Jhep11(2020)008

JHEP11(2020)008 Springer June 24, 2020 : ) symmetry we November 4, 2020 September 30, 2020 : R , : 2 Received Published , Accepted a Published for SISSA by https://doi.org/10.1007/JHEP11(2020)008 and Gary Shiu b [email protected] , . 3 Toshifumi Noumi ) symmetry we ﬁnd a simple condition on the two Wilson coeﬃcients a 2006.06696 R ; The Authors. Black Holes, Black Holes in String Theory, String Duality, Supersymmetry c

d, d Positivity bounds coming from consistency of UV scattering amplitudes are , [email protected] Department of Physics, Kobe University, Kobe 657-8501, Japan E-mail: [email protected] Department of Physics, UniversityMadison, of WI Wisconsin-Madison, 53706, U.S.A. b a Open Access Article funded by SCOAP ArXiv ePrint: these symmetries gives corrections which vanish identically, as expected forKeywords: BPS states. and Duality inherited from the UV theoryaxion on theory. higher-derivative operators for Using Einstein-Maxwell-dilaton- find black that hole thermodynamics, thepreserved for weak O( a gravity preserved conjecture SL( which then ensures the does positivity of follow correctionsthe to from null the energy charge-to-mass positivity ratio condition and bounds. alone. that follows We from find For that a imposing supersymmetry on top of either of Abstract: not always sufficient toMaxwell. prove Additional the ingredients about weakof the gravity UV parameter conjecture may space for be which theoriesconjecture. necessary are to beyond naïvely In exclude Einstein- in this those conflict paper regions with we the explore predictions the of consequences of the imposing weak gravity additional symmetries Gregory J. Loges, Duality and supersymmetry constraintsgravity on conjecture the weak JHEP11(2020)008 ] 3 4 8 11 31 , 30 ] for attempts to 36 , 35 , 33 7 ]. Then, it is natural to expect 2 2 16 ] (see also [ 5 , 4 – 1 – 7 ] for a review of the WGC and other swampland conjectures. 29 , 28 ]. In its mild from, it is enough to have the extremality 1 6 ] which follow from consistency of UV scattering amplitudes may 14 ]). See also [ ]. For this reason, considerable efforts have been devoted toward a 34 27 33 – 15 1 when combined with additional UV properties such as modular invari- ) ) ) ) 1 R R 15 ; ; R R ]. While the mild form alone does not display the full predictive power for , , 14 ] it can lead to stronger forms of the WGC such as the sublattice WGC [ d, d d, d – 2 32 SL(2 O( SL(2 O( – 30 An important issue in this context is to identify consistency conditions necessary to The power of the WGC lies with its constraints on low energy physics (most notably in constraining 4.1 Leading-order thermodynamics4.2 4.3 4.4 Supersymmetry 2.1 2.2 1 play a crucialMaxwell role theory in under demonstrating reasonableuse assumptions the positivity [ conjecture. bounds to constrain It the is charged particle indeed spectrum theaxion at inflation case low models energies). [ in Einstein- demonstrate the conjecture.bound may At be low captured energies, byfollows higher-derivative if corrections operators, the so effective to couplings that satisfy the thethat a positivity mild black certain bounds form inequality hole [ [ of extremality the WGC bound of charged black holesincluded be [ less stringent whenphenomenology, corrections to the classicalance action [ are and the tower WGC [ proof of the mild form of the WGC as a basis of the web of WGCs. 1 Introduction The weak gravity conjecture (WGC) positsits the mass, existence in of a appropriate state units with [ charge larger than 5 Discussion A Positivity bounds in gravitational theories 3 Leading-order solution 4 Higher-derivative corrections via thermodynamics Contents 1 Introduction 2 Symmetries of the low-energy effective action JHEP11(2020)008 ) 2 R ≥ , (2.1) 2 (1.1) N , ) symmetry: both R a ; F · a d, d F 1 2 − 2 H 1 12 ) symmetry respected by the higher- − . Throughout we use reduced Planck R ρ ]. We also study implications of , ν 2 we recall the realizations of the SL( 5 ) 2 40 F 2 – ∂φ µρ 38 φF ) symmetry and an O( we present the leading order, dyonic solutions; ] for previous discussions of the mild form of – 2 – + 4( ν R 3 , R φ∂ 2 10 µ , φ ]. Of course, such a term never exists in isolation and ∂ 2 9 7 − , 7 g e − √ x 4 d Z 1 2 = ], for a few hand-picked choices of UV completion one can show that 7 I . ], there are regions of parameter space in which the axion weak gravity = 1 ) symmetries for the two-derivative EMda action and impose these symmetries 37 we use black hole thermodynamics to compute the corrections to the extremal R N ; 4 πG d, d 8 This paper is organized as follows: in section In this paper we consider the implications of imposing such additional structures on the A similar scenario is present for the Einstein-axion-dilaton theory, where, as discussed However, there are some low-energy effective theories for which positivity bounds are two U(1)s for simplicity. Inboth discussing the between two string symmetries itstring and is frame, Einstein useful to frame go back-and-forth and between