Quantum improved charged black holes

Akihiro Ishibashi,1,2,* Nobuyoshi Ohta ,3,2,† and Daiki Yamaguchi1,‡1Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502, Japan2Research Institute for Science and Technology, Kindai University, Higashi-Osaka,

Osaka 577-8502, Japan3Department of Physics, National Central University, Zhongli, Taoyuan 320317, Taiwan

(Received 26 July 2021; accepted 17 August 2021; published 15 September 2021)

We consider quantum effects of gravitational and electromagnetic fields in spherically symmetric blackhole spacetimes in the asymptotic safety scenario. Introducing both the running gravitational andelectromagnetic couplings from the renormalization group equations and applying a physically sensiblescale identification scheme based on the Kretschmann scalar, we construct a quantum-mechanicallycorrected, or “quantum improved” Reissner-Nordstrom metric. We study the global structure of thequantum improved geometry and show, in particular, that the central singularity is resolved, being generallyreplaced with a regular Minkowski core, where the curvature tensor vanishes. Exploring cases with moregeneral scale identifications, we further find that the space of quantum improved geometries is divided intotwo regions: one for geometries with a regular Minkowski core and the other for those with a weaksingularity at the center. At the boundary of the two regions, the geometry has either Minkowski-, de Sitter-,or anti-de Sitter core.

DOI: 10.1103/PhysRevD.104.066016

I. INTRODUCTION

The occurrence of a spacetime singularity under com-plete gravitational collapse is a robust prediction of generalrelativity, as the singularity theorem asserts [1]. Generalrelativity is therefore not a complete theory of gravity as itbreaks down at spacetime singularities. The resolution ofspacetime singularities is one of the main purposes ofconstructing quantum theory of gravity.The asymptotic safety scenario is one of such attempts to

formulate a consistent quantum theory of gravity, based onthe renormalization group flow that controls coupling con-stants [2]. For reviews of this approach and related refer-ences, see [3,4]. As an application to black holes, Bonannoand Reuter [5] have studied possible consequences of theasymptotic safe gravity on the Schwarzschild metric, inwhich the ordinary Newton constant in the classical metriccomponent is replaced with the running gravitationalconstant obtained from the renormalization group equation.Such a quantum-mechanically corrected black hole isreferred to as the “renormalization group improved” or

quantum improved black hole. Since the running Newtonconstant acquires a certain scale dependence, the globalstructure of the quantum improved Schwarzschild blackhole becomes quite different from its classical counterpart,admitting an inner (Cauchy) horizon besides the blackhole event horizon. It was shown, in particular, that thecentral curvature singularity of the classical Schwarzschildmetric is resolved to be a regular center, the neighbor-hood of which is well described by the de Sitter metric. Inthis regard, the global structure of the quantum improvedSchwarzschild black hole is similar to that of the Bardeenblack hole [6].Although the appearance of the Cauchy horizon raises

new issues such as the breakdown of the strong cosmiccensorship, the resolution of a classical singularity is afascinating aspect anticipated for the asymptotically safegravity to be a successful candidate of quantum theoryof gravity. Subsequently further analyses of quantumimproved black holes have been made, and a number ofbasic properties, such as the horizon structure, the thermo-dynamic laws, and Hawking temperature, have beenstudied in detail [7–16]. For instance, the analysis wasgeneralized to include the rotation (angular momentum)parameter and a cosmological constant (either positive ornegative), with the latter replaced with the running cos-mological constant [13]. It was found that the curvaturesingularity of the quantum improved Kerr–(anti)-de Sitter[(A)dS] geometry becomes less divergent compared to theclassical counterpart but is not fully removed.

*akihiro@phys.kindai.ac.jp†ohtan@ncu.edu.tw‡daichanqg@gmail.com

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

PHYSICAL REVIEW D 104, 066016 (2021)

2470-0010=2021=104(6)=066016(13) 066016-1 Published by the American Physical Society

So far, most of the analyses of quantum improved blackholes have focused on quantum effects of pure gravity,rather than combined effects of gravity and other funda-mental forces. However, when considering quantum effectsaround a singularity, it is natural to ask whether funda-mental forces other than gravity play any significant role.It is far from obvious if, among all, gravity is the mostdominant force to shape the structure of singularities.The contribution of matter fields to the evolutions ofthe cosmological constant and Newton constant hasbeen studied in [17–20]. Even within classical framework,including, for example, electromagnetic interaction drasti-cally modifies the global structure of the otherwise vacuumSchwarzschild solution by rendering the central singularitytimelike hidden inside the Cauchy horizon, as the classicalReissner-Nordstrom metric exhibits. In view of this, it is ofconsiderable interest to explore if and how quantum effectsof the electromagnetic field together with gravity affect thespacetime structure especially near singularities in thecontext of the asymptotic safety [21–23]. For other relatedapproach to the resolution of singularities, see [24,25], andalso, e.g., [26,27].In this paper, we consider the problem of singularity

resolution by examining yet another quantum improvedblack hole including the running gravitational as well asUð1Þ gauge couplings: that is, the “quantum improvedReissner-Nordstrom metric,” as a first step toward under-standing fundamental interactions all together in theasymptotic safety scenario. For this purpose, we need toaddress the following two issues: First, when taking intoaccount quantum effects of the electromagnetic field,one has to deal with Landau poles, which involve the(logarithmic) divergence of the running Uð1Þ gauge cou-pling at a finite momentum scale. To handle this problemwe employ the method developed by Eichhorn andVersteegen [23], which exploits the ultraviolet comple-tion for Uð1Þ gauge theory induced from asymptotic safequantum gravity. Second, one has to determine how toidentify the cutoff scale with physical distance. See [28] fordiscussions on issues with such identification. So far,several different schemes for the scale identification havebeen proposed. In this paper we employ the scheme basedon the Kretschmann scalar considered by Pawlowskiand Stock [13]. This has the advantage that it is a diffeo-morphism invariant scheme and is also easy to compare thecase of classical vacuum (e.g., Schwarzschild) solutionwith that of nonvacuum (e.g., Reissner-Nordstrom) solu-tion. With these strategies at our hand, we will construct aquantum improved Reissner-Nordstrom metric. We showthat the singularity of our quantum improved Reissner-Nordstrom black hole is in general resolved so that in theneighborhood of the center the curvature tensor vanishes.Namely, the quantum improved Reissner-Norstrom blackhole generally admits the “Minkowski core.” For compari-son, we also consider the type of scale identification in

