††institutetext: Rudolf Peierls Centre for Theoretical Physics, University of
Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

# The Monodromic Axion-Photon Coupling

Prateek Agrawal  and Arthur Platschorre

###### Abstract

We consider the general form of the axion coupling to photons in the axion-
Maxwell theory. On general grounds this coupling takes the form of a
monodromic function of the axion, which we call $g(a)$, multiplying the Chern-
Pontryagin density $F\widetilde{F}$ of the photon. We show that the non-
linearity of $g(a)$ is a spurion for the shift symmetry of the axion. In this
context, when $g(a)\neq\mathbb{Z}a$, the linearized coupling of the axion
$g^{\prime}(a)$ is not quantized and there is a correlated mass term for the
axion. Singularities in $g(a)$ due to the fast rearrangement of degrees of
freedom are shown to have corresponding cusps and singularities in the axion
potential. We derive the general form of $g(a)$ for the QCD axion, axions with
perturbatively broken shift symmetries and axions descending from extra
dimensions. In all cases, we show that there is a uniform general form of the
monodromic function $g(a)$ and it is connected to the axion potential.

##  1 Introduction

The axion-Maxwell Lagrangian describes the low-energy physics of one of the
most compelling new physics candidates, the axion, and its experimentally
important coupling to photons. The discovery of the axion-photon interaction
will not just be a discovery of a new particle, but can provide deep insights
into the structure of the standard model. The QCD axion elegantly explains the
non-observation of CP violation in the strong sector Peccei:1977hh ;
Weinberg:1977ma ; Wilczek:1977pj . Axions can solve the cosmological puzzle of
dark matter Preskill:1982cy ; Abbott:1982af ; Dine:1982ah  and may appear as
dark energy Marsh:2012nm ; PhysRevD.83.123526 . The axion-photon coupling can
provide access to the fundamental unit of electric charge Agrawal:2017cmd ;
Fraser:2019ojt ; Agrawal:2019lkr ; Agrawal:2020euj  and test simple models of
Grand Unification Agrawal:2022lsp . Axions have a strong interplay with ideas
in quantum gravity Heidenreich:2021yda ; Reece:2023czb  and string theory
Svrcek:2006yi ; Arvanitaki:2009fg ; Cicoli:2012sz .

A large part of this wealth of information derives from the special nature of
the axion-photon coupling and the associated symmetries and redundancies. In
this work, we derive the general form of this coupling that is ideally suited
to study the quantization of the axion-photon coupling, the physics of axion
domain walls and strings and the symmetry structure in axion-Maxwell theory.

We argue that the general low-energy axion-Maxwell Lagrangian takes the form,

$\displaystyle\mathcal{L}_{\rm axMax}$
$\displaystyle=-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}F_{a}^{2}\partial_{\mu}a\partial^{\mu}a-V(a)+\frac{g(a)}{16\pi^{2}}F_{\mu\nu}\widetilde{F}^{\mu\nu}\,.$
(1)

For convenience we have chosen a basis where $F_{a}$, the fundamental period
of the axion, as well as the electromagnetic gauge coupling $e$ are included
in the kinetic term. The function $g(a)$ is a monodromic function, defined by
the property,

$\displaystyle g(a+2\pi)$ $\displaystyle=g(a)+2\pi n\,.$ (2)

The integer $n$ is the monodromic charge of the function $g(a)$. This property
of the monodromic function arises from the discrete gauge symmetry of the
axion, $a\to a+2\pi$, under which the path integral weight, $e^{iS}$, is
required to be invariant.

It is an extremely important fact that the monodromy of $g(a)$ does not imply
that the perturbative coupling of the axion to photons around the CP
conserving point $a=0$ is quantized. Indeed, the coupling for canonically
normalized fields,

$\displaystyle g_{a\gamma\gamma}$ $\displaystyle=\frac{\alpha_{\rm em}}{\pi
F_{a}}g^{\prime}(a)|_{a=0},$ (3)

which can be an arbitrary number for a non-linear monodromic function $g(a)$.

This resolves a small puzzle in the QCD axion coupling to photons, as was
noted in Agrawal:2017cmd  and further discussed in Fraser:2019ojt . On one
hand, we usually justify the non-quantized couplings of the axion by invoking
the mixing with the pion. On the other hand, for all values of the axion the
pion remains heavy and can stay integrated out, leaving an apparent non-
monodromic function. The resolution to the non-quantization of the coupling
therefore should appear in the low-energy axion-Maxwell theory without needing
to invoke the pion. Indeed, this is achieved by a monodromic non-linear
function $g(a)$.

The form of the coupling $g(a)$ generated by the anomaly between $U(1)_{\sf
PQ}$ and $U(1)_{\rm em}^{2}$ is $g(a)=na$. This form of $g(a)$ is protected by
the continuous shift symmetry of the axion, which also protects the axion from
getting a mass. Both the potential $V(a)$ and a non-linear $g(a)$ are
therefore spurions for the axion continuous shift symmetry breaking
Agrawal:2017cmd ; Fraser:2019ojt . This gives a precise sense in which the
deviation from quantization of axion couplings and the generation of a mass
are linked. Thus, while the monodromy of $g(a)$ follows from topology, the
special case of $g(a)=na$ additionaly requires the presence of a continuous
global shift symmetry.

In general we expect the size of the two spurions for the same symmetry to be
commensurate. If the axion-photon coupling is nearly quantized, then we can
express the degree of non-quantization as

$\displaystyle g(a)-na=zf(a),$ (4)

with $z\ll 1$ and $f(a)$ an $O(1)$ periodic function. The estimate for the
mass of the axion is

$\displaystyle m^{2}$ $\displaystyle\sim z\frac{\Lambda^{4}}{F_{a}^{2}}\,.$
(5)

where $\Lambda$ is the UV cutoff of the effective theory consistent with the
coupling $g(a)$ (e.g. for the QCD axion $\Lambda\simeq\Lambda_{\rm QCD}$). We
emphasize that this is a heuristic estimate and the actual correlation may be
different in specific examples. However this correlation highlights the point
that if there is an axion that is parametrically lighter than its naively
expected mass, that also corresponds to a coupling to photons that is very
nearly an integer. Similarly, if an axion coupled to photons picks up a mass
it generically also picks up a non-linear $g(a)$ coupling to photons
Fraser:2019ojt .

The general periodic function $f(a)$ in equation (4) can be expanded in
Fourier modes. In some cases only the first few terms in the expansion
dominate. This is simply the expected contribution from axion-dependent
perturbative corrections to non-topological quantities like $\alpha_{\rm em}$.
However, in many cases, including the case of the QCD axion, the final form of
$g(a)$ requires the sum over the entire Fourier tower, and it is interesting
that a closed form for $g(a)$ can be derived.

The functional coupling $g(a)$ elucidates many interesting physics points. As
mentioned above, it captures the correct monodromy in the axion-Maxwell
Lagragian when all other fields can be integrated out for all values of the
axion. In cases where this is not possible (e.g. when some particles become
light at some value of the axion field) $g(a)$ also captures fast
rearrangement of degrees of freedom through its singularities at isolated
points. This correlates with cusps and singularities in the axion potential at
the same point, and interesting dynamics induced on an axion domain wall.

Phenomenologically, the full non-linear form of $g(a)$ is most relevant for
scenarios where the axion traverses an $O(1)$ fraction of its field range.
This is certainly true for axion strings and domain walls, and sharp features
in $g(a)$ can affect axion emission from these objects. It can also be true
for dense axion objects, like axion miniclusters or superradiant axion clouds
surrounding rotating black holes.

The fact that in many simple models the whole Fourier tower needs to be summed
up to get the relevant $g(a)$ highlights another interesting point. For
effective field theories involving compact fields the standard polynomial
basis might not be the most convenient basis to work in.

This paper is organised as follows. In section 2, the general properties of
$g(a)$ and the symmetries of the axion-Maxwell Lagrangian in the presence of
$g(a)$ are discussed, together with the connection between the mass and non-
quantization of $g(a)$. Section 3 discusses the QCD axion and the
corresponding axion-photon coupling. In section 4, the important case of case
of perturbative shift symmetry breaking is introduced and shown to share many
features of the QCD axion. The final section 5 is entirely devoted to axion
potentials and photon couplings in the presence of a tower of states.

##  2 Axion-Maxwell Theory

In this section, we discuss the general properties and symmetries of the
axion-Maxwell Lagrangian $\mathcal{L}_{\rm axMax}$ in the presence of an
effective axion-photon coupling $g(a)$.

$\displaystyle\mathcal{L}_{\rm axMax}$
$\displaystyle=-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}F_{a}^{2}\partial_{\mu}a\partial^{\mu}a-V(a)+\frac{g(a)}{16\pi^{2}}F_{\mu\nu}\widetilde{F}^{\mu\nu}\,.$
(6)

Here $F_{a}$ is the fundamental period of the axion, and we have normalized
the gauge field such that the electric charge of the electron is -1.

###  2.1 Quantization

The function $g(a)$ is a monodromic function, defined by the property,

$\displaystyle g(a+2\pi)$ $\displaystyle=g(a)+2\pi n\,.$ (7)

Here $n$ is the monodromic charge of the function $g(a)$, which is usually
taken to be an integer in order for the path integral weight $e^{iS}$ to be
invariant under the identification $a\equiv a+2\pi$.

