), i.e., the scale at which the tower of N E 2 j 1 ( − )/ eg2 M D 2 2 charged states appears, j,q Pl (5.21)  → (D−2)/2, lim QG gM (5.17) (D−2)/2 N→∞ Pl using an integral approximation for E gMPl and ( − )/ E eM D 2 2. Thus, applying the species bound, we where the dependence on k goes away because rG  dG . Pl Similar results hold for other large rank simple groups. Thus obtain 5(D−2) for larger nonabelian groups, small gauge couplings become 1 2 2(D+1) SU(2) × U(1) :   e D+1 g D+1 M , (5.22) increasingly powerful constraints on the validity of effective QG Pl field theory. up to order one factors and the dependence on the sublattice The matching of gauge theory and quantum gravity cutoffs index. Note that if e ∼ g, we can understand this result using continues to hold, as in the cases we have already considered. the logic of the previous section: we have one raising operator In particular, the quadratic Casimir C2(R) and the Dynkin (from the SU(2) factor) and two Cartan generators (one from index I (R) of R are given by each factor), so the total number of states up to the nth rung of the ladder scales as n3, and the bound is given by setting ( ) =  · (  +  ), C2 R Q R Q R 2Q0 nG = 3in(5.15). C2(R) dim(R) As before, the Landau pole bounds from the tower I (R) = , (5.18) dG of charged states parametrically coincide with QG.For instance, for the U(1) ( ˆ )  + ( ) | |2 ( )  f Q R |  | G 2 hence C2 R Q R and I R Q R at large D−2 dG g2 j2+e2q2≤E2/M   Pl Q R. Thus, 2 D−4 2 λU(1)(E) ∼ e E q (2 j + 1) j,q  ( )2≤ 2 Q R nQmax ˆ   − f (Q R(n))  + 5 D+1 λ (E)  g2 E D 4 |Q (n)| G 2 − 1 E E gauge R ∼ e2 E D 4 ∼ , dG 3 2 (D−2)/2 5(D−2)/2 n1,...,nr ≥0 e g 2 G MPl eg MPl r + + + − Q G G 2 EnG D 2 (5.23) ∼ g2 E D−4 max ∼ , r D−2 k G (nG +2) rG nG 2 which reproduces equation (5.22). A similar result holds for k g MPl (5.19) the SU(2) factor. (D−2)/2  Note that the above discussion assumes gMPl  (D−2)/2   which gives parametrically the same bound as above. We QG and eMPl QG. Even if both gauge couplings give a simpler argument for this in Sect. 5.4. are small, this need not be true if one is much smaller than the 123 337 Page 12 of 33 Eur. Phys. J. C (2018) 78 :337

( − )/  D 2 2 fi (qˆi )  + other. Only gauge group factors with WGC scale gMPl I (R) = I (R ) dim R ∼ |q | i 2 i i j d i below the quantum gravity scale contribute to our bounds on j=i i   QG. This is discussed further in Sect. 7.2.  r × f (qˆ ) |q | j , (5.28)  More generally, we consider a gauge group G = U(1) 0 × j j j p j=i i=1 Gi for simple Gi with gauge couplings gi and an 2 r0 × r0 abelian gauge kinetic matrix τij (generalizing 1/e using the estimate below (5.18), so that below the mass scale for a single U(1)). Irreps are labeled by (q , R ,...,Rp) for E 0 1    U(1)r0 charges q and G representation R , corresponding to 0 i i ( ) fi (qˆi )  + tot  r0  | | i 2 ri  the highest weight vector QR = (q0, q1,...,qp). Superex- Ii d q0 qi d qi ||Q||

  1  + − (n+2)(D−2)  ( ) − / n D 2 ( + − ) Q E   ( τ) 1 2 ni 2 n D 2 , 2 D−4 2 QG det gi MPl (5.27) λgauge(E) ∼ e E dq q nE (q), i 0 Q(E) 14 D−2 which generalizes (5.15). λgrav(E) ∼ G N E dq nE (q), (5.31) Estimating the size of loop corrections in the gauge theory 0 leads to parametrically the same bound, as in all our previous where Q(E) is the largest charge in the spectrum for masses examples. For instance, focusing on one of the non-abelian below E. The average charge of all the particles with mass factors Gi ,wehave less than E is  Q(E) 2 13 dq q nE (q) To perform the integral, it is useful to reimagine the integral over the q2 = 0 . (5.32) = +  E Q(E) ri components of q(i) as an integral over ni ri i components of dq nE (q) some fictitious vector by passing to spherical coordinates, factoring out 0 the angular integral (which we drop along with other numerical factors) We will see that for a large family of smooth functions nE (q) and passing back to rectangular coordinates. In this way, the integral 2 which cut off at q = Q(E) the average charge q E is Q over reduces to a straightforward spherical integral. parametrically of the same order as Q(E)2. This means that 14 It is relatively straightforward to see how the sublattice data should  2 appear in this expression. For instance, if 0 includes only ki times e 2 ni ri ni λ (E) ∼ q  λ (E) the weight lattice of Gi , then we should replace g with k g . More gauge 2 E grav i i i G N E generally, the index |0|/|G | of the sublattice 0 within G , i.e., the fraction of  sites which lie on  , will appear as an extra factor inside e2 Q(E)2 G 0 ∼ λ (E). the parentheses. 2 grav (5.33) G N E 123 Eur. Phys. J. C (2018) 78 :337 Page 13 of 33 337   ( )  / (D−2)/2 However, the (s)LWGC requires Q E E eMPl , example, higher multiplicities for the near-extremal states and in particular if the constraint is nearly saturated for E should not change the conclusion. (D−2)/2 ( ) ∼ / (D−2)/2 More generally, q2 ∼ Q(E)2 when n (q) is not eMPl then Q E E eMPl .By(5.33)this E E ( − )/ | | ( ) ( ) implies λ (E) ∼ λ (E) for E eM D 2 2, and too sharply peaked at q Q E . For instance, if nE q gauge grav Pl ∼ ( − ) Q ( ) ∼ Q 2 ( ) ∼ in particular gauge theory and gravitational loop corrections exp Q q , then 0 dq nE q 0 dq q nE q exp(Q), without a Q2 enhancement in the second integral. become large at parametrically the same scale QG. Having understood the consequences, we now give argu- This includes the caveat we have made above: if there are λ ( ) ments why typically q2 ∼ Q(E)2 up to order-one factors. large numbers of neutral particles, they correct grav E but E λ ( ) We begin with a simple example: suppose that all particles not gauge E and hence spoil gauge-gravity unification. of a given mass have the same |q| and |q|=E/E0 increases linearly with energy, as in an (s)LWGC saturating tower for ( ) ρ( ) := dN 6 Gauge-gravity unification implies the Weak Gravity a single U 1 gauge group. Let E dE be the density of states. We then have Conjecture nE (q) = E0ρ(|q|E0)(E −|q|E0), (5.34) So far we have argued that the sLWGC leads naturally (with some caveats) to a single scale at which gauge forces and where (x) is the Heaviside step function, so that gravity become strong, suggesting a unification of forces.  / E E0 2ρ(| | ) We now consider the converse case, where we demand such 2 0 dq q q E0 q E =  a unification and explore its consequences. More generally, E/E0 dqρ(|q|E0) 0 we will assume that gauge couplings become strong at or E    1 dE E 2ρ(E ) below the quantum gravity scale, allowing for the possibil- = 0 . 2 E (5.35) ity of gauge theories which emerge from non-gravitational E dEρ(E) 0 0 dynamics, as demonstrated, e.g., by Seiberg duality. Stated For a minimal sLWGC saturating spectrum, and in real alternately, we assume that there cannot be any weakly cou- quantum gravities that behave like this, such as Kaluza pled gauge bosons at the quantum gravity scale.16 Klein theory, ρ(E) is asymptotically a constant and we  2  1 ( / )2 = 1 ( )2 ρ( ) obtain q E 3 E E0 3 Q E .Evenif E 6.1 Basic argument grows asymptotically the large E part of the integral is  2 ( )2 ρ( ) enhanced and q E is yet closer to Q E .Onlyif E The requirement that gauge forces become strong (λgauge ∼ fallsoffas1/E or faster does this conclusion change, 1) at or below the quantum gravity scale is the requirement but the sLWGC sets a lower bound ρ(E)  1/(kE0) for that a sublattice spacing k, so such a falloff is incompatible 1  ∼ D−4 q2,    with it. 2 gauge i for gauge QG. (6.1) e | < For more complicated gauge groups, the spectrum is dif- i mi gauge ferent because near-extremal particles which are charged  / To derive the ordinary WGC from this, consider the particle under other gauge group factors can have q E e ( / ) =: (D−2)/2 of largest charge-to-mass ratio q m max zmax among all MPl , as can the lower weights in nonabelian irreps. For the particles with mass below  . For every i we have instance, consider n U(1)s with gauge couplings e ,...,e gauge 1 n q2 < z2 m2 and so and an LWGC saturating spectrum. We find i max i

 −  e2q2≤E2/M D 2 n−1 1 − i i  Pl 2 2  D 4 2 2 1 E gaugezmax mi n (q ) = 1 ∼ − e2q2 e2 E 1 D−2 1 1 | < e2 ...en i mi gauge q2,...,qn MPl − −  D 2 2 ( ) ∝ ( ( )2 − 2) n 1 , gaugezmax N gauge Q1 E q1 2 (5.36)  D−2 2 ( )  −  QG zmax N QG where Q (E)2  E2/ e2 M D 2 . This gives q2  1 1 Pl 1 E  2 D−2. 1 ( )2 zmax MPl (6.2) n+2 Q1 E . The same result still holds when there are non- abelian factors in the rest of the gauge group Ð with n equal to m2 <2 the total rank plus half the total number of roots, as above Ð In the second line we use i gauge to place an upper   as well as with arbitrary sublattice spacings.15 As in our first bound on the sum, in the third line we use gauge QG, and in the last line we apply the species bound − − N( )D 2  M D 2. Rearranging the last inequality, we 15 QG QG Pl If q1 is a Cartan charge of a nonabelian factor then nE (q1) can take a more complicated functional form, but the qualitative behavior is similar. 16 We explore some situations where these assumptions fail in Sect. 7. 123 337 Page 14 of 33 Eur. Phys. J. C (2018) 78 :337

