Corrections to Hawking Radiation from Asteroid Mass Primordial Black Holes I. Formalism of Dissipative Interactions in Quantum Electrodynamics Makana Silva1 2Gabriel Vasquez1 2yEmily Koivu1 2zArijit Das1 2xand Christopher M. Hirata1 2 3

2025-05-06 0 0 911.01KB 30 页 10玖币

侵权投诉

Corrections to Hawking Radiation from Asteroid Mass Primordial Black Holes:

I. Formalism of Dissipative Interactions in Quantum Electrodynamics

Makana Silva,1, 2, ∗Gabriel Vasquez,1, 2, †Emily Koivu,1, 2, ‡Arijit Das,1, 2, §and Christopher M. Hirata1, 2, 3, ¶

1Center for Cosmology and Astroparticle Physics, The Ohio State University,

191 West Woodruﬀ Avenue, Columbus OH, 43210, USA

2Department of Physics, The Ohio State University,

191 West Woodruﬀ Avenue, Columbus OH, 43210, USA

3Department of Astronomy, The Ohio State University,

140 West 18th Avenue, Columbus OH, 43210, USA

(Dated: October 4, 2022)

Primordial black holes (PBHs) within the mass range 1017 −1022 g are a favorable candidate for

describing the all of the dark matter content. Towards the lower end of this mass range, the Hawking

temperature, TH, of these PBHs is TH&100 keV, allowing for the creation of electron – positron

pairs; thus making their Hawking radiation a useful constraint for most current and future MeV

surveys. This motivates the need for realistic and rigorous accounts of the distribution and dynamics

of emitted particles from Hawking radiation in order to properly model detected signals from high

energy observations. This is the ﬁrst in a series of papers to account for the O(α) correction to the

Hawking radiation spectrum. We begin by the usual canonical quantization of the photon and spinor

(electron/positron) ﬁelds on the Schwarzschild geometry. Then we compute the correction to the

rate of emission by standard time dependent perturbation theory from the interaction Hamiltonian.

We conclude with the analytic expression for the dissipative correction, i.e. corrections due to the

creation and annihilation of electron/positrons in the plasma.

I. INTRODUCTION

Primoridal black holes (PBHs) are possible relics that could provide insights into the physics of the earliest moments

of the Universe [1–5]. There are several proposed formation mechanisms for PBHs, such as collapse of non-Gaussian

ﬂuctuations in the Early Universe, tunneling through some scalar potential, etc. [2, 6–10]. Primordial black holes are

an interesting candidate for dark matter (DM) since, although some new physics at a high energy scale is required

to form them, the PBH scenario does not require any new long-lived particles to be added to the Standard Model

[11]. A wide range of observational constraints have removed diﬀerent mass ranges as candidates for dark matter (see

recent summaries [12, 13]), leaving a mass range of about 1017–1022 g as a possible candidate to describe all of the

dark matter content.

There are several probes for detecting PBHs based on their associated physical properties. From a gravitational

perspective, PBHs could be detected through their lensing eﬀects on various bright back ground sources, e.g. mi-

crolensing, gravitational wave detection, etc. [12, 14–20]. Another method involves quantum processes near the black

hole horizon that were predicted by Hawking [21] known as Hawking radiation. This is the predicted eﬀect that black

holes would radiate away (“evaporation”) by the emission of radiation and other particles. The foundation of this

argument lies in how we deﬁne the vacuum state; prior to the formation of the black hole (ﬂat spacetime) we deﬁne

a vacuum state with zero occupation of particles, but after a black hole forms, the vacuum near the horizon becomes

a thermal state (due to the existence of an accelerating frame of reference near the horizon, i.e. Unruh eﬀect) from

which particles can arise and escape to inﬁnity [21–24]. The Hawking radiation of a black hole is dependent on its

mass: TH= 1/(8πM), where THis the Hawking temperature of the radiation in units where G=kB=c=~= 1,

showing that lower mass black holes radiate at higher temperatures than more massive ones. This process places

constraints on the lower mass range of PBHs due to lifetime of PBHs with MP BH .5×1014 g being comparable to

the age of the Universe, i.e. evaporated away [4, 25]. For PBHs of MP BH .1017 g, their Hawking radiation would

be in the γ-ray regime, making it a novel and eﬀective probe for direct detection [26, 27].

