arX

iv:2

006.

0138

8v2

[gr

-qc]

28

Aug

202

0

Scattering of massless scalar field by charged dilatonic black holes

Yang Huang 1∗ and Hongsheng Zhang 1,2†1 School of Physics and Technology, University of Jinan, 336,

West Road of Nan Xinzhuang, Jinan 250022, Shandong, China2 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing, 100190, China

Wave propagations in the presence of black holes is a significant problem both in theoretical andobservational aspects, especially after the discovery of gravitational wave and confirmation of blackholes. We study the scattering of massless scalar field by a charged dilatonic black hole in frameof full wave theory. We apply partial wave method to obtain the scattering cross sections of thescalar field, and investigate how the black hole charge affects the scalar scattering cross sections.Furthermore, we investigate the Regge pole approach of the scattering cross section of the dilatonicblack hole. We find that in order to obtain results at the same precision, we need more Regge polesas the black hole charge increases. We compare the results in the full wave theory and results inthe classical geodesic scattering and the semi-classical glory approximations, and demonstrate theimprovements and power of our approach.

I. INTRODUCTION

Black holes are among the most striking predictionsin modern physics. Black hole, as a mathematical so-lution of the field equation in General Relativity (GR)or alternative theories of gravity, has been studied manyyears. In fact, black hole may be the most thoroughlystudied object before its discovery. For years the astro-physical black hole is carefully called black hole candi-date, or more directly X-ray source. Things changed in2016 because of the discovery of gravitational wave frombinary black hole [1]. After that several gravitationalwave events have been reported [2], the observation fromevent horizon telescope (EHT) presents the second clear-cut evidence for the existence of black hole [3].Among several aspects of investigations of black hole,

scattering and absorption from black holes takes a fun-damental status, which has been studied more than 50years [4]. This subject attracts increasing attentions dueto its relevance with many interesting phenomena, suchas glory, rainbow and superradiant scatterings [5–7]. Thescattering of scalar (s = 0), Dirac (s = 1/2), electromag-netic (s = 1) and gravitational (s = 2) waves by blackholes in GR have been thoroughly explored [8–16]. Thesestudies have been extended to scattering of plane wavesby black holes beyond GR [17–21] and ultra-compact ob-jects [22–27].In the previous studies of the scattering of waves by

black holes, the most frequently approach is based onthe partial wave expansion. This method is natural andstraightforward, but it suffers from divergence for smallscattering angles due to the Coulomb characteristic ofthe potential. This problem can be handled by the se-ries reduction method ([28], see also [12, 15]) or by theComplex Angular Momentum (CAM) techniques, which

∗ sps huangy@ujn.edu.cn† sps zhanghs@ujn.edu.cn

is first applied to the black hole scattering problem byAndersson and Thylwe [29, 30]. Recently, the CAM the-ory has been successfully applied to the scattering of fun-damental fields by black holes and ultra-compact objects[22, 31, 32].In this paper, we study the scattering of massless

scalar field by the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black hole, a spherically chargeddilatonic solution of the low energy limit of heteroticstring theory in four dimensions, which was first foundby Gibbons and Maeda in [33] and independently ob-tained by Garfinkle, Horowitz, and Strominger in [34] afew years later.GMGHS black hole has been investigated in theoreti-

cal and observational aspects. Particle trajectory aroundGMGHS black hole was investigated in [35]. Late-timeevolution of a charged massless scalar field around theGMGHS was studied in [36]. Hawking radiation of blackholes with such a singular horizon was seriously studiedin [37]. Strong gravitational lensing by GMGHS blackhole was explored in [38], which showed that there arefew observational differences between Schwarzchild andGMGHS black holes for strong lensing. Accretion disksaround GMGHS black hole was studied in [39]. Last butnot least, the (in)stability under charged scalar pertur-bations and the existence of scalar clouds of the GMGHSblack hole were studied in [40–46].In Einstein frame, the line element of the GMGHS so-

lution is given by

ds2 = −F (r)dt2+F (r)−1dr2+r2G(r)(

dϑ2 + sin2 ϑdϕ2)

,(1)

with

F (r) = 1− 2Mr, and G(r) = 1− Q

2

Mr, (2)

whereM and Q are the mass and charge of the black holerespectively. The Maxwell field and dilaton field read,

FM = Q sinϑdϑ ∧ dϕ, (3)

http://arxiv.org/abs/2006.01388v2mailto:sps_huangy@ujn.edu.cnmailto:sps_zhanghs@ujn.edu.cn

2

and,

e−2φ = e−2φ0(

1− Q2

Mr

)

, (4)

respectively. φ0 denotes the value of the dilaton φ atspacelike infinity. φ0 = 0 implies an asymptotic flat man-ifold.The event horizon is located at r = 2M . The area of

the sphere goes to zero when r = Q2/M and the sur-

face is singular. For Q < Qmax ≡√2M , the singularity

is enclosed by the event horizon. In the extremal caseQ = Qmax, and the singularity coincides with the hori-zon. It is convenient to introduce the normalized chargeq = Q/Qmax.The reminder of this paper is organized as follows. In

Sec. II the geodesic scattering in the GMGHS spacetimeis analyzed, from which we compute the classical scat-tering cross sections and glory parameters. In Sec. III,we describe the scattering of massless scalar field by theGMGHS black hole, focusing on computing the scatter-ing cross sections via the partial wave method and theRegge pole approximation. Our numerical results arepresented in Sec. IV and we conclude the paper in Sec.V.