2 2 D−2  have e zmax MPl 1, which has the form of the original and (2) perturbative string theory, where gauge bosons and Weak Gravity Conjecture. It is slightly strengthened, since gravitons share a common origin as excitations of the string. the superextremal particle we have found is below the quan- In the former case, the number of KK modes up to a scale E tum gravity cutoff. (However, see Sect. 7 below for excep- is N(E) ∼ ERwhere R is the compactification radius. Thus, tions in which this argument does not apply.) by a familiar calculation (this is essential the same situation From the constraint (6.1) we can also obtain statements as that in Sect. 3.2): about the spectrum as a whole. For instance, we can rewrite E D−2 it as λ ( ) ∼ ( ) ,λ ( ) ∼ 2 D−4 ( )3 . grav E D−2 ER gauge E e E ER 1 D−4 2 MPl ∼  N( ) q  e2 gauge gauge gauge (6.5) 1 D−2 2  M q  , (6.3) − 2 Pl gauge / 2 = ( / ) 2 D 2 λ ( ) ∼ gauge However, 1 e 1 2 R MPl , hence grav E λgauge(E). Below the compactification scale 1/R, λgauge = 0 again using the species bound. We can rearrange this result (there are no charged particles), but λgrav = 0, so the match- in the suggestive form ing begins near the compactification scale, exactly where we

2 2 2 D−2 expect the forces to unify. More complicated KK examples   e q  M . (6.4) gauge gauge Pl behave in the same way, as can be seen using, e.g., the general arguments of Sect. 5.4. This is in itself an interesting WGC-like statement that The case of perturbative string theory is more complicated, bounds the strong coupling scale in terms of the average since gauge fields can have several different origins, from charge of particles with mass below that scale. Since every both open and closed strings, and in the latter case from both < weakly coupled particle has m gauge, all of their masses the NSNS and RR sectors. Since the graviton lives in the are bounded in terms of the average charge; for instance (6.4) NSNS closed string sector, we expect gauge fields from this  implies that the particles lighter than gauge are, on average, sector to unify with it at the string scale. In Sect. 7.1 we will superextremal. argue that indeed λgauge ∼ λgrav at the string scale for NSNS sector gauge bosons (except those with no charged particles 6.2 Comparisons at lower energies at or below this scale; see Sect. 7.2 for related caveats.) With this motivation, such as it is, we proceed to com- We have seen that gauge-gravity unification, in the sense pare λgauge(E) and λgrav(E) at scales parametrically below defined above, has very interesting consequences. We can the quantum gravity scale and derive the consequences of obtain even stronger statements if we assume that the certain simple assumptions. First, suppose that λgauge(E)  strengths of gauge and gravitational interactions unify below λgrav(E) at some particular scale E  QG. This means that the quantum gravity scale. However, to do so we need to   specify what exactly this means. ( ) 2  2 2  2 2 ( ) 2 G N N E E e qi e zmax E mi We argued in Sect. 3 that λgauge(E) and λgrav(E) Ð defined i|m E i|m E in (3.7) and (3.8), respectively Ð are useful heuristics which i i  2 2 ( ) ( ) 2, estimate the fractional size of gauge theory and gravity loop e zmax E N E E (6.6) corrections at a scale E coming from light particles (parti- cles with mass below E). When λ  1orλ  1 by essentially the same reasoning as in Sect. 6.1, where gauge grav ( ) the corresponding loop expansion breaks down, though of zmax E is the largest charge-to-mass ratio among the parti- ( ) 2 course it could break down at a lower scale for other rea- cles lighter than E. Dividing by N E E , we conclude that sons. However, at scales where the λs are small, they have no there is a superextremal particle lighter than E, and the WGC clear physical interpretation. They do not even represent the is satisfied. fractional size of all loop corrections, but only those coming A more intriguing statement arises if we assume from particles lighter than E; the contributions from heavy λgauge(E) ∼ λgrav(E) for E  E0, (6.7) particles and/or higher dimensional operators are typically much larger, though still suppressed by powers of E/. which is the heuristic notion of gauge-gravity unification at Nonetheless, in situations where we expect gauge and weak coupling that we motivated above, with unification      gravitational forces to unify below the quantum gravity scale, scale E0 QG. This means that for E0 E QG we have we find λgauge(E) ∼ λgrav(E) beginning at the expected uni- fication scale. There are two principal examples of this: (1)  2 2 ∼ ( ) 2. KaluzaÐKlein theory, where the graviphoton shares a com- e qi G N N E E (6.8) |  mon origin with the graviton at the compactification scale, i mi E 123 Eur. Phys. J. C (2018) 78 :337 Page 15 of 33 337

Here N(E) is a monotonically increasing function of E.Let Nonetheless, λ(E) does not have the same connection to us assume that there are sufficiently many particles above E0 unification in the simple examples that we discussed above.   that we can approximate these functions as continuous. If we While λgauge(E) ∼ λgrav(E) in KK theory, this continues differentiate both sides with respect to E, we can rewrite the below the compactification scale even though at low ener- ( 2)/ = derivative of the left-hand side in terms of d q dE gies the common origin of the graviphoton and graviton is d( q2)/dN · dN/dE and divide through by dN/dE to not evident. In perturbative string theory, it is difficult to even obtain: defineλ(E), in part because the two-point function is an off-    shell quantity. The results we get depend on whether we count ( 2) 2 d q 2 d log E states above the string scale (and how we count them). Both e ∼ G N E 1 + 2 , (6.9) dN d log N of these examples illustrate thatλ(E) is a UV-sensitive quan-  tity, whereas λ(E) depends only on the light spectrum and / ≥ d( q2) where d log E d log N 0. Notice that dE E has the infrared couplings. While the precise physical interpretation interpretation as the total squared charge contributed by par- of λ(E) remains unclear, it is arguably both a better measure λ( ) ticles with mass in a range E near E, and as a result of unification than E as well as a better behaved quantity d( q2)/dN can be interpreted as the average q2 of the in effective field theory. particles with mass near E. In other words, the condition (6.7) implies that for all states with mass approximately E, 6.3 Product groups we have The above discussion applies equally well to an arbitrary 2 2  2, e q m≈E G N m (6.10) gauge group as to the case of a single gauge boson or a simple gauge group. In particular, the one-loop correction (3.3)to m which is to say that the average particle of mass near is the gauge boson propagator is unaffected by the presence superextremal. Hence the relation (6.7) implies the existence of other gauge group factors, except that we must allow for tower E of a of superextremal resonances at energies above 0. kinetic mixing between photons, as we did above in Sect. 5.3. This is an sLWGC-like statement.

The close relationship between (6.7) and the sLWGC, at 1 a b μν L =− τab(F )μν(F ) , (6.11) least parametrically, suggests the intriguing possibility that kin 4 there is some sharp property of, e.g., the high energy behav- τ ior of scattering amplitudes in quantum gravities that has for some positive definite gauge coupling matrix ab (gener- / 2 the same close relationship to the sLWGC. As we discussed alizing 1 e in the single-photon case). Loop corrections give in the introduction, the sLWGC itself is somewhat poorly an energy-dependent correction to the gauge boson propaga- defined in theories with strong coupling, since it refers to tor scaling as single particles but these may be unstable. An alternate def-  τ (1−loop)( ) ∼ D−4. inition in terms of the S-matrix could address this issue, but ab E qiaqibE (6.12) | < at present it remains unclear whether any sharpened version i mi E of the heuristic λ(E) can be extracted from the S-matrix. We The direct analogue of λ (E) is a matrix λa (E) = leave further exploration of this idea to the future. gauge c τ abτ (1−loop)( ) Note that if we define a variant λ(E) which includes the bc E , but a more straightforward approach relies a fractional size of loops of heavy particles as well as the light on a choice of direction n in charge space. particles accounted for in λ(E) Ð as discussed further in Specifically, we can adapt our preceding arguments to the   kinetically mixed case as follows: the derivation of the WGC “Appendix A.1” Ð then λgauge(E)  λgrav(E) leads to a different conclusion than above: there must be a superex- from the condition (6.1) that a gauge interaction is strong at the scale QG carries through with the replacements tremal particle with mass between E and QG. Moreover,   λgauge(E) ∼ λgrav(E) for E  E0 does not have the same strong implications. Because heavy particles typically give 1 → nanbτ , q → naq . (6.13) the dominant contribution toλ(E), this parametric matching e2 ab i ia does not directly constrain lighter particles, and no sLWGC- like statement follows. None of our previous results were In other words, if any particular linear combination of U(1)s is sensitive to the distinction between λ(E) and λ(E), which strong at the quantum gravity scale, we deduce the existence illustrates the more speculative nature of the present section: of a particle charged under that linear combination. If we gauge-gravity unification at weak coupling is a concept with impose that gauge couplings are strong, in the sense of the no obvious definition, and we could have chosen a different condition (6.1), for all possible choices of na, then we obtain   one, such as λgauge(E) ∼ λgrav(E). the convex hull condition for the product gauge theory. 123 337 Page 16 of 33 Eur. Phys. J. C (2018) 78 :337

Similarly, the arguments of Sect. 6.2 comparing λgrav(E) and the kinetic terms become to λ (E) may be rephrased in terms of the condition gauge 2 1 P 1 2 1 2 L =− + Lμν + Hμν . (6.18) 2 2 2 + 2 2 1 1 4 eA eB eA P eB  q2, 2 2 (6.14) e G N E Now, a particle of charge (qA, qB ) under the original sym- metries couples to the linear combination a form suitable for making the replacements (6.13). Again, the arguments go through once we select a direction na in qA Aμ + qB Bμ = (PqA + qB) Lμ charge space. 2 − 2 qAeA PqBeB + Hμ. (6.19) e2 + P2e2 6.4 Higgsing A B As expected, the unbroken gauge field couples to a U(1) with It has been pointed out that the WGC and most of its known integer charges Q = PqA + qB while the heavy eigenstate stronger variants are not automatically preserved under Hig- in general can couple to irrational charges. gsing ([52,53], see also [41]). In other words, given an effec- A particle of charge (qA, qB ) is superextremal with respect tive field theory which apparently satisfies the variant of the to the un-Higgsed theory if WGC in question and which contains a light charged scalar, the effective field theory obtained by giving a vev to the scalar 2 ≤ γ 2 2 + 2 2 D−2, m eAqA eBqB MPl (6.20) may not satisfy the same variant of the WGC (or even the WGC itself). Of course, this does not imply the same state- where γ is a dimension-dependent factor, see, e.g., [35]. In ment about effective theories with quantum gravity comple- the Higgsed theory, it has a diagonal charge Q = PqA + qB tions: if the WGC variant in question is correct then these and couples via the diagonal coupling must satisfy non-trivial additional constraints which ensure 2 that it remains true after Higgsing. In many concrete exam- 1 = P + 1 , 2 2 2 (6.21) ples, this is the case, and the WGC/sLWGC remain true after eD eA eB Higgsing. 2 ≤ γ 2 2 D−2 Below, we review why the WGC and its lattice variants so it is superextremal if m eD Q MPl . Suppose that = + are not automatically preserved under Higgsing. We then a particle of charge Q PqA qB is extremal in the un- discuss to what extent our arguments above are affected by Higgsed theory, i.e., saturating (6.20). Whether it is superex- , these subtleties. Other recent discussions of Higgsing and tremal or not in the Higgsed theory depends on qA qB . = − the WGC include [34,54]. Attempts to exploit this kind of Putting qB Q PqA into (6.20) and completing the square, loophole for large field axion inflation include [17,32]. we find:

We start with the very simple case of two abelian gauge 2 e2 e2 e2 bosons, A and B, which are unmixed and do not couple to 2 = γ 2 2 D−2 + γ A B − D D−2. m eD Q MPl 2 qA P 2 Q MPl massless dilatons: eD eA (6.22) L =−1 1 2 + 1 2 . 2 Aμν 2 Bμν (6.15) 4 eA eB Thus, the particle is extremal in the Higgsed theory if and only if Suppose they are Higgsed to the diagonal by a scalar field of ( , − ) 2 2 charge 1 P for integer P, so that the linear combination eD eD qA = P Q, which is equivalent to qB = Q; Hμ = Aμ − PBμ becomes heavy. Then the light field that 2 2 eA eB is not Higgsed is (6.23) 2 + 2 eA Bμ PeB Aμ Lμ = . (6.16) otherwise, it is subextremal. In other words, extremality 2 + 2 2 eA P eB is preserved if the charge vector (qA, qB) is parallel to Pe2 /e2 , e2 /e2 . For other charged particles, Higgsing In terms of the light and heavy eigenstates the original fields D A D B makes them less extremal. are The ordinary WGC (i.e., the convex hull condition [36]) is 2 2 equivalent to the requirement that for every site in the charge eA PeB 2 Aμ = PLμ+ Hμ, Bμ = Lμ − Hμ , lattice (qA, qB) ∈ Z there is a superextremal multiparticle e2 + P2e2 e2 + P2e2 A B A B state with charge (rqA, rqB) for some rational r > 0. If (6.17) 2 / 2 2 / 2 2 / 2 eA eB is rational then so are eD eA and eD eB , and 123 Eur. Phys. J. C (2018) 78 :337 Page 17 of 33 337

However, if the convex hull condition is satisfied by a finite number of particles18 in the un-Higgsed theory then the WGC is automatically satisfied in the Higgsed theory. This is because the above argument produces multiparticle states which are arbitrary close to extremal if we take qA, qB large / 2 / 2 with qA qB a rational approximant to PeB eA. If a finite number of particles generate all of these multiparticle states then at least one of these particles must be superextremal, and the WGC is satisfied. This is depicted graphically in Fig. 3. It is not too hard to generalize this argument to the case of N > 2 photons and/or kinetic mixing between the photons. It is convenient to canonically normalize: Fig. 2 Higgsing in a lattice with e2 /e2 irrational. If the direction A B orthogonal to the Higgsed particle (shown in red) does not intersect  2  2 1 a 2 m A a any lattice points, then the WGC (and sLWGC) need not be satisfied in L =− (Fμν) − q Aμ . (6.25) q2 a the resulting theory 4 a a where q ∈  is the charge of the Higgs field and  ⊂ RN a Q Q is the charge lattice. We decompose into heavy and light fields Pe2 /e2 e2 /e2 D A , D B ∈ Z2 gauge fields Hμ and La : ( 2 / 2 , 2 / 2 ) ( 2 / 2 , 2 / 2 ) μ gcd PeD eA eD eB gcd PeD eA eD eB a =ˆa + a i , (6.24) Aμ q Hμ ei Lμ (6.26) ˆ := /| | | |2 := δab a 17 where qa qa q for q qaqb and ei is chosen to is a lattice site. The ordinary WGC in the un-Higgsed a a b satisfy qae = 0 and δabe e = δij. The superextremality theory then implies that there is a superextremal multipar- i i j conditions before and after Higgsing are ( 2 / 2 , 2 / 2 ) ticle state of charge rPeD eA reD eB for some rational 2 ≤ γδab D−2 2 ≤ γδij ˜ ˜ D−2, r > 0. By the above reasoning, this multiparticle state is also m Qa Qb Mpl and m Qi Q j Mpl superextremal in the Higgsed theory, which implies the exis- (6.27) tence of a superextremal charged particle, hence the WGC is ˜ = a preserved. respectively, where Qi ei Qa is the charge after Higgsing. 2 / 2 Suppose a particle, charge Qa, is extremal before Higgsing, The situation is similar for the sLWGC. When eA eB is rational, the lattice vector (6.24) generates a one-dimensional then sublattice of the charge lattice. The intersection of this sub- 2 = γδij ˜ ˜ D−2 + γ( ˆa)2 D−2, m Qi Q j Mpl Qaq Mpl (6.28) lattice with the two-dimensional sublattice of superextremal δij a b = δab −ˆa ˆb charged particles required by the un-Higgsed sLWGC is a where we use ei e j q q . Thus, the particle is a one-dimensional sublattice, and for each site on this sublat- extremal after Higgsing if and only if Qaqˆ = 0; otherwise it tice (corresponding to a one-dimensional sublattice of the is subextremal. By the same arguments as above, if the plane Higgsed charged lattice) we obtain a superextremal charged orthogonal to qa contains an N − 1 dimensional sublattice particle in the Higgsed theory, hence the sLWGC is pre- of the charge lattice Q then the WGC and the sLWGC are served. each preserved under Higgsing, whereas if not then in general 2 / 2 stronger constraints are needed in the un-Higgsed theory to If on the other hand eA eB is irrational, then even if the LWGC is satisfied in the un-Higgsed theory, i.e., if for every satisfy the (sL)WGC in the Higgsed theory. If the convex hull 2 (qA, qB ) ∈ Z there is a superextremal charged particle, the condition is satisfied by a finite number of particles before ordinary WGC in the Higgsed theory may not hold. In partic- Higgsing, it is still satisfied after Higgsing.19 ular, if the charged particles are all extremal (or subextremal) Note that the condition that the charge lattice Q inter- in the un-Higgsed theory then there are no superextremal sects the plane orthogonal to qa in an N − 1 dimensional 2 / 2 charged particles in the Higgsed theory precisely because sublattice generalizes the requirement that eA eB is rational, there are no charge vectors in the charge lattice parallel to 2 / 2 , 2 / 2 18 PeD eA eD eB , and the WGC is violated. This is depicted There are concrete examples of (supersymmetric) quantum gravities graphically in Fig. 2. for which the convex hull condition cannot be satisfied by a finite number of particles, see [35,41]. 19 Our results disagree slightly with [53], which concluded that the ordinary WGC and the sLWGC are automatically preserved under Hig- 17 Note that gcd is naturally defined for rational arguments via gsing. As shown above, this is not the case in general, though the ordi- p , q = 1 ( , ) / ( , ) / ( , ) gcd r s rs gcd ps qr so that x gcd x y and y gcd x y nary WGC is automatically preserved if the convex hull condition is are always coprime integers for any x, y ∈ Q. satisfied by a finite number of particles. 123 337 Page 18 of 33 Eur. Phys. J. C (2018) 78 :337

Fig. 3 Preservation of the WGC under Higgsing for a theory with a finitely generated convex hull. Since the convex z2 hull condition is satisfied in the direction e⊥ orthogonal to the z3 Higgsed particle (shown in red, with charge-to-mass vector z0), z1 we necessarily have either z0 |z1 ·e⊥|≥1or|z2 ·e⊥|≥1. This ensures that one of these two particles will still satisfy the convex hull condition after Higgsing

z4

z6

z5

since in that simple example the charge lattice is generated the gauge group were unbroken Ð even if m A lies above the 20 by (eA, 0) and (0, eB ) with qa = (eA, −PeB ), so we require WGC scale Ð so long as m A  QG. non-trivial solutions to (meA, neB ) · (eA, −PeB ) = 0, i.e., In particular, gauge-gravity unification in the sense of 2 / 2 = / , eA eB Pn m, for rational m n. As in this simple exam- Sect. 2 is unaffected by Higgsing. We could reach the same ple, in general charge lattices which satisfy this property are conclusion by ignoring the massive gauge bosons entirely dense in the set of all charge lattices. and focusing on some U(1) in the Cartan of the unbroken This means that the WGC is not necessarily preserved group. If the sLWGC is satisfied in the un-Higgsed theory, the under Higgsing, assuming the original theory exactly satu- arguments given above imply that it must be at least approxi- rated the WGC bound. The same argument applies to the mately satisfied in the Higgsed theory. The general argument sLWGC: a theory which saturates the sLWGC can in prin- of Sect. 5.4 can then be applied, regardless of additional mul- ciple be Higgsed to a theory that violates it. This is not a tiplicities which arise from sLWGC constraints coming from counterexample to the WGC or sLWGC, however: it simply the enhanced gauge group in the UV. shows that stronger constraints must be imposed on the orig- Conversely, if we assume that the gauge forces in the un- inal theory to ensure that these bounds are not violated after Higgsed theory become strong at or below the quantum grav- Higgsing. ity scale, we can apply the arguments of Sect. 6.1 to this un- It is also worth noting that although the WGC can in prin- Higgsed theory. Equation (6.2) then ensures that the WGC ciple be violated, it will still be approximately true in the will be (approximately) satisfied for this theory. Furthermore, Higgsed theory. In the above example, we may choose qA, the masses of the superextremal particles will be below QG, / ≈ 2 / 2 qB such that qA qB PeB eA to arbitrarily good preci- and there are only a finite number of such particles. This sion. If the WGC is satisfied in the un-Higgsed theory, then means that the convex hull condition will be (approximately) there must exist some (possibly multiparticle) state with these satisfied by a finite number of particles in the un-Higgsed charges (or a multiple thereof), and this will reduce upon Hig- theory, which implies that it will also be (approximately) gsing to a (possibly multiparticle) state that approximately satisfied in the Higgsed theory. satisfies the WGC bound. The same statement is true for the As before, we reach the same conclusion if we consider sLWGC, except we demand that these multiparticle states be aU(1) in the Cartan of the unbroken gauge group, ignoring single particle states or resonances. the broken generators. If we assume that this U(1) becomes We now consider the effect of Higgsing on our arguments strongly coupled below QG,(6.2) again ensures that the about UV cutoffs. If the scale of at which the gauge group WGC will be (approximately) satisfied for this U(1). If we is Higgsed is well below the quantum gravity scale, m A  further assume that λgauge(E) ∼ λgrav(E) for this U(1) over QG, then from a UV perspective we can treat the gauge group as unbroken, and we still expect a tower of charged 20 (D−2)/2 However, we need to be cautious about identifying any massive vec- states to appear near the WGC scale, eMPl . Heavier tor with an enhanced gauge group. One case in which this is obviously particles in such a tower generally dominate λgauge and λgrav, incorrect is KK theory with a higher dimensional photon, for which hence the conclusions about UV cutoffs are the same as if there is a tower of graviphoton-charged massive vectors, but no corre- sponding nonabelian gauge group. 123 Eur. Phys. J. C (2018) 78 :337 Page 19 of 33 337 some range of energies E  E0, as in Sect. 6.2,(6.10) ensures sublattice index in a real quantum gravity. We now give a the existence of a tower of superextremal resonances with suggestive argument that this is unlikely to occur. energy above E0. As before, suppose that the LWGC is saturated in the UV Thus, the relationship between unification in the sense theory. In the above example we needed a Higgs field with of Sect. 2 and the (sL)WGC is largely unaffected by subtleties parametrically large charge to obtain a parametrically large related to Higgsing. There is, however, an important caveat keff . To quantify how large this charge is, observe that if the to keep in mind when considering bounds on the quantum Higgs field were extremal, it would have mass gravity scale, such as (3.12), (5.15), or (5.27): the sublat- e2 e2 tice index can change upon Higgsing. Hence, even if k ∼ 1 2 = γ A B D−2. mHiggs;ext 2 MPl (6.32) in the UV theory, our arguments do not exclude k 1in eD the infrared theory, leading to weaker constraints on the UV If we demand that this lies below the quantum gravity scale, cutoff. QG, then we obtain the constraint To illustrate this, consider the two-photon example dis- e e D−2 cussed above. Assume for simplicity that the UV theory A B 2   , MPl QG (6.33) contains an LWGC-saturating tower of extremal particles. eD For fixed Q = PqA +qB there is an approximately extremal where we drop the order-one factor γ . Combining this with 2/ 2 ≤ + ε2 particle in the IR theory (i.e., with m mext 1 for the UV constraint ε  ( − ) 1) whenever 1 2 D 2 D   (eAeB) D M (6.34) P − εe /e P + εe /e QG Pl A B Q ≤ q ≤ A B Q (6.29) 2 + 2 / 2 A 2 + 2 / 2 from (5.27), we obtain P eA eB P eA eB 4(D−1)(D−2)  / = 2(D−1) 3(D−2) has an integer solution. If P eA eB then qA 0isasolu- ( ) D D  2 , eAeB MPl eD MPl (6.35) tion for any Q, and keff  1 (the LWGC is approximately satisfied after Higgsing). On the other hand, if P eA/eB hence ( − ) 3(D−2) then q = Q/P is a solution for any Q ∈ PZ (with 1 2 D 2 1 A D D−1 2(D−1) QG  (eAeB) D M  e M , (6.36) ε ≥ eA/(PeB )) and keff  P (the sLWGC is approximately Pl D Pl satisfied with sublattice index P). and the naive k  1 constraint is enforced in the infrared / Thus, for P 1 and P eA eB, it is possible for the theory. sublattice index to be parametrically larger in the Higgsed Note that this is not quite the same as enforcing that keff ∼ 21 theory. Accounting for the change in sublattice index, the 1. Rather, we merely showed that under these assumptions “infrared” constraint the constraint (6.30) holds with keff set to 1. To illustrate the 3(D−2) = = = 1 2(D−1) difference, consider, e.g., the case eA eB e and D 4.   ( ) D−1 , QG keff eD MPl (6.30) Then, the constraint that mHiggs;ext  QG is follows automatically in the UV theory. To see this, note that − / e2(P2 + 1)  e ⇒ P  e 1 2, (6.37) eD  eA/P for P eA/eB and eD  eB for P  eA/eB   from (6.21), where keff P and keff 1 in these two limits, so for e  1 we can have keff  P 1, but nonetheless 3/2 respectively. Thus, (6.30) follows from either eD  e/P  e , so that 1 3(D−2) 1 3(D−2) / ( − ) ( − ) 1/2 1 3   D−1 2 D 1   D−1 2 D 1 QG  e MPl  e MPl, (6.38) QG eA MPl or QG eB MPl (6.31) D in the two limits.22 and the IR k ∼ 1 constraint is enforced. The reason for Of course, since it is not possible to determine the sublat- this discrepancy is that there are more near-extremal charged tice index in the deep infrared, the constraint (6.30)isofno states in the infrared theory than the minimal ones required practical use unless we can assume that keff is not too large. by the sublattice index keff ; in this example, for instance, ( )  Thus, it would be very interesting to determine whether Hig- there are O n charged particles with charge q keff n. gsing can lead to a very large (or even parametrically large) It is not difficult to generalize this line of reasoning to the case of N photons and arbitrary kinetic mixing, using the 21 These are the same as the conditions that the IR WGC scale notation of (6.25) and following. The UV constraint on QG (D−2)/2 (D−2)/2 eD MPl is parametrically below the UV WGC scales eA MPl from (5.27) can be written as (D−2)/2 and eB MPl ,asin[52,53], though the sublattice index is not nec- (N+2)(D−2) 1 ( + − ) essarily the same as the ratio between these two scales. + − 2 N D 2 QG  |Q| N D 2 M , (6.39) 22 (D−2)/2 (D−2)/2 Pl When the UV WGC scales eA MPl and eB MPl are both below QG, the UV theory enforces the stronger constraint QG  for k ∼ 1, where Q is the UV charge lattice and |Q| is the ( )1/D 2(D−2)/D  eAeB MPl ,but(6.31) holds regardless. volume of the fundamental domain of Q. We can assume 123 337 Page 20 of 33 Eur. Phys. J. C (2018) 78 :337 that the Higgs charge qa is primitive Ð i.e., not a non-unit Does gauge-gravity unification arise in such theories? If multiple of another charge in the charge lattice Q Ð since we naively compute λgauge(E) and λgrav(E) in a weakly cou- otherwise we can choose a Higgs field with a smaller charge pled string theory for energies E above the string scale, we | |=| |/| | and the same effect. In this case, Q˜ Q q , where find that both grow very rapidly, a simple consequence of the  ρ( ) ∼ ( / ) Q˜ is the charge lattice after Higgsing. The assumption that Hagedorn density of states E exp E TH . However, mHiggs;ext  QG becomes states with higher charge come with lower multiplicities, and  2∝ 2 ∝ 2 D−2 D−2 as a consequence q E, even though qmax E . Thus, γ | | 2  (| |/| |) 2   , λ ( )  λ ( ) q MPl Q Q˜ MPl QG (6.40) well above the string scale gauge E grav E . We illustrate this in the simple example of ten-dimensional so that combining with (6.39)gives heterotic string theory, with spectrum determined by the con- (N+2)(D−2) 1 2(N+D−2) ditions   | | N+D−2 M QG Q Pl α (N+1)(D−2) 2 1 2 1 2(N+D−3) m = NL + Q − 1 = NR. (7.1)  | | N+D−3 , Q˜ MPl (6.41) 4 2 ∈ Z and the IR k ∼ 1 constraint follows from the UV k ∼ 1 Here NL,R ≥0 count left and right-moving oscillators, ( ) constraint. Thus, to parametrically violate these constraints and each comes with an associated multiplicity dL NL and d (N ), equal to the multiplicity at the Nth level of the open using Higgsing we need mHiggs;ext QG. R R bosonic string and the open superstring, respectively. Thus, This can be interpreted in two ways. Firstly, we need α the number states at a given mass-level m2 = N is mHiggs  QG in order to describe the Higgsing in effective 4 field theory, hence to violate the k ∼ 1 constraints the Higgs Q2≤2(N+1) field must be very superextremal. Secondly Ð following the 2 d(N) = dL (N + 1 − Q /2) dR(N). (7.2) arguments of Sect. 6.1, in particular (6.4) Ð if weakly cou- Q∈ pled gauge theory and gravity emerge from the same strong coupling scale QG then to violate the k ∼ 1 constraints the To estimate this, we use the asymptotic formulae [55, ¤2.3, Higgs field must have a charge much larger than the aver- 5.3] √ √ age charge of other particles within the effective field theory. π π e4 n e 8n In fact, in typical examples only a few light particles will d (n) ∼ , d (n) ∼ , (7.3) L n27/4 R n11/4 be very superextremal, with the rest of the spectrum near- extremal or subextremal, implying that the Higgs field has a up to order-one constants. Thus, √ charge which is much larger than almost every other particle  e4π N+1−Q2/2 in the effective field theory. d(N) ∼ d (N) d16 Q R 2 27/4 Q2≤2N (N + 1 − Q /2) Although these suggestive arguments do not rule anything √ π out, they show that violating the k  1 constraints on   4π N− √ Q2 QG e N through Higgsing in a way consistent with our other assump- ∼ d (N) d16 Q R N 27/4 tions about emergence from a UV cutoff requires the Higgs √ √ e2π(2+ 2) N field to have peculiar properties such as a charge that is much ∼ , (7.4) larger than most or all of the other particles in the effective N 11/2 field theory. For this reason Ð and because we know of no where in the second step we use the fact that the integrand is quantum gravities with a very large sublattice index Ð we dominated by the region Q2  O(N 1/2). This agrees with, expect that the the UV cutoff bounds such as (3.12), (5.15), e.g., [55, ¤6.4]. or (5.27) with k ∼ 1 are never parametrically violated. Thus, √ √ 2π(2+ 2) N α λ ( ) ∼ 2 e , = 2. grav E gs / N E (7.5) 7 Caveats N 3 2 4 To compute λgauge, we follow the same steps but count states 7.1 String theory at weak coupling 2 weighted with Q1 for some particular Cartan generator√Q1.  2 1 By a straightforward calculation we obtain Q1 2π N We have discussed examples in which the spectrum of par- as well as ticles approximately saturates sLWGC bounds. This is char- √ √ e2π(2+ 2) N α acteristic of Kaluza-Klein theories, for example, and related λ (E) ∼ g2 , N = E2. gauge s 2 (7.6) quantum gravities such as large volume compactifications N 4 of M-theory. Of course, other examples of weakly coupled Therefore, both λgrav and λgauge grow rapidly above the string gauge theory can arise in string theory at gs  1. scale, but λgauge grows slightly less rapidly. 123 Eur. Phys. J. C (2018) 78 :337 Page 21 of 33 337

The same conclusion should follow for NSNS charges and perhaps not possible at all Ð to distinguish between gauge in an arbitrary string theory. A rough argument is as fol- forces, gravitational forces, and other interactions; λgauge and lows: the modular invariance argument of [41,42] implies λgrav as we have defined them become meaningless. Whether := − 1 2 that the multiplicities depend only on tL,R L,R 2 QL,R some improved notion can be found is beyond the scope of (and some additional discrete data). Thus, if the multiplici- this paper. ties in the neutral√ sector, QL = Q R =√ 0, are Hagedorn then Nonetheless, we can still ask whether unification occurs at a t a t dL (tL ) ∼ e L L and dR(tR) ∼ e R R are Hagedorn. We or below the string scale. We again consider ten-dimensional find the density of states heterotic string theory as an example. We have

Q2 ≤2( +1) ∼ 2/ 8, 2 ∼ 2/ 6, L,R G N gs Ms g gs Ms (7.10) d( ) = d ( − Q2 /2) d ( − Q2 /2), L L R R which gives Q∈ λ ∼ 2( / )8,λ ∼ 2( / )6, (7.7) grav gs E Ms gauge gs E Ms (7.11) or below the string scale, up to numerical constants. Thus, well    below the string scale λgrav  λgauge, but at the string scale a −Q2 /2+a −Q2 /2 ( ) ∼ L L R R r d e d Q itself λgauge ∼ λgrav, at least parametrically in gs  1.  √ Similarly, if we compactify heterotic string theory on a ( + ) − a√L 2 − a√R 2 aL aR QL Q R r ∼ e 4 4 d Q rectangular p-torus with radii R1,...,Rp s and no Wil- √ son lines then ( + ) ∼ e aL aR , (7.8) g2 g2 G ∼ s , g2 ∼ s , 2 N 8 ... 6 ... to leading order. Redoing this calculation weighted by Q1 Ms R1 Rp Ms R1 Rp for some left-moving charge Q1, we find g2 g2 ∼ s , (7.12) √  i 8 2 2 α M R R1 ...Rp Q2∼ , = E2, (7.9) s i 1 a 4 L where gi denotes the gauge coupling of the KK photon asso- 23 and likewise for right-moving charges, hence λgauge(E)  ciated to the Ri circle. Below the compactification scale, λgrav(E) far above the string scale, as before. we find What should we make of this? It is clear that in string the- 2 8−p 2 6−p λ ∼ gs E ,λ ∼ gs E , ory there is a sense in which both gauge theory and gravity are grav 8 gauge 6 (7.13) M R1 ...Rp M R1 ...Rp emergent. What has really broken down is our ability to use s s simple, field-theoretic one-loop arguments to discuss the rel- with no KK modes contributing, and thus nothing charged ative strength of gauge theory and gravity. One way to argue under the KK photons. Above the compactification scale but this is that the particles running in loops, for mass well above below the string scale, we find the string scale, are not particles at all: they are extended λ ∼ g2(E/M )8,λ ∼ g2(E/M )6, objects, and their couplings should involve form factors. It is grav s s gauge s s λ ∼ 2( / )8, a familiar property of closed string worldsheet perturbation KK,i gs E Ms (7.14) theory that naive quantum field theoretic expectations about by counting KK modes. Here λKK ∼ λgrav as expected from the behavior of loops are modified due to modular invariance. the general argument of Sect. 5.4, but still λgauge λgrav.At It is unclear what energy scale we should call QG in the string scale, however, we find λgauge ∼ λgrav ∼ λKK ∼ weakly coupled string theory. It is tempting to say that it is the 2 gs , and the forces “unify” in the sense of Sect. 6.2. string scale, since quantum field theory breaks down there. On the other hand, consider type I string theory in ten Such an identification has been argued for in the context of dimensions, with gauge fields in the open string sector. In / 2 the species bound, with an effective number of species 1 gs this case, [56,57]. In other words, the explosive Hagedorn growth of ∼ 2/ 8, 2 ∼ / 6, the density of states may translate into an effectively finite G N gs Ms g gs Ms (7.15) number of degrees of freedom from the point of view of black which gives hole evaporation or of loop corrections to the Planck mass. On the other hand, above we have taken  to be the λ ∼ 2( / )8,λ ∼ ( / )6, QG grav gs E Ms gauge gs E Ms (7.16) energy at which a theory can no longer be viewed as weakly coupled in any sense. When g  1, string theory is still s 23 In addition, there are p gauge bosons associated to the reduction of weakly coupled at the string scale Ð it is simply not a field the Kalb-Ramond B-field, but the lightest charged particles are well theory. Above the string scale it is no longer straightforward Ð above the string scale; these are discussed further in the next section. 123 337 Page 22 of 33 Eur. Phys. J. C (2018) 78 :337

λ ∼ λ ∼ 2 so that at the string scale gauge gs grav gs , with The lightest charged object is the D0 brane which has mass 4 ∼ / similar results upon toroidal compactification and in T-dual gMPl Ms gs (up to order-one factors), so there are no toroidal orientifolds of type II string theories with D-branes. charged particles below QG ∼ Ms. Thus, in these examples There are many other string theory examples of a similar nature where the charged objects are BPS branes wrapped λ ( ) ∼ 2,λ(closed)( ) ∼ 2, grav Ms gs gauge Ms gs on cycles. For instance, consider type II string theory com- ( ) pactified on a circle of radius R. The Kalb-Ramond B-field λ open (M ) ∼ g . (7.17) gauge s s generates a photon whose charged states are wound strings. The lightest of these has mass of order RM2, which for a We expect this simple result to generalize to a broad class of s large torus R  is well above the string scale. A para- perturbative string theories with charged particles at or below s metrically similar scaling arises in the WGC bound for gauge the string scale. In such examples λ  λ at the string gauge grav fields on D7 branes in approximately isotropic, large-volume scale Ð regardless of whether the gauge bosons come from compactifications of the IIB string. closed or open strings Ð implying the Weak Gravity Conjec- In these examples, the WGC scale is above the string scale ture up to order-one factors by the arguments of Sect. 6.2. and so the simple perturbative field theory arguments we Gauge fields in the RR sector are a potential exception have given throughout the paper do not apply. The under- to (7.17), but in this case the charged objects are wrapped lying gauge theories may still be thought of as emerging, D-branes. If the cycle in question is large in string units then in some sense, from the quantum gravity scale. In the case the wrapped branes are heavy, and don’t appear in the low- of the D0 brane this becomes manifest in the g 1 limit energy effective field theory (see Sect. 7.2 for further discus- s where they are simply KK modes; in the case of winding sion). However, in K3 and Calabi-Yau compactifcations of strings, the U(1) symmetry arises from the B-field which is type II string theory, shrinking two-, three- and four-cycles part of the supergravity multiplet, and which is T-dual to a can appear at singular points in the moduli space while main- graviphoton in toroidal examples. It may be worthwhile to taining a large overall volume. These can lead to light RR search for a modified version of our arguments that can apply charged states, as in, e.g., [43]. We won’t attempt to address to examples like these. For now, we simply highlight them the issue of unification in such cases in this paper, but as a shortcoming of our approach. we discuss the related issue of ultralight charged particles in four dimensions in Sect. 7.3. 7.3 Logarithmic running and ultralight particles

7.2 Heavy spectra In our previous discussion we have ignored the possibility of large logarithms. This is justified in many cases. Consider Throughout this paper, we have assumed that there are for example the one-loop beta function of a KK photon in charged particles with masses below the quantum gravity four dimensions: scale. We can heuristically motivate this assumption in a μR four dimensional weakly coupled gauge theory by noting 1 1 b  μR  R2 M2 − n2 log , (7.20) that the WGC scale eMPl is below the Planck scale, and the 2(μ) Pl π 2 e 2 8 = n sLWGC suggests that charged particles must appear at or n 1 below this scale. However, this argument is too naive, and for some order-one constant b, where for simplicity we fails even in the case of two weakly coupled photons A, B assume no massless particles in the original five-dimensional  3  with freely adjustable couplings eA,B.IfeA eB eB theory. We have then the WGC scale eB MPl for the second gauge theory lies 1/3 N above QG  e MPl, so there is no reason for B-charged N N 3 N A n2 log ∼ − + ..., whereas particles to appear below the quantum gravity scale. n 9 12 (D−2)/2 n=1 In fact, eM  QG occurs frequently in real quan- Pl N tum gravities, and in many such cases there are no charged N 3 N 2 n2 ∼ + + ..., (7.21) particles below QG. A simple example of this is the RR pho- 3 2 n=1 ton C1 in ten-dimensional type IIA string theory, for which at large N, so we obtain the same behavior at large N Ðup 2 ∼ / 6, 8 ∼ 8/ 2, g 1 Ms MPl Ms gs (7.18) to order-one constants Ð regardless of whether we include the log or omit it. This is because for most terms in the sum so that the logarithm is not large: the KK modes become increasing dense near the cutoff on a logarithmic scale even as they are 4 ∼ / . gMPl Ms gs Ms (7.19) spaced evenly on a linear scale. 123 Eur. Phys. J. C (2018) 78 :337 Page 23 of 33 337

For this reason, we expect that logarithmic corrections can To do so, we need to define λgauge(E) properly in the be consistently neglected to leading order in many of our cal- presence of large logarithms. Recall that in an abelian four- culations. However, there are certain circumstances in which dimensional gauge theory, we had previously this is not the case. For instance, in a four dimensional theory e2  with a light charged particle electric forces are screened at λ (E) := q2. (7.26) gauge 16π 2 i large distances. As explained in the introduction, if there are i: mi  2/ 2  light particles in four dimensions. In Sect. 6.2, we saw that or smaller for any E mi .Itfollowsthat q m m≤E /( 2 2 ) < perturbative gauge-gravity unification, in the form 1 eIR MPl , i.e., the average particle with m E is superex- tremal and the WGC is satisfied. λgauge(E) ∼ λgrav(E) for E  E0 (7.25) 24 We keep the loop factor to be consistent with the loop factor which has similar consequences to the sLWGC. It is interesting to λ appears in (7.22). We should also keep the loop√ factor in grav, roughly κ2 2  ask what this means in situations with ultralight particles in λ (E) = E d(i),whereκ := 8πG = 1/M is the grav 16π2 i:mi

2 Next, suppose that λgauge(E)  λgrav(E) above some Conjecture follows automatically (in the usual form m    2 2 2 λ ( )  λ ( ) scale E0 QG. Following the same steps as in Sect. 6.2, eIRq MPl), because by assumption gauge E0 grav E0 . we obtain The mechanism for this is conceptually simple: if there are  ( 2) 2 charged particles light enough to substantially renormalize 2 d q E 2 d log E M  + 2E e2 versus e2(E ) then these charged particles are necessarily Pl dN e2(E) d log N IR 0   superextremal. 1  b × − i q2 , (7.31) By turning around the above arguments, its easy to see e2(E) 16π 2 i i: mi

2 • The seesaw mass of right-handed neutrinos, MN ∼ y × even be absolutely stable due to a discrete gauge symmetry . 6 × 1014 GeV × 0 1eV. like baryon triality [66]. More generally, approximate flavor mν √ • The string scale Mstring ∼ gs MPl/ V where V is the symmetries (e.g. arising from massive U(1) gauge bosons volume of the internal six dimensions in string units. obtaining stringy Stückelberg masses) could provide addi- • The QCD axion decay constant fa, which is around tional spurions suppressing the decay rate. Hence we cannot 1012 GeV for conventional axion cold dark matter sce- make absolute statements, except that the discovery of a very small gauge coupling in nature could be consistent with the narios but could be larger in√ other scenarios. • The SUSY-breaking scale F0 ∼ m3/2 MPl, which is sLWGC only if additional structure exists to protect the pro- larger than 6 × 1011 GeV if we demand that gravitinos ton. decay before BBN (and can be even larger in sequestered Nonabelian gauge groups with small couplings have been scenarios). advocated in several cosmological contexts, which we will now summarize. We should be careful about drawing too-hasty conclusions about which of these scales must be below QG in a con- 8.1 Nonabelian dark interacting with dark matter sistent theory. Any scale that we treat as the mass scale of a weakly coupled particle, including MGUT, MN, and Mstring, Nonabelian dark radiation interacting with dark matter has are bounded above by QG. However, expectation values the unusual property that, due to the t-channel scattering dia- 2 of fields are not obviously constrained in this way; for one gram, the scattering rate  ∼ T . This is the same scaling as familiar example, consider the transplanckian field ranges the Hubble rate during a radiation-dominated era, and so the in large-field inflation, which may have potential tensions scattering does not decouple. This can lead to striking cos- with quantum gravity but certainly do not with effective field mological consequences even for small couplings [67]. Large  theory. The Hubble constant during inflation, Hinf , should couplings would predict sizable deviations from CDM cos- also be below QG as it corresponds to the curvature scale mology that have not been observed, so any phenomeno- of space. However, the energy density in a spacetime is less logically viable version of this scenario potentially has an obviously bounded. On the other hand, in many concrete sce- sLWGC constraint. narios, such as a natural inflation model where the potential Studies have found that tensions in CMB data (involving σ Vinf is generated by confinement [60], there will be such a the values of the Hubble constant and 8) may be partially −4 bound. Similarly, in many axion theories, for instance of the relaxed in such a model with gauge couplings g ∼ 2 × 10 KSVZ type, there are physical particle masses of order fa [68Ð70], though early Lyman-alpha data diminishes the sig- which must be below QG. nificance [71]. (Likelihoods are not yet available to test this One general consideration for QG is the extremely strong model with more recent Lyman-alpha data.) If this scenario experimental constraint on proton decay. Processes like p → is true, then for an SU(2) gauge theory we would have + + e π 0 or p → K ν arise from dimension-six operators of 1/2 16 QG  g MPl ≈ 3 × 10 GeV. (8.2) the type QQQL or ucucdcec in the Standard Model. Super- Kamiokande has constrained the lifetime of these processes This would be in modest tension with GUT unification or with 1/4 to be larger than about 1034 years [61]. If we write the oper- the value of V in a just-around-the-corner detection of r,  ators suppressed simply by QG, we obtain a bound since both scenarios involve physics very near the scale QG. Theories with larger SU(N) groups would have stronger con- 16 QG  2 × 10 GeV ≈ MGUT. (8.1) straints. However, similar cosmological phenomenology can be obtained in models with larger couplings but with only This means that in a quantum gravity theory with no addi- a fraction of dark matter interacting with dark radiation tional structure, proton decay requires a large QG and [70,72]. Further observations of the matter power spectrum at is inconsistent with the existence of very weakly coupled smaller scales would be needed to distinguish the signatures gauge theories. In some theories, this bound becomes much of these models. stronger. For instance, in the context of theories with approx- The sLWGC tower would involve particles with non- imate supersymmetry, we have dimension-five proton decay abelian gauge charges beginning at a mass scale of gMPl ∼ from superpotential operators (after using R-parity to forbid 5 × 1014 GeV, which is heavy enough relative to the Hubble the dimension-four terms). In such theories the bound would scale during inflation as to not be an obvious problem on its require not only a large QG but also a sufficiently large scale own. On the other hand, if r is relatively large these particles, of supersymmetry breaking [62Ð65]. However, it is worth with masses only an order-one factor above the Hubble scale, keeping in mind that certain quantum gravity theories may might leave detectable imprints in non-Gaussianities. contain special structure that allows QG to be much smaller We emphasize that in this case, the tension would be than this naive estimate. In the extreme limit, the proton could between the interacting nonabelian dark radiation scenario 123 337 Page 26 of 33 Eur. Phys. J. C (2018) 78 :337 and other, unrelated physics, like GUTs. Experimental con- firmation of this cosmological scenario would not, in itself, disprove the sLWGC.

8.2 Chromonatural inflation

There are few known cosmological mechanisms for gener- ating detectable primordial tensor modes. The most famil- iar is large-field inflation. In recent years another sce- nario, chromonatural inflation [73], has claimed to generate detectable tensor modes via a different mechanism [74Ð76]. Tensor modes arise as the product of a classical gauge field background and perturbations in the gauge field. The classi- cal gauge field background spontaneously breaks the prod- Fig. 4 (s)LWGC UV cutoff bound as a function of g and N (solid con- uct of spatial rotations and nonabelian gauge symmetries to tours, labeled by representative physical scenarios). The dashed lines a diagonal, as first suggested in the related theory of gauge- are benchmark choices of g in two particular cosmological scenarios flation [77]. that favor small nonabelian gauge couplings Aside from kinetic terms, the Lagrangian for chromonat- ural inflation includes different variant of the model has χ serving as a spectator λ φ 4 a aμν field while another field drives inflation [83]. Recently it − μ [1 + cos(χ/f )] − χ Fμν F . (8.3) 8 f has been claimed that such a theory can produce detectable tensor modes even for very low-scale inflation, with a Hub- Here χ is an axion field, and its coupling to the gauge ble scale just above that constrained by BBN [84]. However, fields (which are approximately static) generates an effec- accomplishing this requires exponentially small gauge cou- tive friction term that modifies the evolution compared to plings, and so we expect the sLWGC constraint to be even standard slow-roll inflation. Demanding that the model gives more severe for such a scenario than for the minimal real- rise to inflation with sufficiently many e-folds and matches ization of chromonatural inflation. We will not consider the the observed values of the scalar power spectrum amplitude constraint on this alternative scenario in detail here. P ∼ 2×10−9 and of the spectral index n tightly constrains s s Our results suggest that the sLWGC is in modest, but not the available parameter space [75,76,78]. In fact, the mini- decisive, tension with chromonatural inflation. Another field mal version of the model predicts a primordial tensor signal theoretic concern with this theory is that if χ is a compact r that is too large, but a modified Higgsed version of the the- axion field of period 2π f (as suggested by the choice of ory is compatible with data without qualitatively changing cosine potential), then the coupling λ is actually quantized: the properties of the model [79]. The upshot of the fit to data is that the standard chromonat- g2 −6 λ = n , n ∈ Z. (8.4) ural inflation benchmark models have g ∼ 10 ,asmall π 2 1/2 4 number that is approximately obtained as Ps multiplied by order-one numbers. (For a full explanation, we refer Given that the fit to inflationary phenomenology prefers ∼ −6 λ ∼ readers to the literature.) The model parametrically favors g 10 and 100, this requires a large integer of 1/2 order 1015 to appear in the theory! This suggests that a fairly μ ∼ g MPl, the same scaling as QG in the (most opti- extreme version of axion monodromy must appear in the UV mistic) SU(2) case. For instance,√ all benchmark points in completion of chromonatural inflation. Since the sLWGC Table 1 of [79]haveμ = g/50MPl, and are thus only puts the UV cutoff of the theory just overhead, it will be diffi- marginally compatible with μ  QG. A different set of phenomenologically consistent parameters can be obtained cult to explain the origin of this large dimensionless number if the axion starts very close to the minimum of its potential. from smaller input parameters. We will not undertake this In this regime chromonatural inflation matches onto the ear- challenge here. lier gauge-flation model [80Ð82]. This regime accommodates larger values of g ∼ 10−3,butμ must be taken significantly 8.3 Summary of phenomenological consequences larger: μ ≈ 0.1MPl. Such a large μ is in clear tension with sLWGC constraints on even mildly small gauge couplings. In Fig. 4, we show contours of the largest QG allowed by We emphasize, however, that the original gauge-flation sce- the sLWGC as a function of g and N for SU(N) gauge theo- nario did not invoke a χ field, and so one could consider other ries. The dashed horizontal lines correspond to the approxi- possible UV completions that may evade WGC bounds. A mate size of couplings of interest for dark matterÐdark radi- 123 Eur. Phys. J. C (2018) 78 :337 Page 27 of 33 337 ation interactions and chromonatural inflation. We see that scales we can imagine: the sLWGC has the potential to put a variety of interest- 19 b E (4/3)n f 10 GeV ing high-scale physics in tension with small gauge couplings log  log ∼ 2.4 n , (8.5) π 2 π 2 −33 f that may be of phenomenological interest. On the other hand, 8 m 8 10 eV some high-scale physics, like a conventional QCD axion with where we consider n f Dirac fermions for definiteness and 12 decay constant fa ≈ 10 GeV, is relatively safe; we would put the ratio of the Planck scale to the present-day Hubble need a theory to predict a small gauge coupling of order 10−7 scale into the log. This means that unless we have a large or smaller to have tension between the sLWGC and the PQ- number of light charged particles, or light particles with para- breaking scale. metrically large charge, the screening effect is negligible for To be interesting from the viewpoint of sLWGC con- small gauge couplings e  1. Even if we choose n f and/or straints on QG, gauge couplings have to be quite small; q to be large, we cannot completely evade constraints. For 2 for instance, merely demanding that the nonabelian gauge instance, if we fix q ∼ 1, then we need n f  1/e to gen- theory’s confinement scale be smaller than the Hubble scale erate significant screening, but then the species bound gives   −1/2  of our today is not sufficient to derive an inter- QG n f MPl eMPl, which is a much stronger bound esting bound. Cases of interest generally arise in cosmol- then the one we were trying to evade! If instead we fix n f ∼ 1 ogy and come from tight constraints on the size of gauge then we need q  1/e to generate significant screening, but couplings. For instance, nonabelian gauge preheating [85] then qe  1, and the light particles have O(1) couplings! requires small couplings, of order 10−4 or smaller, because Thus, for all practical purposes we can ignore screening otherwise the gauge fields’ interactions with each other back- effects when placing phenomenological constraints on weak react to shut off resonant particle production. On the other gauge couplings. hand, this scenario has potential difficulties purely within effective field theory, since it assumes that higher-dimension operators play a crucial role in the scalar coupling to gauge 9 Conclusions fields but not in the scalar potential. Another theory that would be interesting to explore from We have argued that towers of charged particles generically the sLWGC viewpoint is gaugid inflation [86], which relies lead to low cutoffs on both gauge theory and gravity, and not on a nonabelian group but on a U(1)3 gauge theory. that for towers of approximately WGC-saturating particles, The product of spatial rotations and rotations among the 3 these cutoffs are parametrically the same. This suggests that gauge fields is broken to the diagonal. In a theory of free in sufficiently weakly coupled gauge theories, we can con- gauge fields, a rotation among the 3 gauge fields is a symme- cretely understand the emergence of the gauge theory from try (though one without a gauge-invariant Noether current). the quantum gravity scale: the size of the gauge field kinetic However, the existence of a charge lattice explicitly breaks term is parametrically determined by loops of the tower of the symmetry. Since the scenario relies on higher-dimension charged particles. operators built out of the gauge fields, one would need some We have also shown some interesting converse statements sort of discrete symmetry to ensure that these operators (at to this, most notably that if we assume an approximate match- least approximately) respect the appropriate symmetry after ing between the gauge theory and gravitational cutoffs then integrating out the particles whose existence the sLWGC the Weak Gravity Conjecture follows. Yet stronger state- demands. (In other words, in the presence of a charge lattice, ments follow if we make additional assumptions concerning the full SO(3) rotation symmetry must be an accidental sym- “unification” between gauge and gravitational forces, though metry, enforced by some smaller discrete symmetry.) Again, the physical meaning of these assumptions is not complete we will defer further consideration of this model for future clear at present. work. There are a number of open questions remaining. As Finally, we note that the bounds on QG can in princi- emphasized in Sect. 7, there are simple examples of quantum ple be relaxed if there is a substantial screening effect from gravity theories for which our arguments do not apply. These light charged particles, since in four dimensions Ð as we include perturbative heterotic strings at gs  1, D0 branes, argued in Sect. 7.3 Ð the gauge coupling renormalized near and winding strings. In these cases we should not necessarily the WGC scale E0 ∼ e(E0)MPl is what should appear in the trust the perturbative field theory calculations we have per- bounds, rather than the infrared gauge coupling. However, formed, but there may be generalizations of our arguments. phenomenological constraints do not depend on the infrared Another concern arises from cases with ultralight charged gauge coupling either, but rather the gauge coupling renor- particles in four dimensions, as in the conifold example, malized at some finite scale, albeit possibly a very low scale. which suggests that the sLWGC must be modified in the Due to the logarithm and the loop factor the screening effect context of running couplings. These shortcomings should be is not very large, even if we choose the largest hierarchy of explored more fully in the future. 123 337 Page 28 of 33 Eur. Phys. J. C (2018) 78 :337

Throughout this paper we have treated gauge couplings A.1 Loops of heavy particles as constants, but in quantum gravity theories we expect cou- plings to be determined by the expectation values of mod- Particles with mass m > E run in loops, but have contri- uli fields. The original Swampland conjectures suggest that butions that may be expanded as a series in powers of E2. whenever we find a tower of particles becoming light, we This suggests that rather than equation (3.7) we could have should expect a logarithmic divergence in distance in the defined  moduli space [2,3]. Recently these moduli space conjectures − λ (E) := e2 E D 4 I (i) have appeared in work on the WGC and its connection to gauge i: m 3(D−2)/2 m E do not change our logic. Second, we will argue that eMPl loop corrections to the n-point functions do not lead to lower  cutoffs than those we have discussed. which is the same scaling as for λgauge(E). 123 Eur. Phys. J. C (2018) 78 :337 Page 29 of 33 337

  More generally, if we assume a smooth function N(M) that λgauge(E) ∼ λgrav(E) over a range of energies E  E0 is characterizing the total number of particles of mass below far less constraining that the analogous statement with λ(E)s, M, with average squared charge Q2(M) for the particles of since in typical examplesλ(E) is dominated by particles near   mass near M,wehave: the cutoff, and λgauge(E) ∼ λgrav(E) for E   follows if these particles are near extremal, regardless of the light spec-   trum. δλ ( ) ∼ 2 dN D−4 grav E G N E dM M Thus, in this specific context we need to be more careful E dM   about which loop corrections we are considering. This is pre-  2 2 dN 2 D−6 δλgauge(E) ∼ e E dM Q (M)M . (A.6) cisely because the notion of gauge-gravity unification at weak E dM coupling has no obvious definition. We argue in Sect. 6.2 that the notion λ (E) ∼ λ (E) has the right behavior  gauge grav From these results we immediately see that δλgrav(E) ∼ in examples and has nice properties in effective field theory,  2 2 2 δλgauge(E) whenever Q (M) ∼ G N M /e , which is the but do not attempt to motivate it from first principles. case for a tower of approximately WGC-saturating particles. In this manner, one can rewrite all the arguments of Sects. 5 A.2 Higher-point loop amplitudes and 6 in terms of the modified λgauge(E) and λgrav(E) includ- ing contributions of particles with m E. Rather than focusing on loop corrections to the off-shell pho- Notice that because we only apply these formulas at ton and graviton propagators, we can consider the behavior E  , these modifications to the definition of the λ(E)s of on-shell S-matrix elements. We would like to argue that actually dominate over the formulas used elsewhere in the the cutoff  at which the loop expansion of the S-matrix paper. However, they are conceptually somewhat murkier; breaks down is the same as the cutoff we have inferred from as discussed in Sect. 3.1, for most of the paper we assume two-point functions. We will only discuss the case of a simple that we can study an effective theory at the scale E with- U(1) tower for brevity. out necessarily committing ourselves to assumptions about Consider, as an illustration, the 2 → 2 photon scatter- much heavier particles, which are difficult to distinguish from ing amplitude. In a D-dimensional theory this amplitude has higher-dimension operators. scaling dimension 4 − D. We show several contributions It is not surprising that our conclusions about the cut- to the amplitude in Fig. 5. At tree level, graviton exchange 2 off scale do not change when we include particles with produces an amplitude with parametric scaling E G N or E  m   in loops. The reason is that as E → , 2 2−D E MPl . At one loop, there is a contribution from charged almost all of the particles we should consider have masses particles of mass m and charge q.IfE m, this scales as below E, for which the formulas used in the bulk of the paper e4q4 E D−4, while if E  m, we obtain an Euler-Heisenberg apply. Furthermore, particles of mass near E can be treated as Lagrangian with four field strengths so the amplitude scales either heavy or light without any parametric difference in the as e4q4 E4m D−8. If we are interested in behavior near the formulas. We never consider particles parametrically heavier cutoff E ∼ , we need only consider the m  E case, and than , and so the modifications above become irrelevant at we have energies near the cutoff. Hence, all results in the paper based  4 D−4 4 4 D−4 5 M1−loop e E q e E Q(E) on ascertaining when λgauge(E) ∼ 1orλgrav(E) ∼ 1 should ∼ ∼ 2 2 be unaffected by the above modifications. Mtree E G N E G N One place where our arguments require more careful atten- E D−1 ∼ . (A.7) tion to the definition of the λ(E)s is in Sect. 6.2, when we 3(D−2)/2 eMPl studied the consequences of λgauge(E)  λgrav(E) at an energy E below the cutoff and of λgauge(E) ∼ λgrav(E) Hence we see that the energy E at which the 1-loop contri- over a range of energies E  E0. These arguments did not bution is of the same order as the tree-level contribution is assume that the λ(E)swereO(1). We can produce modified the familiar scale (3.12) versions of these arguments, making use of the form (A.6) 3(D−2) 1 2(D−1)  ∼ D−1 . where appropriate. However, there is an interesting concep- e MPl (A.8) tual difference. The assumption that λgauge(E)  λgrav(E) is valid at some energy E was argued in Sect. 6.2 to imply Do higher loops change the result? At two loops we have the existence of a particle with m  E obeying the original multiple contributions; from the lower diagrams in Fig. 5 we WGC (up to order-one factors). With the modified defini- see that tion, the assumption thatλ (E)  λ (E) now implies   2 gauge grav M ∼ 6 2D−8 6 + 4 2D−6 2 . that the original WGC is obeyed by a particle with mass 2−loop e E q e G N E q E  m  . Moreover, the ostensibly stronger statement (A.9) 123 337 Page 30 of 33 Eur. Phys. J. C (2018) 78 :337

Fig. 5 Contributions to four-photon scattering: at tree level, through graviton exchange at order G N ; at one loop, through electromagnetic inter- 4 6 4 actions with charged particles at order e ; at two loops, from one diagram of order e and one of order e G N

    2 Writing q6 ∼ Q(E)7 and q2 ∼ Q(E)6, we see that on the case where G is simple, but similar techniques can be the second term grows more quickly with energy, so that extended to arbitrary G. We say that a sublattice  ⊆ G is primitive if it is not a M 4 2D−6 ( )6 2−loop ∼ e G N E Q E multiple of another sublattice. Any sublattice can be written 2 Mtree E G N as a multiple of a primitive sublattice, hence it suffices to 6 − E classify primitive Weyl-invariant sublattices. ∼ e4 E2D 8 . (A.10) 6 3(D−2) We initially assume that G is simply connected. A lattice e MPl  is a Weyl-invariant sublattice of G if and only if Again, we read off that the scale where this becomes order-   ∨  ∨  one is the familiar scale (3.12). ∀Qα ∈ , Qα ·  ⊆ Z and (Qα · )Qα ⊆ , (B.1) Similar results can be extracted for higher-point diagrams.  ∨  2  For instance, the tree-level n-photon scattering amplitude where Qα := 2Qα/Qα is the coroot associated to Qα.The 2 n/2−1 through graviton exchange scales as E G N . The one- first condition requires that  is a sublattice of the weight loop n-photon scattering through charged particles scales lattice, whereas the second is equivalent to Weyl invariance, n D−n n n ∼ ( )n+1 ∼  as e E q . Again writing q Q E since the Weyl group reflection associated to Qα takes ( /( (D−2)/2))n+1  E eMPl , we derive the same cutoff for the    ∨   scale at which the one-loop contribution becomes compet- Q → Q − (Qα · Q)Qα. (B.2) itive with the tree-level result. Once again one can readily check that there is a class of 2-loop diagrams (containing Note that the coroots are the roots of the Langlands dual two closed loops of charged particles) that become of the group G∨, whose weight lattice is the dual of the G weight  same order at the same scale . Thus, we believe our results lattice and which is simple if and only if G is simple.  to be quite robust against the specific choices of the loop Thus, for each root Qα there is a positive integer kα such  ∨ diagrams we have considered to obtain them. that Qα · = kαZ. Since G is simple, the Weyl group relates any two roots of the same length, and we can have at most two distinct ks, klong and kshort. Moreover, the short roots B Weyl-invariant sublattices generate the root lattice, and likewise the short coroots Ð corresponding to the long roots Ð generate the coroot lattice, ˜ In this appendix, we classify Weyl invariant sublattices of implying that klong divides kshort. Therefore,  := /klong ˜ ˜ the weight lattice G of a compact Lie group G. We focus with klong = 1 and kshort = kshort/klong ∈ Z is a Weyl 123 Eur. Phys. J. C (2018) 78 :337 Page 31 of 33 337    =    invariant sublattice, implying that klong 1 if and only if  H+ := (a ,...,a ) a ∈{ , }, a ≡ , is primitive.25 1 n  i 0 1 i 0mod2 Using the second condition in (B.1), we find that  ⊆ i long (B.4)  for primitive , where long ⊆ G is the sublattice gen- erated by the long roots. Thus, primitive Weyl-invariant sub- where the latter two are equivalent for n = 2. lattices correspond to Weyl invariant subgroups of the finite We can summarize the above classification as follows. group Zˆ :=  / . Not all such subgroups corre- G G long For each H ⊆ Z(G), there is a primitive Weyl-invariant spond to primitive lattices, since  ⊆  does not imply long sublattice corresponding to the weight lattice of G/H.In k = 1,26 but every primitive Weyl-invariant sublattice long addition, for G not simply laced and G = US p (2k),  corresponds to a unique Weyl-invariant subgroup of Zˆ . long G is a primitive Weyl-invariant sublattice. Finally, for G = To compute Zˆ and the Weyl group action on it, we note G US p (2k) Ðforwhich is twice the weight lattice Ð that there is a simply laced Lie group Gˆ with root lattice long there is a primitive Weyl-invariant sublattice of the form  and weight lattice  , where G = Gˆ if and only if G long G (2Z)k ∪[(2Z)k + (1,...,1)] in a basis where the weight  =  ( ˆ ) ˆ is simply laced. Since long root G , ZG is the center lattice is Zk and the root lattice is the index-two sublat- ( ˆ ) ˆ  Z G of G. Precisely when G is not simply laced, the Weyl tice {(n ,...,n )| n =∼ 0mod2}. Thus, besides the ˆ 1 k i i group WG acts non-trivially on ZG , with the non-trivial action G/H weight lattices, there is exactly one additional primi- generated by the reflections associated to the short roots. This tive Weyl-invariant sublattice for each non-simply-laced G. φ : W → (Zˆ ) action is a homomorphism G Aut G , hence it is When G is not simply connected, the weight lattice G is  := / φ  ˜ encoded by WG WG ker . The elements of WG are a sublattice of the weight lattice  ˜ of the universal cover G, ˆ G outer automorphisms of G. and the smallest multiple of each primitive Weyl-invariant ˆ ˆ  It is now straightforward to compute G, ZG , and WG for   sublattice of G˜ which lies inside G is a primitive Weyl- G all simply connected, simple Lie groups : invariant sublattice of G . All primitive Weyl-invariant sub-  ˆ ˆ ˆ  lattices take this form, since any primitive sublattice of G G ZG G G ZG WG  ( ) Z ( ) ( )k Zk is a multiple of a primitive sublattice of G˜ . SU n n US p 2k US p 2 2 Sk Spin(4k) Z2 ⊕ Z2 Spin(4k + 1) Spin(4k) Z2 ⊕ Z2 Z2 Spin(4k + 2) Z4 Spin(4k + 3) Spin(4k + 2) Z4 Z2 E Z G SU(3) Z Z 6 3 2 3 2 References E7 Z2 F4 Spin(8) Z2 ⊕ Z2 S3 E8 −− 1. N. Arkani-Hamed, L. Motl, A. 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