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Beyond WIMPs: the Quark (Anti) Nugget Dark Matter
Ariel Zhitnitsky1,a
1Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada
Abstract. We review a testable dark matter (DM) model outside of the standard WIMP
paradigm. The model is unique in a sense that the observed ratio Ωdark ≃Ωvisible for
visible and dark matter densities ﬁnds its natural explanation as a result of their common
QCD origin when both types of matter (DM and visible) are formed during the QCD
transition and both are proportional to single dimensional parameter of the system, ΛQCD.
We argue that the charge separation eﬀect also inevitably occurs during the same QCD
transition in the presence of the CP odd axion ﬁeld a(x). It leads to preferential formation
of one species of nuggets on the scales of the visible Universe where the axion ﬁeld a(x)
is coherent. A natural outcome of this preferential evolution is that only one type of the
visible baryons (not anti- baryons) remain in the system after the nuggets complete their
formation. Unlike conventional WIMP dark matter candidates, the nuggets and anti-
nuggets are strongly interacting but macroscopically large objects. The rare events of
annihilation of the anti-nuggets with visible matter lead to a number of observable eﬀects.
We argue that the relative intensities for a number of measured excesses of emission
from the centre of galaxy (covering more than 11 orders of magnitude) are determined
by standard and well established physics. At the same time the absolute intensity of
emission is determined by a single new fundamental parameter of the theory, the axion
mass, 10−6eV ≲ma ≲10−3eV. Finally, we comment on implications of these studies for
the axion search experiments, including microwave cavity and the Orpheus experiments.
1 Introduction
This talk is mostly based on recent paper [1]. It is generally assumed that the Universe began in a
symmetric state with zero global baryonic charge and later, through some baryon number violating
process, evolved into a state with a net positive baryon number. As an alternative to this scenario
we advocate a model in which “baryogenesis” is actually a charge separation process in which the
global baryon number of the Universe remains zero. In this model the unobserved antibaryons come
to comprise the dark matter in form of the dense heavy nuggets, similar to the Witten’s strangelets
[2]. Both quarks and antiquarks are thermally abundant in the primordial plasma but, in addition to
forming conventional baryons, some fraction of them are bound into heavy nuggets of quark matter.
Nuggets of both matter and antimatter are formed as a result of the dynamics of the axion domain
walls [1, 3, 4], some details of this process will be discussed later in the text.
An overall coherent baryon asymmetry in the entire Universe is a result of the strong CP violation
due to the fundamental θ parameter in QCD which is assumed to be nonzero at the beginning of the
ae-mail: arz@phas.ubc.ca
arXiv:1611.05042v1  [hep-ph]  15 Nov 2016

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QCD transition1. This source of strong CP violation is no longer available at the present epoch as a
result of the axion dynamics, see original [6–8] and more recent papers [9–16] on the subject. Were
CP symmetry to be exactly preserved an equal number of matter and antimatter nuggets would form
resulting in no net “baryogenesis". However, CP violating processes associated with the axion θ(x)
term in QCD result in the preferential formation of one type of species. This preference is essentially
determined by the dynamics of coherent axion ﬁeld θ(x) at the initial stage of the nugget’s formation.
Asymmetric production of the nuggets directly translates into asymmetry between visible baryons
and anti-baryons because the total baryon charge is a conserved quantity in this framework. If more
anti-nuggets are produced in the system then less conventional anti-baryons remain in the system.
These remaining anti-baryons in the plasma then annihilate away leaving only the baryons whose
antimatter counterparts are bound in the excess of anti-nuggets and thus unavailable to annihilate.
One should emphasize that the resulting asymmetry is order of one. It is not sensitive to a relatively
small magnitude of the axion mass2 nor to the relatively small magnitude of the axion ﬁled θ(x) ∼
(10−2 −10−4) at the beginning of the QCD transition as long as it remains coherent on the scale of the
Universe, see [1] for the details. This is precisely the main reason of why the visible and dark matter
densities must be the same order of magnitude
Ωdark ≈Ωvisible
(1)
as they both proportional to the same fundamental ΛQCD scale, and they both are originated at the
same QCD epoch. In particular, if one assumes that the nuggets and anti-nuggets saturate the dark
matter density today than the observed matter to dark matter ratio Ωdark ≃5 · Ωvisible corresponds to a
speciﬁc proportion when number of anti-nuggets is larger than number of nuggets by a factor of ∼3/2
at the end of nugget’s formation. This would result in a matter content with baryons, quark nuggets
and antiquark nuggets in an approximate ratio
|Bvisible| : |Bnuggets| : |Bantinuggets| ≃1 : 2 : 3,
(2)
with no net baryonic charge. If these processes are not fundamentally related the two components
Ωdark and Ωvisible could easily exist at vastly diﬀerent scales.