To be more precise, the quantization of the monodromy depends on the global
structure of the gauge group Aharony:2013hda ; Tong:2017oea . If the smallest
allowed representation has physical electric charge $eq$, then the
electromagnetic instanton number $I=\frac{1}{16\pi^{2}}\int F\widetilde{F}$ is
valued in $\frac{\mathbb{Z}}{q^{2}}$. In such a model, the monodromic charge
$n$ can take values in $q^{2}\mathbb{Z}$. Correspondingly, colourless magnetic
monopoles can have a physical minimum magnetic charge $q_{m}=\frac{2\pi}{eq}$
by Dirac quantization.

The quantization of the monodromic charge $n$ can also be shown by several
connected topological arguments similar to Dirac’s argument for quantization
of electric charge in $U(1)$ gauge theory. In a theory with a $U(1)$ gauge
field and an axion discrete gauge shift symmetry, both magnetic monopoles and
axion strings exist as twisted sectors. In quantum field theory the cores of
these objects may be singular, but in presence of gravity these singularities
will be behind a horizon.

Consider a magnetic monopole with minimum magnetic charge
$q_{m}=\frac{2\pi}{eq}$ scattering on a trajectory through an axion string
loop Sikivie:1984yz . For the purposes of this thought experiment, it does not
matter if the axion has a mass or not. Along the trajectory, the monopole sees
a monodromy of the axion as $g(\theta+2\pi)-g(\theta)$. Through the Witten
effect, this implies that the monopole electric charge shifts by $\Delta
q_{e}=-\frac{q_{m}e^{2}}{4\pi^{2}}(g(\theta+2\pi)-g(\theta))=-\frac{ne}{q}$.
By Dirac-Zwanziger quantization of dyons $\Delta q_{e}\in eq\mathbb{Z}$ and
therefore the monodromic charge $n\in q^{2}\mathbb{Z}$.

There is another argument for the quantization in this example that does not
directly rely on the Witten effect. As the monopole traverses the axion string
loop, electric charge is carried from the magnetic monopole to the axion
string by the Goldstone-Wilczek current Goldstone:1981kk , which is a purely
bulk effect following from the axion-photon coupling. The current can be
integrated to calculate the total charge exchanged between the monopole and
the axion string (see e.g. Agrawal:2020euj ). The charge carriers on the axion
strings are zero modes of bulk fields (e.g. PQ fermions), and therefore are
also quantized.

In the remainder of this paper we shall focus on theories in which the
smallest unit of charge is that of the electron such that $g(a)$ has integer
monodromy, but we shall briefly return to this issue when we discuss the QCD
axion and the allowed standard model representations. Results for other
minimal charges can be recovered by the appropriate multiplication.

###  2.2 Symmetries of $\mathcal{L}_{\rm axMax}$

We begin by reviewing the symmetry structure of this theory (see e.g.
Choi:2022fgx ; Brennan:2020ehu ; Hidaka:2020iaz ; Cordova:2022ieu  for further
details) in the limit of a massless axion and $g(a)=a$.

In fact, in the even simpler limit where the axion coupling to photon is
turned off, we have the following symmetry structure. The Maxwell theory is
well-known to have two global one-form symmetries, the electric
$U(1)^{(1)}_{e}$ and magnetic $U(1)^{(1)}_{m}$, under which Wilson lines and
‘t Hooft lines transform respectively Gaiotto:2014kfa . These symmetries act
on the photon and the dual photon by a shift by a closed one-form,

$\displaystyle A$ $\displaystyle\xrightarrow{U(1)^{(1)}_{e}}A+c^{(1)},\quad
dc^{(1)}=0\,,$ (8) $\displaystyle\tilde{A}$
$\displaystyle\xrightarrow{U(1)^{(1)}_{m}}\tilde{A}+\tilde{c}^{(1)},\quad
d\tilde{c}^{(1)}=0\,.$ (9)

Equivalently, non-contractible Wilson and ‘t Hooft loops transform by a $U(1)$
phase under the respective symmetries. In the absence of charged matter, these
symmetries above are clearly symmetries of the Maxwell Lagrangian. In the real
world we know the electric symmetry to be emergent below the electron mass,
and it is strongly believed that the magnetic symmetry will also be broken
completely Heidenreich:2021xpr .

The massless axion Lagrangian has an ordinary global $U(1)^{(0)}_{\sf PQ}$
symmetry, the usual continuous shift symmetry of the massless axion,

$\displaystyle a\xrightarrow{U(1)^{(0)}_{\sf PQ}}a+c^{(0)},\quad
dc^{(0)}=0\,,$ (10)

as well as a two-form symmetry $U(1)^{(2)}$ which measures the axion winding
number, under which axion string worldsheets are charged. This symmetry is a
shift symmetry of the dual two-form field $B$,

$\displaystyle B\xrightarrow{U(1)^{(2)}}B+c^{(2)},\quad dc^{(2)}=0\,.$ (11)

or equivalently a phase rotation of a non-contractible axion string
worldsheet. The symmetry structure at this level is thus,

$\displaystyle U(1)^{(0)}_{\sf PQ}\times U(1)^{(2)}\times U(1)^{(1)}_{e}\times
U(1)^{(1)}_{m}\,.$ (12)

A linear axion-photon coupling in the Lagrangian introduces mixed anomalies
between the $U(1)^{(0)}_{\sf PQ}$ and the one-form symmetries of Maxwell
theory, as well as an ABJ anomaly Adler:1969gk ; Bell:1969ts  for the axion
shift symmetry,

$\displaystyle\partial_{\mu}j^{\mu}_{\sf PQ}$
$\displaystyle=\frac{1}{16\pi^{2}}F_{\mu\nu}\widetilde{F}^{\mu\nu}\,.$ (13)

Therefore, from this point of view it is somewhat mysterious which symmetry is
formally responsible for protecting the axion from getting a mass. One
argument could be that if we are working on $\mathbb{R}^{4}$, then there are
no Abelian instantons and the RHS does not produce any physical effect. In
particular, we can define the PQ charge on a fixed time slice,

$\displaystyle{\sf Q}$ $\displaystyle=\int d^{3}x\left(j^{0}_{\sf
PQ}-\frac{1}{8\pi^{2}}\epsilon^{ijk}A_{i}\partial_{j}A_{k}\right)\,.$ (14)

The charge $\sf{Q}$ defined on $\mathbb{R}^{3}$ is gauge invariant, so it
looks like we can rescue the shift symmetry of the axion if we are content to
work on $\mathbb{R}^{4}$.

However, this argument is a bit too quick. We cannot use the same argument for
potential UV contributions to the axion mass. The topology of spacetime seen
by the Abelian gauge field can change in the UV, both in extra-dimensional
theories and 4D theories, a simple example being the ‘t Hooft-Polyakov
monopole. It will be much more useful to find a symmetry and associated
spurions that parametrize both UV and IR mass generation effects on general
manifolds. This is especially valuable for the case of axions where we expect
at least quantum gravitational effects to generate a mass.

On a general manifold there does not exist a gauge-invariant charge $\sf{Q}$.
This is most easily seen if our spatial slice is $S^{1}\times S^{2}$ with
magnetic flux $m=\frac{1}{2\pi}\int_{S^{2}}F_{23}$ on the $S^{2}$. Performing
a large gauge transformation $A_{1}\rightarrow A_{1}+2\pi$ on the compact
$S^{1}$ shifts the charge $Q$ by $m$. The operator implementing the $U(1)_{\sf
Q}$ symmetry, $\exp(i\alpha{\sf Q})$ with $\alpha\in[0,2\pi)$ is not gauge-
invariant under this transformation. A more modern viewpoint is that the
introduction of topologically non-trivial backgrounds can be captured by
turning $U(1)^{(0)}_{\sf PQ}$ into an unbroken non-invertible symmetry
Choi:2022fgx ; Cordova:2022ieu ; Shao:2023gho . In the cases that such
background fluxes or instantons become dynamical, the symmetry is explicitly
broken and the axion is expected to get a mass. This is the case when the
axion in addition to the photon is also coupled to a non-Abelian gauge theory
or when magnetic monopoles are dynamical.

In the presence of a general effective axion-photon coupling $g(a)$, the
conservation equation (13) of $U(1)^{(0)}_{\sf PQ}$ is modified to

$\partial_{\mu}j^{\mu}_{\sf{PQ}}=\frac{g^{\prime}(a)}{16\pi^{2}}F_{\mu\nu}\widetilde{F}^{\mu\nu}$
(15)

and no such conserved PQ charge (equation (14)) exists unless $g^{\prime}(a)$
is an integer. General forms of $g^{\prime}(a)$ therefore explicitly break
$U(1)^{(0)}_{\sf PQ}$. It is this sense in which $g(a)$ can parametrize both
the UV dynamical topology changes as well as other dynamical sources of
$U(1)^{(0)}_{\sf PQ}$ breaking. The non-linearity of $g(a)$ therefore acts as
a spurion for the $U(1)^{(0)}_{\sf PQ}$ shift symmetry.