This particular mass range (“asteroid mass”) and lower would have a Hawking temperature TH&100 keV (TH∼

(1016 g/MP BH ) MeV), well within the γ-ray regime of the electromagnetic spectrum, making their Hawking radiation

a point of interest for observations by MeV observatories such as AMEGO or SMILE [28–31]. For the higher regime of

∗silva.179@osu.edu

†vasquez.119@osu.edu

‡koivu.1@osu.edu

§das.241@osu.edu

¶hirata.10@osu.edu

arXiv:2210.01914v1 [gr-qc] 4 Oct 2022

TH(i.e. lower end of the asteroid mass regime), PBHs are able to emit electron – positron (e+e−) pairs, making them

the dominate contributor to the detectable Hawking radiation and allowing for constraints on PBH mass distributions.

Since the emission of e+e−would lead to the increase of ﬂux of 511 keV lines in any MeV survey and PBHs occupy

spherical halos around galaxies, the 511 keV emission line near and away from the the Galactic center would be an

ideal method of constraint [32]. DeRocco and Graham [33] was able to place constraints on PBH abundance as a

DM candidate by using 511 keV lines from MeV surveys of the Galactic Bulge from INTEGRAL [34], though the

underlying assumption was the rate of production of e+e−was not enough to form a plasma around a PBH (similar

to [35]), thus allowing for the escape of positrons to be annihilated at some further distance. This extended travel of

the positron requires knowledge of its propagation through the ISM in order to carefully account for excess 511 keV

lines in various regions of the Galactic Bulge. Recently, Coogan et al. [26] showed that upcoming MeV telescopes

could directly detect asteroid mass PBHs within dark matter halos. The dependence of the particles created by PBHs

at TH&100 keV motivates the need to have careful and rigorous models of the distributions of particles in order to

use intermediate-energy leptons [36] and the 511 keV positron annihilation line [33, 37, 38] to constrain PBH models

of DM.

Although Hawking radiation is often described as a blackbody, the emission spectrum is modiﬁed by a “graybody”

factor related to the energy-dependent cross section for the black hole to absorb a particle. The standard calculation

treats this by the partial wave expansion for non-interacting particles [39, 40]. Since then, there have been many

investigations into the case of interactions and secondary particles in various approximations. Page [41] showed that

when charged particles are emitted, the black hole develops an opposite charge, leading to an electric potential that

alters the emission of further charged particles, and an O(α) change to the charged particle emission rate, where

α≈1/137 is the ﬁne structure constant. At TH&20 MeV, strongly interacting particles can be produced, and there

have been investigations of hadronization [42, 43]. Also most of the particles produced at these higher energies are

unstable, leading to secondary particles from their decay; in some regimes, these processes can even dominate the

neutrino spectrum [44]. Several papers have discussed the possibility that at very high TH, the density of emitted

particles might result in a high optical depth and form a “photosphere” or a region of relativistic ﬂuid ﬂow followed

by decoupling [45–49], although subsequent work incorporating the special relativistic kinematics of the outgoing

particles showed a much smaller optical depth and no photosphere [35]. Page et al. [50] considered the O(α) inner

bremsstrahlung emission accompanying the emission of charged particles and showed that this process could be a

signiﬁcant contribution to the low-energy spectrum. Coogan et al. [26] speciﬁcally show that the secondary spectrum

(lower energy photons) of Hawking radiation from PBHs in the MeV range would be the most signiﬁcant contributor to

the detection of this spectrum of radiation, showing that the discovery reach for upcoming MeV surveys will increase by

an order of magnitude by including the secondary spectrum. This shows that understanding the secondary spectrum

is vital for any sort of MeV survey that hopes to make direct detection of Hawking radiation from PBHs. This has also

motivated study of other quantum electrodynamics (QED) contributions to the emitted spectrum such as annihilation

of e+e−pairs [51].