Unlike conventional dark matter candidates, dark-matter/antimatter nuggets are strongly interact-
ing but macroscopically large. They do not contradict the many known observational constraints on
dark matter or antimatter for three main reasons [18]:
• They carry a huge (anti)baryon charge |B| ≳1025, and so have an extremely tiny number density;
• The nuggets have nuclear densities, so their eﬀective interaction is small σ/M ∼10−10 cm2/g, well
below the typical astrophysical and cosmological limits which are on the order of σ/M < 1 cm2/g;
• They have a large binding energy such that the baryon charge in the nuggets is not available to
participate in big bang nucleosynthesis (bbn) at T ≈1 MeV.
To reiterate: the weakness of the visible-dark matter interaction in this model due to the small ge-
ometrical parameter σ/M ∼B−1/3 rather than due to the weak coupling of a new fundamental ﬁeld
to standard model particles. It is this small eﬀective interaction ∼σ/M ∼B−1/3 which replaces the
conventional requirement of suﬃciently weak interactions for WIMPs.
1It is known that that the QCD transition is actually a crossover rather than a phase transition [5]. In context of the present
paper the important factor is the scale ∼170 MeV where transition happens rather than its precise nature.
2The axion’s mass ma(T) as a function of the temperature has been computed using the lattice simulations [17] with very
high accuracy in the region well above the QCD transition.

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A fundamental measure of the scale of baryogenesis is the baryon to entropy ratio at the present
time
η ≡nB −n ¯B
nγ
≃nB
nγ
∼10−10.
(3)
If the nuggets were not present after the transition the conventional baryons and anti-baryons would
continue to annihilate each other until the temperature reaches T ≃22 MeV when density would
be 9 orders of magnitude smaller than observed. This annihilation catastrophe, normally thought
to be resolved as a result of “baryogenesis," is avoided in our proposal because more anti-baryon
charges than baryon charges are hidden in the form of the macroscopical nuggets and thus no longer
available for annihilation. Only the visible baryons (not anti-baryons) remain in the system after
nugget formation is fully completed.
In our proposal (in contrast with conventional models) the ratio η is determined by the formation
temperature Tform at which the nuggets and anti-nuggets basically have completed their formation
and below which annihilation with surrounding matter becomes negligible. This temperature is de-
termined by many factors: transmission/reﬂection coeﬃcients, evolution of the nuggets, expansion of
the universe, cooling rates, evaporation rates, the dynamics of the axion domain wall network, etc. In
general, all of these eﬀects will contribute equally to determining Tform at the QCD scale. Technically,
the corresponding eﬀects are hard to compute as even basic properties of the QCD phase diagram at
nonzero θ are still unknown. However, an approximate estimate of Tform is quite simple as it must be
≈170MeV
θ
T
µ
QGP
CS
Hadron
1
2
3
Tform ≈41MeV
Tc
(Phase Unknown)
Figure 1. The conjectured phase diagram. The plot is taken from [1].
Possible cooling paths are denoted as path 1, 2 or 3. The phase diagram is in
fact much more complicated as the dependence on the third essential
parameter, the θ is not shown as it is largely unknown. It is assumed that the
ﬁnal destination after the nuggets are formed is the CS region with Tform ≈41
MeV, µ > µc and θ ≈0, corresponding to the presently observed ratio (3), see
text for the details.
expressed in terms of the gap ∆∼100 MeV when the colour superconducting phase sets in inside
the nuggets. The observed ratio (3) corresponds to Tform ≃41 MeV which is indeed a typical QCD
scale slightly below the critical temperature Tc ≃0.6∆when colour superconductivity (CS) sets in. In
diﬀerent words, in this proposal the ratio (3) emerges as a result of the QCD dynamics when process
of charge separation stops at Tform ≃41 MeV, rather than a result of baryogenesis when a net baryonic
charge is produced.
2
Quark (anti) nugget DM confronting the observations
While the observable consequences of this model are on average strongly suppressed by the low
number density of the quark nuggets ∼B−1/3 as explained above, the interaction of these objects with
the visible matter of the galaxy will necessarily produce observable eﬀects. Any such consequences
will be largest where the densities of both visible and dark matter are largest such as in the core of
the galaxy or the early universe. In other words, the nuggets behave as a conventional cold DM in the
environment where density of the visible matter is small, while they become interacting and emitting
radiation objects (i.e. eﬀectively become visible matter) when they are placed in the environment with
suﬃciently large density.
