Thermodynamics of 4D Dilatonic Black Holes and the Weak Gravity Conjecture
PPrepared for submission to JHEP
KOBE-COSMO-19-15, MAD-TH-19-07
Thermodynamics of 4D Dilatonic Black Holes andthe Weak Gravity Conjecture
Gregory J. Loges, a Toshifumi Noumi b and Gary Shiu a a Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA b Department of Physics, Kobe University, Kobe 657-8501, Japan
E-mail: [email protected] , [email protected] , [email protected] Abstract:
Taking a thermodynamic perspective, we study the weak gravity conjecture inthe context of 4D Einstein-Maxwell-dilaton theory. We find closed-form expressions for thecorrected thermodynamic quantities in the presence of four-derivative terms in the action,and in particular the charge-to-mass ratio and entropy, for several families of solutionsof special magnetic-to-electric charge ratio or dilaton coupling constant. Assuming thatdyonic black holes themselves are the conjectured charged states, this places constraints onthe Wilson coefficients of the theory which we show are satisfied under mild assumptionson the UV theory. a r X i v : . [ h e p - t h ] S e p ontents Q e = Q m h = 1 h = 1 / h = / Mass Corrections 23
Classically, black holes have a well-defined temperature and thermodynamic entropy andare the source of several puzzles, such as the information paradox and the origin of theirarea-law entropy. String theory has offered explanations and explicit microscopic countingof states which give rise to the area-dependence for these solutions [1], a property whichis seemingly mysterious when viewed from the perspective of classical Einstein gravity.Similarly, questions surrounding the infinite-density singularities of the classical theory areexpected to be resolved by higher-energy physics, be it through stringy or, more generally,quantum gravity effects. One may systematically characterize these effects using the tech-niques of effective field theory, where a derivative expansion captures those features of the– 1 –V theory which are relevant for lower-energy processes. Of course, the challenge is to sumover the infinitely many higher derivative corrections in situations where this is necessary,such as in addressing the aforementioned cosmic singularities. String theory provides a UVcomplete framework in which such summation can at least in principle be carried out, asmanifest e.g. in the Regge behavior of string scattering amplitudes.Black holes also feature in the swampland program [2], where they play a central role inthought experiments which provide a motivation for the weak gravity conjecture (WGC) [3].For a recent review of the swampland program, including the WGC, see [4, 5]. The WGCstates that effective theories which have a UV completion including gravity must containa state with charge-to-mass ratio greater than one, in appropriate units. It was quicklynoted that black holes themselves may satisfy the conjecture if higher-derivative correctionsincrease the charge-to-mass ratio of extremal black hole states from their classical value ofone [6]. For black holes at least, the WGC is equivalent to requiring that two identicalextremal black holes repel one another, | F EM | > | F grav | + | F scalar | , (1.1)and hence have the decay to smaller black holes being kinematically allowed (in general theWGC and “Repulsive Force Conjecture” are not equivalent [7], but such distinction is notrelevant for our discussion). In 4D with Coulombic forces, this becomes M < M Pl ( Q e + Q m − Q φ ) , (1.2)where M represents the black hole’s mass and Q e , Q m and Q φ represent its electric, mag-netic and scalar charges, respectively. The weak version of the WGC has been demonstrated,under some assumptions , in Einstein-Maxwell theory [8–11].It is natural to extend these arguments to Einstein-Maxwell-dilaton (EMd) theory,which arises as the low-energy theory of both KK reduction and string theory. With amassless dilaton the low-energy effective theory now includes more degrees of freedom thanEinstein-Maxwell theory, and one cannot integrate out the dilaton to work in an EFT ofonly gravitons and photons. In this paper, we leverage the thermodynamic properties ofblack holes to derive bounds on the Wilson coefficients of the low-energy effective theory forseveral important EMd black hole solutions and show that the bounds are satisfied undergeneric assumptions on the UV theory.This paper is organized as follows. In section 2 we review 4D EMd black holes so-lutions and their thermodynamics. In section 3 we outline the technique used to extracthigher-derivative corrections to all thermodynamic quantities. In section 4 we present thecorrections to the charge-to-mass ratio and entropy which are relevant for the WGC. Insection 5 we show that the derived bounds are indeed satisfied under mild assumptions onthe UV theory, and we conclude in section 6. See our comment in section 6. – 2 –
Review of Einstein-Maxwell-dilaton Black Hole Solutions
We begin by reviewing static, dyonic black hole solutions of 4D EMd theory, for which wewrite the action as I = (cid:90) d x √− g (cid:20) M Pl R − M Pl ∂φ ) − e − λφ (cid:0) F (cid:1)(cid:21) . (2.1)Here and in what follows we use the compact notation ( ∂φ ) = ∂ µ φ∂ µ φ and ( F ) = F µν F µν .Going forward we will set M Pl = 1 / πG N = 1 . The exponential coupling constant, λ , maytake on any real value, and it will be convenient to introduce the associated constant h = λ ∈ (0 , . Several special values for λ are of note: λ = 0 ( h = 2 ) gives Einstein-Maxwell theory with a decoupled dilaton; λ = 1 / ( h = 1 ) corresponds to the low-energyeffective action of string theory; λ = 3 / ( h = 1 / ) corresponds to the KK reduction ofEinstein gravity from 5D to 4D, where the radion plays the role of the dilaton.The action/equations of motion of (2.1) enjoy two dualities: ( λ, φ ) → ( − λ, − φ ) , ( F µν , (cid:101) F µν , φ ) → ( (cid:101) F µν , − F µν , − φ ) , (2.2)where (cid:101) F µν ≡ e − λφ (cid:15) µνρσ F ρσ . Under the second duality, which we refer to as electromag-netic duality, electric and magnetic charges are interchanged as ( q e , q m ) → ( q m , − q e ) .A spherically-symmetric, static solution of the equations of motion is [12, 13] d s = − f ( r ) d t + d r f ( r ) + r ( H e H m ) h dΩ , (2.3) f ( r ) = ( H e H m ) − h (cid:18) − ξr (cid:19) , (2.4) e − λφ = (cid:18) H e H m (cid:19) − h , (2.5) F (2) = q e r H − e H − hm d t ∧ d r + q m sin θ d θ ∧ d ϕ , (2.6)where the functions H α ( r ) ( α = e, m and α = m, e ) satisfy r dd r (cid:20) r (cid:18) − ξr (cid:19) H (cid:48) α ( r ) H α ( r ) (cid:21) = − h − q α H − α H − hα . (2.7)Imposing the boundary conditions H α → for r → ∞ and H α > for r → ξ , this solutionis well-behaved outside the outer horizon, r > ξ . For convenience we have set φ ∞ = 0 ; aconstant shift to φ is compensated for by a rescaling of the electric and magnetic charges.While closed-form solutions of equation (2.7) do not exist for general λ , one can showthat series