This is the ﬁrst in a series of papers whose ultimate goal is to compute the O(α) correction to the Hawking radiation

from a Schwarzschild black hole. The current analysis of the photon spectrum from PBHs was investigated to O(α)

in Coogan et al. [26] showing at lower photon energies, the secondary spectrum becomes the dominant contributor to

the total spectrum (see Fig. 2 in Coogan et al. [26]). However, this photon emission calculation was performed using

ﬂat spacetime inner bremsstrahlung formulae [52], leading to the purpose and motivation of this series of papers: to

compute the O(α) correction to the Hawking radiation but on a curved spacetime, i.e., the Schwarzschild metric. Given

the historical challenges and controversies in understanding interacting particles resulting from Hawking radiation,

we believe that a full quantum treatment on the curved background is a necessary step in order to put the calculation

of inner bremsstrahlung and related O(α) eﬀects on a rigorous foundation.

This paper is organized as follows. In Sec. II we lay out our conventions for the spacetime metric and spinor

algebra used in the quantization of the electromagnetic and spinor ﬁelds (electron and positrons). In Sec. III and

Sec. IV, we work through the canonical quantization of the electromagnetic and spinor ﬁelds. In Sec. V, we compute

the interaction Hamiltonian between spinors and photons in terms of annihilation and creation operators and mode

overlap integrals. In Sec. VI, we re-express our results in the interaction picture as appropriate for a time dependent

perturbation theory treatment, and arrive at our expressions for the dissipative O(α) contribution to the photon

emission of the Hawking radiation. Finally, we conclude and discuss follow up works in Sec. VII.

II. CONVENTIONS

We use the + + +−signature for the metric. We use Greek indices αβ to indicate any spacetime index; and Latin

indices ijk to select only spatial indices; and Latin indices ABC to denote Dirac spinor indices.

We use the tortoise coordinate r?to write the Schwarzschild metric:

ds2=1−2M

rdr2

?+r2dθ2+r2sin2θ dφ2−1−2M

rdt2,(1)

where rand r?are related by

r?=r+ 2Mln r−2M

2Mand dr

dr?

= 1 −2M

r.(2)

We have r?≈rat large radius, but at the horizon r→2Mwhereas r?→ −∞. In general rappears in most of our

expressions, but r?is a better choice of independent variable for numerical calculations near the horizon since one

can avoid cancellations in the expression 1 −2M/r.

We use the overdot symbol ˙ to indicate partial derivatives with respect to t, and the prime 0to indicate partial

derivatives with respect to r?.

The electron mass is µand the black hole mass is M. The speed of light, Newton’s gravitational constant, the

reduced Planck’s constant (“~”), and the permittivity of the vacuum (“0”) are set equal to unity.

The Dirac matrices written in an orthonormal basis are denoted by ˜

γµ, and satisfy the usual anti-commutation

relation {˜

γµ,˜

γν}= 2ηµν I4×4. We use the representation of the Dirac matrices, in 2 ×2 form,

γi=0σi

σi0and ˜

γ4=iI2×20

0−iI2×2,(3)

where σiare the Pauli matrices and I2×2is the 2 ×2 identity. This follows the convention of Brill & Wheeler [53];

note that due to the signature, ˜

γ4is anti-Hermitian whereas ˜

γiis Hermitian. We deﬁne the usual adjoint spinor

appearing in the Dirac Lagrangian ¯

ψ=ψ†β, where

β=−i˜

γ4=I2×20

0−I2×2.(4)

In this convention, the action of quantum electrodynamics is

SQED =Z¯

ψ(−γµDµ−µ)ψ−1

4Fµν Fµν √−g d4x, (5)

where γµ=aµα˜

γαis the Dirac γ-matrix in covariant notation; aµαis the 4 ×4 matrix of vierbein components; and

the covariant derivative acting on the electron spinor is

Dµ=∂µ−Γµ−ieAµ.(6)

This expression contains two corrections to the partial derivative: the 4 ×4Γµmatrix, which encodes the rotation

of the vierbein when we move in direction µ(Eq. 8 of Ref. [53]); and the U(1) gauge transformation term −ieAµfor

electron charge e. The electromagnetic term includes the ﬁeld strength tensor, Fµν =∂µAν−∂νAµ.