###  2.3 General properties of $g(a)$

We have seen that the general non-linear function $g(a)$ breaks the (non-
invertible) axion shift symmetry. Therefore, we expect a general connection
between $g(a)$ and the potential for the axion $V(a)$. Indeed,
$g^{\prime}(a)\notin\mathbb{Z}$ implies a mass for the axion. Similarly, a
potential for the axion $V(a)$ and a quantized axion-photon coupling will flow
to a non-quantized $g(a)$ with the same monodromy.

In the examples considered in this paper, the connection between the potential
$V(a)$ and axion-photon coupling $g(a)$ is best provided by the repackaging of
a real parameter $z$ together with the axion $a$ into a complex quantity,

$\mathcal{Z}=ze^{ia}\,.$ (16)

The real and imaginary parts of powers of $\mathcal{Z}$ respectively
contribute to the CP even potential $V(a)$ and CP odd effective axion-photon
coupling $g(a)$, providing the connection between the two. The duality
$\mathcal{Z}\rightarrow\frac{1}{\mathcal{Z}}$ leaves both the potential and
$g(a)$ invariant. Such a repackaging of the parameters in the case of
instanton contributions to the axion potential $z=e^{-S_{\mathrm{inst}}}$ was
already noted in contributions to the superpotential in Dine:1987bq .

Common to these examples is a prototypical axion-photon coupling $g(a)$ that
can be expressed as a contour integral,

$g(a)=\mathrm{Im}\int_{C}\frac{d\mathcal{Z}}{\mathcal{Z}}\frac{1-\mathcal{Z}}{1+\mathcal{Z}}\,,$
(17)

where the contour $C$ is an arc at radius $z$ of angular size $a$. The
monodromic charge $n$ can be extracted from equation (17) by the poles of the
integrand that are included in the closed contour at radius $z$. The poles for
this particular function are located at $\mathcal{Z}=0$ and $\mathcal{Z}=-1$
with respective residues $1$ and $-2$, giving a monodromic charge that is
$n=\mathrm{sign}(1-z)$.

The effective axion-photon coupling can be extracted from equation (17) by
performing the contour integral over the arc $C$,

$g(a)=2\arctan{\left(\frac{1-z}{1+z}\tan{\frac{a}{2}}\right)}+2\pi\mathrm{sign}(1-z)\Theta(a-\pi)\,,$
(18)

where $\Theta$ is the Heaviside function. The full profile of $g(a)$ is
plotted for several relevant parameter values in figure 1. The function $g(a)$
can be decomposed into a monodromic part $na$ and a periodic part, the latter
captures the explicit breaking of the continuous axion shift symmetry.

The feature most relevant to current experiments is the slope of the effective
axion-photon coupling around the minimum $a=0$ of the potential,

$g^{\prime}(0)=\frac{1-z}{1+z}\,.$ (19)

Under the transformation $z\rightarrow\frac{1}{z}$, the slope and monodromy of
$g(a)$ swap signs, which is a reflection of the $\mathcal{Z}\to
1/{\mathcal{Z}}$ duality mentioned above.

There are three values of the real parameter $z$ that are interesting. At the
points $z=\\{0,\infty\\}$, $g^{\prime}(a)\in\mathbb{Z}$ and the axion shift
symmetry is restored. In our examples, the axion potential also vanishes for
these values of $z$. The function $g(a)$ does not have a well-defined limit as
$z\to 1$, it changes discontinuously across $z=1$. In this limit,
$g^{\prime}(0)=k\in\mathbb{Z}$, but the axion shift symmetry is not restored,
and the monodromy is not equal to $k$.

Common to our examples will be the restoration of a $\mathbb{Z}_{2}$ discrete
symmetry at $z=1$, which has an anomaly with electromagnetism. The anomaly is
captured in the low-energy effective theory by $g(a)$ changing discontinuously
across $z=1$. Furthermore, the different profiles $\lim\limits_{\ {z\to
1^{-}}}g(a)$ and $\lim\limits_{\ z\to 1^{+}}g(a)$ are both discontinuous at
the point $a=\pi$, describing a fast rearrangement of degrees of freedom and
restoration of a $U(1)$ symmetry. This discontinuity in $g(a)$ at $a=\pi$ is
reproduced at the same point by a singularity in the potential $V(a)$ or its
derivatives.

Figure 1: The effective axion-photon coupling $g(a)$ for the prototypical
example in equation (18) at values $z=\\{0.01,0.5,0.99,1.01,2,100\\}$ showing
that $g(a)$ jumps across $z=1$ and further becomes discontinuous as $z\to
1^{\pm}$ at $a=\pi$.

##  3 The QCD axion

We study QCD in the two flavour approximation $N_{f}=2$ coupled to the axion
with Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle=\frac{1}{2}F_{a}^{2}(\partial
a)^{2}-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2g^{2}_{s}}\mathrm{Tr}\left(G_{\mu\nu}G^{\mu\nu}\right)$
$\displaystyle\quad+\sum_{i=1}^{2}\overline{\Psi}_{i}\left(i\not{D}-m_{i}\right)\Psi_{i}+\frac{Na}{8\pi^{2}}\mathrm{Tr}\left(G_{\mu\nu}\widetilde{G}^{\mu\nu}\right)+\frac{Ea}{16\pi^{2}}F_{\mu\nu}\widetilde{F}^{\mu\nu},$
(20)

where $F_{a}$ is the fundamental period of the axion, $E$ is the primordial
anomaly of $U(1)_{\sf PQ}$ with $U(1)_{\rm em}$ and
$N\in\frac{1}{2}\mathbb{Z}$ is the anomaly coefficient of $U(1)_{\sf PQ}$ with
QCD.111Here we have used the unfortunate standard convention making $N$ in
general half-integer.

The condition on $E$ in order for the axion to have $2\pi$ periodicity depends
on the the chosen subgroup $\Gamma=1,\mathbb{Z}_{2},\mathbb{Z}_{3}$ or
$\mathbb{Z}_{6}$ of the standard model gauge group $SU(3)\times SU(2)\times
U(1)/\Gamma$ Tong:2017oea . In this paper we take $\Gamma=\mathbb{Z}_{6}$ in
order for the electron to have the minimum quantum of electric charge. With
this choice, a sufficient condition for axion $2\pi$ periodicity is

$E-\frac{2N}{3}\in\mathbb{Z}\,.$ (21)

The axion-gluon coupling explicitly breaks the PQ shift symmetry of the axion.
We therefore expect the low-energy effective axion theory to have a potential
$V(a)$ and generate an effective axion-photon coupling $g(a)$. The symmetries,
phases and domain walls of this theory have been well-studied using the chiral
Lagrangian DiVecchia:2017xpu ; Dvali:1996xe ; Gabadadze:2000vw ;
PhysRevD.32.1560  and anomaly matching Gaiotto:2017tne ; Anber:2020xfk .

The mass for the axion and its coupling to photons have been calculated at
high precision GrillidiCortona:2015jxo ,

$\displaystyle m_{a}^{2}$
$\displaystyle=\frac{m_{\pi}^{2}f_{\pi}^{2}N^{2}}{F_{a}^{2}}\left(\frac{4}{z+\frac{1}{z}+2}+\ldots\right),\qquad
g_{a\gamma\gamma}=\frac{\alpha_{\rm em}}{\pi
F_{a}}\left(E-\frac{5}{3}N-\frac{1-z}{1+z}N+\ldots\right),$ (22)

where $z=\frac{m_{u}}{m_{d}}$ measures the isospin breaking of $SU(2)_{V}$ and
$\ldots$ denote higher order terms in the chiral Lagrangian. Note that as is
conventional we have written the coupling $g_{a\gamma\gamma}$ in the canonical
basis for both axions and photons.

The usual explanation for the irrational contribution proportional to
$\frac{1-z}{1+z}$ is that it arises from the mixing with the pion. This is
certainly true, but raises a minor puzzle. In the effective theory, we can
integrate out the pion and for all values of the axion, the pion degree of
freedom is heavy and the EFT is valid. Therefore, the quantization of the
monodromy of the axion-photon coupling should be visible in the effective
theory.

The resolution to this puzzle has been discussed in Agrawal:2017cmd  and
Fraser:2019ojt  and arises exactly through the monodromic function $g(a)$. As
we will show below, the axion coupling to photons can be packaged in this
functional form, such that $g(a)$ has integer monodromy under the axion
discrete gauge symmetry, but $g^{\prime}(0)$ can be irrational. We review the
calculation of the axion potential in the Chiral Lagrangian and derive the
form of $g(a)$ relevant for the QCD axion below.

