††institutetext: Deutsches Elektronen Synchrotron, Notkestrasse 85, 22607
Hamburg, Germany

# Photophilic hadronic axion from heavy magnetic monopoles

Anton V. Sokolov  Andreas Ringwald<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose a model for the QCD axion which is realized through a coupling of
the Peccei-Quinn scalar field to magnetically charged fermions at high
energies. We show that the axion of this model solves the strong CP problem
and then integrate out heavy magnetic monopoles using the Schwinger proper
time method. We find that the model discussed yields axion couplings to the
Standard Model which are drastically different from the ones calculated within
the KSVZ/DFSZ-type models, so that large part of the corresponding parameter
space can be probed by various projected experiments. Moreover, the axion we
introduce is consistent with the astrophysical hints suggested both by
anomalous TeV-transparency of the Universe and by excessive cooling of
horizontal branch stars in globular clusters. Assuming infrared Abelian
dominance in QCD, we show that the leading term for the cosmic axion abundance
is not changed compared to the conventional pre-inflationary QCD axion case
for much of the allowed parameter space.

DESY 21-046

##  1 Introduction

The Standard Model (SM) of particle physics is a very successful theory. Its
structure alone predicts many low energy symmetries which were never disproved
by any experiment, such as for instance baryon number conservation or time
reversal symmetry of quantum electrodynamics. Not all of the possible
symmetries of the theory can be however inferred from the structure of the SM,
in particular this is the case of time reversal symmetry of Quantum Chromo-
Dynamics (QCD). Namely, there is a special free parameter $\bar{\theta}$ in
the SM which indicates whether this symmetry holds. Fortunately, there is also
an experimentally accessible observable proportional to $\bar{\theta}$ – the
neutron electric dipole moment (EDM). While any measured value of this
observable would call for some explanation in terms of a more fundamental
theory, it is especially challenging that the measurements of the neutron EDM
reveal it to be consistent with zero with an unprecedented precision of
$10^{-26}~{}e\cdot\text{cm}$ Abel:2020gbr . The question of why QCD is
symmetric under time reversal constitutes the core of the so-called strong CP
problem. As science aims to explain what we observe, one is tempted to
hypothesize a new model where the neutron EDM is constrained to be practically
zero. In particular, one of the ideas proposed is to drive this observable to
zero dynamically by introducing a new pseudoscalar particle called axion,
which is a pseudo Goldstone boson associated to spontaneous breaking of
anomalous Peccei-Quinn (PQ) symmetry Peccei:1977hh ; Peccei:1977ur ;
Weinberg:1977ma ; Wilczek:1977pj . The great advantage of this mechanism is
that the introduction of the axion can naturally solve not only the strong CP
problem, but also a much more pressing problem of missing mass in the
Universe, i.e. the axion is a perfect candidate for dark matter
Preskill:1982cy ; Abbott:1982af ; Dine:1982ah .

Details of particular axion models can vary. The first axion model proposed,
which is the PQWW model Peccei:1977hh ; Peccei:1977ur ; Weinberg:1977ma ;
Wilczek:1977pj , identified the axion field with a phase of the Higgs in a
two-Higgs-doublet model (2HDM) and was ruled out experimentally soon after the
proposal. Then the KSVZ Kim:1979if ; Shifman:1979if  and DFSZ Dine:1981rt ;
Zhitnitsky:1980tq  axion models were constructed, which were called invisible,
because interactions of the corresponding axion particles with the SM are very
faint. Such faint they are that even after four decades of exploration the
parameter space of these models is still largely terra incognita. Appeal of
the invisible models is their simplicity: the DFSZ model exploits the 2HDM
just as in the case of the PQWW axion but the axion is now identified with the
phase of a new SM-singlet complex scalar field which couples to the Higgses at
high energies; while the KSVZ model exploits coupling of a new SM-singlet
complex scalar field, the phase of which is identified with the axion, to a
new heavy quark. Over the years, there have been attempts of constructing
axion models which would be more "visible" than the DFSZ and KSVZ models,
however it always turned out that simplicity was to be sacrificed. For
example, in the clockwork axion model Farina:2016tgd , in order to get an
enhancement of the axion-photon coupling by six orders of magnitude compared
to the KSVZ model, one has to introduce at least 13 new scalar fields. A
similar enhancement by six orders of magnitude in all couplings to SM
particles is achievable in the $\mathbb{Z}_{\mathcal{N}}$ axion model
Hook:2018jle ; DiLuzio:2021pxd , but it requires ${\mathcal{N}}=45$ copies of
the SM. Although quite non-minimal from the theory side, such an enhancement
would allow one to explain some uneven astrophysical observations concerning
cooling of the horizontal branch stars in globular clusters Ayala:2014pea  and
anomalous TeV-transparency of the Universe DeAngelis:2008sk ; Horns:2012fx ,
not to mention that such photophilic axions can be well probed experimentally
in the nearest future.

Motivation of this work is then to build a photophilic axion model which
possesses the advantages listed above, but which involves a minimal number of
new fields and representations. We show that this can be done by introducing a
SM-singlet complex scalar field which couples to magnetic monopoles at high
energy. In particular, we proceed as follows. First, in section 2, we briefly
review the current status of Abelian as well as non-Abelian magnetic monopoles
and discuss the charge quantization condition. Then, in section 3, we describe
our axion model and outline the solution to the strong CP problem it provides,
proceeding to section 4, where we compute the low energy effective axion
Lagrangian. Finally, we discuss phenomenology and cosmology of the model in
section 5.

There is yet another motivation for our study, which is to broaden the current
understanding of axion models and of possible implementations of the PQ
mechanism. The model we present has qualitative features which no other axion
model possesses: in particular, it has an increased sensitivity to the
structure of infrared (IR) QCD, its axion-gluon coupling is not automatically
standard, and it predicts magnetic monopoles. Besides new experimental
prospects, it can well provide a basis for novel insights on the structure of
the ultraviolet (UV) theory, which is finally to give a more fundamental
description of nature than the SM does.

##  2 Abelian and non-Abelian magnetic monopoles

As it was shown by Dirac Dirac:1931kp , the observed quantization of charge in
electrodynamics can be elegantly explained by adding a magnetic monopole to
the theory. The consistency condition for a theory with both electric and
magnetic currents is:

$eg=2\pi n\,,\;\,n\in\mathbb{Z}\,,$ (1)

where $e$ is the elementary electric charge and $g$ is the magnetic charge of
the monopole. Local Lagrangian quantum field theory (QFT) of Dirac magnetic
monopoles was constructed later by Zwanziger PhysRevD.3.880 . In order to
obtain a consistent local theory, both electric and magnetic four-potentials
($A_{\mu},B_{\mu}$) had to be introduced which are sourced by magnetically and
electrically charged particles, respectively. It turned out that the resulting
Lagrangian possesses an $SO(2)$ symmetry, which rotates charges ($e,g$) and
four-potentials ($A_{\mu},B_{\mu}$) in the electric-magnetic plane, which is
broken to a $\mathbb{Z}_{2}$ symmetry exchanging electric and magnetic
quantities in the full quantum theory: $(e,g)\rightarrow(g,-e)$ and
$(A_{\mu},B_{\mu})\rightarrow(B_{\mu},-A_{\mu})$. Moreover, the total gauge
group of this theory is $U_{\text{e}}(1)\times U_{\text{g}}(1)$, electric
charges transforming in a representation of the "electric" $U_{\text{e}}(1)$
group, while magnetic charges transform in a representation of the "magnetic"
$U_{\text{g}}(1)$ group. Due to the condition (1) the theory is essentially
non-perturbative, the asymptotic Dyson series being not well-defined and the
corresponding Feynman diagrams as well as the Lagrangian itself losing Lorentz
covariance. Of course, failure of our perturbative techniques does not mean
that the theory is by itself inconsistent and indeed, it was formally shown by
Brandt, Neri and Zwanziger PhysRevLett.40.147 ; PhysRevD.19.1153  using the
path-integral approach that observables of the Zwanziger theory are Lorentz
invariant if the Dirac condition (1) is satisfied. Note that this analysis was
performed both in the case where electric and magnetic particles are complex
scalars as well as in the case where they are all Dirac fermions. In the case
where electric and magnetic particles include both fermions and scalars, it is
known that the Zwanziger effective theory is not enough for an adequate
description of the low energy phenomena, the most famous example being proton
decay due to the Rubakov-Callan effect Rubakov:1981rg ; Callan:1982au .
Although spectacular, such violations of the decoupling principle will not
concern us in this work, since we will exploit solely fermionic electric and
magnetic charges in our model.