The relevant phenomenological features of the resulting nuggets are determined by properties of
the so-called electro-sphere as discussed in original refs. [19–24]. These properties are in principle,

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calculable from ﬁrst principles using only the well established and known properties of QCD and
QED. As such the model contains no tunable fundamental parameters, except for a single mean baryon
number ⟨B⟩which itself is determined by the axion mass ma as we already mentioned.
A comparison between emissions with drastically diﬀerent frequencies from the centre of galaxy
is possible because the rate of annihilation events (between visible matter and antimatter DM nuggets)
is proportional to the product of the local visible and DM distributions at the annihilation site. The
observed ﬂuxes for diﬀerent emissions thus depend through one and the same line-of-sight integral
Φ ∼R2
Z
dΩdl[nvisible(l) · nDM(l)],
(4)
where R ∼B1/3 is a typical size of the nugget which determines the eﬀective cross section of inter-
action between DM and visible matter. As nDM ∼B−1 the eﬀective interaction is strongly suppressed
∼B−1/3 as we already mentioned in the Introduction. The parameter ⟨B⟩∼1025 was ﬁxed in this pro-
posal by assuming that this mechanism saturates the observed 511 keV line [19, 20], which resulted
from annihilation of the electrons from visible matter and positrons from anti-nuggets. It has been
also assumed that the observed dark matter density is saturated by the nuggets and anti-nuggets. It
corresponds to an average baryon charge ⟨B⟩∼1025 for typical density distributions nvisible(r), nDM(r)
entering (4). Other emissions from diﬀerent bands are expressed in terms of the same integral (4),
and therefore, the relative intensities are completely determined by internal structure of the nuggets
which is described by conventional nuclear physics and basic QED. We present a short overview of
these results below.
Some galactic electrons are able to penetrate to a suﬃciently large depth of the anti-nuggets.
These events no longer produce the characteristic positronium decay spectrum (511 keV line with
a typical width of order ∼few keV accompanied by the conventional continuum due to 3γ decay)
but a direct non-resonance e−e+ →2γ emission spectrum. The transition between the resonance
positronium decays and non-resonance regime is determined by conventional physics and allows us
to compute the strength and spectrum of the MeV scale emissions relative to that of the 511 keV
line [21, 22]. Observations by the Comptel satellite indeed show some excess above the galactic
background consistent with our estimates.
Galactic protons incident on the anti-nugget will penetrate some distance into the quark matter
before annihilating into hadronic jets. This process results in the emission of Bremsstrahlung photons
at x-ray energies [23]. Observations by the Chandra observatory apparently indicate an excess in x-
ray emissions from the galactic centre. Hadronic jets produced deeper in the nugget or emitted in the
downward direction will be completely absorbed. They eventually emit thermal photons with radio
frequencies [24–26]. Again the relative scales of these emissions may be estimated and is found to be
in agreement with observations.
We conclude this brief overview on observational constraints of the model with the following
remark. This model which has a single fundamental parameter (the mean baryon number of a nugget
⟨B⟩∼1025, corresponding to the axion mass ma ≃10−4 eV), and which enters all the computations
is consistent with all known astrophysical, cosmological, satellite and ground based constraints as
highlighted above. Furthermore, in a number of cases the predictions of the model are very close to
the presently available limits, and very modest improving of those constraints may lead to a discovery
of the nuggets. Even more than that: there is a number of frequency bands where some excess
of emission was observed, and this model may explain some portion, or even entire excess of the
observed radiation in these frequency bands.

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3 Five crucial ingredients of the proposal.
In this section we explain the crucial elements of the proposal. The detail discussions for each ingre-
dient can be found in original paper [1].
3.1 NDW = 1 domain walls
First important element of this proposal is the presence of the topological objects, the axion domain
walls [27]. As we already mentioned the θ parameter is the angular variable, and therefore supports
various types of the domain walls, including the so-called NDW = 1 domain walls when θ interpolates
between one and the same physical vacuum state with the same energy θ →θ + 2πn. The axion
domain walls may form at the same moment when the axion potential get tilted, i.e. at the moment Ta
when the axion ﬁeld starts to roll due to the misalignment mechanism. The tilt becomes much more
pronounced at the transition when the chiral condensate forms at Tc. In general one should expect that
the NDW = 1 domain walls form once the axion potential is suﬃciently tilted, i.e. anywhere between
Ta and Tc.
One should comment here that it is normally assumed that for the topological defects to be formed
the Peccei-Quinn (PQ) phase transition must occur after inﬂation. This argument is valid for a generic
type of domain walls with NDW , 1. The conventional argument is based on the fact that few physi-
cally diﬀerent vacua with the same energy must be present inside of the same horizon for the domain
walls to be formed. The NDW = 1 domain walls are unique and very special in the sense that θ inter-
polates between one and the same physical vacuum state. Such NDW = 1 domain walls can be formed
even if the PQ phase transition occurred before inﬂation and a unique physical vacuum occupies entire
Universe [1].
It has been realized many years after the original publication [27] that the axion domain walls,
in general, demonstrate a sandwich-like substructure on the QCD scale Λ−1
QCD ≃fm. The arguments
supporting the QCD scale substructure inside the axion domain walls are based on analysis [28] of
QCD in the large N limit with inclusion of the η′ ﬁeld. It is also supported by analysis [29] of
supersymmetric models where a similar θ vacuum structure occurs. The same structure also occurs in
CS phase where the corresponding domain walls have been explicitly constructed [30].
3.2
Spontaneous symmetry breaking of the baryon charge
Second important element is that in addition to this known QCD substructures [28–30] of the axion
domain walls expressed in terms of the η′ and gluon ﬁelds, there is another substructure with a similar
QCD scale which carries the baryon charge. Precisely this novel feature of the domain walls which
was not explored previously in the literature will play a key role in our proposal because exactly this
new eﬀect will be eventually responsible for the accretion of the baryon charge by the nuggets. Both,
the quarks and anti-quarks can accrete on a given closed domain wall making eventually the quark
nuggets or anti-nuggets, depending on the sign of the baryon charge. The sign is chosen randomly
such that equal number of quark and antiquark nuggets are formed if the external environment is CP
even, which is the case when fundamental θ = 0.
Indeed, in the background of the domain wall, the physics essentially depends on two variables,
(t, z). One can show that in this circumstances the induced baryon charge N for a single fermion in
the axion domain wall background may assume any integer value N, positive or negative. The total
baryon charge B accumulated on a nugget is determined by the degeneracy factor in vicinity of the
domain wall [1]
N =
Z
d3x ¯Ψγ0Ψ = −(n1 + n2),
B = N · g ·
Z d2x⊥d2k⊥
(2π)2
1
exp( ϵ−µ
T ) + 1.
(5)

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In this formula g is appropriate degeneracy factor, e.g. g ≃NcNf in CS phase and µ is the chemical
potential in vicinity of the domain wall.
The main point of this section is that the domain walls generically will acquire the baryon or
anti-baryon charge. This is because the domain wall tension is mainly determined by the axion ﬁeld
while the QCD substructure leads to small correction factor of order ∼ΛQCD/ fa ≪1. Therefore,
the presence of the QCD substructure with non vanishing N , 0 increases the domain wall tension
only slightly. Consequently, this implies that the domain closed bubbles carrying the baryon or anti
baryon charge will be copiously produced during the transition as they are very generic conﬁgurations
of the system. Furthermore, the baryon charge cannot leave the system during the time evolution as
it is strongly bound to the wall due to the topological reasons. The corresponding binding energy per
quark is order of µ and increases with time as shown in [1]. One can interpret this phenomenon as a
local spontaneous symmetry breaking eﬀect, when on the scales of order the correlation length ξ(T)
the nuggets may acquire the positive or negative baryon charge with equal probability. This is because
the sign of N in eq. (5) may assume any positive or negative values with equal probabilities.
3.3 Kibble-Zurek mechanism
Next important ingredient of the proposal is the Kibble-Zurek mechanism which gives a generic pic-
ture of formation of the topological defects during a phase transition, see original papers [31], review
[32] and the textbook [33]. In our context the Kibble-Zurek mechanism suggests that once the axion
potential is suﬃciently tilted the NDW = 1 domain walls form. The potential becomes much more
pronounced when the chiral condensate forms at Tc. After some time after Ta the system is domi-
nated by a single, percolated, highly folded and crumpled domain wall network of very complicated
topology. In addition, there will be a ﬁnite portion of the closed walls (bubbles) with typical size of
order correlation length ξ(T), which is deﬁned as an average distance between folded domain walls at
temperature T. It is known that the probability of ﬁnding closed walls with very large size R ≫ξ is
exponentially small.