II. CLASSICAL AND SEMI-CLASSICAL

SCATTERING

A. Geodesic scattering

Under the condition of short wavelength approxima-tion, classical geodesic scattering provides acceptable re-sults of the differential scattering cross section [17, 47].In this subsection, as the first step, we investigate thegeodesic scattering in the GMGHS spacetime. The La-grangian associated to the null geodesics is given by

L = 12gµν ẋ

µẋν = 0, (5)

where ẋµ = dxµ/dλ, and λ is an affine parameter. Sincethe background spacetime is spherically symmetric, weshall only consider geodesics on the equatorial planewithout loss of generality. Substituting the GMGHS met-ric into Eq.(5), we obtain the orbit equation

(

du

dϕ

)2

= U(u),

U(u) = (1− ur−)2[

1

b2−(

1− ur+1− ur−

)

u2]

,

(6a)

(6b)

where u = 1/r, r+ = 2M , r− = 2Mq2, and b is the

impact parameter.From Eq.(6a) one sees U(u) ≥ 0 along the orbit. There

is a critical value of the impact parameter b = bc, bywhich the geodesic may be deflected at any angle. The

critical impact parameter can be obtained by solvingequations U(u) = 0 and U ′(u) = 0. The result is:

bcM

=

√

27− q4 − 18q2 + (9− q2)√

(1− q2) (9− q2)2

.

(7)For b < bc, U(u) is always positive outside the black hole,and a photon coming from infinity (i.e., u = 0 initially)will finally be absorbed by the black hole. In the scatter-ing scenario b > bc, the photon will not be absorbed bythe black hole, but will back to infinity after crossing theturning point u = u0, at which U(u0) = 0. In this case,the deflection angle of the geodesic is given by

Θ(b) = 2

∫ u0

0

du√

U(u)− π. (8)

We apply the numerical integration to obtain the deflec-tion angle. In the weak-field limit (b≫M), it is possibleto find the analytic expression of Θ, i.e.,

Θ ≈ 4Mb

+3π

4

(

5− 4q2 − 43q4)

M2

b2. (9)

Note that this equation is slightly different from that ofthe Reissner-Nordström black hole [48], and the first termis the term of a typical strong lensing.Given the relation between b and the deflection angle,

the classical differential scattering cross section is pre-sented by

dσ

dΩ=

b

sin θ

∣

∣

∣

∣

db

dθ

∣

∣

∣

∣

. (10)

The effects of q on the classical scattering cross sectionis exhibited in Fig. 1. This figure shows that for moder-ate angles, the classical scattering cross section decreasesas the black hole charge q increases. But this effect isnot obvious for small values of θ. The interpretation ofthis result is as follows: combining Eqs.(9) and (10), weobtain

dσ

dΩ≈ 16M

2

θ4+

3πM2

4θ3

(

5− 4q2 − 43q4)

. (11)

This equation shows clearly that the black hole charge qdoes not contribute to the dominant term of the classicalscattering cross section for small values of θ.

B. Glory scattering

The glory approximation of the scalar scattering crosssections is [6]

dσ

dΩ= 2πωb2g

∣

∣

∣

∣

db

dθ

∣

∣

∣

∣

θ=π

J20 (ωbg sin θ), (12)

where J0(x) is the Bessel function of the first kind, andbg is the impact parameter of backscattered geodesics

3

q=0.0

q=0.5

q=0.9

20 40 60 80 100 120 140 160 1800

1

2

3

4

scattering angle �(deg)

log10(d

�

/d

�

M-2)

FIG. 1. Classical scattering cross section of GMGHS blackholes, with q = 0, 0.5 and 0.9. The black solid line is givenby Eq.(11), with q = 0.

bg

bg2|db/d� �=π

0.0 0.2 0.4 0.6 0.8 1.0

3.0

3.5

4.0

4.5

5.0

5.5

q

UnitsinM

FIG. 2. Glory scattering parameters as functions of q.

(θ = π). There are multiple values of bg correspond-ing to the multiple values of the deflection angle, i.e.,θ = π + 2nπ (with n = 0, 1, 2, · · · the number of timesthat the null geodesic rotates around the black hole),but the major contribution to the differential scatteringcross section comes from the n = 0 case [17]. Althoughthe semi-classical glory approximation (12) is expectedto be valid at high frequencies (ωM ≫ 1), it is still ingood agreement with the numerical results for interme-diate frequencies, i.e., ωM ∼ 1 [48].According to Eq.(12), we need to determine bg and

|db/dθ|θ=π in order to obtain the glory scattering crosssection. Given a q, we obtain bg by numerically solvingthe equation Θ(b) = π, where Θ(b) is given in Eq.(8).Then, it is straightforward to obtain |db/dθ|θ=π via thefinite difference formula. Numerical results of bg andb2g|db/dθ|θ=π are presented in Fig.2. As is shown in theplot, both bg and b

2g|db/dθ|θ=π decrease monotonically

as the black hole charge q increases. These results indi-cate that with the increase of q, (i) interference fringesget wider, since the interference fringe width is inverselyproportional to bg; (ii) the backscattered flux intensitydecreases.

III. WAVE SCATTERING

A. Massless scalar field in the GMGHS spacetime

We consider a massless scalar field propagating in aGMGHS spacetime, which obeys the Klein-Gordon equa-tion

∇µ∇µΦ = 0. (13)

The background spacetime is stationary and sphericallysymmetric. Thus, we write Φ = Rωl(r)Ylm(ϑ, ϕ)e

−iωt,where Ylm(ϑ, ϕ) are the scalar spherical harmonics. Theradial function Rωl(r) obeys the radial equation

∆d

dr

(

∆dRωldr

)

+[

G(r)2ω2r4 −∆l(l + 1)]

Rωl = 0,

(14)where we have introduced a new function ∆ = (r −r+)(r − r−). By introducing ψωl(r) = Rωl(r)/r, we canrewrite the radial equation in the following form

d2

dr2∗ψωl(r) +

[

G(r)2ω2 − Vl(r)]

ψωl(r) = 0, (15)

where the tortoise coordinate is defined by dr∗ =f(r)−1dr, and f(r) = ∆/r2, while the potential is givenby

Vl(r) = f(r)

[

f ′(r)

r+l(l + 1)

r2

]

. (16)