We denote quantum numbers of modes as follows:

1. For photon modes, the energy is denoted ωand the angular quantum numbers are denoted by `and m. Parity is

denoted by (p)∈ {(e),(o)}(“even” or “odd” sector; also described as “electric” or “magnetic” when discussing

atomic or nuclear transitions, or “polar” or “axial” sectors [54]).

2. For electron modes, the energy is denoted by hand the total (orbital+spin) angular momentum and parity are

denoted by j,m, and p=±1. We may for shorthand use the combination k=s(j+1

2) instead of writing both

jand p, where the sign of kis s= (−1)j−1/2p.

3. For both types of modes, we use Xto denote a mode associated with the “in from ∞” (“in”) channel, or “up

from the horizon” (“up”) channel, as depicted in Fig. 1. We may also use the alternative basis of the “out” or

“down” modes.

4. Since we will be doing non-linear interactions, we will often work with multiple electron or photon modes. Mode

indices may be primed or subscripted to distinguish them.

We deﬁne the triangle inequality condition:

∆(j1, j2, j3) = 1j1+j2≥j3and j3+j1≥j2and j2+j3≥j1

0 otherwise ,(7)

and deﬁne (−1)p= +1 for (p)=(e) and (−1)p=−1 for (p)=(o).

FIG. 1. Representation of the diﬀerent scattering states from the eﬀective potential (curve labeled V(r∗)) coming from the

black hole (BH) horizon (“up”) and coming in from ∞(“in”) as well as the equivalent scattering basis for radiation falling

towards the BH near the horizon (“down”) and radiation leaving the black hole to ∞(“out”).

III. THE ELECTROMAGNETIC FIELD

We ﬁrst review the electromagnetic ﬁeld in the Schwarzschild spacetime. This is described by an Schr¨odinger-like

equation with a Regge-Wheeler potential [55, 56], which is implemented in commonly used tools such as BlackHawk to

compute Hawking radiation for spin 1 massless particles (see Eq. 3.33 of [57]). We work through the derivation here,

both for completeness and to make explicit our gauge choice and the expression for potentials in terms of annihilation

and creation operators.

Because we want to write an interaction Hamiltonian, we will follow the canonical quantization method, and we also

work in terms of the vector potential Aµrather than the ﬁeld components Fµν (or the closely related Newman-Penrose

variables Φ0and Φ2[58–61]). This approach has been used for some problems in black hole spacetimes, although with

a diﬀerent gauge choice than we will take here – e.g., the modiﬁed Feynman gauge [62], the gauge choices based on

null vectors [63, 64] (which extend naturally to the Kerr case), or the zero scalar potential gauge At= 0 [65] (although

this leads to the same mode functions for the vector potential because it is the same as the Coulomb gauge in the

case where there are no charges). It is also used for the Proca equation [66, 67] (but in that case the Lorenz condition

Aµ;µ= 0 holds and we cannot impose a diﬀerent gauge choice).

A. Angular decomposition

The electromagnetic ﬁeld action is SEM =R−1

4Fµν Fµν √−g d4x, where the ﬁeld tensor is Fµν =∂µAν−∂νAµ.

Written in terms of components, the Lagrangian LEM =dSEM/dt is

LEM =Z∞

−∞ Z2π

0Zπ

0n1

2(Φ0+˙

Ar?)2+1−2M/r

2r2h(∂θΦ + ˙

Aθ)2+1

sin2θ(∂φΦ + ˙

Aφ)2i

−1−2M/r

2r2h(∂θAr?−A0

θ)2+1

sin2θ(∂φAr?−A0

φ)2i−(1 −2M/r)2

2r4sin2θ(∂θAφ−∂φAθ)2osin θ dθ dφ r2dr?

1−2M/r ,

(8)

where Φ ≡ −At. It is convenient to break down the vector potential in a multipole expansion: we write

Φ(t, r?, θ, φ) =

∞

`=0

m=−`

Φ`,m(t, r?)Y`,m(θ, φ),

Ar?(t, r?, θ, φ) =

∞

`=0

m=−`

A(r),`,m(t, r?)Y`,m(θ, φ),

Aθ(t, r?, θ, φ) =

∞

`=1

m=−`A(e),`,m(t, r?)∂θY`,m(θ, φ)−A(o),`,m(t, r?)1

sin θ∂φY`,m(θ, φ),and

Aφ(t, r?, θ, φ) =

∞

`=1

m=−`A(e),`,m(t, r?)∂φY`,m(θ, φ) + A(o),`,m(t, r?) sin θ ∂θY`,m(θ, φ).(9)

Here Φ`,m and A(r),`,m are the usual multipole moments of the scalar potential and radial part of the vector potential.

