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JHEP06(2023)014
Published for SISSA by Springer
Received:February 2, 2023
Accepted:May 15, 2023
Published:June 5, 2023
Naturally astrophobic QCD axion
Marcin Badziakaand Keisuke Harigayab,c,d,e
aInstitute of Theoretical Physics, Faculty of Physics, University of Warsaw,
ul. Pasteura 5, PL-02-093 Warsaw, Poland
bDepartment of Physics, University of Chicago,
5720 South Ellis Avenue, Chicago, IL 60637, U.S.A.
cEnrico Fermi Institute, University of Chicago,
933 East 56th Street, Chicago, IL 60637, U.S.A.
dKavli Institute for Cosmological Physics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, U.S.A.
eKavli Institute for the Physics and Mathematics of the Universe (WPI),
The University of Tokyo Institutes for Advanced Study, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
E-mail: mbadziak@fuw.edu.pl,kharigaya@uchicago.edu
Abstract: We present a QCD axion model where the couplings of the axion to nucleons,
electrons, and muons are naturally suppressed because of the appropriate choice of the
Peccei-Quinn charges of the Standard Model fermions. We reexamine next-to-leading order
corrections to the couplings of the axion with nucleons and photons and show that the axion
decay constant may be as small as 107GeV. It is also possible to suppress the coupling with
the photon so that the decay constant is even smaller and minimal axiogenesis works. In
this scenario, the axion has a mass above 1 eV and may be directly detected via absorption
of axion dark matter. Flavor-violating axion couplings are generically predicted in our
model, but we show that they may be naturally and sufficiently suppressed. We discuss the
implications of the hints for anomalous cooling in several stellar environments to our model.
Keywords: Axions and ALPs, Rare Decays
ArXiv ePrint: 2301.09647
Open Access,c
The Authors.
Article funded by SCOAP3.https://doi.org/10.1007/JHEP06(2023)014
JHEP06(2023)014
Contents
1 Introduction 1
2 Axion couplings 3
2.1 Quark and nucleon couplings 3
2.2 Lepton couplings 7
2.3 Photon coupling 8
2.4 Flavor violation 8
3 UV completions 10
3.1 Extra fermions 10
3.2 Extra scalars 12
4 Stellar cooling anomalies 12
5 Minimal axiogenesis 14
6 Summary and discussion 16
1 Introduction
The strong CP problem may be solved by the Peccei-Quinn (PQ) mechanism [1,2] that
predicts a hypothetical particle called the QCD axion [3,4]. The axion is also a good dark-
matter (DM) candidate [5–7]. In typical QCD axion models, phenomenologically viable
axion masses are much below the eV scale. The strongest upper bounds on the QCD axion
mass come from astrophysics. While the structure of axion couplings varies among different
models, in most models an axion-nucleon coupling is present, leading to an upper bound
from the observations of the neutrino burst in SN1987A [8–13] and the cooling of neutron
stars [14–22]. For example, in the minimal KSVZ [23,24] and DFSZ models [25,26], these
observations lead to an upper bound on the axion mass of O(10−2)eV or, equivalently, to
a lower bound on the axion decay constant of fa&O(109)GeV.
There are theoretical and phenomenological motivations to consider values of famuch
smaller than that allowed in minimal QCD axion models. On the theoretical side, the axion
quality problem is relaxed for smaller decay constants [27–30]. On the phenomenological
side, low fais preferred to explain the observed baryon asymmetry simultaneously with the
observed DM abundance in the axiogenesis scenario [31] in which axion DM is produced
via the kinetic misalignment mechanism [32]. Such a scenario predicts the QCD axion
mass around the eV scale that can be detected using optical haloscopes [33,34], which are
based on absorption of DM, such as the LAMPOST experiment [35].
– 1 –
JHEP06(2023)014
All of the above motivate construction of QCD axion models in which axion-nucleon
couplings are strongly suppressed. Such models have been proposed and dubbed astro-
phobic axion models in [36]. Astrophobic axion models to deserve their name should also
avoid astrophysical constraints on the other axion couplings, especially the axion-electron
coupling that also leads to a lower bound on fa∼ O(109)GeV from White Dwarfs (WDs)
unless the axion does not couple to electrons at tree level [37–39]. Astrophobic axion mod-
els proposed in [36] generalize the DFSZ model by introducing flavor non-universal PQ
charges that allow for very small axion-nucleon couplings by fine-tuning. In such models
axion-lepton couplings are generically not suppressed but the bound on the axion-electron
coupling from WDs can be satisfied at the cost of additional tuning of the model parame-
ters. A generalization of this setup with three Higgs doublets was proposed in [40] in which
simultaneous suppression of the axion couplings to nucleons and electron is achieved by a
single tuning of parameters.
The goal of this paper is to construct natural astrophobic axion models in which the
suppression of astrophysically relevant axion couplings does not require tuning of param-
eters but stems from the PQ charge assignment of the Standard Model (SM) fermions.
Indeed, if the up and down quarks have PQ charges of 2and 1, respectively, and there is
no QCD anomaly of the PQ symmetry beyond that from the up and down quarks, axion-
nucleon couplings are suppressed. Vanishing PQ charges of the electron and muon also
ensure that the axion does not couple to them at the tree level.
Unlike KSVZ and DFSZ axions, astrophobic axions are generically flavor violating.
The axion-down-strange coupling, which is strongly bounded from kaon decay [41,42],
can be naturally suppressed if the PQ charges of down and strange quarks are the same.
