Heating up PecceiQuinn scale

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Heating up Peccei–Quinn scale

Sabir Ramazanov and Rome Samanta

CEICO, FZU-Institute of Physics of the Czech Academy of Sciences,

Na Slovance 1999/2, 182 00 Prague 8, Czech Republic

May 30, 2023

Abstract

We discuss production of QCD axion dark matter in a novel scenario, which assumes

time-varying scale of Peccei-Quinn symmetry breaking. The latter decreases as the

Universe’s temperature at early times and eventually stabilises at a large constant

value. Such behavior is caused by the portal interaction between the complex ﬁeld

carrying Peccei-Quinn charge and a Higgs-like scalar, which is in thermal equilibrium

with primordial plasma. In this scenario, axions are eﬃciently produced during the

parametric resonance decay of the complex Peccei-Quinn ﬁeld, relaxing to the minimum

of its potential in the radiation-dominated stage. Notably, this process is not aﬀected

by the Universe’s expansion rate and allows to generate the required abundance of

dark matter independently of an axion mass. Phenomenological constraints on the

model parameter space depend on the number density of radial ﬁeld ﬂuctuations, which

are also generically excited along with axions, and the rate of their thermalization

in the primordial plasma. For the ratio of radial ﬁeld and axion particles number

densities larger than ∼0.01 at the end of parametric resonance decay, the combination

of cosmological and astrophysical observations with the CAST limit conﬁnes the Peccei-

Quinn scale to a narrow range of values ∼108GeV, — this paves the way for ruling

out our scenario with the near future searches for axions.

1 Introduction

Axions, pseudo-Nambu-Goldstone bosons of spontaneously broken Peccei-Quinn symme-

try [1,2,3,4], provide an elegant resolution of the strong CP problem in Quantum Chromo-

dynamics (QCD). Instanton eﬀects endow an axion with a mass ma, inversely proportional

to the scale of Peccei-Quinn symmetry breaking fP Q. The latter must be set well above

the Higgs ﬁeld expectation value vSM to warrant small axion couplings to Standard Model

arXiv:2210.08407v3 [hep-ph] 26 May 2023

(SM) particles. There are two benchmark axion setups realising the hierarchy fP Q ≫vSM :

the Kim-Shifman-Veinshtein-Zakharov (KSVZ) scenario and its variations [5,6], plus the

Dine-Fischler-Srednicki-Zhitnitski (DFSZ) type of models [7,8]. For substantially large fP Q,

axions are stable on cosmological time scales, and thus can be regarded as promising dark

matter candidates, if produced abundantly in the early Universe [9,10,11]. In the present

work, we are primarily interested in the case of QCD dark matter axions.

While very weakly coupled to matter ﬁelds, axions are actively searched for in astro-

physical and cosmological backgrounds, as well as with Earth-based facilities [12]. The

search programs strongly depend on the range of values of fP Q, which in turn is lim-

ited by the choice of axion production mechanisms. Most commonly, axions are produced

through vacuum realignment [9,10,11] or decay of topological defects [13,14] in the range1

1011 GeV ≲fP Q ≲1012 GeV. For such large values of fP Q, axions are unlikely to have a

sizeable impact on stellar evolution. In particular, their eﬀect on the duration of neutrino

burst from the Supernova 1987A suggests the rough limit fP Q ≳4×108GeV [15]. Further-

more, anomalous cooling of horizontal branch stars favours relatively small fP Q ∼108GeV

corresponding to axion masses ma∼5·10−2eV [16,17]. These considerations motivate de-

vising new mechanisms capable of generating axions in the range of parameters interesting

for astrophysics, cf. Refs. [18,19,20,21,22,23].

We propose a novel scenario of QCD axion genesis at high temperatures, which leads

to the correct dark matter abundance for the masses ma≳10−2eV. We assume that

the scale fP Q decreases with the Universe’s temperature Tat suﬃciently early times, i.e.,

fP Q(t)∝T(t), and later on, stabilises at a constant value f0

P Q. This is achieved by the

portal coupling of the complex ﬁeld Scarrying Peccei-Quinn charge to a thermal scalar ϕin

the primordial plasma; see Section 2. The scale f0

P Q is related to variance of the ﬁeld ϕby

fP Q ∝ ⟨ϕ†ϕ⟩1/2. Thus, if the ﬁeld ϕis in the spontaneously broken phase at low temperatures

with the expectation value vϕ, one has f0

P Q ∝vϕ. At very early times ⟨ϕ†ϕ⟩1/2∼T, and we

achieve the desired temperature dependence of the scale fP Q. (See also Ref. [24] discussing

diﬀerent forms of Peccei-Quinn scale’s temporal variation with a similar goal of altering axion

parameter space.). We would like to remark that the generic idea of spontaneous symmetry

breaking caused by thermal ﬂuctuations of hot primordial plasma is rather old [25]. It was

later implemented in various contexts: topological defects [26,27,28], baryon asymmetry

generation [29], dark matter production [30], and conformal ﬁeld theories [31]. The present

work continues this research line by applying the idea to axion models.

