Black Hole Extremality in Nonlinear Electrodynamics: A Lesson for Weak Gravity and Festina Lente Bounds

We study black hole extremality in nonlinear electrodynamics motivated by the Weak Gravity Conjecture (WGC) and the Festina Lente (FL) bound. For illustration, we consider the Euler-Heisenberg model and the Dirac-Born-Infeld model in asymptotically flat spacetime, de Sitter spacetime, and anti-de Sitter spacetime. We find that in all cases the extremal condition enjoys a certain monotonicity expected by the WGC. This provides evidence for the conjecture beyond the leading order corrections to the Einstein-Maxwell theory. We also study how light charged particles modify the mass-charge relation of Nariai black holes in de Sitter spacetime and discuss possible implications for the FL bound. Besides, we point out an interesting similarity between our black hole analysis and gravitational positivity bounds on scattering amplitudes.


Introduction
Thermodynamic properties of black holes play a central role in the study of quantum gravity. In the context of the Swampland Program [1], more specifically, various thought experiments on charged black holes have been performed to explore possible quantum gravity constraints on the charged particle spectrum. See, e.g., [2][3][4] for review articles.
A famous example for such swampland conditions is the Weak Gravity Conjecture (WGC) [5], which predicts existence of a charged state whose mass-to-charge ratio is smaller than unity in an appropriate unit. The conjecture is equivalent to requiring that all the black holes have to decay unless they are protected by (super)symmetries. Applying this to macroscopic extremal black holes, which have zero temperature and thus cannot decay through the standard Hawking radiation mechanism, implies existence of the WGC state.
More recently, an interesting Swampland condition called the Festina Lente (FL) bound was proposed based on thought experiments about charged black holes in de Sitter spacetime [35]. As depicted in Fig. 2, black holes in de Sitter space have an upper bound on the mass for a given charge to fit inside the cosmological horizon. By postulating that the black holes saturating the bound (Nariai black holes) decay into neutral black holes, rather than naked singularities, a lower bound m ≳ √ gqM Pl H on the mass of charged Figure 3: In flat space, the WGC implies that the mass-to-charge ratio of extremal black holes are lowered by quantum corrections and the correction is monotonic with respect to the charge Q. We expect similar monotonicity in dS and AdS as well 1 .
particles was proposed [35], where m and q are the mass and charge of the particle, and g, M Pl and H are the gauge coupling, the reduced Planck mass and the Hubble constant.
In contrast to the WGC, the FL bound has to be satisfied by all the charged particles to avoid fast discharge processes that lead to naked singularities. See, e.g., [36][37][38] for its phenomenological implications.
The other motivation is in the FL bound. The FL bound was proposed by postulating that Nariai black holes do not decay into naked singularities. While the original argument about discharge processes through the Schwinger mechanism highly depend on the charged black hole spectrum, light charged particles may modify the black hole solutions. Thus, it is of great interests how light charged particles saturating the FL bound modify the black hole solutions and more specifically the Nariai curve. Based on this motivation, we use the Euler-Heisenberg model to study how light charged particles modify the Nariai curve. Interestingly, we find for magnetic black holes that the Nariai curve is flattened in the presence of light charged particles. This motivates further studies of the Schwinger mechanism of Nariai black holes and sharpening the FL bound. This paper is organized as follows: In Sec. 2, we first review the charged black hole solutions of the Einstein-Maxwell theory and introduce a general procedure to calculate the mass-to-charge ratio of extremal and Nariai black holes in nonlinear electrodynamics. In Sec. 3, we study the extremal curve in the EH model and the DBI model in asymptotically flat spacetime. In Sec. 4 we extend the analysis to the nonzero cosmological constant. In particular, for asymptotically de Sitter case, we study how the Nariai curve is modified by light charged particles and discuss its possible implications for the FL bound. In Sec. 5 we discuss similarity between our black hole analysis and gravitational positivity bounds on scattering amplitudes. Then we conclude the paper in Sec. 6 with an outlook for future work. Some technicalities and our notation are collected in Appendices.

Generality
In this section we review charged black hole solutions in the Einstein-Maxwell theory and then present general construction of charged black holes in nonlinear electrodynamics.