which the cutoff momentum is given by the general inversepower law and show that in the space of quantum improvedReissner-Nordstrom solutions, the geometries with eitherde Sitter(dS) core or anti-de Sitter(AdS) core appear as theboundary between the region for regular solutions and thatfor weakly singular solutions.This paper is organized as follows. In Sec. II, we briefly

review the quantum improved Schwarzschild spacetimestudied in Ref. [5], which is regular at the center with a dScore. We also discuss how the running Newton constantdepends on the cutoff momentum scale k, and how toidentify the scale k with the inverse of the distance r fromthe center of spherical symmetry. In Sec. III, we studypossible combined effects of the Uð1Þ gauge field togetherwith quantum gravity on the Reissner-Nordstrom metric.We derive both the running gravitational and Uð1Þ gaugecouplings from the renormalization group equations. Wefind that our quantum improved Reissner-Nordstrom space-time in general possesses a regular core at the center. As thefinal topic of Sec. III, we consider the scale identificationscheme of the inverse power radial dependence and studythe condition for quantum improved solutions with regularcenter and that for weakly singular solutions. Section IV isdevoted to summary and discussions.

II. QUANTUM IMPROVED SCHWARZCHILDBLACK HOLE

In this section we briefly review the result of Ref. [5].The Schwarzschild spacetime is the unique sphericallysymmetric vacuum solution of the Einstein equation. TheSchwarzschild metric is written as

ds2 ¼ −fðrÞdt2 þ 1

fðrÞ dr2 þ r2ðdθ2 þ sin2θdϕ2Þ; ð2:1Þ

where the metric function is given by

fðrÞ ¼ 1 −2G0M

r; ð2:2Þ

with the mass parameter M and the Newton constant G0.Hereafter, the metric function fðrÞ is called the lapsefunction.In Ref. [5], quantum gravitational effects in the

Schwarzschild spacetime have been studied, by using therenormalization group method and introducing the runningNewton constant. The first step is to derive the scale-dependent Newton constant GðkÞ by considering theevolution of scale-dependent effective gravitational actionby means of an exact renormalization group equation. Theevolution of the dimensionless Newton constant GðkÞ ≔k2GðkÞ is governed by

kdGdk

¼ βðGðkÞÞ; ð2:3Þ

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where the beta function βðGÞ is given by Eq. (2.13) ofRef. [5]. This has two (i.e., IR and UV) fixed points. Byinspecting the UV attractive fixed point, the dimensionfulNewton constant GðkÞ is obtained, with the identificationG0 ¼ Gðk ¼ 0Þ, as

GðkÞ ¼ G0

1þ ωG0k2; ð2:4Þ

where ω is the constant which can essentially be identifiedwith the inverse of the UV fixed point (see discussionaround Eqs. (2.21)–(2.25) of Ref. [5]).The next step is to identify the cutoff scale k with the

radial distance r from the center of the spherical symmetry.Such a scale identification is schematically written as

k ¼ ξ

dðrÞ ; ð2:5Þ

where dðrÞ denotes an appropriate function of r and ξ anumerical constant. So far, several different choices of dðrÞhave been proposed. For example, one can choose dðrÞ asthe proper distance from an observer to the center, along astraight (non-null) radial path. In this case, one obtainskðrÞ ∝ 1=r3=2 in the UV regime.The idea of constructing the quantum improved

Schwarzschild metric is to obtain the position-dependentNewton constant GðrÞ ¼ GðkðrÞÞ by inserting a certainchoice of the scale identification (2.5) into the runningNewton constant GðkÞ, and then to replace the Newtonconstant G0 in the classical lapse function (2.2) by theposition-dependent counterpart GðrÞ. By doing so, oneobtains the quantum improved lapse function

fðrÞ ¼ 1 −2Mr

GðrÞ: ð2:6Þ

Note that as will be described below, instead of replacingthe classical gravitational constant G0 in the metric func-tion, one can also take the replacement ofG0 in the Einsteinequations or G0 in the Einstein-Hilbert action.For the choice of dðrÞ as the proper radial distance,

one finds that dðrÞ ¼ r½1þOð1=rÞ� at large r and dðrÞ ¼r3=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=ð9G0MÞp þOðr5=2Þ near the center r ¼ 0 and

therefore that the position-dependent Newton constantbecomes GðrÞ ≃G0 − ωξ2G2

0=r2 þOð1=r3Þ at large r

andGðrÞ ≃ 2r3=9ωξ2G0M near r ¼ 0. Hence, in particular,

the quantum improved lapse function behaves near thecenter as

fðrÞ ¼ 1 −4

9ωξ2G0

r2 þOðr3Þ: ð2:7Þ

Thus, the quantum improved lapse function is regular nearthe center and describes the de Sitter geometry withthe curvature length l ¼ ð3=2Þξ ffiffiffiffiffiffiffiffiffi

ωG0

p. Such a near-center

region is called the “de Sitter core.” Note that accordingto Ref. [5], for some cases of interest, by comparingthe behavior of GðrÞ at large r with the results of theperturbative quantization of the Einstein gravity, the valueof ωξ2 can be determined as ωξ2 ¼ 118=ð15πÞ.The choice of dðrÞ determines the r dependence of the

momentum scale k. However, such a scale identification isnot unique, in particular, for pure gravity systems. So far,several different scale identifications have been considered.In the above example [5], dðrÞ is taken as the proper radialdistance. Another diffeomorphism invariant procedure is toexpress dðrÞ in terms of the curvature scalars, such as thescalar curvature, or the Kretschmann scalar of either theclassical geometry or the corresponding quantum improvedgeometry. We list some examples of dðrÞ relevant to oursubsequent analyses and their behavior in the UV regime inTable I. For more detailed list, see Ref. [13]. See alsoRef. [29] for discussion on the use of various curvatureinvariants in quantum improvements of spherically sym-metric and axially symmetric black hole spacetimes andtheir coordinate (in)dependence.Let us briefly summarize the prescription for construct-

ing quantum improved geometry. For more details, see,e.g., [5,13,16,30].Step 1. Derive the scale-dependent coupling constants

from exact renormalization group (RG) equations:

e.g., k dðk2GÞdk ¼ βðk2GÞ for the running gravita-

tional constant GðkÞ.Step 2. Find a cutoff scale identification k ¼ kðxÞ and

convert the scale dependence of the couplings intothe position dependence: e.g., GðkÞ → GðxÞ ≔GðkðxÞÞ. For static spherically symmetric geom-etries, it is natural to assume that k be a functiononly of the area radius r. As briefly shown inTable I, there have been several different schemesfor scale identifications, which may roughly beclassified into two types:

TABLE I. Behavior of dðrÞ in the short distance limit. Numerical factors are dropped.