The effective Lagrangian for the photon, the QCD axion $a$ and the pion
$\pi^{0}$, can be written as,

$\displaystyle\mathcal{L}$ $\displaystyle=\frac{F_{a}^{2}}{2}\ (\partial
a)^{2}+\frac{f_{\pi}^{2}}{2}\
(\partial\pi^{0})^{2}-V(a,\pi^{0})+{\left(E-\frac{5}{3}N\right)\frac{a}{16\pi^{2}}F\tilde{F}+\frac{\pi^{0}}{16\pi^{2}}F\tilde{F}}\,,$
(23)

with a potential $V(a,\pi^{0})$ given by

$V(a,\pi^{0})=f_{\pi}^{2}m_{\pi}^{2}\left(1-\cos{\frac{2Na}{2}}\cos{\pi^{0}}+\frac{1-z}{1+z}\sin{\frac{2Na}{2}}\sin{\pi^{0}}\right)\,.$
(24)

In this basis, the two discrete gauge symmetries involving the axion and the
pion are implemented by $(a,\pi^{0})\rightarrow(a+2\pi,\pi^{0}+2N\pi)$ and
$\pi^{0}\rightarrow\pi^{0}+2\pi$. The potential has characteristic eigenvector
directions which reverse roles when the sign of $1-z$ flips, which will be
important to our discussion throughout.

We would like to study the low-energy limit of this theory. In the limit that
$f_{\pi}\ll F_{a}$, the pion is much more massive than the axion and can be
integrated out. If this can be done consistently at every value of the axion
$a$, then axion domain walls are completely describable within the effective
field theory. There can in general be additional domain walls (perhaps
metastable or unstable) that also involve rearrangements of heavy degrees of
freedom or new massless states appearing on the domain walls. These domain
walls are not described completely within the EFT. We show that the function
$g(a)$ captures this exact behaviour.

An axion domain wall $a\rightarrow a+2\pi$ in this particular basis of the
potential (equation (24)) requires a pion domain wall
$\pi^{0}\rightarrow\pi^{0}+n\pi$ with $n\in 2N\mathbb{Z}$. For $0\leq z<1$,
the most energetically favourable domain wall is
$\pi^{0}\rightarrow\pi^{0}-2N\pi$ with the $\pi^{0}\rightarrow\pi^{0}+2N\pi$
domain wall having an additional tension $\Delta
T\propto\sqrt{|\frac{1-z}{1+z}|}$. For $1<z<\infty$, the roles of the two
domain walls are reversed.

For any $z\geq 0$, to first order in $\frac{f_{\pi}}{F_{a}}$, we can integrate
out the pion using its equation of motion,

$\frac{\partial
V}{\partial\pi^{0}}=0\implies\pi^{0}=-\arctan{\left(\frac{1-z}{1+z}\tan{\frac{2Na}{2}}\right)}-\pi\mathrm{sign}(1-z)\sum_{k=1}^{2N}\Theta\left(a-(2k-1)\frac{\pi}{2N}\right)\,.$
(25)

Here the $\Theta$ Heaviside-function ensures that the axion domain wall is
smooth and heavy pion degrees of freedom are not excited.

This yields the effective Lagrangian as,

$\mathcal{L}=\frac{F_{a}^{2}}{2}(\partial
a)^{2}-V(a)+\frac{g(a)}{16\pi^{2}}F_{\mu\nu}\tilde{F}^{\mu\nu},$ (26)

with

$\displaystyle V(a)$
$\displaystyle=-f_{\pi}^{2}m_{\pi}^{2}\sqrt{1-\frac{4z}{\left(1+z\right)^{2}}\sin^{2}\left(\frac{2Na}{2}\right)}\,,$
(27) $\displaystyle g(a)$
$\displaystyle=Ea-\frac{5}{3}Na-\arctan{\left(\frac{1-z}{1+z}\tan{\frac{2Na}{2}}\right)}-\pi\mathrm{sign}(1-z)\sum_{k=1}^{2N}\Theta\left(a-(2k-1)\frac{\pi}{2N}\right)\,.$
(28)

We see the prototypical example of the function $g(a)$ – it has a monodromy
under axion shift symmetry given by

$g(a+2\pi)=g(a)+2\pi\left(E-\frac{2N}{3}-N\left(1+\mathrm{sign}(1-z)\right)\right)\,.$
(29)

The monodromic charge
$\left(E-\frac{2N}{3}-N\left(1+\mathrm{sign}(1-z)\right)\right)\in\mathbb{Z}$
by equation (21). For the specific choice $\frac{E}{N}=\frac{8}{3}$, the
monodromy vanishes when $0\leq z<1$ and is $2N$ when $z>1$.

The slope around the axion minimum for a generic $z$ is irrational and the
axion-photon coupling at this point is given by

$g_{a\gamma\gamma}=\frac{\alpha_{\mathrm{em}}}{\pi
F_{a}}g^{\prime}(0)=\frac{\alpha_{\mathrm{em}}}{\pi
F_{a}}\left(E-\frac{5}{3}N-\frac{1-z}{1+z}N\right)\,.$ (30)

Note that both the potential $V(a)$ and $g(a)$ depend on the axion-pion mixing
parameter $z$ and that $g(a)$ is quantized exactly in the limit
$z\rightarrow\\{0,\infty\\}$ when the mass vanishes. Under the transformation
$z\leftrightarrow\frac{1}{z}$, the effective potential is left unaltered and
the monodromy of $g(a)$ changes by $2N$.

In the isospin restoring limit $z\rightarrow 1$, the potential (equation (24))
has an additional $\mathbb{Z}_{2}\subset SU(2)_{V}$ pion parity
$(-1)^{N_{\pi}}$ symmetry that sends $\pi^{0}\rightarrow-\pi^{0}$ and the
tension difference between the domain walls $\pi^{0}\rightarrow\pi^{0}\pm
2N\pi$ goes to zero. The profiles and monodromies for $\lim\limits_{\ {z\to
1^{-}}}g(a)$ and $\lim\limits_{\ z\to 1^{+}}g(a)$ differ as this
$\mathbb{Z}_{2}$ is broken by the Wess-Zumino-Witten term. Additionally, in
the limit $z\rightarrow 1$, the pion shift symmetry is restored at $a=\pi$ and
the pion becomes massless. The potential $V(a)$ has a corresponding cusp at
this point and $g(a)$ is discontinuous due to the massless pion jump. This
cusp and the discontinuity at $a=\pi$ is an accidental restoration of the pion
shift symmetry at $a=\frac{\pi}{2N}$ for $N_{f}=2$ at this order in the chiral
Lagrangian and is resolved at higher orders Smilga:1998dh .

In the next few sections, we shall see several physical systems with the same
prototypical $g(a)$ but different potentials $V(a)$.

##  4 Perturbative PQ breaking

It is very instructive to compare the breaking of the axion shift symmetry by
QCD effects to a perturbative form of shift symmetry breaking. The simplest
such model is a massive charged Dirac fermion $\Psi$ coupled to a $U(1)$ gauge
field and an axion $a$ coupled through a chiral mass term, resulting in a
Lagrangian of the form

$\mathcal{L}=i\overline{\Psi}\not{D}\Psi-f\overline{\Psi}e^{ia\gamma_{5}}\Psi-
m_{\Psi}\overline{\Psi}\Psi\,.$ (31)

This Lagrangian has an axion shift symmetry $a\rightarrow a+c$ and
$\Psi\rightarrow e^{-i\frac{c}{2}\gamma_{5}}\Psi$, which is explicitly broken
by the mass term $m_{\Psi}$. We therefore expect to generate both a $g(a)$ and
$V(a)$ and the low-energy effective Lagrangian should be of the form equation
(6). We shall see that this simple model captures a lot of the features of the
QCD axion.

###  4.1 The effective potential

In order to calculate $g(a)$ and $V(a)$ in this simple model, we compute the
effective action by integrating out the massive fermion,

$iS_{\mathrm{eff}}=\mathrm{Tr}\ln{\left[i\left(i\not{D}-m_{\Psi}-fe^{ia\gamma_{5}}\right)\right]}\,.$
(32)

This yields an effective potential for a constant axion $a$ as

$V(a)=2i\mathrm{Tr}\ln{\left[\partial^{2}+m_{\Psi}^{2}+2m_{\Psi}f\cos{a}+f^{2}\right]}\,.$
(33)

This is simply the Coleman-Weinberg potential following from a particle with
an effective mass

$m(a)^{2}=m_{\Psi}^{2}+2m_{\Psi}f\cos{a}+f^{2}=(m_{\Psi}+f)^{2}\left(1-\frac{4}{\frac{1}{z}+z+2}\sin^{2}\left(\frac{a}{2}\right)\right)\,,$
(34)

where we have defined the parameter

$z=\frac{f}{m_{\Psi}}\,.$ (35)

The Coleman-Weinberg potential $V(a)$ is

$V(a)=-c_{1}m(a)^{2}-\frac{m(a)^{4}}{16\pi^{2}}\ln{\frac{m(a)^{2}}{c_{2}^{2}}}\,,$
(36)

where $c_{1}$ and $c_{2}$ are renormalization scheme-dependent quantities.

This potential shares many features of the QCD axion. For instance, the
potential becomes axion independent when $z\rightarrow\\{0,\infty\\}$ as this
is when either $f$ or $m_{\Psi}$ are zero, the latter being the shift symmetry
restoring limit in which the axion can be rotated into a $F\widetilde{F}$
term.

In the limit $z=1$, the effective mass of the particle $m(a)$ vanishes at the
chiral symmetry restoring point $a=\pi$ and should not be integrated out. This
is reflected by a singularity in $V^{\prime\prime\prime\prime}$ at $a=\pi$.