Instead, special flavor symmetry can suppress the coupling without assuming the same
charges of the down and strange quarks. We also study other flavor-violating couplings
and show that they are also sufficiently small.
It is also possible to suppress the axion-photon coupling. This accidentally occurs if
the electromagnetic anomaly coefficient of the PQ symmetry, E, is twice larger than that of
QCD anomaly, N, as pointed out in [43]. The decay constant may be then below 107GeV
and as small as 106GeV. The minimal axiogenesis, which suffers from the overproduction
of axion DM by kinetic misalignment in the KSVZ and DFSZ models, becomes successful.
We also discuss the implications of the hints for anomalous cooling in several stellar
environments [44–46] that can be explained by the axion-electron and/or photon couplings.
If the electron coupling is generated by the quantum correction from the axion-gluon or
axion-Wcoupling, the cooling anomalies can be explained if E/N = 2. It is also possible to
have a small tree-level axion-electron coupling, for which the best-fit value can be obtained
even if E/N 6= 2.
We present UV completions of natural astrophobic axion models that include vector-
like fermions or Higgs doublets, which can be thought of as generalized KSVZ and DFSZ
models, respectively.
The rest of the article is organized as follows. In section 2, we discuss the coupling of the
axion with SM particles at an effective field theory level and show that the decay constant
may be naturally O(106−7)GeV. In section 3, we discuss UV completions of the setup.
– 2 –
JHEP06(2023)014
Stellar-cooling anomalies and axiogenesis are discussed in sections 4and 5, respectively.
Section 6gives summary and discussion.
2 Axion couplings
In this section, we discuss the coupling of the axion to the SM particles in the effective field
theory with the SM particles and the axion. UV completions are discussed in section 3. We
consider a class of models such that the SM fermions are charged under the PQ symmetry
and the phases of the Yukawa couplings depend on the axion field [47,48]. Unlike the
models in refs. [49,50], we do not attempt to explain the flavor hierarchy solely by the
PQ symmetry. Rather, we only require that the model does not require any unnatural
structure. We denote the PQ charge of a left-handed Weyl fermion fas Qf.
Without much loss of generality, we consider the case where only the right-handed
fermions ¯u,¯c,¯
t,¯
d,¯s,¯
b,¯e,¯µ, and ¯τare charged under the PQ symmetry. In the limit
where the Yukawa couplings are diagonal, even if some of the left-handed fermions are PQ-
charged, we may take a linear combination of the PQ symmetry and baryon and lepton
symmetry of each generation to make the left-handed fermions neutral under the PQ sym-
metry. The generality is lost for flavor-violating axion couplings. As we will see, however,
PQ-charged left-handed fermions typically lead to larger flavor-violating coupling unless
the PQ charge is generation independent, for which the PQ charge of left-handed fermions
can be removed by combining it with flavor-universal baryon or lepton symmetry. Thus,
models where only right-handed fermions are PQ-charged are the most conservative ones.
2.1 Quark and nucleon couplings
If Q¯u/Q ¯
d= 2 and QCD anomaly beyond that from ¯uand ¯
dis absent or cancelled, the
axion-nucleon coupling is suppressed. This can be most easily seen in the basis where the
up and down quark masses depend on the axion field. In this basis, the kinetic mixing
between the axion and the pion is absent. Also, since mu/md'0.5, the axion-dependent
quark mass is isospin singlet to the leading order in 1/fa, and the axion-pion mass mixing
is suppressed.
Let us explicitly see the suppression of the nucleon coupling, including quantum cor-
rections. We assign Q¯u= 2 and Q¯
d= 1. We also assign Q¯
t= 0. Otherwise, one-loop RG
correction from the axion-top coupling generates the axion-up and -down coupling that
contributes to the axion-nucleon coupling [51–54], and the axion is no longer astrophobic
unless that coupling is canceled by fine-tuning [55].
When the QCD anomaly comes only from the up and down quarks, after removing the
axion field from the fermion mass terms by chiral rotation, the axion-fermion couplings at
a UV scale are ∂µa
faX
f
cff†¯σµf, cf=−Qf
3.(2.1)
Here the factor of 3comes from the QCD anomaly coefficient of 3. In this basis, the axion
also couples to the gluon. Below the electroweak symmetry breaking scale, it is convenient
– 3 –
JHEP06(2023)014
gu
A−gd
A1.2723(23)
Nf= 2 + 1 + 1 Nf= 2 + 1
gu
A+gd
A0.34(5) 0.44(4)
δs0.059(8) 0.044(9)
δc0.0065(39) 0.0092(39)
δb0.0045(12) 0.0063(15)
z=mu/md0.465(24) 0.485(19)
w=mu/ms0.023(1) 0.024(1)
Table 1. Numerical values of the constants that determine the axion-nucleon coupling. In all the
plots presenting numerical results we use the average of the central values of the Nf= 2 + 1 + 1
and Nf= 2 + 1 data unless stated otherwise.
to write the interaction in terms of the axial current of Dirac fields ψ,
∂a
2faX
ψ
Cψ¯
ψγµγ5ψ, Cu=−c¯u, Cd=−c¯
d, Ce=−c¯e,··· .(2.2)
Following the computation in [56,57], we find that the axion-nucleon couplings are given by
∂µa
2faX
N=p,n
CN¯
Nγµγ5N, (2.3)
Cp−Cn=gu
A−gd
ACu−Cd−1−z
1 + z+w,
Cp+Cn=gu
A+gd
A0.95 (Cu+Cd)+0.05 −1 + z
1 + z+w−2δ,
δ=X
i=s,c,b
δiCi+m2
π
m2
η0
fπ
mN
√6z
(1 + z)2×G.