With the advent of the Standard Model (SM) of particle physics, the Dirac
condition (1) was extended Englert:1976ng  to include all possible types of
magnetic charges $\vec{Q}_{M_{i}}$ in the theory:

$\exp\left(i\sum\limits_{i=1}^{r}\vec{Q}_{M_{i}}\,\vec{\mathcal{H}}\right)=1\,,$
(2)

where $\mathcal{H}_{k}\equiv e_{k}\cdot h_{k}$ are Cartan generators of the
Lie algebra $\mathcal{G}$ of rank $r$ of the gauge group multiplied by the
corresponding electric charges $e_{k}$. In case of a non-Abelian gauge theory,
$e_{k}$ are equal to the gauge couplings of the theory. For the SM, at low
energies, we have $\mathcal{G}=\mathfrak{su}(3)\oplus\mathfrak{u}(1)$, which
means that a magnetically charged particle has generally Abelian as well as
non-Abelian magnetic charges. In this theory, the minimal magnetic charge
corresponding to the electromagnetic subgroup, is still $g=2\pi/e$, although
there are now fractionally charged quarks. The reason is that quarks interact
strongly with the monopole that has a color magnetic charge, compensating the
would-be observable phase which results from the electromagnetic interaction.
In particular, for a down-type quark the quantization condition (2) can be
written as:

$\xi
g_{s}t_{3}+\zeta\sqrt{3}\,g_{s}t_{8}-\frac{e}{3}\,g=2\pi\cdot\mathrm{diag}\left(n_{1},n_{2},n_{3}\right),$
(3)

where $\xi,\zeta\in\mathbb{R}$, $n_{1},n_{2},n_{3}\in\mathbb{Z}$,
$t_{3}=\lambda_{3}/2$, $t_{8}=\lambda_{8}/2$; $\lambda_{a}$ are Gell-Mann
matrices; $g_{s}$ is the strong coupling. Coexistence of a monopole with
charged leptons requires $eg=2\pi m,\,m\in\mathbb{Z}$. Then Eq. (3) can be
solved with respect to the coefficients $\xi,\zeta$:

$\xi=\frac{2\pi}{g_{s}}\cdot\left(2n_{1}+n_{3}+m\right),\,\,\,\zeta=-\frac{2\pi}{g_{s}}\cdot\left(n_{3}+\frac{m}{3}\right)\,.$
(4)

Note that the quantization condition for up-type quarks is satisfied
automatically as long as Eq. (3) holds, for their electric charges differ by
one elementary charge $e$ from those of the down-type quarks. One can see that
$m=1$, which corresponds to the minimal Dirac magnetic charge, is still
possible, although magnetic monopole must carry non-Abelian magnetic charge as
well. The latter is not necessary in the case $m=3$ where viable solutions
include $\xi=\zeta=0$, which means vanishing non-Abelian magnetic charge.

Having discussed QFT of the Abelian magnetic monopoles and the generic
quantization condition pertinent to both Abelian and non-Abelian magnetic
charges, let us outline the status of the theory of the latter. First, we note
that the condition (2) can be expressed in a simple way using the language of
the Lie group theory. In particular, Goddard, Nuyts and Olive Goddard:1976qe
showed that the condition (2) in a theory with gauge group $G$ can be regarded
as a one-to-one correspondence between the magnetic charges of monopoles in
this theory and the weights of the Langlands dual gauge group $G^{V}$, which
is now also known as the GNO group. For example, the gauge group of
electromagnetism is self-dual in this sense: $\left(U(1)\right)^{\\!V}=U(1)$;
and the GNO group corresponding to the gauge theory of QCD can be inferred
from the following identity: $\left(SU(3)/\mathbb{Z}_{3}\right)^{\\!V}=SU(3)$.
Based on the derived relation between magnetic charges and the dual gauge
group $G^{V}$, which is completely analogous to the relation between electric
charges and the gauge group $G$, Goddard, Nuyts and Olive suggested that
magnetic monopoles of a gauge theory with a group $G$ generally transform in
the representations of the group $G^{V}$. The above conjecture, known as the
GNO conjecture, obviously holds in the case of the Abelian group $G=U(1)$, for
which the Zwanziger theory discussed earlier in this section can be
constructed. The GNO conjecture for the non-Abelian monopoles, in its stronger
form known as the Montonen-Olive conjecture Montonen:1977sn , has recently
been proven by Kapustin and Witten Kapustin:2006pk  for a twisted $\cal{N}$
$=4$ supersymmetric Yang-Mills (YM) theory. In this work we assume that the
GNO conjecture holds for the gauge theory of QCD as well, inspired by the
findings of Hong-Mo, Faridani and Tsun Chan:1995xr  that the classical
(nonsupersymmetric) YM equations possess a generalized dual symmetry similar
to the electric-magnetic $\mathbb{Z}_{2}$ symmetry of the Zwanziger theory
mentioned above. Let us also note, that although non-Abelian magnetic charges
are often introduced as emergent from spontaneous breaking of some larger
gauge symmetry, the results by Goddard, Nuyts and Olive do not depend on such
a construction and can be as well stated for generic magnetic monopoles
defined in the fiber bundle framework of Wu and Yang Wu:1975es . We will thus
consider magnetic particles as fundamental in this work, leaving aside the
questions concerning their possible inner structure.

For concreteness, in the next sections we limit ourselves to the two minimal
magnetic charge assignments: a pure Abelian magnetic monopole with a charge
$6\pi/e$ and a non-Abelian color-magnetic monopole with an Abelian magnetic
charge $2\pi/e$, which correspond respectively to the cases $m=3$ and $m=1$
discussed after Eq. (4). For the non-Abelian case, we will consider only
magnetic charges transforming in the fundamental representation of $SU(3)$
with the coupling constant $2\pi/g_{s}\,$, bearing in mind that the higher
representation GNO monopoles are unstable due to the Brandt-Neri-Coleman
analysis Brandt:1979kk ; Coleman:1982cx .

##  3 Solution to the strong CP problem

Suppose there exist a vector-like fermionic magnetic monopole
$\psi={\psi}_{L}+{\psi}_{R}$ which transforms under an anomalous PQ symmetry
$U(1)_{PQ}$ Peccei:1977hh ; Peccei:1977ur  and a complex scalar field $\Phi$
which breaks the PQ symmetry spontaneously at some high energy scale $v_{a}$.
As discussed in the previous section, we consider minimal magnetic charge
assignments corresponding either to the Abelian (electromagnetic) monopole or
to the non-Abelian (color-magnetic) one. In the former case we assume that
$\psi$ transforms in a fundamental representation of the QCD gauge group, i.e.
it is a new quark. As far as we do not consider the electromagnetic
interaction, such model with a new quark is an exact analog of the KSVZ axion
model and thus it provides a solution to the strong CP problem in the same way
the KSVZ model does. The aim of this section is then to show that the model
with the non-Abelian color-magnetic monopole solves the strong CP problem as
well. The high-energy Lagrangian in this case includes the following terms:

$\mathcal{L}\;\supset\;i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+\bar{\psi}\gamma^{\mu}C_{\mu}\psi+y\left(\Phi\,\bar{\psi}_{L}{\psi}_{R}+\text{h.c.}\,\right)-\lambda_{\Phi}\left(\left|\Phi\right|^{2}-\frac{v_{a}^{2}}{2}\right)^{\\!\\!2},$
(5)

where $C_{\mu}$ is a connection on a GNO group $SU(3)$ multiplied by the
corresponding magnetic coupling: $C_{\mu}=g_{\text{m}}\,t_{a}\,C^{a}_{\mu}$.
In the broken phase, there exists a pseudo Goldstone boson $a$ (axion), which
can be introduced via the polar decomposition of the PQ scalar field
$\Phi=\frac{1}{\sqrt{2}}\left(v_{a}+\sigma\right)\cdot\exp{(-ia/v_{a})}$ near
the vacuum. Let us dispose of the axion dependence in the Yukawa term by
performing a chiral rotation of the fermions
$\psi\rightarrow\exp{(ia\gamma_{5}/2v_{a})}\cdot\psi$. Omitting the terms
containing a heavy radial field $\sigma$, one then obtains:

$\mathcal{L}\;\supset\;i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+\bar{\psi}\gamma^{\mu}C_{\mu}\psi+\frac{yv_{a}}{\sqrt{2}}\,\bar{\psi}\psi-\frac{\partial^{\mu}a}{2v_{a}}\,\bar{\psi}\gamma_{\mu}\gamma_{5}\psi+\mathcal{L}_{\text{F}},$
(6)

where $\mathcal{L}_{\text{F}}$ is a Fujikawa contribution coming from the
transformation of the fermion measure in the path integral, i.e. the density
of the index of the Dirac operator
$\gamma^{\mu}D_{\mu}=\gamma^{\mu}(\partial_{\mu}-C_{\mu})$. By the Atiyah-
Singer index theorem, the latter is equal to the characteristic class of the
GNO group bundle, so that:

$\mathcal{L}_{\text{F}}=-\frac{a}{16\pi^{2}v_{a}}\,\textrm{tr}\,C_{\mu\nu}\tilde{C}^{\mu\nu},$
(7)

where $C_{\mu\nu}$ is the curvature of the GNO group connection and
$\tilde{C}_{\mu\nu}=\epsilon_{\mu\nu\lambda\rho}\,C^{\lambda\rho}/2\,$,
$\,\epsilon_{0123}\equiv 1$.

