# A mass for the dual axion

Arthur Platschorre<EMAIL_ADDRESS>Rudolf Peierls Centre
for Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK

###### Abstract

In this article we study a modification of axion physics in which the dual
axion acquires a mass. This mass explicitly breaks the shift symmetry of the
dual axion. The potential breaking of this shift symmetry poses a dual axion
quality problem. When the dual axion acquires a mass, the axion gets eaten and
becomes the longitudinal degree of freedom of a massive vector field. In this
phase, axion strings are screened and far-separated instanton configurations
are exponentially suppressed. This confinement of instantons corresponds to
the worldline action of a particle-like soliton traveling between the
instantons analogous to Abrikosov/Nielsen-Oleson vortex solitons that stretch
between confined magnetic monopoles in a superconductor. We calculate the cost
of this additional worldline suppression and provide several models in which
both the confined instantons and confining worldline are dynamical.

##  I Introduction

Among the many beyond the Standard Model (BSM) extensions, the axion, a
(pseudo)-Nambu-Goldstone boson, has received ever-growing attention in recent
decades. The QCD axion would provide a simple explanation of the strong CP
problem Peccei:1977hh ; Wilczek:1977pj ; Weinberg:1977ma , axions would
provide an elegant solution to several cosmological questions such as the
nature of dark matter Preskill:1982cy ; Abbott:1982af ; Dine:1982ah  or dark
energy Marsh:2012nm ; PhysRevD.83.123526  and axions are expected to be
ubiquitous in string theory compactifications Svrcek:2006yi ;
Arvanitaki:2009fg .

Most of the desired properties of an axion rely on the existence of an
(approximate) continuous shift symmetry, making the axion generically very
light. In quantum gravity, such global symmetries are expected to be absent
Banks:1988yz ; Kallosh:1995hi ; Banks:2010zn . The axion’s shift symmetry is
expected to be either broken or gauged.

The breaking of the axion’s shift symmetry by sensitivities to UV physics and
a potential loss of control over the axion mass provides a quality problem for
parametrically light axions. Standard arguments for the QCD axion require such
(misaligned) sources to have either extremely suppressed couplings or to be
generated at a (very) high order in operators (for a recent review see
Hook:2018dlk ). This has been dubbed the QCD axion quality problem
Kamionkowski:1992mf ; Dine:1986bg .

Recent investigations have re-examined the QCD axion quality problem from the
perspective of the dual of the QCD axion Kallosh:1995hi ; Quevedo:1995ep ;
Burgess:2023ifd ; Dvali:2022fdv ; Sakhelashvili:2021eid ; Choi:2023gin , the
associated two-form Kalb-Ramond field Kalb:1974yc . From this dual
perspective, breaking the axion shift symmetry corresponds to a gauged shift
symmetry of the dual axion.

In this article, we consider the second option allowed by quantum gravity, a
gauged shift symmetry of the axion. The axion gets ‘eaten’ and becomes the
longitudinal degree of freedom of a massive vector field. Generically, such
axions arise in string theory in anomaly cancellation mechanisms in either
compactifications of heterotic string theory Svrcek:2006yi  or type II
intersecting brane models Ibanez:2001nd ; Blumenhagen:2006ci ;
Antoniadis:2002qm , resulting in massive vector fields with a stringy mass
scale.

In the presence of this massive vector field, the vacuum acts as a
superconductor. Electric field lines can end in the vacuum and field lines
emanating from electrically charged particles are screened at long distances.
Magnetic field lines are forced into flux tubes and magnetic monopoles are
confined.

This motivates a general question: What happens to instantons and strings
coupled to the axion in this superconducting phase?

As the axion’s degree of freedom is gauged, we turn to the dual description.
From the dual perspective, a gauged shift symmetry of the axion corresponds to
a mass for the dual axion. We will briefly review this dual perspective and
potential mass generating mechanisms in Section II. A potential loss of
control over the dual axion mass due to sensitivities to UV physics presents a
dual axion quality problem and dramatically alters the interactions of the
axion with axion strings and instantons.

Strings and instantons coupled to the axion are respectively ‘electrically’
and ‘magnetically’ charged under the dual axion. Analogous to a massive vector
field, a massive dual axion implies that the ‘electric flux’ lines of the dual
axion can end in the vacuum and the dual axion profile around strings is
screened at large distances. The dual axion’s ‘magnetic flux’ lines
surrounding instantons are forced into ‘flux tubes’ and instantons are
confined. This phase should be contrasted to a phase in which instead the
axion is massive; strings are confined by axion domain walls and the axion
field around instantons is screened at large distances.

These effects are a natural consequences of gauging the axion shift symmetry.
The axion profile around axion strings is compensated by that of the vector
field and the strings become non-interacting at large distances. Interactions
with their environment can come from coupling to other long-range fields such
as electromagnetism or gravity. Their phenomenology is that of gauge strings
and we connect back to this point of view in Section III.

The gauged axion shift symmetry also implies that instantons coupled to the
axion exist only in charge neutral dipole configurations and any far-separated
instanton configurations are exponentially suppressed. The absence of isolated
instantons in such theories was already noted in Heidenreich:2020pkc ;
Heidenreich:2021xpr . In Section IV, we show that this confinement of
instantons corresponds to the worldline action of a particle-like soliton
traveling between the instantons. This soliton is analogous to
Abrikosov/Nielsen-Oleson vortex solitons that stretch between confined
magnetic monopoles in a superconductor. We proceed to calculate the cost of
this additional worldline suppression.

The particle-like soliton is electrically charged under the massive vector
field. This completes the duality of the superconductor; magnetic monopoles of
the massive vector field are confined by ‘electrically’ charged strings under
the massive dual axion. The ‘magnetically’ charged instantons of the dual
axion are confined by electrically charged particles under the massive vector
field.

The confinement of instantons and their contributions to the path integral
presents interesting model-building opportunities. A non-exhaustive list of
models in which both the confined instantons and confining worldline are
dynamical is provided in Section IV. These include adaptations of models of
particle confinement and the Green-Schwarz mechanism Green:1984sg . We comment
on the required spectrum of particles in such simple models, the connection to
the massless up quark solution and the relevance for the strong CP problem.

In other areas of the literature, axions with gauged shift symmetries have
been discussed in field theoretic contexts as dark matter candidates
Coriano:2018uip ; Coriano:2010py , at the LHC Irges:2007yha ; Coriano:2009zh ;
Coriano:2010nf ; Coriano:2008pg ; Coriano:2005own ; Dror:2017nsg ;
Dror:2018wfl ; Dror:2017ehi ; Dror:2017nsg , as mixing with ordinary axions
Shiu:2015uva ; Shiu:2015xda ; Choi:2019ahy ; Berg:2009tg ; Fraser:2019ojt , in
brane-bulk scenarios Cheng:2001ys  or as solutions to the strong CP problem
Aldazabal:2002py .

##  II Massive dual axion

We begin by briefly reviewing the axion, the dual axion and the massive dual
axion theory. Special importance is placed on the effect of this dual axion
mass on the symmetries of the massless theory. Studying the fate of these
symmetries will illuminate the fate of the charged objects of the theory,
axion strings and instantons, in the superconducting phase.

###  II.1 Massless axion

A massless axion is described by a compact scalar $a\equiv a+2\pi$ with a
Lagrangian of the form,

$\mathcal{L}=\frac{F_{a}^{2}}{2}\left(\partial_{\mu}a\right)^{2}\,.$ (1)

Here $F_{a}$ is the fundamental period of the axion. This axion can be
realized from the spontaneous breaking of a $U(1)^{(0)}$ symmetry111The
notation used here $G^{(p)}$ is a standard convention adopted from the
generalized symmetries literature for a symmetry group $G$ acting on gauge
invariant objects of space-time dimensionality $p$. For our purposes, $p$ will
always align with the number of indices on the associated gauge field that
transforms under the shift., in which case the $a\equiv a+2\pi$ identification
is an emergent gauge symmetry below the spontaneous breaking scale.

The Lagrangian (1) has a well-known non-linear $U(1)^{(0)}$ shift symmetry
under which the axion transforms as

$a\rightarrow a+c^{(0)},\quad\partial_{\mu}c^{(0)}=0\,.$ (2)

This shift symmetry is generated by the associated current

$j^{(0)}_{\mu}=F_{a}^{2}\partial_{\mu}a$ (3)

and is conserved by the equations of motion of (1).

Instantons and other space-time configurations that are sources of a non-zero
divergence of $j^{(0)}_{\mu}$ are said to be electrically charged under the
axion. This follows identical conventions for the photon and the divergence of
its field strength. The effect of such a source on the theory can be probed by
an $e^{ia}$ (Wilson point) insertion in the path integral. The symmetry is
explicitly broken if such sources have a finite creation action (are
dynamical), such as gauge instantons.

In four space-time dimensions, the Lagrangian (1) also has an additional
$2$-form $U(1)^{(2)}$ symmetry Gaiotto:2014kfa ; Brennan:2020ehu ;
Choi:2022fgx ; Heidenreich:2021xpr  associated with the conserved current

$j^{(2)}_{\mu\nu\rho}=\frac{1}{2\pi}\epsilon_{\mu\nu\rho\lambda}\partial^{\lambda}a\,.$
(4)

The charge associated with the current (4) measures the winding number
(monodromy) of the axion field around charged strings. In the frame of the
axion, the conservation of this current is a topological condition ($dda=0$)
Heidenreich:2023pbi . This should be contrasted with the conservation (3) of
the axion shift symmetry by the equations of motion. This $U(1)^{(2)}$
symmetry will be associated with the shift symmetry of the dual $2$-form of
the axion and will therefore be referred to as the dual axion shift symmetry.

Axion strings and other space-time configurations that are sources of a non-
zero divergence of $j_{\mu\nu\rho}^{(2)}$ are said to be magnetically charged
under the axion. The insertion of a probe (non-dynamical) charged string
worldsheet can be performed in the frame of the axion by excising a tube
around the worldsheet and demanding a non-zero winding number on any curve
linking this worldvolume. This requires the introduction of an unphysical
Dirac sheet attached to the string. The $U(1)^{(2)}$ symmetry can be broken by
the existence of dynamical strings with a finite creation action around which
$dda\neq 0$.

The axion shift symmetry $U(1)^{(0)}$ and dual axion shift symmetry
$U(1)^{(2)}$ share a mixed ’t Hooft anomaly Brennan:2020ehu . Gauging one of
the symmetries explicitly breaks the other. This mixed anomaly is key to
understanding the often inverted behaviour of instantons and axion strings
under deformations of the massless axion theory. If one of these symmetries is
a gauge symmetry, sources that would break this symmetry are confined into
charge neutral dipole configurations. The mixed anomaly implies that the other
symmetry is explicitly broken. Sources that break that symmetry source a field
profile that is screened at large distances. If the $U(1)^{(0)}$ symmetry is a
gauge symmetry, this corresponds to the confinement of instantons and
screening of axion strings. If the $U(1)^{(2)}$ symmetry is a gauge symmetry,
axion strings are confined and the axion profile around instantons is
screened.

###  II.2 Dual axion

In four space-time dimensions, the theory (1) of a massless compact scalar is
dual Svrcek:2006yi  to that of a massless anti-symmetric two-index tensor
field $B_{\mu\nu}$ with a Lagrangian

$\mathcal{L}=\frac{1}{12f_{a}^{2}}H^{\mu\nu\rho}H_{\mu\nu\rho},\quad
H_{\mu\nu\rho}=\frac{1}{2}\partial_{[\mu}B_{\nu\rho]}$ (5)

and a gauge identification

$B_{\mu\nu}\equiv B_{\mu\nu}+\partial_{[\mu}\Lambda_{\nu]}\,,$ (6)

for an arbitrary massless $U(1)$ gauge field $\Lambda_{\nu}$.

The normalization of the fundamental period of the dual $f_{a}=2\pi F_{a}$ is
chosen Reece:2018zvv  such that the worldsheet $\Sigma$ of axion strings with
unit winding number couples to $B$ as $\int_{\Sigma}B$. The inversion of the
coupling strength $F_{a}\leftrightarrow\frac{1}{f_{a}}$ is sometimes called a
‘weak/strong’ duality and is a general feature of dualizations Burgess:2023ifd
.

A massless $2$-form with gauge identification (6) has $1$ d.o.f. matching that
of the original massless axion. The Lagrangian (5) also has both a
$U(1)^{(0)}$ and $2$-form $U(1)^{(2)}$ symmetry. Under the $U(1)^{(2)}$ shift
symmetry, the dual axion $B$ transforms as

$B_{\mu\nu}\rightarrow
B_{\mu\nu}+c^{(2)}_{\mu\nu},\quad\partial_{[\mu}c^{(2)}_{\nu\rho]}=0\,.$ (7)

In this dual frame, the generating current

$j^{(2)}_{\mu\nu\rho}=\frac{1}{2f_{a}^{2}}\partial_{[\mu}B_{\nu\rho]}\,$ (8)

and is conserved by the equations of motion. This should be contrasted with
the $U(1)^{(0)}$ symmetry in this frame, which is a topological condition and
generated by the current

$j^{(0)}_{\mu}=\frac{1}{4\pi}\epsilon_{\mu\nu\rho\lambda}\partial^{\nu}B^{\rho\lambda}\,.$
(9)

Configurations that are a source of non-zero $U(1)^{(2)}$ current are
electrically charged under the dual axion. The effect of such sources on the
theory can be probed by an $e^{i\int B}$ (Wilson sheet) insertion along their
worldsheet. Configurations that are a source of non-zero $U(1)^{(0)}$ current
are magnetically charged under the dual axion. Non-dynamical configurations
with $U(1)^{(0)}$ charge can be created in this frame by excising a small
volume around a space-time point and demanding a non-zero $U(1)^{(0)}$ charge
measured by any three sphere linking this point. This requires the
introduction of an unphysical Dirac point (line in space-time).

###  II.3 Massive dual axion

The aim of this section is to study the effect of mass contributions to the
dual axion $B$. To this end, we add a mass $m$ to Lagrangian (13) through a
generalization of the Proca/Stückelberg mechanism by adding a one-form $U(1)$
vector field $\tilde{A}$ as,

$\mathcal{L}=\frac{1}{12f_{a}^{2}}H_{\mu\nu\rho}H^{\mu\nu\rho}-\frac{m^{2}}{4f_{a}^{2}}\left(B_{\mu\nu}-\frac{f_{a}}{m}\partial_{[\mu}\widetilde{A}_{\nu]}\right)^{2}\,.$
(10)

Massive $2$-form fields with gauge identification (6) have 3 d.o.f. and their
properties have been studied in for instance Cecotti:1987qr ; Kuzenko:2020zad
; Townsend:1981nu ; Hell:2021wzm ; Smailagic:2001ch  and in the context of
cosmology in Capanelli:2023uwv .

Several mass generating mechanisms which lead to a low-energy Lagrangian (10)
exist for massless two-forms in field/string theories. Any strings coupled to
the dual axion with either a finite creation action or which can be wrapped
around compact $2$-cycles of the manifold give a mass to (components of) the
dual axion. Such cycles can be absent in the IR, but the topology of spacetime
seen by the dual axion can change in the UV, both in extra-dimensional
theories and $4$D theories. A four dimensional mass for the dual axion can
also be generated by the dimensional reduction of higher dimensional theories
with couplings of the form $B^{2}F^{N}$ where $F$ is the field-strength of
some $p$-form field. Activating fluxes $\int F=n$ on the internal/compactified
cycles of the manifold will generate a mass $m\sim n^{N}$ Capanelli:2023uwv .
Such fluxes are standard in KKLT Kachru:2003aw  or Large Volume Scenarios
Balasubramanian:2005zx  in string theory. Further examples of massive dual
axions are those that participate in anomaly cancellation mechanisms such as
the Green-Schwarz mechanism Green:1984sg . In string theory, massive dual
axions occur for instance in compactifications of heterotic string theory
Svrcek:2006yi  or type II intersecting brane models Ibanez:2001nd ;
Blumenhagen:2006ci ; Antoniadis:2002qm .

For completeness, we briefly mention that the massive dual axion theory can be
thought of as the far IR realisation of a gauged $1$-form $U(1)^{(1)}$ shift
symmetry of $\tilde{A}$ that is spontaneously broken by the formation of a
string condensate. This should be contrasted with theories in which the axion
is massive and the path integral receives contributions from an instanton
liquid instead.

###  II.4 Massive vector field

Strings are ‘electrically’ coupled to the dual axion and understanding their
phenomenology in a phase in which the dual axion has a mass can be
straightforwardly done from the massive dual axion perspective (10), as will
be done shortly in Section III.

Instantons are ‘magnetically’ coupled to the dual axion and understanding
their fate is inevitably tied to understanding the fate of the original
massless axion. The Lagrangian (10) is a dual description of several well-
known models, one of which is a massive vector field, which has ‘eaten’ the
axion. We briefly review these dualization steps here.

We first proceed by dualizing the vector field $\widetilde{A}\rightarrow A$,
which yields the BF-theory,

$\mathcal{L}=\frac{1}{12f_{a}^{2}}H_{\mu\nu\rho}H^{\mu\nu\rho}-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{1}{4\pi}B_{\mu\nu}\widetilde{F}^{\mu\nu}\,.$
(11)

The coupling, field strength and dual field strength are normalized as

$e=\frac{2\pi m}{f_{a}}\,,\quad
F_{\mu\nu}=\partial_{[\mu}A_{\nu]}\,,\quad\widetilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}\,.$
(12)

The BF coupling $B_{\mu\nu}\widetilde{F}^{\mu\nu}$ between the dual axion and
vector field is a topological term in contrast to the kinetic coupling in (10)
involving the dual vector field. This is another manifestation of duality;
dynamic/kinetic couplings in one frame become topological in a dual frame
(11).

The topological coupling in (11) implies that the axion is eaten by the vector
field $A$. This can be seen by also dualizing the dual axion $B\rightarrow a$,
which yields the Proca/Stückelberg action of a massive vector field,

$\mathcal{L}=-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{F_{a}^{2}}{2}\left(\partial_{\mu}a-A_{\mu}\right)^{2}\,.$
(13)

The original axion becomes embedded as the longitudonal mode of a $U(1)$ gauge
field $A$ matching the 3 d.o.f. of theory (10). The vacuum acts as a
superconductor and the fate of instantons in this phase is discussed in
Section IV.

##  III Axion strings

We proceed by studying the fate of axion strings and instantons in the
superconductor, starting with axion strings.

Strings can be electrically coupled to the dual axion $B_{\mu\nu}$. If such a
string is described by a current density $J^{\mathrm{string}}_{\mu\nu}$, then
this coupling is facilitated by the interaction

$\mathcal{L}\supset\frac{1}{2}B^{\mu\nu}J^{\mathrm{string}}_{\mu\nu}\,.$ (14)

Such strings act as sources of field strength $H_{\mu\nu\rho}$ ‘electric flux’
lines.

In the presence of a mass for the dual axion, these flux lines can end in the
string condensate and become screened. This can be observed by studying the
equations of motion of (10) in the gauge $d\tilde{A}=0$, which become

$\partial^{\rho}H_{\rho\mu\nu}=-m^{2}B_{\mu\nu}+f_{a}^{2}J^{\mathrm{string}}_{\mu\nu}\,.$
(15)

The mass $m$ suppresses any static long-distance non-zero $B_{\mu\nu}$ fields
and far-separated strings become non-interacting.

This can be confirmed by studying a string that has unit charge and is
oriented in the $z$-direction of a cylindrical coordinate system
$(t,z,r,\phi)$ such that $J^{\mathrm{string}}_{tz}=\delta(r)$. We can solve
for the profile around a charged string by inverting (15) as

$B_{\mu\nu}=\frac{f_{a}^{2}}{\partial^{2}+m^{2}}\left(J^{\mathrm{string}}_{\mu\nu}-\frac{1}{m^{2}}\partial^{\rho}\partial_{[\mu}J^{\mathrm{string}}_{\nu]\rho}\right)\,.$
(16)

Solving around such a profile yields a radially decreasing dual axion field

$B_{tz}=-\frac{f_{a}^{2}}{2\pi}K_{0}(mr)\,.$ (17)

Here $K_{i}(x)$ is the i’th modified Bessel function. For distances
$r\gg\frac{1}{m}$, the dual axion field is screened exponentially.

The monodromy around axion strings is no longer conserved as the $U(1)^{(2)}$
shift symmetry (7) of the dual axion is explicitly broken by the mass. The
winding number of a string at a given radius $r$ is given by integrating (8)
around the string, which for profile (17) implies

$Q_{\mathrm{wind}}=mrK_{1}(mr)\,.$ (18)

Close to the string, the winding number is constant. For distances
$r\gg\frac{1}{m}$, the dual axion field is screened and the winding number is
exponentially small.

In the dual Proca theory (13) of a superconductor, the screening of the dual
axion field profile is understood as follows. The axion profile winds around
the string $da\neq 0$. Far away from the string core, the axion profile is
compensated by a gauge field such that the axion kinetic term
$\frac{F_{a}^{2}}{2}\left(\partial_{\mu}a-A_{\mu}\right)^{2}$ vanishes. The
tension of the string due to the winding of the axion only receives
contributions from the string core and therefore far-separated strings become
non-interacting.

The phenomenology of such strings is that of gauge strings and the screening
of the dual axion profile does not imply that axion strings in a
superconductor are unobservable. As the kinetic term vanishes, the magnetic
flux carried by the string is non-zero: $\int A=\int da$. This magnetic flux
results in an Aharanov-Bohm phase for any charged particle winding around the
string.

Further interactions of gauge strings with their environment can come from
coupling to other long-range fields such as electromagnetism or gravity.

One of the earliest phenomenological signatures of such gauge strings focused
on their gravitational signal, which is set by the string’s tension $T\sim
F_{a}^{2}$. Oscillating loops of such strings produce gravitational waves
PhysRevD.31.3052 ; PhysRevD.42.354  with the total power $P$ being radiated by
oscillating loops of strings given by $P\sim GF_{a}^{4}$ where $G$ is Newton’s
constant. Additionally, such strings can form seeds for cosmological
perturbations Kibble:1980mv ; Zeldovich:1980gh ; Vilenkin:1981iu . As a string
of length $R$ has a mass $RF_{a}^{2}$, the gravitational effects of such
strings are only observable for very large scales $F_{a}$ Witten:1984eb .

A secondary, non-gravitational interaction of the gauge string with the
environment can come from the physics at the core of the string. This requires
a description of the physics at the core of the string and a distinction has
to be made between a string formed in an abelian Higgs mechanism and a so-
called fundamental or Stückelberg string Reece:2018zvv . Both admit
Lagrangians (10) far away from the core, but only the former string has a core
at which the shift symmetry is restored.

Zero modes can reside at the cores of both strings. These zero modes can carry
electric charge, making the string superconducting Witten:1984eb . If such a
string passes through galactic magnetic fields, large currents can be induced
on the string. These currents in turn create a large magnetic field, which
could be observed through radiation or through the synchotron radiation of
trapped charged particles.

Stückelberg strings are still accompanied by both an axion and a radial mode,
but the vanishing point of this radial mode lies at an infinite distance in
field space. There is no symmetry restoration at the core, but instead local
effective field theory is expected to break down near the core of the string
Reece:2018zvv ; March-Russell:2021zfq  and their phenomenology might be vastly
different.

##  IV Instantons

Instantons can be electrically coupled to the axion $a$. In theories in which
the axion shift symmetry is gauged, instantons no longer couple to just axion
insertions but instead couple to the gauge-invariant operator

$\mathrm{Exp}\left[ia-i\int_{C}A\right]\,.$ (19)

where $C$ is any curve extending from the instanton to infinity and
$e=\frac{m}{F_{a}}$. The presence of this additional Wilson line specifies the
worldline of a charged (non-dynamical) particle stretching from the instanton.
The space-time process describes the creation/decay of the particle at the
instanton. The additional worldline cost of this charged particle to the
partition function $Z$ of any isolated instanton confines this instanton. Only
gauge invariant combinations in which the worldline $C$ stretches between
different instantons contribute to the vacuum to vacuum partition function.
Instantons will still contribute to amplitudes in which the Wilson lines can
end on the boundary, such as those that involve the confining particles in the
initial or final states.

This confinement of instantons is analogous to the confinement of particles by
Abrikosov/Nielsen-Oleson vortex solutions Abrikosov:1956sx ; Nielsen:1973cs
stretching between magnetic monopoles in a superconductor or between electric
charges in models with a massive dual photon Hook:2022pcf . An artistic
impression for the case of a single instanton and an instanton-anti-instanton
dipole is provided in figure 1.

IspacetimeAIIspacetime Figure 1: Left: Space-time diagram of an instanton (I)
(red) confined by particle-like soliton worldlines (orange) describing the
creation/decay of such particles at the instanton vertex. Right: Space-time
diagram of an instanton (I) (red) and anti-instanton (AI) (blue) configuration
with a particle-like soliton worldline (orange) stretched between the
instantons describing the creation and subsequent decay of the particle.

Confinement of instantons is not unique to (10) and has a long history. One of
the earliest examples of instanton confinement-deconfinement transitions
occurs in $2$D (space-time dimensions) in the well-known $XY$ model. This
model has the coarse-grained description of a simple massless compact scalar.
Vortices or strings of this compact scalar are instantons in $2$D and are
confined by a logarithmic potential, forming bound vortex pairs. Increasing
the temperature allows for a transition to a gas of free unbound vortices
above a critical temperature described by the infinite order
Berezinskii–Kosterlitz–Thouless Berezinsky:1970fr ; Kosterlitz:1974nba  phase
transition.

Confinement of instantons in the presence of massless fermions also played an
important role in the early studies of the vacuum structure of QCD
Callan:1976je ; Callan:1977gz . In his seminal paper, ’t Hooft showed
tHooft:1976snw  that the existence of normalizable zero mode solutions to the
massless Dirac equation implied the vanishing of the single instanton
partition function in the presence of massless fermions. The exchange of
massless fermions gives rise to strong logarithmic interactions between
instantons and anti-instantons, leading to logarithmic confinement
Callan:1977gz ; Lee:1979sm . Instanton and anti-instantons form closely bound
dipole pairs and vacuum tunneling effects become negligible in the absence of
chiral symmetry-breaking sources. Instantons would instead contribute to
correlation functions that include the fermion zero mode, such as a chiral
condensate.

Confinement of instantons is crucial for extended particle-like field
configurations that enjoy topological protection, like vortices and monopoles.
Smooth deformations of their field profile in time cannot change the
topological index associated with these solitons, and their number is
conserved. However, singular field configurations can describe quantum
tunneling events between these distinct topological sectors. Therefore, an
additional prerequisite to the number conservation of such solitons is the
confinement of the associated instantons. This is most pronounced for
topological defects that lack core singularities, such as Skyrmions. For
instance, unconfined instantons exist in $3$D in the $\mathbb{C}P^{1}$ model
describing a $S^{2}$ valued field $\phi$ on a compact space $S_{2}$
Polyakov:1975yp , where instantons describe the sudden decay of the Skymrion
and violation of Skyrmion number. However, not all such defects are confined.
Hopfions Faddeev:1975tz ; Faddeev:1996zj , non-trivial maps $\pi_{3}(S^{2})$
from a compact space $S^{3}$ to a $S^{2}$ valued field do have an associated
conserved charge in $4$D space-time. This is due to the logarithmic
confinement of the associated Hopf instantons. A general discussion on the
stability of topological defects due to instanton confinement and the
renormalization flow of instanton interactions can be found in the recent
paper Nikolic:2023uxw .

Let us briefly comment that this instanton suppression cannot solve the axion
quality problem of a secondary massless axion. In the presence of multiple
species of instantons, a diagonal combination of instantons is uncoupled to
the axion with a dual mass. This diagonal combination will remain unsuppressed
and can give a mass to a secondary axion.

###  IV.1 Instanton gas

Confinement forces instantons into gauge-invariant dipole configurations such
as the instanton-anti-instanton dipole depicted in figure 1. Analogous to flux
tubes stretching between confined particles producing a linear potential, the
existence of the worldline of the charged particle implies an additional cost
to the instanton-anti-instanton partition function. We proceed by calculating
this additional cost.

In a system consisting of a gas of instanton-anti-instanton configurations, in
which the worldline would be dynamical, this worldline would relax to the
shortest distance between any instanton-anti-instanton pair. In the massive
dual axion theory (10), both the worldline and the instantons are non-
dynamical probe objects. We will rectify this in the next subsection by
introducing additional degrees of freedom to (10) in order to make both the
instantons and worldlines dynamical, but for now we wish to describe only this
steady state situation in which the worldline has relaxed to the shortest
distances between the instantons.

There is an equivalent formulation of the coupling (19) first introduced by
Zwanziger Zwanziger:1970hk  for the analogous electromagnetic coupling of
photons to magnetic monopoles, which will be useful to describe this
instanton-anti-instanton dipole. In this formulation, one introduces axion
shift symmetry sources by an instanton density $J_{\mathrm{inst}}$. To each
source, one attaches a Dirac line described by a unit vector $n^{\mu}$. The
coupling between the density and the axion is then achieved by the appropriate
addition of the interaction Lagrangian

$\mathcal{L}_{\mathrm{int}}=\left(a-\frac{n^{\mu}}{n\cdot\partial}A_{\mu}\right)J_{\mathrm{inst}}\,.$
(20)

In the limit $m\rightarrow 0$ keeping $F_{a}$ fixed, the coupling $e$ of the
photon vanishes and one returns to a theory of a compact scalar $a$, the
axion, coupled to instantons. When $m\neq 0$, the Dirac line becomes physical
and is identified with the worldline of the non-dynamical particle. Since the
worldline of the particle is non-dynamical, the interaction (20) is Lorentz
violating.

In the gauge $a=0$, the system in the Zwanziger formulation (20) is described
by an effective non-conserved current

$J^{\mu}=\frac{n^{\mu}}{n\cdot\partial}J_{\mathrm{inst}}\,,$ (21)

coupled to massive vector field $A_{\mu}$ with corresponding equations of
motion in Euclidean space-time

$A^{\mu}=\frac{g^{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{m^{2}}}{-\partial^{2}+m^{2}}e^{2}J^{\nu}\,.$
(22)

The current (21) describes the worldline of a charged particle in the
direction $n^{\mu}$ that emanates from a non-zero instanton density
$J_{\mathrm{inst}}$. This should be compared to the equivalent Wilson line
formulation (19) which requires the introduction of a non-conserved current
$J^{\mu}$ along the curve $C$ coupled to the massive gauge field $A^{\mu}$
that emerges from an insertion of $e^{ia}$.

We are interested in the field configuration in Euclidean space-time
describing instanton-anti-instanton dipole configuration with a separation
length $R$ in time. In a spherical coordinate system $(t,r,\phi,\theta)$, such
a configuration is described by the instanton density
$J_{\mathrm{inst}}=\delta(x^{\mu}+\frac{R}{2}\hat{t})-\delta(x^{\mu}-\frac{R}{2}\hat{t})$
and corresponds to an effective current (21) of the form

$J^{0}=\delta(r)\mathrm{Rect}\left(\frac{t}{R}\right)$ (23)

in which the Dirac line $n=\hat{t}$ stretches between the instantons and
$\mathrm{Rect}\left(\frac{t}{R}\right)$ is a unit box with size $R$ centered
at $t=0$. This current describes the existence of a charged particle that sits
at the origin for a time $R$.

In the limit $R\rightarrow\infty$, the profile for $J^{\mu}$ is that of a
charged particle sitting at the origin for all time and one has the ordinary
screened solution (in the normalization of (13)) for a charged particle,

$A^{0}=\frac{e^{2}}{4\pi r}e^{-mr}\,.$ (24)

At a finite distance $R$, we can calculate the contribution to the partition
function $Z$ due to such an instanton-anti-instanton configuration. Given an
effective current $J^{\mu}$, such an evaluation of the partition function
amounts to completing the square of (13) as,

$\ln{Z}=-\frac{e^{2}}{2}J_{\mu}\left(\frac{g^{\mu\nu}-\frac{\partial^{\mu}\partial^{\nu}}{m^{2}}}{-\partial^{2}+m^{2}}\right)J_{\nu}\,.$
(25)

In the Wilson line formulation (19) and the gauge $a=0$, this simply amounts
to calculating the expectation value of $e^{i\int_{C}A}$ along the contour $C$
stretched between the instantons.

We proceed by plugging in the explicit expression (23) for $J^{\mu}$,

$\ln{Z}=-\frac{e^{2}}{2}\int_{-\frac{R}{2}}^{\frac{R}{2}}dt\int_{-\frac{R}{2}}^{\frac{R}{2}}dt^{\prime}\
G(t-t^{\prime})\,.$ (26)

Here, $G(t)$ is the Green’s function of the massive vector field. Plugging in
the expression for this Green’s function in momentum space, introducing the
three momentum $\vec{k}$ as $k_{\mu}=(k_{0},\vec{k})$ and performing the
temporal integrals gives

$\ln{Z}=-2e^{2}\int\frac{dk^{0}}{2\pi}\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{\sin^{2}{\left(\frac{k_{0}R}{2}\right)}}{k_{0}^{2}}\frac{1-\frac{k_{0}^{2}}{m^{2}}}{k^{2}+m^{2}}\,.$
(27)

Performing the temporal momentum integrals, keeping only the $R$-dependent
pieces and splitting the partition function for later convenience yields

$\ln{Z}=-\frac{e^{2}}{2}S-\frac{e^{2}}{(2\pi)^{2}}\frac{K_{1}(mR)}{mR}\,,$
(28)

where $S$ is dominated by the action of the charged soliton,

$S=\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\left(\frac{R}{\vec{k}^{2}+m^{2}}+\frac{e^{-R\sqrt{\vec{k}^{2}+m^{2}}}}{\left(\vec{k}^{2}+m^{2}\right)^{\frac{3}{2}}}\right)\sim
R\Lambda\,.$ (29)

From the partition function (28) it is clear what the interpretation of the
addition of the current $J_{\mathrm{inst}}$ to the system is. In the limit
$mR\ll 1$ keeping $F_{a}$ fixed, the second term in (28) dominates
$\ln{Z}\sim-\frac{1}{F_{a}^{2}R^{2}}$. This simply describes a massless axion
propagating between the instanton and anti-instanton.

In the opposite limit $mR\gg 1$ keeping $F_{a}$ fixed, the massive photon is
coupled to the current $\frac{n^{\mu}}{(n\cdot\partial)}J_{\mathrm{inst}}$.
This current describes a unit box or equivalently, the insertion of a charged
particle for a time $R$. The partition function of such a configuration scales
as $\ln{Z}\sim-e^{2}R\Lambda$ where the UV-cutoff $\Lambda$ is the electric
self-energy of the charged particle. The instantons are linearly confined.

Similar to the confinement of colour, the presence of instantons with a finite
action $e^{-S_{0}}$, reduces the effective range of the confining
contribution. In the presence of such instantons, the worldline of the charged
particle can be broken by a dipole instanton-anti-instanton pair. Such a
breaking becomes favourable whenever the distance
$e^{2}R\gtrsim\frac{S_{0}}{\Lambda}$ and therefore the confining contribution
is only of finite range. This length is further reduced by virtual instanton
fluctuations Nikolic:2023uxw .

We briefly mention that these results could have equivalently been obtained
from a dimensional reduction of ordinary particle confinement. Loops of
particles along the extra dimension are instantons in the reduced theory. The
partition function $\ln Z$ in $4$D of these instantons is the potential
$V_{5}$ in the $5$D theory of the looping particle. If the particles are
charged under a gauge field in $5$D, this potential can either be Coulomb
$V_{5}(R)\sim\frac{1}{R^{2}}$ for a massless photon or confining $V_{5}(R)\sim
R$ for a massive dual photon Hook:2022pcf .

###  IV.2 Confinement of instantons

Both the instanton and confining worldline can be made dynamical by
introducing additional degrees of freedom to the massive theory (10). These
confining degrees of freedom come in two flavours, bosons and fermions.

####  IV.2.1 Bosonic confinement

Simple models of confinement of instantons by worldlines of bosons can be
obtained in $3$D. We will focus on the confinement of magnetic monopoles,
which are instantons in this dimension and are associated with a non-trivial
second homotopy index $\pi_{2}$. As $\pi_{2}(G)=1$ for any compact, connected
Lie group $G$ Cartan , such instantons only form when a gauge group is
spontaneously broken $G\rightarrow H$ and are associated with the index
$\pi_{2}(G/H)$. Subsequent breaking of $H$ will always eventually result in
confinement of such instantons as they should be absent from the full breaking
$\pi_{2}(G/1)=1$.

Consider simple two-step breaking models in which a gauge group $G$ breaks as
$G\rightarrow H_{1}\rightarrow H_{2}$. We will focus on models in which
instantons that form in the first breaking step confine due to meta-stable
particles that form in the second breaking step $H_{1}\rightarrow H_{2}$.
Particles formed in the second breaking step are classified by their
topological index $\pi_{1}(H_{1}/H_{2})$. In the larger gauge group $G$, such
configurations are described by the topological index $\pi_{1}(G/H_{2})$ and
for each particle there exists a mapping

$\pi_{1}(H_{1}/H_{2})\rightarrow\pi_{1}(G/H_{2})$ (30)

If this mapping has a non-trivial kernel, then two distinct particles $p_{1}$
and $p_{2}$ associated with distinct non-trivial loops in $H_{1}/H_{2}$ can be
continuously connected in $G/H_{2}$. In such cases, there exists an instanton
associated with an element in $\pi_{2}(G/H_{1})$ describing a surface in
$G/H_{2}$ that connects the two loops. The winding configurations are said to
be able to unwind in the larger gauge group $G$ and the instanton allows for
the particle transition/decay $p_{1}\rightarrow p_{2}$. If a particle is
mapped to the identity in $\pi_{1}(G/H_{2})$, then the instanton describes the
decay of the particle to the vacuum. A full account of the decays of
metastable topological defects in any dimension can be found in
Preskill:1992ck .

If the instanton is absent from the total breaking $\pi_{2}(G/H_{2})=1$, then
the instanton only exists as an endpoint or kink or twist on the worldline of
meta-stable particle(s). Simple examples of such models are breaking patterns

$G\rightarrow U(1)\rightarrow 1$ (31)

for compact, connected and simply connected Lie groups $G$. Particles form
during the second breaking step $\pi_{1}(U(1))=\mathbb{Z}$, but can unwind in
the larger simply connected gauge group as $\pi_{1}(G)=1$. The
decays/transitions of these particles are facilitated by instantons. The
instantons are absent from the full breaking and instead are confined as the
end-points of such particle worldlines. Similar considerations apply to
instantons confined on the worldlines of non-Abelian particles such as in
Tong:2003pz ; Shifman:2004dr .

A well-known example is magnetic monopoles (instantons in this dimension) that
form in $SU(2)\rightarrow U(1)\rightarrow 1$ breaking patterns by subsequently
one adjoint and one fundamental scalar. During the second breaking by the
fundamental scalar, non-perturbative particles form
$\pi_{1}(U(1))=\mathbb{Z}$, which are time-independent configurations in which
the fundamental scalar winds. In the full breaking pattern, all such winding
configurations are trivial $\pi_{1}(SU(2))=1$ and can therefore be unwound by
instanton insertions $\pi_{2}(SU(2)/U(1))=\mathbb{Z}$. These instantons are ’t
Hooft-Polyakov monopoles tHooft:1974kcl ; Polyakov:1974ek  that form during
the first breaking of $SU(2)\rightarrow U(1)$ by the adjoint scalar. These
instantons are absent in the full breaking $\pi_{2}(SU(2))=1$ and only occur
at the end-points of the worldlines of the non-perturbative particles.

If the second breaking in (31) is instead to a discrete subgroup $Z_{N}\subset
U(1)$, then the instantons become kinks or twists on the worldline of
particles. A well-known example is magnetic monopoles, which are kinks on
particle worldines in $SU(2)\rightarrow U(1)\rightarrow\mathbb{Z}_{2}$ double
breaking models PhysRevLett.55.2398  by two orthogonal adjoints. During the
second breaking $U(1)\rightarrow\mathbb{Z}_{2}$ by an adjoint $\phi$,
particles form $\pi_{1}(U(1)/\mathbb{Z}_{2})=\mathbb{Z}$ as winding
configurations of $\phi$. In the full breaking, there are only two
topologically inequivalent winding configurations:
$\pi_{1}(SU(2)/\mathbb{Z}_{2})=\mathbb{Z}_{2}$. The magnetic monopole formed
during the first breaking connects the worldlines of these configurations. No
closed loop can be formed and the instanton is confined.

Both of these models provide examples of completions of the massive dual axion
Lagrangian (10) in $3$D in which both the instanton and worldline are
dynamical. We identify the axion with the magnetic dual of the $U(1)$ photon
of these models. After the first breaking step, monopoles (instantons) form
which are magnetically charged and couple to the axion/dual photon. A gas of
these instantons would have given a mass $m\sim e^{-S_{\mathrm{mon}}}$ to the
axion/dual photon Polyakov:1976fu .

However, this is prevented by the second breaking step at scales $v$, where
particles associated with the winding of the second scalar $\phi$ form
attached to the monopole. These particles come attached to any ’t Hooft vertex
associated with the monopole and the monopole becomes confined. The photon
(dual axion) obtains a mass from eating the second scalar $\phi$. The
partition function of any pair of instantons separated by a space-time
distance $R$ is suppressed by the action cost of the charged particle
travelling between the instantons. As the non-perturbative particle has mass
$v$ (which includes the self-energy), this results in a partition function
suppression of $\ln{Z}\sim-vR$.

Extending such models of instanton confinement to $4$D, one runs into issues.
In this dimension, we consider instantons associated with the topological
index $\pi_{3}$ and again wish to explore the confinement of such instantons
by particle worldlines. This time instantons already exist; $\pi_{3}(G)\neq 0$
for a generic non-Abelian gauge group $G$. In two-step breaking models, we
need to follow both such instantons and possible additional instantons created
in the breaking associated with the index $\pi_{3}(G/H_{1})$.

Associated with the first breaking $G\rightarrow H_{1}$, there is a fibration
$H_{1}\rightarrow G\rightarrow G/H_{1}$ and a long exact sequence Csaki:1998vv
in homotopy

$\cdots\rightarrow\pi_{3}(H_{1})\rightarrow\pi_{3}(G)\rightarrow\pi_{3}(G/H_{1})\rightarrow\pi_{2}(H_{1})=1\rightarrow\cdots$
(32)

The exactness of the first three entries of the sequence implies that the
instantons of $G$ either descend to an instanton in the unbroken gauge group
$H_{1}$ or form a subset of the instantons associated with non-trivial
elements of the index $\pi_{3}(G/H)$. In fact, the exactness of the last three
entries and the vanishing index $\pi_{2}(H_{1})=1$ implies that no other
instantons are created during the breaking process,

$\pi_{3}(G/H_{1})\equiv\pi_{3}(G)/\pi_{3}(H_{1})\,.$ (33)

All instantons descend from those of $G$ and split into instantons associated
with either the unbroken part of the gauge group $\pi_{3}(H_{1})$ or the
broken part $\pi_{3}(G/H_{1})$.

A simple example involves the breaking of $SU(2)\rightarrow U(1)$ by an
adjoint scalar. The fibration associated with the breaking is the Hopf
fibration $S^{1}\rightarrow S^{3}\rightarrow S^{2}$. The low-energy gauge
group $U(1)$ has no instantons, instead
$\pi_{3}(SU(2)/U(1))=\mathbb{Z}=\pi_{3}(SU(2))$.

What happens to these instantons during the second breaking $H_{1}\rightarrow
H_{2}$? Particle-like defects can form during the second breaking, this time
associated with the topological index $\pi_{2}(H_{1}/H_{2})$. In the larger
gauge group $G$, the worldlines of particles that can end on instantons are
associated with the kernel of the mapping

$\pi_{2}(H_{1}/H_{2})\rightarrow\pi_{2}(G/H_{2})\,.$ (34)

However, the kernel of such a mapping must always be trivial Preskill:1992ck
as $\pi_{2}(H_{1})=1$ for all compact finite-dimensional Lie groups $H_{1}$.
Any magnetic monopole associated with $\pi_{2}(H_{1}/H_{2})$ remains
absolutely stable when embedded into a larger gauge group $G$. If instantons
associated with the index $\pi_{3}(G/H_{1})$ do connect particle worldlines,
then these worldlines are topologically equivalent with respect to the index
$\pi_{2}(H_{1}/H_{2})$.

Even-though the topological index of instantons is protected against
confinement, there could still be dynamical reasons for the absence or
confinement of instantons. Upon higgsings, gauge instantons in $4$D in the
broken part of the gauge group become constrained instantons Affleck:1980mp .
By a generalization of Derrick’s theorem Derrick:1964ww , such classical
instanton solutions do not exist in the presence of a non-trivial scalar
Manton:2004tk  in the absence of any scale-fixing source Affleck:1980mp . Due
to the Higgs phase of the bulk, any instanton solution with a non-trivial
scalar profile wants to shrink to minimize its action.

So what happens to instantons associated with $\pi_{3}(G/H_{1})$ during the
first breaking? Quantum corrections, such as asymptotic freedom, can prefer
larger instanton solutions. A competition between these two effects can lead
to constrained instantons Affleck:1980mp ; approximate instanton solutions, to
still dominate the functional integral tHooft:1976snw  and such instantons are
unconfined.

Instantons can also survive the Higgs phase of the bulk by residing as moduli
degrees of freedom on the worldvolume of larger host solitons Eto:2004rz . The
action of these instantons diverges with the infinite extent of the host
soliton, but there is no topological protection against compactifying the
worldvolume. If the host soliton worldvolume is compact, any charge associated
with the host vanishes, and only the instanton number remains.

Examples of instantons residing on host solitons are worldlines of magnetic
monopoles that carry a $U(1)$ modulus rotor degree of freedom. A twist or
winding of this rotor degree of freedom on the magnetic monopole loop carries
instanton number Jackiw:1975ep  through their Hopf invariant Bruckmann:2002jm
; Jahn:1999wx . Such examples include $SU(2)$ gauge theories spontaneously
broken to $U(1)$, which admit ’t Hooft-Polyakov monopoles as
$\pi_{2}(SU(2)/U(1))=\mathbb{Z}$ and instantons
$\pi_{3}(SU(2)/U(1))=\mathbb{Z}$. These instantons live on host magnetic
monopole worldlines. Since the $U(1)$ modulus descends from an unbroken $U(1)$
gauge symmetry in the bulk, the modulus does not admit a potential and the
instanton is diluted along the monopole worldline. The action for a loop of
such a magnetic monopole with Schwinger length $\ell$ includes a cost
$m_{M}^{2}\ell$ for a monopole with mass $m_{M}$, which is balanced against a
gradient energy from the winding of the instanton $\sim\frac{1}{\ell}$ around
the loop. Contributions of such instantons to the axion mass were recently
calculated in Fan:2021ntg .

Additional examples of instantons on host solitons exist in $2$D where
$Q$-lumps are twists on $Q$-kinks Abraham:1992qv . In $4$D, $\mathcal{N}=2$
SQCD models in the Higgs phase Hanany:2004ea  have Yang-Mills instantons that
reside as lumps on the worldsheet of non-Abelian vortex strings Tong:2003pz ;
Eto:2004rz  or as stabilized Skyrmions on domain wall worldsheets Eto:2005cc .
Further examples can be found in Nitta:2013vaa ; PhysRevD.87.066008 ;
Nitta:2022ahj  and references therein. We should mention that the exact
relation (see for instance Brower:1996js ) between these hosted instantons and
the constrained instantons of Affleck:1980mp  is still an open problem and
deserves further study.

####  IV.2.2 Fermionic confinement

Instantons can become confined by the existence of fermion zero modes. By the
Atiyah-Singer index theorem Atiyah:1963zz , the difference between the number
of left and right chiral solutions to the massless Dirac equation on an even-
dimensional compact space is given by the instanton number. In his seminal
paper, ’t Hooft showed tHooft:1976snw  that the existence of normalizable zero
mode solutions to the massless Dirac equation in the background of an
instanton implied the vanishing of the single instanton partition function.
Instead, the instanton will contribute to correlation functions that include
the fermion zero mode, such as the chiral condensate.

Instanton-anti-instanton dipoles will receive contributions to their partition
function from both the exchange of gluons and massless fermions. The gluonic
exchange gives rise to dipole-dipole interactions $\ln{Z}\sim-\frac{1}{R^{4}}$
where $R$ is the distance between the instanton and anti-instanton
Callan:1977gz ; Schafer:1996wv . The exchange of massless fermions gives rise
to strong logarithmic interactions $\ln{Z}\sim-\ln{R}$ between the instanton
and anti-instanton, leading to logarithmic confinement Callan:1977gz ;
Lee:1979sm .

Coupling such logarithmically confined instantons to a gauged axion results in
the well-known Green-Schwarz mechanism Green:1984sg  and another realization
of a superconductor in which both the confined instantons and confining
particles are dynamical.

Consider coupling the massive dual axion or gauged axion to instantons. These
instantons can be those of the massive vector field or possibly an external
gauge group $G$, which we take to have traceless generators for simplicity.
This can be done by a coupling222The allowed values of $c_{a}$ and $c_{G}$
depend on the spacetime manifold and the global gauge group, see Choi:2023pdp
; Reece:2023iqn . For simplicity, we pick a spin manifold and the product
group $U(1)^{0}\times G$, which implies
$(3c_{a},c_{G})\in\mathbb{Z}\times\mathbb{Z}$

$\mathcal{L}\supset
c_{a}\frac{a}{16\pi^{2}}F\widetilde{F}+c_{G}\frac{a}{16\pi^{2}}\mathrm{tr}G\widetilde{G}\,.$
(35)

This coupling is anomalous under the gauged $U(1)^{0}$ shift symmetry
$a\rightarrow a+\lambda$. In order to cancel the anomaly, one introduces a set
of massless Weyl fermions charged under both the non-Abelian gauge group $G$
and the gauged axion shift symmetry $U(1)^{0}$. One can choose the anomalous
variation of the fermion measure to be invariant under $G$ transformations.
Under variations $\lambda$ of $U(1)^{(0)}$, the total fermion measure333In
theories of gravity there would also be a gravitational anomaly $\sim\
\mathrm{tr}R\wedge R$ where $R$ is the Riemann curvature $2$-form. then
changes as

$D\Psi\rightarrow D\Psi\ \mathrm{exp}\left[-\frac{1}{16\pi^{2}}\int d^{4}x\
\lambda\left(c_{a}F\widetilde{F}+c_{G}\mathrm{tr}G\widetilde{G}\right)\right]\,.$
(36)

The constants $c_{a}$ and $c_{G}$ are determined by the exact fermion content.
If the $i$-th Weyl-fermion $i$ sits in representations $(q_{i},r_{i})$ of
$U(1)^{0}\times G$, then the total change of the measure is

$c_{a}=\frac{1}{3}\sum_{i}q_{i}^{3}\mathrm{dim}(r_{i})\,,\quad
c_{G}=\sum_{i}q_{i}I(r_{i})$ (37)

where $I(r_{i})$ is the Dynkin index of representation $r_{i}$ normalized such
that $I(\square)=1$ for the fundamental representation. If the fermion content
can be chosen to cancel the anomalous variations of (35), then the gauge
symmetry is non-anomalous. The instantons associated with a non-zero total
instanton number
$I=\frac{c_{a}}{16\pi^{2}}F\widetilde{F}+\frac{c_{G}}{16\pi^{2}}\mathrm{tr}G\widetilde{G}$
are then confined by the massless fermions and will only contribute to
amplitudes involving these massless fermions in the initial/final states.

One of the simplest examples is $c_{a}=0$ and the gauge group $G=SU(2)$ with
gauge field $B_{\mu}$ and field strength $G_{\mu\nu}$. The massive dual axion
is coupled to the instantons as

$\mathcal{L}\supset\frac{F_{a}^{2}}{2}\left(\partial_{\mu}a-A_{\mu}\right)^{2}+c_{G}\frac{a}{16\pi^{2}}\mathrm{tr}G\widetilde{G}\,.$
(38)

This theory is anomalous and requires the introduction of a set of four
massless left-handed Weyl-fermions $\psi_{i}$ in the fundamental
representation of $G$. We pick the charges of these fermions under $A^{\mu}$
to be the Fermat charges $q_{i}=\\{3,4,5,-6\\}$, such that the constants
$c_{a}=0$ and $c_{G}=6$. The fermionic Lagrangian is

$\mathcal{L}\supset\sum_{i}\psi^{\dagger}_{i}\overline{\sigma}_{\mu}\left(i\partial^{\mu}+B_{\mu}+q_{i}A_{\mu}\right)\psi_{i}\,.$
(39)

The gauge field $A^{\mu}$ is coupled to the chiral current $j^{\mu}_{5}$,
which satisfies the equations of motion

$j^{\mu}_{5}=\sum_{i}q_{i}\psi^{\dagger}\overline{\sigma}_{\mu}\psi\,,\quad\partial_{\mu}j^{\mu}_{5}=\frac{c_{G}}{16\pi^{2}}\mathrm{tr}G\widetilde{G}\,.$
(40)

The non-zero divergence implies that instantons act as a source for this
chiral current. Every instanton vertex comes with associated fermion
worldlines with a total non-zero chiral charge (see figure 1) and is confined.

Classically, one can calculate the worldline contribution to the partition
function of an instanton-anti-instanton dipole pair separated by a distance
$R$ in time by studying the equations of motion (40), which for such a
configuration are

$\partial_{\mu}j^{\mu}_{5}=\delta(x^{\mu}+\frac{R}{2}\hat{t})-\delta(x^{\mu}-\frac{R}{2}\hat{t})\,.$
(41)

In a spherical coordinate system $(t,r,\phi,\theta)$, these equations are
solved by

$j^{\mu}_{5}=\delta(r)\mathrm{Rect}\left(\frac{t}{R}\right)\delta_{\mu,0}+\chi_{\mu}\,,$
(42)

where $\chi_{\mu}$ has zero divergence $\partial_{\mu}\chi^{\mu}=0$ and
represents the contribution of a compact particle worldline.

The first part of this current was already studied in (23) and integrating out
the gauge field $A^{\mu}$ would lead to linear confinement, this time by
chirally charged fermion zero modes. However, the vacuum contains massless
fermions; the Wilson line interpretation of the fermion breaks down and
quantum corrections may be large. A full calculation of the partition function
would require both the fermion profile in an instanton-anti-instanton
background and a control over these quantum corrections. We will not attempt
this here.

Instanton can become unconfined if the fermion zero mode Wilson lines attached
to the instanton vertex can end in the bulk. For instance, an interaction with
strength $g$ between the four fermions can be added of the form

$\mathcal{L}\supset-
g_{1}e^{-ic_{G}a}\left(\psi_{1}\psi_{2}\right)\left(\psi_{3}\psi_{4}\right)+\mathrm{h.c.}\,,$
(43)

or two two-point interactions

$\mathcal{L}\supset-
g_{2}e^{-i(q_{1}+q_{2})a}\psi_{1}\psi_{2}-g_{3}e^{-i(q_{1}+q_{2})a}\psi_{3}\psi_{4}+\mathrm{h.c.}$
(44)

The colour and Lorentz indices are contracted pair-wise using Levi-Civita
symbols.

The zero modes associated with the gauge-variant instanton vertex
$e^{ic_{G}a}$ can end in the bulk by an insertion of the term above. The
instanton no longer couples to the gauged axion and is unconfined.

A second way to unconfine such instantons is the introduction of a second
axion $b^{\mu}$, whose shift symmetry is also gauged,

$\mathcal{L}\supset\frac{F_{b}^{2}}{2}\left(\partial_{\mu}b-A_{\mu}\right)^{2}\,.$
(45)

Instantons can now be coupled to the gauge-invariant combination $a-b$ and
become unconfined. This can be done either by directly altering (35) by
$a\rightarrow a-b$ or adding a fermion mass of the form (44) with
$a\rightarrow b$. The gauge-invariant combination $a-b$ can then receive a
potential in an instanton liquid. A second set of instantons coupled to the
orthogonal gauge-variant combination $a+b$ would however still be confined.

In the gauge $a=0$, we recognize this simple toy model (39) as a massive
vector field coupled to an anomalous current involving massless fermions. The
dynamics confining the instantons in these simple models are equivalent to
those of the massless up quark and its solution to the strong CP problem. We
encourage further model-building efforts exhibiting instanton confinement
beyond such simple toy models, particularly those in which the confining
dynamics do not require the introduction of additional degrees of freedom.

The addition of the massive vector fields coupled to the non-conserved current
can potentially lead to linear confinement due to the additional self-energy
of the charged particles, but also leads to phenomenological signatures
including amplitudes involving a longitudinal mode emission of the massive
vector field proportional to the ratio (energy of the process/vector mass).
For Standard Model currents, such couplings have been extensively studied in
the literature (see Dror:2017nsg ; Dror:2018wfl ; Dror:2017ehi  and references
therein) and include currents broken at tree level such as axial currents
broken by fermion masses or tree-level conserved currents broken by a chiral
anomaly, such as the SM baryon number or lepton number currents. Both
couplings lead to FCNC and rare flavour-changing meson decays and can be
constrained by missing energy signatures or visible decays depending on the
specific model Dror:2017nsg . By the Goldstone boson equivalence theorem, such
couplings are equivalent to axion-like FCNC and constrained in
Izaguirre:2016dfi . The non-observation of such processes would imply that
such couplings can only exist for small couplings $e$ or large masses $m$.

##  V Discussion

We have studied a modification of axion physics in which the dual axion
acquires a mass. In such a superconducting phase, axion strings are screened
and far-separated instanton configurations are exponentially suppressed.
Dipole configurations of instantons received an additional exponential
suppression proportional to the distance between the individual instantons,
corresponding to a charged particle travelling between the instantons. We have
considered several models of such confinement with both dynamical instantons
and dynamical worldlines. Much about the confinement of particles in non-
supersymmetric theories is still poorly understood and we do not expect this
to be an exhaustive list of instanton confinement models either.

The dynamics confining instantons and their contributions to the path integral
in some of the simple models presented here were equivalent to the massless up
quark and its solution to the strong CP problem. It would be interesting to
further extend the list of such models to also include confinement of
instantons akin to particle colour confinement in QCD in which the confining
flux tubes are dynamically generated and do not require the introduction of
additional degrees of freedom. Useful insight in such models of particle
confinement has been gained in the past by considering supersymmetric QCD. In
these theories, Seiberg and Witten Seiberg:1994rs  were able to demonstrate
that, in a pure $SU(2)$ gauge theory, confinement of electric monopoles could
be described by magnetic monopole condensation. Furthermore, by adding matter
multiplets, they showed that it is possible to interpolate between a Higgs
phase in which quarks are condensed to a phase in which magnetic monopoles
condense Seiberg:1994aj . It would be interesting to apply similar techniques
to instanton confinement. Given the complexity of such an endeavor, we leave
this as an open problem for future studies.

## Acknowledgments

I would like to thank Prateek Agrawal, Junwu Huang, Ethan Carragher, Matthew
Reece and especially Mario Reig for helpful discussions. Any errors are my
own. This paper was inspired by analogous considerations of a dual photon mass
in the recent paper Hook:2022pcf . Arthur Platschorre is supported by a STFC
Studenship No. 2397217 and Cultuurfondsbeurs No. 40038041 made possible by the
Pieter Beijer fonds and the Data-Piet fonds.

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