solutions of the form H α ( r ) = 1 + P (1) α r + P (2) α r + P (3) α r + · · · (2.8)– 3 –lways converge on (2 ξ, ∞ ) [12]. In addition, there is a first-integral of equation (2.7) whichwhen evaluated at r → ∞ gives [13] ( P (1) e ) + ( P (1) m ) + 2( h − P (1) e P (1) m + 2 ξ ( P (1) e + P (1) m ) − h − ( q e + q m ) = 0 . (2.9)Along with equation (2.7), this allows one to solve for the coefficients P ( k ) α order-by-orderin terms of P (1) α , ξ , h and the q α alone. The entire series is then fixed by prescribing aphysical parameter, such as the mass or temperature. In section 4 we will focus only onthose cases where the functions H α take a particularly manageable form. In writing down the solution to the equations of motion we have introduced the parameters ξ , P ( k ) e and P ( k ) m ; we would like to interpret these in terms of the physical properties of theblack hole. Ultimately the full solution may be determined by specifying only the blackhole’s charges and temperature.Metric singularities occur at r = 0 and r = 2 ξ , which set the locations of the hori-zons. Extremality thus corresponds to the limit ξ → , and requiring the regularity of theEuclidean section at the outer horizon gives the black hole temperature as T = f (cid:48) (2 ξ )4 π = 18 πξ (cid:2) H e (2 ξ ) H m (2 ξ ) (cid:3) − h . (2.10)The areas of the inner and outer horizons are A − = lim r → + πr (cid:2) H e ( r ) H m ( r ) (cid:3) h ,A + = 16 πξ (cid:2) H e (2 ξ ) H m (2 ξ ) (cid:3) h . (2.11)The Hawking-Bekenstein entropy, S = A + / G N = 2 πA + , thus vanishes in the extremallimit only when A + → A − = 0 . Using equations (2.10) and (2.11) one finds that thetemperature and entropy are related according to T S = 4 πξ , (2.12)so that at least one of T and S must vanish at extremality in the classical solution.In discussing the thermodynamics of these black holes we will need the electric andmagnetic potentials which are conjugate to their respective charges. Using equation (2.7),the 4-potential is A (1) = (cid:20) hq e r (cid:18) − ξr (cid:19) H (cid:48) e ( r ) H e ( r ) + A h (cid:21) d t + q m ( W − cos θ ) d ϕ , (2.13) With this parametrization the Einstein-Maxwell ( h = 2 ) solution is H e ( r ) = (cid:0) Pr (cid:1) q eq e + q m and H m ( r ) = (cid:0) Pr (cid:1) q mq e + q m , which takes the usual Reissner-Nordström form after the change of coordinates r → r − P . Note that only the product H e H m = 1 + Pr appears in the metric and field strength. – 4 –here the constants A h and W may be fixed by a gauge choice. From this we may read offthe black hole’s (gauge-invariant) electric potential and the analogous magnetic potentialrelative to infinity, Φ ≡ A t (cid:12)(cid:12) ξ − A t (cid:12)(cid:12) ∞ = hP (1) e q e , Ψ = hP (1) m q m . (2.14)Although the factors of q α seem misplaced, recall that the P (1) α are functions of the q α aswell. By expanding the metric solution at infinity one finds that the gravitational mass is M = 4 π (cid:2) ξ + h ( P (1) e + P (1) m ) (cid:3) . (2.15)Equation (2.9) along with the dilatonic charge, defined via φ = q φ /r + · · · , allows one towrite the mass as M = (8 πξ ) + 2 (cid:0) Q e + Q m − Q φ (cid:1) , (2.16)where we have introduced the rescaled charges Q α = 4 πq α ; we will use both q α and Q α throughout the remaining sections. Very nicely, the above expression for the mass is in-dependent of the coupling λ . Note, however, that Q φ = − πhλ ( P (1) e − P (1) m ) is not anindependent parameter; in particular, one always has M > . In the extremal limit ξ → the WGC bound (1.2) is saturated, but it remains to be seen whether higher-derivativecorrections will decrease the mass and lead to a strict inequality, or increase the mass andlead to such black holes not satisfying the bound.To have a controlled classical solution on which to consider higher-derivative correctionswe should require T < ∞ and S > at extremality. Pure electric or magnetic black holeshave vanishing area for any h (cid:54) = 2 , and in fact have diverging temperature if h < , sothat we will have to treat these cases with care. The curiosity of infinite-temperature blackholes was pursued in [14], where it was argued that one should morally think of thesesolutions as elementary particles and that while the temperature is formally infinite, therate of thermal radiation emission does go to zero as a result of an infinite mass gap. Aqualitative difference between the h < and h > regimes for single-charge black holeswas also found in [15] when considering connections between the weak gravity conjectureand cosmic censorship. Thermodynamic ideas have recently been pursued in calculating corrections to extremalblack holes’ charge-to-mass or angular momentum-to-mass ratio [16–18]. Here we outlinehow one may use the Euclidean action to compute thermodynamic quantities, and in par-ticular their higher-derivative corrections, for any finite temperature black hole solution.For a recent thorough discussion, see [17]. We are ultimately interested in extremal blackholes, and so when executing the described procedure we will find it convenient to workwith the temperature as an expansion parameter.– 5 –ollowing the usual procedure, temporarily restrict attention to r < R when making aWick rotation to Euclidean time, ( it ) ∼ ( it ) + β , with β = 1 /T the inverse temperature.Eventually the R → ∞ limit will be taken. The action in equation (2.1) does not lead toa well-posed variational problem; boundary terms are required to ensure that the actionis stationary under all variations of the metric and vector potential which vanish on ∂ M .The appropriate choice is to include a Gibbons-Hawking-York (GHY) term, which leads to I R E , [ g, A, φ ] = − (cid:90) M d x √ g (cid:20) R −
12 ( ∂φ ) − e − λφ (cid:0) F (cid:1)(cid:21) − (cid:73) ∂ M d Σ √ h (cid:0) K − K (cid:1) , (3.1)where K is the trace of the extrinsic curvature on ∂ M . Since the GHY term at r = R diverges in the infinite-volume limit we have subtracted off the analogous quantity for flatspace, K , to regularize the action and ensure that it remains finite in the infinite-volumelimit.The action I R E , is invariant under all variations of the metric which vanish on ∂ M . Inaddition, the appropriate boundary conditions for the Maxwell field are to prescribe A (cid:107) onthe boundary, i.e. fix the electric potential at the horizon and the magnetic charge. Thismay be seen by rewriting the Maxwell term as − (cid:90) M d x √ g (cid:20) − e − λφ (cid:0) F (cid:1)(cid:21) = − (cid:90) M d x √ g (cid:104) ∇ µ (cid:0) e − λφ F µν (cid:1) A ν (cid:3) + 12 (cid:73) ∂ M d Σ √ h (cid:0) n µ e − λφ F µν A ν (cid:1) . (3.2)The surface term is invariant under variations of the 4-potential which vanish at the horizon,which amounts to fixing A t and A ϕ . Thus when working directly with the Euclidean actionit is appropriate to work in the grand canonical ensemble, in which the temperature, electricpotential and magnetic charge are the independent quantities [19, 20].After