The key point for our proposal is there existence of these ﬁnite closed bubbles made of the axion
domain walls. Normally it is assumed that these closed bubbles collapse as a result of the domain
wall pressure, and do not play any signiﬁcant role in dynamics of the system. However, as we already
mentioned in Introduction the collapse of these closed bubbles is halted due to the Fermi pressure
acting inside of the bubbles. Therefore, they may survive and serve as the dark matter candidates. The
percolated network of the domain walls will decay to the axion in conventional way as discussed in
[34–37]. Those axions (along with the axions produced by the conventional misalignment mechanism
[35, 38]) will contribute to the dark matter density today. The corresponding contribution to dark
matter density is highly sensitive to the axion mass as Ωdark ∼m−1
a , and it is not subject of the present
work. Instead, the focus of the present work is the dynamics of the closed bubbles, which is normally
ignored in computations of the axion production. Precisely these closed bubbles, according to this
proposal, will eventually become the stable nuggets and may serve as the dark matter candidates.
The nugget’s contribution to Ωdark is not very sensitive to the axion mass, but rather, is determined by
the formation temperature Tform as explained in Introduction.
3.4 Colour Superconductivity
There existence of CS phase in QCD represents the next crucial element of our scenario. The CS has
been an active area of research for quite sometime, see review papers [39, 40] on the subject. The
CS phase is realized when quarks are squeezed to the density which is few times nuclear density. It

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has been known that this regime may be realized in nature in neutron stars interiors and in the violent
events associated with collapse of massive stars or collisions of neutron stars, so it is important for
astrophysics.
The force which squeezes quarks in neutron stars is gravity; the force which does an analogous
job in early universe during the QCD transition is a violent collapse of a bubble of size R ∼ξ(T)
formed from the axion domain wall as described in Section 3.3 above. If number density of quarks
trapped inside of the bubble (in the bulk) is suﬃciently large, the collapse stops due to the internal
Fermi pressure. In this case the system starts to oscillate and the quarks in the bulk may reach the
equilibrium with the ground state being in a CS phase. As we advocate in [1] this is very plausible
fate of a relatively large size bubbles of size R ∼ξ(T) made of the axion domain walls which were
produced after the QCD transition.
Indeed, one can numerically solve the equations describing the evolution of the nuggets. A typical
solution describes an oscillating bubble with frequency ω ∼ma. The bubble is slowly decreasing its
radius R(t) with a characteristic dumping scale τ. The time evolution of such a bubble can be well
approximated as follows
R(t) = Rform + (R0 −Rform)e−t/τ cos ωt,
(6)
where R0 is initial size of a bubble R0 ∼ξ(T), while Rform is a ﬁnal bible’s size when formation
is almost complete. In formula (6) parameter τ represents a typical damping time- scale which is
expressed in terms of the axion mass ma and ΛQCD. It turns out that numerically τ is of order of
cosmological scale τ ∼10−4s. This numerical value is fully consistent with our anticipation that
the temperature of the Universe drops approximately by a factor of ∼3 or so during the formation
period. During the same period of time the chemical potential µ inside the nugget reaches suﬃciently
large values when the CS sets in. Therefore, our phenomenological analysis in Section 2 (where DM
nuggets are treated as very dense objects in CS phase) is supported by present studies on formation
and time evolution of the nuggets made of the axion domain walls and ordinary quarks in CS phase.
3.5 Coherent CP odd axion ﬁeld
If θ vanishes, then equal number of nuggets and anti-nuggets would form. However, the CP violating
θ parameter (the axion ﬁeld), which is deﬁned as value of θ at the moment of domain wall formation
generically is not zero, though it might be numerically quite small. Precisely the dynamics of the
coherent axion ﬁeld θ(t) leads to preferences in formation of one species of nuggets, as argued in
details in [1]. This sign-preference is correlated on the scales where the axion ﬁeld θ(t) is coherent,
i.e. on the scale of the entire Universe at the moment of the domain wall formation. In other words,
we assume that the PQ phase transition happened before inﬂation. One should emphasize that this
assumption on coherence of the axion ﬁeld on very large scales is consistent with formation of NDW =
1 domain walls, see section 3.1.
Precise dynamical computations of this CP asymmetry due to the coherent axion ﬁeld θ(t) is a
hard problem of strongly coupled QCD at θ , 0. It depends on a number speciﬁc properties of the
nuggets, their evolution, their environment, modiﬁcation of the hadron spectrum at θ , 0, etc. All
these factors equally contribute to the diﬀerence between the nuggets and anti-nuggets. In order to
eﬀectively account for these coherent eﬀects one can introduce an unknown coeﬃcient c(T) of order
one as follows
Bantinuggets = c(T) · Bnuggets, where |c(T)| ∼1,
(7)
where c(T) is obviously a negative parameter of order one.