In our scenario, axions appear through parametric resonance when the radial part of

the Peccei-Quinn complex scalar |S|relaxes to the minimum of its spontaneously broken

potential (Section 3). Compared to Refs. [18,19,32], we consider the situation when the

1For simplicity, in this work we identify the axion decay constant faand Peccei-Quinn symmetry breaking

scale fP Q. See the remark in the beginning of Section 4.

initial amplitude of oscillations is smaller than fP Q. This is a natural option in our case,

given huge values of fP Q at large temperatures and initial conditions for the ﬁeld Sset by

cosmic inﬂation. As a result, axion creation proceeds in the regime, which is qualitatively

similar to a narrow parametric resonance, cf. Refs. [20,21]. Typically, eﬃciency of a narrow

parametric resonance is limited by the Universe’s expansion [33,34] diluting enhancement

due to Bose condensation of produced particles. Such a limitation does not take place in our

case: in the radiation-dominated Universe, the time-decrease of Peccei-Quinn scale eﬀectively

compensates for the redshift of axion physical momenta. As a result, axion abundance

grows exponentially and quickly reaches values necessary to explain dark matter (Section 4).

Notably, this can be fulﬁlled for essentially any value of low energy Peccei-Quinn scale f0

P Q.

Besides axions, ﬂuctuations of the radial ﬁeld |S|serve as a promising source of non-trivial

phenomenology. Contrary to conventional scenarios, where the ﬁeld |S|is superheavy, in our

case typical values of the radial ﬁeld are lying in the keV- and MeV-range, and thus particles

|S|can be abundantly produced in the stellar cores or in supernovae explosions. Furthermore,

excitations of the ﬁeld |S|generated along with axions during the parametric resonance, may

impact the Big Bang Nucleosynthesis (BBN) and the Cosmic Microwave Background (CMB);

see Subsection 5.1. This impact is unacceptably large, if the energy density of radial ﬁeld

particles generated in this way is comparable to that of axions at production, and if these

radial ﬂuctuations survive till BBN. However, for the values f0

P Q ≃(a few) ×108GeV,

one can eﬃciently thermalize radial ﬂuctuations in the plasma and avoid the conﬂict with

BBN and CMB data (Subsection 5.2). Yet for so low values of f0

P Q, in the near future our

scenario can be probed with helio-/haloscopes [35,36], detailed study of stellar evolution, or

a combination of both.

2 The Model

Dynamics of the complex ﬁeld Scarrying Peccei–Quinn charge is described by the La-

grangian:

L=∂µS†∂µS+g2S†Sϕ†ϕ−λSS†S2.(1)

We assume the following properties of the scalar ﬁeld (or scalar multiplet) ϕ: it is in ther-

mal equilibrium with hot primordial plasma and has a non-zero expectation value at low

temperatures:

⟨ϕ†ϕ⟩=v2

2(T≲vϕ).(2)

It is tempting to identify ϕwith the SM Higgs ﬁeld, but we will show later that this option

is quite restrictive, and therefore we choose to keep the discussion generic. Note that we do

not introduce a bare tachyonic mass for the ﬁeld Sinto the Lagrangian (1). Nevertheless,

Peccei-Quinn symmetry is spontaneously broken, once we ﬁx the sign of the coupling between

the ﬁelds ϕand S:

g2>0.(3)

With this choice, the ﬁeld Sacquires a non-zero expectation value at low temperatures given

P Q =1

√2β·vϕ

g(T≲vϕ).(4)

Here we introduced the notation

β≡λS

g4.(5)

Convenience of the parameter βis as follows: while the self-interaction constant λSand

the portal coupling g2can be very small in the phenomenologically most interesting part

of parameter space, the ratio (5) is typically close to unity (cf. Refs. [27,28,30]). The

upperscript ‘0’ in Eq. (4) means that we are working in the low temperature limit. Being

interested in the regime f0

P Q ≫vϕ, we should require that λS≪g2. Such small values

are theoretically consistent: the lower bound on λSfollowing from the stability of two-ﬁeld

system reads λSλϕ≥g4/4, where λϕis the self-interaction constant of the ﬁeld ϕ, so that

β≥1

4λϕ

.(6)

From perturbative unitarity we have λϕ≲1 and hence β≳1.