Einstein-Maxwell theory
Consider the Einstein-Maxwell theory in four dimensions with a cosmological constant Λ: where g e is the (electric) gauge coupling constant and the Newton constant G is related to the reduced Planck mass M Pl as G = 1 8πM 2

Pl
. We use G and M Pl interchangeably depending on the context for notational simplicity. The Einstein equation is where G µν := R µν − 1 2 g µν R is the Einstein tensor and T µν is the energy-momentum tensor of the matter sector. In the Einstein-Maxwell theory, the energy-momentum tensor reads The Maxwell equation and the Bianchi identity are where the dual field strength is defined byF µν := 1 2 ϵ µνρσ F ρσ . ϵ µνρσ is the Levi-Civita tensor on the curved spacetime normalized such that ϵ 0123 = −(−g) −1/2 . In other words, it is related to the Levi-Civita symbol ε µνρσ (ε 0123 = −1 ) as ϵ µνρσ = (−g) −1/2 ε µνρσ . For details of the anti-symmetric tensors and symbols, see Appendix B.
We consider static and spherically symmetric black holes with either electric or magnetic charges. Let us employ the following ansatz of the metric in the polar coordinates 2 : Then, for magnetic black holes, the Gauss law says where n is the quantized integer charge of the black hole. Without loss of generality, we assume n ≥ 0 throughout the paper. Note that this field configuration is not modified by higher derivative corrections. The magnetic flux on the two-dimensional sphere reads On the other hand, for electric black holes, the Maxwell equation says n r 2 dr ∧ dt , (2.8) where k e = g 2 e /4π is the (electric) Coulomb constant and n is the quantized integer charge. The electric flux on the two-dimensional sphere reads where * 4 is the four-dimensional Hodge star. More explicitly, In contrast to the magnetic case, the field configuration (2.8) is modified in the nonlinear electrodynamics, essentially because definition (2.9) of the electric charge is modified in the presence of higher derivative corrections.
It is also instructive to note 1 4g 2 e F µν F µν = − g 2 e n 2 32π 2 r 4 = − k e n 2 8πr 4 for electric black holes (2.11) and 1 4g 2 e F µν F µν = g 2 m n 2 32π 2 r 4 = k m n 2 8πr 4 for magnetic black holes, (2.12) which manifests the electric-magnetic duality. Here we introduced the magnetic gauge coupling g m = 2π ge and the magnetic Coulomb constant k m = g 2 Especially in figures, we sometimes parameterize the black hole charge in the unit of gauge couplings as Q e := g e n , Q m := g m n . (2.13) In this paper, we discuss electric and magnetic black holes separately, so that we suppress the subscripts e, m of g e,m , k e,m , and Q e,m to simplify the notation as g, k, and Q, as long as it is obvious from the context.
With the above gauge field configurations, the Einstein equation simply reduces to where the Euler operator θ r := r ∂ ∂r counts the exponent of r. f (r) is then determined as where the integration constant M is identified with the black hole mass.

Black hole extremality
Location of the black hole horizon and the cosmological horizon is determined by the equation f (r) = 0. This means that when a horizon is located at r = r H , the black hole mass M is written as a function of r H and n: When there exist multiple horizons, the extremal condition for horizon degeneracy reads Asymptotically flat geometry. First, let us consider the asymptotically flat geometry, i.e., Λ = 0. Generically, there are two positive real solutions for f (r) = 0 corresponding to the Cauchy horizon and the event horizon. When the two horizons degenerate, its location r H is determined by (2.19) Asymptotically de Sitter geometry. Next we consider asymptotically de Sitter (dS) geometry, i.e., Λ > 0. Generically, f (r) = 0 has three positive real solutions corresponding to the two black hole horizons and the cosmological horizon. The condition for the horizon degeneracy reads For n < (4GkΛ) −1/2 , it has two positive real solutions

Nonlinear electrodynamics
Let us move on to black holes in nonlinear electrodynamics with the following action 3 : where L(F, G) is a function of F and G defined by To separate the cosmological constant from the gauge field sector, we assume L(0, 0) = 0. We also assume that L(F, G) is analytic at F = G = 0 and the Lagrangian is parity invariant. The Einstein equation reads with the energy-momentum tensor, The equation of motion for the Maxwell field and the Bianchi identity are