This key relation of this framework (7)
unambiguously implies that the baryon charge in form of the visible matter can be also expressed in

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terms of the same coeﬃcient c(T) ∼1 as follows Bvisible = −(Bantinuggets + Bnuggets). Using eq. (7) the
expression for the visible matter Bvisible can be rewritten as
Bvisible ≡

Bbaryons + Bantibaryons

= −[1 + c(T)] Bnuggets = −
"
1 +
1
c(T)
#
Bantinuggets.
(8)
The same relation can be also represented in terms of the measured observables Ωvisible and Ωdark at
later times when only the baryons (and not anti-baryons) contribute to the visible component3
Ωdark ≃
 1 + |c(T)|
|1 + c(T)|
!
· Ωvisible at T ≤Tform.
(9)
One should emphasize that the relation (8) holds as long as the thermal equilibrium is maintained,
which we assume to be the case. Another important comment is that each individual contribution
|Bbaryons| ∼|Bantibaryons| entering (8) is many orders of magnitude greater than the baryon charge hidden
in the form of the nuggets and anti-nuggets at earlier times when Tc > T > Tform. It is just their total
baryon charge which is labeled as Bvisible and representing the net baryon charge of the visible matter
is the same order of magnitude (at all times) as the net baryon charge hidden in the form of the nuggets
and anti-nuggets.
The baryons continue to annihilate each other (as well as baryon charge hidden in the nuggets)
until the temperature reaches Tform when all visible anti-baryons get annihilated, while visible baryons
remain in the system and represent the visible matter we observe today. It corresponds to c(Tform) ≃
−1.5 as estimated below if one neglects the diﬀerences in gaps in CS and hadronic phases, see footnote
3. After this temperature the nuggets essentially assume their ﬁnal form, and do not loose or gain
much of the baryon charge from outside. The rare events of the annihilation between anti-nuggets and
visible baryons continue to occur. In fact, the observational excess of radiation in diﬀerent frequency
bands, reviewed in section 2, is a result of these rare annihilation events at present time.
The generic consequence of this framework represented by eqs. (7), (8), (9) takes the following
form for c(Tform) ≃−1.5 which corresponds to the case when the nuggets saturate entire dark matter
density today:
Bvisible ≃1
2 Bnuggets ≃−1
3 Bantinuggets,
Ωdark
≃
5 · Ωvisible,
(10)
which is identically the same relation (2) presented in Introduction. The relation (10) emerges due to
the fact that all components of matter, visible and dark, proportional to one and the same dimensional
parameter ΛQCD, see footnote 3 with a comment on this approximation. In formula (10) Bnuggets and
Bantinuggets contribute to Ωdark, while Bvisible obviously contributes to Ωvisible. The coeﬃcient ∼5 in
relation Ωdark ≃5 · Ωvisible is obviously not universal, but relation (1) is universal, and very generic
consequence of the entire framework, which was the main motivation for the proposal [3, 4].
4
Implications for the axion search experiments
The goal of this section is to comment on relation of our framework and the direct axion search
experiments [9–16]. We start with the following comment we made in section 2: this model which
has a single fundamental parameters (a mean baryon number of a nugget ⟨B⟩∼1025 entering all the
computations) is consistent with all known astrophysical, cosmological, satellite and ground based
constraints as reviewed in section 2. For discussions of this section it is convenient to express this
3In eq. (9) we neglect the diﬀerences (due to diﬀerent gaps) between the energy per baryon charge in hadronic and CS phases
to simplify notations. This correction obviously does not change the main claim of this proposal stating that Ωvisible ≈Ωdark.

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single normalization parameter ⟨B⟩∼1025 in terms of the axion mass ma ∼10−4 eV as these two
parameters directly related as discussed in the origin paper [3]. The corresponding relation between
these two parameters emerges because the axion mass ma determines the wall tension σ ∼m−1
a . At
the same time parameter σ enters the expression for equilibrium, which itself determines the size of
the nuggets, Rform (and therefore B ∼R3
form) when the formation is complete at temperature Tform.
The lower limit on the axion mass, as it is well known, is determined by the requirement that
the axion contribution to the dark matter density does not exceed the observed value Ωdark ≈0.23.