Here Nis a Dirac field. The values of gu
A−gd
A,gu
A+gd
A,δi,z=mu/md, and w=mu/ms
are shown in table 1. Here we used the up-to-date lattice data for Nf= 2 + 1 + 1 and 2+1,
with Nfbeing the number of flavors, summarized in [58]. Since the errors of two cases
are comparable, we use the average of them. Cu,d,s,c,b are those at a UV scale of 107GeV,
although the change of the coefficients δiby using a different UV scale is negligible in com-
parison with the errors. The uncertainties in the coefficients δs,c,b are dominated by those
of the nucleon matrix element of the strange axial current (gs
A), that of charm (gc
A), the RG
evolution of the axion-quark couplings and the nucleon matrix element of isospin-singlet
up and down axial current (gu+d
A), respectively. Since Cu= 2/3,Cd= 1/3, and z'0.5,
the axion-nucleon coupling is indeed suppressed. The astrophysical constraints are often
expressed in terms of dimensionless combinations gaNN ≡CNmN/fa.
We included O(mu/ms)effect and an O(m2
π/m2
η0)term in δthat are not included in the
previous literature on astrophobic axions. These are usually negligible, but are important
for out setup where the leading-order axion-nucleon coupling naturally vanishes. The last
– 4 –
JHEP06(2023)014
¯u¯
d
Qf2 1
Table 2. The PQ charges Qfof the SM fermions in the minimal model. Fermions other than ¯u
and ¯
dare PQ neutral. There should be no QCD anomaly of the PQ symmetry beyond that from
the up and down quarks.
0.47 0.48 0.49 0.50 0.51 0.52
-0.02
-0.01
0.00
0.01
0.02
z
δ
107GeV
8×106
6×106
4×106
2×106
Minimal model
with |G|<3
Figure 1. The lower bound on the decay constant fafrom the cooling of neutron stars for given z
and δ. The expected value of δin the minimal model in table 2is shown by the blue band.
term in δoriginates from the axion-dependent quark mass in the basis where the axion-pion
and axion-eta mixing from the quark mass is removed. To the reading order in 1/fa, this
becomes the coupling of the axion to the SU(3)-singlet pseudo-scalar quark bilinear. We
may estimate the induced axion-nucleon coupling from the axion-eta’ mixing
θaη0∼m2
π
m2
η0
√6z
(1 + z)2
fπ
fa
(2.4)
and the zero-momentum limit of the eta’-nucleon coupling gη0NN . The constant Gin
eq. (2.3) is expected to be O(gη0N N ). Theoretical estimations based on the sum rule and
the Goldberger-Treiman relation find gη0NN =O(1) [59,60].
In figure 1, we show the lower bound on fafrom the cooling of neutron stars, gapp <
1.5×10−9and gann <1.2×10−9[22], as a function of zand δ. The uncertainty on gu
A+gd
A
affects the lower bound on faby O(10)%. In the minimal model, only ¯uand ¯
dare PQ
charged, as shown in table 2, so δis determined by the last term. The blue band shows
the expectation for such a case assuming |G|<3. For the allowed range of z,famay be
below 107GeV, or even below few 106GeV.
In the minimal model, Q¯s6=Q¯
d. As we will see in section 2.4, the flavor-violating axion-
down-strange coupling is generically too large unless special flavor structure is imposed.
We thus also consider models with Q¯s=Q¯
d= 1 and allow any SM quarks except for the
– 5 –
JHEP06(2023)014
-3-2-1 0 1 2 3
-3
-2
-1
0
1
2
3
Qc
_
Qb
_
107GeV
8×106
6×106
Qc
_+Qb
_+1=0
z=0.48
-3-2-1 0 1 2 3
-3
-2
-1
0
1
2
3
Qc
_
Qb
_
107GeV
8×106
6×106
4×106
2×106
Qc
_+Qb
_+1=0
z=0.49
-3-2-1 0 1 2 3
-3
-2
-1
0
1
2
3
Qc
_
Qb
_
107GeV
8×106
6×106
4×106
Qc
_+Qb
_+1=0
z=0.5
-3-2-1 0 1 2 3
-3
-2
-1
0
1
2
3
Qc
_
Qb
_
107GeV
8×106
Qc
_+Qb
_+1=0
z=0.51
Figure 2. The lower bound on the decay constant fafrom the cooling of neutron stars for Q¯s= 1
and G= 0. The dashed lines show the case where the QCD anomaly of the PQ symmetry is
determined by the SM quarks.
top quark to have non-zero PQ charges, as shown in table 3. If only SM quarks contribute
to the QCD anomaly, Q¯c+Q¯
b=−1is required. If there are extra colored particles
that contribute to the QCD anomaly, as in some of the UV completion introduced in
section 3, they can take different values. In these models, it is possible that some of the
SM fermions are not PQ charge eigenstates and their charges are only effective ones that
are not quantized; see section 3.
In figure 2, we show the lower bound on fafrom the cooling of neutron stars as a
function of (Q¯c, Q¯
b)for Q¯s= 1 and G= 0. For reasonable choice of Q¯cand Q¯
b,famay be
below 107GeV, and as small as 2×106GeV for z'0.49.
– 6 –
JHEP06(2023)014
-3-2-1 0 1 2 3
-3
-2
-1
0
1
2
3
Qc
_
Qb
_
107GeV
8×106
6×106
4×106
2×106
4×106
Qc
_+Qb
_+1=0
z=0.49
δc=0.012
-3-2-1 0 1 2 3
-3
-2
-1
0
1
2
3
Qc
_
Qb
_
107GeV
8×106
6×106
Qc
_+Qb
_+1=0
z=0.49
δc=0.004
Figure 3. Same as figure 2, but with different δc.