Magnetic black holes
We begin by magnetic black holes. Again we employ the following ansatz of the metric: As we mentioned in the previous subsection, the gauge field configuration is unchanged from the Einstein-Maxwell case due to the Gauss law and the charge quantization: where n is the quantized integer charge. We then have Then, the Einstein equation reduces to which determines f (r) as Here the integration constant M is again interpreted as the black hole mass. Note that in the large r regime r → ∞, the last term is subdominant compared to the first three terms because we assumed that L(F, G) is analytic and vanishes at F = G = 0.
To identify the extremal condition, it is convenient to express the black hole mass as a function of the horizon radius r H and the charge n: Then, the condition for horizon degeneracy reads

Electric black holes
Legendre transformation. Next we consider electric black holes. In contrast to the magnetic case, higher derivative operators modify definition of the electric charge and the Gauss law accordingly. To handle this modification systematically, it is convenient to perform a Legendre transformation of the form, An explicit form of the two-form field P µν conjugate to F µν is so that the equation of motion (2.28) corresponding to the modified Gauss law is simply We find that in terms of P µν the equation of motion takes the same form as the standard Gauss law and also it does not depend on the choice of the function L(F, Q) explicitly. This is why the Legendre transformation (2.36) makes the analysis more tractable. Also, in terms of L(F, G), the Hamiltonian type operator H is given by Inverse Legendre transformation. In our analysis, we need to perform the inverse Legendre transformation afterwards. By analogy with F and G, let us introduce If we think of H as a function of P and Q, F µν is given by and correspondingly the Lagrangian L reads We can also write the energy-momentum tensor (2.27) in terms of H, P, and Q as where we used the following identities in four dimensions: Black hole solutions. Now we are ready to construct black hole solutions. We employ the static and spherically symmetric ansatz (2.29) of the metric and solve the modified Maxwell equations (2.28) that are written in terms of P µν and H as follows: For electric black holes, P µν is specified by Eq. (2.45) as with a quantized integer charge n. Correspondingly, we have Nonzero components of the energy-momentum tensor (2.43) are . (2.49) Then, the Einstein equation reduces to Here the integration constant M is interpreted as the black hole mass. As before, we write the black hole mass as a function of the horizon radius r H , which gives the following condition for horizon degeneracy: Note that the above results for electric black holes are reproduced by a simple replacement , 0) in the corresponding magnetic results. In particular the charge relation is Q m = g m n → Q e = g e n.

Asymptotically flat black holes in nonlinear electrodynamics
In this section we study the extremal condition of asymptotically flat black holes in the nonlinear electrodynamics. For illustration, we consider the Euler-Heisenberg model and the Dirac-Born-Infeld model, and confirm the monotonicity expected by the WGC.

Euler-Heisenberg black holes
The Euler-Heisenberg (EH) model [83][84][85] is an effective field theory (EFT) after integrating out a minimally coupled charged particle at the one-loop level. It is applicable when the electromagnetic fields are nearly constant at the Compton scale of the charged particle. See the last paragraph of this subsection for validity of this approximation. Also except in Sec. 5 we assume that gravitational corrections are subdominant. A concrete form of the effective Lagrangian after integrating out a charged scalar/fermion is 4 where g e is the electric gauge coupling and m is the mass of the electrically charged particle integrated out. We also introduced To ignore the Schwinger effect [85] and work with static black hole solutions, our EH analysis focuses on magnetic black holes, leaving electric black holes for future work.
As we discussed in the previous section, we have G = 0 for magnetic black holes, so that what we practically need is a concrete form of the function L(F, 0). Noticing where we rescaled the integration variable as s → s √ 2F . Note that its explicit form up to four-derivatives is given by The mass-to-charge ratio of extremal black holes for m = 10 −5 M Pl and g e = 1 are given in Fig. 5, where the upper/lower panel shows the result for the scalar/fermion case. The blue and orange curves are for the full EH analysis and the four-derivative model (i.e., Eq. (3.5) with truncation of the O(F 3 ) terms), respectively. We confirm the expected monotonicity for both cases. More interestingly, we find that the correction to the extremal condition in the EH model is milder than the four-derivative model, which offers a concept of ultraviolet (UV) completion in the context of black hole extremality. Also note that the orange curve damps rapidly around Q = g m n ∼ 10 10 , beyond which the four-derivative truncation does not work as we discuss at the end of the subsection. It is also useful to notice that the Cauchy horizon disappears at Q ∼ 10 10 for extremal black holes in the model after four-derivative truncation. Therefore, there is no solution for the horizon It is also useful to see g e -and m-dependence of the extremal condition. Fig. 6 shows the g e -dependence of the mass-to-charge ratio of extremal black holes for m = 10 −5 M Pl . The upper/lower panel is for the scalar/fermion loop. The correction is larger for a larger electric gauge coupling. Fig. 7 shows the m-dependence for g e = 1. Again, the upper/lower panel is for the scalar/fermion loop. The correction is larger for a smaller mass, but the tilt in the small Q region is insensitive to the mass, since the logarithmic behavior is associated with running of the gauge coupling and small Q corresponds to high energy.
Validity of the Euler-Heisenberg EFT. To close the EH analysis, we elaborate on for which charge range the use of the EH Lagrangian (3.1) is justified and the full order analysis without four-derivative truncation is needed. First, when deriving the EH Lagrangian, the electromagnetic field F µν is assumed to be nearly constant at the Compton scale of the charged particle integrated out, which is schematically given by g m n/M Pl ∼ Q/M Pl is larger than the Compton length ∼ 1/m of the charged particle, i.e., Q ≫ M Pl /m. Also, in the EH model, higher derivative corrections appear schematically in the form, F(F/m 4 ) n (n = 1, 2, . . .). Therefore, the derivative expansion does not work and the full-order analysis of the EH model is required when |F| ≳ m 4 . For extremal magnetic black holes, this condition reads n 2 /r 4 H ≳ m −4 , which is equivalent to Q ≲ g e (M Pl /m) 2 . See Fig. 8 for a summary of the paragraph.