At the event horizon r = 2M , the purely ingoing wavesolution of the radial equation is

ψ(r) ∼ e−iκωr∗ , (17)

where κ = 1 − Q2/2M2. At spatial infinity r → ∞, theradial solution behaves as

ψ(r) ∼ A(−)l (ω)e−iωr∗ +A(+)l (ω)e

iωr∗ . (18)

Here the coefficients A(−)l (ω) and A

(+)l (ω) are complex

amplitudes of the ingoing and outgoing waves, respec-tively. Given these amplitudes, the S-matrix elementsSl(ω) is given by

Sl(ω) = ei(l+1)πA

(+)l (ω)

A(−)l (ω)

. (19)

There are two types of poles of Sl(ω): (i) for l ∈ N, thepoles of Sl(ω) in the complex plane of ω give the quasinor-mal frequencies, and they are associated to the dynamicsof scalar perturbations around the GMGHS black hole;(ii) Regge poles are the poles of1 Sλ−1/2(ω) in the firstand third quadrants of the complex plane of λ = l+ 1/2for ω ∈ R, and they can be expressed as λ = λn(ω) withn = 1, 2, 3, · · · .

1 Here Sλ−1/2(ω) denotes the analytic extension of Sl(ω).

4

B. Scattering cross section

The differential scattering cross section is given by

dσ

dΩ= |f(ω, θ)|2. (20)

where f(ω, θ) the scattering amplitude, which is ex-pressed by the following partial wave series

f(ω, θ) =1

2iω

∞∑

l=0

(2l + 1) [Sl(ω)− 1]Pl(cos θ). (21)

In order to derive the scattering cross section via thepartial wave series, we need to compute Sl(ω) by solvingthe radial equation (15) (or equivalently Eq.(14)) withrespect to the boundary conditions given in Eqs.(17) and(18). We apply the Runge-Kutta method to solve thisproblem numerically. Note that the sum in Eq.(21) doesnot convergent very quickly for small values of θ. Thus,we employ the method developed in [28], and first appliedto the black hole scattering problem in [12], to improvethe convergence of the sum.

C. Regge pole approximation

According to the CAM theory, the scattering ampli-tude can be split into a background integral and a sumover Regge poles [29, 30]

fP(ω, θ) = −iπ

ω

∞∑

n=1

λn(ω)rn(ω)

cos [πλn(ω)]Pλn(ω)−1/2(− cos θ),

(22)where Pλ−1/2(x) denotes the analytic extension of theLegendre polynomials Pl(x), and rn(ω) are the residuesof the matrix Sλ−1/2(ω) at λ = λn(ω), they are given by

rn(ω) = eiπ[λn(ω)−1/2]

A(+)λ−1/2(ω)

ddλA

(−)λ−1/2(ω)

λ=λn(ω)

, (23)

where the complex amplitudes A(−)λ−1/2(ω) and A

(−)λ−1/2(ω)

are defined from the analytic extension of Eq.(18). Itwas shown that the contribution from the backgroundintegral is negligible for high frequencies [31]. Hence, thesum over Regge poles (22) provides a good approximationof the scalar differential cross section.Here, we aim to construct the Regge pole approxima-

tion of the scalar scattering cross section of the GMGHSblack hole. To this end, it is necessary to determine theRegge poles λn(ω) and the corresponding residues rn(ω)(see Eq.(22)). We apply the continued fraction methodto determine the Regge poles. By definition, Regge poles

are zeros of A(−)λ−1/2(ω) in the complex λ plane, they can

be obtained by numerically solving the following equation

0 = β0 −α0γ1β1−

α1γ2β2−

α2γ3β3−

· · · , (24)

n=1

n=2

n=3

n=4

n=5

0.0 0.2 0.4 0.6 0.8 1.06

8

10

12

14

16

q

Re

[ λn(ω

)]

n=1

n=2

n=3

n=4

n=5

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6

q

I�

[λn(�

)]

FIG. 3. Real and imaginary parts of the first five Regge poles(with ωM = 3.0) as functions of q.

or alternatively, by solving the n-th inversion of this equa-tion. Here αn, βn and γn are functions of (ω, λ), and theyare given in Eqs.(11)-(13) of Ref.[45]. Figure.3 shows thereal and imaginary parts of the first five Regge poles asfunctions of q. We see that both real and imaginary partsof λn decrease monotonically with the increase of q.To get the residues, we apply a method that has been

used in Ref.[49] to compute the quasinormal mode exci-tation factors. This method is based on the formalism de-veloped by Mano, Suzuki, and Takasugi (MST) [50], see

also [51–53]. First we compute A(+)λ−1/2(ω) and A

(−)λ−1/2(ω)

at λ = λn(ω). Then, we consider λ = λn(ω) + δ and

compute A(−)λ−1/2(ω) at the new value of λ. Finally, the

derivative of A(−)λ−1/2(ω) with respect to λ is given by

d

dλA

(−)λ−1/2 =

A(−)λn+δ−1/2

−A(−)λn−1/2δ

. (25)

In our calculation we choose δ = 10−7, i.e., we differen-tiate along the real axis; as a check, we also differentiatealong the pure-imaginary axis and find that they are con-sistent in high precision. In Appendix. A, we present thedetails of how to solve Eq.(14) and compute amplitudes

A(+)λ−1/2(ω) and A

(−)λ−1/2(ω) via the MST method.

In Fig. 4, we present the path followed by the real andimaginary parts of the first two residues rn(ω), as q varies

5

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

Re[r1(ω)]

Im[r1(ω

)]

-10 -5 0 5 10

-10

-5

0

5

10

Re[r2(ω)]

Im[r2(ω

)]

FIG. 4. Path of the first two residues rn(ω) as q varies in the range 0 ≤ q ≤ 0.999. The red and blue points correspond toq = 0 and q = 0.999, respectively. We set the frequency ωM = 3.0.

in the range 0 ≤ q ≤ 0.999. Once we have obtained λn(ω)and rn(ω), we can build up the Regge pole approximationof the scattering amplitude from Eq.(22).