¯u¯c¯
t¯
d¯s¯
b
Qf2Q¯c0 1 1 Q¯
b
Table 3. The PQ charges Qfof the SM quarks that avoid the flavor bound without relying on
special flavor structure. If only SM quarks contribute to the QCD anomaly, Q¯
b+Q¯c+ 1 = 0 is
required for the axion-nucleon coupling to be suppressed.
In figure 2, the central values for δs,c,b and G= 0 are assumed. Using different values do
not change how low facan be for given zonce scanned over (Q¯c, Q¯
b), but the preferred set
of (Q¯c, Q¯
b)changes. In figure 3, we show the same plots as figure 2but with different δcthat
has the largest fractional error. The coefficient δsalso has a large absolute error. Changing
it within the error shifts the contours of the lower bound on faby ±0.008/(0.008(4)) to
the Q¯cdirection. Changing gu
A+gd
Awithin the error is much less important and leads to
a shift in the contour by O(0.1) in the Q¯cdirection. The effect of different Gcan be also
estimated in the similar manner. Numerically, the contribution of Gto δis 0.001 ×G, so
the effect of its uncertainty is subdominant.
The constraint from the cooling of neutron stars may be relaxed if the heating of
neutron stars by the decay of magnetic fields in them is significant [22]. The bound from
SN1987A, however, gives a similar constraint [13].
2.2 Lepton couplings
The axion-lepton couplings are also given by eqs. (2.1) and (2.2). To suppress the coupling
with the electron, Q¯eshould be zero. We also assume that Q¯µ= 0; otherwise the lower
bound on fafrom SN1987A is O(108)GeV [61]. This also suppresses the muon-electron
flavor-violating axion coupling. So the only possible PQ-charged lepton is the right-handed
tau. In the minimal model, Q¯τ= 0.
– 7 –
JHEP06(2023)014
¯u¯c¯
t¯
d¯s¯
b¯e¯µ¯τ
Qf2Q¯c0 1 Q¯s−(Q¯c+Q¯s)00−Q¯c
Table 4. The PQ charges Qfof the SM fermions that lead to naturally suppressed axion-nucleon
and axion-photon couplings. Here it is assumed that the QCD and electromagnetic anomaly of the
PQ symmetry solely comes from the SM fermions.
2.3 Photon coupling
The axion-photon coupling is given by
−gaγγ
8aµνρσ Fµν Fρσ , gaγ γ =α
2πfa
Cγ,
Cγ=E
N−CQCD
γ, CQCD
γ=2
3
4 + z+w
1 + z+ 0.06(2),(2.5)
where E/N is the electromagnetic anomaly coefficient of the PQ symmetry relative to the
QCD anomaly. Here we used the estimation in [62] with keeping zand wunfixed. For
the central values of these parameters, CQCD
γ'2.07(4). The observations of the stellar
population in globular clusters gives [63]
fa>1.8×107GeV × |Cγ|.(2.6)
For E/N 6= 2,fashould be above O(107)GeV.
To have fa=O(106)GeV, as required for successful minimal axiogenesis discussed
in section 5, it is necessary to suppress the axion-photon coupling. This can be naturally
achieved by E/N = 2 [43], which leads to |Cγ|=O(0.1) and fa&106GeV.1Interestingly, in
the minimal model, if there is no electromagnatic anomaly of the PQ symmetry beyond that
given by ¯uand ¯
d,E/N = 2 is satisfied. More generically, if the QCD and electromagnetic
anomaly of the PQ symmetry is only given by the SM fermions, the PQ charges of them
are constrained, as shown in table 4.
2.4 Flavor violation
The setup generically leads to flavor violation. Let us take the first two generations of the
up-type quarks. The Yukawa interactions are given by
Hq1q2
y11e−2iθ y12 e−iQ¯cθ
y21e−2iθ y22 e−iQ¯cθ
¯u
¯c
+ h.c., (2.7)
where q1and q2are the first- and second-generation quark doublets, respectively. We may
remove the axion from the mass matrix by the rotation of ¯uand ¯c,
−∂µθ¯u†¯c†
2
Q¯c
¯σµ
¯u
¯c
+Hq1q2
y11 y12
y21 y22
¯u
¯c
+ h.c.(2.8)
1The result for the axion-photon coupling neglecting the effects of the strange quark, i.e., with w= 0,
found in [56], Cγ'E/N −1.92(4), is significantly different from that in [62]. However, for E/N = 2, a
very similar lower bound on fais obtained.
– 8 –
JHEP06(2023)014
The quark mass matrix is diagonalized by the rotation of (u, c)and (¯u, ¯c). Unless the PQ
charge of ¯uis equal to that of ¯c, i.e., Q¯c= 2, the rotation introduces an axion- ¯u-¯ccoupling.
The flavor violation is suppressed in the following structure that is consistent with generic
flavor symmetry that explains yuyc[64],
y11 'yu, y22 'yc, y21 =uyu, y12 =cyc, u,c ∼θ12 ∼0.1,(2.9)
where θ12 is the CKM mixing between the first and second generations. The rotation angle
θ¯u¯cbetween ¯uand ¯cto diagonalize the mass matrix is
θ¯u¯c= (u+c)yu
yc
=O(10−4).(2.10)
With this suppression, the lower bound on the axion decay constant from D-meson decay
is O(104)GeV. Here we used the constraint derived in [42] based on [65]. Note that the
bound is only O(105)GeV even if u,c =O(1), so a flavor symmetry that explains the
Cabibbo angle is not necessary as far as the flavor violation is concerned. Assuming similar
suppression of flavor-violating axion-bottom couplings, the lower bound on fafrom B-
meson decay [42,66] for Q¯
b6=Q¯sis also O(105)GeV.
