PHYSICAL REVIEW D 15 JANUARY 1998VOLUME 57, NUMBER 2

One-loop supergravity corrections to black hole entropy and residual supersymmetry

Renata Kallosh, J. Rahmfeld, and Wing Kai WongPhysics Department, Stanford University, Stanford, California 94305-4060

~Received 18 August 1997; published 19 December 1997!

We study the one-loop corrections to the effective on-shell action ofN52 supergravity in the backgroundof the Reissner-Nordstro¨m black hole. In the extreme case the contributions from the graviton, gravitino, andphoton to the one-loop corrections to the entropy are shown to cancel. This gives the first explicit example ofthe supersymmetric nonrenormalization theorem for the on-shell action~entropy! for BPS configurations whichadmit Killing spinors. We display the residual supersymmetry of the perturbations of a general supersymmetrictheory in a bosonic BPS background.@S0556-2821~98!00902-3#

PACS number~s!: 04.65.1e, 04.70.Bw, 04.70.Dy

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It was shown by Gibbons and Hawking@1# for theReissner-Nordstro¨m black holes and in@2,3# for dilatonblack holes that the Bekenstein-Hawking entropy of blaholes can be found by evaluating the Euclidean on-shelltion of the theory in the semiclassical approximation. Tfirst quantum corrections to the entropy can be foundevaluating the one-loop partition function in the black hobackground.

The arguments in favor of a supersymmetric nonrenmalization theorem for the on-shell action~entropy! for con-figurations which admit Killing spinors were presented@4,2,5#. The proof is based on Berezin rules of integrationsuperspace and the fact that Killing spinors can be relateisometries in fermionic directions in superspace. The splest configuration constitutes the extreme ReissnNordstrom ~RN! black hole of pureN52 supergravity@6#.The existence of Killing spinors for this solution was estalished in@7#. However, there was no explicit one-loop calclation available to support or disprove the theorem of nrenormalization of the entropy, not even in the RN case.

The situation with the one-loop corrections in theN52theory was obscured by the existence of the so-called cformal anomalies@8#. The trace anomaly of the one-looon-shell supergravity in a gravitational background is givby the following expression:

T5gmn^Tmn&5gmn]

]gmnH ~detD3/2!2

detD2detD1J

5A

32p2* Rmnld* Rmnld. ~1!

HereDs denotes the contribution from positive and negathelicity states of spins52, two spins53/2, and spins51fields of the fullN52 supergravity multiplet. The coefficienA is known for all fields interacting with gravity.

The one-loop counterterm is proportional to the Eunumber of the manifold,Sone loop5(1/e)Ax. The fields ofN52 supergravity include a graviton, two gravitini, andvector field. The anomaly coefficientA5 11

12 of pure N52supergravity does not vanish. This means that in apurelygravitational background the contributions from spin 2, twspin 3/2, and spin 1 fields of the fullN52 supergravity

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multiplet do not cancel in the first loop. Although thosbackgrounds do not admit Killing spinors, and hence arerelevant for the proving or disproving the nonrenormaliztion theorem, there was not much incentive in calculatone-loop corrections in pureN52 supergravity in the background of the extreme Reissner-Nordstro¨m black hole. Onthe other hand, in theN54 theory where the conformaanomalies are absent, the calculations may be much mcomplicated and they have not been done either. In thisper, we will see that the one-loop contributions within tsupergravity multiplet cancel in a background admittingKilling spinor.

Meanwhile from a completely different perspective,study of quasinormal modes of various black holes wasveloped~see@11#!. Quasinormal modes of black holes prvide an opportunity to identify black holes when large-sclaser-interferometric detectors for gravitational waves willavailable. The formalism was applied in the study of blaholes with general dilaton coupling constanta in @9#. In mostcases only numerical calculations are possible. By applythose methods to the RN black hole Onozawa, MishimOkamura, and Ishihara@10# discovered a curious fact: Thresonant frequencies ofs52 waves with multipole indexj scoincide with those ofs51 waves with multipole indexj s21 in the extremal limit. This would be very difficult toestablish by looking directly into the standard form of twave equations in generic, nonextreme RN as given inbook of Chandrasekhar@11# in Chap. 5, Eq.~270!. The po-tential in the wave equation fors51,2 ~photons and gravi-tons! is defined there in terms of two different functionsf 1and f 2 and two different parameters1 q1 andq2:

Vs656qs

d fs

dr*1qs

2f s21@As~As22!# f s , s51,2. ~2!

Here

f s5D

r 3@~As22!r 1qs#~3!

1The notation forq1 andq2 is reversed in@11#.