There is a number of uncertainties in the corresponding estimates. We shall not comment on these
subtleties by referring to the review papers [9–16]. The corresponding uncertainties are mostly due
to the remaining discrepancies between diﬀerent groups on the computations of the axion production
rates due to the diﬀerent mechanisms such as misalignment mechanism versus domain wall/string
decays. If one assumes that the dominant contribution to the axion density is due to the misalignment
mechanism than the estimates suggest that the axion of mass ma ≃2 · 10−5 eV saturates the dark
matter density observed today, while the axion mass in the range of ma ≥10−4 eV contributes very
little to the dark matter density.
There is another mechanism of the axion production when the Peccei-Quinn symmetry is broken
after inﬂation. In this case the string-domain wall network produces a large number of axions such
that the axion mass ma ≃10−4 eV may saturate the dark matter density, see relatively recent estimates
[35–37] with some comments and references on previous papers. Our original remark here is that the
NDW = 1 domain walls can be formed even if the PQ phase transition occurred before inﬂation and a
unique physical vacuum occupies entire Universe, see section 3.1 with comments and references.
Axion Coupling |ga | (GeV-1)
Axion Mass mA (eV)
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-5
10-4
10-3
Cavity Experiments
ADMX
Solar
Expected sensitivity
of proposed technique
A
B
C
D
KSVZ
DFSZ
Axion Dark Matter
Figure 2. Cavity / ADMX experimental constraints on the axion
mass shown in green. The expected sensitivity for the Orpheus axion
search experiment [15] is shown by blue regions “A", “B", “C" and
“D". In particular, experiment “B", covers the most interesting region
of the parametrical space with ma ≃10−4 eV corresponding to the
nuggets with mean baryon charge ⟨B⟩≃1025 which itself satisﬁes all
known astrophysical, cosmological, satellite and ground based
constraints as discussed in section 2. The plot is taken from [15].
The main lesson to be learnt from the present work is that in addition to these well established
mechanisms previously discussed in the literature there is an additional contribution to the dark matter
density also related to the axion ﬁeld. However, the mechanism which is advocated in the present work
contributes to the dark matter density through formation of the nuggets, rather than through the direct
axion production. The corresponding mechanism as argued in section 3.5 always satisﬁes the relation
Ωdark ≈Ωvisible, and, in principle is capable to saturate the dark matter density Ωdark ≈5Ωvisible by
itself for arbitrary magnitude of the axion mass ma as the corresponding contribution is not sensitive
to the axion mass in contrast with conventional mechanisms mentioned above. A precise coeﬃcient
in ratio Ωdark ≈Ωvisible is determined by a parameter of order one, |c(T)| ∼1, which unfortunately is
very hard to compute from the ﬁrst principles, as discussed in section 3.5.
Our choice for ma ≃10−4 eV which corresponds to ⟨B⟩∼1025 is entirely motivated by our
previous analysis of astrophysical, cosmological, satellite and ground based constraints as reviewed
in Section 2. As we mentioned in Section 2 there is a number of frequency bands where some excess
of emission was observed, and this model may explain some portion, or even entire excess of the
observed radiation in these frequency bands. Our normalization ⟨B⟩∼1025 was ﬁxed by eq.(4) with

EPJ Web of Conferences
assumption that the observed dark matter is saturated by the nuggets. The relaxing this assumption
obviously modiﬁes the coeﬃcient c(T) as well as ⟨B⟩.
Interestingly enough, this range of the axion mass ma ≃10−4 eV is perfectly consistent with
recent claim [41],[42] that the previously observed small signal in resonant S/N/S Josephson junction
[43] is a result of the dark matter axions with the mass ma ≃1.1 · 10−4 eV. We conclude this section
on optimistic note with a remark that the most interesting region of the parametric space with mean
baryon charge ⟨B⟩≃1025 which corresponds to ma ∼10−4 eV might be tested by the Orpheus axion
search experiment [15] as shown on Fig. 2.
Conclusion. Future directions.
First, we want to list the main results of the present studies, while the comments on possible future
developments will be presented at the end of this Conclusion.
1. First key element of this proposal is the observation (5) that the closed axion domain walls are
copiously produced and generically will acquire the baryon or anti-baryon charge. This phenomenon
of “separation of the baryon charge" can be interpreted as a local version of spontaneous symmetry
breaking. This symmetry breaking occurs not in the entire volume of the system, but on the correlation
length ξ(T) ∼m−1
a
which is determined by the folded and crumpled axion domain wall during the
formation stage. Precisely this local charge separation eventually leads to the formation of the nuggets
and anti-nuggets serving in this framework as the dark matter component Ωdark.