Kretschmann Radial path Geodesic path

Schwarzschild (Classical) r3=2 r3=2 r3=2

Schwarzschild (Quantum) r3=4 r r3=4

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(i) Based on curves toward a singular point as theUV limit: e.g., k ∝ 1=dðrÞ with dðrÞ being theproper radial distance.

(ii) Based on invariant curvature scalars: e.g.,such as R, Rα

βRβα, or K ¼ RμναβRμναβ with

k ∝ K1=4.One may use the relevant geometric quantities above withrespect either to the classical geometry or to the quantumimproved geometry [13].Step 3. Perform quantum improvement. There are three

approaches:(i) Improved solutions. One considers a classical

solution and replaces the couplings appearingin the classical solution with the position-dependent ones: e.g., G0 in the classical lapsefunction fðrÞ is replaced with GðrÞ.

(ii) Improved equations of motion. One replacesthe couplings with the position- dependent onesat the level of equations of motion: e.g., oneneeds to solve the improved Einstein equations,Gμν ¼ 8πGðxÞTμν − ΛðxÞgμν, where ΛðxÞ isthe running cosmological constant and wherethe running of matter couplings are ignored.

(iii) Improved action. One replaces the couplingswith the position-dependent ones at the level ofaction, derives the modified field equationsfrom the quantum improved action, and thensolves the modified field equations. For in-stance, the quantum improved Einstein-Hilbertaction is given by the modified Lagrangian,L ¼ 1

16πGðxÞ fR − 2ΛðxÞg.These three approaches make, in principle,

different predictions. For instance, the equa-tions of motion derived from the quantumimproved Einstein-Hilbert action can containthe derivatives of GðrÞ, while the quantumimproved Einstein equation does not. Theimprovement at the level of action is expectedto be most natural and have highest predictivepower. The improvement at the level of sol-utions is the simplest approach but is thought tobe applicable only when the quantum improvedmetric is not drastically different from theclassical counterpart.

In the next section, we will consider quantum improve-ment of the Reissner-Nordstrom geometry. With theelectromagnetic coupling e0 in addition to G0, we followSteps 1 and 2. (We set Λ ¼ 0.) Then, with two position-dependent couplings eðrÞ, GðrÞ at hand, the hardest part ofthe job would be to perform Step 3 if one takes theapproach of either (ii) the improved Einstein equations, orof (iii) the improved Einstein-Hilbert-Maxwell action. Inthis paper, we shall take (i) the improved solution approach,following [5].

III. QUANTUM IMPROVED REISSNER-NORDSTROM BLACK HOLE

In this section, we study quantum effects of both gravityand electromagnetic fields on the static spherically sym-metric charged black hole geometry. For this purpose, wetake, as our classical background, the Reissner-Nordstrommetric of the form (2.1) with the classical lapse function

fðrÞ ¼ 1 −2G0M

rþ G0e20

r2; ð3:1Þ

where e0 is the classical Uð1Þ coupling. We start withderiving the running Newton constantGðkÞ and the runningUð1Þ gauge coupling eðkÞ from the functional renormal-ization group equations. Each of these two runningcouplings acquires a certain dependence on the radialdistance r through a scale identification with the momen-tum cutoff k. As a consequence, we obtain a quantum-mechanically improved Reissner-Nordstrom metric. Westudy the global structure—in particular, the central singu-larity—of the quantum-mechanically improved Reissner-Nordstrom black hole.

A. Running couplings

When considering quantum effects of electromagneticinteraction, one must deal with Landau poles. For thispurpose we employ the method recently proposed byEichhorn and Versteegen [23], in which the UV completionforUð1Þ gauge theory is successfully achieved by using thefunctional renormalization group in Uð1Þ-coupled gravitytheory. Although the running of the cosmological constantΛ is also considered in Ref. [23], in this paper, we restrictour attention to the Newton and Uð1Þ gauge couplings.Then, our beta functions for running gravitational andUð1Þgauge couplings are given as follows (see also [21]):

kdGdk

¼ 2G

�1 −

1

4παG

�; ð3:2Þ

kdedk

¼ 1

4πe

�b4π

e2 − G

�; ð3:3Þ

where α; b are the parameters which specify the fixedpoints G�; e� as

G� ¼ 4πα; e2� ¼ ð4πÞ2 αb: ð3:4Þ

Note that for the Schwarzschild background case, one mayidentify α with ω in Eq. (2.4) as

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ω ¼ 1

4πα: ð3:5Þ

One may specifically determine the value of b in terms oftheUð1Þ charges ðQDi; QSiÞ of the matter (Dirac and scalar)fields:

b ¼ 4

3

Xi

ðQDiÞ2 þ1

3

Xi

ðQSiÞ2; ð3:6Þ

where the subscripts D and S stand for Dirac and forcharged scalar fields, respectively, and i denotes the kindsof the fields. In the context of the Standard Model inparticle physics, the fixed-point value of the Uð1Þ gaugecoupling is given with the value b ¼ 41=6 (see Eq. (4.1) inRef. [23]). If one chooses

NV ¼ 1þ 2ND −NS

2; ð3:7Þ

with NV; ND; NS being the numbers of vector, Dirac, andscalar fields, respectively, then the cosmological constantmay be set to zero consistently with the renormalizationgroup equation (namely it is a fixed point). Though thisrelation is not satisfied in the Standard Model in whichNV ¼ 12; ND ¼ 45=2, and NS ¼ 4, there may be addi-tional particles like dark matter and we can expect that thismay be satisfied in particle physics models beyond theStandard Model.We will find that the singularity is resolved only for

vanishing cosmological constant, and then the condition(3.7) could be even regarded as an interesting criterionto select theories. The difference of the rhs and lhs ofEq. (3.7) for the Standard Model is 32, which requires 64additional scalar fields as dark matter. Another examplethat is closer to satisfying the condition (3.7) is the minimalsupersymmetric standard model where we have additionalsupersymmetric partners, ΔND ¼ ð12þ 4Þ=2 and ΔNS ¼94 (we must include a second Higgs doublet in super-symmetric theory). This is close to satisfying (3.7) [thedifference of the rhs and lhs of Eq. (3.7) reduces to 1]but not precisely. We could also consider SUð5Þ grandunified theory, where we have 24 gauge bosons, and twoHiggs fields in the representations of 24 and 5 to breakSUð5Þ gauge symmetry down to SUð3Þ × SUð2Þ ×Uð1Þand to give masses to quarks and leptons, respectively. Herewe have NV ¼ 24, ND ¼ 45=2, and NS ¼ 34, and thedifference is 5. In this case, we could consider an additional(complex) scalar multiplet in the 5 representation of SUð5Þfor the dark matter. With these considerations and otherpossibilities in mind, we assume the relation (3.7) in thispaper. This also simplifies our analysis.By comparing (3.3) with the renormalization group

equation for the Uð1Þ coupling in [23], we get the relation

1

α¼ −2ND − NS þ 4NV þ 29

3: ð3:8Þ

Upon eliminating NV by use of the relation (3.7), wehave α ¼ 1=ð11þ 2ND − NSÞ. One may expect α to takesome value much smaller than 1: α ≪ 1. For example, ifwe take ND ¼ NS ¼ 1 for simplicity, then α ¼ 1=12. Forb ¼ 41=6, ND ¼ 45=2, and NS ¼ 4, we have α ¼ 1=52,e2� ¼ ð4πÞ2α=b ¼ 24π2=533 ≈ 0.444. However, when oneis concerned with the UV regime very close to the(classical) central singularity, it is far from obvious thatone can adequately estimate model parameters by the low-energy degrees of freedom. Rather one may anticipatesome nontrivial contributions from physics beyond theStandard Model. Taking account of such possibility as wellas the developing nature of the asymptotic safety scenario,in this paper we assume that α takes a wider parameterrange: 0 < α≲ 1.Integrating (3.2) from a reference scale k0 to k and using

the relation GðkÞ ¼ k2GðkÞ, we obtain the running Newtonconstant

GðkÞ ¼ 4παGðk0Þ4παþ ðk2 − k20ÞGðk0Þ

: ð3:9Þ

In what follows, we assume k0 ¼ 0 and set G0 ¼Gðk0 ¼ 0Þas in the Schwarzschild background case. Then, ourrunning Newton constant becomes

GðkÞ ¼ 4παG0

4παþG0k2: ð3:10Þ

Note that this is precisely the same as (2.4) with theidentification (3.5). The asymptotic behaviors of GðkÞ atshort and large distances are

GðkÞ ≃ 4πα

k2ðk → ∞Þ; GðkÞ ≃G0ðk → 0Þ: ð3:11Þ

As for the Uð1Þ gauge coupling, by introducing thefollowing functions

PðkÞ ≔ αG0k4παþG0k2

; QðkÞ ≔ bð4πÞ2k ¼ α

e2�·1

k;

ð3:12Þ

we can express (3.3) in the form of the Bernoulli equation,

dedk

þ PðkÞe ¼ QðkÞe3: ð3:13Þ

Integrating (3.13), we obtain the formal solution:

1

e2ðkÞ ¼ exp

�2

ZdkPðkÞ

��C0 − 2

ZdkQðkÞ

× exp

�−2

ZdkPðkÞ

��; ð3:14Þ

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where C0 is an integration constant. A similar formula isalso obtained in [21]. To proceed further, we need to treattwo cases, 0 < α < 1 and α ¼ 1, separately.

1. Case α ≠ 1ðα > 0ÞIn this case, inserting (3.12) in the solution (3.14), we

obtain the Uð1Þ gauge coupling

1

e2ðkÞ ¼ C0ð1þDk2Þα þ 1

e2�

α

ð1 − αÞ ð1þDk2Þ

× Fð1; 1 − α; 2 − α; 1þDk2Þ; ð3:15Þ

where F denotes the hypergeometric function, and here andhereafter we use the abbreviation D ≔ G0=4πα for nota-tional brevity. Applying the linear transformation formulaof the hypergeometric function [31], we can express(3.15) as

1

e2ðkÞ ¼ Cð1þDk2Þα þ 1

e2�· F

�1; α; 1þ α;

1

1þDk2

�;

ð3:16Þ

where we have defined a new constant C involving theintegration constant C0 in (3.14). We will see that this isrelated to the magnitude of the electric charge at lowenergy. From this expression, we can immediately read offthe UV behavior k → ∞ as

1

e2ðkÞ ≃ C ·Dα · k2α þ 1

e2�þOðk2ðα−1ÞÞ: ð3:17Þ

Note that when α > 1, the term of Oðk2ðα−1ÞÞ becomeslarger than 1=e2�, but is anyway subleading.We can further transform the solution (3.16) into the

form

1

e2ðkÞ ¼ C · ð1þDk2Þα

þ α

e2�·X∞n¼0

ðαÞnðn!Þ2

�ψðnþ 1Þ − ψðαþ nÞ

− log

�Dk2

1þDk2

���Dk2

1þDk2

�n

; ð3:18Þ

where ψ denotes the digamma function andðαÞn ≔ Γðαþ nÞ=ΓðαÞ. From this expression, we can readoff the IR behavior k → 0 as

1

e2ðkÞ ≃ C −α

e2�½γ þ ψðαÞ� þ α

e2�· log

�1þDk2

Dk2

�; ð3:19Þ

where γ ¼ −ψð1Þ is the Euler constant. Although thelogarithmic divergence appears in the IR expression above,one can now single out the arbitrary (integral) constant C

and may fix its value in terms of the classical counterpart e0of the Uð1Þ coupling.Some comments on the constant C are in order. First,

note that given a finite positive value of α, the second termof the rhs of (3.19) is finite (as α½γ þ ψðαÞ� ¼ −1þOðα2Þfor small α) and the third term is positive. This third termgrows for small k and makes (3.19) positive for sufficientlysmall k whatever the value of the constant C is. It thenappears from the UV expression of (3.17) that the solution(3.16) may change its sign for negative C when varying kfrom 0 to ∞. However this should not happen because,according to the differential equation (3.3), its solutionstops running when it comes close to zero before it changesthe sign. The point where 1=e2 becomes zero correspondsto a singularity, beyond which the solution cannot beextended. The correct interpretation is that

(i) for negative C in which the solution appears tochange sign, the Uð1Þ coupling diverges and therunning stops there. This is what is known asLandau pole;

(ii) for positive C in which the solution does not changeits sign, we have the Uð1Þ coupling for the wholerange of 0 < k < ∞, and it becomes zero asymp-totically;

(iii) for special critical boundary condition C ¼ 0, itbecomes constant asymptotically.

Let us define the Uð1Þ coupling at the energy k ¼kL ≡ 1 GeV by

1

e20¼ Cð1þDk2LÞα þ

1

e2�F

�1; α; 1þ α;

1

1þDk2L

�:

ð3:20Þ

For the choice of α ¼ 1=52, e2� ¼ 24π2=533, and theNewton constant G0 ¼ 6.7 × 10−39 GeV, we find thecorresponding values betweenC and e20 as given in Table II.For comparison, we also show the correspondence for

α ¼ 2=3 and e2� ¼ 24π2=533 in Table III. This is to becompared to the standard value e2 ¼ 1=137 ¼ 0.00730.

TABLE II. The corresponding values of C and e20 for α ¼ 1=52and e2� ¼ 24π2=533.

C −0.7 −0.4 −0.1 0 0.1 0.4 0.7

e20 0.189 0.179 0.170 0.167 0.164 0.156 0.149

TABLE III. The corresponding values of C and e20 for α ¼ 2=3and e2� ¼ 24π2=533.

C −0.7 −0.4 −0.1 0 0.1 0.4 0.7

e20 0.00738 0.00737 0.00735 0.00734 0.00734 0.00732 0.00731

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Figure 1 is the plot of the Uð1Þ running coupling (3.15)for the case of α ¼ 1=52 and e2� ¼ 24π2=533, in which wecan see that the Uð1Þ coupling diverges for larger valuesthan that corresponding to C ¼ 0ðe20 ¼ 0.167Þ, whereas itconverges to zero asymptotically for the values below that.For the critical value of e20 ¼ 0.167, the coupling goes tothe finite value (see the green curve in Fig. 1).The behavior of the Uð1Þ coupling for α ¼ 2=3 is more

or less the same as the case for α ¼ 1=52, but it diverges atlower energies than those for α ¼ 1=52 when C < 0.

2. Case α= 1

Integrating (3.3) with the parameter value α ¼ 1 [henceb ¼ ð4πÞ2=e2�], we find the general solution:

1

e2ðkÞ ¼�1þ G0

4πk2��

C0 −1

e2�· log

�G0k2

4π þG0k2

��;

ð3:21Þ

where again C0 denotes an integration constant. Noting thatγ þ ψðα ¼ 1Þ ¼ 0, one can see that the IR behavior of

e2ðkÞ is the same as (3.19) with C replaced by C0 and α setto α ¼ 1. We again define the Uð1Þ coupling at the energyscale k ¼ kL ≡ 1 GeV by

1

e20¼

�1þ G0

4πk2L

��C0 −

1

e2�· log

�G0k2L

4π þG0k2L

��: ð3:22Þ

The UV behavior of the Uð1Þ coupling for positive C0 is

1

e2ðkÞ ≃ C0

G0k2

4π: ð3:23Þ

For negative C0, it diverges at some energy and the theoryis not well defined beyond this energy. Note that for α ¼ 1,there is no critical case in contrast to α ≠ 1.We find the corresponding values between C0 and e20 in

Table IV. We plot the behaviors of the above coupling(3.21) in Fig. 2 for the case of Table IV.

B. Kretschmann scale identification and quantumimproved geometries

Now that we have obtained the running couplings GðkÞand eðkÞ, we next relate the cutoff scale k to the radialdistance r from the center, thereby obtaining the position-dependent coupling constants GðrÞ and eðrÞ. The key stepis to choose an appropriate scale identification kðrÞ. Whenwe consider the Schwarzschild metric as our background,any curvature scalar composed of the Ricci tensor cannotbe used to define the scale identification kðrÞ, since theRicci tensor vanishes identically as a classical vacuumsolution. In this case, the use of the Kretschmann scalar

FIG. 1. The behaviors of e2ðkÞ for α ¼ 1=52 ande2� ¼ 24π2=533. The lines correspond to e20 ¼ 0.179, 0.170,0.167, 0.164, 0.156, and 0.149 from top to bottom.

TABLE IV. The corresponding values of C0 and e20 for α ¼ 1

and e2� ¼ 24π2=533.

C0 −0.7 −0.4 −0.1 0 0.1 0.4 0.7

e20 0.00493 0.00492 0.00492 0.00491 0.00491 0.00490 0.00490

FIG. 2. e2ðkÞ for decreasing e0 from top to bottom as given in Table IV. Left for the range of k ¼ 10 ∼ 3 × 1020 GeV, right fork ¼ 1017 ∼ 3 × 1020 GeV.

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K ≔ RμναβRμναβ appears to be most appropriate, as it is adiffeomorphism invariant quantity of momentum dimen-sion four, as discussed in Ref. [13]. The scale identificationbased on the Kretschmann scalar is given by

k4 ¼ χ4KðrÞ; ð3:24Þ

where χ is a numerical constant. Note that if one employsthe quantum Kretschmann scale identification in which Kis computed with respect to the quantum improved metric,the value of χ may be constrained as briefly discussed inRef. [13]. Note also that when considering an asymptoti-cally de Sitter or AdS spacetime, one may need to subtractthe asymptotic value Kð∞Þ from (3.24).For the Reissner-Nordstrom metric, the Ricci tensor no

longer vanishes. Therefore, we have, in principle, a largervariety of scale identifications, being able to includethose composed of the Ricci tensor and scalar curvature.Nevertheless, we employ, as our scale identification kðrÞ,the above definition (3.24) based on the Kretschmannscalar with respect to the classical Reissner-Nordstrommetric. By doing so, we can directly and clearly comparepossible consequences of our case with those of thequantum improved vacuum black hole cases previouslystudied in Refs. [5,13]. We will discuss more generalchoices of scale identification and possible conse-quences later.For the classical Reissner-Nordstrom metric (2.1) with