axion and 2-form field. Start in charge-to-mass ratio, and weunits: conclude in section 2 Symmetries of theLet low-energy us effective begin action by recalling the two-derivative action for EMda theory: we will work with higher derivative operators, as expected for BPS states. and O( on the higher-derivative terms; inin section section In particular, we will workcan with be an present SL( incorrespond the to two-derivative EMda S- action and and T-duality,supersymmetry respectively in in [ 4D these effective string setups.helpful to actions make these We corrections to find extremality identically that zero the even in puzzling the presence terms of mentioned nontrivial earlier are UV theory for theaxion WGC (EMda) as applied theory tothe (see, extremal WGC e.g., black in holes [ thedescend in presence from Einstein-Maxwell-dilaton- of the a UVnull dilaton). theory, energy when conditions, Extra combined are with symmetries then either in strong scattering the enough positivity to effective bounds demonstrate action the or which WGC for this system. recently in [ conjecture is violated, even when positivityextra bounds are structure taken on into account. the UVderivative There, terms, imposing theory, greatly namely constrains an the SL( formfollows of from the the corrections and positivity ensures bounds. the axion WGC positivity bounds are accounted forpresumably [ this negative contribution never dominatesAs over discussed other (positive) in contributions. [ indeed this puzzling term is never problematic. not sufficient on their ownof to prove extremal the black positivity holes. ofwhere corrections This to the occurs, the term for charge-to-mass example, ratio in the Einstein-Maxwell-dilaton theory, contributes to the charge-to-mass ratio in a way directly at odds with the WGC once JHEP11(2020)008 + of θ H (2.6) (2.7) (2.8) (2.9) (2.2) (2.3) (2.4) (2.5) . (2.10) = ) α ( τ and , O ) , G and , − a θ a − F a e d . F F ? · ) φ a a 4 d Re ( . e F · + ∞ θ F = as well. . a Z )] 1 2 cτ F a − a ( H φ + a F 2 = 2 πp · a F − ) transformation at 7→ . a e F R − ∧ a − F − , · 1 1 2 a = 4 gives 2 a F F a ( a Z F − ) F P H τ 2 φ . 2 gives = + +2 2 1 H a F − , ! , ( φ H Im [ e µν a a and ) 4 for antisymmetric tensors g d 1 2 a P Q − − a R φ e ). 1111 F e 2 , G ,P θ i 2 − α e πq τ · is summed over. Here we have introduced ) 2 ) ) induces 1 2 ) ! 12 µ − ± τ ) according to ) − ,F 2 a τ G → a , R = +1 SL(2 = 4 b − τ∂ d 2.11 H ∂θ c d a b , F ≡ τF µ ( 2 a µν 2 ∈ +

∧ ) – 3 – + ∂ φ 2 g = 1 2(Im 1 2 Q 4 0123 ! G e a ∂φ Re ( cτ aτ 7→ − + 16(Im = 1 2 ?H − 2( d φ c d a b ∞ ! R ± term in ( 4 a 7→ − a Z a and

+ = − − F 2 g P Q e σ ) cτ R 1111 ), while integrating out − 1 2

··· = 2 α µ 0123 ∂φ √ g − 2.2 7→ ( e x H 2( − F 4 − σ 1 Z √ d τ − ρσ ··· F and the index µ x 1 2 F R Z 4 a G d 1 2 ⊃ F g , τ + Im ≡ I µνρσ Z − ∧ = µν a 1 2 H √ g I g reproduces ( · τ F A x − = 4 θ 7→ G − d √ I Re ), these transform under SL( ) 1 2 B µν R Z ) symmetry is best discussed in Einstein frame, where by defining ) symmetry acts nonlinearly on the fields as g , 2.7 = R Z 1 2 R , , = d 2 2 = µν = e F H a I and SL(2 Q φ It is worth noting that the transformations of the fields are altered in the presence of higher-derivative 2 2 − These ensure that the equations of motion are invariant under the SL( We will make use of the rescaled charges terms in the action: for example, the and using ( where and is present atdefined the by level of the equations of motion. Electric and magnetic charges are The SL( The SL( ie the action becomes where 2.1 Integrating out the short-hand the same rank. Going to string frame via Dualizing to an axion is accomplished via where JHEP11(2020)008 ). = 0 d Upon 2.11 (2.13) (2.11) (2.12) 4 ) with )-symmetric R . In reducing , , R ; 0 i ; H 2   , d . p 0 E d, d d 3 ). We will not consider (1) µ ) when reducing to α R cdab -dimensional dilaton, R A ; ) is any rational function of ; + k α 0 d np + ) 2 mn = , d B | 0 τ )-invariant action. B ( τ d d, d j − R µ (4 + ; abdc r ) τ∂ (2) µ n ρ α µ ν d, d A ∂ + = b ) | n 2 ) due to non-perturbative effects. Im- F + d for the α Z , where , (1) ν F , µρ 2 bacd · 2 ) + A Φ − a α   2 + +2 c ) mn = pn F F τ F τ p B ν )( G µ )( ) symmetry we have in mind this family of p τ mn m ( (1) ν − R b abcd ) symmetry to, among others, the coefficient τ∂ 4 τ∂ G four-derivative operators may be written as ; (1) µ A µ F (1) µ µ R G 3 – 4 – ∂ · , A 4 ∂ A ( ) ( 2 mp d, d q − τ 1 a ab + B , α F α (1) ν ( m + (Im 1 4 ] A mn (2) ν − − ) p ) ) pq abcd τ G A is a total derivative in 4D. For the action to be real the (2) ν m ( τ τ (1) ν G α j m 2 A p 2 A r µ ) R (1) [ − )-preserving, , (1) µ τ mp A R + A   , log det + (Im + (Im G and 2 dimensions the symmetry is enhanced to O( , α -dimensional indices and 2 − m + ) 1 2 ) (Im d d ba h (2) µ m µν (1) µ +2 g + µν α 4+ A F A g B 4 Ric − φ = )(       is the Eisenstein series of weight four. However, being non-perturbatively generated 2 − (4 + √ − 4 ab 2 . When we refer to O( x = = = F 4 G α ( ˆ ˆ ) ν ν d is now related by the SL( Φ = 2 2 M ν ˆ ˆ µ µ ) + 3 R ) symmetry is known to be broken to SL( g ) symmetry is best discussed in string frame with the 3-form τ A ; B Z R . to 4 dimensions on a torus, one finds a collection of KK scalars and U(1) , R ) , d 2 2 ; = = Riem d (Im gauge fields in d, d F I 2 k ) ( + 2 d, d τ 2 ∆ E ( ) O( 4 + ∂φ∂φF F j , d Using hats for A complete set of SL( -invariant and r ∂φ The SL( With j 3 4 ( + 1 this extension here. posing this lesse.g. restrictive symmetry onthe the four-derivativesuch action terms will allows be for highly a suppressed. wider range of terms, d symmetries which appear when reducingfour-derivative on terms an we arbitrary discuss torus.even below lower The are dimensions. O( invariant under O( the decomposition The O( from gauge fields which collectreducing themselves further into a to three, manifestly two O( or one dimension(s), the symmetry is O( allowed terms and withnot regions ruled of out parameterterm by space positivity in bounds. conflictof with Note the that WGC that the2.2 are coefficient of the previously-noted Wilson coefficients must be real and have the following symmetries: All-told there are 12 realAs parameters controlling mentioned the in Wilson coefficients the of introduction, equation ( without the imposed symmetry there are far more where JHEP11(2020)008 = and mn (2.18) (2.19) (2.14) (2.15) (2.16) (2.17) T ) sym- G , Ω R Q ; , H 0 . F Ω P , d ) 0 µν N ) , respectively. 2 F d σµ Q ) F 0 N H H 8 F α ( ( F , H → P ρσ O O − MN M , F ˆ N 0 ρ + H + F ˆ ν α 16 ˆ µ M µν ·F Q Q νρ N H F N , ˆ ρ F F M 1 4 ˆ F ν ) = ( ·F NQ P ˆ ! µ M F µν η − M P F ) H F pn F α, β F N H . MN MP G N ( 1 = 12 mn R µ δ η F F correction to the Bianchi identity φ ∂ G 1 8 mp -dimensional gravitational Chern- N 2 1 M − ) ) Q MN M B − and − d δ β F + MN Φ e F F ·F ( , one finds (using tree-level equations δ ˆ H µ 1 4 0) P qn θ ≡ φ R µ O ∂ , M 4 NP ∂ 0 B F − Φ ! 8 δ (4 + − α F → ˆ N NQ µ pq e 2 η ) pn ∂ Tr( MN 0 2 1 mn F G H MQ δ δ B 1 8 δ + NQ ∂θ M 1 2 − ) = ( MP ( mp δ 1 2 ! + mp 2 – 5 – η φ F R 0 n B 2 mn − 4 1 8 G − δ E 0 e m MP α, β H − φ Φ δ 1 2 −

η ( 2 + − 1 − n NQ 12 µνρσ 1 = − 16 mn e δ 0 , m g e R 2 − NP . This is manifestly invariant under δ G ) 2 δ − α − N )

MP MN δ √ ρσ N ∂φ µνρσ N MQ H = = ∂φ 1 8 + 2 x F δ 2( R ∧ F d F 1 2 N + . We will consider the background with internal scalars − M α e 4+ M +4( ) MN MN F − d µν M · F η R A R R PQ H φ ; δ F R 2 Z M φ ] g NQ − 2 MN 1 2 δ F − d, d − η 45 MN µνρσ g e 1 2 δ – MN = so that √ MP η O( g e R − MN -dimensional action, x I δ 41 − 1 4 φ ) 4 − √ ∈ 6 1 8 = d d θ η = 0 B − √ x β Ω 1 4 4 e x Z N d 4 + + mn + + 1 2 F (4 + d = d for ], which with the above choice for internal scalars read Z B M = Z 45 F term arises from dimesionally reducing a H = F I 2 1 . Ω I β and R In going to Einstein frame and dualizing H = ∆ I mn Simons term, and so shouldfor be accompanied by an of motion) having introduced the notation The bosonic and heterotic stringsThe have There are only twometry independent [ four-derivative terms which respect the O( where F → δ to the 4D form [ bring the and the matrices JHEP11(2020)008 (3.3) (3.1) (3.2) (2.20) , , we need only ) solution. Such R 2 4 are determined by 2 ; ϕ 0 d ∂θ > d, d ϑ , respectively, so that . a 2 ξ O ) κ Q ξ + N 2 = 2 ·F + sin + 2 F r P 2 N 2 M ϑ κ F . ( , d F ·F ) 2 ) ξ and κ ) ξ N M 2 , = κ F + 2 ) ∂φ∂φ ·F + 2 2 2 ξ + = 0 2 κ . M MN r κ δ r )( F + 2 MN )( )( ξ , and the constants 1 η ξ φ ξ 2 ) 6 φ κ κ θ 4 πp − ( + 2 – 6 – − e + + 2 + NQ 1 e O 1 8 r 1 δ 1 = 4 κ , p κ ( κ − + − ( ) 2 ( P + ( ξ , MP π π π N 2 η N 2 ) N r 2 F 1 + 2 ·F and d 16 F κ = 4 = = 8 . From the metric we may read off the mass, temperature 1 1 ) M 0 M − M ξ κ S + T − πq ϕ . t , ( F M 2 f F r F d d 1 → R κ )( . − 1 + PQ ϑ , ξ = 4 1 δ κ 2 r 1 2 3 MN = κ t ( q Q η κ κ ∂φ∂φ r cos + MN d 2 2 + MN η q ) r + + p f , η φ r 4 r r ( − − − ∂φ 1 MN − 32 ( δ e φ = 0 = = = = φ 4 1 4 4 ) = 1 2 2 φ θ φ − 6 r 2 s − e A A ( e β − − d 1 2 f e e + + + 2 + 2 ) transformation, but does not represent the most general O( R , We will consider dyonic black holes, where the solutions are regular even in the extremal 2 The physical charges are The inner andextremality outer corresponds to horizons areand located entropy: at limit, with constant axion forSL( ease of calculation. Thissolutions can are be given arranged by via an appropriate 3 Leading-order solution The higher-derivative operators discussedto in extremal the black previousthe holes; section uncorrected will for black induce the corrections holeextremality thermodynamic condition. solution arguments in of order section to compute the leading corrections to the We have omitted thefor higher-derivative terms the solution involving of the section axion because they all vanish JHEP11(2020)008 a Ψ (4.2) (4.3) (4.4) (4.1) contains ∂ 5 I in the grand canonical . ) G 2 , α a ( , P O ) d . The contribution 2 a α Φ T 2 0)] + − + Ψ , P , a a , and may even include fields which ∂ ,P, 2(1 Φ I X a Φ dΦ + + a Q T, 2 I Q ( − Φ all denotes higher-derivative terms with their X T − [ 2 − I E + ∆ – 7 – TS T I 1 ∆ 0 d − I ) leads to S the electric potentials at the outer horizon and = ) = M − )] = 3.1 I ∞ ≡ = ,P = Φ r G G | , P, α ) d Φ T, a t = ( A E T, G ( − ( TI X ), we discuss briefly the determination of the black hole’s ther- [ − E I ξ 3.1 ], and the boundary terms associated with the other terms in the bulk =2 r | 47 ) a t . ) A 0 − α . ( 4 O ≥ is the two-derivative action and = ( in the grand canonical ensemble using only the leading-order solution: 0 D a ) ] for a more complete discussion of these ideas. I α Φ 46 ( The strength of this approach lies in there being no need to find solutions to the One may evaluate the Euclidean action to find the free energy The full four-derivative action can be written as This fails in 3D even for pure Einstein gravity, where the fall-off of boundary counter-terms is less than to the action [ O 5 T M 2 two-derivative Euclidean action for ( those in 4.1 Leading-order thermodynamics For the solution ofmodynamic ( properties via thetop Euclidean of action. which we This will compute give corrections the due leading to behavior, the on four-derivative terms. Evaluating the The dynamical fields arevanish at collectively denoted by with the analogous magnetic potential.may Via find straightforward the thermodynamic mass manipulations as one a function of the chargesperturbed and equations temperature in of the motion.to canonical ensemble. Namely, the Euclidean action may be reliably evaluated ensemble, given by where corresponding Wilson coefficients, collectively denotedall by boundary terms and isform required will for a not well-defined be variational principle: relevant to the our details discussion. of its action The vanish Gibbons-Hawking-York in term the contributes infinite-volume limit. See [ horizon, even at extremality. This ensures that the derivative expansion4 is under control. Higher-derivative corrections viaWe thermodynamics will leverage black holeratio thermodynamics of to extremal black compute holes, corrections and to so begin the by charge-to-mass recalling the key ingredients to this procedure. For large enough charges the black hole is large and the curvatures are small at the outer JHEP11(2020)008 , . T (4.8) (4.9) (4.5) (4.6) (4.7) (4.10) . The 2 ) 0 2 ) (other and , Φ → p P , 3 , − w 2.11 ) , T q 2 Q (1 w QP T 2 Φ 2 2 ) 2 2 ) 2 ) ) Φ + − 2 2 2 Φ p Φ − Φ Φ (1 3 − 2 3 ) − term in ( 3 3 ) − − 2 P w ) ) T . 2 (1 2 2 2 (1 Φ 2 p Φ (1 (1 + 2 Φ Φ or to thermodynamically in favor of P 2 2 1111 Φ w q − Φ − q , , ξ Φ Φ α − − . − QP q p w (1 − q, p (1 − − 2 2 2; 7; (1 (1 Q w w w 2 , 2 2 T x and ··· T 2 2 2 T T in the canonical ensemble, with 2 2 2 2 z ) = ) = ) = ) = P ) 1 κ P P P ) + − , F z 1 2 2 κ 2; 7; + 2; 7; 1) ) , 1; 6; 1; 6; , y 2 2 , , T, Q, P T, Q, P T, Q, P T, Q, P 2 − ( ( 3 1 1 P T, QT (2 ) ( 2 1 Φ( Ψ( 2 1 8 z x 1 1 F w F Φ 2 F F )( 2 − 2 2 ,S y 3 – 8 – = − (1 + 2Φ ) ) 2 z ] T 2 2 2 ! p − 2 4 ) 2 T − Φ Φ 4 w 3(1 x 2 ) Φ 2 ,M , ) 2 2 2 − − Φ P P Φ T ! ! Φ 3 2 − 2 2 and ) + 2 ) = ( ) = 1 ) ) − ! ! 2(2 2(2 − 2 P 1111 2 2 ) upon eliminating 2 x 2 2 ) ) T (1 2 2 ( α Φ ) ) 2 2 ) (1 Φ Φ T 2 2 2 2 y, z T T f 2 T 2 − Φ 3.3 5(1 P ( 3 4 2 2 − π 2 Φ Φ T − − − Φ 2 ) ) 2 P w 2 Φ Φ − 2 2 P 1 Φ T − − 2 − (1 (1 P 128 Φ Φ 2 3(1 , (1 Φ QT, P T 1111 1111 (1 (1 2 P − − 2 2 ( − − + α α 3(1 Φ 2 [3 1 1 Φ Φ w expansion which begins 2 2 − − 2 2 T

π π 0 1 1 Φ − 5(1 5(1 − − PT Φ 2 = 1

Φ T 1 1 T 64 64 − PT q 1 Φ T 1 T w + + + + ) ) = ) = ) = ) = y, z > R ) = ) = ) = , ,P ,P ,P ,P Φ Φ Φ Φ ,P ,P ,P being the root of the quintic Φ Φ Φ T, T, T, T, SL(2 ) ( ( ( T, T, T, S Ψ( Q ( ( M y, z Ψ( Q ( M from the metric in equation ( 4.2 For the sake ofcorrections example have we a present similar briefly form). the In results the for grand the canonical ensemble, we find This root is identifiedother by branches its of giving solutions aunstable correspond positive configurations. to mass different The which signs expressions remains for above finite exactly for match those found by reading off w with a small We have used from which it follows that in the grand canonical and canonical ensembles we have JHEP11(2020)008 (4.14) (4.15) (4.16) (4.11) (4.12) (4.13) ) P Q ) 2 q P Q 2 q w − w − , − 2 − QT ) )] QT (1 2 , 2 q (1 q 5; 8; 1 ) Φ ) q w , 2 q w )] 2 5; 6; 1 q 3 w 2 3 q , w − ) w ) 1 − 2 w q 2 1 − − (1 − T w Φ F 1 2 QT 3 q (1 + 3 QT 2 . (1 F Φ (1 − 2 q − w , q 2 q QT w ) QT 1; 6; 1 (1 + 3 . w − (1 w 1; 6; 1 (1 , q ) q P Q 2 , 1 1111 − ) 2 q )] ) − w 1 QT − w 2222 2 q T α 2 q 2 q 2 q w − 2 2 2 1 3 α w w w ) w 1 − − π 1 F P T 2 − )] F 2 2 q . − 3 q − 2 QT + ( q 2 Φ )[4 QT 2; 7; 1 w QT (1 2 1; 6; 1 q 128 w 22 , (1 3 q i (1 , − 84 ) q P w 1; 6; w 2 3 q 2 q α q (1 + 3 1 )[4 w 2; 6; 1 2 , P Q P Q w w q QT w , w (1 q 1 − 1 Φ − 5 2 1 2 1 w w − QT QT P Q 1111 − − F − − − F Φ 1 T (1 + 3 − 2 α − 1111 2 1 3 q 3 – 9 – − 5(1 q 2 + − F ) ) ) α P Q − F 2 w 2 w (1 q π 2 QT q 2 q 2 2 2 3 − w π w 2; 6; 1 ) w ) − ) 1; 6; 1 ! QT T 1; 6; 1 − QT 2 , q 2 2; 6; 1 p 2 q , QT 128 1111 5; 10; 1 2 , 3 2 5 1 , 64 − − 2 q )[4 1 w Φ , 3; 6; 1 w ) w 1 α 1 2 q P q , 2 3 − 5 2 w − 2 − 1 (1 (2 1 − 1 w − ) w Φ 1 π T QP T q 3 q 1 q 3; 8; 1 − F 2 q 1 F 2 ) )[4 F , 2 w F − w w 1 2 (1 2 2 q − q 64 F w 2 2; 7; + 3 2 P 2 (1 ) 3(1 2 F , ) (2 w w p 4 4 2 2 q − q 2 (1 )(8 ) 1 4 QT 5(1 ) 2 2 q q T − − ) w w QT QT F 1111 2 (2 1 − w w 2 P w P Q − q α − 2 F Φ + + 2 + + 1 ( 1122 1111 2 5(1 − − w 5(1 p + −