In order to see that such a model provides a solution to the strong CP
problem, we invoke Abelian gauge fixing introduced by ’t Hooft tHooft:1981bkw
. In the Abelian gauges there arise singularities corresponding to effective
color magnetic currents which result in the violation of the non-Abelian
Bianchi identities (VNABI) Bonati:2010tz . The time reversal violating term of
the QCD action can then be expanded as follows:

$\displaystyle\mathcal{S}_{\text{QCD}}\;\supset\;\frac{\bar{\theta}g_{s}^{2}}{32\pi^{2}}\int
d^{4}x\,\sum_{a=1}^{8}G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}\;=\;\frac{\bar{\theta}g_{s}^{2}}{32\pi^{2}}\,\times\qquad\qquad\qquad\qquad\qquad\qquad\quad$
$\displaystyle\left\\{\int
d^{4}x\,\,\epsilon_{\mu\nu\lambda\rho}\,\partial^{\mu}\\!\\!\\!\sum_{a,b,c\,=1}^{8}\\!\\!\left(A_{a}^{\nu}\,G_{a}^{\lambda\rho}-\frac{1}{3}\,g_{s}f_{abc}\,A_{a}^{\nu}A_{b}^{\lambda}A_{c}^{\rho}\right)-2\int
d^{4}x\,\sum_{a=1}^{8}A^{a}_{\nu}\left(D_{\mu}\tilde{G}^{\mu\nu}\right)_{\\!a}\right\\},\qquad\;$
(8)

where $G_{\mu\nu}^{a}$ ($A_{\mu}^{a}$) are components of the non-Abelian field
strength tensor $G_{\mu\nu}$ (four-potential $A_{\mu}$) of QCD, $f_{abc}$ are
$\mathfrak{su}(3)$ structure constants, $\bar{\theta}$ is QCD vacuum angle,
$\tilde{G}_{\mu\nu}^{a}=\epsilon_{\mu\nu\lambda\rho}\,G^{a\,\lambda\rho}/2\,$.

Let us consider the first term on the right-hand side of Eq. (3). Since all
the singularities characteristic of the Abelian ’t Hooft gauges arise in the
diagonal part of the gluon field, i.e. in the components $A_{\mu}^{3}$ and
$A_{\mu}^{8}$, the terms of the integrand which contain solely off-diagonal
fields can be safely integrated with the use of the Stokes theorem:

$\displaystyle\int
d^{4}x\,\,\epsilon_{\mu\nu\lambda\rho}\,\partial^{\mu}\\!\\!\\!\sum_{a,b,c\,=1}^{8}\\!\\!\left(A_{a}^{\nu}\,G_{a}^{\lambda\rho}-\frac{1}{3}\,g_{s}f_{abc}\,A_{a}^{\nu}A_{b}^{\lambda}A_{c}^{\rho}\right)=\qquad\qquad\qquad\qquad\qquad\quad$
$\displaystyle\int
d^{4}x\,\,\epsilon_{\mu\nu\lambda\rho}\,\partial^{\mu}\sum_{\alpha=3,8}\,\sum_{b,c\,=1}^{8}\left(A_{\alpha}^{\nu}\mathscr{G}_{\alpha}^{\lambda\rho}+2\,g_{s}f_{\alpha
bc}\,A_{\alpha}^{\nu}A_{b}^{\lambda}A_{c}^{\rho}\right)+\int\limits_{\Omega_{\infty}}dS^{\mu}\,\mathfrak{K_{\mu}}\left[A_{\text{off-
diag}}\right]\,,\qquad\quad$ (9)

where
$\mathscr{G}_{\alpha}^{\mu\nu}=\,\partial^{\mu}\\!A^{\nu}_{\alpha}\,-\,\partial^{\nu}\\!A^{\mu}_{\alpha}$
($\alpha=3,8$) are Abelian field strength tensors. As it is derived both from
theoretical considerations Kondo:2014sta  and lattice calculations
Amemiya:1998jz , in the Abelian gauges off-diagonal gluons obtain finite mass,
which means that the functional $\mathfrak{K_{\mu}}\left[A_{\text{off-
diag}}\right]$ vanishes at the surface at infinity, $\Omega_{\infty}$. For the
same reason the integrand in Eq. (3) proportional to
$\partial^{\mu}(A_{\alpha}^{\nu}A_{b}^{\lambda}A_{c}^{\rho})$ is restricted to
arbitrarily small surfaces around the singularities after application of the
Stokes theorem and finally integrates to zero due to regularity of the off-
diagonal fields.

Equation (3) can now be rewritten in the following way:

$\displaystyle\int
d^{4}x\,\sum_{a=1}^{8}G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}\,=\,\int
d^{4}x\sum_{\alpha=3,8}\mathscr{G}_{\mu\nu}^{\alpha}\tilde{\mathscr{G}}^{\alpha\,\mu\nu}+\qquad\qquad\qquad\qquad\qquad$
$\displaystyle 2\int
d^{4}x\left(\sum_{\alpha=3,8}A_{\alpha\,\nu}\partial_{\mu}\tilde{\mathscr{G}}_{\alpha}^{\mu\nu}-\sum_{a=1}^{8}A_{a\,\nu}\left(D_{\mu}\tilde{G}^{\mu\nu}\right)_{\\!a}\right)\\!.\;$
(10)

Let us show that the VNABI, $D_{\mu}\tilde{G}^{\mu\nu}$, is diagonal in color
space, so that the second row in Eq. (3) equals to zero. First, note that the
only contribution to VNABI comes from singularities, where topological defects
associated with the monopoles hamper commutation of partial derivatives, so
that in the expression for a commutator of covariant derivatives,

$\displaystyle\left[D_{\rho},D_{\lambda}\right]\,=\,-iG_{\rho\lambda}\,+\,\left[\partial_{\rho},\partial_{\lambda}\right]\,,$
(11)

the second term on the right does not vanish. After taking advantage of Eq.
(11) and Jacobi identities for partial as well as covariant derivatives, the
expression for VNABI can be simplified Suzuki:2017lco :

$D_{\mu}\tilde{G}^{\mu\nu}=\,\,\frac{1}{2}\,\epsilon_{\mu\nu\rho\lambda}\left[D^{\mu},G^{\rho\lambda}\right]=\,\,\frac{1}{2}\,\epsilon_{\mu\nu\rho\lambda}\left[\partial^{\rho},\partial^{\lambda}\right]A^{\mu}=\partial_{\rho}\tilde{\mathscr{G}}^{\rho\nu},$
(12)

where in the last step only diagonal gluons survive. One can see that the
diagonal form of VNABI is ensured by its linearity in the $A_{\mu}$ field. We
note that the second term on the right-hand side of Eq. (3) is then nothing
but a manifestation of the Witten effect Witten:1979ey : QCD monopoles are
dyons with color electric charges proportional to the vacuum angle
$\bar{\theta}$.

Due to the identities Eqs. (3) and (12) the CP violating term of the QCD
Lagrangian reduces in the Abelian gauges to

$\frac{\bar{\theta}g_{s}^{2}}{32\pi^{2}}\sum_{\alpha=3,8}\mathscr{G}_{\mu\nu}^{\alpha}\tilde{\mathscr{G}}^{\alpha\,\mu\nu}\,,$
(13)

which involves now only Abelian four-potentials. By the analogous
transformation of the Fujikawa contribution (7) to the axion Lagrangian (6),
i.e. choosing the same Abelian gauge in the GNO gauge group, one obtains the
term for the interaction of the axion with the Abelian dual four-potentials:

$\mathcal{L}_{\text{F}}=-\frac{ag_{\text{m}}^{2}}{32\pi^{2}v_{a}}\sum_{\alpha=3,8}C_{\mu\nu}^{\alpha}\tilde{C}^{\alpha\,\mu\nu},$
(14)

where the axion field is assumed to be constant and homogeneous, since this is
a vacuum expectation value of it which is a key to the PQ mechanism. Now that
we have abelianized the relevant terms, we are in the realm of the Zwanziger
theory, so that the electric and magnetic four-potentials can be related due
to the dual $\mathbb{Z}_{2}$ symmetry111The existence of the SM quarks – given
the absence of their magnetic partners – obviously violates the electric-
magnetic symmetry of this $U(1)^{2}$ Zwanziger-like theory. However, these
quarks are known to be massive. This means they have no relevance for the
instanton vacuum effects which are responsible for the generation of the
$\bar{\theta}$-term,
$C_{\mu\nu}^{\alpha}=\tilde{\mathscr{G}}^{\alpha}_{\mu\nu}$, which yields:

$\mathcal{L}_{\text{QCD}}\;\supset\;\frac{v_{a}g_{s}^{2}\,\bar{\theta}+ag_{\text{m}}^{2}}{32\pi^{2}v_{a}}\sum_{\alpha=3,8}\mathscr{G}_{\mu\nu}^{\alpha}\tilde{\mathscr{G}}^{\alpha\,\mu\nu}\,.$
(15)

Physically, this is just an instantiation of the fact that the $U(1)$ electric
and magnetic fields enter the expressions (13) and (14) symmetrically, as
products $\vec{E}\cdot\vec{B}$. The standard PQ mechanism is now in order:
redefinition of the pseudo Goldstone axion field $a\rightarrow
a-v_{a}\bar{\theta}\,g_{s}^{2}/g_{\text{m}}^{2}$ absorbs the
$\bar{\theta}$-term into the axion-gluon term and subsequent application of
the Vafa-Witten theorem Vafa:1984xg  ensures $\langle a\rangle=0$. The strong
CP problem is thus solved.

##  4 Calculation of the effective Lagrangian

Let us return to the original Lagrangian (5) and derive the corresponding low
energy physical phenomena. For that, we use a linear decomposition of the PQ
field, $\Phi=v_{a}+\sigma+ia$, where $a$ is a pseudo Goldstone axion
field.222$\sigma$ and $a$ introduced here are different from the fields
denoted by the same letters in Sec. 3, but there should be no confusion, since
different notations are restricted to different sections. Below the PQ scale,
the field $\sigma$ decouples and we are left with the Lagrangian involving
axion and heavy monopoles:

$\mathcal{L}\;\supset\;i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+\bar{\psi}\gamma^{\mu}C_{\mu}\psi+\frac{yv_{a}}{\sqrt{2}}\,\bar{\psi}\psi+\frac{iy}{\sqrt{2}}\,a\bar{\psi}\gamma_{5}\psi\,,$
(16)

where $C_{\mu}$ now also includes the electromagnetic four-potential and
corresponds in general to the connection on either of the two GNO gauge
groups, Abelian or non-Abelian, discussed in the end of Sec. 2. The aim of
this section is to integrate out the heavy field $\psi$. The beauty of the
pseudoscalar interaction is that in this case the calculations can be done
exactly, without the need of perturbative expansion in the coupling constant.
In order to get an effective Lagrangian at low energy we use the proper time
method Schwinger:1951nm  developed by Schwinger. The effective pseudoscalar
current is

$\displaystyle
J_{a}=i\left\langle\,C\left|\bar{\psi}(x)\gamma_{5}\psi(x)\right|C\right\rangle=-\,i\,\frac{yv_{a}}{\sqrt{2}}\int_{0}^{\infty}ds\,e^{-isy^{2}v_{a}^{2}/2}\;\textrm{tr}\left[\langle
x|\gamma_{5}e^{-i\hat{H}s}|x\rangle\right],$ (17)

with the proper time Hamiltonian

$\hat{H}=-\left(\mathrlap{\mkern-1.7mu\not{\phantom{\hat{p}}}}\hat{p}-\mathrlap{\mkern-1.7mu\not{\phantom{C}}}C(\hat{x})\right)^{2}=-\left(\hat{p}_{\mu}-C_{\mu}(\hat{x})\right)^{2}+\frac{1}{2}\,\sigma^{\mu\nu}C_{\mu\nu}(\hat{x})\,,$
(18)

where $\sigma_{\mu\nu}=\frac{i}{2}\left[\gamma_{\mu},\gamma_{\nu}\right]$,
$\;C_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}+\left[C_{\mu},C_{\nu}\right]\,$
and $\mathrlap{\mkern-1.7mu\not{\phantom{a}}}a\equiv a_{\mu}\gamma^{\mu}$.

First, our goal is to evaluate the matrix element entering Eq. (17), which
modulo $\gamma_{5}$ denotes the probability amplitude of returning to the same
point $x^{\mu}$ in Minkowski space after proper time $s$. Note that since we
are interested in the phenomenology at energies much less than the PQ scale
$v_{a}$ and the fluctuations of heavy fields $\psi$ are possible only at the
spatial and temporal extent $\sim v_{a}^{-1}$, external gauge fields in the
following calculation can be considered constant. Our calculation of the
pseudoscalar current then closely follows that performed by Schwinger
Schwinger:1951nm , although we are considering generic non-Abelian GNO group
connection instead of the electromagnetic four-potential. We solve the
Heisenberg equations of motion in a constant field $C_{\mu\nu}$,

$\displaystyle\frac{d\hat{\pi}_{\mu}}{ds}=i\left[\hat{H},\hat{\pi}_{\mu}\right]=2\,C_{\mu\nu}\hat{\pi}^{\nu}\,,$
(19)
$\displaystyle\frac{d\hat{x}_{\mu}}{ds}=i\left[\hat{H},\hat{x}_{\mu}\right]=2\,\hat{\pi}_{\mu}\,,$
(20)

and find the generalized momentum $\hat{\pi}_{\mu}=\hat{p}_{\mu}-C_{\mu}$ and
position $\hat{x}_{\mu}$ as a function of proper time $s$:

$\displaystyle\hat{\pi}_{\mu}(s)=e^{2s\,C_{\mu\nu}}\,\hat{\pi}^{\nu}(0)\,,$
(21)
$\displaystyle\hat{x}_{\mu}(s)=\hat{x}_{\mu}(0)+2\left(C^{\mu\lambda}\right)^{-1}\cdot
e^{sC^{\lambda\nu}}\sinh{sC_{\nu\rho}}\cdot\hat{\pi}^{\rho}(0)\,.$ (22)

Next, with the use of Eqs. (21) and (22) we rewrite the Hamiltonian (18) in
terms of position operators $\hat{x}_{\mu}(s)$ and $\hat{x}_{\mu}(0)$:

$\hat{H}\,\supset\;-\,\frac{1}{4}\left(\sinh{sC_{\kappa\lambda}}\right)^{-1}C_{\lambda\nu}C^{\nu\rho}\left(\sinh{sC^{\rho\sigma}}\right)^{-1}\times\left[\hat{x}_{\kappa}(s),\hat{x}^{\sigma}(0)\right]\;+\;\frac{1}{2}\,\sigma_{\mu\nu}C^{\mu\nu}\,,$
(23)

leaving only the terms that do not vanish after taking the matrix element
$\langle x(0)|\hat{H}|x(s)\rangle\propto\langle x(0)|x(s)\rangle$. Note that
the exponents coming from Eqs. (21), (22) contract into the identity matrix
due to antisymmetricity of the field strength tensor $C_{\mu\nu}$. The
commutator in Eq. (23) is easily calculated with the help of Eq. (22) and
canonical commutation relations. Since the Hamiltonian (23) is a generator of
proper time translations, one can write now a differential equation for the
sought-after matrix element:

$i\partial_{s}\langle x(0)|x(s)\rangle=\langle
x(0)|\hat{H}|x(s)\rangle=\langle
x(0)|x(s)\rangle\times\left\\{\frac{i}{2}\,C_{\mu\nu}\coth{sC^{\mu\nu}}\;+\;\frac{1}{2}\,\sigma_{\mu\nu}C^{\mu\nu}\right\\}.$
(24)

The solution is:

$\langle
x(0)|x(s)\rangle=A\,\frac{\textrm{pf}\,C_{\alpha\beta}}{\textrm{pf}\sinh{sC_{\alpha\beta}}}\cdot\exp\left(-\frac{is}{2}\sigma_{\mu\nu}C^{\mu\nu}\right),$
(25)

where $A=-i/(4\pi)^{2}$ is an integration constant which is calculated by
matching with the elementary case of vanishing field strength
$\mathscr{G}^{\alpha}=0$. A skew-symmetric four-by-four matrix has two pairs
of opposite sign eigenvalues, which we denote as $\pm\Lambda_{1}$,
$\pm\Lambda_{2}$ in the particular case of $C_{\alpha\beta}$. The trace
entering Eq. (17) can be now rewritten in the following form:

$\textrm{tr}\left[\langle
x|\gamma_{5}e^{-i\hat{H}s}|x\rangle\right]=-\,\frac{i}{16\pi^{2}}\,\textrm{tr}_{c}\left[\frac{\Lambda_{1}\Lambda_{2}}{\sinh{s\Lambda_{1}}\sinh{s\Lambda_{2}}}\times\textrm{tr}_{\gamma}\left\\{\gamma_{5}\exp\left(-\frac{is}{2}\sigma_{\mu\nu}C^{\mu\nu}\right)\right\\}\right],$
(26)

where we have explicitly separated traces over colour ($\textrm{tr}_{c}$) and
spinor ($\textrm{tr}_{\gamma}$) indices. Sums over the spinor indices can be
performed using simple algebraic relations, namely
$(\sigma_{\mu\nu}C^{\mu\nu})^{2}=8I_{1}+8i\gamma_{5}I_{2}$,
$\gamma_{5}^{2}=1$,
$\textrm{tr}\,\gamma_{5}=\textrm{tr}\,\sigma_{\mu\nu}=\textrm{tr}\,\gamma_{5}\sigma_{\mu\nu}=0$,
where $I_{1}\equiv C_{\mu\nu}C^{\mu\nu}/4$,
$I_{2}\equiv\epsilon_{\mu\nu\lambda\rho}\,C^{\mu\nu}C^{\lambda\rho}/8\,$:

$\textrm{tr}_{\gamma}\left\\{\gamma_{5}\exp\left(-\frac{is}{2}\sigma_{\mu\nu}C^{\mu\nu}\right)\right\\}=4i\,\mathrm{Im}\cosh{sX}=4\sinh{\left(s\,\frac{X+X^{*}}{2}\right)}\sinh{\left(s\,\frac{X-X^{*}}{2}\right)}\,,$
(27)

where $X\equiv i\sqrt{2}\cdot\sqrt{I_{1}+iI_{2}}$. Quite conveniently, by
solving the characteristic equation for the matrix $C_{\alpha\beta}$, which
has the form $\Lambda^{4}+2I_{1}\Lambda^{2}-I_{2}^{2}=0$, one can infer that

$\Lambda_{1}=\frac{X+X^{*}}{2}\,,\quad\Lambda_{2}=\frac{X-X^{*}}{2}\,,$ (28)

and the overall expression for the current simplifies into

$J_{a}=\frac{iyv_{a}}{4\sqrt{2}\pi^{2}}\,\textrm{tr}_{c}(I_{2})\int_{0}^{\infty}ds\,e^{-isy^{2}v_{a}^{2}/2}=\frac{1}{16\sqrt{2}\pi^{2}yv_{a}}\,\epsilon_{\mu\nu\lambda\rho}\,\textrm{tr}_{c}\\!\left(C^{\mu\nu}C^{\lambda\rho}\right).$
(29)

Finally, we calculate the trace over color indices and expand in terms of the
electromagnetic and color gauge fields:

$J_{a}=\frac{1}{8\sqrt{2}\pi^{2}yv_{a}}\times\begin{dcases}-\,3g_{1}^{2}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{g_{m}^{2}}{2}\,G^{a}_{(d)}{{}_{\mu\nu}}\tilde{G}^{a\,\mu\nu}_{(d)}\,,\\\
-\,3g_{2}^{2}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{g_{s}^{2}}{2}\,G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}\,,\end{dcases}$
(30)

where summation over $a=1\dots 8$ is implied; $g_{1}=2\pi/e$ and
$g_{2}=6\pi/e$ – we separated the two cases discussed in the end of Sec. 2,
corresponding to the stable non-Abelian monopole and the minimal Abelian one,
respectively. We also introduced notation for the dual gluon fields
$G^{a}_{(d)}{{}_{\mu\nu}}$, which are components of the connection on the
color GNO group, and expressed the dual electromagnetic field strength tensor
in terms of the conventional one using
$F_{(d)}{{}_{\mu\nu}}=\tilde{F}_{\mu\nu}$, which obviously holds for constant
fields, since the vacuum Maxwell equations are dual-invariant. The effective
axion Lagrangian is then given by the following expression:

$\mathcal{L}_{\text{eff}}\;\supset\;\frac{y}{\sqrt{2}}\,aJ_{a}\;=\;\frac{a}{16\pi^{2}v_{a}}\times\begin{dcases}-\,\frac{3}{4\alpha^{2}}\,e^{2}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{1}{8\alpha_{s}^{2}}\,g_{s}^{2}\,G^{a}_{(d)}{{}_{\mu\nu}}\tilde{G}^{a\,\mu\nu}_{(d)}\,,\\\
-\,\frac{27}{4\alpha^{2}}\,e^{2}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{1}{2}\,g_{s}^{2}\,G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}\,,\end{dcases}$
(31)

where we introduced the fine-structure constant $\alpha=e^{2}/4\pi$ and its
QCD analogue $\alpha_{s}=g_{s}^{2}/4\pi$.

##  5 Phenomenology

Let us introduce the axion decay constant $f_{a}=4\alpha_{s}^{2}v_{a}$
($f_{a}=v_{a}$), for the case of the non-Abelian (Abelian) monopole. Using the
dual symmetry of a Zwanziger-like theory describing diagonal gluons, we obtain
the relation between the magnetic and electric $U(1)$ field strength tensors
$\mathscr{G}^{\alpha}_{(d)}=\tilde{\mathscr{G}}^{\alpha}$ ($\alpha=3,8$). The
effective Lagrangian Eq. (31) can then be rewritten in the following form:

$\mathcal{L}_{\text{eff}}\;\supset\;\begin{dcases}-\,\frac{1}{4}\left(g^{0}_{a\gamma}\right)_{1}a\,F_{\mu\nu}\tilde{F}^{\mu\nu}\,-\,\frac{ag_{s}^{2}}{32\pi^{2}f_{a}}\;G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}+\mathcal{L}_{\text{off}}\,,\\\
-\,\frac{1}{4}\left(g^{0}_{a\gamma}\right)_{2}a\,F_{\mu\nu}\tilde{F}^{\mu\nu}\,+\,\frac{ag_{s}^{2}}{32\pi^{2}f_{a}}\;G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}\,,\end{dcases}$
(32)

where

$g^{0}_{a\gamma}=\begin{dcases}3\alpha_{s}^{2}/\left(\pi\alpha
f_{a}\right)\,,\\\ 27/\left(4\pi\alpha f_{a}\right)\,,\end{dcases}$ (33)

is a coupling of axion to photons. For convenience, we separated some axion-
gluon interactions into $\mathcal{L}_{\text{off}}$, which is given by the
following expression:

$\displaystyle\mathcal{L}_{\text{off}}$
$\displaystyle=\,\frac{ag_{s}^{2}}{32\pi^{2}f_{a}}\times\left(G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}-\sum_{\alpha=3,8}\mathscr{G}_{\mu\nu}^{a}\tilde{\mathscr{G}}^{a\,\mu\nu}\,+\,\left(A\rightarrow
A_{(d)}\right)\right)=\qquad\qquad$
$\displaystyle-\,\frac{g^{2}_{s}\,\partial^{\mu}a}{16\pi^{2}f_{a}}$
$\displaystyle\\!\\!\epsilon_{\mu\nu\rho\lambda}\times\left\\{\sum_{i,j,k\,\in\,\mathcal{I}_{\text{off}}}\\!\\!\left(A_{i}^{\nu}\,\partial^{\,\rho}A_{i}^{\lambda}+\frac{1}{3}\,g_{s}\,f_{ijk}A_{i}^{\nu}A_{j}^{\lambda}A_{k}^{\rho}\right)+\right.$
(34)
$\displaystyle\qquad\qquad\qquad\qquad\left.\sum_{\alpha=3,8}\;\sum_{j,k\,\in\,\mathcal{I}_{\text{off}}}g_{s}f_{\alpha
jk}A_{\alpha}^{\nu}A_{j}^{\lambda}A_{k}^{\rho}\;+\,\left(A\rightarrow
A_{(d)}\right)\,\right\\}\,,$

where $\mathcal{I}_{\text{off}}=[1;7]/\\{3\\}$333By this notation we mean all
integers from 1 to 7 excluding 3.. Note that each of the interactions
presented in Eq. (5) contains two or three off-diagonal (dual) gluon fields.
Restricting our study in what follows to the field of low energy QCD, we
neglect contribution from these terms. The reason are strong indications
Suzuki:1989gp ; Stack:1994wm ; Amemiya:1998jz ; Kondo:2014sta  of Abelian
dominance in QCD below the energies of $1~{}\text{GeV}$, which means that the
processes involving off-diagonal gluons are suppressed in the IR. Moreover, as
we show in Appendix A, the term (5) is exactly zero in the classical
approximation. Let us note, however, that in the future it would be very
interesting to study if the quantum effects can generate non-zero
$\mathcal{L}_{\text{off}}$, because, although such effects are expected to be
small in IR, they would be a very distinctive feature of the model we discuss.

The effective Lagrangian Eq. (32) without the term $\mathcal{L}_{\text{off}}$
has the form of the conventional axion effective Lagrangian. As we will show,
however, the corresponding axion particle has couplings with SM particles
which differ a lot from the ones calculated in DFSZ and KSVZ models. In
particular, the coupling to photons $g_{a\gamma}$ is enhanced by many orders
of magnitude compared to the conventional models. Namely, after the standard
chiral rotation of quarks

$q\rightarrow\exp\left(i\gamma_{5}\,\frac{aM_{q}^{-1}}{2f_{a}\textrm{tr}M_{q}^{-1}}\right)\cdot
q,\quad M_{q}=\textrm{diag}\left(m_{u},m_{d}\right),$ (35)

which eliminates the $G\tilde{G}$ term, is performed, one finds that the
coupling to photons is

$g_{a\gamma}=g^{0}_{a\gamma}-\frac{\alpha}{3\pi
f_{a}}\left(\frac{4m_{d}+m_{u}}{m_{d}+m_{u}}\right)\simeq g^{0}_{a\gamma},$
(36)

so that it is practically not affected by the quark masses. In the
conventional notation used to parameterize the strength of the axion-photon
coupling,

$g_{a\gamma}=\frac{\alpha}{2\pi f_{a}}\cdot\frac{E}{N}\,,$ (37)

our model predicts

$\frac{E}{N}=\begin{dcases}6\alpha_{s}^{2}/\alpha^{2}\,,\\\
27/(2\alpha^{2})\,,\end{dcases}$ (38)

so that the coupling gets increased by 5-6 orders of magnitude. Bearing in
mind that the standard expression for the axion mass,

$m_{a}=\frac{m_{\pi}f_{\pi}\sqrt{m_{u}m_{d}}}{\left(m_{u}+m_{d}\right)f_{a}}\,,$
(39)

is derived from the conventional axion-gluon coupling and thus holds in our
case automatically as long as $\mathcal{L}_{\text{off}}$ is small, we plot the
axion-photon coupling as a function of axion mass and decay constant in Fig. 1
together with the hints and existing as well as projected constraints from
various experiments and astrophysical observations.444Hints and most
constraints are discussed in detail in Ref. DiLuzio:2020wdo . We present
updated astrophysical constraints from Ref. 2020arXiv200808100A  together with
the constraints derived from Chandra data on NGC 1275 Reynolds_2020  and
projected constraints from advanced LIGO Nagano:2019rbw . For reference, we
show axion-photon couplings in KSVZ models with heavy fermions in one
representation of the SM gauge group DiLuzio:2017pfr  and in DFSZ model.

Figure 1:  Axion-photon coupling as a function of axion mass and decay
constant for various axion models together with the existing and projected
(dashed lines) constraints on the corresponding parameter space from
experiments as well as from astrophysical data. Astrophysical hints are also
shown. The dash-dotted line corresponds to the model of this work with the
non-Abelian monopole where the IR strong coupling $\alpha_{s}$ value
calculated in the AdS/QCD framework is adopted. The line in the center of the
vertically hatched band corresponds to the model with the minimal Abelian
monopole. For further discussion, see main text.

In Fig. 1, possible values for the axion-photon coupling in the model with the
non-Abelian monopole are organized in a vertically hatched band, while the
model with the minimal Abelian monopole yields a single line inside this band.
The band denotes the uncertainty we estimate for the model with the non-
Abelian monopole, which is associated to the dependence of the first line of
Eq. (33) on the strong coupling $\alpha_{s}$ in the IR. The state of the art
in studies of the behavior of the latter was discussed in detail in a recent
review Deur:2016tte , where it was shown that there exists a definition of
$\alpha_{s}$ in the IR, which is analytic, independent of the choice of
renormalization scheme or gauge, universal, based on first principles and IR-
finite (see Table 5.4 in Ref. Deur:2016tte ). This choice of definition for IR
$\alpha_{s}$ corresponds to the so-called effective charges $\alpha_{g_{1}}$,
$\alpha_{F_{3}}$ and $\alpha_{\tau}$, which are directly related to the
observables of low energy QCD. The measurements show that the IR strong
coupling $\alpha_{s}$ defined in such a way freezes at low energies. The
freezing behavior of IR $\alpha_{s}$ is also supported by the success of the
AdS/QCD technique in the description of hadron properties Erlich:2005qh .
Moreover, the value of the IR strong coupling calculated in AdS/QCD,
$\alpha_{\text{AdS}}\left(0\right)=\pi$, is consistent with the values
$\alpha_{g_{1}}(0)$ and $\alpha_{F_{3}}(0)$ 555Although the effective charge
$\alpha_{\tau}(0)$ is different, it is known that it contains an unsubtracted
pion pole.. All this convinces us to assume that the AdS/QCD value of IR
strong coupling is a relevant one, that is why we highlight the corresponding
values of $g_{a\gamma}$ in Fig. 1 with a dash-dotted line. However, bearing in
mind that low energy QCD is still largely terra incognita, we allow for
uncertainty in $\alpha_{s}$ which results in a band in Fig 1 where the lower
edge $\alpha_{s}(0)=0.7$ is chosen. Such choice is suggested by the
observation in Ref. Deur:2016tte  that most of the values of $\alpha_{s}(0)$
in the literature are clustered around $\alpha_{s}(0)\sim 3$ (close to the
AdS/QCD value) and $\alpha_{s}(0)\sim 0.7$, not taking into account the
decoupling solution $\alpha_{s}(0)=0$ disfavored for a number of reasons
Fischer:2008uz ; Kugo:1979gm . Let us note as well that too large values of
$\alpha_{s}$ are disfavored by calculations in Ref. Liao:2008jg , where it was
shown that the magnetic coupling (i.e. the coupling inverse to $\alpha_{s}$)
never gets too small in pure $SU(2)$ gluodynamics, these results being
extended to the pure $SU(3)$ case in Ref. Bonati:2013bga .

Finally, let us mention that there is yet another source of uncertainty in our
predictions, both for the models with Abelian and non-Abelian monopoles, which
is associated with the $U(1)$ magnetic charges of the monopoles. Whereas we
consider them to be minimal in each of the model, they are in principle not
constrained by the stability arguments. This means that $g_{a\gamma}$ can be
further increased in Fig. 1 for the models of this work.

Next, let us consider axion couplings with matter, $g_{ai}\equiv
C_{ai}m_{i}/f_{a}\,$, where $m_{i}$ is the mass of fermion $i$, which
correspond to the following terms in the effective Lagrangian:

$\mathcal{L}_{\text{eff}}\supset
C_{ai}\frac{\partial_{\mu}a}{2f_{a}}\,\bar{\psi}_{i}\gamma^{\mu}\gamma_{5}\psi_{i}\,.$
(40)

As electrons do not carry PQ charge in the model we consider, the axion-
electron coupling $g_{ae}$ is generated radiatively Chang:1993gm ;
Srednicki:1985xd :

$g_{ae}=g^{0}_{a\gamma}\cdot\frac{3\alpha}{2\pi}m_{e}\ln\frac{f_{a}}{m_{e}}\,,$
(41)

where we took into account that the term associated to the axion-pion mixing
is negligibly small compared to the leading contribution. We find that the
experiments and astrophysical observations probing axion-electron interactions
do not yield new constraints on the model. Indeed, the CAST bound
Anastassopoulos:2017ftl  on the axion-photon coupling, $g_{a\gamma}<0.66\cdot
10^{-10}~{}\text{GeV}^{-1}$, constrains the phenomenologically viable region
for axion-electron coupling: $g_{ae}<1.2\cdot 10^{-16}\ln f_{a}/m_{e}$. This
constraint is stronger than any existing or projected bound from interaction
with electrons. As to the interactions of the axion with nucleons, it turns
out that contributions from radiatively generated axion-quark couplings are
non-negligible and actually enhance axion-nucleon couplings with respect to
the conventional DFSZ case in much of the parameter space. One can find that
the coefficients $C_{ap}$ and $C_{an}$ are