Supernovae observations impose a lower bound on the Peccei–Quinn scale f0

P Q, i.e., f0

P Q ≳

108GeV. As it follows from Eq. (4), for not very large expectation values vϕ, we deal with

a feebly coupled Peccei–Quinn ﬁeld S:

g≃7·10−6

√β·vϕ

1 TeV· 108GeV

P Q !.(7)

Furthermore, the scalar Sis rather light:

S=p2λSf0

P Q =v2

√2βf0

P Q ≈7 MeV

√β·vϕ

1 TeV2

· 108GeV

P Q !.(8)

This fact clearly demonstrates a drastic diﬀerence between our setup and more conventional

axion models, where the expectation value of the ﬁeld Sis set as a bare parameter in the

Lagrangian and the mass MSis assumed to be of the order of Peccei-Quinn symmetry

breaking scale fP Q.

Most interesting for our further discussion is the behaviour of the ﬁeld Sat temperatures

T≫vϕ. Notably, the ﬁeld Sis in the spontaneously broken phase also at those times. Its

expectation value is sourced by thermal ﬂuctuations of the scalar (multiplet) ϕdescribed by

the dispersion:

⟨ϕ†ϕ⟩(t) = NT 2(t)

24 (T≳vϕ),(9)

where Nis the total number of relativistic degrees of freedom associated with ϕat temper-

atures T≳vϕ; note that N= 4 in the case of Higgs doublet. Using Eqs. (1), (5), and (9)

we get the time-dependent expectation value fP Q(t):

fP Q(t) = sN

24β·T(t)

g(T≳vϕ),(10)

which redshifts ∝1/R with the scale factor R(t). The temperature Tcan reach large values

in the primordial Universe; given also that the constant gis tiny, we end up with a picture of

Peccei–Quinn symmetry breaking scale gliding from sub-Planckian values to relatively small

values as the Universe cools down.

3 Parametric resonance

The best-known way of producing axions is through the misalignment mechanism, which

leads to the correct dark matter abundance in the range f0

P Q ≃1011 −1012 GeV. For much

smaller/larger values of f0

P Q, coherent axion oscillations triggered by vacuum realignment

give a negligible/too large contribution to dark matter [9,10,11]. Axions are also emitted

by global topological defects [13,14], i.e., cosmic strings and domain walls. The latter are

formed by the Kibble-Zurek mechanism, provided that the complex Peccei-Quinn ﬁeld is set

to zero in the post-inﬂationary Universe (commonly by the thermal mass), and the symmetry

gets spontaneously broken later on. In our case, the light ﬁeld Sminimally coupled to gravity

gets oﬀset from zero by superhorizon perturbations ampliﬁed during inﬂation. Furthermore,

the thermal mass of the ﬁeld Sacquired through the interaction with the scalar multiplet ϕ

is tachyonic and thus cannot stabilise the ﬁeld Sat zero. We conclude that global strings

are not formed in our scenario2.

Time-dependence of the Peccei–Quinn scale enables another axion production mechanism

through the decay of the ﬁeld |S|coherent oscillations in the parametric resonance regime.

These oscillations occur because the ﬁeld Sis initially displaced from its expectation value

fP Q in the post-inﬂationary Universe. Axions produced in this decay can constitute all

of the dark matter in the Universe for virtually any value of f0

P Q, in particular as low as

P Q ≲1011 GeV, for which the misalignment mechanism is ineﬃcient. Notably, it is possible

to describe the decay semi-analytically, at least in the small amplitude regime. We will

2One can avoid this issue by assuming the coupling of the ﬁeld Sto the Ricci scalar ∼ R|S|2, but we

choose to proceed with the minimal coupling in what follows.

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HeatingupPeccei–QuinnscaleSabirRamazanovandRomeSamantaCEICO,FZU-InstituteofPhysicsoftheCzechAcademyofSciences,NaSlovance1999/2,18200Prague8,CzechRepublicMay30,2023AbstractWediscussproductionofQCDaxiondarkmatterinanovelscenario,whichassumestime-varyingscaleofPeccei-Quinnsymmetrybreaking.Thelatterdecr...

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