IV. RESULTS

In this section, we present a selection of the numericalresults for the differential scattering cross sections of theGMGHS black hole.Figure 5 shows comparisons of the numerical scalar

scattering cross sections of the GMGHS black holes withthe classical and semi-classical glory approximations (seeEqs.(10) and (12)). We observe that the glory approxi-mation fits well with the numerical results for large angles(θ & 160◦), while the classical approximation fits well thesmall angle region (θ . 20◦). In the intermediate range,only full wave scattering treatment yields accurate result.In Fig. 6, we compare the scattering cross sections for

different values of the black hole charge (q = 0.3, 0.6, 0.9)with ωM = 1.0 (top), 2.0 (middle), and 3.0 (bottom).For comparison, we also plot the Schwarzschild (q = 0)case in each subplots of this figure. From the plots inFig. 6, we find that (i) the effect of q on the scatteringcross section at small angles is negligible, and (ii) theinterference fringe width increases with the increase ofthe black hole charge q for given values of ωM , or with thedecrease of ωM for given values of q. These results canbe understood as follows: (i) From Eq.(11), we see thatfor small angles, the scattering cross section is dominatedby the term 16M2/θ4, the black hole charge q only affectsthe cross section in the subdominant term proportionalto M2/θ3; (ii) The glory approximation given in Eq.(12)implies that for large angles (θ ≈ 180◦), the interferencefringe width is proportional to 1/(bgω). In addition, fromFig.2 we see that bg decreases monotonically with theincrease of q.Figure 7 shows the scalar scattering cross sections of

the GMGHS black holes at low frequencies. In this case,the scattering cross section does not oscillate along the

ωM=3.0, q=0.3

Partial wave

Classical

Glory

20 40 60 80 100 120 140 160 180-1

0

1

2

3

4

scattering angle θ(deg)

log10(dσ/dΩM

-2)

ωM=3.0, q=0.6

Partial wave

Classical

Glory

20 40 60 80 100 120 140 160 180-1

0

1

2

3

4

scattering angle θ(deg)

log10(dσ/dΩM

-2)

ωM=3.0, q=0��

Partial wave

Classical

Glory

20 40 60 80 100 120 140 160 180-1

0

1

2

3

4

scattering angle θ(deg)

log10(dσ/dΩM

-2)

FIG. 5. Comparison of scattering cross sections with thegeodesic and glory approximations for q = 0.3 (top), 0.6 (mid-dle), and 0.9 (bottom). We set ωM = 3.0.

θ-axis, and the scattering cross section decreases withthe frequency. Furthermore, in the limit ωM → 0, thescattering cross section of the GMGHS black hole is given

6

ωM=1.0

Schwarzschild

q=0.3

q=0.6

q=0.9

20 40 60 80 100 120 140 160 180-1

0

1

2

3

4

scattering angle θ(deg)

log10(dσ/dΩM

-2)

ωM=2.0

Schwarzschild

q=0.3

q=0.6

q=0.9

20 40 60 80 100 120 140 160 180-1

0

1

2

3

4

scattering angle θ(deg)

log10(dσ/dΩM

-2)

ωM=3.0

Schwarzschild

q=0.3

q=0.6

q=0.9

20 40 60 80 100 120 140 160 180-1

0

1

2

3

4

scattering angle θ(deg)

log10(dσ/dΩM

-2)

FIG. 6. Comparison of scattering cross sections with ωM =1.0 (top), 2.0 (middle), and 3.0 (bottom), for different valuesof the black hole charge.

by [54]

limωM→0

(

1

M2dσ

dΩ

)

=1

sin4(θ/2). (26)

Although the glory approximation fits the scatteringcross section remarkably well for large angles, there isa quantitative difference between the amplitudes of thebackscattered wave (θ = 180◦) obtained via the partialwave method and the glory approximation [17, 48]. InFig. 8, we compare the amplitudes of the backscatteredwave obtained via the glory approximation and the par-tial wave method. We see that the results obtained viapartial wave method oscillate around the semi-classical

q=0.3

ωM=0.1

ωM=0.05

ωM=0.001

A������

0 20 40 60 80 100 120 140 160 180-1

0

1

2

3

4

5

6

scattering angle θ(deg)

log10(dσ/dΩM

-2)

q=0.9

ωM=0.1

ωM=0.05

ωM=0.001

����������

0 20 40 60 80 100 120 140 160 180-1

0

1

2

3

4

�

6

scattering angle θ(deg)

log10(dσ/dΩM

-2)

FIG. 7. The low frequency behavior of the scattering crosssection for q = 0.3 (top) and 0.9 (bottom). The solid greenlines are given by the analytic formula (26). This figure showsthat in the low frequency limit ωM → 0, the scattering crosssection of the GMGHS black hole is independent on q.

Glory

ωM=1.0

ω�=2.0

ω�=3.0

0.0 0.2 0.4 0.6 0.8 1.010

1�

20

2

30

3!

q

dσ/dΩ/(ωM3)

θ=π

FIG. 8. Amplitudes of the backscattered waves (θ = 180◦) asfunctions of q for different values of ωM . The solid green lineis given by the glory approximation.

glory result.Let us now focus on the Regge pole approximation

of the scattering cross section. In Fig. 9, we comparethe partial wave results of the scattering cross sectionwith its Regge pole approximation at low frequencies,i.e. ωM ≤ 0.3. This plot shows clearly that the dif-

7

ference between the partial wave result and the Reggepole approximation decreases with the increase of ωM .We evaluate the relative error of the Regge pole approx-imation in the range 20◦ ≤ θ ≤ 180◦, and find that forωM = 0.1 the maximum relative error is about 42.6%,whereas for ωM = 0.3 is only about 2.1%. This is con-sistent with the results in [31] that the contribution fromthe background integral for high frequencies is negligible.In Fig.10, we plot the scattering cross section obtained

from the partial wave method and the Regge pole ap-proximation with and without the background integral2.We can see that taking into account the contribution fromthe background integral improves the Regge pole approx-imation drastically at low frequencies.Figure 11 compares the scattering cross sections ob-

tained by the partial wave method and the Regge poleapproximation for q = 0.3. The frequency is ωM = 3.0.We see that the Regge pole approximation (with only 5Regge poles) is very close to the result obtained via par-tial wave method for a wide range of angles (θ & 120◦),and the Regge pole approximation can be improved bytaking into account more Regge poles in the sum (22); Bysumming over 50 Regge poles, we get the result that isindistinguishable from the partial wave result for θ & 20◦.In Fig. 12, we compare the Regge pole approximation

with the partial wave result of the scattering cross sec-tion of a near extremal GMGHS black hole (q = 0.999).Again, the Regge pole approximations fit the partial waveresult well for large angles. However, we find that withthe increase of q, more Regge poles should be taken intoaccount in order to obtain results at the same preci-sion. As is shown Fig. 12, summing over 50 Regge polesyields the result in good agreement with the partial wavemethod for θ & 80◦.Figure 13 compares the amplitudes of the backscat-

tered wave obtained via the partial wave method andRegge pole approximation for ωM = 3.0. The plot showsthat summing over a few Regge poles captures most ofthe features of the the backscattered wave. With only4 Regge poles, the agreement is excellent for q . 0.7.When q increases, more Regge poles should be includedto describe the backscattered waves. This is consistentwith the result in Fig. 12.

V. CONCLUSION

Scattering from black holes is a fundamental and im-portant problem in astrophysics and theoretical physics.Because of its ultrastrong attractive interaction, blackholes stimulate several novel phenomena which are neverseen in scattering experiment in lab, for example blackhole shadow, glory etc. Some objects, for example the

2 We apply the series reduction technique introduced in [31] toimprove the convergence of the background integral.

ωM=0.1

Partial wave

S"# o$%& R'()* p+,-. (n=1,...,20)

20 4/ 67 89 100 120 :;< =>? @BC0.0

DEF

1.0

GHJ

2.0

KLN

3.0

scattering angle θ(deg)

log10(dσ/dΩM

-2)

ωM=0.2

Partial wave

OPQ TUVW XYZ\] ^_`ab (c=1,...,20)

20 de fg hi 100 120 jkl mrs tuv0.0

wxy

1.0

z{|

2.0

}~

3.0

scattering angle θ(deg)

log10(dσ/dΩM

-2)

ωM=0.3

Partial wave

(=1,...,20)

20 100 120  0.0

¡¢£

1.0

¤¥¦

2.0

§¨©

3.0

scattering angle θ(deg)

log10(dσ/dΩM

-2)

FIG. 9. Comparison of the scattering cross sections obtainedvia the partial wave method and the Regge pole approxima-tions for ωM = 0.1 (top), 0.2 (middle), and 0.3 (bottom). Weset q = 0.

gravitational wave, neutrino and the expected dark mat-ter particles, only have extremely weak interaction withordinary matters. Thus it is difficult to observe theirscattering phenomena with ordinary matters. On thecontrary, they strongly interact with black holes and yieldconspicuous phenomena [55, 56]. To study the scatter-ing from black holes is very helpful to explore the elusiveobjects, including gravitational wave, neutrino and darkmatters.In this paper we discuss the scattering of a massless

scalar field by the GMGHS black hole. Using the partialwave method, we numerically computed the scalar scat-tering cross sections, and compared the numerical results

8

ωM=0.1

ª background integral

Partial wave

Sum over Regge poles «n=1,...,20)

Sum over Regge poles (n=1,...,20)

20 40 60 80 100 120 140 160 180

0.5

1.0

1.5

2.0

2.5

3.0

scattering angle θ(deg)

log10(dσ/dΩM

-2)

FIG. 10. Comparison of the partial wave result of the scat-tering cross section with its Regge pole approximation withand without the background integral. The parameters are thesame as those in Fig. 9 (with ωM = 0.1).

ωM=3.0

Partial wave

Sum over Regge poles (n=1,...,5)

Sum over Regge poles (n=1,...,10)

Sum over Regge poles (n=1,...,50)

20 40 60 80 100 120 140 160 180-1

0

1

2

3

scattering angle θ(deg)

log10(dσ/dΩM

-2)

FIG. 11. Comparison of the scattering cross sections obtainedvia the partial wave method and the Regge pole approxima-tions. The black hole charge is q = 0.3.

ωM=3.0

Partial wave

Sum over Regge poles (n=1,...,20)

Sum over Regge poles (n=1,...,30)

Sum over Regge poles (n=1,...,50)

20 40 60 80 100 120 140 160 180-1

0

1

2

3

scattering angle θ(deg)

log10(dσ/dΩM

-2)

FIG. 12. Comparison of the scattering cross sections obtainedvia the partial wave method and the Regge pole approxima-tions. The black hole charge is q = 0.999.

Partial wave

Sum over Regge poles (n=1)

Sum over Regge poles (n=1,2)

Sum over Regge poles (n=1,...,3)

Sum over Regge poles (n=1,...,4)

Sum over Regge poles (n=1,...,5)

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

q

dσ/dΩ/(ωM3)

θ=π

FIG. 13. Comparison of the amplitudes of the backscatteredwaves obtained by the partial wave method and the sum overRegge poles.

with the geodesic and glory approximations. We summa-rize the main results as follows:First, the scattering cross section is well approximated

by the classical results obtained via the geodesic analysisfor small angles, and the effects of the black hole chargeis negligible in this case (see Eq.(11)).Second, for large scattering angles, the black hole

charge has a significant effect on the scattering cross sec-tion, which is well described by the glory approximation.We showed that the interference fringe width increaseswith the increase of the black hole charge, which is pre-dicted by the glory approximation. However, the ampli-tude of the backscattered wave is not equal to the gloryamplitude. We showed that the numerical result oscil-lates around the glory amplitude when q increases, seeFig. 8.We also construct the scattering cross sections of the

GMGHS black holes by sum over Regge poles. We findelegant agreement of the Regge pole analysis and the par-tial wave results, especially for intermediate and large an-gles. Finally, we show that when q increases, it is neces-sary to include more Regge poles to derive the scatteringcross section.

ACKNOWLEDGMENTS

We thank the anonymous reviewer for their helpfulcomments and suggestions. Y. Huang wishes to thankYuan-Xing Gao for helpful discussions. This work is sup-ported by the National Natural Science Foundation ofChina (NSFC) under grant Nos.11805166 and 11575083,as well as Shandong Province Natural Science Founda-tion under grant No. ZR201709220395.

9

Appendix A: Computing A(+)λ−1/2(ω) and A

(−)λ−1/2(ω)

In this appendix, we present the details of how to solve the radial equation (14) and compute amplitudes A(+)λ−1/2(ω)

and A(−)λ−1/2(ω) via the MST method [50–52].

1. Near horizon solution in series of hypergeometric functions

It is convenient to express the radial equation in terms of the dimensionless variables

x = −r − r+κr+

; κ = 1− Q2

2M2; ǫ = 2Mω, (A1)

where r+ = 2M . Then, the radial equation (14) becomes

x(1 − x)d2R

dx2+ (1 − 2x)dR

dx+ UR = 0, (A2)

where

U =ǫ2

x+ l(l+ 1)− (1 + 2κ) ǫ2 + (2 + κ)κǫ2x− κ2ǫ2x2. (A3)

For simplicity, we have omitted the subscripts ”ωl” of the radial function. The ingoing-wave radial solution has thefollowing form

Rin(x) = eiκǫx(−x)−iǫpin(x). (A4)

Substituting Eq.(A4) into Eq.(A2), one finds the differential equation of pin(x)

x(1 − x)p′′in + [1− 2iǫ− (2− 2iǫ)x] p′in + [l(l+ 1) + iǫ(1− iǫ)] pin = 2iκǫ [xpin − x(1− x)p′in] + ǫ(ǫ− iκ)pin, (A5)

where a prime denotes d/dx. The left-hand side of Eq.(A5) is in the form of a hypergeometric equation. In the limitǫ→ 0, a solution of Eq.(A5) that is finite at x = 0 is given by

pin(ǫ→ 0) = F (−l, l+ 1; 1;x). (A6)

For a general value of ǫ, the solution of Eq.(A5) may be expanded in a series of hypergeometric functions with ǫ beinga kind of expansion parameter. The essential point of the MST formalism is to introduce the so-called renormalizedangular momentum ν. By adding the term [ν(ν + 1)− l(l+ 1)] pin to both sides of Eq.(A5), one obtains

x(1 − x)p′′in + [1− 2iǫ− (2 − 2iǫ)x] p′in+ [ν(ν + 1) + iǫ(1− iǫ)] pin= 2iκǫ [xpin − x(1− x)p′in] + [ν(ν + 1)− l(l+ 1) + ǫ(ǫ− iκ)] pin.

(A7)

Clearly, for arbitrary value of ν, the above equation is equivalent to Eq.(A5). The trick is to consider the right-handside of Eq.(A7) as a perturbation, and look for a formal solution specified by the index ν in a series expansion form

pνin(x) =

∞∑

n=−∞

a νn pn+ν(x), (A8)

where

pn+ν(x) = F (n+ ν + 1− iǫ,−n− ν − iǫ; 1− 2iǫ;x). (A9)

Then, we look for the eigenvalue of ν such that the series expansion is convergent. We also require that in the limitǫ → 0, the generalized angular momentum tends to l. This eigenvalue problem could be solved via the continuedfraction method. It is easy to show that the hypergeometric functions pn+ν(x) satisfy the recurrence relations

xpn+ν =−(α+ n)(n+ γ − β)

(α− β + 2n)(α− β + 2n+ 1)pn+ν+1 +2(α+ n)(n− β) + (α+ β − 1)γ(α− β + 2n− 1)(α− β + 2n+ 1)pn+ν

− (n− β)(α − γ + n)(α− β + 2n− 1)(α− β + 2n)pn+ν−1,

(A10)

10

x(1 − x)p′n+ν =(α+ n)(n− β)(n+ γ − β)

(α− β + 2n)(α− β + 2n+ 1)pn+ν+1 +(α+ n)(n− β)(α + β − 2γ + 1)(α− β + 2n− 1)(α− β + 2n+ 1)pn+ν

− (α+ n)(n− β)(α − γ + n)(α− β + 2n− 1)(α− β + 2n)pn+ν−1.

(A11)

where α = ν+1− iǫ, β = −ν− iǫ and γ = 1− 2iǫ. Substituting Eq.(A8) into Eq.(A7) and using the above recurrencerelations, we obtain a three-term recurrence relation among the expansion coefficients a νn

α νn aνn+1 + β

νn a

νn + γ

νn a

νn−1 = 0, (A12)

where

α νn =iκǫ(n+ ν + 1− iǫ)(n+ ν + 1 + iǫ)2

(n+ ν + 1)(2n+ 2ν + 3),

β νn = (n+ ν)(n + ν + 1)− l(l+ 1) + ǫ2(κ+ 1) +κǫ4

(n+ ν)(n+ ν + 1),

γ νn = −iκǫ(n+ ν − iǫ)2(n+ ν + iǫ)

(n+ ν)(2n+ 2ν − 1) .

(A13a)

(A13b)

(A13c)

It is useful to introduce

Rn(ν) ≡a νna νn−1

and Ln(ν) ≡a νna νn+1

, (A14)

they could also be expressed by continued fractions

Rn(ν) = −γ νn

β νn + ανn Rn+1(ν)

= − γνn

β νn −α νn γ

νn+1

β νn+1−α νn+1γ

νn+2

β νn+2−· · · ,

Ln(ν) = −α νn

β νn + γνn Ln−1(ν)

= − ανn

β νn −α νn−1γ

νn

β νn−1−α νn−2γ

νn−1

β νn−2−· · · .

(A15a)

(A15b)

The solution of the three-term recurrence relation is ”minimal” (i.e., the series expansion in Eq.(A8) is convergent)if and only if the renormalized angular momentum ν satisfies

β νn + ανn Rn+1(ν) + γ

νn Ln−1(ν) = 0. (A16)

Note that the value of n of this equation is arbitrary, it is convenient to set n = 0 to solve for ν. Note also thatEq.(A16) contains an infinite number of roots: (i) if ν is a solution of Eq.(A16), ν ± 1 is also a solution; (ii) if ν isa solution, −ν − 1 is also a solution. However, not all of these solutions can be used to build up the radial solutionwe want. By definition, we must have ν → l (or ν → −l − 1) in the limit ǫ → 0. Thus, we choose ν such that itapproaches to l as ǫ→ 0.Given the eigenvalue of ν, it is straightforward to compute the series coefficients a νn from the three-term recurrence

relation (A8). From Eqs.(A13), one finds α−ν−1−n = γνn and γ

−ν−1−n = α

νn so that a

−ν−1−n obeys the same recursion

relation as a νn does. This implies that if we take aν0 = a

−ν−10 = 1, we have a

νn = a

−ν−1−n . From Eq.(A16), we have

limn→∞

na νna νn−1

= − limn→−∞

na νna νn+1

=iκǫ

2. (A17)

Combining the large n behavior of hypergeometric functions, one finds

limn→∞

na νn pn+νa νn−1pn+ν−1

= − limn→−∞

na νn pn+νa νn+1pn+ν+1

=iκǫ

2

[

1− 2x+(

(1− 2x)2 − 1)1/2

]

. (A18)

This implies that the series of hypergeometric functions (A8) converges in all over the complex plane of x except for|x| = ∞. Thus, we have obtained the ingoing-wave radial solution which is valid for |x| < ∞. We note a property ofthe hypergeometric function

F (a, b; c;x) =Γ(c)Γ(b − a)Γ(c− a)Γ(b) (1− x)

−aF

(

a, c− b; a+ 1− b; 11− x

)

+ (a↔ b) . (A19)

11

This enables us to express the radial solution as

Rin = R ν0 +R−ν−10 , (A20)

where

R ν0 = eiκǫx(−x)−iǫ(1−x)iǫ+ν

∞∑

n=−∞

a νnΓ(1− 2iǫ)Γ(2n+ 2ν + 1)

Γ(n+ ν + 1− iǫ)2 (1−x)nF

(

−n− ν − iǫ,−n− ν − iǫ;−2n− 2ν; 11− x

)

.

(A21)Here, we have used the property a νn = a

−ν−1−n . Clearly, Eq.(A20) explicitly exhibits the symmetry of R

in under theinterchange ν ↔ −ν − 1.

2. Far region solution in series of Coulomb wave functions

The solution in the form of series of hypergeometric functions discussed in Appendix A1 is convergent at any finitevalue of r. However, it does not converge at infinity. The analytic solution convergent at infinity was obtained byLeaver as a series of Coulomb functions [57]. Here, we follow the procedure in Ref.[52] to obtain the radial solutionof Eq.(14) that is convergent at infinity.First, we define a variable z = ω

(

r −Q2/M)

= κǫ(1− x), and introduce the following form

RC = z−1(

1− κǫz

)−iǫ

f(z). (A22)

Then the radial equation (14) becomes

z2f ′′ +[

z2 + 2ǫz − ν(ν + 1)]

f = κǫz (f ′′ + f) + κǫ(1− 2iǫ)f ′ +[

l(l + 1)− ν(ν + 1)− ǫ2]

f, (A23)

where a prime denote d/dz. As in Eq.(A7), we have introduced the renormalized angular momentum ν in thisequation. We denote the solution specified by the index ν as fν(z), and expand it in terms of Coulomb wave functionsas

fν(z) =

∞∑

n=−∞

b νn Fn+ν(z), (A24)

where the Coulomb wave function is given by

Fn+ν(z) = e−iz(2z)n+νz

Γ(n+ ν + 1 + iǫ)

Γ(2n+ 2ν + 2)Φ(n+ ν + 1 + iǫ, 2n+ 2ν + 2; 2iz), (A25)

and Φ denotes the regular confluent hypergeometric function. It can be shown that the Coulomb wave functions givenin Eq.(A25) satisfy the following recurrence relations

1

zFn+ν =

n− ã+ 2b̃(b̃+ n)(2b̃+ 2n− 1)

Fn+ν+1 +i(b̃− ã)

(b̃+ n− 1)(b̃+ n)Fn+ν +

ã+ n− 1(b̃ + n− 1)(2b̃+ 2n− 1)

Fn+ν−1, (A26)

F ′n+ν =(b̃+ n− 1)(ã− 2b̃− n)(b̃+ n)(2b̃+ 2n− 1)

Fn+ν+1 +i(b̃− ã)

(b̃+ n− 1)(b̃+ n)Fn+ν +

(ã+ n− 1)(b̃+ n)(b̃+ n− 1)(2b̃+ 2n− 1)

Fn+ν−1, (A27)

where ã = ν + 1 + iǫ and b̃ = ν + 1. Substituting Eq.(A24) into Eq.(A23) and using the above recurrence relations,one finds a three-term recurrence relation among the expansion coefficients b νn

α̃ νn bνn+1 + β̃

νn b

νn + γ̃

νn b

νn−1 = 0, (A28)

with

α̃ νn = i(n+ ν + 1 + iǫ)