If Q¯
d6=Q¯s, axion- ¯
d-¯scoupling is introduced. Applying the same analysis as the ¯u-¯c
case, the expected suppression is θ12 yd/ys=O(10−2). The lower bound on fafrom kaon
decay [41,42] would be then 1010 GeV, which is much stronger than the astrophysical
bounds. This bound is avoided by Q¯s=Q¯
d. In the minimal model, however, Q¯s= 0 6=Q¯
d.
The flavor violation can be suppressed by imposing a particular flavor symmetry and its
breaking. For example, we may impose U(1)d×U(1)ssymmetry with charges q1(1,0),
q2(0,1),¯
d(−1,0), and ¯s(0,−1) so that the coupling of the axion with the down and strange
quarks becomes flavor diagonal. The CKM mixing comes from the up-type Yukawa that
explicitly breaks this symmetry. Using the up-type Yukawa as a spurion, one can show
that the axion-down-strange coupling is suppressed by θ12y2
cyd/ys∼10−7, and the lower
bound on fais O(105)GeV.
If Q¯τ6= 0, flavor-violating axion-tau-muon and electron couplings are introduced.
Assuming the similar suppression as the quark sector, the strongest constraint comes from
the axion-tau-muon coupling that is suppressed only by yµ/yτ=O(0.1); the 2-3 MNS
mixing is O(1). Using the bound derived in [67] based on [68], the lower bound on fafrom
tau decay is O(105)GeV. Interestingly, the strongest constraint on the axion-tau-muon
coupling comes from cosmology due to hot axions produced in flavor-violating tau decays
in the early Universe. The Planck constraint on dark radiation [69] leads to the lower
bound on faof O(106)GeV [70].
If the left-handed quarks have non-universal PQ charges, the flavor-violating axion
couplings are suppressed only by the CKM angle and the lower bound on fabecomes
stronger. For example, if Qq16=Qq2, the lower bound on fafrom kaon decay is O(1011)GeV.
Even with flavor symmetry that eliminates axion-strange-down coupling, axion-charm-up
coupling is unavoidable, and the lower bound on fais O(107)GeV. If Qq36=Qq2, the lower
bound on fafrom B-meson decay is O(106)GeV. Although this is not as stringent as the
astrophysical bound, the constraint is much stronger than the case with Q¯
b6=Q¯sand
Qq3=Qq2.
– 9 –
JHEP06(2023)014
3 UV completions
In this section, we present UV completions of the axion-dependent Yukawa couplings.
They may be understood as higher dimensional couplings between the SM fields and the
PQ breaking field P, which can be generated by the exchange of heavy fermions or scalars.
We denote the vacuum expectation value of Pas vPand the phase direction of Pas θP.
3.1 Extra fermions
Let us first discuss the axion-dependent up Yukawa. It may be UV completed by intro-
ducing vector-like fermions that have the same gauge charge as the right-handed up quark,
U−1and ¯
U0, where the subscripts denote the PQ charges. The quark that is the domi-
nant component of the right-handed up quark is denoted as ¯u2. The Yukawa couplings
consistent with the PQ symmetry are
yHq ¯
U0+λ0P¯
U0+λ2P†¯u2U−1+ h.c., (3.1)
where we assume that the PQ charge of Pis 1. Assuming λ0λ2, we may integrate out
the heavy fermions ¯
U0and U−1to obtain an effective coupling
−yue−2iθPHq¯u, yu=yλ2
λ0
,(3.2)
where ¯uis the right-handed up quark that is dominantly ¯u2. In this setup, the heavy
fermions ¯
U0and U−1contribute to the QCD anomaly of the PQ symmetry. Also, the small
up Yukawa coupling is not explained by the ratio between the PQ symmetry and some
other higher mass scale, but rather by small couplings yλ2, which can be understood by
some flavor symmetry.
One may consider a more Froggatt-Nielsen-like PQ model by introducing U−1,¯
U1,
¯
U0, and U0. If their Dirac masses Mare larger than the PQ symmetry-breaking scale,
the up Yukawa coupling is suppressed by v2
P/M2. In this case, the extra fermions do not
contribute to the QCD anomaly.
The extra fermions may have the same gauge charges as doublet quarks. Let us
introduce Q−2and ¯
Q1and the couplings
yHQ−2¯u+λ−2P Q−2+λ0P†q¯
Q1+ h.c.(3.3)
After integrating out Q−2and ¯
Q1assuming λ−2λ0, we obtain
−yue−2iθPHq¯u, yu=yλ0
λ−2
.(3.4)
In this setup, the extra fermions contribute to both the QCD and weak anomaly of the
PQ symmetry. It is also possible to make the model more Froggatt-Nielsen like by further
introducing Q−1and ¯
Q2.