having regularized the boundary term at r = R we may safely take the infinite-volume limit and consider I E , [ g, A, φ ] ≡ lim R →∞ I R E , [ g, A, φ ] . (3.3)The GHY term at infinity contributes to the Euclidean action even in this limit [19]. Eval-uating on the EMd solution ( g , A , φ ) gives I E , [ g, A, φ ] = 4 πβ (cid:0) ξ + hP (1) m (cid:1) = β (cid:0) M − Q e Φ + Q m Ψ (cid:1) . (3.4)The Smarr-like formula, M = 2 T S + Q e Φ + Q m Ψ , implies that we may write I E = βG ≡ β ( M − T S − Q e Φ) , (3.5)where G is the free energy. Importantly, this relationship between the free energy andEuclidean action remains true even when higher-derivative corrections are included, as longas S is taken to be the Wald entropy [21]. Using the first law, d M = T d S + Φ d Q e + Ψ d Q m , d G = − S d T − Q e dΦ + Ψ d Q m , (3.6)– 6 –e may read off S = − (cid:18) ∂ G∂T (cid:19) Φ ,Q m , Q e = − (cid:18) ∂ G∂ Φ (cid:19) T,Q m , Ψ = (cid:18) ∂ G∂Q m (cid:19) T, Φ . (3.7)These relations allow one to compute thermodynamic quantities in the presence of higher-derivative corrections, to which we turn next. Effective theories allow one to systematically parametrize the effects of UV physics in termsof a small number of numerical coefficients. The values of these Wilson coefficients aredetermined by the UV theory, up to field redefinitions.In string frame we assume an action of the form I = (cid:90) d x √− g e − λφ (cid:2) L ( g, A, ∂φ ) + L h.d. ( g, A, ∂φ ) (cid:3) , (3.8)this structure being motivated by the low-energy string effective action at leading orderin g s . Returning to Einstein frame, the most general collection of parity-preserving, four-derivative terms for 4D EMd theory may be written as α i I i ≡ (cid:90) d x √− g (cid:104) α e − λφ (cid:0) F (cid:1) + α e − λφ (cid:0) F (cid:101) F (cid:1) + α e − λφ (cid:0) F F W (cid:1) + α e − λφ ( R GB )+ α e − λφ ( ∂φ ) + α e − λφ ( ∂φ ) (cid:0) F (cid:1) + α e − λφ ( ∂φ∂φF F ) (cid:105) , (3.9)where we have used the compact notation ( F F W ) = F µν F ρσ W µνρσ , ( R GB ) = R µνρσ R µνρσ − R µν R µν + R , ( ∂φ ) = ( ∂ µ φ∂ µ φ ) , ( ∂φ∂φF F ) = ∂ µ φ∂ ν φF µρ F νρ , (3.10)and where W is the Weyl tensor. We have chosen to parametrize these in terms of R GB ,rather than W , so that we may more easily compare with previous Einstein-Maxwell resultsin the λ → limit where the Gauss-Bonnet term becomes topological. Note that the firstduality of equation (2.2) is maintained, while the electromagnetic duality is broken. Wewill treat electric and magnetic solutions separately in section 4.Upon adding these higher-derivative terms the equations of motion which need to besolved are altered, leading to perturbed solutions for g , A and φ . We favor, however, a thermodynamic approach which has recently been used in [17] and [18]. Equations (3.5)and (3.7) are used along with the following fact: I E [ g, A, φ ] = I E [ g, A, φ ] + O ( α i ) . (3.11)We emphasize that g , A and φ are the EMd solutions without higher derivative corrections,whereas I E is the full Euclidean action including the higher-derivative terms. For a proofof the above in Einstein gravity we refer the reader to Ref. [17].Given equations (3.7) and (3.11), corrections in the grand canonical ensemble are (cid:18) ∂ G∂α i (cid:19) T, Φ ,Q m = T I E ,i , (3.12)– 7 –uclidean Action Grand Canonical
Fixed: T, Φ , Q m Canonical
Fixed:
T, Q e , Q m Microcanonical
Fixed:
M, Q e , Q m WGC: z ext > WGC: ∆ S (cid:12)(cid:12) z =1 > Figure 1 : Relationship between different ensembles and the weak gravity conjecture. (cid:18) ∂ S∂α i (cid:19) T, Φ ,Q m = − (cid:18) ∂ ( T I E ,i ) ∂T (cid:19) Φ ,Q m ,α j , (3.13) (cid:18) ∂ Q e ∂α i (cid:19) T, Φ ,Q m = − T (cid:18) ∂ I E ,i ∂ Φ (cid:19) T,Q m ,α j , (3.14) (cid:18) ∂ Ψ ∂α i (cid:19) T, Φ ,Q m = T (cid:18) ∂ I E ,i ∂Q m (cid:19) T, Φ ,α j , (3.15) (cid:18) ∂ M∂α i (cid:19) T, Φ ,Q m = T I E ,i − T (cid:18) ∂ ( T I E ,i ) ∂T (cid:19) Φ ,Q m ,α j − T Φ (cid:18) ∂ I E ,i ∂ Φ (cid:19) T,Q m ,α j , (3.16)where I E,i denote the Euclidean versions of the terms in the four-derivative action, equa-tion (3.9). One may then transition to different ensembles, as outlined in figure 1, byinverting to leading order in α i . The weak gravity conjecture is directly a statement aboutthe charge-to-mass ratio in the canonical ensemble, where at fixed temperature ( T = 0 ) weexpect Q/M ∼ z ext > .One also expects that at fixed mass and charge, the entropy of z = 1 black holesincreases in the presence of higher derivative corrections. Using standard thermodynamicmanipulations, the corrections to the entropy in the microcanonical ensemble are (cid:18) ∂ S∂α i (cid:19) M,Q e ,Q m = − (cid:18) ∂ ( βG ) ∂α i (cid:19) T, Φ ,Q m = − I E ,i ( T, Φ , Q m ) , (3.17)where T and Φ are implicitly functions of M , Q e and Q m . We find that these correctionsare never O ( α i ) for extremal black holes, in agreement with the Supplemental Materialof [8]. We turn now to the execution of the procedure outlined in section 3. There are severalspecial choices of the exponential coupling constant and/or charges for which equation (2.7) The first equality is found by using the definition of the free energy, equation (3.6) and the tripleproduct rule along with (cid:18) ∂ S∂α i (cid:19) T, Φ ,Q m = (cid:18) ∂ S∂M (cid:19) Q e ,Q m ,α i (cid:18) ∂ M∂α i (cid:19) T, Φ ,Q m + (cid:18) ∂ S∂Q e (cid:19) M,Q m ,α i (cid:18) ∂ Q e ∂α i (cid:19) T, Φ ,Q m + (cid:18) ∂ S∂α i (cid:19) M,Q e ,Q m . – 8 – ζ ∞ Figure 2 : Charge/dilaton coupling parameter space for black holes with a massless dilaton.The black lines denote those regions covered by sections 4.1, 4.3, 4.4 and 4.5. The Einstein-Maxwell case is shown in gray.is exactly solvable. We will use five such closed-form solutions of equation (2.7) which allowus to calculate corrections analytically; (i) pure magnetic, (ii) pure electric, (iii) equal-charge, (iv) dyonic with string theory coupling ( h = 1 ), and (v) dyonic with 5D KKreduction coupling ( h = 1 / ). Corrections to the mass for the case of the magnetically-charged Garfinkle-Horowitz-Strominger black hole ( h = 1 ) have been computed in [22]. Seefigure 2 for the complete parameter space.In presenting the following consistency conditions, it is worth noting that α and α aresmall when gravitational effects are negligible, and that unitarity of the S -matrix requiresthat α , α , α and α are all positive. We will discuss these statements further in section 5.It is convenient to introduce the following notation when discussing corrections in thecanonical ensemble: X ≡ X (cid:12)(cid:12) α j =0 , δ i X ≡ X (cid:18) ∂ X∂α i (cid:19) T,Q e ,Q m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α j =0 , ζ ≡ Q m Q e . (4.1)In this section we present only the thermodynamics of the EMd solutions and those cor-rections which are relevant for the weak gravity conjecture. For clarity we work only withpositive charges; negative charges may be accounted for with the addition of appropriateabsolute values.