2. Number density of nuggets and anti-nuggets and their size distributions will not be identically
the same as a result of the coherent (on the scale of the Universe) axion CP -odd ﬁeld. We parameter-
ize the corresponding eﬀects of order one by phenomenological parameter c(T) ∼1. It is important
to emphasize that this parameter of order one is not fundamental constant of the theory, but, calcula-
ble from the ﬁrst principles. In practice, however, such a computation could be quite a challenging
problem when even the QCD phase diagram is not known. The fundamental consequence of this
framework, Ωdark ≈Ωvisible, which is given by (1) is universal, and not sensitive to any parameters
as both components are proportional to ΛQCD. The observed ratio (2), (10) corresponds to a speciﬁc
value of c(Tform) ≃−1.5 as discussed in section 3.5.
3. Another consequence of the proposal is a natural explanation of the ratio (3) in terms of the
formation temperature Tform ≃40 MeV, rather than in terms of speciﬁc coupling constants which
normally enter conventional “baryogenesis" computations. This observed ratio is expressed in our
framework in terms of a single parameter Tform when nuggets complete their formation. This param-
eter is not fundamental constant of the theory, and as such is calculable from the ﬁrst principles. In
practice, however, the computation of Tform is quite a challenging problem as explained in the original
paper [1]. Numerically, the observed ratio (3) corresponds to Tform ≃40 MeV which is indeed slightly
below the critical temperature TCS ≃60 MeV where the colour superconductivity sets in. The relation
Tform ≲TCS ∼ΛQCD is universal in this framework as both parameters are proportional to ΛQCD. As
such, the universality of this framework is similar to the universality Ωdark ≈Ωvisible mentioned in
previous item. At the same time, the ratio (3) is not universal itself as it is exponentially sensitive to
precise value of Tform due to conventional suppression factor ∼exp(−mp/T).
4. The only new fundamental parameter of this framework is the axion mass ma. Most of our
computations (related to the cosmological observations, see section 2.) however, are expressed in
terms of the mean baryon number of nuggets ⟨B⟩rather than in terms of the axion mass. However,
these two parameters are unambiguously related as explained in the text.
5. This region of the axion mass ma ≃10−4 eV corresponding to average size of the nuggets
⟨B⟩≃1025 can be tested in the Orpheus axion search experiment [15] as shown on Fig. 2.

CONF12
We conclude with few thoughts on future directions within our framework. It is quite obvious
that future progress cannot be made without a much deeper understanding of the QCD phase diagram
at θ , 0. In other words, we need to understand the structure of possible phases along the third
dimension parametrized by θ on Fig 1. Due to the known “sign problem", the conventional lattice
simulations cannot be used at θ , 0. Another possible development from the “wish list" is a deeper
understanding of the closed bubble formation. Presently, very few results are available on this topic.
The most relevant for our studies is the observation made in [11] that a small number of closed
bubbles are indeed observed in numerical simulations. However, their detail properties (their fate,
size distribution, etc) have not been studied yet. A number of related questions such as an estimation
of correlation length ξ(T), the generation of the structure inside the domain walls, the baryon charge
accretion on the bubble, etc, hopefully can be also studied in such numerical simulations.
One more possible direction for future studies from the “wish list" is a development some QCD-
based models where a number of hard questions such as: evolution of the nuggets, cooling rates,
evaporation rates, annihilation rates, viscosity of the environment, transmission/reﬂection coeﬃcients,
etc in unfriendly environment with non-vanishing T, µ, θ can be addressed, and hopefully answered.
All these and many other eﬀects are, in general, equally contribute to our parameters Tform and c(T) at
the ΛQCD scale in strongly coupled QCD. Precisely these numerical factors eventually determine the
coeﬃcients in the observed relations: Ωdark ≈Ωvisible given by eq. (9) and η given by eq. (3).
Last but not least: the discovery of the axion in the Orpheus experiment [15] would conclude a
long and fascinating journey of searches for this unique and amazing particle conjectured almost 40
years ago. Such a discovery would be a strong motivation for related searches of “something else" as
the axion mass ma ≃10−4 is unlikely to saturate the dark matter density observed today. We advocate
the idea that this “something else" is the “quark nuggets" (where the axion plays the key role in entire
construction) which could provide the principle contribution to dark matter of the Universe as the
relation Ωdark ≈Ωvisible in this framework is not sensitive to the axion mass.
Acknowledgments
This work was supported in part by the National Science and Engineering Research Council of
Canada.
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