DBI black holes
The Dirac-Born-Infeld (DBI) model was first introduced to remove divergence associated with the self-energy of charged particles [86,87]. It also provides a low-energy EFT of D-branes, which provides an illustrative example for the nonlinear electrodynamics with a well-motivated UV origin. In four dimensions, a concrete form of the Lagrangian is  where Λ DBI is the brane tension that characterizes the nonlinearity. Note that its derivative expansion up to four-derivatives is which we use when comparing our results with the four-derivative analysis in the literature.
In particular, H(x, 0) = −L(−x, 0) reflects the electromagnetic duality of the DBI model. As we mentioned at the end of Sec. 2.2.2, this shows that the DBI analysis for electric black holes is essentially the same as the magnetic black holes, so that we focus on the magnetic case in the following.
Black hole extremality. The algorithm to identify the extremal condition is the same as previous examples. First, we solve the condition (2.35) for horizon degeneracy. In the DBI model, the condition reads where we used Q = g m n = 2πn/g e to parameterize the magnetic charge. Also, we kept the cosmological constant Λ general for later reference. For the asymptotically flat case Λ = 0, the solution for Eq. (3.9) is given by (3.10) This shows that horizon degeneracy occurs only for Q ≥ (2GΛ 2 DBI ) −1 . More comments on this point will be given shortly in the last paragraph of the subsection.
Next we evaluate the mass of the extremal black hole for a given charge n using the mass formula (2.34). In the DBI model, the mass formula reads where 2 F 1 is the Gauss hypergeometric function. Again we kept the cosmological constant Λ general for later reference. Substituting Eq. (3.10) and Λ = 0 into Eq. (3.11) gives the extremal condition. Fig. 9 shows the extremal curve of the DBI model with Λ DBI = 10 −5 M Pl (blue curve), where we confirm the expected monotonicity. Also, we find that the correction to the extremal condition in the DBI model is milder than the four-derivative model (orange curve). Similarly to the EH case, this shows that the DBI model provides a UV completion of the four-derivative model.
Phase structure of horizons. As we mentioned, Eq. (3.10) shows that the horizon degeneracy does not occur for Q ≤ (2GΛ 2 DBI ) −1 , which is in sharp contrast to black holes in the Einstein-Maxwell theory. To elaborate on this feature, it is convenient to take a closer look at the shape of the function f (r) defining the horizon. In particular, it turns out that the sign of f (+0) is crucial. More explicitly, f (r) behaves in the limit r → +0 as where we introduced the critical mass for a given charge Q by In particular, there is no regime with two horizons. In the small Q region and above the critical curve, the horizon structure of the DBI black holes is similar to the Schwarzschild one. To summarize this feature, it is useful to draw a phase diagram given in Fig. 11. There, we find that the extremal curve and the critical curve intersect at Q = (2GΛ 2 DBI ) −1 .