$\displaystyle C_{ap}=-0.47-0.39\,\delta c_{d}+0.88\,\delta c_{u}\,,$ (42)
$\displaystyle C_{an}=-0.02+0.88\,\delta c_{d}-0.39\,\delta c_{u}\,,$ (43)

where the numerical coefficients were calculated in diCortona:2015ldu  and the
radiatively generated quark couplings read as follows:

$\displaystyle\delta
c_{u}=g^{0}_{a\gamma}f_{a}\cdot\frac{8\alpha}{27\pi}\ln\frac{f_{a}}{m_{N}}\,,$
(44) $\displaystyle\delta
c_{d}=g^{0}_{a\gamma}f_{a}\cdot\frac{\alpha}{54\pi}\ln\frac{f_{a}}{m_{N}}\,,$
(45)

where $m_{N}$ is the nucleon mass. Constraints on axion-neutron interactions
are more stringent than constraints on interactions with protons. We plot
$g_{an}$ as a function of axion mass and decay constant in Fig. 2 together
with the constraint from neutron star cooling Beznogov:2018fda  and the
projected reach of the CASPEr Wind experiment JacksonKimball:2017elr . For
reference, we show the neutron-axion coupling in DFSZ models, the range of
which is constrained by the requirement of perturbative unitarity of the
Yukawa couplings of SM fermions Chen:2013kt . Note that the slope of the DFSZ
band in Fig. 2 is different from the slope of the band corresponding to the
axion model of this paper. The difference arises because, in the DFSZ case,
one obtains a linear dependence of the coupling on the axion mass,
$g_{an}\propto m_{a}$, characteristic of the tree-level couplings to quarks,
while in the case of our model the linear dependence is superseded by a
nonlinear one, $g_{an}\propto m_{a}\ln\left(\mathrm{const}/m_{a}\right)$, due
to the radiative origin of the coupling.

Figure 2:  Axion-neutron coupling as a function of axion mass and decay
constant for various axion models together with the existing and projected
(dashed lines) constraints on the corresponding parameter space from
experiments as well as from astrophysical data. The dash-dotted line
corresponds to the model of this work with the non-Abelian monopole where the
IR strong coupling $\alpha_{s}$ value calculated in the AdS/QCD framework is
adopted. The line in the center of the vertically hatched band corresponds to
the model with the minimal Abelian monopole. For further discussion, see main
text.

In Fig. 2, we show also the CAST bound Anastassopoulos:2017ftl  which is
translated to a constraint on the axion-neutron coupling with the use of Eqs.
(43-45). Uncertainty in the prediction of the axion-neutron coupling in the
axion models of this work comes from the uncertainty in the prediction of the
axion-photon coupling, the latter being discussed at length above.

Finally, let us discuss if the axions we propose can comprise dark matter. In
order to avoid the cosmological magnetic monopole problem Zeldovich:1978wj ;
Preskill:1979zi , i.e. overproduction of monopoles during the hot Big Bang
epoch, we will set their masses and therefore the axion decay constant $f_{a}$
to be larger than the reheating temperature. This means that we have to deal
only with the pre-inflationary scenario of axion dark matter production, which
hinges upon the misalignment mechanism Abbott:1982af ; Dine:1982ah ;
Preskill:1982cy . Note that while Abelian dominance suggests that the low
temperature axion mass $m_{a}(f_{a})$ is given approximately by the familiar
expression for the standard QCD axion, at higher temperatures, $T\gtrsim
1~{}\text{GeV}$, the axion mass can differ significantly from the standard
case. The cosmic axion abundance resulting from the misalignment production
mechanism $\rho_{a}^{\text{mis}}$ is inversely proportional to the square root
of the axion mass at the moment where oscillations of the axion field start:

$\rho_{a}^{\text{mis}}\propto\frac{f_{a}}{\sqrt{m_{a}(T_{\text{roll}})}}\cdot\mathcal{F}\,(T_{\text{roll}})\,,$
(46)

where $T_{\text{roll}}$ is the temperature at which
$m_{a}(T_{\text{roll}})=3H(T_{\text{roll}})$, $H$ being the Hubble expansion
rate, and $\mathcal{F}$ a fixed function of temperature. Due to Abelian
dominance, we expect that $\rho_{a}^{\text{mis}}$ does not change too much
with respect to the conventional QCD axion models if
$T_{\text{roll}}<1~{}\text{GeV}$. The latter condition can be recast into the
form $m_{a}(\text{GeV})<3H(\text{GeV})$, which yields
$f_{a}>10^{12}~{}\text{GeV}$ assuming the axion mass at $1~{}\text{GeV}$ is
not much off the values given in Borsanyi:2016ksw . Combining it with the CAST
bound, we see that in much of the allowed parameter space axions produced via
the misalignment mechanism have approximately the same abundance as axions
with the same mass in KSVZ and DFSZ-like models. The case $f_{a}\lesssim
10^{12}~{}\text{GeV}$ is more difficult: in order to infer the abundance of
cosmic axions in the model discussed, one has to calculate the axion mass as a
function of temperature in the energy range where there is no Abelian
dominance. We leave a more thorough investigation of axion cosmology in our
model for future work.

##  6 Discussion

In this work, we introduced a new hadronic axion model which involves a very
heavy vector-like fermion magnetically charged under either the full non-
Abelian symmetry of the low energy SM or only its electromagnetic subgroup. We
showed that both cases can realize the PQ mechanism and thus provide a
solution to the strong CP problem. We found that both cases lead to a very
interesting phenomenology. Although we assumed Abelian dominance in our
discussion of phenomenology of the model with the non-Abelian monopole, it is
easy to see that the functions $g_{a\gamma}(f_{a})$, $g_{ae}(f_{a})$ and
$g_{an}(f_{a})$ are independent of this assumption: the latter two couplings
are generated at 1-loop through the coupling to photons $g_{a\gamma}$ while
the axion-photon coupling is completely dominated by the $aF\tilde{F}$ term,
see Eq. (36). The quantities which are sensitive to the axion interactions
with off-diagonal gluons are the axion mass $m_{a}(f_{a})$, the coupling to
the nuclear EDM $g_{d}(f_{a})$ and, in some part of the parameter space, the
couplings with protons and pions. If there exist non-vanishing quantum
corrections to the term $\mathcal{L}_{\text{off}}$ (Eq. (5)), the model with
the non-Abelian monopole constitutes a counterexample to the assertion of
universality of axion-gluon coupling and EDM coupling in QCD axion models. If
such corrections are not too small, difference in EDM coupling with respect to
the conventional QCD axion models can offer an exciting opportunity of
distinguishing our axion model from the other QCD axion models in experiments
such as CASPEr Electric JacksonKimball:2017elr .

It is especially intriguing that the model of the QCD axion we discuss is
consistent with the astrophysical hints suggested both by anomalous TeV-
transparency of the Universe DeAngelis:2008sk ; Horns:2012fx  and by excessive
cooling of horizontal branch stars in globular clusters Ayala:2014pea , see
Fig. 1. Moreover, Fig. 1 shows that the parameter space of our model is to be
probed in the future by many experiments and astrophysical observatories,
namely ALPS II Bahre:2013ywa , BabyIAXO Abeln:2020ywv , IAXO Armengaud:2014gea
and Fermi-LAT Meyer:2016wrm . Meanwhile, advanced LIGO Nagano:2019rbw , KLASH
2017arXiv170706010A  and ABRACADABRA Kahn:2016aff  experiments have all the
chances to discover the cosmic abundance of such axions. As to the experiments
which probe the interactions of axion with neutrons, one can see in Fig. 2
that although the projected reach of the CASPEr Wind experiment is not enough
to probe the QCD axion model we propose, the gap between theory and experiment
is way smaller than in the case of DFSZ axions.

The model we discussed is peculiar in yet another way. Suppose that the axion
is found through its coupling with photons and that investigation of its EDM
coupling shows preference for the model involving non-Abelian monopole. Then
one can infer the IR strong coupling $\alpha_{s}(0)$ by Eq. (33). This would
be an independent experimental hint for the coupling constant $\alpha_{s}(0)$
which could refine other determinations.

Needless to say, it would be valuable to construct a UV completion to the
model discussed. Presence of heavy magnetic monopoles in the spectrum of the
UV theory can influence Z-boson physics, possibly providing an additional
opportunity for probing the model of this work experimentally. An interesting
question regarding the UV completion is whether the magnetic charges we
discuss can emerge from some Grand Unified theory via the ’t Hooft-Polyakov
construction tHooft:1974kcl ; Polyakov:1974ek . Note that the model of this
work requires magnetic charges to be carried by fermionic particles. The
latter can arise as systems of magnetically and electrically charged bosons
Goldhaber:1976dp , e.g. as pairs of identical dyons. Fermionic monopoles
naturally arise in supersymmetric theories.