(n+ ν + 1− iǫ)ανn ,

β̃ νn = βνn ,

γ̃ νn = −i(n+ ν − iǫ)(n+ ν + iǫ)

γ νn ,

(A29a)

(A29b)

(A29c)

12

where α νn , βνn and γ

νn are defined in Eqs.(A13). Clearly, the recurrence relation (A28) is transformed to the one in

Eq.(A12) if we redefine the coefficients as

b νn = (−i)n(ν + 1− iǫ)n(ν + 1 + iǫ)n

a νn , (A30)

where (x)n ≡ Γ(x + n)/Γ(x). Since the recurrence relation obtained for the Coulomb expansion case is identical tothe one for the hypergeometric case, the renormalized angular momentum ν from both solutions are the same. Thisis crucial in matching these two solutions which we will discuss later. From Eqs.(A17) and (A30), we find

limn→∞

nb νnb νn−1

= limn→−∞

nb νnb νn+1

=κǫ

2. (A31)

By combing the large n behavior of the Coulomb wave functions, we have

limn→∞

b νn Fn+νb νn−1Fn+ν−1

= limn→−∞

b νn Fn+νb νn+1Fn+ν+1

=κǫ

z. (A32)

This implies that the series of Coulomb wave functions (A24) converges at z > κǫ or equivalently r > r+. By usingthe formula

Φ(a, c; z) =Γ(c)

Γ(c− a)eiaπΨ(a, c; z) +

Γ(c)

Γ(a)eiπ(a−c)exΨ(c− a, c;−z), (A33)

and Eq.(A30), we can rewrite Eq.(A22) as

RνC = RνC in +R

νC out, (A34)

with

RνC in =2νeiπ(ν+1+iǫ)

Γ(ν + 1− iǫ)Γ(ν + 1 + iǫ)

e−izzν+iǫ(z − κǫ)−iǫ∞∑

n=−∞

ina νn (2z)nΨ(n+ ν + 1 + iǫ, 2n+ 2ν + 2; 2iz),

(A35)

RνC out =2νe−iπ(ν+1−iǫ)eizzν+iǫ(z − κǫ)−iǫ

∞∑

n=−∞

in(ν + 1− iǫ)n(ν + 1 + iǫ)n

a νn (2z)nΨ(n+ ν + 1− iǫ, 2n+ 2ν + 2;−2iz),

(A36)where RνC in and R

νC out are the ingoing wave and outgoing wave solutions at infinity, respectively. And Ψ is the

irregular confluent hypergeometric function. At large |z|, Ψ has the following asymptotic behavior

Ψ(a, c; z) → z−a as |z| → ∞. (A37)

Thus, at infinity, RνC in and RνC out behave as

RνC in = Aνinz

−1e−i(z+ǫ ln z), (A38)

RνC out = Aνoutz

−1ei(z+ǫ ln z), (A39)

where

Aνin = 2−1−iǫei

π

2(ν+1+iǫ)Γ(1 + ν − iǫ)

Γ(1 + ν + iǫ)

∞∑

n=−∞

a νn , (A40)

Aνout = 2−1+iǫe−i

π

2(ν+1−iǫ)

∞∑

n=−∞

(−1)n (ν + 1− iǫ)n(ν + 1 + iǫ)n

a νn , (A41)

Note that there exist another independent solution which is obtained by replacing ν with −ν − 1. It is given by

R−ν−1C = −ie−iπνsinπ(ν + iǫ)

sinπ(ν − iǫ)RνC in + ie

iπνRνC out. (A42)

13

3. Matching of the two solutions

We have obtained two types of solutions R ν0 and RνC of Eq.(14). Note that both of them could be expressed in

terms of z = κǫ(1 − x), and they are convergent for a wide range of z, i.e., κǫ < z < ∞. Using the large z behaviorof the (confluent) hypergeometric functions, one finds that R ν0 must be proportional to R

νC , i.e.,

R ν0 = KνRνC . (A43)

The constant factor Kν is obtained by comparing the asymptotic behavior of Rν0 and R

νC . It is given by

Kν =eiκǫ(2κǫ)−ν−pip

Γ(1− 2iǫ)Γ(p+ 2ν + 2)Γ(p+ ν + 1 + iǫ)3

×(

∞∑

n=p

(−1)nΓ(n+ p+ 2ν + 1)(n− p)!

Γ(n+ ν + 1 + iǫ)2

Γ(n+ ν + 1− iǫ)2 aνn

)

×(

p∑

n=−∞

(−1)n(p+ 2ν + 2)n(p− n)!

(ν + 1− iǫ)n(ν + 1 + iǫ)n

a νn

)−1

,

(A44)

where p is an arbitrary integer and Kν is independent of the choice of p. This can be used as a check of the calculation.Thus, the ingoing wave solution (A4) could also be expressed the Coulomb functions, i.e.,

Rin = KνRνC +K−ν−1R

−ν−1C =

(

Kν − ie−iπνsinπ(ν + iǫ)

sinπ(ν − iǫ)K−ν−1)

RνC in +(

Kν + ieiπνK−ν−1

)

RνC out. (A45)

From Eq.(18), the radial function has the form

R(r) ∼ A(−)λ−1/2( r

2M

)−1−i2Mω

e−iωr +A(+)λ−1/2 × (ω → −ω) (A46)

as r → ∞. Here, we have used l = λ−1/2. Finally, comparing this equation with Eq.(A45) and using z = ω(r−Q2/M),we obtain

A(−)λ−1/2 = ǫ

−1e−iǫ ln ǫ(

Kν − ie−iπνsinπ(ν + iǫ)

sinπ(ν − iǫ)K−ν−1)

Aνin, (A47)

A(+)λ−1/2 = ǫ

−1eiǫ ln ǫ(

Kν + ieiπνK−ν−1

)

Aνout. (A48)

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