We next discuss the down Yukawa. We may introduce D0and ¯
D0that has the same
gauge charge as the right-handed down quark and the couplings and the mass term
yH†q¯
D0+λP †D0¯
d1+MD0¯
D0+ h.c.(3.5)
– 10 –
JHEP06(2023)014
Assuming MλvP, we may integrate out D0and ¯
D0to obtain
−yde−iθPH†q¯
d, yd=yλvP
M.(3.6)
In this setup, the small down Yukawa may be understood by small vP/M. Also, the extra
fermions do not contribute to the QCD anomaly. We may instead introduce D−1and ¯
D0
and the couplings and the mass
yH†q¯
D0+λP ¯
D0D−1+MD−1¯
d1+ h.c.(3.7)
Assuming λvPM, we may integrate out D−1and ¯
D0to obtain
−yde−iθPH†q¯
d, yd=yM
λvP
.(3.8)
In this setup, the small down Yukawa may be understood by small M/vP. The extra
fermions contribute to the QCD anomaly. It is straightforward to construct UV completion
of the down Yukawa by Q−1and ¯
Q1, and that by Q−1and ¯
Q0.
Similar UV completion can be straightforwardly constructed for other Yukawa cou-
plings.
We note that the UV completion by extra fermions can produce axion-SM fermions
couplings that are not quantized if mass parameters of the theory are not hierarchically
different from each other. For example, let us consider the UV completion in eq. (3.5)
beyond the approximation MλhPi. We replace Pwith vPeiθPand remove θPfrom the
mass term by the rotation of ¯
d1. A linear combination of ¯
D0and ¯
d1, which we denote as
¯
D, obtains a large Dirac mass qM2+λ2v2
Ppaired with D0. The relation between (¯
d, ¯
D)
and (¯
d1,¯
D0)is given by
¯
d1
¯
D0
=
cosαsinα
−sinαcosα
¯
d
¯
D
,tanα=λvP
M.(3.9)
The coupling of θPwith ¯
doriginates from that with ¯
d1and is given by
−∂µθP¯
d†¯σµ¯
d×cos2α, (3.10)
which is not quantized. This is because the SM right-handed down quark is a linear combi-
nation of fermions with different PQ charges. If MλvPor MλvP, for which the SM
right-handed down quark is almost a PQ charge eigenstate, the axion-down coupling is al-
most quantized. Generically, if a SM fermion fis a linear combination of fiwith PQ charges
Qfiwith coefficients ci, the axion-fermion coupling is determined by an effective PQ charge
Qeff,f =X
i|ci|2Qfi.(3.11)
Unless fis almost a PQ charge eigenstate, the axion-fermion coupling is no longer quan-
tized.
– 11 –
JHEP06(2023)014
The suppressed axion-nucleon coupling requires Cu'2/3and Cd'1/3. To achieve
this without fine-tuning, it is crucial that the up and down quarks are nearly PQ charge
eigenstates. To suppress the flavor-violating axion-down-strange coupling, electron cou-
pling, and muon coupling, the strange quark, electron, and muon should also be approx-
imately PQ charge eigenstates. The charm, bottom, and top quark and the tau do not
have to have quantized coupling with the axion, so they may be a mixture of different PQ
charge eigenstates. Irrational (Q¯c, Q¯
b)in figures 2and 3may be understood in this way.
This generically requires coincidence of masses, but in the UV completion in eqs. (3.1)
and (3.3), the masses are given by the PQ breaking field and the coincidence is required
for dimensionless constants rather than for energy scales.
3.2 Extra scalars
Let us discuss the up Yukawa. We introduce a scalar field that has the same gauge charge
as the SM Higgs and has a PQ charge of −2,H−2. The interactions and masses consistent
with the PQ symmetry is
yH−2q¯u+λP 2H−2H†+ h.c.−M2|H−2|2.(3.12)
Assuming M2λv2
P, we may integrate out H−2to obtain
yue−2iθPHq¯u, yu=yλv2
P
M2.(3.13)
Other Yukawa couplings can be obtained in a similar manner by introducing scalar
fields that have the same gauge charge as the SM Higgs and appropriate PQ charges. In
this setup, the gauge anomaly of the PQ symmetry solely comes from the SM fermions.
The minimal model can be UV-completed by three Higgses; H,H−2, and H1.
If the SM Higgs is not nearly a zero-PQ charge eigenstate, in the low energy EFT after
integrating out heavy Higgses, the axion couples to the Higgs current. This may be removed
by the hypercharge rotation proportional to the axion field without gauge transformation
on gauge fields, but axion-fermion current couplings proportional to the hypercharge are
induced. This gives Cu/Cd6= 2 and Ce6= 0, so the axion would be no longer astrophobic.
It is necessary that the SM Higgs is nearly a zero-PQ charge eigenstate. This means that
the O(1) top yukawa should be PQ neutral.
Notice also that the previously proposed astrophobic axion models [36,40] also in-
troduce extra Higgses but introduce a non-vanishing PQ charge for the SM Higgs so they
require fine-tuning to suppress the axion-nucleon coupling. This is in contrast to our models
where the suppression is obtained by a discrete choice of the PQ charges of the SM fermions.
4 Stellar cooling anomalies
Let us also discuss stellar-cooling anomalies that have been observed in several stellar
environments and can be explained if the axion couples to electrons [46]. The axion-
electron coupling required to explain these anomalies is Ce'2×10−3fa/(107GeV), with
the SM disfavored by more than 3σ[71]. Such a value prefers models with either tree-level
– 12 –
JHEP06(2023)014
axion-electron coupling and/or small fa. This cannot be achieved in the minimal KSVZ
model while in the minimal DFSZ model explanation of these anomalies is in tension
with stringent astrophysical constraints on axion-nucleon couplings or the perturbativity
of Yukawa couplings [46]. On the other hand, the stellar-cooling anomalies can be perfectly
explained in nucleophobic models with tree-level axion-electron couplings [36,40,71–73].