###  4.2 The effective axion-photon coupling

Just as with the QCD axion, one expects to generate an effective axion-photon
coupling $g(a)$ from shift symmetry breaking. Such terms can be calculated by
first taking a derivative of the effective action (equation (32)) with respect
to $a$ to obtain

$\frac{\delta S_{\mathrm{eff}}}{\delta
a}=-\mathrm{Tr}\left(\frac{\gamma_{5}fe^{ia\gamma_{5}}}{i\not{D}-m_{\Psi}-fe^{ia\gamma_{5}}}\right)=\mathrm{Tr}\left(\frac{\gamma_{5}f(ie^{ia\gamma_{5}}\not{D}+m_{\Psi}e^{ia\gamma_{5}}+f)}{\not{D}^{2}+m(a)^{2}}\right)\,.$
(37)

The trace is both over spinor indices as well as the implicit momentum
integrals and this time we keep both a constant axion $a$ and a constant field
strength $F_{\mu\nu}$, such that
$\left(\not{D}\right)^{2}=D^{2}-\frac{1}{2}\sigma_{\mu\nu}F^{\mu\nu}$ with
$\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. By matching equation
(37) with the same derivative of the low-energy effective axion-Maxwell action
(Eq. (6)), we find $g^{\prime}(a)$.

Since $g^{\prime}(a)$ is CP even, we expand equation (37) to second order in
$F$ and keep only CP odd terms as

$\frac{\delta S_{\mathrm{eff}}}{\delta
a}|_{\mathrm{odd}}=\frac{1}{4}\mathrm{Tr}\left(\gamma_{5}\frac{f^{2}+m_{\Psi}f\cos{a}}{\left(D^{2}+m(a)^{2}\right)^{3}}\sigma^{\mu\nu}F_{\mu\nu}\sigma^{\alpha\beta}F_{\alpha\beta}\right)\,.$
(38)

The trace can now be reduced by using the gamma matrix identity
$\mathrm{Tr}\left(\gamma^{5}\sigma_{\mu\nu}\sigma_{\alpha\beta}\right)=-i4\epsilon_{\mu\nu\alpha\beta}$
and inserting a trace over four momenta yields

$\frac{\delta S_{\mathrm{eff}}}{\delta
a}|_{\mathrm{odd}}=2i\int\frac{d^{4}p}{(2\pi)^{4}}\frac{f^{2}+m_{\Psi}f\cos{a}}{\left(p^{2}-m(a)^{2}\right)^{3}}F\widetilde{F}\,.$
(39)

The momentum integrals are convergent and rewriting in terms of the order
parameter $z$ yields

$\frac{\delta S_{\mathrm{eff}}}{\delta
a}|_{\mathrm{odd}}=\frac{1}{16\pi^{2}}\left(\frac{z^{2}+z\cos{a}}{1+2z\cos{a}+z^{2}}\right)F\widetilde{F}\,.$
(40)

Comparing this result to the effective low-energy action of the axion
(equation (6)) allows for the matching

$g^{\prime}(a)=\frac{z^{2}+z\cos{a}}{1+2z\cos{a}+z^{2}}\,.$ (41)

Integrating this function yields the effective low energy axion-photon
coupling as

$g(a)=\frac{1}{2}a-\mathrm{arctan}\left(\frac{1-z}{1+z}\tan{\frac{a}{2}}\right)-\mathrm{sign}(1-z)\pi\Theta(a-\pi)\,.$
(42)

Similar to the potential, $g(a)$ captures many features of the symmetries of
the Lagrangian in its fully summed form. For instance, in the limit $z=\infty$
or $m_{\Psi}=0$, $g(a)$ becomes $aF\widetilde{F}$. When $z=0$, the effective
coupling to photons vanishes as $f=0$. Thus in both cases $g(a)$ becomes of
the form $\mathbb{Z}a$ when the axion becomes massless as predicted by general
symmetry arguments.

Similar to the QCD axion, in the limit $z\rightarrow 1$, there is an apparent
restoration of a $\mathbb{Z}_{2}$ symmetry that acts on the fields as

$\Psi(t,x)\rightarrow\gamma^{0}\Psi(t,-x)\qquad
A_{\mu}(t,x)\rightarrow(-1)^{\mu}A_{\mu}(t,-x)\qquad a(t,x)\rightarrow
a(t,-x)\,,$ (43)

where $(-1)^{\mu}=1$ if $\mu=t$ and $-1$ otherwise. This symmetry leaves the
Lagrangian invariant in this limit up to the change of the Chern-Pontryagin
density $F\widetilde{F}$. Correspondingly, the profiles for $\lim\limits_{\
{z\to 1^{-}}}g(a)$ and $\lim\limits_{\ z\to 1^{+}}g(a)$ differ and the
monodromies are respectively $0$ and $1$. Both profiles of $g(a)$ also have a
discontinuous jump at the chiral symmetry restoring point $a=\pi$ where the
fermion becomes massless.

##  5 Axions from extra dimensions and $g(a)$

A class of interesting axions is those that descend from gauge theories and
higher form fields in extra dimensions. These are particularly motivated both
due to the fact that they arise generically in string compactifications as
well as due to high quality global symmetry that descends from a gauge
symmetry in the bulk.

In these models, axion potentials arise from charged objects wrapping internal
cycles, which appear as instantons in the $4$D theory, see e.g. Dine:1986zy ;
Dine:1987bq ; Becker:1995kb ; Witten:1996bn ; Ooguri:1996me . Alternatively,
this potential can be thought of as arising from the axion dependence of a KK
tower of states which undergoes spectral flow as axion $a\rightarrow a+2\pi$.
A similar effect arises from a tower of dyonic states in axion-Maxwell theory
Fan:2021ntg .

In this section we bridge the relation between these two sources for the axion
potential through an instructive example. In doing so, we show that massive
charged fermions with additional compact degrees of freedom coupled to the
axion generate a similar axion potential $V(a)$ and an effective axion-photon
coupling $g(a)$.

In appendix A, we reformulate the results of this section in terms of an
effective worldline formalism of a chargded massive $4$D fermion with
additional compact degrees of freedom coupled to the axion. In doing so, we
derive the effective axion-Euler-Heisenberg Lagrangian to all orders in
constant $a$ and $F$.

###  5.1 5D instantons

We consider a $U(1)$ gauge theory with gauge field $A$ in 5D Euclidean space
($g_{\mu\nu}=\delta_{\mu\nu}$) with a massive charged fermion $\Psi$, with the
fifth dimension $y$ compactified on a circle of radius $R$,

$\displaystyle S$ $\displaystyle=\int d^{4}x\int
dy\,\left[-\frac{1}{4e^{2}}F_{MN}F^{MN}-\Psi^{\dagger}\left(\not{D}+m\right)\Psi\right]\,.$
(44)

The axion is identified with a Wilson loop $\int dyA$ around the compact extra
dimension in almost axial gauge as

$A_{5}(x,y)=\frac{a(x)}{2\pi R}\,.$ (45)

In any theory with a compact dimension, the modes of the particle can be
understood in terms of a tower of states (KK modes). In the present theory,
this leads to a description of the $5$D fermion as a tower of electrically
charged massive $4$D fermions with axion-dependent masses.

An alternative and more useful representation for our purposes is in terms of
winding modes of the fermion around the compact dimension. Non-local loops of
the fermion around the compact dimension appear as instanton effects in $4$D,
giving a mass to the axion. In such a formulation, the axion dependence of the
theory can be put into a twisted boundary condition for the fermions Arkani-
Hamed:2007ryu ,

$\displaystyle\Psi(y+2\pi nR)$ $\displaystyle\simeq e^{in(\pi-a)}\Psi(y)\,,$
(46)

in which we have also given the fermion additional anti-periodic boundary
conditions to align the minimum of the potential with $a=0$.

The Green’s function on the compactified space $G$ in the presence of an axion
can similarly be decomposed as a sum over twisted flat space Green’s functions
$D_{F}$ as

$G(x,y)=\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}D_{F}(x,y+2\pi nR)\,.$ (47)

Thus, only fermion propagators that loop around the extra dimensions are
sensitive to an axion background, and can hence generate a potential for the
axion and an effective axion-photon $g(a)$222On flat space, integrating out
$5D$ fermions generates a level $\frac{1}{2}$-Chern-Simons term. In order to
still have an axion gauge symmetry, the action therefore implicitly includes a
primordial level $\pm\frac{1}{2}$ Chern-Simons term, which won’t concern us
for the rest of the calculation. . This effect is suppressed by the small
spacelike propagator for heavy $\Psi$ to loop around the extra dimension
$z=e^{-2\pi Rm}$, such that the instanton contributions to the axion effective
action can be packaged in the complex number quantity,

$\mathcal{Z}=e^{-2\pi Rm}e^{ia}\,.$ (48)

The real and imaginary parts of powers of $\mathcal{Z}$ respectively
contribute to the CP even potential $V(a)$ and CP odd effective axion-photon
coupling $g(a)$. Similar contributions were noted to the superpotential in
Dine:1987bq .

The equivalence between a tower of states (e.g. dyons) and instantons is
exactly given by the equivalence between the KK mode and winding modes
formulation Cheng:2002iz  (of which equation (47) is a special example). The
relation between the two is provided by Poisson resummation Fan:2021ntg ,

$\sum_{n=-\infty}^{\infty}s\left(n-\frac{a}{2\pi}\right)=\sum_{k=-\infty}^{\infty}e^{-ika}S(k)\qquad,\qquad
S(k)=\int_{-\infty}^{\infty}dx\ e^{-i2\pi kx}s(x)\,.$ (49)

We proceed to calculate both the potential $V(a)$ and effective axion-photon
coupling $g(a)$ by evaluating the effective action (Eq. (44)) after
integrating out the massive charged fermions in the winding mode basis. An
alternative derivation of both $V(a)$ and $g(a)$ using KK modes can be found
in the appendix B.

###  5.2 Calculation of $V(a)$

In order to obtain an effective action for the axion and photons, we integrate
out the fermions to obtain

$e^{S_{\rm eff}[a,A]}=\int D\Psi D\overline{\Psi}\ e^{S[a,A,\Psi]}\,.$ (50)

This yields an effective action

$S_{\rm
eff}[a,A]=S[a,A]+\mathrm{Tr}\left(\log{\left(-\not{D}-m\right)}\right)\,.$
(51)

A simple way to calculate $V(a)$ is to take a derivative of the effective
action (51) with respect to a constant axion $a$ and set the photon field to
zero. This yields

$\left(2\pi R\right)\frac{\delta S_{\mathrm{eff}}}{\delta
a}\supset-\mathrm{Tr}\left(\gamma_{5}G\right)\,.$ (52)

The compact Green’s function $G$ can be expanded in terms of twisted flat
space Green’s functions as

$\mathrm{Tr}\left(\gamma_{5}G\right)=\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}\mathrm{Tr}\left(i\gamma_{5}\frac{e^{n2\pi
R\partial_{5}}}{\not{D}+m}\right)\,.$ (53)

Multiplying top and bottom by the same factor, one arrives at

$\mathrm{Tr}\left(\gamma_{5}G\right)=\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}\mathrm{Tr}\left(i\gamma_{5}\frac{e^{n2\pi
R\partial_{5}}(\not{D}-m)}{\left(\not{D}\right)^{2}-m^{2}}\right)\,.$ (54)

This allows us to calculate the $4$D potential by equation (52) as

$\frac{\partial V}{\partial
a}=4\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}\int\frac{dp^{5}}{(2\pi)^{5}}\frac{\
p_{5}e^{in2\pi Rp_{5}}}{p^{2}+(p_{5})^{2}+m^{2}}\,.$ (55)

Note that this average of momentum in 5D is non-zero due to the discrete
nature of the momenta. Integrating these out333The $n=0$ term is axion
independent and vanishes due to CP symmetry. yields

$\frac{\partial V}{\partial
a}=-4\sum_{n=1}^{\infty}(-1)^{n}\sin{\left(na\right)}\int\frac{dp^{4}}{(2\pi)^{4}}e^{-n|2\pi
R|\sqrt{p^{2}+m^{2}}}\,.$ (56)

Integrating over momenta and with respect to the axion $a$ yields the
effective potential

$V(a)=\frac{m^{2}}{(2\pi
R)^{2}}\sum_{n=1}^{\infty}\frac{1}{\pi^{2}n^{3}}e^{-n2\pi|Rm|}(-1)^{n}\cos{\left(na\right)}\left(1+\frac{3}{2\pi|Rm|n}+\frac{3}{\left(2\pi
Rmn\right)^{2}}\right)\,.$ (57)

This potential is known as the one generated by a four-dimensional particle
with a rotor degree of freedom coupled to the axion Fan:2021ntg  and was also
discussed in various limits in Arkani-Hamed:2007ryu ; Sundrum:2005jf ;
Cheng:2002iz ; Arkani-Hamed:2003xts . Various other representations of this
potential are recorded in the appendices in equation (A.12) and equation
(B.2).

We see that for the 5D instantons, the spurion is parametrized by the
parameter $z=e^{-2\pi Rm}$ with the symmetry $z\leftrightarrow\frac{1}{z}$
leaving the potential invariant and implementing the $-m$ to $m$ domain wall.

At the symmetric point $z=1$, the 5D fermion becomes massless and the
Lagrangian has an apparent $\mathbb{Z}_{2}$ ($5$D parity) symmetry, which is
broken by the topological Chern-Simons term. In this limit, at the point
$a=\pi$, the lightest $4$D fermion in the tower becomes massless and a $4$D
$U(1)$-chiral symmetry is restored, meaning that this fermion should not have
been integrated out. This is reflected by a singularity of $V^{\prime\prime}$
at $a=\pi$.

###  5.3 Calculation of $g(a)$

In this section, we calculate the effective axion-photon coupling resulting
from a charged massive fermion with additional compact degrees of freedom
coupled to the axion. The existence of such a $g(a)$ was already well-known in
the context of finite temperature field theory in various dimensions see e.g.
Dunne:1996yb ; Sisakian:1997cp  and references therein.

We can calculate $g(a)$, the effective axion-photon coupling, by taking a
derivative of the effective action (equation (51)) with respect to the axion,
which will contain a term of the form

$i\frac{g^{\prime}(a)}{16\pi^{2}}F\tilde{F}\subset(2\pi R)\frac{\delta
S_{\mathrm{eff}}}{\delta a}\,.$ (58)

The calculation of $g(a)$ will proceed along similar lines as in section 4. We
keep both a background constant axion $a$ and a constant background field of
the zero KK mode of the photon $F^{(0)}_{\mu\nu}$ which we simply write as
$F_{\mu\nu}$. In these circumstances,
$\left(\not{D}\right)^{2}=D^{2}-\frac{1}{2}\sigma_{\mu\nu}F^{\mu\nu}$.

Returning to the effective action in equation (52) and keeping only the
relevant CP odd terms as

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta
a}|_{\mathrm{odd}}=mi\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}\mathrm{Tr}\left(\gamma_{5}\frac{\
e^{n2\pi R\partial_{5}}}{\left(\not{D}\right)^{2}-m^{2}}\right)\,.$ (59)

Expanding this to second order in $F$, one obtains

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta
a}|_{\mathrm{odd}}=i\frac{m}{4}\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}\mathrm{Tr}\left(\gamma_{5}\frac{\
e^{n2\pi
R\partial_{5}}}{D^{2}-m^{2}}\sigma^{\mu\nu}F_{\mu\nu}\sigma^{\alpha\beta}F_{\alpha\beta}\right)\,.$
(60)

Using the identity
$\mathrm{Tr}\left(\gamma^{5}\sigma_{\mu\nu}\sigma_{\alpha\beta}\right)=-4\epsilon_{\mu\nu\alpha\beta}$
one arrives at

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta
a}|_{\mathrm{odd}}=2mi\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}\int\frac{dp^{5}}{(2\pi)^{5}}\frac{\
e^{in2\pi Rp^{5}}}{\left((p^{5})^{2}+p^{2}+m^{2}\right)^{3}}F\widetilde{F}\,.$
(61)

The integrals over momenta are convergent. Performing the integrals yields

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta
a}|_{\mathrm{odd}}=\frac{i}{32\pi^{2}}\mathrm{sign}(m)\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}e^{-2\pi|Rmn|}F\widetilde{F}\,.$
(62)

By comparing with equation (58), we find that

$g^{\prime}(a)=\frac{\mathrm{sign}(m)}{2}\sum_{n=-\infty}^{\infty}e^{in(a+\pi)}e^{-2\pi|Rmn|}\,.$
(63)

This sum can be explicitly calculated and yields a $g^{\prime}(a)$ of the form

$g^{\prime}(a)=\frac{1}{2}\frac{\sinh{2\pi Rm}}{\cosh{2\pi Rm}+\cos{a}}\,.$
(64)

Integrating this with respect to $a$ yields our final result for $g(a)$ and
adding a primordial $\pm\frac{1}{2}$-level Chern-Simons term (see foonote 2)
yields

$g(a)=\pm\frac{1}{2}a+\arctan{\left(\frac{1-z}{1+z}\tan{\left(\frac{a}{2}\right)}\right)}+\pi\mathrm{sign}(1-z)\Theta(a-\pi)\,.$
(65)

In the limit $R\rightarrow\infty$, this reproduces the well-known result,

$g(a)=\frac{1}{2}\left(\pm 1+\frac{m}{|m|}\right)a\,.$ (66)

Similar to the QCD axion, there is an apparent $\mathbb{Z}_{2}$ ($5$D parity)
restoration as $m\rightarrow 0$ or $z\rightarrow 1$ in equation (44). The
profiles and monodromies of $\lim\limits_{\ {z\to 1^{-}}}g(a)$ and
$\lim\limits_{\ z\to 1^{+}}g(a)$ differ however due to the gauge-parity
anomaly. In the same limit $z\rightarrow 1$, there is a jump in both profiles
of $g(a)$ at $a=\pi$ due to the lightest fermion in the tower becoming
massless.

In the 5D theory, a domain wall describing the $-m\rightarrow m$ transition
has a massless chiral fermion on it and describes anomaly inflow consistent
with our $4$D analysis.