α 2 α − 2 ) w q 2 h 2 1 2 P Q + (1 2(2 (1 + q q T π 2 3 QP q p Φ 2 w − 5(1 − limit at fixed charges gives the corrected extremal charge-to-mass ratio: + 2 2 T w w Q w 1 × × × × P 64 − α 2 1 p 0 ( 1 + + 1212 p + ) = ) = ) = ) = 126 → 5 α 1 11 84 α α T ) = + + (4 − ,P = 1 + Φ T, Q, P T, Q, P T, Q, P T, Q, P ( ( S T, Φ( Ψ( ext ( M z S Taking the In the canonical ensemble, we find JHEP11(2020)008 a (4.20) (4.17) (4.18) (4.19) = 0) , such as )-invariant under the is positive, s, t shows that R ( , 2 ext 1) = 0 M z M , = 0) by considering a A ext s, t 3 = (0 ( z s for more discussion v M , and , A 2 i 0) s , Disc = ) is clear from the symmetry 2 , so that their corresponding s abcd . πi τ 0 = (1 d 2 α 2 0 ↔ s , u ≤ ) 1 ≥ 2 µν + 4 ∞ 22 0 α η , s s Z ) is more clearly seen. 2 + , α and 2222 cd = s 2 1 δ abcd . + ] from each terms in the SL( 11 α P ab α 4 0 α 0 ab α ↔ s µν 34 x δ α d ext g − 1 ≥ ↔ v z − − −∞ + c Z and u Q b 0 and v ) = ) = ) = 16( 1122 = 0) a – 10 – = φ ∓ d ≥ α u P ∓ b is also positive. 2 A 2 3 separately. There the invariance of x s, t θθθθ ∓ c gives s α ↔ ( ( 2 φA 1 A ) M = 0) X + ± a a,b,c,d Q x − ± b 3 M M 1 s A in . In particular, choosing − A α ( s, t a , 2 ( ± a Im 1111 s ) = , v M A s α a ( M d u = (1 s M ∞ 0 v πi s d φφφφ 2 , ( Z ) C 2 π are each positive. For the four-photon amplitudes, the crossing- I M , x = ext z = (1 line. u encircles the origin and is deformed to two integrations along cuts beginning = 0) = P C = s, t ( Q , and . (Rescaled) individual contributions to 0 M ] : the contributions at infinity are dropped, having assumed the Froissart bound. By ≥ 2 0 s s Assuming gravitational effects are subdominant (see appendix One may obtain positivity bounds on the coefficients present in [ ± 1212 symmetric combinations must be positive for allα real From these we may readcontributions off that to at the optical theorem the imaginaryshowing part that of the the crossing-symmetric coefficient amplitude of on this point), one has the following forward scattering amplitudes: The contour preserved electromagnetic duality ( scattering amplitudes aroundappears the in background the asymptoticcrossing-symmetric region forward amplitude, of we the have black [ holes considered here. For Figure 1 action. The preservedabout electromagnetic the duality ( Each contribution is shown in figure JHEP11(2020)008 2.2 (4.23) (4.24) (4.25) (4.26) (4.21) (4.22) ) 2 2 F . ( 2 a ) e F , with components ∂φ · ( 4 2 2 a z , F 1 is nearly fixed. F z 2 A ) transformations this + ∆ ± ext R ) z ; 2 1 = + 3∆ F ( 4 1 N d, d 2 F ) e z F · ∂φ ( 0 z , M β > 2 + 3∆ F z ) 2 2 2 1 + ∆ ). ) ∆ . F F 2 P P Q Q 2 1 ( β 1 MN F F z .) Up to O(                  η ( 1 − − θ 1 2 + 4.16 z ∆ 1 2 ...... A A α 0 0 1 8 ) symmetry is far more constraining than ∆ A A z ∂φ∂φF ± ± 1 2 ) symmetry the gravitational four-derivative R z − → − ∂φ∂φF ; ∆                  R 5; 6; 1 1; 6; 1 ) z − α θ ; 2 , , 2 – 11 – and ) = 1 1 F d, d 2 2 2∆ . That is, we should identify )( + 4∆ F d, d 1 1 2 M 1 3 )( 2 − F F F 2 1 F A 2 2 ( 2 = 1 + a 2 in equation ( F z F ( F 2 = 1 z ∆ RF · ext 4 z 1 4 z ∆ RF a ≥ x z ∆ F − + ∆ ext 2 2 − ) ) 1 2 z = 2 2 2 1 1 1 F F F + 1 ( ( N z z 1 F RF ∆ ∆ 1 ·F z 1 2 1 8 may be absorbed into ∆ RF M a z − − e F F − ∆ · ) 2 1 2 a R E MN z ; F δ . The choice x ± d, d = = ∆ ), the four-derivative terms are controlled by only two undetermined coefficients in such a way that β α O( z z R , M The extremal charge-to-mass ratio may be written as It is necessary to make connection between the gauge fields discussed in section ∆ ∆ 2 A where (The sign of is accomplished with and the interplay between positive and negative contributions to and the gauge fields solution ofof section For those theories withterms a are preserved always of O( the sameamong order the as Wilson coefficients, those we involving do theat not gauge have fields. our positivity bounds Without disposal. from a scatteringSL( hierarchy amplitudes However, since the O( is enough to conclude that 4.3 for all real JHEP11(2020)008 are , β (4.31) (4.32) (4.33) (4.34) (4.35) (4.36) (4.37) (4.38) (4.27) (4.28) (4.29) (4.30) z ∆ q p q p − − q p and − α z ): ∆ 6; 10; 1 6; 10; 1 , , 2 4; 8; 1 3 4 , 3 1 1 F . F 1 2 2 ) F . This WGC bound implies q 2 | 4 4 q p q p β q q + pq 2 3 − p 4 − ( ≥ | − + 2 p − 3 3 5 α 2 , , ± pq , pq q p 6; 8; 1 q p , is positive, in agreement with other 6; 10; 1 , , , = 23 , α − 1 q q p p , , or − 1 , , β q p + 67 − 0 q p z q p 1 − − 2 2 1 − F ∆ q q q p p q ≥ − F q p − 2 2 2 2 4; 8; 1 p ) p β − − , q − – 12 – q 6; 10; 1 1 4 4; 8; 1 4; 8; 1 , ± , p , , 6; 8; 1 2 6; 8; 1 ) 1 3 3 + 8 4; 6; 1 , α , q , + 337 + 217 2 2 F 1 p 3 2; 6; 1 2; 6; 1 1 2 1 1 4; 8; 1 ( q q , , 1120 F + , 2 p 1 3 3 F F 4 1 1 2 1 4 1 p p p 2 2 p F ) ) ( F 24 2 F q q ) ) 1 1 2 p 20 1 2 ) q q 5 F F ) ) 2 2 q F + + ) ) symmetry present the combinations 2 2 ) ) q q ) 2 q + + , . 