Black holes in dS and AdS
In this section we extend the flat space analysis of the previous section to black holes with a nonzero cosmological constant. We discuss asymptotically de Sitter (dS) spacetime in Sec. 4.1 and then asymptotically anti-de Sitter (AdS) spacetime in Sec. 4.2.

de Sitter black holes
Our task here is basically the same as the flat space case: We use the same algorithm to identify the condition ( Figure 11: Phase diagram of DBI black holes in flat spacetime for Λ DBI = 10 −7 M Pl black hole mass using the mass formula (2.34). A new feature here is that there exists a cosmological horizon in addition to black hole horizons, so that our focus will be more on how the Nariai curve is modified in the nonlinear electrodynamics.

Euler-Heisenberg black holes in de Sitter
We begin by the EH model with a positive cosmological constant Λ > 0. To avoid Schwinger effects, we consider magnetic black holes as before. Generically there exist two solutions for the condition (2.35) for horizon degeneracy, where the smaller/larger horizon radius corresponds to the extremal/Nariai black hole. We solve the condition (2.35) numerically and then evaluate the corresponding mass numerically using the formula (2.34), drawing the extremal and Nariai curves.
For illustration, we set the cosmological constant as Λ = (10 −15 M Pl ) 2 . Fig. 12 shows the mass-to-charge ratio of extremal black holes in the EH model for scalar/fermion loop (upper/lower) with the mass m = 10 −5 M Pl (blue curve) in comparison with the Einstein-Maxwell theory (green curve) and the four-derivative model (orange curve). Recall that the charge-to-mass ratio of extremal black holes in the Einstein-Maxwell theory is not constant in dS because of the curvature effects. Therefore, what we expect is the monotonicity of the correction to the extremal condition, rather than the extremal curve itself (see also Fig. 3). Indeed we confirm the monotonicity of the correction in the EH model (and also in the four-derivative model). Besides, the correction in the EH model is milder than the four-derivative model, similarly to the flat space case.
In Fig. 13, we show how the "shark fin" shape surrounded by the extremal curve and the Nariai curve is modified in the EH model for (orange curve), 5 in comparison to the Einstein-Maxwell theory (dashed curve). We find that the Nariai curve is flattened by the nonlinear effects and this correction is larger for the lighter charged particle. We also find that this feature is more significant for the fermion loop than the scalar loop. Note that if we make the gauge coupling g e smaller, the shark fin shape approaches to the Einstein-Maxwell one similarly to the flat spacetime case.
validity of Euler-Heisenberg model in de Sitter. Similar to the flat case, we go into detail on for which charge range the use of the EH Lagrangian (3.1) is justified and the full order analysis is needed in de Sitter case. First of all, in EH model charged particles are not dynamical. In de Sitter space time this condition means that charged particle are not excited by de Sitter background, which reads m > √ Λ. For extremal black holes, the rest conditions are the same as the flat space case, but for Nariai black holes, the valid energy region is different. First, in EH model, the electromagnetic field F µν is assumed to 5 The EH model is applicable when the Compton length of the charged particle is smaller than the Hubble scale m ≳ Λ 1/2 . Even though m = 10 −15 M Pl is marginal to this bound, we show the result for illustration.  be nearly constant at the Compton scale of the charged particle integrated out, which is given by 1 m ∂F ∂r ≪ |F |. For Nariai magnetic black holes, this condition is satisfied when the compton length of charged particle integrated out is smaller than de Sitter radius. Also, we need full order analysis without four derivative truncation when the inequality |F| ≳ m 4 is satisfied. For Nariai magnetic black holes, this condition reads n 2 Λ 2 ≳ m −4 , which is equivalent to Q ≳ m 2 geΛ . See Fig. 14   Implications for the FL bound. We close our EH analysis on de Sitter by discussing possible implications for the Festina Lente (FL) bound [35], which was originally proposed based on thought experiments about decay of Nariai black holes: In the presence of electrically charged particles, electric Nariai black holes decay by emitting radiation due to Schwinger effects. If the discharge process is too fast compared to the energy loss, the black hole may decay into a naked singularity outside the shark fin. By postulating that this process is prohibited, Ref. [35] proposed a lower bound on the mass of charged particles that has to be satisfied by all charged particles, where the Hubble constant H is related to the positive cosmological constant Λ as H ∼ Λ 1/2 and q is the integer charge of the particle. Now let us recall our results showing that light electrically charged particles may flatten the Nariai curve of magnetic black holes by the nonlinear effects of the EH model. We expect that a similar phenomenon will happen for electric black holes too. If it is indeed the case, we need to revisit the original FL argument about the Nariai black hole decay based on the black hole spectrum modified by backreaction from the light charged particles prohibited by the bound. We leave this issue for future work.
A more direct relation of our analysis and the FL bound can be found along the line of the argument in Ref. [36,88] which fixed the O(1) coefficient of the bound as we summarize below: In four dimensions, the electric WGC for a unit charge requires existence of a charged particle satisfying the bound, with an O(1) coefficient α. Since the WGC particle has to satisfy the FL bound, combining Eq. (4.2) and Eq. (4.3) with q = 1 gives It is similar to the condition that magnetic black holes with a unit charge can exist in de Sitter spacetime, which is given by in the Einstein-Maxwell theory. Ref. [36,88] fixed the O(1) coefficient of the FL bound as α = 6 1/4 by postulating that the two conditions (4.4)-(4.5) match with each other. Now let us recall our analysis showing that the shark fin shape of magnetic black hole, especially the maximum charge of magnetic black holes, is modified in the presence of electrically charged particles. Therefore, the bound (4.5) is modified in the presence of light charged particles prohibited by the FL bound and therefore the O(1) coefficient may be modified by such backreaction. It would be interesting to explore further along the line of this consideration to sharpen the FL bound.

DBI black holes in de Sitter
Next we consider the DBI model with a positive cosmological constant Λ > 0. Thanks to the electromagnetic duality of the DBI model, the analysis of electric black holes is essentially the same as the magnetic case, so that we again focus on magnetic black holes. First, the condition (3.9) for horizon degeneracy generically has two positive real solutions: where the plus/minus sign corresponds to the Nariai/extremal condition. Note that the solution corresponding to the extremal condition can be used also in the AdS analysis, whereas that for the Nariai condition becomes complex when the cosmological constant is negative. Substituting it into Eq. (3.11) gives the mass-charge relation of the Nariai/extremal black holes. Also, similarly to the flat space case, there exists a critical mass (3.13) beyond which the number of horizons changes because of the sign flip of f (+0).
For illustration, we again consider Λ = (10 −15 M Pl ) 2 . First, Fig. 15 shows the massto-charge ratio of extremal black holes in the DBI model with the mass m = 10 −5 M Pl . Similarly to the EH case, we confirm the monotonicity of the correction to the extremal condition and also find that the correction in the DBI model is milder than the fourderivative model.    critical curves, respectively. Each black hole on the critical curve has the minimum mass for the given charge.

Anti-de Sitter black holes
Finally, we consider black holes in AdS. Since there is no cosmological horizon and therefore there is no Nariai black hole, the results for AdS are qualitatively similar to the flat space case. The algorithm to derive the extremal condition is the same as previous examples. Therefore we just provide final plots for the extremal condition. For illustration, we set the cosmological constant as Λ = −(10 −15 M Pl ) 2 . Fig. 17 and Fig. 18 show the mass-to-charge ratio of the extremal black holes in the EH model and the DBI model, respectively. There we confirm the monotonicity of the correction to the extremal condition.

Black hole analogue of gravitational positivity
In this section we point out an interesting similarity between our black hole analysis and positivity bounds on scattering amplitudes [39,40], especially in gravity theories. In Sec. 3.1 we evaluated the mass-to-charge ratio, of extremal magnetic black holes in the Euler-Heisenberg model with Λ = 0. While the Einstein-Maxwell theory provides a good approximation as long as the black hole charge is large enough, the nonlinearity becomes important once the charge becomes as small as the critical value Q * ∼ g e (M Pl /m) 2 (see also Fig. 8). For sufficiently small Q ≪ Q * , the correction to the mass-to-charge ratio (5.1) in the EH model scales logarithmically as Here and in what follows we do not care about O(1) factors, even though we care the sign. Physically, the logarithmic behavior corresponds to the running of the gauge coupling induced by the charged particle. More quantitatively, the energy scale E associated with the electromagnetic fields near the horizon reads where we used Q = g m n, g m ∼ 1/g e , and r H ∼ Q/M Pl . Hence we can think of Q −1/2 as a measure of the energy scale in the Planck unit.
While the EH model captures non-gravitational corrections to the Einstein-Maxwell theory from charged particles, there exist gravitational corrections as well. For example, four-derivative operators schematically of the form F 2 R are generated at one loop. Their contribution to the extremal condition is (see, e.g., Ref. [18]) where we emphasize that the gravitational correction is positive. In the spirit of the WGC, let us postulate that the mass-to-charge ratio of extremal black holes has to be smaller than unity. Then, we obtain the following bound 6 : (5.5) 6 Here we implicitly assumed that the charged particle satisfies the WGC bound, having QED in mind.
Otherwise, the gravitational correction dominates over the non-gravitational one and then the total correction to the mass-to-charge ratio of extremal black holes become positive even in the large Q region, where the four-derivative model is applicable.
Interestingly, the bound can be rephrased in terms of the energy scale (5.3) as E ≲ g e mM Pl , (5.6) which is reminiscent of the cutoff energy scale suggested by gravitational positivity bounds in QED [89]. 7 Caveat. While the above observation is interesting and suggestive, a caveat is needed: When the bounds (5.5)-(5.6) are saturated, the black hole radius is comparable to the Compton length of the charged particle, r H ∼ Q/M Pl ≃ g e /m, so that we cannot justify the use of the EH model. However, we expect that the logarithmic behavior (5.2) still holds even in this regime because it is related to the running of the gauge coupling. It would be desirable to reformulate our analysis in terms of running couplings from the Wilsonian EFT perspective, which we leave for future work.
Interpretation. Given this caveat, we interpret that requirement of the WGC type bound µ ≤ 1 for extremal black holes with arbitrary charge Q provides the black hole analogue of improved positivity bounds [90][91][92]: To explain this, it is convenient to compare our EH analysis with the four-derivative analysis in the literature. As we explained in Sec. 3.1, the four-derivative analysis is valid only for sufficiently large charge Q ≫ Q * or in other words only in the low-energy limit. On the other hand, our EH analysis is applicable even for smaller charge Q ≲ Q * , so that we can test the WGC type inequality ∆µ = ∆µ EH + ∆µ grav ≤ 0 for a wider range of Q. If we employ the WGC type bound ∆µ ≤ 0 as a criterion for consistent gravity theories, one may ask up to which value of Q the bound is satisfied and how to modify the theory such that ∆µ ≤ 0 is satisfied for all Q. Since Q is associated with energy, this is equivalent to identifying the cutoff scale and asking how to UV complete the theory. This is the same philosophy as the improved positivity bounds, which provide an energy-scale-dependent bound useful for identifying the cutoff scale. Indeed, our EH analysis implies the same cutoff scale as gravitational positivity in QED.

Conclusion
In this paper, we studied the extremal condition of charged black holes in nonlinear electrodynamics beyond the four-derivative corrections. More specifically, we considered the Euler-Heisenberg model and the DBI model in asymptotically flat spacetime, de Sitter spacetime, and anti-de Sitter spacetime. In all cases, we confirmed the monotonicity of the correction to the mass-to-charge ratio of extremal black holes, which supports the black hole version of the Weak Gravity Conjecture. Our analysis took into account all orders in the derivative expansion, so that its applicability is not limited to the large black hole limit or in other words the low-energy limit. Indeed, we found that the corrections in the 7 To be precise, positivity bounds in the presence of gravity hold only approximately, at least in the present technology. See [18,19,[41][42][43][44][45][46][47][48][49][50][51][52] for recent discussion. The cutoff scale E ≲ √ gemM Pl follows under the assumption that the allowed negativity does not dominate over the negative gravitational contribution. Euler-Heisenberg model and the DBI model are milder than the four-derivative model, which offers a concept of the UV completion in the black hole context.
Our analysis for asymptotically de Sitter black holes is relevant to the Festina Lente bound too. We used the Euler-Heisenberg model to demonstrate that the Nariai curve for magnetic black holes is flattened by light (electrically) charged particles. This is relevant to the argument in Ref. [36] that fixed the O(1) coefficient of the bound. Moreover, if a similar flattening by light charged particles happens for electric black holes, we need to revisit original discussion motivating the bound. It would be interesting to study electric black holes in the Euler-Heisenberg model, appropriately taking into account Schwinger effects captured by the imaginary part of the effective Lagrangian.
Besides, we found an interesting similarity between our black hole analysis and positivity bounds on scattering amplitudes. In the spirit of the black hole WGC, we postulated that the mass-to-charge ratio of extremal magnetic black holes is smaller than unity µ ≤ 1 for arbitrary charge Q and then the Euler-Heisenberg analysis implied a cutoff energy scale ∼ √ g e mM Pl similar to the one implied by gravitational positivity bounds in QED [89]. This observation would be useful when sharpening positivity bounds in the presence of gravity. It would be interesting to collect more evidences for the correspondence in more realistic models along the line of Refs. [89,[93][94][95]. Such an interplay between the black hole thermodynamics and the S-matrix bootstrap would broaden our global view of the bootstrap in gravity theories.

