Quantum gravitational theories enjoy a natural cutoff, the Planck scale Mpl. When curvature is of the order of Mpl2 the Einstein term in the action becomes strong and the quantum fluctuations render the classical picture of gravity inadequate. However, the actual scale in which the quantum gravity effective action can become invalid may be smaller. Indeed, it was argued in [1–4] that if there are light species of particles present in the theory, the consistency of black hole entropy with the effective field theory (EFT) description requires the breakdown of the classical picture at a lower scale called the “species scale”, Λs. It was proposed in [5] that the mechanism for this is that the higher-order corrections to the gravitational action become strong at the scale Λs instead of Mpl as one may naively expect. Even though the introduction of Λs was motivated by the appearance of the tower of light states, surprisingly the mass scale of this tower, m, does not seem to be directly captured by Λs. Indeed, the EFT does not directly encode the mass scale of the tower. For example, when the spectrum of light states is dominated by Kaluza–Klein (KK) modes coming from decompactifying a d-dimensional theory to D dimensions, the species scale corresponds to the Planck mass in the higher-dimensional theory ΛsD-2∼Mpld-2Rd-D, and not the KK mass scale mKK=1/R. In other words, the number of extra dimensions D-d is harder to detect from the low energy EFT. Note that a sufficiently high precision measurement of the gravitational amplitudes can always determine mKK. However, as explained in the Appendix, the corrections to the EFT coefficients at energy scale μ≪Λs that are controlled by mKK are always subleading, either to the massless threshold contribution or to the terms that are powers of the species scale. One way the lower-dimensional theory can find out about this scale is in the study of black holes. When black holes get smaller than R, the black hole becomes thermodynamically unstable due to the Gregory-Laflamme transition [6]: the larger black holes correspond to black branes wrapped around the extra dimensions, but the smaller ones are localized in the extra dimensions. This transition to a new black hole solution marks a new scale in the lower-dimensional theory, which we denote as ΛBH. A similar phenomenon occurs when we go to regions in field space with light fundamental strings. In this case, the species scale can be identified with the Hagedorn temperature which is proportional to the string scale Λs∼Ms. However, as we consider high-temperature black holes, there is evidence that before we get to the Hagedorn temperature, there is another scale where the black holes undergo a transition to a more stable state, the Horowitz-Polchinski solution, which we identify as ΛBH. These two types of towers are the only types expected in the large-distance limits in quantum gravity [7]. Motivated by these observations, we propose a new Swampland condition: For all quantum gravitational theories, there is a temperature scale, ΛBH, which is the lowest scale at which the black hole describable by the EFT becomes unstable due to a transition into a more stable saddle: ΛBH≲Λs≲Mpl. Moreover, this scale marks the existence of a phase transition from the black hole well-described by the low-energy EFT to a more stable solution which is not visible to that EFT. We explain how this is related to the fact that massive weakly coupled states, such as towers, are omitted from the dynamical spectrum of the low-energy EFT. In particular, we propose that in the context of type IIA string theory a transition analogous to the Horowitz-Polchinski transition continuously deforms to the Gregory-Laflamme transition as we move from weak string coupling to strong coupling.

The organization of the rest of this paper is as follows. In Sec. II, we review the species scale. In Sec. III, we review the Gregory-Laflamme transition and the Horowitz-Polchinski transition. In Sec. IV, we conclude by proposing a new Swampland condition.

In the EFT approach to quantum gravity in d≥4 spacetime dimensions, the leading terms are given by the Einstein action. The EFT is defined with a natural energy scale μ to describe backgrounds with curvature scales R∼μ2. Additionally, there will be higher-order terms which are suppressed for low-curvature regions. The EFT is expected to break down when these higher-order terms become of the same order as the leading term. Then, the coefficients of these higher-order terms, denoted as a(ϕi,μ), can depend on the massless scalar fields ϕi that parametrize the moduli space MQG as well as μ. Hence, the higher-order corrections appearing in the action can be written as Scorr,d⊃∫ddx-g[∑nan(ϕi,μ)On(R)],(1)where On(R) denotes 2n-dimensional operators.¹

Note that our convention for the coefficients of higher-derivative operators differs from the definition in [8] by power of Planck mass.