The simplest case for our purpose is the magnetically charged black hole, as here
Φ = Q e = 0 and what we call the grand canonical and canonical ensembles are one and the same. With Q e = 0 , equation (2.7) has solutions H e ( r ) = 1 , H m ( r ) = 1 + P m r , (4.2) Complete corrections are available upon request. – 9 –here hP m ( P m + 2 ξ ) = q m . This classical solution has area, temperature and horizondilaton at extremality given by A − = 0 , T → (cid:40) h > ∞ h < , e − λφ ∼ (cid:18) rq m (cid:19) − h → . (4.3)Despite having vanishing classical area, the derivative expansion is well-behaved due tothe suppression from higher-and-higher powers of e − λφ . For example, terms of the form e − (2+4 k ) λφ ( F ) k +1 have an expansion parameter e − λφ ( F ) ∼ q m − rq m + · · · (4.4)at extremality near the horizon. As long as the magnetic charge is large enough, thederivative expansion is under control.While the interpretation of infinite temperature in the classical solution is suspect, wenote that the regions h > and h < are treated identically in the following thermodynamicapproach. Indeed, defining τ h − ≡ Q m T / √ h , the EMd solution has G = √ hQ m (cid:20) − h − h τ − τ + · · · (cid:21) , (4.5) S = Q m τ − h h (cid:20) h h − τ + · · · (cid:21) , (4.6) Ψ = √ h (cid:20) − τ − h + 18( h − τ + · · · (cid:21) , (4.7) M = √ hQ m (cid:20) − h h τ + 3 − h h − τ + · · · (cid:21) . (4.8)The extremal limit is τ → for all h (cid:54) = 1 . For h = 1 the series expansions break down andextremality becomes T → Q − m . In the canonical ensemble, we find δ M = − h q m (cid:20) − h )(4 h + 1)2 h ( h − τ + · · · (cid:21) , (4.9a) δ M = − h q m (cid:20) (11 − h ) (4.9b) + (2 − h )[873 − h + 202 h − − h )(2 h −
3) log τ ]6 h ( h − τ + · · · (cid:21) ,δ M = − h (2 − h )10 q m (cid:20) (5 − h ) (4.9c) − − h + 4432 h − h + 286 h − h (2 − h )(8 − h + 2 h ) log τ h ( h − τ + · · · (cid:21) ,δ M = − h (2 − h ) q m (cid:20) − h + 143 h − h (2 − h ) log τ h ( h − τ + · · · (cid:21) , (4.9d) δ M = − h (2 − h )20 q m (cid:20) − h + 101 h − h (2 − h ) log τ h ( h − τ + · · · (cid:21) , (4.9e)– 10 –nd δ M = δ M = 0 . In the extremal limit the charge-to-mass ratio, z = √ hQ m /M , is z ext = 1 + h q m (cid:20) α + 13 (11 − h ) α + (2 − h )(5 − h ) α + 18 (2 − h ) α + 12 (2 − h ) α (cid:21) . (4.10)Note that while smooth at h = 1 , this result does not directly apply for h = 1 since thereour expansion in τ is ill-defined. For h (cid:54) = 1 , the weak gravity conjecture is then C h (cid:54) =1 mag ( α i ) ≡ α + 13 (11 − h ) α + (2 − h )(5 − h ) α + 18 (2 − h ) α + 12 (2 − h ) α > . (4.11)Note that as functions of h the coefficients of each α i are of definite sign, with the α and α contributions to C h (cid:54) =1 mag always being positive. As a check, this reduces to the Einstein-Maxwell condition for magnetically charged black holes for λ → : C h (cid:54) =1 mag ( α i ) (cid:12)(cid:12) h =2 = 2 α + α > . (4.12)To obtain the entropy in the microcanonical ensemble, we must invert M = √ hQ m (cid:18) − h q m C h (cid:54) =1 mag ( α i ) + 2 − h h τ + · · · (cid:19) . (4.13)in favor of τ . When z = √ hQ m /M = 1 , we find τ ≈ h C h (cid:54) =1 mag ( α i )5(2 − h ) q m , (4.14)which leads to ∆ S (cid:12)(cid:12) z =1 = Q m h (cid:32) h C h (cid:54) =1 mag ( α i )5(2 − h ) q m (cid:33) − h (cid:2) O ( α i ) (cid:3) . (4.15)This leading shift to the entropy comes entirely from evaluating S at nonzero temperature. For a massless dilaton the classical solution has H e ( r ) = 1 + P e r , H m ( r ) = 1 , (4.16)where hP e ( P e + 2 ξ ) = q e . The area, temperature and horizon dilaton at extremality are A − = 0 , T → (cid:40) h > ∞ h < , e − λφ ∼ (cid:16) q e r (cid:17) − h → ∞ . (4.17)In contrast with the magnetic case, the diverging dilaton spoils the derivative expansion,since higher-derivative operators are enhanced near the outer horizon, e.g. e − λφ (cid:0) F (cid:1) (cid:29) e − λφ (cid:0) F (cid:1) . (4.18) If h = 2 , then one goes to the next order in τ and finds τ ∼ √ a i . There would then be an O ( √ α ) correction to the entropy, as in [8]. – 11 –his divergence may be avoided by stabilizing the dilaton with a mass m φ (cid:38) | λ | M (for massesbelow this the solution approaches that of a massless dilaton near the outer horizon andthe divergences survive). The classical solution now takes the form [23, 24] d s = − f ( r ) d t + d r f ( r ) + R ( r ) dΩ , (4.19) f ( r ) = (cid:18) − M πr + q e r (cid:19) − λ q e m φ r + · · · , (4.20) R ( r ) = r (cid:32) − λ q e m φ r + · · · (cid:33) , (4.21) φ = − λ q e m φ r + · · · , (4.22) F tr = q e r (cid:32) − λ q e m φ r + · · · (cid:33) . (4.23)We have checked that running the thermodynamic procedure reproduces the same leading-order corrections as integrating out φ at tree-level. The Wilson coefficients for the resultingEinstein-Maxwell theory are α (cid:48) = α + λ m φ + O (cid:16) m φ (cid:17) , α (cid:48) , , = α , , + O (cid:16) m φ (cid:17) , α (cid:48) , , = 0 . (4.24)We may thus immediately write down the charge-to-mass ratio, z = √ Q e /M , at extremal-ity, z ext = 1 + 25 q e (cid:0) α (cid:48) − α (cid:48) (cid:1) = 1 + 25 q e (cid:0) α − α (cid:1) + 2 λ q e m φ + · · · . (4.25)The weak gravity conjecture is then C el ( α i ; m φ ) ≡ α (cid:48) − α (cid:48) = 2 α − α + λ m φ > , (4.26)and the entropy is corrected as ∆ S (cid:12)(cid:12) z =1 = 4 πQ e √ (cid:113) C el ( α i ; m φ ) + O ( α i , m − φ ) . (4.27) Q e = Q m For black holes of equal electric and magnetic charges the equations of motion have thefollowing solution: H e ( r ) = H m ( r ) = (cid:18) Pr (cid:19) /h = 1 + Phr + (1 − h ) P h r + · · · (4.28)where P ( P + 2 ξ ) = q e = q m ≡ q . In fact, since the dilaton profile is trivial, this isa solution of Einstein-Maxwell theory for which g µν and A µ are both independent of λ .– 12 –he usual dyonic, Reissner-Nordström solution is found after the change of coordinates r → r − P . One is then faced with computing corrections due to only ( F (cid:101) F ) and R GB inEinstein-Maxwell theory. The charge-to-mass ratio, z = 2 Q/M , at extremality is simply z ext = 1 + 2 α q , (4.29)so that the weak gravity conjecture is C Q e = Q m ( α i ) ≡ α > . (4.30)The entropy of an extremal black hole is corrected as ∆ S (cid:12)(cid:12) z =1 = 8 πQ √ √ α + O ( α , α ) . (4.31)We will use these particularly simple expressions as a cross-check on the remaining twocases. h = 1 For λ = 1 / ( h = 1 ) the exponential coupling corresponds to that found in the low-energyeffective action of string theory. Here we consider dyonic solutions, for which the extremallimit is T → . Equation (2.7) has for solutions H e ( r ) = 1 + P e r , H m ( r ) = 1 + P m r , (4.32)where P α ( P α + 2 ξ ) = q α . The classical area, temperature and dilaton at the horizon are allwell-behaved for extremal solutions: A − = 4 πq e q m , T → , e − λφ → q e q m . (4.33)In particular, the derivative expansion is intact as long as both charges are nonzero. TheEMd solution has G = 1 − Φ T + Q m T − Φ ) , (4.34) S = 1 − Φ T − Q m − Φ ) , (4.35) Q e, = Φ T − Q m Φ T (1 − Φ ) , (4.36) Ψ = Q m T − Φ , (4.37) M = 1 T − Q m Φ T (1 − Φ ) . (4.38)Inverting Q e, in favor of Φ gives Φ = 1 − Q m T − Q m (2 Q e + Q m ) T + · · · . (4.39)– 13 –n the canonical ensemble, the above leads to G = Q m (cid:20) Q e T + 18 Q e ( Q e + Q m ) T + · · · (cid:21) , (4.40) S = Q e Q m (cid:20) Q e + Q m ) T + 38 ( Q e + Q m ) T + · · · (cid:21) , (4.41) Ψ = 1 − Q e T − Q e ( Q e + 2 Q m ) T + · · · , (4.42) M = ( Q e + Q m ) (cid:20) Q e Q m T + 18 Q e Q m ( Q e + Q m ) T + · · · (cid:21) . (4.43)The electromagnetic duality of the classical solution is evident. Corrections to the mass inthe canonical ensemble take the form δ i M = − q e q m M i ( ζ ) + O ( T ) , (4.44)where M ( ζ ) = (1 − ζ )(8 + 103 ζ − ζ − ζ + 3 ζ ) + 60 ζ (1 − ζ ) log ζ ζ )(1 − ζ ) , (4.45a) M ( ζ ) = 2 ζ (1 + ζ ) , (4.45b) M ( ζ ) = − (1 − ζ )(39 − ζ − ζ + 334 ζ − ζ ) + 60 ζ (1 − ζ + 6 ζ + ζ ) log ζ ζ )(1 − ζ ) , (4.45c) M ( ζ ) = − (1 − ζ )(71 + 111 ζ − ζ + 331 ζ − ζ ) + 60 ζ (4 − ζ + 4 ζ + ζ ) log ζ ζ )(1 − ζ ) , (4.45d) M ( ζ ) = (1 − ζ )(3 + 178 ζ + 478 ζ + 178 ζ + 3 ζ ) + 60 ζ (1 + ζ )(1 + 5 ζ + ζ ) log ζ ζ )(1 − ζ ) , (4.45e) M ( ζ ) = (1 − ζ )(9 + 299 ζ + 239 ζ − ζ − ζ ) + 60 ζ (2 + 6 ζ − ζ ) log ζ ζ )(1 − ζ ) , (4.45f) M ( ζ ) = − − ζ )(1 + 28 ζ + ζ ) + 60 ζ (1 + 3 ζ + ζ ) log ζ ζ )(1 − ζ ) . (4.45g)These functions are in fact all finite for ζ → , as seen in figure 3. The charge-to-massratio, z = ( Q e + Q m ) /M , at extremality is thus z ext = 1 + 25 q e q m α i M i ( ζ ) . (4.46)The weak gravity conjecture for general ζ is then C h =1 dyon ( α i ; ζ ) ≡ α i M i ( ζ ) > . (4.47)We note that the α , α and α contributions to C h =1 dyon are always positive, while the α contribution is always negative. The equal-charge and magnetic limits agree with those– 14 – igure 3 : The functions M i ( ζ ) , with solid and dashed lines indicating positive and negativevalues respectively. Only M is nonzero for ζ → .found above (the factor of ζ is due to our definition of M i ): lim ζ → C h =1 dyon ( α i ; ζ ) = C Q e = Q m ( α i ) , lim ζ →∞ (4 ζ ) C h =1 dyon ( α i ; ζ ) = C h (cid:54) =1 mag ( α i ) (cid:12)(cid:12) h =1 . (4.48)The magnetic limit here should not be taken too seriously, since the extremal limit is notcaptured by the expansions of section 4.1 and there is no reason to expect that the extremaland h → limits commute.For the entropy corrections, inverting M = ( Q e + Q m ) (cid:18) − q e q m C h =1 dyon ( α i ; ζ ) + 18 Q e Q m T + · · · (cid:19) (4.49)for T when z = ( Q e + Q m ) /M = 1 gives T ≈ π (cid:113) C h =1 dyon ( α i ) √ Q e Q m . (4.50)The entropy correction is then ∆ S (cid:12)(cid:12) z =1 = 4 π √ Q e + Q m ) (cid:113) C h =1 dyon ( α i ) + O ( α i ) . (4.51)As before, the leading contributions comes only from ∆ S . h = 1 / With λ = 3 / ( h = 1 / ) the exponential coupling corresponds to the KK reduction ofEinstein gravity on M × S , with the radion playing the role of the dilaton. In this caseequation (2.7) has for solutions H e ( r ) = 1 + P e r + P (2) e r , H m ( r ) = 1 + P m r + P (2) m r , (4.52)– 15 –here the coefficients are the positive solutions of q α = P α ( P α + 2 ξ )( P α + 4 ξ ) P e + P m + 4 ξ , (4.53) P (2) α = P e P m ( P α + 2 ξ )2( P e + P m + 4 ξ ) . (4.54)The classical area, temperature and horizon value of the dilaton at extremality are A − = 4 πq e q m , T → , e − λφ → q e q m , (4.55)so that the derivative expansion is well-behaved for non-vanishing charges. Defining T ≡ Q m T Φ(2Φ − , the EMd solution has G = Q m Φ √ − (cid:20) − T + Φ T + · · · (cid:21) , (4.56) S = Q m − / (cid:2) − T + · · · (cid:3) , (4.57) Q e, = Q m (2Φ − / (cid:20) − T + Φ ) T + · · · (cid:21) , (4.58) Ψ = Φ √ − (cid:20) − T + 3Φ T + · · · (cid:21) , (4.59) M = 2 Q m Φ (2Φ − / (cid:20) − T + 12 (1 + 3Φ ) T + · · · (cid:21) . (4.60)To work in the canonical ensemble it is convenient to introduce (cid:101) T ≡ ( Q e Q m ) / T (cid:113) (cid:0) Q / e + Q / m (cid:1) . (4.61)With this the EMd solution in the canonical ensemble has G = Q m √ (cid:113) ζ − / (cid:20) (cid:0) ζ − / (cid:1) (cid:101) T + · · · (cid:21) , (4.62) S = Q e Q m (cid:104) (cid:0) ζ / + ζ − / (cid:1) (cid:101) T + · · · (cid:105) , (4.63) Φ = 1 √ (cid:113) ζ / (cid:20) − ζ / (cid:101) T − (cid:0) ζ / (cid:1) (cid:101) T + · · · (cid:21) , (4.64) Ψ = 1 √ (cid:113) ζ − / (cid:20) − ζ − / (cid:101) T − (cid:0) ζ − / (cid:1) (cid:101) T + · · · (cid:21) , (4.65) M = 1 √ Q / e + Q / m ) / (cid:20) (cid:101) T + · · · (cid:21) . (4.66)Again, electromagnetic duality is manifest in the classical solution.Much like the h = 1 case, we may write the mass corrections and charge-to-mass ratioin the canonical ensemble as δ i M = − q e q m M i ( ζ ) + O ( T ) , z ext = 1 + 25 q e q m α i M i ( ζ ) , (4.67)– 16 – igure 4 : The functions M i ( ζ ) , with solid and dashed lines indicating positive and negativevalues respectively. Only M is nonzero for ζ → .where the functions M i ( ζ ) are plotted in figure 4: the expressions are left to appendix A.Despite the functional form of the M i being quite different from those of the M i , there isa striking similarity between their behavior as ζ varies. From these mass corrections, weobtain the consistency condition required by the weak gravity conjecture: C h = dyon ( α i ; ζ ) ≡ α i M i ( ζ ) > , (4.68)where again the α , α and α contributions to C h = dyon are always positive and the α con-tribution is always negative. The equal-charge and magnetic limits agree with those foundabove: lim ζ → C h = dyon ( α i ; ζ ) = C Q e = Q m ( α i ) , lim ζ →∞ (16 ζ ) C h = dyon ( α i ; ζ ) = C h (cid:54) =1 mag ( α i ) (cid:12)(cid:12) h = . (4.69)Inverting M = 1 √ (cid:0) Q / e + Q / m (cid:1) / (cid:18) − q e q m C h = dyon ( α i ; ζ ) + 12 (cid:101) T + · · · (cid:19) for (cid:101) T when z = ( Q / e + Q / m ) / / √ M = 1 gives (cid:101) T ≈ (cid:118)(cid:117)(cid:117)(cid:116) C h = dyon ( α i ; ζ )5 q e q m . (4.70)The entropy correction is then ∆ S (cid:12)(cid:12) z =1 = 4 π √ (cid:0) ζ / + ζ − / (cid:1)(cid:114) Q e Q m C h = dyon ( α i ; ζ ) + O ( α i ) , (4.71)with the leading term coming from ∆ S only.– 17 – .6 Comments on Entropy Corrections In all of the cases considered above we have found entropy corrections to extremal blackholes which are not O ( α i ) , which we may interpret as arising from the stretching of theblack hole horizon. For example, before the introduction of higher-derivative correctionsthe dyonic, h = 1 case at extremality has f ( r ) = r q e q m − ( q e + q m ) r q e q m + · · · , (4.72)i.e. the degenerate horizon is at r = 0 . O ( α i ) corrections to f ( r ) lead to ∆ r ∼ √ α i . Inour thermodynamic approach the temperature is tied to the separation between the twohorizons via equation (2.10), and so we are still able to capture this square-root behaviorin equation (4.51) even without computing perturbations to the metric. We may reconcilethese observations with equation (3.17) by noting that I E ,i ( M, Q e , Q m ) often diverges as ( z − − / , so that one must go to the next order in α i , z = 1 + O ( α i ) , to achieve a finiteanswer. The O ( α − hi ) behavior in the magnetic case stems from f ( r ) not being quadraticnear r = 0 when h < . Having obtained consistency conditions on the Wilson coefficients of the higher-derivativeterms for EMd theory, we now turn to the task of showing that these conditions are satisfiedunder generic assumptions on the UV theory. For this discussion we find it useful toreinstate factors of M Pl . Before moving to particular examples, it is worth noting thatrenormalization group effects from graviton, photon and dilaton loops lead to the runningof the α i . These Wilson coefficients have dimensions which may be computed perturbatively,with only one or more being most important as one runs to the deep IR. Presently, however,we will show that for a number of generic UV completions of the EMd EFT the weak formof the WGC holds. Assuming that gravitational effects are subdominant, unitarity requires that both α and α be non-negative. Here we will show that a similar condition applies also to α and α : α , α ≥ . (5.1)We use the spinning polynomial basis, P s n ( x ) , of [25] and assume that no exchangedparticles are massless. Factorization implies the following form for two of the forwardhelicity amplitudes: M ( φφφφ ) = (cid:88) n (cid:104) g n m n − s P s n (1) + g n m n + s P s n (1) + a n + b n s (cid:105) (5.2) = (cid:16) (cid:88) n g n m n (cid:17) s + · · · , – 18 – ( φA ± φA ∓ ) = (cid:88) n (cid:104) g ± n m n − s P ± ∓ s n (1) + g ± n m n + s P ± ∓ s n (1) + a n + b n s (cid:105) (5.3) = (cid:16) (cid:88) n g ± n m n s n + 1 s n (cid:17) s + · · · . The potentially dangerous contribution from intermediate spin-0 particles is avoided sincein that case the coupling g ± n is forbidden by locality. On the other hand, the higher-derivative terms of (3.9) generate M ( φφφφ ) ∝ α M Pl s + · · · , M ( φA ± φA ∓ ) ∝ α M Pl s + · · · , (5.4)with positive constants of proportionality. From these we may match the s coefficientsand read off α ∝ (cid:88) n g n m n ≥ , α ∝ (cid:88) n g ± n m n s n + 1 s n ≥ . (5.5)It is interesting to note that the positivity of α implies a contribution from α ( ∂φ∂φF F ) to the charge-to-mass ratio which is always negative. Conceivably this contribution to z could dominate and lead to a violation of the WGC. This shows that in general unitarityof the S-matrix is not sufficient to ensure that the weak form of the WGC holds. This issimilar to the situation in [10], where fine-tuning of non-minimal couplings allows for thepossibility of violating the WGC when running the Wilson coefficients into the deep IR,but this is not borne out in examples. In the EMd EFT the graviton, photon and dilaton are all massless. In UV completionsfor which the next-lightest fields are neutral (pseudo)scalars, integrating out these fields attree-level generates nonzero α , α , α and α . Specifically, if one has L χ = −
12 ( ∂χ ) − m χ χ + χf χ e − λφ (cid:0) F (cid:1) + g χ M Pl χ e − λφ ( ∂φ ) , L a = −
12 ( ∂a ) − m a a + af a e − λφ (cid:0) F (cid:101) F (cid:1) , (5.6)then one finds α = 2 M Pl m χ f χ , α = 2 M Pl m a f a , α = 2 g χ M Pl m χ , α = 4 g χ M Pl m χ f χ . (5.7)With these we find that all of the consistency conditions as required by the WGC aresatisfied: C h (cid:54) =1 mag ( α i ) = M Pl m χ (cid:20) M Pl f χ + (2 − h ) g χ (cid:21) > , (5.8a) C el ( α i ; m φ ) = 4 M Pl m χ f χ + λ M Pl m φ > , (5.8b)– 19 – igure 5 : Examples of 1-loop diagrams leading to higher-derivative interactions. C Q e = Q m ( α i ) = 2 M Pl m a f a > , (5.8c) C h =1 dyon ( α i ; ζ ) = 2 M Pl m a f a M ( ζ ) (5.8d) + 2 M Pl m χ f χ (cid:34) M ( ζ ) + (cid:18) g χ f χ M Pl (cid:19) M ( ζ ) + 2 (cid:18) g χ f χ M Pl (cid:19) M ( ζ ) (cid:35) > , C h = dyon ( α i ; ζ ) = 2 M Pl m a f a M ( ζ ) (5.8e) + 2 M Pl m χ f χ (cid:34) M ( ζ ) + (cid:18) g χ f χ M Pl (cid:19) M ( ζ ) + 2 (cid:18) g χ f χ M Pl (cid:19) M ( ζ ) (cid:35) > . These rely on the following facts, M ( ζ ) > ∀ ζ , M ( ζ ) > ∀ ζ , M ( ζ ) + x M ( ζ ) + 2 x M ( ζ ) ≥ ∀ x, ζ , M ( ζ ) + x M ( ζ ) + 2 x M ( ζ ) ≥ ∀ x, ζ , (5.9)the last two of which are nontrivial: x , M ( ζ ) and M ( ζ ) may be of either sign. If the lowest-lying states in the UV theory are charged scalars and/or fermions, then theleading contributions to the α i are generated by one-loop diagrams, such as those in fig-ure 5. Assuming weak coupling at the scale of these charged states, we need consider onlyelectrically-charged particles. Writing z ∼ q e m for the charge-to-mass ratio of such a particle,we may estimate | α , | ∼ max {O (1) , O ( z ) , O ( z ) } , | α , , | ∼ max {O (1) , O ( z ) } , | α , | ∼ O (1) . (5.10)For z (cid:29) , electromagnetic effects dominate and unitarity ensures that α and α arepositive. Of course, if z (cid:29) then the WGC is already satisfied without considering blackhole states, but in this limit we do find that the consistency conditions are satisfied: C h (cid:54) =1 mag ( α i ) ≈ α > , (5.11a)– 20 – el ( α i ; m φ ) ≈ α + λ m φ > , (5.11b) C Q e = Q m ( α i ) = α > , (5.11c) C h =1 dyon ( α i ; ζ ) ≈ α M ( ζ ) + α M ( ζ ) > , (5.11d) C h = dyon ( α i ; ζ ) ≈ α M ( ζ ) + α M ( ζ ) > , (5.11e)where we have used that M , M , M and M are positive for all ζ . The above conditionsensure that these black hole states are unstable and can decay to smaller dyonic black holes,as was the original motivation for the WGC. Suppose now that the UV theory has no low-lying states, but rather towers of higher-spin,Regge states accompanying the graviton, photon and dilaton. Writing Λ QFT < M Pl for thescale at which quantum field theory breaks down, one would expect the following hierarchy: | α , , , , | ∼ O (cid:18) M Pl Λ QFT (cid:19) , | α | ∼ O (cid:18) M Pl Λ QFT (cid:19) , | α | ∼ O (1) . (5.12)By itself such a hierarchy is not enough to guarantee that the WGC conditions are satisfied,even if supplemented with α , α , α , α ≥ . While the contributions to the charge-to-mass ratio from the α , α and α terms are always positive, the α contribution is alwaysnegative and the α contribution changes sign with ζ larger or smaller than one. It isenough, however, to have max { α , α , α } (cid:38) | α | , α ≥ . (5.13)Such an inequality is found in open string-like UV completions, where the Regge states ofthe photon are open string states, while the Regge states of the graviton and dilaton areclosed string states. Since g s ∼ g open , the contributions from each sector to the Wilsoncoefficients are then [ α , ] open ∼ M Pl g s M s , [ α , ] closed ∼ [ α , , , , ] open ∼ [ α , , , , ] closed ∼ M Pl M s . (5.14)Given g s (cid:28) , α and α dominate and the WGC conditions are satisfied, just as in (5.11). Here we quickly check that the Wilson coefficients derived from compactifying heteroticstring theory down to 4D satisfy the conditions found for h = 1 . In string frame the O ( α (cid:48) ) heterotic string action reads [26] I = M (cid:90) d x √− G e − (cid:104) R + 4( ∂ Φ) − F MN F MN + α (cid:48) (cid:16) R MNLP R MNLP + 34 ( F MN F MN ) + 34 ( F MN (cid:101) F MN ) (cid:17)(cid:105) . (5.15)– 21 –imensionally reducing to 4D on M = M × X and rescaling the dilaton to φ = √ leads to I = (cid:90) d x √− g e −√ φ (cid:104) R + ( ∂φ ) −
14 ( F )+ α (cid:48) (cid:16) R µνρσ R µνρσ + 34 ( F ) + 34 ( F (cid:101) F ) (cid:17)(cid:105) , (5.16)where M Pl = M vol( X ) = 1 . In Einstein frame the above becomes I = (cid:90) d x √− g (cid:104) R −
12 ( ∂φ ) − e −√ φ ( F ) + α (cid:48) (cid:16) e − √ φ ( F ) + 74 e − √ φ ( F (cid:101) F ) + e −√ φ R GB + 2 e − √ φ ( F )( ∂φ ) (cid:17)(cid:105) . (5.17)In particular, ( α , α , α , α , α , α , α ) = α (cid:48) (cid:0) , , , , , , (cid:1) , (5.18)which indeed ensure C h =1 dyon ( α i ; ζ ) > for all ζ . In this paper we have calculated higher-derivative corrections to Einstein-Maxwell-dilatonblack holes for a variety of choices for electric charge, magnetic charge and dilaton cou-pling constant. Motivated by the swampland program and the weak gravity conjecture inparticular, we found constraints on the Wilson coefficients of the effective theory whichensure that the charge-to-mass ratio of black holes increases from its classical value. Forelectrically charged black holes perturbative control is lost due to the classically vanishingarea