While the axion-photon coupling is not necessary to explain these anomalies, the best-
fit value of the axion-photon coupling is |Cγ| ' fa/(6 ×107GeV)[71]. For typical values of
E/N,fais preferred to be between 107and 108GeV while for E/N = 2,fa' O(106)GeV is
preferred. Such small fais viable only in models with suppressed axion-nucleon couplings.
In this range of fathe best-fit is obtained for Ce' O(10−2)if E/N 6= 2 or Ce' O(10−3)
if E/N = 2.
In our setup, a small tree-level electron coupling Ctree
earises if the electron or the
SM Higgs contains a small fraction of a state with a non-zero PQ charge, as discussed in
section 3. We note that Ctree
e6= 0 generically leads to µ→ea decay and the lower bound on
fafrom this decay, found in [67] based on [74,75], is O(Ctree
e107(109)GeV)with (without)
suppression of the flavor violation by ye/yµ. Thus, when Ctree
eoriginates from the effective
PQ charge of the right-handed electron the flavor violation does not lead to any relevant
constraint for values of Ctree
eand fathat explain stellar-cooling anomalies. On the other
hand, if the flavor-violation comes from the effective PQ charge of the left-handed electron,
the value of fathat explains the stellar-cooling anomalies is still comparable with the lower
bound on fafrom µ→ea. The latter scenario is expected to be tested with future muon
beam experiments [67] such as MEG II [76] and Mu3e [77,78]. A better sensitivity in near
future can be obtained from the µ→eaγ process at MEG II if a new trigger is implemented
in a dedicated run [79].
It is also possible to generate an axion-electron coupling by quantum corrections [80].
In the minimal model, the RG correction between faand the QCD scale ΛQCD is pro-
portional to the up or down yukawa coupling and is negligible. The quantum correction
between the QCD scale and meis given by
Ce'3α2
4π2Cγlog ΛQCD
me'2×10−5Cγ,(4.1)
which is too small to explain the cooling anomaly.
In non-minimal models, the quantum correction can be larger. For example, an axion-
Wboson coupling that can arise from heavy SU(2)- and PQ-charged fermions gives
Ce'9
2
α2
2
16π2cWlog fa
mW'3×10−4cW
log(fa/mW)
10 ,(4.2)
where cWis the weak anomaly of the PQ symmetry relative to the QCD anomaly and
we used the RGE in [55]. Here we assumed a RGE correction from fato maximize the
correction; generically the correction starts from the masses of heavy fermions responsible
for the weak anomaly, which may be smaller than faif the coupling of them with the PQ
breaking field is small. Comparing this with the result in [71], one can see that this electron
coupling can explain the cooling anomaly within 2σwhen E/N 6= 2 and fa=O(107)GeV,
– 13 –
JHEP06(2023)014
and 1σwhen E/N = 2 and fa=O(106)GeV. Such small values of faare viable only for
astrophobic axions.
An axion-gluon coupling arising from heavy PQ-charged colored fermions also gener-
ates Cethrough a two-loop correction involving the top Yukawa coupling to the axion-
Higgs coupling. It is of the similar size as the one from the axion-Wboson coupling with
cW∼1[53] and can explain the cooling anomalies.
The quantum corrections involving the hypercharge gauge interaction tend to be small.
The RGE correction between faand mWis given by
Ce'15
2
α2
1
16π2cY+8
3(c¯u+c¯c) + 2
3(c¯
d+c¯s+c¯
b)log fa
mW
'5×10−5cY+8
3(c¯u+c¯c) + 2
3(c¯
d+c¯s+c¯
b)log(fa/mW)
10 ,(4.3)
where cYis the hypercharge anomaly of the PQ symmetry relative to the QCD anomaly.
For E/N 6= 2, this electron coupling cannot explain the cooling anomaly. For E/N = 2,
the anomaly can be explained within 2σ.
5 Minimal axiogenesis
In this section, we discuss the compatibility of the astrophobic axion model with the baryo-
genesis scenario from the rotation of the axion field and the electroweak sphaleron process,
dubbed as minimal axiogenesis [31].
If the radial direction of the PQ-breaking field Pis flat, as naturally occurs in super-
symmetric theories, the radial direction may take on a large field value in the early universe.
Then higher order terms in the potential of Pbecomes important. We assume that some
of them explicitly violate the PQ symmetry, so that rotation of Pin field space is initiated
by the potential gradient to the angular direction, as in the Affleck-Dine mechanism [81].
It is also possible to first initiate the rotation of other scalar fields, such as squarks and
sleptons, and transfer the angular momentum of them to P[82]. In this case, the potential
of Pdoes not have to be flat.
The angular momentum of Pcorresponds to a non-zero PQ charge. The PQ charge
is partially transferred into particle-antiparticle asymmetry of SM particles via the cou-
pling of the axion with them and the SM interactions. The asymmetry is converted into
baryon asymmetry via the electroweak sphaleron process. At the equilibrium, the baryon
asymmetry nBnormalized by the entropy density sis given by
nB
s≡YB=cBTEW
fa2
YPQ,(5.1)
where TEW '130 GeV [83] is the temperature below which the electroweak sphaleron
process becomes ineffective, YPQ =˙
θf 2
a/s is the PQ charge density normalized by the
– 14 –
JHEP06(2023)014
entropy density, and cBis a model-dependent constant given by [84]2
cB' − 21
158 +12
79cW+X
i18
79cqi−21
158c¯ui−15
158c¯
di+25
237c`i−11
237c¯ei
=−18 + 45(Q¯s+Q¯
b) + 63Q¯c+ 22Q¯τ
1422 +12
79cW.(5.2)
In the second equality, we have imposed the astrophobic conditions.