2 q q p p R q ) ) + 133 = + 363 + ) ) ( ( ; p p ) ) q q + + 1 q q 4 + 4 6 q ( ( α p − − q q + q p p p p 3 4 4 p z p + + 4 1 ( ( p p + − 6 p p p p q q q ( d, d + + ( ( 3 4 ( follows from the null energy condition alone. ∆ p p 41 2 (17 4 2 2 4 p p q q 4 p p ( ( p p | q p p ( ( 2 ( 2 ( p 2 q q 5 420 210 1260 q β p p p p then amounts to 5 5 5 5 210 210 35 − − − − + 2 + 280 − − − 1 ≥ | ======α ≥ ) ) ) 2 2 4 4 2 1 1 2 1 2 2 ) 2 ) 2 2 2 1 2 F F F F F E 2 2 1 2 2 1 1 F F F z z z ext ( ( F F )( ( ( 2 2 z 2 ) ) ∆ ∆ ∆ RF RF 1 z z z z F ( ∂φ ∂φ ∆ ∆ ( ( ∂φ∂φF ∆ ∆ z z z z ∆ ∆ ∆ ∆ Requiring that the coefficientconsiderations of such the as Gauss-Bonnet stringthe term, theory condition examples and entropy arguments. As we now show, However, with thequite simple: O( (any term not present vanishes). The individual terms are (see figure JHEP11(2020)008 ν (4.40) (4.41) (4.42) (4.39) k ) 2 µ k α ( 2 O ) + . ∂φ β ( z corrections from ρ µν ) ∆ g b r α 1 2 F ( . With the choice and O − (4) a tρ µν α φ F T z ν rr ∆ g φ∂ tt and to µ g ∂ i z − (2) µν , ∆ T q E + 2 0 . ν , + 2 0 k , , ρ µ , the null energy condition requires (4) µν k µ 0 rr T b r k ν g F b ≥ can possibly give nonzero contribution to k + are potentially nonzero. Thus on this back- √ µ F ν , · k a rρ k (2) tt µν (4) µν a F µ g 2 T a ϑϕ T – 13 – k F ) rr − F g = , but there are also implicit µν µν ∂θ q g ( T α + D µν 1 4 ρ and µν T = g − b t 2 1 µ F ρ a tr terms of k F − b ν ) a tρ θ F α F ν ( has contributions from both the two- and four-derivative terms tt and many terms drop out of . O g a µρ θ∂ ) ) µ µν F − 2 α ∂ T ( α . (Rescaled) individual contributions ( φ ab ab O 4 δ δ O e φ φ = 2 2 2 1 − − on the corrected background. e e + φ, θ Figure 2 = = 0 + = (2) µν ν T k terms are explicitly of order µ k (4) µν (2) , so that µν T For our purposes it will suffice to work with the spherically-symmetric background with On any background and for any null vector q remains diagonal and only T = µν The last equality follows fromg the spherical symmetry ofground the corrected only solution, the for explicit which the leading contribution vanishes: The evaluating p where the stress tensor in the action: JHEP11(2020)008 is in (4.47) (4.45) (4.46) (4.43) (4.44) 2 ) with A 3.1 , ) νβ i g F 4 . F µα µν ) , must vanish. Similarly, F 2 T . 1 i ( 2 F α ) ext ( . 2 . 2 + 3 1111 z while the left-hand side has ∇ , ) 0 F ) να 2 ( α β µν ) − − 1 gravity multiplet and 2 2 ≥ T 2 ∇ φ F A F 1 8 4 1111 ) + ( a γν α µν − 1 ) which places the gauge fields in ξ α φ µ F 8 T = 2 have a negative coefficient and so − − 2 A . ) 2 − 1 1 2 ) 1 + 1 βγ a ) F 2 − 2 A βγ N 4.23 κ b 1) q 4 A F − F + 1 r F ) + 1 ( )( + ∓ + − α A 2 2 r 2 RFF 1 ( A r ) 2 a αβ p F F ( ( (1 2 ( a ν ( q 2 µν )( F , p 8 α 2 F T 1 F σ as expected, while the lower sign places the 5 ) ) 4 ( F 1 4 ab 2 ( is in the a µν ∼ M β the only contribution which does not vanish is a µ δ ∼ M να F T ) 1 + – 14 – F 1 µ ( ∓ ) = 1 8 − 2 A k − F − ) α φ 2 µαβγ α 1 2 h − = 1 + a να ) + µ ext F R 2 − a ( 1 and the corrections are F z µν φ 1 6 ) case this is easy to check: the heterotic string has F h F α ext F − F ( = (2 term proportional to µν α a R z + 4 supersymmetry we expect there to be no correction to h 8 1 − T 2 ν ; 2 F a µ h ) + k 1 2 s a 2 − − F b + − h µ F 8 ) h ( F d, d k , then − a ( 2 = a 2 − F , which leads to µν F M β 2 M RFF µν T φ, θ µν N ≥ RF ) )( T RFF g 2 2 µν a 1 ab RFF 2 1 µν T δ F F 2 h ( ( ( , T − β µν µν µν =1 X g a g g α ± . In contracting with , so that 1 ) = = = = = 0 κ ]. For the O( ) we may gain some insight by using relations between scattering amplitudes , 8 α ) 4 2 6 ) 2 R 2 = iθ F (4) 2 − µν µν , a F 2 , gravity multiplet giving 2 F T T RFF ± )( µν 0 , which coincides exactly with the WGC bound. ( µν κ 2 1 α 0 16 T φ T F : ( µν ≥ ν = 4 = T and term generated by the higher-derivative terms, and so k β ± ) = ( µ For SL( 2 q N φ k s ∓ = µν α α, β But the right-hand sideno has an for gauge fields in vector multiplets giving positive corrections to for fields in thethe same vector supermultiplet. multiplet If with ( The top sign correspondsthe to the choice in equation ( 4.4 Supersymmetry We can also askconsidered what above. restrictions With supersymmetrythe places charge-to-mass on ratio of topthe BPS of WGC states, the in much two [ like symmetries was found for quantum corrections to That is, the null2 energy condition requires that These terms we mayp simply evaluate on thefrom leading-order the solution last of term equation in ( T JHEP11(2020)008 (4.48) ) symme- R ; 2 s d, d , so that it must 2222 11 α α ) = 8 − 2 A − 2 A , the relation + 2 ± A φ ) and underlying O( + 2 ) symmetry we have found that the R A R , ( ; 2 and are nonzero, giving positive correction to 2 term proportional to d, d 2 A that follows from the null energy condition 2 ∼ M α s ) β − φ – 15 – and − and φ 22 α + α φ , + φ ( 2222 α are vital in canceling the positive contributions from other M ) symmetry, positivity bounds are then enough to demon- R , ) = 2 2 u + ∂φ∂φF F 2 t ( 1 ]. The work of GL and GS is supported in part by the DOE grant DE- α . For the vector multiplet with . All told, only 37 + 8 = 0 = 0 2 1 s 11 2 α α α 4 only when fields in the vector multiplet are turned on. Imposing SUSY on top of these two symmetries, we have found that the extremal The leading EMda action can have both an SL( ext related topic [ SC0017647 and the Kellett Awardby of JSPS the KAKENHI University Grant of Numbers Wisconsin. JP17H02894acknowledge TN and the JP20H01902. is hospitality supported TN of in andNSF part GS the PHY-1748958) gratefully Kavli while Institute part for of this Theoretical work Physics was (supported completed. by assumption on the UV theory. We leave this interesting questionAcknowledgments to future work. We would like toGS thank thank Yuta Hamada Stefano and Andriolo, Yu-tin Tzu-Chen Huang Huang for and useful Hirosi discussion. Ooguri TN for and collaboration in a charge-to-mass ratio is not correctedcharge alone, when as black expected holes for carryfrom BPS the terms states. gravity such Importantly, multiplet the as photon (always)terms. negative It contributions would be interesting,ering however, the to implications consider of what SUSY mileage alone one on the can four-derivative get terms, from again consid- as an intermediate strate the WGC inWGC general. requires In a imposing relationship the between alone. O( Whether the nullmore (or general other) settings energy is condition a implies question the worth mild further form exploration. of the WGC in whenever the dominant higher-derivative terms respect the symmetry intry. question. Imposing each of these individuallyterms on to the a action small restricts handful theafter which allowed higher-derivative then having succumb to imposed other SL( considerations. We have shown that which control the relativeappealing sizes to and scattering positivity signs bounds, but ofspace there these remain in allowed which terms regions corrections of may the to be parameter WGC. extremal Rather partially black than holes constrained working are with by atthe a odds particular form with UV of the theory expectations the insymmetries of order higher-derivative the to inherited corrections, determine one from more may finely the consider UV the as implications an of using intermediate assumption. Our analysis holds 5 Discussion The Einstein-Maxwell-dilaton-axion theory hasterms a which large correct number the of possible action in four-derivative the effective action framework. The Wilson coefficients be that imposes z with now only the right-hand side having an JHEP11(2020)008 ), is A.5 (A.3) (A.4) (A.5) (A.6) (A.1) (A.2) has no others M and the full others 6 M , , ] about positivity in the Regge limit: . 4 . 2 ··· ··· fixed + < s + . fixed 2 0 ) 0 2 , φ u ) µ fixed t < t < + 0 ] employed an assumption that s, t φ∂ 2 ( 4 µ t ∂ with ( t < + with α only. As a result, 2 s others s s + M with φ α grav -channel exchange appearing in the first µ t s ) necessary for the following argument. M ) + + 2 φ∂ , which is the main obstruction to deriving a µ t for large graviton exchange satisfying the bound ( 4 for large A.6 s, t ∂ 2 2 u Pl ( s 1 2 ), this in turn guarantees that coefficient, ref. [ – 16 – M 2 + 2 2 − grav 4 for large s A.3 t stu < s R < s ∼ − M | + 2 | ) 2 Pl ) 4 2 Reggeized s M s, t ) = < s s, t ( 2 in the Regge limit: Pl | ( g ) 1 s, t 2 M ( − 2 grav s, t coefficient. others √ ( M < s 2 x s |M 4 ) = |M |M d s, t Z ( . In particular the graviton ) = t M S + would not satisfy the bound ( s is the graviton exchange Reggeized by the higher spin tower and ( is Reggezied by the higher spin states and bounded as − grav = others grav u M Here it is crucial torather separate than theM simple graviton exchange. Otherwise, the subtracted amplitude Combined with the assumption ( massless poles in particular. M The massless graviton pole appears in To derive a rigorous bound on the The higher spin states are not necessarily particles appearing in tree-level exchange. They can also be 1. 2. 6 multi-particle states appearing in loops, where angular momenta of relative motion play the role of spins. the following conditions are satisfied: where contributions from other states. More explicitly, the decomposition is performed such that the graviton exchange is Reggeizedamplitudes by is an bounded infinite as tower of higher spin states Then they decomposed the amplitude as where term behaves in the forward limitpositivity as bound on the where the dots standamplitudes for follows higher from this derivative EFT terms. as The IR expansion of scalar four-point In this appendixbounds we in review gravitational theories. and Forto elaborate illustration, four let on derivatives us (generalization the consider to a argument more scalar-graviton general in EFT EFTs up ref. is straightforward): [ A Positivity bounds in gravitational theories JHEP11(2020)008 ] . ]. ), 4 of 8 : 34 6 A.6 (A.7) (A.8) (A.9) s, t, u T (A.10) (A.11) (A.12) Regge does not M . Also one 0 , , α > 4) 2 ), is subdominant. − u/ 