(3.1), we have

KCRNðrÞ ¼8G2

0

r8ð6M2r2 − 12Me20rþ 7e40Þ; ð3:25Þ

where we identify e0 with the low-energy value of theUð1Þcoupling defined in (3.20). Inserting (3.25) into (3.24), wefind the Kretschmann scale identification

k2ðrÞ ¼ 2ffiffiffi2

pχ2G0

r4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6M2r2 − 12Me20rþ 7e40

q: ð3:26Þ

We can immediately see the behavior of k2ðrÞ at short andlarge distances as

k2ðrÞ ≃ 2ffiffiffiffiffi14

pχ2G0e20r4

ðr → 0Þ;

k2ðrÞ ≃ 4ffiffiffi3

pχ2G0Mr3

ðr → ∞Þ: ð3:27Þ

Substituting (3.26) into the running couplings GðkÞ andeðkÞ found in the previous subsection, we have theposition-dependent couplingsGðrÞ and eðrÞ, and therefore,through (3.1), we obtain the quantum improved lapsefunction

fðrÞQRN ¼ 1 −2Mr

GðrÞ þ GðrÞe2ðrÞr2

: ð3:28Þ

To see the behavior of the quantum improved lapsefunction near the center r ¼ 0, we note that at shortdistance, r → 0:

GðrÞ ≃ AðαÞr4 þOðr5Þ; AðαÞ ≔ 2παffiffiffiffiffi14

pχ2G0e20

:

ð3:29Þ

1. Case α ≠ 1

It follows from (3.17) together with (3.27) that at shortdistance, r → 0:

e2ðrÞ ≃ 4πBðαÞr4α þOðr4αþ1Þ; ð3:30Þ

where the constant BðαÞ is

1

BðαÞ ≔ 4π

� ffiffiffiffiffi14

pχ2G2

0e20

2πα

�α

C; ð3:31Þ

with C given by (3.20).Inserting (3.29) and (3.30) into (3.28), we find that the

quantum improved lapse function fðrÞQRN behaves near thecenter as

fðrÞQRN ≃ 1 − 2MAðαÞr3 þ AðαÞBðαÞr4αþ2: ð3:32Þ

Since α > 0, not only the function fðrÞQRN but also thegeometry itself is regular at the center with all curvaturescalars vanishing there, as we will discuss later.At large distances, r → ∞, it follows from (3.19),

e2 ≃e2�

α logð1=Dk2Þ : ð3:33Þ

Thus, together with (3.11) and (3.27), the large distancebehavior is expressed as

fðrÞQRN ≃ 1 −2MG0

rþG0e2�

r2·

1

α logðπαr3= ffiffiffi3

pχ2G2

0MÞ :

ð3:34Þ

This renders the metric asymptotically flat in the sense thatthe resultant quantum improved geometry approaches theSchwarzschild spacetime at large distances. However, thegeometry does not asymptotically approach the Reissner-Nordstrom spacetime since, with the logarithmic depend-ence, the Coulomb potential term decays slightly fasterthan its classical counterpart.Finally, one can check that as in the classical counterpart,

the quantum improved Reissner-Nordstrom black hole can

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also have either two horizons (the event and inner Cauchyhorizons), one degenerate horizon, or no horizon dependingupon the parameter values. This can be seen in the plot ofthe quantum improved lapse function depicted in Fig. 3. Wecan also find that for some cases of interest, the magnitudeof the surface gravity of the inner Cauchy horizon is largerthan that of the event horizon, as can be read off fromFig. 4. This implies that the mass inflation [32,33] can alsooccur for quantum improved Reissner-Nordstrom blackhole, rendering its Cauchy horizon unstable.

2. Case α= 1

Let us turn to the exceptional case α ¼ 1. At shortdistance, it follows from (3.11) and (3.23), GðkÞ ≃ 4π=k2

and e2ðkÞ ≃ 4π=C0G0k2. Together with (3.27), one has

GðrÞ ≃ 2πffiffiffiffiffi14

pχ2G0e20

r4; e2ðrÞ ≃ 2πffiffiffiffiffi14

pχ2G2

0C0e20r4:

ð3:35Þ

Therefore at short distance, one obtains

fðrÞQRN ¼ 1 −4πMffiffiffiffiffi

14p

χ2G0e20r3 þ 2π2

7χ4G30C0e40

r6: ð3:36Þ

Thus, again the quantum improved geometry itself isregular at the center r ¼ 0.At large distances, one can find that fðrÞQRN behaves the

same way as (3.34) with α ≠ 1. Again, it is asymptoticallyflat, but is not precisely the same way as the classicalReissner-Nordstrom black hole, due to the logarithmicdependence.We plot the lapse function in Fig. 5 for χ ¼ 1, e2� ¼

24π2=533, e20 ¼ 0.00491 and various masses. Dependingon the parameter values, the quantum improved metricadmits either two horizons, one degenerate horizon, or nohorizon. We see that the qualitative behavior is similar tothe α ≠ 1 case.

C. More general scale identificationsand the UV limit

So far, we have restricted our attention to the scaleidentification (3.24) based on the Kretschmann scalar(3.25) with respect to the classical metric, and found thatour quantum improved geometry is regular at the center. Itis natural to ask what happens, in particular near the center,if a different scale identification scheme is chosen. Here wediscuss possible quantum improved black hole geometriesin more general setting.Let us first briefly recapitulate some basic properties

of the curvature tensor for static, spherically symmetricgeometry. Consider the general form of the lapse function,

fðrÞ ¼ 1 − λrν; ð3:37Þ

where λ and νð≠ 0Þ are some constants. It is immediate tofind that the scalar curvature and the Kretschmann scalarbecome

FIG. 3. fðrÞQRN in the Planck unit G0 ¼ 1 and with χ ¼ 1,α ¼ 1=52, e2� ¼ 24π2=533, e20 ¼ 0.164ðC ¼ 0.1Þ, for the increas-ing mass M from top to bottom: M ¼ 4 (blue: no horizon), 4.93(orange: extremal horizon), 6 (green: two horizons).

FIG. 4. The surface gravity as a function of the mass M for thequantum improved Reissner-Nordstrom black hole with the sameparameter values as those in Fig. 3: G0 ¼ 1, χ ¼ 1, α ¼ 1=52,e2� ¼ 24π2=533, e20 ¼ 0.164ðC ¼ 0.1Þ. The upper (blue) curvecorresponds to the surface gravity of the event horizon and thelower (orange) curve to minus the surface gravity of the innerCauchy horizon. The absolute value of the surface gravity of theCauchy horizon is larger than that of the even horizon.

FIG. 5. fðrÞQRN in the Planck unit G0 ¼ 1 with χ ¼ 1,e2� ¼ 24π2=533, e20 ¼ 0.00491, for the increasing mass M fromtop to bottom: M ¼ 0.6 (blue: no horizon), 0.85 (orange:extremal horizon), 1.2 (green: two horizons).

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R ¼ λðνþ 1Þðνþ 2Þrν−2;K ¼ λ2ðν4 − 2ν3 þ 5ν2 þ 4Þr2ν−4: ð3:38Þ

Thus, if ν > 2, those curvature scalars vanish at thecenter. In fact, when ν > 2, not only these curvature scalarsbut all components of the Riemann tensor vanish at thecenter. This can easily be seen by introducing the standardorthonormal frame feðαÞg ¼ f−fðrÞ1=2dt; fðrÞ−1=2dr; rdθ;r sin θdϕg and checking that all components of the curva-ture tensor with respect to the basis feðαÞg are written as

RðαÞðβÞðγÞðδÞ ¼ crν−2; ð3:39Þ

with c being some constant. With all curvature com-ponents vanishing at the center, the case with ν > 2 canbe said to possess the Minkowski core. The marginalcase ν ¼ 2 can be said to admit the dS core if λ > 0and the AdS core if λ < 0. When 0 < ν < 2, althoughthe lapse function (3.37) is regular at r ¼ 0, the geo-metry itself becomes singular at the center. In the case0 < ν < 2, however, the central singularity is less divergentthan those appearing in the cases ν < 0 such as theSchwarzschild and Reissner-Nordstrom metrics. In thissense, the case 0 < ν < 2 may be said to be weaklysingular at the center.Now let us assume that our scale identification (2.5) is

given by the following power law in the UV regime:

k ∼ξ

rp; ð3:40Þ

with a positive constant p. The classical Reissner-Nordstrom Kretschmann scale identification (3.26) corre-sponds to p ¼ 2. For a general p > 0, one may have tointroduce an appropriate dimensionful parameter to justifythe scale identification (3.40), but here we leave aside suchan issue for simplicity.One may view that quantum improved charged black

holes can be characterized by—besides the classical back-ground parameters ðG0M; e0Þ—two quantum improvingparameters ðp; αÞ which specify the scale identificationand the fixed points of the couplings, respectively. Forthe scale identification (3.40), one can find that therunning couplings behave in the UV as GðrÞ ∼ r2p from(3.11) and e2 ∼ r2pα from (3.17), and hence find that thecorresponding quantum improved lapse function behavesin the UV as

fðrÞ ∼ 1 − λþr2p−1 þ λ−r2ðpαþp−1Þ; ð3:41Þ

where λ� are both positive constants. Roughly speaking,λþ > 0 describes the strength of quantum correctedgravitational interaction and λ− > 0 that of quantumcorrected electromagnetic interaction. For example, in

the Kretschmann scale identification, λþ ¼ 2MAðαÞ, λ− ¼AðαÞBðαÞ [see (3.32)].Comparing (3.41) with (3.37) and inspecting (3.38) or

(3.39), one can find that the regularity at the center isguaranteed when 2p − 1 ≥ 2; 2ðpαþ p − 1Þ ≥ 2. Moreprecisely, one can classify the quantum improved geom-etries as follows (see also Fig. 6.):

(i) p > 32; α > 2

p − 1: Minkowski core

(ii) p ¼ 32; α > 1

3: dS core

(iii) p > 32; α ¼ 2

p − 1: AdS core

(iv) p ¼ 32; α ¼ 1

3: either Minkowski-, dS-, or AdS core

(v) 0 < p < 32; α > 0: weak singularity.

Note that at the corner p ¼ 3=2, α ¼ 1=3, the geometryadmits dS core for λþ > λ−, AdS core for λþ < λ−, orMinkowski core for λþ ¼ λ−. The dS core appears onlywhen p ¼ 3=2, which corresponds to the power of thescale identification with the radial proper distance, as wellas the Kretschmann scalar with respect to the classicalSchwarzschild metric, as shown in Table I, while the AdScore appears for all higher power 3=2 < p < 2. Theappearance of the AdS core is a characteristic propertywhen the running Uð1Þ coupling is included. The geom-etries become weakly singular whenever 0 < p < 3=2.This can be the case, for example, when dðrÞ of (2.5) isidentified with the luminosity distance or the affineparameter of the ingoing null geodesic, for which p ¼ 1.Note that by “weak singularity” we mean that theKretschmann scalar of the quantum improved metric isless divergent than that of the classical counterpart in theshort distance limit. This can easily be seen by comparing(3.41) with (3.1) and by noting that 2p − 1 > −1,2ðpαþ p − 1Þ > −2, provided p > 0; α > 0.

FIG. 6. The colored (orange) region in the ðp; αÞ plane isdefined by p > 3=2, α > 2=p − 1, in which a quantum improvedblack hole is allowed to have a regular center of Minkowski core.The lower boundary (thick red curve) along α ¼ 2=p − 1corresponds to AdS core and the left boundary (green half-line)along p ¼ 3=2 to dS core. At the corner Að3=2; 1=3Þ, one haseither de Sitter core, AdS core, or Minkowski core, dependingupon the parameter values. The rest (light-blue) corresponds toweakly singular geometries. The dashed blue interval ðp ¼2; 0 < αÞ corresponds to the case analyzed under the Kretsch-mann scale identification in the previous subsection.

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IV. SUMMARY AND DISCUSSION

In this paper, we have studied the quantum improvementof the charged spherically symmetric black hole spacetimesin the asymptotic safety scenario. We have first derivedthe running gravitational and electromagnetic couplingsfrom the renormalization group equations for the vanishingcosmological constant, and then obtained the position-dependent running couplings by introducing the scaleidentification based on the Kretschmann scalar. Repla-cing the classical coupling constants in the metric functionwith the position-dependent running counterparts, we haveconstructed the quantum improved Reissner-Nordstromblack hole, which involves the parameters α and e� thatcharacterize the fixed points of the running gravitationaland Uð1Þ gauge couplings. The global structure of ourquantum improved Reissner-Nordstrom black hole is moreor less the same as the classical counterpart, but in theUV limit, the central singularity is fully resolved, beingreplaced with a regular Minkowski type core, thanks tothe combined effects of both quantum gravitational andelectromagnetic interactions. The number of the horizons inthe quantum improved geometry can be either two (eventand Cauchy horizons) at a maximum, one (degenerateextremal horizon), or zero (no horizon). As for the IR limit,the geometry at large distances is asymptotically flat but notthe same way as its classical counterpart.Our analysis has been made for the case in which the

cosmological constant Λ is zero (which is a fixed point).One might expect that cosmological constant would notsignificantly affect the short distance behaviors. However,as shown in [13] for the quantum improved Schwarzschild–(A)dS case, the quantum improvement with nonvanishingcosmological constant does not fully remove the central

singularity, in contrast to our case as well as the quantumimproved Schwarzschild case [5]. So, this could be evenan interesting criterion to select particle content of thetheory. Our case with vanishing Λ has also the advantageto allow us to find the analytic solution to the renorma-lization group equation, making the analysis easier andclearer.The results should be compared with the pure gravity

case, i.e., the quantum improved Schwarzschild black hole,for which the UV geometry admits the dS core, but not theMinkowski core or AdSc ore, under the Kretschmann scaleidentification or the scale identification based on the properradial length. Our results imply whether or not a classicalcurvature singularity is resolved, and what type of thecenter core appears depends upon what types of inter-actions are involved. It would be interesting to generalizethe present analysis to include Yang-Mills gauge fields, aswell as matter fields coupled to the gauge fields.In this paper we have employed the Kretschmann scale

identification as a physically sensible scale identificationscheme. It is natural to ask how the singularity resolutiondepends on the type of scale identification. As brieflymentioned in Sec. II, there are several different scaleidentification schemes, which are classified into two types:(i) one based on the proper radial distance of a certain curveof either timelike or spacelike (or a null curve for which theluminosity distance can be considered) that approaches thecentral singularity, and (ii) the other based on the invariantcurvature scalars. The former (i) might be interpreted asan observer-dependent scheme by viewing the selectedcurve as the world line of some observer approaching thesingularity. For the latter (ii), one can consider a number ofdifferent combinations of the invariant curvature scalars,

fR;RαβRβ

α; RαβRβ

γRγα; Rα

βRβγRγ

δRδα; Rαβ

γδRγδαβ; Rαβ

γδRγδρσRρσ

αβ;…g:

For more detailed list of invariant curvature scalars, see,e.g., Appendix A of [29]. The Kretschmann scalar,K ¼ RμναβRμναβ, is the simplest choice of (ii), which canapply to the case of vacuum (Rαβ ¼ 0) classical geometries.One can employ more complicated curvature invariantssuch as those composed of three curvature tensors. It isclear that the radial dependence kðrÞ of the cutoff mo-mentum depends on the choice of scale identification.As a simple generalization, we have examined the case in

which the cutoff momentum scale has the inverse powerradial dependence k ∼ 1=rp with p > 0. We have surveyedthe two-dimensional parameter space ðp; αÞ and found thatin the ðp; αÞ plane, the quantum improved geometries witheither dS core or AdS core appears at the boundary betweenthe region for regular geometries with Minkowski core andthat for weakly singular geometries. The dS core is formed

when the scale identification is made with p ¼ 3=2, whichcorresponds to the power of the classical SchwarzschildKretschmann scalar. However, the dS core is possible onlywhen 1=3 < α. In contrast, the AdS core can occur for3=2 < p < 2 and α < 1=3, including the value α ¼ 1=52.It should also be emphasized that the AdS core is possibleonly when the Uð1Þ gauge coupling is involved.Now to see more on the choice of scale identification and

its physical consequences, let us consider the metricfunction, (3.41), with negative ν < 0 so that near the centerfðrÞ ∼ r−jνj as in the classical Schwarzschild ν ¼ −1 andReissner-Nordstrom ν ¼ −2 case. The proper radial dis-tance toward the singularity is given as ∼rðjνjþ2Þ=2 (compareTable I) and the luminosity distance ∼r for null geodesic,whereas the curvature invariant Kn ¼ Rαβ

γδ � � �Rρσαβ that

consists of n curvature tensors behaves typically as

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∼r−nðjνjþ2Þ near the center1 [see (3.39)]. Therefore, startingeven from the same classical geometry, if a differentscale identification is employed, the resultant quantumimproved geometry can differ. For example, the quantumimproved Reissner-Nordstrom geometry has a regularcenter with p ¼ 2 for the Kretschmann and proper dis-tance scale identification,2 while it has a weak singu-larity with p ¼ 1 for the scale identification with theluminosity distance along a null geodesic curve. In par-ticular, based on curvature invariants, it appears alwayspossible to find a scale identification that attains a suffi-ciently large p required for the singularity resolution,provided that a suitable dimensionful parameter is avail-able. However, apart from the singularity resolution, thereseems to be—at least within the scope of the presentpaper—no persuasive reason to discard scale identificationsthat leads to a weak singularity. In this regard, it should alsobe noted that although resolving singularity is an attractivefeature, if all types of singularities are to be resolved, thensuch a quantum gravity theory may suffer from pathologi-cal features. For example, the present scheme for quantumimprovement can straightforwardly be applied to thenegative mass Schwarzschild solution. If the singularityof the negative mass Schwarzschild solution is resolved,then admitting a regular negative energy solution, such a

quantum gravity theory would not allow any stable lowest-energy solution [34]. Further analyses are needed to clarifythe relation between possible choice of scale identificationsand underlying physics.The appearance of the AdS core suggests the tantalizing

possibility that the present results may be incorporated intothe scenario recently proposed by Ref. [35], in which theasymptotic safe scaling regime can be connected to stringtheory in the deep UV, where AdS background is morenaturally realized. It would be interesting to pursue thisdirection further.

ACKNOWLEDGMENTS

We thank the Yukawa Institute for Theoretical Physics,at Kyoto University, where part of this work was carriedout during the YITP-W-20-09 workshop on “10th Inter-national Conference on Exact Renormalization Group 2020(ERG2020).” We would also like to thank Chiang-MeiChen for valuable discussions. This work was supportedin part by JSPS KAKENHI Grants No. 20K03938,No. 20K03975, and No. 15K05092 (A. I.), and in partby Grants No. 16K05331, No. 20K03980, and No. MOST110-2811-M-008-510 (N. O.).

Note added.—After this paper appeared in arXiv, it waspointed out that there is a closely related paper [36] whichdiscusses RG improvement of RN–(A)dS black hole.The main focus of the paper [36] is on the solution withnonvanishing cosmological constant, resulting in no reso-lution of the central singularity.

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2This appears to be always the case when one setsk ¼ ξ=rp ∝ fproper radial distanceg−1 ∝ ðKnÞ1=2n.

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