A Details of numerical analysis
We provide technical details on the approximation and numerical calculation used in the Euler-Heisenberg analysis. First, in order to derive an analytic approximation of the EH Lagrangian (3.4), we divide the integration range of the second term into 0 ≤ s ≤ 1 and 1 ≤ s ≤ ∞, and perform Taylor expansion of the integrand in each range. In the range 0 ≤ s ≤ 1, we expand the integrand in s around s = 0 up to the fourth/sixth order for scalar/fermion loop. On the other hand, in the range 1 ≤ s ≤ ∞, we expand in e −s = 0 around e −s = 0 (s = ∞) up to the fifth/third order for scalar/fermion loop. In Fig. 19 the original integrand (green dashed), the expansion around s = 0 (blue), and that around s = ∞ (orange) are compared for the parameter choice √ F m 2 = 10 5 . There we find a good agreement between the original integrand and our analytic approximation in each range.

B Anti-symmetric symbols and tensors
In this appendix, we summarize our convention of the anti-symmetric tensors and symbols. Although we give the explicit form in four-dimensional case, it is straight forward to extend it to the general D-dimension.
First, let us introduce the vierbein (vielbein) by g µν = η ab e a µ e b ν , (B.1) where µ, ν, . . . are the indices of the curved geometry and a, b, . . . denote those of the local Lorentz frame. We use the mostly plus convention for the Minkowski metric η ab = diag(−1, +1, . . . , +1). The volume factor is √ −g = − det g µν = det e a µ =: e . (B.2) On the local Lorentz frame, we introduce the anti-symmetric tensor as up to a normalization factor α. Its standard choice is α = ±1. 8 In addition, let us introduce ϵ µνρσ and ϵ µνρσ by the action of the vierbein on these anti-symmetric tensor as S p denotes the p-dimensional symmetric group and σ is its element. sgn(σ) is the signature of the element σ. The overall minus signature comes from the normalization (B.3). In addition, it is worth commenting on the following relation: In this paper, the dual field strength is defined bỹ We note that G = 1 4 F µνF µν depends on the metric only through the volume factor √ −g and the variation by the metric is Hodge dual. The Hodge dual of the p-form field ω p is given by using the above antisymmetric symbol as * D ω p = e p!(D − p)! ω ν 1 ···νp ε ν 1 ···νpµ 1 ···µ D−p dx µ 1 ∧ · · · ∧ dx µ D−p , (B. 13) where * D denotes the D dimensional Hodge star. This satisfies the following equations: * D * D ω p = α 2 (−1) p(D−p)+1 ω p , * D 1 = e D! ε µ 1 ···µ D dx µ 1 ∧ · · · ∧ dx µ D = −α √ −gd D x . (B.14) Using the Hodge star, the kinetic term of the p-form field is written as where F p+1 = dω p .

C Schwinger effect
In this paper we focus on magnetic black holes in order to ignore the Schwinger effect.
Here we explain why the Schwinger effect cannot be neglected for electric black holes in the regime of our interests by comparing the charge loss rate with the black hole radius.
Near the black hole horizon, the pair production rate per unit volume is Γ = 2 Im L = g 2 e Q 2 64π 3 r 4 Since the charge loss rate per unit volume is −g e Γ, the charge loss rate of the black hole is estimated asQ The corresponding decay rate of the black hole charge reads The Schwinger effect is negligible if it is sufficiently small compared to the curvature scale r −1 H . Below, we examine this condition for extremal and Nariai black holes, respectively.
Extremal black hole. The curvature of the extremal black hole with the charge Q is (C.4)