Energy conservation and axion back-reaction in a magnetic eld Srimoyee Sen1Lars Sivertsen1

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Energy conservation and axion back-reaction

in a magnetic ﬁeld

Srimoyee Sen,1Lars Sivertsen,1

1Department of Physics and Astronomy, Iowa State University, Ames IA 50011

E-mail: srimoyee08@gmail.com,lars@iastate.edu

Abstract: Axion clumps in an external magnetic ﬁeld can emit electromagnetic ra-

diation which causes them to decay. In the presence of a plasma, such radiation can

become resonant if the clump frequency matches the plasma frequency. Typically, the

decay or back-reaction of the clump is ignored in the literature when analyzing such

radiation. In this paper we present a self consistent, semi-analytic approach which

captures axion back-reaction using energy conservation. We ﬁnd that inclusion of

back-reaction changes the clump frequency over time enabling clumps with a range of

diﬀerent initial frequencies to become resonant at some point in their time evolution.

arXiv:2210.01149v2 [hep-ph] 21 Oct 2022

Contents

1 Introduction 1

2 The axion-photon equations and backreaction 4

3 The axion proﬁle 8

3.1 Long wavelength instability 12

4 electromagnetic radiation 13

5 Applications 15

5.1 Strategy for computing back-reaction 18

5.2 Results 19

5.3 Decay time-scales and regime of validity of our approach 26

6 Conclusion 30

1 Introduction

Axions are pseudo-scalar particles originally introduced to solve the strong CP prob-

lem in QCD [1–3]. In many models, the axion (and axion-like particles (ALP)) interact

weakly with standard model particles and are therefore considered to be suitable candi-

dates for dark matter [4–18]. Due to their bosonic statistics, axions can form coherently

oscillating Bose-Einstein condensates (BEC). These BECs are a consequence of axions

clumping in space owing to their gravity and or their self interaction [19–21]. The

clumps are well described by a localized spatial proﬁle for the axion ﬁeld, which can be

obtained by solving the classical equations of motion for it [19,20,22–24]. Since axion

number is not conserved, the clumps can decay via scalar radiation. This causes the

clumps to have a ﬁnite lifetime. However, this decay can be signiﬁcantly suppressed

for certain axion potentials [19,22,25] giving rise to long-lived clumps.

Axions can also couple to electromagnetic ﬁelds and send out electromagnetic radi-

ation through several diﬀerent processes which in turn can cause the clumps to decay.

For example, an axion particle can decay to two photons which allows axion clumps to

radiate via spontaneous or stimulated emission. For ultralight axion clumps however,

– 1 –

the former is highly suppressed [26] whereas the latter isn’t. The decay time for stim-

ulated emission can be small, for example of the order of a few seconds for ultralight

axions of mass 10−11 eV. Similarly, in the presence of a background electromagnetic

ﬁeld, an axion particle can convert to a photon. This causes an axion clump in a

electromagnetic (EM) ﬁeld to emit EM radiation. To see how this comes about, one

has to solve the classical equations of motion for the axion and the electromagnetic

ﬁelds simultaneously. Maxwell’s equations in this case get augmented with an oscillat-

ing current source term originating from the axion clump, which acts like an antenna

contributing to EM radiation.

All of these processes take away energy from the axion clump depleting the clump.

This in turn reduces the EM radiation coming out of these clumps. Typically when

considering electromagnetic radiation in these problems, one considers the axion am-

plitude to be independent of time ignoring the back-reaction of the radiation on the

axion clump itself. This is justiﬁed when axion-photon coupling is weak so long as one

is only interested in the leading order EM radiation coming out of the clump. One can

estimate the axion decay timescale by taking a ratio of the total initial energy stored in

the axion clump to the total radiated power. For example, in [27] it is mentioned that

for an axion photon coupling of Cβ

πfawhere Cis an order 1 number and fais the axion

decay constant, the decay time of an axion clump with axion mass main the presence

of a uniform background magnetic ﬁeld Bis given by

τ∼πfa

Cβ 2ma

3B2.(1.1)

This procedure may give approximately correct estimates for the decay time for the

axion clumps. However, there are several interesting and phenomenologically exciting

features of axion radiation which cannot be captured in this method. For example,

consider the ﬁndings of [26,28] where it was shown that the EM radiation eﬃciency of

an axion clump in external EM ﬁeld depends on a combination of the clump frequency

and its spatial extent. Clumps that were too large or too small compared to the

wavelength of the radiation would radiate minimally, whereas there would be eﬃcient

(resonant) radiation when the two values were close. Strictly speaking, this conclusion

is accurate only when the decay of the axion clump (axion back-reaction) is ignored.

As observed in [19,23] the decay of axion clumps causes their frequency, their spatial

extent and the wavelength of EM radiation to change over time. This makes it possible

to imagine scenarios where a poorly radiating clump at a certain instant in time can

become resonant at a later instant by altering its frequency and spatial extent as

it radiates. Similarly, a clump that is radiating resonantly at a certain instant in

time can move out of resonance as time passes. This dynamical change in resonance

– 2 –

condition cannot be captured in an analysis which ignores back-reaction where the

axion amplitude and its frequency are assumed to be constant in time. Our goal in this

paper is to address this by taking into account axion back-reaction. Our framework is

speciﬁcally designed to address EM radiation from clumps in a background magnetic

ﬁeld. It should be possible to extend this approach to stimulated emission, which

however is beyond the scope of this paper.