1063 © 1997 The American Physical Society

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1064 57RENATA KALLOSH, J. RAHMFELD, AND WING KAI WONG

and qs ,As ,D are defined below. The graviton and photperturbations are encoded in functionsZ1 andZ2 @11# whichsatisfy the radial equations2

DsZs~r ![F d2

dr*2

1v22Vs~r !GZs~r !50, ~4!

where

dr

dr*5

D

r 2, D5r 222Mr 1Q2, ~5!

Vs5D

r 5FAsr 2qs14Q2

r G , ~6!

q153M2A9M214Q2~A222!, ~7!

q253M1A9M214Q2~A122!, ~8!

As5 j s~ j s11!, ~9!

j s5 l 1s ~ l 50,1,2, . . . !. ~10!

HereM andQ are the mass and charge of the black hole, aj s is an angular multipole index of the perturbation. In tSchwarzschild caseQ50, s51 ands52 correspond to theelectromagnetic and gravitational perturbations, respectivFor nonvanishing charge, however,Z1 and Z2 are linearcombinations of graviphoton and graviton perturbationsthe same multipole indexj s . Schematically, this can be understood from the quadratic action

S~2!51

2S Dgd2S

dgdgDg12Dg

d2S

dgdADA1DA

d2S

dAdADA

1DCd2S

dCdCDC D , ~11!

where the derivatives are evaluated using the backgroconfiguration. For Schwarzschild black holesd2S/dgdAvanishes and no diagonalization of the above actionneeded. However, whenQÞ0, the decoupled variablesZ1andZ2 are linear combinations ofDg andDA.

The spins5 32 contribution comes purely from the grav

itino and obeys Eq.~4! with the potential@12,13#

V3/25G2dT1

dr*, G5

D

r 6~lr 212Mr 22Q2!, ~12!

2The radial equations here describe the axial perturbationsVs

2[Vs . The polar perturbations have potentialsVs1 . We will rely

on the fact established in@11# that the quasinormal modes for axiand polar perturbations for gravitational waves~and for electromag-netic ones! are identical and therefore we will study here only tmechanism of cancellation in the one-loop approximation of aperturbation from spin 1, 3/2, 2. However, a more detailed analof this point would be desirable.

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T151

F22QF dF

dr*2lAl211G , ~13!

F5r 6

D1/2G, l5S j 3/22

1

2D S j 3/213

2D , ~14!

j 3/25 l 13

2~ l 50,1,2, . . . !. ~15!

The one-loop corrections to the partition function for tnonextreme RN black hole can be calculated using thwave equations. The total answer will be given by the cotribution of each field in the multiplet for eachl50,1,2, . . . :

I one loop5~eiW!one loop5)l 50

` H ~detD3/2!2

detD2detD1J . ~16!

Here, we have doubled the contribution from the axial pturbations to include the polar perturbations. Those havesame quasinormal frequencies and transmission and abtion coefficients~up to a phase! @11#, which leads us to ex-pect that they will give the same contributions to the onloop action as the axial modes.

For black holes which are far from extreme there isreason to expect any cancellation between different sfields. The potentials are completely different. In particulif one would takeQ50 corresponding to the Schwarzschiblack hole, it is reasonable to expect that the conformanomaly expression in Eq.~1! will be reproduced. This ispossible since the Euler number of the Schwarzschild blhole equals 2. It is difficult, however, to get anything elfrom the study of the expression for the one-loop correctiofor the nonextreme black holes.

Looking into either form of the potential for spin one anspin two waves as given in Eqs.~2! and ~6! for nonextremeblack holes, as well as on the gravitino potential, it is difcult to see how they could be related in the extreme limit.this limit at M51 we get

q15412 j 1 , q25222 j 2 . ~17!

But even at the extremeV1 is different fromV2 for all l :

V12V252D

r 5~22r !~ l 12!. ~18!

However,numerical calculationsin @10# have shown that thequasinormal frequency trajectories of the photon, gravitiand graviton with increasing charge meet at the same pin the limit of maximal chargeQ5M . Moreover, in @14#later the numerical calculations were extended to higherrap-idly damped modesof nearly extreme black holes and it wafound that theyspiral into the value of the extreme blachole as the charge increases. The behavior of the trajectoriewas found to change dramatically for the chargeM>Q.0.9M .

To explain this curious behavior of quasinormal modfor the various spin waves in the background of the neextreme RN black holes, the authors of@10# and @15# were

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57 1065ONE-LOOP SUPERGRAVITY CORRECTIONS TO BLACK . . .

able to reorganize the form of the potentials in the waequations in such a way as to explain their previous resultan analytic way for the extreme black holes withM5Q51 with horizon atr 51. First of all the potential for allthree fields is now given in terms of only one function ofr :

f 5r 21

r 2. ~19!