##  Appendix A Axion-gluon coupling in the classical approximation

In this Appendix, we show that the axion-gluon coupling in the model with a
heavy non-Abelian monopole preserves its universality in the classical
approximation, i.e. it is given by the expression:

$-\,\frac{ag_{s}^{2}}{32\pi^{2}f_{a}}\;G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu},$
(47)

so that $\mathcal{L}_{\text{off}}=0\,$ in Eq. (32), at least classicaly. We
use the formalism of loop space variables pioneered by Polyakov
Polyakov:1980ca  and developed with the focus on the electric-magnetic dual
symmetry of the YM theory by Hong-Mo, Faridani and Tsun Chan:1995xr . Central
object of the formalism is the parallel phase transport along the loop
$\xi(s),\,s\in\left[0,2\pi\right]$ from one point $s_{1}$ to another $s_{2}$:

$\Phi_{\xi}(s_{2},s_{1})=P_{s}\exp{\left(ig_{s}\int_{s_{1}}^{s_{2}}ds\,A_{\mu}\left(\xi(s)\right)\dot{\xi}^{\mu}(s)\right)},$
(48)

where $P_{s}$ is the Dyson ordering. Loop derivative of the holonomy defines
the Polyakov variables:

$F_{\mu}[\xi|s]=\frac{i}{g_{s}}\Phi_{\xi}^{-1}\left(2\pi,0\right)\cdot\frac{\delta\Phi_{\xi}\left(2\pi,0\right)}{\delta\xi^{\mu}(s)}\,,$
(49)

which are known to constitute a valid set for a full description of the YM
field Chan:1985ix ; Chan:1985xc . It was shown in Ref. Chan:1995xr  that
another complete set of variables is better suited for dealing with the
electric-magnetic dual symmetry of the classical YM theory, namely:

$E_{\mu}\left[\xi|s\right]=\Phi_{\xi}\left(s,0\right)F_{\mu}\left[\xi|s\right]\Phi_{\xi}^{-1}\left(s,0\right),$
(50)

which can be connected to the local quantities by the expression:

$\omega^{-1}\\!\left(x\right)\,\tilde{G}_{\mu\nu}\,(x)\;\omega\left(x\right)=\frac{2}{N}\,\epsilon_{\mu\nu\rho\sigma}\int\delta\xi
ds\,E^{\rho}\left[\xi|s\right]\dot{\xi}^{\sigma}(s)\,\dot{\xi}^{-2}(s)\,\delta(x-\xi(s))\,,$
(51)

where $\omega\left(x\right)$ is an arbitrary local $SU(3)$ matrix and $N$ is a
normalization factor. The dual (magnetic) variables $E^{(d)}_{\mu}$ were shown
to be related to the electric ones $E_{\mu}$ in the pure YM theory in the
following way:

$\omega^{-1}\\!\left(\eta(t)\right)E^{(d)}_{\mu}\left[\eta|t\right]\,\omega\left(\eta(t)\right)=\frac{2}{N}\,\epsilon_{\mu\nu\rho\sigma}\dot{\eta}^{\nu}(t)\int\delta\xi
ds\,E^{\rho}\left[\xi|s\right]\dot{\xi}^{\sigma}(s)\,\dot{\xi}^{-2}(s)\,\delta(\xi(s)-\eta(t))\,,$
(52)

while the inverse transformation is:

$\omega\\!\left(\eta(t)\right)E_{\mu}\left[\eta|t\right]\,\omega^{-1}\left(\eta(t)\right)=-\frac{2}{N}\,\epsilon_{\mu\nu\rho\sigma}\dot{\eta}^{\nu}(t)\int\delta\xi
ds\,E^{(d)\,\rho}\left[\xi|s\right]\dot{\xi}^{\sigma}(s)\,\dot{\xi}^{-2}(s)\,\delta(\xi(s)-\eta(t))\,.$
(53)

Since in the derivation of the axion effective Lagrangian external fields can
be considered constant and homogeneous, as discussed in Sec. 4, we can apply
Eqs. (52) and (53) in order to find the relation between the expression 47 and
its dual analogue, constructed from the GNO group connection, in the classical
theory. The calculation proceeds as follows:

$\displaystyle\int
d^{4}x\,a\\!\left(x\right)G_{\mu\nu}^{a}\\!\left(x\right)\tilde{G}^{a\,\mu\nu}\\!\left(x\right)=2\int
d^{4}x\,a\\!\left(x\right)\textrm{tr}\left\\{\omega^{-1}\\!\left(x\right)G_{\mu\nu}\\!\left(x\right)\omega\left(x\right)\omega^{-1}\\!\left(x\right)\,\tilde{G}_{\mu\nu}\,(x)\;\omega\left(x\right)\right\\}=$
$\displaystyle\qquad\frac{8}{N}\int d^{4}x\,\delta\xi
ds\,a\\!\left(x\right)\textrm{tr}\left\\{\omega^{-1}\\!\left(x\right)\,\tilde{G}_{\mu\nu}\,(x)\;\omega\left(x\right)E^{\mu}\left[\xi|s\right]\right\\}\dot{\xi}^{\nu}(s)\,\dot{\xi}^{-2}(s)\,\delta(x-\xi(s))=$
$\displaystyle\frac{16}{N^{2}}\,\epsilon_{\mu\nu\rho\sigma}\\!\int\delta\eta
dt\,\delta\xi
ds\,a\\!\left(\eta(t)\right)\textrm{tr}\left\\{E^{\rho}\left[\eta|t\right]E^{\mu}\left[\xi|s\right]\right\\}\dot{\eta}^{\sigma}(t)\,\dot{\eta}^{-2}(t)\,\dot{\xi}^{\nu}(s)\,\dot{\xi}^{-2}(s)\,\delta(\eta(t)-\xi(s))=\qquad$
$\displaystyle\qquad\frac{8}{N}\int\delta\eta
dt\,a\\!\left(\eta(t)\right)\textrm{tr}\left\\{E^{\mu}\left[\eta|t\right]\,\omega^{-1}\\!\left(\eta(t)\right)E^{(d)}_{\mu}\left[\eta|t\right]\,\omega\left(\eta(t)\right)\right\\}\dot{\eta}^{-2}(t)=$
$\displaystyle\frac{8}{N}\int\delta\eta
dt\,a\\!\left(\eta(t)\right)\textrm{tr}\left\\{\,\omega\left(\eta(t)\right)E^{\mu}\left[\eta|t\right]\,\omega^{-1}\\!\left(\eta(t)\right)E_{\mu}^{(d)}\left[\eta|t\right]\,\right\\}\dot{\eta}^{-2}(t)=$
$\displaystyle-\frac{16}{N^{2}}\,\epsilon^{\mu\nu\rho\sigma}\\!\int\delta\eta
dt\,\delta\xi
ds\,a\\!\left(\eta(t)\right)\textrm{tr}\left\\{E^{(d)}_{\rho}\left[\eta|t\right]E^{(d)}_{\mu}\left[\xi|s\right]\right\\}\dot{\eta}^{\sigma}(t)\,\dot{\eta}^{-2}(t)\,\dot{\xi}^{\nu}(s)\,\dot{\xi}^{-2}(s)\,\delta(\eta(t)-\xi(s))=\qquad$
$\displaystyle\quad-\int
d^{4}x\,a\\!\left(x\right)G^{a}_{(d)}{{}_{\mu\nu}}\\!\left(x\right)\tilde{G}_{(d)}^{a\,\mu\nu}\\!\left(x\right)$
(54)

where we took advantage of Eqs. (51), (52) and 53, as well as of the cyclic
property of the trace. The last identity follows automatically as far as one
notices that the third and the sixth lines of the Eq. (A) are identical but
for the overall sign and electric-magnetic variables interchange. Now, one can
clearly see that classically we recover the universal axion-gluon coupling
even in the model with the non-Abelian magnetic monopole:

$\mathcal{S}_{\text{eff, classical}}\;\supset\;\int
d^{4}x\;\frac{ag_{s}^{2}}{32\pi^{2}f_{a}}\;G_{(d)}^{a}{{}_{\mu\nu}}\tilde{G}_{(d)}^{a\,\mu\nu}=-\int
d^{4}x\;\frac{ag_{s}^{2}}{32\pi^{2}f_{a}}\;G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}\,.$
(55)

## Acknowlegments

We thank Claudio Bonati and Thomas Biekötter for discussions. A.R.
acknowledges support and A.S. is funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC
2121 Quantum Universe – 390833306.

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