4) n 0 2 coefficient. n Regge α 2 A.9 u/ 2 0 M s − E α 2 Pl M 4)Γ( 4)Γ( t/ × 0 t/ t 0 -th order polynomials of 0 α . n α α , (1) − 1 2 m 2 Regge t O 1 n 2+ in the IR as M 4)Γ( s s 7 =2 4)Γ( ∞ 2 2 , follows at least when the contribution Regge t Pl X n s/ 0 1 0 s/ n,m 2 M 0 Pl M 1 + α a α others 2 Pl 4 M × Γ( − α > u =0 -symmetric M M ' ∞ X Γ( + gravitational effects are subdominant (1) ) × n,m 4 – 17 – 4 O t u stu s, t, u = s, t (1) + ( + + 4 0 enjoys a bound 4 , s t 2 2 stu ≥ O α coefficient, a others 2 coefficients in the second term. Note that the sign of the Pl + type-II 2 α 1 4 term cannot be fixed at least by the present technology. = s M M s M 2 4 coefficient with an unconstrained sign. Based on this, ref. [ (1) coefficient depends on details of Reggeization and its sign α is the only scale characterizing the Reggeization of graviton 2 Pl E O has no massless graviton poles and it is bounded as eq. ( = 1 , we may expand (1) M We have argued that the standard positivity (1) 1 2 O O Regge grav − ), we find others M M M A.2 ) = and an follows under the standard assumptions of the positivity argument [ schematically denotes ] for recent followup] works for on more this detailed generalized study bound. of its dispersion relation and the s, t 0 ( 0 , n , 49 2 2 2 49 , a a E 48 coefficient of the type-II is the Regge slope. Its Regge limit is given by 0 (1) M α Assuming that exchange, we assigned O Regge states and its low-energy expansion is given schematically by where The Reggeization of the graviton exchange is controlled by a mass scale See refs. [ See also ref. [ 3. 7 8 where example for its violation. Consider thefrom tree-level the amplitude 10D of an graviton identical in scalar originating the type II superstring with a simple compactification on where the value ofcannot be the fixed at leastsome by examples the for present technology. such In a the decomposition. restType of this II appendix, we superstring. provide necessarily follow in the presence of gravity. Let us begin by providing an illustrative This is what we meanmay in rephrase the this main conclusion text that by with positive concluded that positivity of the from the gravitational Regge states, i.e., the second term in eq. ( Essentially because positivity of Matching with eq. ( Based on the assumption JHEP11(2020)008 SUSY (A.16) (A.17) (A.18) (A.13) (A.14) (A.15) have the originates = 8 9 | , we cannot appears and N , others | grav and the closed α ) reads 2 α Regge . o grav (1) ) g ). Then, one may M α ) such that O | 2 Pl A.11 s, t ( . A.5 A.4 | M , . Since the latter has a ) disk 1 others coefficient 8 2 s α ) = 0 2 M E others M s ( 1 α 2 s, t Pl s O ( g ) = M + s 1 s, t 4 g ( others u ∼ . coefficient is required by + M 0 4 (higher genus amplitudes) 2 others t s stu > , + M others ) ) + | 4 s s, t , s, t and also the disk amplitude does not contain ( others ( ), e.g., such that 2 , which provides an example for violation of the ) Pl contribution bigger than – 18 – , α 2 α 1 2 M 2 s s s, t A.4 2 ' = 0 ( < s type-II M between the open string coupling α α annulus 1 2 Pl s M ) = g M M annulus s, t ( ) = ∼ ) + Next, as an illustrative example for theories with a positive M 2 s, t / s, t ( 1 s grav ( | ) = g type-II . Note that a vanishing α 0 grav follows also when there exists an intermediate state whose coupling disk ∼ s, t M ( α o M coefficient originating from each amplitude is estimated as M g α > 2 grav s . On the other hand, the same hierarchy being the string scale and the closed string coupling, so that the latter s ) = g M s denotes the eighth and higher order derivatives. In particular, it has a g in general. Indeed, the IR expansion of the amplitude ( s, t ( ) 0 8 coefficient, we conclude that , each term is bounded as E 2 and t . The same hierarchy is expected to hold as long as there is no unusual hierarchy between them. open ( s ) s s α > O M M M ( We assumed that the scale of compactification and the volume of the cycles on which the brane wrapped O 9 , let us consider the case where the scalar originates from a 10D gauge boson in the open gravitational Regge states. Thisthe is the string case, scale e.g., andgraviton when appears the also as dilaton when an is the intermediate stabilized state. well KK below scale isare well below the string scale and the KK same mass scale asfrom gravitational Regge the states. hierarchy Thestring hierarchy coupling thus the positivity of is as small as the gravitational coupling, but whose mass is hierarchically lighter than with dominates over the formerpositive in the weakly coupled regime Note that in this example massive states generating a positive In particular, the where the first two terms arewe the disk ignore and higher annulus amplitudes, genus respectively.negative amplitudes In denoted the following by thegraviton third exchange. term. Therefore, we In may the perform Regge the limit decomposition with ( a (in the 4D language) of the typeOpen II string theories. amplitudes. α string spectrum. 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