##  6 Discussion

We have considered the general properties of the monodromic axion-photon
coupling $g(a)$ and the symmetries of the low-energy axion-Mawell Lagrangian
in the presence of such a coupling. We argued that the non-quantization of
$g^{\prime}(a)$ is a spurion for the axion shift symmetry. The connection
between the axion potential and this coupling has been considered supported by
several examples including the QCD axion, perturbative shift symmetry breaking
and fermions with additional compact degrees of freedom. In all such cases, a
prototypical monodromic function $g(a)$ was derived and could be expanded in
terms of a linear monodromic function with same monodromic charge as $g(a)$
and a periodic function. In some cases only the first few terms in the
expansion of the periodic function dominated. However, in many simple models
the whole Fourier tower needs to be summed up to get the relevant $g(a)$. In
such cases $g(a)$ captured the rearrangement of heavy degrees of freedom
through its singularities at isolate points. This correlated with cusps and
singularities in the axion potential.

There are a number of model building applications of this formulation. Instead
of building effective field theories with polynomial axion couplings, more
general non-linear couplings can arise naturally through the $g(a)$ portal.
This may have interesting avenues for constructing more general natural
potentials for axions. Phenomenologically, the full non-linear form of $g(a)$
is most relevant for scenarios where the axion traverses an $O(1)$ fraction of
its field range. This is certainly true for axion strings and domain walls,
and sharp features in $g(a)$ can affect axion emission from these objects. It
can also be true for dense axion objects, like axion miniclusters or
superradiant axion clouds surrounding rotating black holes.

It will be interesting to study the effective photon coupling for mesons in
the chiral Lagrangian, e.g. for the pion $g(\pi^{0})$ after integrating out
$\eta^{\prime}$. It has been shown Komargodski:2018odf  in the context of a
one-flavor QCD $N$ that degrees of freedom rearrange on the
$\eta^{\prime}\rightarrow\eta^{\prime}+2\pi$ domain wall, leading to a
fractional quantum hall droplet and a potential jump in $g(\eta^{\prime})$. It
will be nice to see this physics captured within the effective field theory.

Lastly, several contributions to $g(a)$ could be considered in the presence of
CP-odd sources of axion shift symmetry breaking such as magnetic monopoles
(with fermions) and non-Abelian instantons.

_Note added_ : While this manuscript was being finalized we became aware of
other studies appearing today Reece:2023iqn ; Choi:2023pdp  which also
consider quantization of the axion-gauge couplings. The main focus of these
works is different from the non-linear coupling to photons highlighted in this
paper.

###### Acknowledgements.

We would like to thank Mario Reig for useful comments on the draft and John
March-Russell, Michael Nee and Thomas Harvey for helpful discussions. PA is
supported by the STFC under Grant No. ST/T000864/1. AP is supported by a STFC
Studenship No. 2397217 and Prins Bernhard Cultuurfondsbeurs No. 40038041 made
possible by the Pieter Beijer fonds and the Data-Piet fonds.

##  Appendix A Axion-Euler-Heisenberg

In this appendix we connect our results back to those of a $4$D worldline
particle with a compact additional degrees of freedom coupled to the axion. We
do this using the Schwinger proper time formalism and consider a fermionic
particle with electrically charged translational modes $x^{\mu}$ and a compact
additional degree of freedom $q$ which is coupled to the axion $a$. In doing
so, we show how a potential $V(a)$ (equation (57)) and effective axion-photon
coupling $g(a)$ (equation (65)) arise in such a wordline formalism. This will
provide us with the low energy effective axion Maxwell field theory to all
orders in $a$ and $F$ by studying the axion-Euler-Heisenberg Lagrangian
resulting from integrating out such fermions. Axial couplings to the
worldlines of particles have been studied in Mondragon:1995ab ; DHoker:1995aat
. The effective Euler-Heisenberg Lagrangian following from loops of fermions
with couplings to non compact psuedoscalar particles has been studied in
Jacobson:2018kso .

We can derive the $4$D wordline formalism of a fermion with additional compact
degrees of freedom coupled to the axion by starting from the $5$D effective
action (51), repeated here for completeness,

$S\supset\mathrm{Tr}\left(\ln\left(-\not{D}-m\right)\right)\,.$ (A.1)

The presence of a psuedoscalar (the axion) implies that the Euclidean
effective action has both a real and imaginary part as the operator $\not{D}$
no longer has a positive definite spectrum. For this reason, the contributions
to the effective action are split into a real and imaginary part as

$S\supset\mathrm{Tr}\left(\ln\left|\not{D}+m\right|\right)+i\mathrm{Tr}\left(\mathrm{Arg}\left(-\not{D}-m\right)\right)\,.$
(A.2)

For our purposes, this split is done by taking a derivative of the effective
action (A.1) with respect to $m^{2}$ as

$\frac{dS}{dm^{2}}\supset\frac{1}{2}\mathrm{Tr}\left(\frac{1}{-\not{D}^{2}+m^{2}}\right)-\frac{1}{2m}\mathrm{Tr}\left(\frac{\not{D}}{-\not{D}^{2}+m^{2}}\right)\,.$
(A.3)

The first term in equation (A.3) can be reformulated using standard techniques
PhysRev.80.440 ; Corradini:2015tik  in terms of a wordline effective action
for a fermion with $4$ translation degrees of freedom $x^{\mu}$ and one
additional compact degree of freedom $q$ as

$S\supset-\frac{1}{2}\int_{0}^{\infty}\frac{ds}{s}e^{-sm^{2}}\int
Dqe^{-\int_{0}^{s}d\tau\left(\frac{1}{4}\dot{q}^{2}-i\dot{q}\frac{a}{2\pi}\right)}\int
Dxe^{-\int_{0}^{s}d\tau\left(\frac{1}{4}\dot{x}^{2}-i\dot{x}_{\mu}A^{\mu}\right)}\mathrm{Spin}[x,A]\,.$
(A.4)

Here we have isolated the compact fifth degree of freedom $q$ of the fermion
that is coupled to the axion and the spin factor is given by

$\mathrm{Spin}[x,A]=\mathrm{Tr}P\mathrm{exp}\left[-\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]\int_{0}^{s}d\tau
F_{\mu\nu}(x(\tau))\right]\,.$ (A.5)

From the effective action it is clear that this term will generate an
effective potential $V(a)$ for the axion $a$ and an effective Euler-Heisenberg
Lagrangian, which shall be calculated in the next section. This effective
potential was studied for loops of bosonic particles with additional rotor
degree of freedoms coupled to the axion in reference Fan:2021ntg .

Importantly however, the second term in equation (A.3) does not vanish when an
axial coupling is present. This term generates an effective axion photon
coupling. We thus see that loops of fermionic particles with an additional
compact degree of freedom $q$ and charged translation modes $x^{\mu}$ can
generate an effective $F\widetilde{F}$ coupling when this additional degree of
freedom is coupled to the axion.

We now proceed to calculate both the real and imaginary contributions (Eq.
(A.2)) to the effective action resulting in an effective axion-Euler-
Heisenberg Lagrangian. We shall do this calculation in terms of the KK mode
decomposition with frequencies $\omega_{n}=\frac{\pi}{2\pi R}(2n+1)$ and
$n\in\mathbb{Z}$.

The first term in equation (A.3) can be rewritten using Schwinger-proper time
as

$\frac{1}{2}\mathrm{Tr}\left(\frac{1}{-\not{D}^{2}+m^{2}}\right)=\frac{1}{2}\int_{0}^{\infty}ds\
e^{-sm^{2}}\mathrm{Tr}\left[\braket{x}{e^{s\not{D}^{2}}}{x}\right]\,.$ (A.6)

We proceed by splitting the covariant derivative
$(\not{D})^{2}=(\not{D}_{4})^{2}+(\not{D}_{5})^{2}$ into the $4$D covariant
derivative $\not{D}_{4}$ and the covariant derivative over the fifth dimension
$\not{D}_{5}$ and have taken the axion $a$ and field strength $F$ to be
constant.