The potentials for the extreme RN solution acquire the fo

V151~ j 111!d f

dr*24 f 31~ j 111!2f 2, ~20!

V3/251S j 3/211

2D d f

dr*24 f 31S j 3/21

1

2D 2

f 2, ~21!

V252 j 2

d f

dr*24 f 31 j 2

2f 2, ~22!

j s5 l 1s ~ l 50,1,2, . . . !, ~23!

and the difference betweenV1 andV2 becomes

V12V252D

r 2~22r !~ l 12!5~ j 11 j 211!

d f

dr*. ~24!

The tortoise coordinater * in terms of which the wave dif-ferential operator is very simple for the extreme RN blahole is

r * 5r 211 ln~r 21!221

r 21. ~25!

Of crucial importance is thatr * maps infinity to infinity andthe horizon to negative infinity:

r→`⇒r *→`, ~26!

r→1⇒r *→2`. ~27!

Also the relation

r *→2r * ⇒r 21→1

r 21~28!

will prove to be very useful.Fortunately for supersymmetry, as we will see later,

wave equation for the radial coordinate has a simple secorder differential operator in tortoise coordinate as givenDs in Eq. ~4!.

These three potentials are related in the following way

V1~r * , j 15 j !5V3/2S r * , j 3/25 j 11

2D5V2~2r * , j 25 j 11!,

~29!

where j is a positive integer. The first equality is obviousEqs. ~20! and ~21!; the potential of the Rarita-Schwingefield is identical to that of the spin one field if we shift thmultipole index by 1/2. The second equality is proved in@15#by using the relationf (r * )5 f (2r * ) which follows from

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Eq. ~28!. ThereforeV2 can be obtained by reflectingV1 orV3/2 about r * 50. This transformation corresponds to thexchange of the horizon and infinity. It has been deducedfrom Eq. ~29! in @15# that a scattering problem for each peturbed field with a corresponding multipole index resultsthe same transmission and reflection amplitudes. Thisalso derived very recently using the unbroken supersymtry of the extreme RN in@16#.

Where does all this bring us with respect to the one-locorrections to the entropy of the extreme black holes? Iffor the strange fact that the potentials for thes52 perturba-tions are related to thes51 perturbations by the inversion othe tortoise coordinate, we would not be able to concluthat there are no corrections. However, the fact that in cculating one-loop Feynman diagrams in the backgroundthe extreme black hole an exchange of the horizon withfinity is needed is rather unusual. Let us look into thclosely. We would like to compare the one-loop contributifor every l for s51 and s52 perturbations. The relevanpart of the path integral is

E dF1dF2expS i E dr* ~F1D1F11F2D2F2! D ~30!

where

D15F d2

dr*2

1v22V1~r * !G , D25F d2

dr*2

1v22V2~r * !G .

~31!

Now we may use the facts that the termd2/dr*2 is even inr *

and thatV1(r * )5V2(2r * ) for the same value ofl . Hence,we conclude that

D1~r * !5D2~2r * !; ~32!

i.e., not only the potential but also the total wave equatiofor spin one and spin two are related by the inversion oftortoise coordinate and therefore by the exchange of therizon and infinity. Thus the quadratic part of the relevaaction fors51 is

S15E2`

1`

dr* F1~r * !D1S d

dr*,r * DF1~r * ! ~33!

and the quadratic part of the relevant action fors52 is

S25E2`

1`

dr* F2~r * !D2S d

dr*,r * DF2~r * !

5E2`

1`

dr* F2~r * !D1S 2d

dr*,2r * DF2~r * !. ~34!

Since the integral overr * extends from2` to 1`, we maychange the integrationr *→2r * variables and get

S25E2`

1`

dr* F2~2r * !D1S d

dr*,r * DF2~2r * !. ~35!

The last step is to consider the change of variablesthe path integral and to integrate instead ofF2(r * ) overF2(2r * ):

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1066 57RENATA KALLOSH, J. RAHMFELD, AND WING KAI WONG

E dF1~r * !dF2~2r * !expS i E dr* @F1~r * !D1F1~r * !

1F2~2r * !D1F2~2r * !# D . ~36!

Introducing the notation

F2~2r * !5F2~r * !, ~37!

we may rewrite the path integral in the form where the cotribution from the gravitons and from the photons is the safor eachl :

E dF1~r * !dF2~r * !expS i E dr* @F1~r * !D1F1~r * !

1F2~r * !D1F2~r * !# D . ~38!

Since the wave equation for the radial mode of the gravitfor eachl coincides with the equation fors51, we concludethat we have now proved the total~axial! cancellation of theone-loop correction to the on-shell action ofN52 super-gravity in the extreme RN background. Assuming thatpolar modes give the same contributions as the axial onefind

I one loop5~eiW!one loop5)l 50

` H ~detD3/2!2

detD2detD1J 5)

l 50

` H detD3/2

detD1J 2

51. ~39!

The nontrivial part of this analysis was the use of the codinate system where the horizon of the extreme black hopushed to infinity which allows a particular change of vaables in the one-loop path integral.

Thus, the cancellation of the one-loop corrections toon-shell action~black hole entropy! found here supports thgeneral argument of the supersymmetric nonrenormalizatheorem@4,2,5#. The theorem states that the extreme blahole entropy does not have corrections in any loop ordesupergravity theory. This is in complete agreement withtopological character of the entropy of black holes with ubroken supersymmetry@17#.

It would be interesting to see whether a similar cancetion can be established in extremal but nonsupersymmeblack holes as suggested in@18#. However, the perturbationequations will be very different and without the hidden heof unbroken supersymmetries it is not clear whether theguments presented in this paper can be repeated.

The striking similarity between the spin 1,32, and spin 2

perturbations in the extremal supersymmetric case certahas a deeper explanation in the unbroken supersymmetthe background. In the present case this was establishe@16#.

We will now set up ageneral framework to relate variouperturbations via the unbroken residual supersymmetry. Asthe Killing spinor, which is the transformation parameter,neither constant nor an arbitrary function in supergravity,are dealing withrigid supersymmetry. We are using here thecondensed notation introduced by De Witt@19# and devel-

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oped in the context of the background field method for gautheories in@20#. The underlying actionS(f,c) depending onbosonic and fermionic fieldsf i andca is invariant under thesupersymmetry transformations

df i5Rai ~f,c!ea,

dca5Raa~f,c!ea, ~40!

whereR denote field-dependent matrices andea are the su-persymmetry parameters. If we are interested in bosonicfermionic perturbationsDf i andDca propagating in a back-ground given byf0 ,c0, it is useful to expand the action anthe supersymmetry transformations.

For a purely bosonic background withc050 the expan-sion of the action is given by

S~f,c!5S~0!~f0 ,c050!1S~2!~Df,Dc,f0 ,c050!1•••,~41!

whereS(2) is the Gaussian action for the perturbations givby

S~2!5 12 Df iS,i j

~0!Df j1 12 DcaS,ab

~0!Dcb. ~42!

This action eventually gives rise to Eq.~4! and also governsthe one-loop corrections. The indices denote derivatives wrespect to bosons (i , j , . . . ) andfermions (a,b, . . . ). Thesuperscript ~0! implies that the relevant expressions aevaluated using the background fields and summation isderstood to include integration.

The supersymmetry transformations become

d~f i1Df i !5Ra~0!iea1Ra,a

~0!iDcaea,

d~ca1Dca!5Ra~0!aea1Ra,i

~0!aDf iea. ~43!

S(f,c) is clearly invariant under the full set of transformtions; however, we are interested in the symmetries ofS(2) ina fixed background. This restricts the transformationsthose which leave the background invariant. The remainsymmetry ofS(2) is a rigid supersymmetry where the tranformation parameters are the Killing spinorseK

A of the back-ground. The resulting symmetry transformations on the pturbations are

dDf i5RA,a~0!iDcaeK

A ,

dDca5RA,i~0!aDf ieK

A , ~44!

where the indexA denotes the Killing spinors, rather than thwhole set of transformation parameters. These equationslate bosonic and fermionic perturbations or quantum flucttions. Hence, this residual supersymmetry controls quancorrections around black holes which admit Killing spinor

Note that the same formalism applies~in any dimension!to local as well as to global supersymmetries. Then, inlocal case the Killing spinors and the residual supersymmtry are rigid, whereas in the global case they are global.

We do not attempt here to make the connection betwthis general formalism and Eqs.~4!–~10! more precise. The

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57 1067ONE-LOOP SUPERGRAVITY CORRECTIONS TO BLACK . . .

advantage of using Eqs.~4!–~10! is that they are very simpleThe advantage of using Eqs.~44! is that they are universaand alwaysconnect solutions for bosonic perturbations withose of fermionic perturbations via Killing spinors. Thisexplains the cancellation of the one-loop corrections toentropy via residual supersymmetry.

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We are grateful to H. Onozawa and N. Anderssonuseful correspondence, and the organizers and participanthe ‘‘Black Hole’’ conference in Banff, and especially GGibbons, G. Horowitz, and E. Poisson for stimulating discsions. The work of R.K., J.R., and W.K.W. is supportedthe NSF grant PHY-9219345.

s

ys.

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