The kinetic energy of the axion rotation is transferred into axion DM density, which
is called the kinetic misalignment mechanism [32]. The number density of the axion na
normalized by the entropy density is
na
s≡Ya=cDMYPQ ,(5.3)
where cDM is an O(1) constant. In the regime where axion DM is produced as a coherent
oscillation of the axion field, which corresponds to fa&1010 GeV [85], numerical and
analytical computations show that cDM '2[32,85]. For lower decay constants, axion DM
is produced via parametric resonance [85,86], and the precise value of cDM is unknown.
In this paper, we take cDM as an unknown O(1) constant, anticipating that it will be
determined by numerical computation in near future.
Requiring that the observed baryon asymmetry be explained by minimal axiogenesis
and axion DM be not overproduced by kinetic misalignment, we obtain an upper bound
on the decay constant,
fa≤2.8×106GeV cB
0.2
1
cDM
.(5.4)
The observed DM density is also explained when the inequality is saturated.
The upper bound is not compatible with the KSVZ and DFSZ axion models, which are
subject to the astrophysical lower bounds of fa>108−9GeV. To overcome this difficulty,
extra baryon or lepton number violations are introduced in [87–92]. Production of helical
hypercharge gauge fields by the tachyonic instability induced by the axion velocity can
produce baryon asymmetry without introducing extra interactions [93], but fine-tuning of
parameters is required.
The astrophobic axion model with E/N = 2 may be consistent with the upper bound
from minimal axiogenesis. Unfortunately, the minimal model has cB'0.01 and is not
compatible with minimal axiogenesis. Non-minimal models may be compatible. For exam-
ple, for (Q¯s, Q¯
b, Q¯c, Q¯τ, cW)=(1,−2,−3,−2,0) and (1,−1,−2,0,−2),cB'0.3and 0.4, so
fa'4and 6×106/cDM GeV is predicted, respectively. If cDM '1, the lower bound on fa
from the cooling of neutron stars is satisfied for zsufficiently close to 0.49.
We note that the QCD axion mass predicted by minimal axiogenesis is around 1 eV,
which is above the range of masses to which future IAXO helioscope [94] will be sensitive.
On the other hand, this is in a range in which optical haloscopes that search for absorption
2The signs of the first two terms are opposite to that in [84]. This is because the sign of the axion-gauge
boson coupling in [84] is opposite to that used in [56] and the literature on astrophobic axions, which stems
from the sign convention of the Levi-Civita tensor.
– 15 –
JHEP06(2023)014
of DM have the best sensitivity to the axion-photon coupling [34], so we expect that exper-
iments such as LAMPOST [35] will probe this scenario. Moreover, in this range of axion
masses, the axion-electron coupling required to explain stellar-cooling anomalies may also
be within the reach of experiments aiming to detect DM via its absorption by molecules [33].
6 Summary and discussion
In this paper, we presented an astrophobic axion model where the couplings of the axion
with nucleons, electrons, and muons are naturally suppressed. The axion decay constant
famay be as low as 107GeV. It is also possible to suppress the coupling with the photon
so that the decay constant is even as small as 106GeV.
We studied the constraint from flavor-violating axion couplings. If the PQ charge of
the strange quark is not the same as that of the down quark, the constraint from kaon decay
requires special flavor structure, although not fine-tuned. If the PQ charges of the strange
and down quarks are the same, generic flavor symmetry that explains the small down and
strange Yukawa couplings significantly suppresses the axion-down-strange coupling and the
constraint from kaon decay is avoided.
The astrophobic axion may have famuch below 1010−12 GeV, for which the axion abun-
dance produced by the misalignment mechanism [5–7] or the decay of cosmic strings [95] is
much below the observed DM abundance. In addition to the kinetic misalignment mecha-
nism discussed in section 5, decay of long-lived domain walls [96–102] or parametric reso-
nance [103–106] can explain the observed DM abundance by axions.
The model can explain the hints for anomalous stellar cooling. The small axion-
electron coupling required for the cooling can be obtained by a radiative correction from
the coupling of the axion with other SM particles or by tree-level mixing of the electron
or Higgs with PQ-charged heavy particles. For the former case, the decay constant needs
to be below O(106−7)GeV. For the latter, the decay constant may be larger. Unless the
(effective) PQ charge of the muon is the same as that of the electron, the axion generically
has a flavor-violating coupling with the electron and muon that can be probed by µ→ea.
The baryon asymmetry of the universe may be explained by minimal axiogenesis.
Unless some of the axion couplings are much larger than naive 1/fa-suppressed ones, the
decay constant should be below 107GeV. This requires the suppression of the axion-photon
coupling, which can be achieved for E/N = 2. The axion has a mass above eV and can be
detected via absorption of axion DM. Since the axion-photon coupling is fixed up to the
dependence on the axion mass, this scenario may serve as a benchmark for experiments
such as LAMPOST that search for absorption of DM. Further requiring that the anomalous
stellar cooling be explained, the axion should have a sizable coupling to the electron, which
helps detection.
Acknowledgments
The work of MB was partially supported by the National Science Centre, Poland, under
research grant no. 2020/38/E/ST2/00243.
– 16 –
JHEP06(2023)014
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited. SCOAP3supports
the goals of the International Year of Basic Sciences for Sustainable Development.
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