The trace over the $4$D covariant derivative in the presence of a constant
field strength $F$ can be calculated using the well-known identity in $4$D
Dunne:2004nc ; Heisenberg:1936nmg ,

$\mathrm{Tr}\left[\braket{x}{e^{\not{D}^{2}_{4}s}}{x}\right]=\frac{1}{64\pi^{2}}\frac{F\widetilde{F}}{\mathrm{Im}\cosh{\left(sX\right)}}\mathrm{Tr}\left[\mathrm{exp}\left(-\frac{s}{2}\sigma_{\mu\nu}F^{\mu\nu}\right)\right]\,,$
(A.7)

and the trace identity as

$\mathrm{Tr}\left[\mathrm{exp}\left(-\frac{s}{2}\sigma_{\mu\nu}F^{\mu\nu}\right)\right]=4\
\mathrm{Re}\cosh{sX}\,,$ (A.8)

with $X$ given by

$X\equiv\sqrt{\frac{1}{2}F^{2}+\frac{i}{2}F\widetilde{F}}\,.$ (A.9)

We can now calculate the action $S$ by plugging these identities into equation
(A.6) and integrating with respect to $m^{2}$ (equation (A.3)) to obtain the
well-known formula for the Euler-Heisenberg Lagrangian. In case of a constant
axion $a$ and field strength $F_{\mu\nu}$ this is

$\mathcal{L}\supset-\frac{1}{32\pi^{2}}\int_{0}^{\infty}\frac{ds}{s}e^{-sm^{2}}\frac{\mathrm{Re}\cosh{sX}}{\mathrm{Im}\cosh{sX}}F\widetilde{F}\frac{1}{\left(2\pi
R\right)}\sum_{n=-\infty}^{\infty}e^{-\left(\omega_{n}-\frac{a}{2\pi
R}\right)^{2}s}\,.$ (A.10)

By expanding this formula in powers of $s$, we can find an alternative
integral representation for the potential of the axion. Observe that to second
order in $s$,

$\frac{\mathrm{Re}\cosh{sX}}{\mathrm{Im}\cosh{sX}}F\widetilde{F}=\frac{4}{s^{2}}+\frac{2}{3}F^{2}+\mathcal{O}(s^{2})\,.$
(A.11)

We recognize the first term as the vacuum energy contribution to the Euler-
Heisenberg Lagrangian. This yields an alternative integral representation for
the potential of the axion of the form

$V(a)=\frac{1}{8\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{3}}e^{-sm^{2}}\sum_{n=-\infty}^{\infty}e^{-\left(\omega_{n}-\frac{a}{2\pi
R}\right)^{2}s}\,,$ (A.12)

and can be rewritten in terms of instanton supressed contributions using
Poisson resummation Fan:2021ntg .

We now proceed to calculate the imaginary part of the effective action
(equation (A.2)) by calculating the contribution of the second term in
equation (A.3). In doing so, we recover an alternative representation for
$g(a)$ and complete the axion-Euler-Heisenberg Lagrangian.

The second term in equation (A.3) can be calculated in a similar manner using
Schwinger proper time

$-\frac{1}{2m}\mathrm{Tr}\left(\frac{\not{D}}{-\not{D}^{2}+m^{2}}\right)=-\frac{1}{2m}\int_{0}^{\infty}ds\
e^{-sm^{2}}\mathrm{Tr}\left[\braket{x}{\gamma_{5}\left(\partial_{5}-i\frac{a}{2\pi
R}\right)e^{\not{D}^{2}s}}{x}\right]\,.$ (A.13)

This term is non-zero due to the discrete nature of the momenta of the
additional degree of freedom.

This term can now be straightforwardly calculated by again splitting the
covariant derivative and using
$(\not{D}_{4})^{2}=D^{2}-\frac{1}{2}\sigma_{\mu\nu}F^{\mu\nu}$ and the trace
identity identity

$\mathrm{Tr}\left[\gamma_{5}\mathrm{exp}\left(-\frac{s}{2}\sigma_{\mu\nu}F^{\mu\nu}\right)\right]=-4\
\mathrm{Im}\cosh{sX}\,.$ (A.14)

Using this latter trace identity and our expression for the $4$D propagator
(equation (A.7)), we can calculate the imaginary contribution to the effective
action as

$\frac{d\mathcal{L}}{dm^{2}}\supset\frac{i}{32\pi^{2}m}\int_{0}^{\infty}ds\
e^{-sm^{2}}F_{\mu\nu}\tilde{F}_{\mu\nu}\sum_{n=-\infty}^{\infty}\left(\omega_{n}-\frac{a}{2\pi
R}\right)e^{-\left(\omega_{n}-\frac{a}{2\pi R}\right)^{2}s}\,.$ (A.15)

Performing the integral with respect to $s$ and then the integral with respect
to $m$, one obtains the contribution to the effective action

$\mathcal{L}\supset\frac{i}{16\pi^{2}}F_{\mu\nu}\widetilde{F}_{\mu\nu}\sum_{n=-\infty}^{\infty}\arctan{\left(\frac{m}{\omega_{n}-\frac{a}{2\pi
R}}\right)}\,.$ (A.16)

By taking a derivative with respect to $a$, we arrive at expression in
equation (B.8) and thus find another representation of $g(a)$ as an infinite
series.

The sum of equations (A.10) and (A.16) provides us with an alternative
worldline formulation resulting in an effective axion-Euler-Heisenberg
Lagrangian providing us with the effective low energy axion Maxwell theory
Lagrangian to all orders in the axion and photon.

##  Appendix B Kaluza-Klein calculation

In this appendix we re-derive the axion potential $V(a)$ (equation (57)) and
the effective axion coupling $g(a)$ (equation (65)) for a tower of fermionic
states in the Kaluza-Klein decomposition basis. We take the fermion to satisfy
anti-periodic boundary conditions such that $\omega_{n}=\frac{\pi}{2\pi
R}\left(2n+1\right)$ in order to align the minimum of the potential with
$a=0$. The axion potential for other boundary conditions can always be
obtained by shifting the axion $a$.

###  B.1 The axion potential

The axion gains an effective potential from interactions with the fermion. We
can calculate this effective potential by taking the axion $a$ to be constant,

$V(a)=-2\sum_{n=-\infty}^{\infty}\int\frac{d^{4}p}{(2\pi)^{4}}\ln{\left(\left(\omega_{n}-\frac{a}{2\pi
R}\right)^{2}+p^{2}+m^{2}\right)}\,.$ (B.1)

The potential in equation (B.1) is well-known Sundrum:2005jf  and the
resulting potential for the axion is

$V(a)=-2\int\frac{d^{4}p}{(2\pi)^{4}}\ln{\left(1+e^{-4\pi RE_{p}}+2e^{-2\pi
RE_{p}}\cos{a}\right)}\,.$ (B.2)

###  B.2 The effective axion-photon coupling

We can calculate $g(a)$, the effective axion-photon coupling, by taking a
derivative of the effective action (equation (51)) with respect to the axion,
which will contain a term of the form

$i\frac{g^{\prime}(a)}{16\pi^{2}}F\tilde{F}\subset(2\pi R)\frac{\delta
S_{\mathrm{eff}}}{\delta a}\,.$ (B.3)

We keep both a constant axion $a$ and constant zero KK mode of the photon
$F_{\mu\nu}$.

Proceeding with the calculation, from equation (51), we have that

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta
a}\supset-\mathrm{Tr}\left(i\gamma^{5}\frac{1}{\not{D}+m}\right)\,.$ (B.4)

For a constant axion we can expand the denominator as

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta
a}\supset-\mathrm{Tr}\left(i\gamma^{5}\frac{\not{D}-m}{D^{2}-\frac{1}{2}\sigma^{\mu\nu}F_{\mu\nu}-m^{2}}\right)\,.$
(B.5)

We proceed to expand the denominator to order $F^{2}$ as

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta
a}\supset-\frac{1}{4}\mathrm{Tr}\left(i\gamma^{5}\left(\not{D}-m\right)\left(\frac{1}{D^{2}-m^{2}}\right)^{3}\sigma^{\mu\nu}F_{\mu\nu}\sigma^{\alpha\beta}F_{\alpha\beta}\right)\,.$
(B.6)

Using the identity
$\mathrm{Tr}\left(\gamma^{5}\sigma_{\mu\nu}\sigma_{\alpha\beta}\right)=-4\epsilon_{\mu\nu\alpha\beta}$
and ignoring higher order $F$ contributions, we find that

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta a}\supset\frac{2im}{2\pi
R}F\widetilde{F}\sum_{n=-\infty}^{\infty}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{\left(\left(\omega_{n}-\frac{a}{2\pi
R}\right)^{2}+k^{2}+m^{2}\right)^{3}}\,.$ (B.7)

Performing the momentum integral yields

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta a}\supset\frac{im}{2\pi
R}F\widetilde{F}\sum_{n=-\infty}^{\infty}\frac{1}{16\pi^{2}}\frac{1}{\left(\omega_{n}-\frac{a}{2\pi
R}\right)^{2}+m^{2}}\,.$ (B.8)

The frequency sum can be done by a method of images as

$\frac{1}{2\pi
R}\sum_{n=-\infty}^{\infty}\frac{1}{\left(\omega_{n}-\frac{a}{2\pi
R}\right)^{2}+m^{2}}=\frac{1}{2|m|}\frac{\sinh{\left(2\pi
R|m|\right)}}{\cosh{\left(2\pi Rm\right)}+\cos{\left(a\right)}}\,.$ (B.9)

Plugging this into equation (B.8), one obtains

$(2\pi R)\frac{\delta S_{\mathrm{eff}}}{\delta a}\supset
i\frac{\sinh{\left(2\pi R|m|\right)}}{\cosh{\left(2\pi
Rm\right)}+\cos{\left(a\right)}}\frac{\mathrm{sign}(m)}{32\pi^{2}}F\tilde{F}\,.$
(B.10)

By comparing this expression with equation (B.3), we see that

$g^{\prime}(a)=\frac{1}{2}\frac{\sinh{\left(2\pi Rm\right)}}{\cosh{\left(\pi
Rm\right)}+\cos{\left(a\right)}}\,,$ (B.11)

which implies

$g(a)=\arctan{\left(\tanh{\left(\pi
Rm\right)}\tan{\left(\frac{a}{2}\right)}\right)}+\pi\mathrm{sign}(m)\Theta(a-\pi)\,.$
(B.12)

This function has the correct $\frac{1}{2}$-level Chern Simons.

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