In the process of constructing our framework we verify the estimate given by Eq.

1.1 while also capturing how clumps can move in and out of resonance with time as they

radiate. Note that, some previous analysis, e.g. [28] has taken into account the eﬀect

of axion back-reaction for suﬃciently large axion-photon coupling where stimulated

emission dominates. Their analysis involves complete numerical simulation of axion-

photon equations of motion as necessitated by a strong axion-photon coupling. Such

numerical simulation is unnecessary when the coupling is weak as is the case in this

paper. Instead, here we develop a semi-analytic perturbative approach to taking into

account axion back-reaction based on energy conservation. This procedure captures

axion clump decay and the resulting decay of electromagnetic radiation in the limit of

weak axion-photon coupling while avoiding the cost of a full axion-photon numerical

simulation.

As stated before, axion clumps can also radiate by emitting scalar(axion) waves.

The stability of relativistic axion clumps against scalar radiation was analyzed in [23]

and it was found that for certain types of axion self interactions the decay is suppressed

suﬃciently so as to produce long living axion clumps. Here the axion decay timescale

is given by τscalar m−1

a. For such clumps, it may be appropriate to neglect the eﬀect

of scalar radiation on the clump and to only consider the eﬀect of electromagnetic

radiation to describe the time evolution. This is the regime where we choose to work

in, i.e. τscalar > τEM > τrad where τEM is the clump decay timescale due to EM radiation

and τrad is the time period of radiated EM waves. As we will see, τrad ∼m−1

a. In

principle, a complete analysis of axion clump decay should include both the eﬀect of

the axion radiation and electromagnetic radiation. Since the goal of this paper is to

describe the eﬀect of electromagnetic radiation and the corresponding back-reaction,

with a few exceptions we mostly restrict ourselves to parameter ranges where scalar

radiation is suppressed compared to the EM radiation.

The organization of this paper is as follows: In the next section we outline the

perturbative semi-analytic approach to taking into account axion back-reaction. In the

following two brief sections, section 3and section 4, we review axion clump solutions and

electromagnetic radiation from them. In the following section, section 5, we highlight

a few processes where the eﬀect of backreaction can have dramatic eﬀects on the time

dependence of radiation, including resonant radiation. This is followed by results which

– 3 –

demonstrate the same. We conclude with a discussion on the decay time-scale of axion

clumps while outlining the regime of validity of our approach.

2 The axion-photon equations and backreaction

In this section we will present the main ideas behind the semi-analytic approach we

take to account for backreaction. Our calculations hold for weak axion-photon coupling

in a regime where the axion decay timescale due to electromagnetic radiation is large

compared to the time period of outgoing radiation. The axion-photon Lagrangian is

given by

L=−1

4Fµν Fµν +Jµ

mAµ+Cβ

4πfa

φµνλρFµν Fλρ +1

2(∂µφ)(∂µφ)−V(φ) + .... (2.1)

In the latter parts of this paper explain the regime of validity of our calculation in

terms of the parameters of this Lagrangian. In the Lagrangian above, φ(x, t) is the

pseudo scalar axion ﬁeld with mass ma,Fµν =∂µAν−∂νAµis the electromagnetic

ﬁeld tensor with corresponding gauge ﬁeld Aµ,V(φ) is the eﬀective axion potential,

and Jµ

mis a background matter current, if any. Furthermore, Cis a model dependent

parameter with C∼1, while fais the axion decay constant, and βis the electromagnetic

ﬁne structure constant. For the QCD axion one has β∼1/137, while for axion-like

particles βcan take other values. The axion mass for a particular potential can be

obtained by ma=∂2V

∂φ2φ=0. If one wants to understand axion dynamics in the absence

of electromagnetic interaction, one has to set C= 0 in the above Lagrangian. As we

know, a ﬁnite number of axion particles can clump together to form axion clumps. Such

clump solutions have been analyzed by several papers [19,20,22,23] in the absence

of electromagnetic coupling. These solitonic solutions can be found by writing down

axion EOM from the above Lagrangian and solving it for a ﬁxed axion number or total

energy. These solutions will play a crucial role in our analysis of axion back-reaction.

If one wants to analyze axion clumps in the presence of electromagnetic coupling,

one needs to write both axion EOM and Maxwell’s equations

– 4 –

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摘要：

Energyconservationandaxionback-reactioninamagneticeldSrimoyeeSen,1LarsSivertsen,11DepartmentofPhysicsandAstronomy,IowaStateUniversity,AmesIA50011E-mail:srimoyee08@gmail.com,lars@iastate.eduAbstract:Axionclumpsinanexternalmagneticeldcanemitelectromagneticra-diationwhichcausesthemtodecay.Inthepresen...

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Energy conservation and axion back-reaction in a magnetic eld Srimoyee Sen1Lars Sivertsen1

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