To ensure the definition (2) of Λs correctly captures the strong-coupling behavior of gravity, one can test it in top-down constructions in string theory. In the corners of the moduli space with a weakly coupled string, we expect the effects of quantum gravity to appear at string mass, i.e., Λs∼Ms∼exp(-ϕ/d-2) where -ϕ is the canonically normalized dilaton in the reduced Planck units. Furthermore, in M-theory corners where there is no weakly coupled string, the quantum gravity effects are expected to be controlled by a higher D-dimensional Planck mass, i.e., Λs∼Mpl,D∼exp(-ϕD-d(D-2)(d-2)) where -ϕ here is the canonically normalized volume modulus in the reduced Planck units. It has been conjectured that these two examples cover all weak-coupling limits in quantum gravity, which are each dubbed the emergent string and the decompactification limits, respectively [7]. The species scale defined in (2) was explicitly shown to satisfy these specific expectations in a diverse set of top-down examples [8].

The above discussion can be illustrated through the instructive example of type IIA supergravity in 10 dimensions [8,9], where the species scale will depend on the dilaton ϕ. The first nonzero higher-curvature correction to the supergravity action appears for the R4 term where the coefficient a4 is computed via the tree-level and one-loop four-graviton scattering amplitudes in type IIA and does not receive any further contributions [10]. Thus, we can identify the species scale with a4(ϕ)-1/6Mpl,10, Λs=1(2π)1/8[3ζ(3)π2e3ϕ/2+e-ϕ/2]-1/6Mpl,10.(3)In this case, the theory has two infinite-distance limits, corresponding to ϕ→±∞. On the one hand, when ϕ→-∞, or equivalently gs≫1, the species scale becomes Λs∼eϕ/62Mpl,10∝Mpl,11,where (Mpl,10/Mpl,11)8=2πe-22ϕ/3. This is precisely the limit at which the theory decompactifies to 11d M-theory. On the other hand, when ϕ→+∞, or equivalently gs≪1, the species scale becomes Λs∼e-ϕ/22Mpl,10∝Ms,where e-22ϕMpl,108∝Ms8. This is precisely the weak-coupling limit at which we have the emergent type IIA string. Therefore, this recovers the expected behavior in the infinite-distance limits of our moduli space. Moreover, note that as expected Λs(ϕ)≲Mpl,10 everywhere in the moduli space.

Many of the unique features of quantum gravity can be traced back to the existence and properties of black holes. In nongravitational theories, the coupling does not have to be related to mass, i.e., couplings are not directly correlated with mass. Therefore, arbitrarily massive states can be weakly coupled (or even free) without any issues. However, in a gravitational theory, every state is coupled to the graviton with a coupling proportional to its mass. Therefore, beyond a mass threshold, the localized states are expected to become strongly coupled, resulting in black holes.

Luckily, we expect the effective theory of gravity including the Einstein term to be a good description of large sized black holes where curvatures are sufficiently small near the horizon. However, as we decrease the size of a neutral black hole we expect the Hawking temperature to rise and beyond some temperature, the calculation of the free energy of the most stable saddle from the EFT becomes incorrect as the EFT does not detect the phase transitions of neutral black holes and also misses the perturbative instabilities associated to these black holes. Let us denote this limiting temperature by T=ΛBH, where the critical size black hole is RBH∼1/ΛBH. Clearly ΛBH≲Mpl. In fact, we would expect ΛBH≲Λs≲Mpl because the action becomes large near the horizon of black holes of size 1/Λs, so the EFT surely breaks down at Λs or below that scale.

From the form of the effective action one may think that ΛBH=Λs because one would naively expect that the only reason the effective action would not be describing the black hole correctly is that the higher-order terms become important. As we will argue, this is not true and generically ΛBH<Λs (Fig. 1). The mechanism for this is that the nature of the black hole described by the IR properties of the EFT is not the stable one; at smaller radii (i.e., higher temperatures) there is another class of solutions which the EFT misses that have lower free energy. So, the black holes undergo a transition at T=ΛBH to these more stable states.

We provide evidence for this in regions of weak-coupling in the QG landscape. These are the asymptotic regions of parameter space where we expect to have either a tower of light KK modes or light fundamental strings [7]. We now turn to these two cases and show that in both there is such a scale ΛBH≲Λs where the EFT incorrectly estimates the free energy of the more stable states to which the black holes with temperatures greater than ΛBH transition.

We note that there can be more than one phase transition in the spectrum of massive states in quantum gravity. An example of this is when there are compact dimensions with parametrically different length scales. In such cases we identify ΛBH as the lowest such scale. The structure of these phase transitions are highly constrained from the top-down perspective and can be motivated using bottom-up arguments in the weak-coupling limits [11]. In particular, the thermodynamics of weakly coupled particle towers at high temperatures has been studied in [12,13] using insights from black hole thermodynamics (see also [11]).

The above discussion in Sec. III suggests that in addition to Mpl and Λs in all the weak coupling regimes of MQG, there exists another scale ΛBH≲Λs≲Mpl which marks the smallest temperature, or the inverse radius of the largest black hole at which the EFT misses the most stable black hole and a phase transition to a more stable state occurs. In both cases the EFT of Einstein action (and higher-order corrections) misses these solutions as it does not incorporate the infinitely many light elements of the tower (the KK tower in the GL case and the light tower of strings manifested by Hagedorn behavior/thermal winding string condensate for the string tower in the HP case).

In particular, in the emergent string limit where Λs is identified with the Hagedorn temperature Λs=TH, the black hole description breaks down at the Horowitz–Polchinski temperature ΛBH=THP where the black holes transition into self-gravitating strings. In the decompactification limit where the d-dimensional quantum gravitational theory decompactifies to a D-dimensional theory, Λs is identified as Mpl,D while the d-dimensional black hole description breaks down at ΛBH∼mKK scale where the d-dimensional black hole transitions into a D-dimensional black hole. In both of these limits, the transition occurs at energies below the species scale, i.e., THP≲TH and mKK≲Mpl,D or equivalently in both cases ΛBH≲Λs. Different from the species scale, neither THP nor mKK is naturally encoded within the action of the EFT. Instead, it comes purely from studying phase transitions of black holes. In other words, this scale arises from the nonanalyticity of black hole entropy and free energy. The black hole transitions into a more stable physical state with lower free energy at temperature ΛBH.

Here we have mainly concentrated on the weak coupling regime where we have seen ΛBH is similar to the mass scale of the light tower. However, their ratio cannot be moduli independent. For example, consider the case where the large extra dimension is a circle in maximal supergravity. In this case, unlike ΛBH which is influenced by higher-derivative corrections, the mass of the KK tower is exactly 1/R and protected by supersymmetry. Due to the corrections to the Einstein action, the ratio of ΛBH/mKK is expected to depend on R. For example, for M-theory on S1, the R4 corrections [10] correct the entropy of the lower and the higher dimensional black holes [24] and the location where they meet which predicts the GL transition point changes. In this case, the 11d black hole has entropy, SM(11d)∼(Mpl,10r)8(rR)[1-13260941(Mpl,11r)6],whereas the 10d black hole in type IIA supergravity theory has entropy, SIIA(10d)∼(Mpl,10r)8[1-804203a4(ϕ)(Mpl,10r)6],where a4(ϕ) is as defined earlier. A rough estimate of ΛBH in the decompactification limit of the type IIA theory can be found by solving for r such that SIIA(10d)=SM(11d). Hence, in the large-volume limit, we obtain ΛBHmKK∝1+a·e-42ϕ+…,(12)where to the leading order a≈269.4. Therefore, the ratio ΛBH/mKK becomes ϕ-dependent at large ϕ. Thus, ΛBH cannot be identified as the mass scale of the lightest state for all moduli.

Note that the above observations hold true in both the weak-coupling regime of the fundamental string and the large-volume regime of compactifications. Consequentially, the notion of this phase transition does not strictly live in the infinite-distance limits of MQG, and ΛBH is expected to be well-defined across the moduli space. Thus, ΛBH should be globally defined in MQG as the temperature at which the Schwarzschild black hole becomes unstable due to the existence of a more stable saddle whose free energy is not seen by the EFT. Therefore, all of this motivates us to propose the following Swampland conjecture.

Any consistent EFT description of d-dimensional quantum gravity must exhibit three scales across its moduli space MQG: 1) the d-dimensional Planck scale, Mpl,d which controls the strength of the Einstein term; 2) the species scale, Λs, where the higher-order gravitational corrections become important; and 3) the black hole scale, ΛBH, where at this temperature, the black hole predicted by EFT undergoes a phase transition to a more stable solution. Furthermore, ΛBH≲Λs≲Mpl,d everywhere in MQG and ΛBH approaches the mass scale of the lightest tower at large distances in field space.

It is amusing to consider the case of type IIA in 10 dimensions for which the moduli space is parametrized by the string coupling (or equivalently the radius of the extra circle). In this case we would expect a behavior shown in Fig. 2. It is interesting to note that in this case we expect the GL transition to be connected continuously to a transition analogous to the HP transition.³

A similar, but distinct, story unfolds in (1+1)d maximally supersymmetric Yang-Mills theory compactified on a generic tori, Tτ2, with appropriate boundary conditions imposed on the fermions [25]. In particular, by varying the complex parameter τ, the analog of the GL transition in gauge theory can then be continuously mapped to the confinement/deconfinement transition.

Note that in discussions regarding the connection between the lightest tower of states and the species scale, the tower is assumed to be weakly coupled. This is because it is not clear how strongly coupled states would contribute to the one-loop running of the Newton’s constant which is calculated perturbatively and are likely irrelevant [8]. An example of such a tower is when a sector localized on a brane has a tower of strongly interacting states with vanishing masses (e.g., tensionless strings). In the limit where the internal space decompactifies, these degrees of freedom decouple from gravity and could form a conformal field theory. Interestingly, in this example, ΛBH which is set by the Gregory-Laflamme scale is also insensitive to this nongravitational tower. This is because the tower is localized in the internal space and are decoupled from the higher dimensional gravity and does not significantly influence the entropies of higher- or lower-dimensional black holes and therefore does not affect the transition temperature.

One could instead hope that mlightest is to be identified with the mass of the lightest tower that becomes light and weakly-coupled in some asymptotic regime of the moduli space. However, as we just explained, the ratio ΛBH/mlightest receives moduli dependent corrections as we move away from the infinite distance limits.

Unlike the Λs where analytic computations are possible in some cases [5,8], it seems that ΛBH is much harder to compute explicitly except in the infinite distance limits. Using the fact that asymptotically ΛBH∼mlightest, it implies from the work [26,27] that at least asymptotically, ∇lnΛs·∇lnΛBH=1d-2.

Since we have a well defined notion of ΛBH and Λs everywhere in moduli space, it is natural to ask if this relation can hold in the interior of the moduli space as well.⁴

Extending the equality of [26,27] into the interior of moduli space has been attempted in [28] with specially chosen BPS scales for 5d N=1 theories. However, these (Dirac-paired) BPS scales are distinct from the scales of interest in this present paper.

In [8,29] evidence was provided for the bound ∇lnΛs·∇lnΛs≲1d-2.This naturally suggests that perhaps the correct relation involving both Λs and ΛBH is also an inequality, instead of the equality which holds only asymptotically, namely ∇lnΛs·∇lnΛBH≲1d-2.(13)

The existence of critical points for Λs would be compatible with this. In particular, this relation, in the large-volume regime of the type IIA theory according to the perturbative corrections given in (12), can be computed as ∇lnΛs·∇lnΛBH=18(1-e-22ϕ+a·e-42ϕ+…)≤18=1d-2,(14)providing an example of this conjecture. Here, note that the perturbative corrections to ΛBH, as we move away from asymptotic limits of moduli space, is parametrically smaller than that of the species scale. Hence, it contributes a subleading effect in the above inequality. In fact, this conclusion remains valid upon compactification of M-theory on Tp, in which case the corrections to ∇lnΛs and ∇lnΛBH still scale as ∼(mKK/Mpl,11)3 and ∼(mKK/Mpl,11)6, respectively. Therefore, as long as |∇lnΛs| in these theories remains below 1/d-2, the inequality (13) is satisfied. It would be important to develop effective tools to compute ΛBH in the interior of moduli space and to investigate the validity of this inequality.

One may ask, what are these three quantum gravity scales in our Universe. For the Dark Dimension scenario, motivated by the Swampland program, these three scales end up being vastly different (see Ref. [30] for a review) and are given by ΛBH∼.01  eV,Λs∼109  GeV,Mpl∼2.4×1018  GeV,which in Planck units are related to the Dark energy Λ∼10-122 by ΛBH∼Λ3/12,Λs∼Λ1/12,Mpl∼Λ0,(and the missing power of Λ above is close to the weak scale ∼Λ2/12 which is also partially explained in the Dark Dimension scenario).