and diverging dilaton at extremality.By considering several generic UV completions of EMd theory we have shown that theconsistency conditions imposed by the WGC are generically satisfied. These checks showthat the charge-to-mass ratio of extremal, dilatonic black holes increases from its classicalvalue for a range of electric, magnetic and dyonic solutions. We have focused on those caseswhere we can obtain closed-form expressions, but much of the parameter space remainsunchecked. For dyonic black holes and general coupling λ , one could check numericallythat similar results hold. Given the similarity of the h = 1 and h = 1 / cases, we expectthat nothing drastically different would be found for general h .Our work provides more nontrivial evidence for the WGC as a general constraint foridentifying quantum gravity-derived EFTs. Even in this more general setting all large blackholes are unstable to decay, either through thermal radiation if at finite temperature orthrough the kinematically allowed emission of a superextremal black hole. For the heteroticstring, the weak form of the WGC pursued here is connected via modular invariance to astrong form where the superextremal states are light [27]. A similar story for the dimensional reduction of 5D Gauss-Bonnet gravity, (cid:82) d x √− g (cid:0) R + αR GB (cid:1) ,leads to α i = α (66 , , − , , , , and C h = dyon ( α i ; ζ ) > for all ζ when α > . – 22 –efs. [10] and [11] have recently shown that one-loop contributions to the Wilson co-efficients generically lead to the weak form of the WGC being satisfied in the deep IR.It would be interesting to investigate this argument with the exponential coupling of thedilaton considered here. In addition, our setup can also be extended to a supersymmetricone by including an axion in addition to the dilaton. In the absence of non-perturbativeeffects the SL(2 , R ) symmetry leads to all dyonic solutions having vanishing classical area,as we had here in the pure electric and magnetic cases. We leave such considerations forfuture work.We conclude with a comment on the difficulty of demonstrating the WGC using posi-tivity bounds for scattering amplitudes. The main obstruction to deriving the mild WGCfrom the positivity bounds is the t -channel graviton exchange ∼ s /t . In [8] it was clarifiedunder which conditions the positivity of the O ( s ) coefficient is justified and thus the mildWGC follows by carefully studying contributions from Regge states. More recently, Ref. [9]proposed a regularization scheme based on compactification of 4D gravitational theory to3D. Even though it was claimed that it leads to the O ( s ) positivity and thus the mildWGC in general setups, several big assumptions are in order: First of all, to remove the t -channel singularity in their scenario, one needs to take the forward limit t → first andthen take the decompactification limit, which is an opposite ordering to obtaining the 4Dbound. Second, this scenario is motivated by a potential non-perturbative UV completionof 3D gravity [28–31]. However, it is far from obvious if the same scenario works in the3D gravitational theory with a 4D origin. For example, if we assume a perturbative UVcompletion of gravity just as sting theory, the KK reduced 3D theory will contain infinitelymany higher-spin Regge states. As demonstrated in [8], the standard derivation of the pos-itivity bounds [32] cannot be justified unless effects of the Regge states are subdominant.Therefore, more studies on Regge states will be encouraged to understand how to resolvethe t -channel singularity and complete the proof of the mild WGC, at least as long as weconsider the string theory type UV completion of gravity. Acknowledgments
TN would like to thank the University of Wisconsin for their hospitality, where a part ofthis work was done. The work of GL and GS is supported in part by the DOE grant de-sc0017647 and the Kellett Award of the University of Wisconsin. TN is supported in part byJSPS KAKENHI Grant Numbers JP17H02894 and JP18K13539, and MEXT KAKENHIGrant Number JP18H04352.
A Dyonic, h = / Mass Corrections
For λ = 3 / ( h = 1 / ) the mass corrections in the canonical ensemble are (see equa-tion (4.67)) δ i M = − q e q m M i ( ζ ) + O ( T ) . (A.1)– 23 –he functions M i ( ζ ) are, introducing x ≡ ζ / , M ( ζ ) = 18 √ x (1 − x ) (1 + x ) (cid:20) (96 − x + x ) (A.2a) − x ) arccosh x √ x − x arcsech x √ − x (cid:21) , M ( ζ ) = 12 x / (1 − x ) (1 + x ) (cid:20) (6 + 8 x + x ) − x arccosh x √ x − (cid:21) , (A.2b) M ( ζ ) = 148 √ x (1 − x ) (1 + x ) (cid:20) (154 − x + 9 x ) − − x ) log x (A.2c) − − x + 4 x ) arccosh x √ x − − − x − x ) arcsech x √ − x (cid:21) , M ( ζ ) = 364 √ x (1 − x ) (1 + x ) (cid:20) (224 − x + 9 x ) − − x ) log x (A.2d) − − x + 16 x ) arccosh x √ x − − − x − x ) arcsech x √ − x (cid:21) , M ( ζ ) = − √ x (1 − x ) (1 + x ) (cid:20) (184 − x − x ) + 40(1 − x ) log x (A.2e) − x − x ) arccosh x √ x − − x − x ) arcsech x √ − x (cid:21) , M ( ζ ) = 1128 √ x (1 − x ) (1 + x ) (cid:20) (136 + 353 x + 6 x ) + 240(1 − x ) log x (A.2f) − x − x ) arccosh x √ x − − x − x ) arcsech x √ − x (cid:21) , M ( ζ ) = 1564 √ x (1 − x ) (1 + x ) (cid:20) x ) + 4(1 − x ) log x (A.2g) − (5 + 4 x − x ) arccosh x √ x − − x − x ) arcsech x √ − x (cid:21) . Principal values should be used for the square-root and inverse hyperbolic functions: branchcuts conspire to make arccosh x/ √ x − and arcsech x/ √ − x smooth, real functions of x > . The functions M i ( ζ ) are plotted in figure 4. References [1] A. Strominger and C. Vafa,
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