The components in the angular (θand φ) directions form a vector on the 2-sphere of constant (t, r?) and have to

be described with two sets of multipole moments: an A(e),`,m component with the same parity as Y`,m (since the

coeﬃcient multiplies ∇Y`,m); and an A(o),`,m component with the opposite parity (since the coeﬃcient multiplies

er?× ∇Y`,m). An equivalent description of the angular components is

Aθ(t, r?, θ, φ)±i

sin θAφ(t, r?, θ, φ) = ∓

∞

`=1

m=−`p`(`+ 1) A(e),`,m(t, r?)±iA(o),`,m(t, r?)±1Y`,m(θ, φ).(10)

The Lagrangian of Eq. (8) can be re-written in terms of these components; with some simpliﬁcation, we ﬁnd

LEM =X

`,m Z∞

−∞nr2

1−2M/r h1

2|Φ0

`,m|2+ Re{Φ0}∗

`,m ˙

A(r),`,m +1

2|˙

A(r),`,m|2i+`(`+ 1)h1

2|Φ`,m|2+ Re Φ∗

`,m ˙

A(e),`,m

2|˙

A(e),`,m|2+1

2|˙

A(o),`,m|2−1

2|A(r),`,m −A0

(e),`,m|2−1

2|A0

(o),`,m|2i−[`(`+ 1)]21−2M/r

2|A(o),`,m|2odr?.

(11)

B. Gauge ﬁxing

To continue the quantization program, we impose a gauge condition. A particularly convenient choice is a gen-

eralization of the Coulomb gauge, which eliminates the cross-terms between the scalar and vector potentials. In

Minkowski spacetime the Coulomb gauge has the disadvantage of breaking Lorentz invariance, but on a Schwarzschild

background this is not a concern. In our case, we choose

K`m ≡ −r2

1−2M/r A(r),`,m0

+`(`+ 1)A(e),`,m = 0,(12)

which turns the combination of the Re{Φ0}∗

`,m ˙

A(r),`,m and Re Φ∗

`,m ˙

A(e),`,m terms in Eq. (11) into a total derivative.

The gauge choice of Eq. (12) can always be achieved by the gauge transformation Aµ→Aµ+∂µχ, where χ=

−˜

H−1

`K`mY`,m(θ, φ) and we deﬁne the radial operator

H`f=−r2

1−2M/r f00

+`(`+ 1)f, (13)

which is positive deﬁnite and Hermitian with respect to the standard inner product (f1|f2)r?=R∞

−∞ f∗

1f2dr?.

C. Sectors of the theory and the radial wave equations

The Lagrangian of Eq. (11) can then be broken down into three parts:

LEM =LEM,Φ+LEM,(e)+LEM,(o),(14)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

CorrectionstoHawkingRadiationfromAsteroidMassPrimordialBlackHoles:I.FormalismofDissipativeInteractionsinQuantumElectrodynamicsMakanaSilva,1,2,GabrielVasquez,1,2,yEmilyKoivu,1,2,zArijitDas,1,2,xandChristopherM.Hirata1,2,3,{1CenterforCosmologyandAstroparticlePhysics,TheOhioStateUniversity,191WestWood...

展开>> 收起<<

Corrections to Hawking Radiation from Asteroid Mass Primordial Black Holes I. Formalism of Dissipative Interactions in Quantum Electrodynamics Makana Silva1 2Gabriel Vasquez1 2yEmily Koivu1 2zArijit Das1 2xand Christopher M. Hirata1 2 3.pdf

共30页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Corrections to Hawking Radiation from Asteroid Mass Primordial Black Holes I. Formalism of Dissipative Interactions in Quantum Electrodynamics Makana Silva1 2Gabriel Vasquez1 2yEmily Koivu1 2zArijit Das1 2xand Christopher M. Hirata1 2 3

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: