MIT-CTP 5644

# The Quality/Cosmology Tension for a
Post-Inflation QCD Axion

Qianshu Lu,1,2 Matthew Reece,3 and Zhiquan Sun4
1 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ
08540, USA
2 Center for Cosmology and Particle Physics, Department of Physics,
New York University, New York, NY 10003, USA
3Department of Physics, Harvard University, Cambridge, MA, 02138, USA
4Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA, 02139, USA

###### Abstract

It is difficult to construct a post-inflation QCD axion model that solves the
axion quality problem (and hence the Strong CP problem) without introducing a
cosmological disaster. In a post-inflation axion model, the axion field value
is randomized during the Peccei-Quinn phase transition, and axion domain walls
form at the QCD phase transition. We emphasize that the gauge equivalence of
all minima of the axion potential (i.e., domain wall number one) is
insufficient to solve the cosmological domain wall problem. The axion string
on which a domain wall ends must exist as an individual object (as opposed to
a multi-string state), and it must be produced in the early universe. These
conditions are often not satisfied in concrete models. Post-inflation axion
models also face a potential problem from fractionally charged relics; solving
this problem often leads to low-energy Landau poles for Standard Model gauge
couplings, reintroducing the quality problem. We study several examples,
finding that models that solve the quality problem face cosmological problems,
and vice versa. This is not a no-go theorem; nonetheless, we argue that it is
much more difficult than generally appreciated to find a viable post-inflation
QCD axion model. Successful examples may have a nonstandard cosmological
history (e.g., multiple types of cosmic axion strings of different tensions),
undermining the widespread expectation that the post-inflation QCD axion
scenario predicts a unique mass for axion dark matter.

###### Contents

  1. 1 Introduction
  2. 2 A ${\mathbb{Z}}_{p}$ symmetry for quality
    1. 2.1 First look
    2. 2.2 Second look: ${\mathbb{Z}}_{p}$ strings
    3. 2.3 ${\mathbb{Z}}_{p}$ embedded in continuous gauge groups
  3. 3 A ${\mathrm{U(1)}}$ symmetry for quality
    1. 3.1 Review of the Barr-Seckel Model
    2. 3.2 Cosmology/quality tension
  4. 4 An ${\mathrm{SU}}(n)\times{\mathrm{SU}}(n)$ symmetry for quality
    1. 4.1 Scalar potential and axion strings
    2. 4.2 Standard Model couplings and Landau pole constraint
  5. 5 Comments on composite axion models
  6. 6 Conclusions
    1. 6.1 Brief discussion of other models
    2. 6.2 Concluding questions

##  1 Introduction

The QCD axion, which dynamically relaxes ${\overline{\theta}}$ and explains
the absence of an observed neutron EDM, has long been the leading contender to
solve the Strong CP problem [1, 2, 3, 4]. Via the misalignment mechanism, a
QCD axion also provides some abundance of dark matter, which could range from
a subdominant contribution to one so large it is observationally ruled out,
depending on the axion mass and cosmological history [5, 6, 7]. For a detailed
introduction to axion physics and the Strong CP problem, see [8, 9, 10]; for
axion cosmology specifically, see [11, 12, 13].

It was appreciated early on that a QCD axion is not truly sufficient to solve
the Strong CP problem: the coupling that allows ${\overline{\theta}}$ to relax
explicitly breaks the axion shift symmetry, so a viable theory should explain
why there are not other shift-symmetry breaking effects that add to the
potential and (generically) displace its minimum. This has come to be known as
the “axion quality problem,” and it is severe. Even Planck-suppressed higher
dimension operators that break a Peccei-Quinn (PQ) symmetry are dangerous, up
to dimension ten or more [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. An axion
model that does not solve the axion quality problem should not be thought of
as a solution to the Strong CP problem, as it requires introducing many
exponentially small parameters rather than than simply assuming
$|{\overline{\theta}}|$ to be small. Solutions to the problem generally either
invoke new gauge symmetries that forbid all of the dangerous operators, or
extra dimensions so that new terms in the axion potential only arise from
nonlocal effects that are exponentially small.

Axion models are divided into two broad categories based on their cosmology,
often referred to as “pre-inflation” and “post-inflation.”111Alternatively, as
an intermediate case, PQ symmetry could have a phase transition during
inflation [24, 25]. We will not discuss this case in detail, but for our
purposes it may be thought of as similar to either pre-inflation or post-
inflation depending on whether or not it introduces a potential domain wall
problem in the late-time universe. In the post-inflation models, there is a
phase transition after the end of inflation in which an approximate global PQ
symmetry is spontaneously broken. The axion emerges as an independent real
scalar field only after this phase transition. In the pre-inflation models,
there is no such phase transition. “Pre-inflation” is something of a misnomer,
because it suggests that there is a PQ symmetry that was realized at some
early time in the history of the universe, and this need not be true. For
example, models in which the axion arises as a mode of an extra dimensional
gauge field (e.g., [26, 27, 28, 29]) offer a compelling solution to the
quality problem, and do not have a PQ phase transition at all. They are
intrinsically of pre-inflation type (as noted in, e.g., [30, 10, 31]).

In this paper, we focus instead on the post-inflation axion scenario. In this
case, the axion generally arises as the phase of a complex (possibly
composite) scalar field, which obtains a vacuum expectation value (VEV) during
the PQ phase transition after inflation. This scenario occurs quite generally
when the axion decay constant is small compared to the inflationary Hubble
scale, $f_{a}<H_{I}/(2\text{p})$. It can also arise even for larger $f_{a}$ in
more model-dependent ways, e.g., if the PQ phase transition temperature is
much lower than $f_{a}$, perhaps because it is triggered by SUSY breaking [32,
33]; or if a sufficiently high temperature is achieved during reheating to
restore the PQ symmetry [34]; or if the heavy lifting effect raises the mass
of the PQ scalar during inflation [35].

The post-inflation scenario has a rich cosmology. During the PQ phase
transition, the value of the axion field becomes randomized in different parts
of the universe. Axion strings, around which the axion field value winds, are
produced through the Kibble-Zurek mechanism. Later, during the QCD phase
transition, the axion acquires a potential and domain walls form that end on
the axion strings. Axion dark matter arises not only through the misalignment
mechanism, but from radiation from the strings after the PQ phase transition
and from the dynamics of the string-domain wall network after the QCD phase
transition.

Domain walls are potentially disastrous for the cosmology of a post-inflation
axion [36], because their energy density redshifts more slowly than that of
matter or radiation and can come to dominate the universe [37]. There are two
mechanisms by which cosmological domain walls can collapse: they can either
have a boundary, ending on a cosmic string[38, 39, 40, 41], or they can be the
boundary of a volume of higher vacuum energy, which exerts a force to collapse
the wall. The former case occurs when the broken discrete symmetry is gauged,
and the latter case occurs when the discrete symmetry is explicitly broken.
Explicit symmetry breaking has been proposed as a solution to the axion domain
wall problem [36, 42, 43, 44], but it has an obvious tension with the quality
problem. PQ-violating terms in the potential must be large enough to cause
domain walls to collapse quickly, but small enough to maintain
$|{\overline{\theta}}|\lesssim 10^{-10}$. A related proposal to solve the
problem is to begin with biased initial conditions, which again can cause
domain walls to collapse [45, 46, 43]. This requires either explicit symmetry
violation at earlier times (again in obvious tension with the quality problem)
or an unconventional cosmology; in any case, it has been argued that the
ensuing domain wall collapse will overproduce axion dark matter [44, 47]. We
will not discuss explicit breaking or biased initial conditions further in
this paper.

In this work, we assume the simplest solution to the axion domain wall
problem: destruction of the axions by a network of cosmic strings that formed
during the PQ phase transition. This mechanism works when a single domain wall
can end on a cosmic string [48]. The number of axion domain walls ending on an
axion string of minimal winding number, often simply called the “domain wall
number,” can be read off from the axion-gluon coupling. Specifically, given a
$2\text{p}$-periodic axion field $\theta(x)$ coupling to gluons via the action

$\frac{k_{G}}{8\text{p}^{2}}\int\theta(x)\,\mathrm{tr}(G\wedge G),$ (1)

we have $k_{G}\in{\mathbb{Z}}$ for topological reasons, and QCD dynamics
generates a potential with the property $V(\theta)=V(\theta+2\text{p}/k_{G})$.
Thus, there are $|k_{G}|$ degenerate minima, and we say the theory has domain
wall number ${N_{\textsc{DW}}}=|k_{G}|$. Hence, the simplest solution to the
axion domain wall problem is to engineer a model with ${N_{\textsc{DW}}}=1$,
and to ensure that axion strings of minimal winding number are produced during
the PQ phase transition.

The post-inflation axion scenario with ${N_{\textsc{DW}}}=1$ is often,
implicitly or explicitly, assumed in discussions of axion phenomenology and
the detection of axion dark matter. In the pre-inflation scenario, a wide
range of axion masses can accommodate the observed dark matter abundance,
depending on initial conditions. The post-inflation axion is often perceived
as more predictive. To quote some recent statements in the literature: “there
is in principle a unique calculable prediction for the axion mass if it is to
make up the complete DM density in such [post-inflation] models, which would
be extremely valuable for experimental axion searches” [49]; “If the PQ
symmetry is broken after the cosmological epoch of inflation, then there is a
unique axion mass $m_{a}$ that leads to the observed DM abundance” [50].
Sophisticated numerical simulations have been undertaken to assess the dark
matter abundance in a post-inflation QCD axion scenario [51, 52, 53, 49, 54,
50], and to predict a specific mass for which axions constitute all of the
dark matter. These computations show that axion strings formed at the PQ phase
transition enter a scaling regime, emitting QCD axion dark matter as the
string network evolves, and then the string-domain wall network tears itself
apart after the QCD phase transition. The axions emitted from string and
domain wall dynamics add to the abundance of dark matter arising from
misalignment. Axion experimentalists take the predictions of such simulations
seriously for determining the optimal mass range to target in their searches.

In this paper, we complicate this picture. We argue that there is a tension
between achieving ${N_{\textsc{DW}}}=1$ and solving the axion quality problem
without introducing other disastrous cosmological problems. We illustrate this
with various examples. While we do not have a general no-go theorem, we also
do not know any example of a QCD axion model without a quality problem that
gives rise to the conventional post-inflation axion cosmology. If there is an
axion theory that solves the quality problem and is not cosmologically
excluded, it seems likely that it also has a non-standard cosmological
history, which would alter the prediction for the axion dark matter abundance.
One message to take away is that experiments searching over a broad mass
range, rather than targeting any theorists’ preference, are vital. Another is
that axion cosmology could differ in a variety of ways from that studied in
existing simulations; for example, there could be a network consisting of
multiple types of axion strings with parametrically different tensions at the
time of the QCD phase transition. Given the computational resources that are
now being deployed to study post-inflation axion cosmology, it would be
worthwhile to carry out a suite of simulations for a wider range of
cosmological scenarios.

This paper is organized as follows. In Sec. 2, we discuss the canonical
solution to the axion quality problem: a discrete ${\mathbb{Z}}_{p}$ symmetry.
We point out a basic cosmological tension. The ${\mathbb{Z}}_{p}$ symmetry
ensures that, at the time of the QCD phase transition, domain walls can form
separating at least $p$ different vacua. When ${\mathbb{Z}}_{p}$ is a gauge
symmetry, these vacua are gauge equivalent and the domain wall can, in
principle, end. However, whether the domain wall can end on a single string,
and whether cosmological dynamics produces such strings, depend on details of
the UV completion. We discuss a specific UV completion in recent literature
where the ${\mathbb{Z}}_{p}$ symmetry is embedded in the center of a
continuous gauge group ${\mathrm{SU}}(N)$ [55]. In Sec. 3, we examine a KSVZ-
type model with a ${\mathrm{U(1)}}$ gauge symmetry to ensure a high-quality
axion and avoid the domain wall problem [19]. We find that the model suffers
from a series of cosmological problems including fractionally charged relics
and Landau poles in the Standard Model couplings below the UV scale. In Sec.
4, we briefly consider a model where the axion quality is protected by a
nonabelian gauge symmetry ${\mathrm{SU}}(n)\times{\mathrm{SU}}(n)$ [56]. The
cosmological considerations are similar to the previous section and we arrive
at qualitatively the same conclusion. In Sec. 5, we comment on the class of
composite axion models, in which we find it is generally hard to avoid the
domain wall problem. We present discussions on other types of models in the
literature in Sec. 6, and pose some open questions on a more holistic approach
to the solution of the axion quality problem, the domain wall problem, and the
cosmological dynamics.

##  2 A ${\mathbb{Z}}_{p}$ symmetry for quality

###  2.1 First look

It is often said that the simplest solution to the axion quality problem is to
impose a ${\mathbb{Z}}_{p}$ discrete symmetry, as first studied in [15]. For
example, if a PQ-charged scalar field $\Phi$ carries charge 1 under such a
symmetry, then all operators of the form $\Phi^{k}/\Lambda^{k-4}$ are
forbidden up to $k=p$. More generally, we will consider $\Phi$ to carry charge
$q$ under the ${\mathbb{Z}}_{p}$ symmetry. Then we define

$d\equiv\gcd(q,p),\qquad{\widetilde{p}}\equiv\frac{p}{d}.$ (2)

In this case, $\Phi^{k}$ terms are forbidden up to $k={\widetilde{p}}$.
Although $q$ is only defined modulo $p$, $d$ and ${\widetilde{p}}$ are well-
defined integers. When $\Phi$ obtains a vacuum expectation value,
${\mathbb{Z}}_{p}$ is spontaneously broken to a residual ${\mathbb{Z}}_{d}$
symmetry that does not act on the axion.

There is a basic tension between such a solution to the quality problem and
the domain wall problem. If the axion arises as the phase of $\Phi$,

$\Phi(x)=\frac{1}{\sqrt{2}}\left(f+s(x)\right){\mathrm{e}}^{{\mathrm{i}}\varphi(x)},$
(3)

then under the generator $z$ of the ${\mathbb{Z}}_{p}$ symmetry
$\Phi\mapsto{\mathrm{e}}^{2\text{p}{\mathrm{i}}q/p}\Phi$ and hence

$z:\qquad\varphi(x)\mapsto\varphi(x)+\frac{2\text{p}q}{p}.$ (4)

(This makes sense as a ${\mathbb{Z}}_{p}$ symmetry action even though $q$ is
only defined mod $p$ because, through its definition as a phase,
$\varphi\cong\varphi+2\text{p}$.) The symmetry (4) allows terms in the
potential of the form $\cos({\widetilde{p}}(\varphi-\varphi_{0}))$, and hence
${\widetilde{p}}$ vacua separated by domain walls. For an exact
${\mathbb{Z}}_{p}$ global symmetry, these are stable domain walls. However, we
don’t expect global symmetries to ever be exact. For a principled mechanism
forbidding higher-dimension operators, we should focus on gauge symmetries.

It is instructive to see how the domain wall number divisible by
${\widetilde{p}}$ arises in a detailed model with a ${\mathbb{Z}}_{p}$ gauge
symmetry. For concreteness, consider a KSVZ-type model with quarks
$Q^{(i)},{\widetilde{Q}}^{(i)}$ carrying ${\mathbb{Z}}_{p}$ gauge charges
$k,-(k+q)$ respectively, with flavor index $i$ spanning a range
$i\in\\{1,\ldots,r\\}$. With this charge assignment, Yukawa terms

$\sum_{i,j}\left(y_{ij}\Phi Q^{(i)}{\widetilde{Q}}^{(j)}+\mathrm{h.c.}\right)$
(5)

are allowed by gauge invariance. (In later such expressions, we will leave the
sum over $i$ and $j$ implicit.) Such a model potentially has a mixed
${\mathbb{Z}}_{p}\text{-}{\mathrm{SU}}(3)_{\textsc{C}}^{2}$ anomaly. In
particular, under a rephasing

$\Phi\mapsto{\mathrm{e}}^{{\mathrm{i}}q\alpha}\Phi,\quad
Q^{(i)}\mapsto{\mathrm{e}}^{{\mathrm{i}}k\alpha}Q^{(i)},\quad{\widetilde{Q}}^{(i)}\mapsto{\mathrm{e}}^{-{\mathrm{i}}(k+q)\alpha}{\widetilde{Q}}^{(i)},$
(6)

the chiral anomaly shifts the action by

$\Delta
I=\frac{1}{8\text{p}^{2}}r\left[k\alpha+(-k-q)\alpha\right]\int\mathrm{tr}(G\wedge
G)=-\frac{rq}{8\text{p}^{2}}\alpha\int\mathrm{tr}(G\wedge G).$ (7)

In order for the ${\mathbb{Z}}_{p}$ transformation to be a valid gauge
symmetry, we require that $\exp[{\mathrm{i}}\Delta I]=1$ when
$\alpha=2\text{p}/p$. The quantization of instanton number
$\frac{1}{8\text{p}^{2}}\int\mathrm{tr}(G\wedge G)\in{\mathbb{Z}}$ then
implies that

$\frac{rq}{p}\in\mathbb{Z},$ (8)

which in turn implies that the number of KSVZ quarks $r$ must be an integer
multiple of ${\widetilde{p}}$. We denote this integer as $s$:

$s\equiv\frac{r}{{\widetilde{p}}}\in{\mathbb{Z}}.$ (9)

Integrating out the heavy modes after $\Phi$ gets a VEV, we obtain an
effective coupling

$\frac{r}{8\text{p}^{2}}\int\varphi(x)\mathrm{tr}(G\wedge G),$ (10)

so the domain wall number $|r|$ associated with $\varphi$ is a multiple of
${\widetilde{p}}$.

###  2.2 Second look: ${\mathbb{Z}}_{p}$ strings

The argument that we have just given is not quite right, although it hints at
an important physical effect. Once we gauge ${\mathbb{Z}}_{p}$, the
identification (4) is a gauge transformation. If all of the vacua separated by
domain walls are gauge equivalent under this transformation, then the domain
wall problem is potentially solved [48]. A minimal domain wall interpolates
between a minimum at some value $\varphi_{0}$ and a neighboring minimum at
$\varphi_{0}+2\text{p}/r$. These two minima are gauge equivalent under
$z^{k}\in{\mathbb{Z}}_{p}$ if

$\frac{2\text{p}}{r}\equiv\frac{2\text{p}kq}{p}\mod
2\text{p}\quad\Rightarrow\quad\frac{2\text{p}}{s{\widetilde{p}}}\equiv\frac{2\text{p}kq}{d{\widetilde{p}}}\mod
2\text{p}.$ (11)

But (for any choice of integer $q$ in its equivalence class) $kq/d$ is an
integer, which means that in order for this expression to hold, $1/s$ must
also be an integer. This is only possible for the special cases $s=\pm 1$;
suppose this is true. Choosing a representative integer $q$ from its
${\mathbb{Z}}_{p}$ equivalence class, we can write $q={\widetilde{q}}d$. The
condition (11) is then equivalent to

$k{\widetilde{q}}\equiv\pm 1\mod{\widetilde{p}}.$ (12)

It is a standard result in modular arithmetic that a solution $k$ exists
precisely when ${\widetilde{p}}$ and ${\widetilde{q}}$ are relatively prime,
which is the case here (by construction). Hence, we conclude that neighboring
minima are gauge equivalent if and only if $|s|=\pm 1$. Indeed, in general,
the ${\mathbb{Z}}_{p}$ orbit generated by (4) includes ${\widetilde{p}}$
distinct gauge equivalent values of $\varphi$, and hence
$|s|=|r|/{\widetilde{p}}$ is the number of distinct gauge orbits of minima.
Thus, $|s|$ is the correct, physical domain wall number in this theory.

We can define a $2\text{p}$-periodic axion field $\theta(x)$ by

$\theta(x)={\widetilde{p}}\varphi(x)=\arg(\Phi(x)^{\widetilde{p}}).$ (13)

In other words, $\theta$ is the phase of the smallest
${\mathbb{Z}}_{p}$-invariant operator constructed from $\Phi$. We will see a
similar pattern in other examples: the $2\text{p}$-periodic axion is often the
phase of the lowest-dimension gauge invariant operator that carries PQ charge
and obtains a vacuum expectation value. In terms of this variable, the
effective coupling (10) is

$\frac{r}{{\widetilde{p}}}\frac{1}{8\text{p}^{2}}\int\theta(x)\mathrm{tr}(G\wedge
G)=\frac{s}{8\text{p}^{2}}\int\theta(x)\mathrm{tr}(G\wedge G),$ (14)

which makes manifest that the true domain wall number is $s$, which could be
$\pm 1$. Thus, a priori, a model with a ${\mathbb{Z}}_{p}$ gauge symmetry
protecting axion quality need not be in conflict with a simple solution to the
axion domain wall problem.

Figure 1: A generic PQ-breaking potential for a complex scalar $\Phi$ with
charge $q$ under a gauged ${\mathbb{Z}}_{p}$ subgroup of
${\mathrm{U(1)}}_{\textsc{PQ}}$. As in the text, we define
$d\equiv\gcd(q,p),{\widetilde{p}}\equiv p/d$. The field
$\theta=\varphi/{\widetilde{p}}$ is $2\text{p}$ periodic, where $\varphi$ is
the phase of $\Phi$. In the illustration, ${\widetilde{p}}=5$. However, at the
time of the PQ phase transition when $\Phi$ obtains a VEV, there is no
dynamical reason for the phase of $\Phi$ to be restricted in the sub-interval
where $\theta\mapsto\theta+2\text{p}$. The cosmic strings produced therefore
wind around the whole $2\text{p}$ period of $\varphi$, which will be connected
to ${\widetilde{p}}$ domain walls after the QCD phase transition.

The trouble is that this argument is kinematic: the symmetries of the problem
allow for the right states to exist to solve the domain wall problem. They do
not, however, guarantee that the dynamics of the theory will solve the domain
wall problem. In particular, at the PQ phase transition, the complex field
$\Phi(x)$ gets a VEV from some potential $V(|\Phi|)$. The Kibble-Zurek
mechanism guarantees that this phase transition will form axion strings around
which the phase $\varphi$ winds from $0$ to $2\text{p}$, as illustrated in
Fig. 1. Equivalently, these are strings for which $\theta(x)$ winds from $0$
to $2\text{p}{\widetilde{p}}$. Such a string can form a junction on which
${\widetilde{p}}$ minimal domain walls end, but it cannot destroy a single
domain wall. In other words, as far as the PQ phase transition is concerned,
we have a theory in which the domain wall number is divisible by
${\widetilde{p}}$.

The ${\mathbb{Z}}_{p}$ gauge symmetry allows for the existence of twist
vortices, dynamical cosmic strings around which fields come back to themselves
only up to a ${\mathbb{Z}}_{p}$ gauge transformation [57]. These objects are
expected to exist in any theory with a ${\mathbb{Z}}_{p}$ gauge symmetry, due
to the absence of global symmetries in quantum gravity [58] (this is one
example of the completeness of the spectrum of quantum gravity [59]). The
existence of these objects, in a theory with ${N_{\textsc{DW}}}=1$, renders
domain walls unstable: there is always some stringlike object (possibly a
composite one) on which a single domain wall can end. Even if these strings
are not populated in the early universe, a domain wall can in principle decay
through the nucleation of a bubble of string within the wall, though this rate
is exponentially suppressed when the string tension is much larger than the
domain wall tension [40], so it does not solve the QCD axion’s domain wall
problem. Instead, we must rely on the cosmological production of appropriate
strings for the wall to end on.

In order for ${\mathbb{Z}}_{p}$ strings to solve the domain wall problem, two
important conditions should be satisfied: there should be a single string on
which a domain wall can end, and these strings should be produced in the early
universe. Neither is guaranteed. For example, consider a minimal domain wall
interpolating between $\varphi=0$ and $\varphi=2\text{p}/{\widetilde{p}}$.
These vacua are equivalent under the gauge transformation by $z^{k}$ with $k$
obeying the condition (12). A ${\mathbb{Z}}_{p}$ string of charge $k$ has the
property that, as we circle the string, the field $\Phi$ maps back to itself
up to a phase $\exp(2\text{p}{\mathrm{i}}kq/p)$. This means that the winding
of $\varphi$ around the $z^{k}$ string is given by

$\oint_{z^{k}}\textrm{d}\varphi=2\text{p}\left(\frac{kq}{p}+n\right),$ (15)

for some integer $n$. There is some choice of $n$ for which this is equal to
the desired $2\text{p}/{\widetilde{p}}$, but whether a string with appropriate
$n$ exists depends on the UV completion of the theory.

(a)

(b)

Figure 2: (a) The potential for $\varphi$ from the phase of $\Phi$, where
$\Phi$ has charge 2 under a ${\mathbb{Z}}_{5}$ gauge symmetry. The action of
the ${\mathbb{Z}}_{5}$ generator $z$ does not map between neighboring vacua in
$V(\varphi)$, but the combination of $z^{3}$ and
$\varphi\mapsto\varphi-2\text{p}$ gauge transformation does. This means that
the object where a single axion domain wall can end on is the bound state of a
$z^{3}$ string and a “$-\varphi$” string (where $\varphi$ winds by
$-2\text{p}$). In isolation, a $z^{3}$ string will have six domain walls
attached to it, five of which can connect to the same $-\varphi$ string to end
the remaining domain wall. Another state on which a single axion domain wall
can end is the bound state of a $z^{2}$ string and a $-\varphi$ string: a
$-\varphi$ string is attached to five domain walls, and four of the five
domain walls are absorbed by the $z^{2}$ string.

To make this more clear, we consider an example: a theory with
${\mathbb{Z}}_{5}$ gauge symmetry under which $\Phi$ has charge 2, and we
suppose that $s=1$. The ${\mathbb{Z}}_{5}$ generator $z$ maps
$\varphi\mapsto\varphi+4\text{p}/5\mod 2\text{p}$. Consider the minimal domain
wall interpolating between $\varphi_{0}$ and $\varphi_{0}+2\text{p}/5$. Modulo
$2\text{p}$, these are related not by the generator $z$ itself, but by
$z^{3}$, because $3\times 2=1+5\cong 1\mod 5$. In (15), then, this example has
$p=5$, $q=2$, $k=3$, and we need $n=-1$ to obtain net winding
$\oint_{z^{3}}\textrm{d}\varphi=2\text{p}\left(3\times
2/5-1\right)=2\text{p}/5$. There may be a model where a string producing such
winding exists as a single object, perhaps even an elementary one. (Because
$z^{3}$ also generates ${\mathbb{Z}}_{5}$, there is no invariant meaning to
our choice of $z$ as a generator, and no reason to favor the case where the
strings with holonomy $z$ are fundamental.) On the other hand, it may be that
the only elementary ${\mathbb{Z}}_{5}$ string implements the operation $z$
such that the winding is $\oint_{z}\textrm{d}\varphi=4\text{p}/5$. In this
case, the $z^{3}$ string could arise by stacking three such $z$ strings on top
of each other, generating winding
$\oint_{z^{3}}\textrm{d}\varphi=12\text{p}/5$. This overshoots the desired
$2\text{p}/5$ winding, and we require an additional shift in $\varphi$ back
$2\text{p}$, as shown in Fig. 2(a). In this case, the desired string on which
a single axion domain wall can end is the bound state of a $z^{3}$ string and
(in an abuse of notation) a $-\varphi$ string where $\varphi$ winds by
$-2\text{p}$. A $z^{3}$ string has six domain walls attached to it, and five
of them are absorbed by the $-\varphi$ string, as illustrated in Fig.
2(b).222Alternatively, because $z^{2}$ is the inverse element of $z^{3}$ in
${\mathbb{Z}}_{5}$, one could stack two anti-$z$ strings, leading to net
winding $-8\text{p}/5$, and one $+\varphi$ string of winding $2\text{p}$ for a
net $2\text{p}/5$. A similar picture results. While $-\varphi$ strings will be
produced by the PQ phase transition following the familiar story of cosmic
string production from a spontaneously broken global ${\mathrm{U(1)}}$
symmetry, the expected abundance for $z^{3}$ strings can only be determined
within a UV completion. Fig. 2(b) is consistent with the general claim that
the domain wall can end when ${N_{\textsc{DW}}}=1$. However, the messy,
composite configuration on which it ends is unlikely to arise dynamically.
More likely, if there is a large abundance of $z^{3}$ strings and $\varphi$
strings, one would form a large, frustrated network and the domain wall
network would not collapse. This is because, effectively, the picture is much
like that for ${N_{\textsc{DW}}}>1$: any given elementary string is attached
to multiple domain walls. In summary, the cosmological consequences of a
discrete gauge symmetry depend on details of the UV completion that cannot be
deduced from the symmetry structure alone.

For the moment, suppose that the requisite ${\mathbb{Z}}_{p}$ strings to
destroy minimal domain walls are somehow produced cosmologically, so that at
the time of the QCD confining transition, the universe is filled with a
network of them. Axion domain walls can end on these strings, solving the
domain wall problem. However, there is an important physical difference from
the standard scenario. Because the ${\mathbb{Z}}_{p}$ strings are not produced
in the PQ phase transition, there is no reason for their tension $\mu$ to be
tied to the scale $f_{a}$ of the axion decay constant. In principle, they
could have a much larger tension. The string network radiates axions
continuously as it evolves, and further emits axions when the string-domain
wall network fragments, all of which enhance the abundance of axion dark
matter. We will follow a recent discussion in [50]. The rate of axion emission
from strings is $\Gamma_{a}\approx 2H\rho_{s}$, with $\rho_{s}$ the energy
density in axion strings. This is directly proportional to the string tension:
$\rho_{s}=\xi\mu/t^{2}$, where $\xi$ is the string length per Hubble volume
and is expected to be only logarithmically sensitive to the string tension or
other detailed physics in the string core. The final axion abundance is
largely determined by this process and proportional to $\Gamma_{a}$. The
estimate of [50] assumes a string tension $\mu_{0}=\text{p}f_{a}^{2}$. If the
${\mathbb{Z}}_{p}$ string tension $\mu$ is much larger, then a first estimate
is that the axion dark matter abundance is enhanced by a factor of
$\mu/\mu_{0}$. (A similar discussion of the role of higher tension strings in
enhancing the axion emission rate appeared in [60], as this paper was being
completed.) This, correspondingly, implies that the correct axion dark matter
abundance is attained for a smaller value of $f_{a}$ and a larger axion mass
than in the conventional axion cosmology. We emphasize that this is just a
crude scaling argument, and the additional physics needed to construct a model
in which the requisite population of ${\mathbb{Z}}_{p}$ strings is produced
cosmologically could further complicate the story.

Before moving on, let us make a brief comment about Peccei-Quinn charges. In
the above discussion, we have never explicitly referred to a
${\mathrm{U(1)}}_{\textsc{PQ}}$ global symmetry. We could do so as follows:
the couplings (5) respect a chiral symmetry under which $Q^{(i)}$ has charge
$p_{Q}=+1$, $\Phi$ has charge $p_{\Phi}=-1$, and ${\widetilde{Q}}^{(j)}$ has
charge $p_{\widetilde{Q}}=0$. (They also respect a “heavy baryon number”
symmetry under which $Q^{(i)}$ has charge $+1$ and ${\widetilde{Q}}^{(j)}$ has
charge $-1$; any linear combination of this and the aforementioned chiral
symmetry could be viewed as a Peccei-Quinn symmetry.) A standard formula is
that the axion-gluon coupling is given by $k_{G}=\sum_{i}2I(r_{i})p_{i}$,
where the sum is over left-handed Weyl fermions with PQ charge $p_{i}$ and
${\mathrm{SU}}(3)_{\textsc{C}}$ representation $r_{i}$. In our example, this
computation gives $k_{G}=r$, as in (10). In order to compute the properly
normalized coupling $k_{G}=r/{\widetilde{p}}=s$ as in (14), then, we should
normalize the PQ charge so that $p_{\Phi}=-1/{\widetilde{p}}$, i.e., it is the
operator $\Phi^{\widetilde{p}}$ that has unit magnitude PQ charge. This may
seem unusual, since we customarily normalize ${\mathrm{U(1)}}$ charges to be
integers so that ${\mathrm{e}}^{{\mathrm{i}}\alpha}\in{\mathrm{U(1)}}$ has a
well-defined action on a field as
$\psi\mapsto{\mathrm{e}}^{{\mathrm{i}}q\alpha}\psi$. However, fractional
normalizations of charges under a global ${\mathrm{U(1)}}$ symmetry are often
a useful convention in the case where a discrete subgroup of ${\mathrm{U(1)}}$
is gauged. A familiar example is baryon number symmetry, where we normalize
quark fields to have baryon number $1/3$, the ${\mathbb{Z}}_{3}$ center of
${\mathrm{SU}}(3)_{\textsc{C}}$ acts on the quarks in the same manner as the
${\mathbb{Z}}_{3}$ subgroup of baryon number, and the gauge invariant baryon
operators have baryon number 1. The utility of fractional PQ charge
assignments will become important in Sec. 5 below.

###  2.3 ${\mathbb{Z}}_{p}$ embedded in continuous gauge groups

We consider an illustrative class of models in a recent work [55], which aims
to produce the required ${\mathbb{Z}}_{p}$ string at the time of the PQ phase
transition. The string tension $\mu$ is then tied with $f_{a}$, and the model
regains the possibility of a precise prediction of the axion mass to produce
the correct dark matter abundance. In this class of models, the axion arises
from a scalar ${\mathcal{S}}$ spontaneously breaking the gauge group
${\mathrm{SU}}(N)$. ${\mathcal{S}}$ lives in either the symmetric or
antisymmetric two-index representation of ${\mathrm{SU}}(N)$, which lead to
different types of models as we will discuss shortly. These models are
realizations of the Lazarides-Shafi idea first established in [48], where the
domain wall problem is solved by embedding the discrete $\mathbb{Z}_{p}$
subgroup of ${\mathrm{U(1)}}_{\mathrm{PQ}}$ in a continuous gauge group.
However, upon closer examination of the models, we find that some subtleties
have been overlooked and the embedding is not necessarily sufficient to solve
the domain wall problem.

The fermion content and charge assignments of the models are universal
regardless of whether ${\mathcal{S}}$ is a symmetric or antisymmetric tensor.
The fermions include ${\mathrm{SU}}(N)$ fundamental quarks $Q$ (in the
$\bm{3}$ of ${\mathrm{SU}}(3)_{\textsc{C}}$) and ${\widetilde{Q}}$ (in the
$\bm{\overline{3}}$) which we can take to have hypercharge $Y=\mp 1/3$, and
${\mathrm{SU}}(N)$ antifundamental leptons $L^{(i)}$ and $\widetilde{L}^{(i)}$
($i=1,2,3$), which we can take to have $Y=0$. These have Yukawa interactions
with ${\mathcal{S}}$,

$\mathcal{L}\supset-
y_{Q}Q{\mathcal{S}}^{*}{\widetilde{Q}}-\left(y_{L}\right)_{ij}L^{(i)}{\mathcal{S}}\widetilde{L}^{(j)}+\mathrm{h.c.}$
(16)

The potential for ${\mathcal{S}}$ is constructed such that it obtains a VEV

$\expectationvalue{{\mathcal{S}}}=v_{{\mathcal{S}}}\left(\mathds{1}_{N/2\times
N/2}\otimes\epsilon\right)$ (17)

when ${\mathcal{S}}$ is in the antisymmetric two-index representation and $N$
is even, which spontaneously breaks ${\mathrm{SU}}(N)$ to
${\mathrm{Sp}}(N),$333When $N$ is odd the symmetry breaking pattern is
different and there is no light axion [61]. and

$\expectationvalue{{\mathcal{S}}}=v_{{\mathcal{S}}}\mathds{1}_{N\times N}$
(18)

when ${\mathcal{S}}$ is in the symmetric two-index representation, which
spontaneously breaks ${\mathrm{SU}}(N)$ to ${\mathrm{SO}}(N)$. In both cases,
we will denote the overall phase of ${\mathcal{S}}$ by $\varphi(x)$; that is,
we take

$\varphi(x)=\frac{1}{N}\arg\det{\mathcal{S}}.$ (19)

Performing a chiral rotation on the heavy quarks to remove $\varphi(x)$ from
the Yukawa terms, we obtain the $\varphi(x)$ coupling to gluons

$\mathcal{L}\supset\frac{N}{8\text{p}^{2}}\int\varphi(x)\mathrm{tr}(G\wedge
G).$ (20)

The domain wall number with respect to a $2\text{p}$ period of $\varphi(x)$ is
thus $N$, i.e., neighboring vacua in the $\varphi(x)$ potential are separated
by $2\text{p}/N$.

From the discussion in Sec. 2.1, we know that this is not the whole story,
since the $N$ vacua could be gauge equivalent. To see if that is the case, we
need to look at the lowest-dimensional ${\mathrm{U(1)}}_{\mathrm{PQ}}$-charged
operator that is consistent with the gauge symmetry of the model. Now we will
look at the antisymmetric and symmetric representations of ${\mathcal{S}}$
separately. When ${\mathcal{S}}$ is in the antisymmetric representation and
$N$ is even, the lowest-dimensional operator is the Pfaffian,
$\mathrm{Pf}\,{\mathcal{S}}=\sqrt{\det{\mathcal{S}}}=|\sqrt{\det{\mathcal{S}}}|{\mathrm{e}}^{{\mathrm{i}}N\varphi(x)/2}$.
The minimal $2\text{p}$-invariant variable with no gauge-equivalent sub-
interval is therefore $\theta(x)=N\varphi(x)/2$, which we will identify as the
axion in our theory. By definition, all cosmic strings must wind by an integer
multiple of $2\text{p}$ in $\theta(x)$. Since

$\mathcal{L}\supset\frac{N}{8\text{p}^{2}}\int\varphi(x)\mathrm{tr}(G\wedge
G)=\frac{2}{8\text{p}^{2}}\int\theta(x)\mathrm{tr}(G\wedge G),$ (21)

we see that the model has a domain wall number of 2, thus the domain wall
problem is not solved. The issue is that even though there is an anomaly-free
${\mathbb{Z}}_{N}$ inside ${\mathrm{U(1)}}_{\textsc{PQ}}$, only the
${\mathbb{Z}}_{N/2}$ subgroup is gauged via its embedding in the center
${\mathbb{Z}}_{N}\subset{\mathrm{SU}}(N)$ by the model construction.

When ${\mathcal{S}}$ is in the symmetric representation, the lowest-
dimensional PQ-breaking operator is the determinant of ${\mathcal{S}}$,
$\det{\mathcal{S}}=|\det{\mathcal{S}}|{\mathrm{e}}^{{\mathrm{i}}N\varphi(x)}$,
for both even and odd $N$. It is tempting to say then that the minimal
$2\text{p}$ periodic variable is $\theta(x)=N\varphi(x)$. However, when $N$ is
even, this is not correct: ${\mathcal{S}}$ being a two-index symmetric
representation of ${\mathrm{SU}}(N)$, it has charge 2 under the
${\mathbb{Z}}_{N}$ center of ${\mathrm{SU}}(N)$ and is invariant under the
${\mathbb{Z}}_{2}$ subgroup of ${\mathbb{Z}}_{N}$. More specifically, under a
$z^{k}\in{\mathbb{Z}}_{N}$ transformation,
$\varphi(x)\mapsto\varphi(x)+4\text{p}k/N$, which cannot relate $\varphi(x)$
with the neighboring vacua $\varphi(x)\pm 2\text{p}/N$ when $N$ is even. The
minimal $2\text{p}$ periodic variable in the model is thus
$\theta(x)=N\varphi(x)/2$. Similar to the antisymmetric case, we have

$\mathcal{L}\supset\frac{N}{8\text{p}^{2}}\int\varphi(x)\mathrm{tr}(G\wedge
G)=\frac{2}{8\text{p}^{2}}\int\theta(x)\mathrm{tr}(G\wedge G),$ (22)

and the domain wall number of the model is again ${N_{\textsc{DW}}}=2$.

When $N$ is odd, there is no ${\mathbb{Z}}_{2}$ subgroup of
${\mathbb{Z}}_{N}$, and a $z^{k}\in{\mathbb{Z}}_{N}$ transformation will be
able to reach all vacua starting from $\varphi(x)$. In this case, we take
$\theta(x)=N\varphi(x)$, and we have ${N_{\textsc{DW}}}=k_{G}=1$. All domain
walls are unstable in this theory. However, referring back to our example of a
${\mathbb{Z}}_{5}$ symmetry in Sec. 2.2, the minimal string for the
${\mathbb{Z}}_{N}$ center cannot destroy a single domain wall, because the
minimal string winds $\varphi$ from 0 to $4\text{p}/N$ rather than
$2\text{p}/N$. The strings that may play a dynamical role in destroying domain
walls are _non-minimal_ , as they are composite objects of $(k+1)$ minimal $z$
strings (or equivalently, $k$ minimal anti-strings), where $N=2k+1$, together
with $\varphi$ strings. We can make this more explicit by studying the
fundamental group involved in the Kibble-Zurek mechanism. The potential for
${\mathcal{S}}$ is built out of the invariants
$\mathrm{tr}({\mathcal{S}}{\mathcal{S}}^{\dagger})$ and
$\mathrm{tr}({\mathcal{S}}{\mathcal{S}}^{\dagger}{\mathcal{S}}{\mathcal{S}}^{\dagger})$.
This potential respects a global symmetry: it is invariant under
${\mathcal{S}}\mapsto U{\mathcal{S}}U^{\top}$ with $U\in\mathrm{U}(N)$, but
the element $U=-\mathds{1}_{N\times N}$ acts trivially on ${\mathcal{S}}$.
Thus, the faithfully acting global symmetry respected by the potential is
$G_{V}=\mathrm{U}(N)/{\mathbb{Z}}_{2}$. After symmetry breaking, the subgroup
of $G_{V}$ leaving the vacuum expectation value (18) invariant is
$H_{V}=\mathrm{O}(N)/{\mathbb{Z}}_{2}$. Part of these groups are gauged in our
theory, but we expect that the topological defects arising when
${\mathcal{S}}$ relaxes to the minimum of the potential are dominantly
determined by $G_{V}$ and $H_{V}$, independently of the gauging. The axion
string configurations formed during the phase transition are determined by
$\pi_{1}(G_{V}/H_{V})$. Using
$\mathrm{U}(N)\cong[\mathrm{SU}(N)\times{\mathrm{U(1)}}]/{\mathbb{Z}}_{N}$, we
can construct noncontractible loops in $G_{V}$ itself as the projections of
the following maps from $[0,1]$ into $\mathrm{SU}(N)\times{\mathrm{U(1)}}$:

$\displaystyle z(t)$
$\displaystyle=\left(\exp\left[-{\mathrm{i}}t\frac{2\text{p}}{N}\begin{pmatrix}1&&&&\\\
&1&&&\\\ &&\ddots&&\\\ &&&1&\\\
&&&&-(N-1)\end{pmatrix}\right],\exp(2\text{p}{\mathrm{i}}t/N)\right),$ (23)
$\displaystyle w(t)$
$\displaystyle=\left(\mathds{1},\exp(\text{p}{\mathrm{i}}t)\right).$ (24)

The first path, $z(t)$, winds from the origin of ${\mathrm{SU}}(N)$ to the
first element of the ${\mathbb{Z}}_{N}$ center, and also winds $1/N$ of the
way around ${\mathrm{U(1)}}$. The ${\mathbb{Z}}_{N}$ quotient in $U(N)$
ensures that this forms a loop in $G_{V}$. The action ${\mathcal{S}}\mapsto
U{\mathcal{S}}U^{\top}$ implies that the corresponding winding of $\varphi$ is
$4\text{p}/N$. The second path, $w(t)$, winds from the identity element of
$U(1)$ to the element $-1$, while remaining constant in ${\mathrm{SU}}(N)$.
The ${\mathbb{Z}}_{2}$ quotient in the definition of $G_{V}$ makes this a
loop. The action on ${\mathcal{S}}$ corresponds to a full $2\text{p}$ winding
of $\varphi$. Both homotopy classes $[z]$ and $[w]$ generate ${\mathbb{Z}}$
subgroups of $\pi_{1}(G_{V})$, but these are not independent, because
$[z]^{N}$ is homotopic to $[w]^{2}$. We conclude that
$\pi_{1}(G_{V})\cong{\mathbb{Z}}$. We have
$\pi_{1}(H_{V})\cong{\mathbb{Z}}_{2}$ and $\pi_{0}(H_{V})\cong 0$ (because the
quotient by ${\mathbb{Z}}_{2}$ identifies the two disconnected components of
$O(N)$). Then the homotopy exact sequence
$\cdots\rightarrow\pi_{1}(H_{V})\rightarrow\pi_{1}(G_{V})\rightarrow\pi_{1}(G_{V}/H_{V})\rightarrow\pi_{0}(H_{V})\rightarrow\cdots$,
together with the fact that the only homomorphism
${\mathbb{Z}}_{2}\to{\mathbb{Z}}$ is 0, implies that
$\pi_{1}(G_{V}/H_{V})\cong\pi_{1}(G_{V})\cong{\mathbb{Z}}$.

The physical meaning of the above analysis is that the axion strings formed by
the Kibble-Zurek mechanism in this model are classified by an integer
topological charge, labeled by the winding of $\varphi$, which comes in units
of $2\text{p}/N$. However, just as in the discussion of Sec. 2.2, this does
not mean that a string of minimal winding is an elementary object. The paths
that achieve minimal winding of $\varphi$ have, from the UV viewpoint,
nontrivial winding in both ${\mathrm{SU}}(N)$ and ${\mathrm{U(1)}}$; they are
composites of the $z(t)$ path (23) (analogous to the $z$ strings of Sec. 2.2)
and the $w(t)$ path (24) (analogous to the $\varphi$ strings of Sec. 2.2). The
string with minimal winding, then, may only exist as a collection of simpler
objects, much like the depiction in Fig. 2(b).

It seems likely that the cosmological dynamics produces primarily simple
strings, $z$ strings generated by the winding (23) and $\varphi$ strings
generated by the winding (24), rather than a single string with minimal
$\varphi$ winding (but much more complicated winding within $G_{V}$). Thus, we
expect that the model has a domain wall problem, unless there is a nontrivial
dynamical mechanism populating the appropriate composite strings. It would be
worthwhile to explicitly construct semiclassical strings solutions of
different winding, and study their relative tensions. The complete dynamics of
string formation may only be definitively answered by dedicated numerical
simulations.

The class of models in Ref. [55] presents one type of embedding of
${\mathbb{Z}}_{p}$ symmetry in a continuous gauge group, and we have discussed
how some nontrivial dynamical questions about the cosmic strings need to be
answered before the domain wall problem can be declared solved. The models do
withstand an examination of other cosmological considerations, such as
existing constraints on fractionally charged relics and avoiding a Landau pole
in the Standard Model couplings below the Planck scale.444There is another
interesting dynamical question about the fate of ${\mathrm{SO}}(N)$ magnetic
monopoles below the confinement scale, briefly discussed in Ref. [55] but
deserving further attention. For odd $N$, ${\mathrm{SO}}(N)$ has trivial
center and admits ${\mathbb{Z}}_{2}$ monopoles thanks to the double cover of
${\mathrm{SO}}(N)$ by the simply connected group $\mathrm{Spin}(N)$. Such
monopoles are produced as topological defects when ${\mathrm{SU}}(N)$ is
higgsed to ${\mathrm{SO}}(N)$. One heuristic argument given in Ref. [55] is
that confinement is dual to higgsing, and so below the ${\mathrm{SO}}(N)$
confinement scale, the magnetic charge is higgsed and monopoles should decay.
On the other hand, because there is no center symmetry, we do not expect any
order parameter for confinement to exist, so it would be surprising if
confinement caused monopoles to decay. The fate of the monopoles is important
for the cosmology of the model, but unrelated to the concerns discussed in
this paper, so we will not attempt to definitively answer the question here.
In the following sections we discuss other promising models that embed
${\mathbb{Z}}_{p}$ symmetry in more complicated symmetry structures. Some of
them have greater success in solving the domain wall problem than the class of
models discussed here. However, it turns out that they suffer from other
cosmological considerations that cannot be simultaneously satisfied with the
axion quality constraint. In other words, the axion quality problem is in
tension with conventional post-inflation axion cosmology.

##  3 A ${\mathrm{U(1)}}$ symmetry for quality

In this section, we review a KSVZ-type model proposed by S. M. Barr and D.
Seckel in Ref. [19]. The authors argue that this model has a high-quality
${\mathrm{U(1)}}_{\textsc{PQ}}$ symmetry and the correct dynamics of string
formation to ensure ${N_{\textsc{DW}}}=1$. In Sec. 3.1, we review the field
content and Lagrangian of the model, and discuss its solution to the quality
problem. We discuss the string formation dynamics and show that the solution
to the domain wall problem is much more complicated than originally claimed in
[19]. We see that this solution reduces to a ${\mathbb{Z}}_{p}$ symmetry
solution in a certain limit. In Sec. 3.2, we show that the model also suffers
from a series of cosmological problems, including fractionally charged relics
and Landau poles in the SM gauge couplings below the UV scale.

###  3.1 Review of the Barr-Seckel Model

The Barr-Seckel model [19] introduces two complex scalars $\Phi_{p}$ and
$\Phi_{q}$, which are singlets under the Standard Model gauge group. The
subscripts indicate their respective charges under an additional
${\mathrm{U(1)}}^{\prime}$ gauge symmetry. The model also introduces $p+q$
copies of left-handed quarks (by which we mean they are in the fundamental
representation of ${\mathrm{SU}}(3)_{\textsc{C}}$) $Q_{0}^{(i)}$, $q$ copies
of left-handed antiquarks (by which we mean they are in the anti-fundamental
representation of ${\mathrm{SU}}(3)_{\textsc{C}}$)
${\widetilde{Q}}_{p}^{(i)}$, and $p$ copies of left-handed antiquarks
${\widetilde{Q}}_{-q}^{(i)}$. They are singlets under
${\mathrm{SU}}(2)_{\textsc{L}}\times{\mathrm{U(1)}}_{\textsc{Y}}$ and the
subscripts indicate their respective ${\mathrm{U(1)}}^{\prime}$ charges. The
multiplicities are necessitated by cancellation of
${\mathrm{SU}}(3)_{\textsc{C}}^{3}$ and
${\mathrm{SU}}(3)_{\textsc{C}}^{2}{\mathrm{U(1)}}^{\prime}$ anomalies.
(Cancellation of the ${\mathrm{U(1)}}^{\prime 3}$ anomaly can be accomplished
with spectator fermions that play no role in the remainder of the discussion.)

We assume that the complex scalars $\Phi_{p}$ and $\Phi_{q}$ have the
appropriate potential to undergo spontaneous symmetry breaking with VEVs
$v_{p}$ and $v_{q}$ respectively. Aside from the relationship between $v_{p}$
and $v_{q}$, the specific form of the scalar potential will not be relevant
for our discussions. We will assume $v_{p}>v_{q}$ without loss of generality.

Given the ${\mathrm{U(1)}}^{\prime}$ charge assignments, the Yukawa
interactions that give the quarks masses after the scalars get VEVs are
therefore

$\mathcal{L}_{\textsc{BS}}\supset(y_{p})_{ij}Q_{0}^{(i)}{\widetilde{Q}}_{p}^{(j)}\Phi_{p}^{\dagger}+(y_{q})_{ij}Q_{0}^{(i)}{\widetilde{Q}}_{-q}^{(j)}\Phi_{q}+\mathrm{h.c.}$
(25)

We will denote the phase degrees of freedom of the two scalars by $\theta_{p}$
and $\theta_{q}$. Removing the $\theta_{p}$ degree of freedom from the Yukawa
terms then requires chiral rotation of $q$ copies of
$Q_{0}{\widetilde{Q}}_{p}$ pairs, while removing the $\theta_{q}$ degree of
freedom requires opposite chiral rotation of $p$ copies of
$Q_{0}{\widetilde{Q}}_{-q}$ pairs. The coupling of the scalar phases to the
gluon is thus given by

$\mathcal{L}_{\textsc{BS}}\supset\frac{q\theta_{p}-p\theta_{q}}{8\text{p}^{2}}\int\mathrm{tr}(G\wedge
G)\equiv\frac{\theta}{8\text{p}^{2}}\int\mathrm{tr}(G\wedge G),$ (26)

and we identify the $2\text{p}$ periodic phase $\theta\equiv
q\theta_{p}-p\theta_{q}$ as the axion degree of freedom that lives in
${\mathrm{U(1)}}_{\textsc{PQ}}$, at energy scale below both $v_{p}$ and
$v_{q}$. The axion decay constant is given by

$f_{a}^{2}=\frac{v_{p}^{2}v_{q}^{2}}{p^{2}v_{p}^{2}+q^{2}v_{q}^{2}}.$ (27)

The definition of $\theta$ makes it manifestly invariant under
${\mathrm{U(1)}}^{\prime}$, as it should be: the two phase rotation symmetries
$\Phi_{p}\to{\mathrm{e}}^{{\mathrm{i}}\theta_{p}}\Phi_{p}$ and
$\Phi_{q}\to{\mathrm{e}}^{{\mathrm{i}}\theta_{q}}\Phi_{q}$ have been
reorganized into an anomaly-free linear combination that we have gauged and
been calling ${\mathrm{U(1)}}^{\prime}$ and an anomalous orthogonal linear
combination which we identified as ${\mathrm{U(1)}}_{\textsc{PQ}}$. All charge
assignments of the new fields are collected in Table 1.

We will assume that $p$ and $q$ are co-prime integers, such that the lowest
order operator that breaks ${\mathrm{U(1)}}_{\textsc{PQ}}$ symmetry is
$(\Phi_{p}^{\dagger})^{q}(\Phi_{q})^{p}$. The correction to the axion
potential induced by this operator is proportional to
$v_{p}^{q}\,v_{q}^{p}/\Lambda^{p+q-4}$, where $\Lambda$ is the UV scale. The
Barr-Seckel model ensures a high quality ${\mathrm{U(1)}}_{\textsc{PQ}}$
symmetry when $p+q$ is sufficiently large. For example, for $v_{p}\sim
v_{q}\sim f_{a}$ and the UV scale set to the Planck mass
$\Lambda=M_{\mathrm{Pl}}$, we need $p+q\geq 10$ for
$10^{10}\;\text{GeV}<f_{a}<10^{12}\;\text{GeV}$ so that $\bar{\theta}$ will
come out sufficiently small.

$\begin{array}[]{ c|c c c c c|c
}&\text{spin}&{\mathrm{SU}}(3)_{\textsc{C}}&{\mathrm{U(1)}}_{\textsc{Y}}&{\mathrm{U(1)}}^{\prime}&\text{copies}&{\mathrm{U(1)}}_{\textsc{PQ}}\\\
\hline\cr Q_{0}&1/2&\bm{3}&0&0&(p+q)&c\\\
\hline\cr{\widetilde{Q}}_{p}&1/2&\bm{\overline{3}}&0&p&q&a-c\\\
{\widetilde{Q}}_{-q}&1/2&\bm{\overline{3}}&0&-q&p&-b-c\\\
\hline\cr\Phi_{p}&0&\bm{1}&0&p&1&a\\\ \Phi_{q}&0&\bm{1}&0&q&1&b\\\
\hline\cr\end{array}$ Table 1: We list the beyond-SM field content of the
Barr-Seckel model [19]. All the fields here are singlets of
${\mathrm{SU}}(2)_{\textsc{L}}\times{\mathrm{U(1)}}_{\textsc{Y}}$. The
specific choice of $c$ does not matter, as long as the combination
$Q_{0}{\widetilde{Q}}_{p}$ has PQ charge $a$ and $Q_{0}{\widetilde{Q}}_{-q}$
has PQ charge $-b$. Different choices of $c$ corresponds to
${\mathrm{U(1)}}_{\textsc{PQ}}\rightarrow{\mathrm{U(1)}}_{\textsc{PQ}}+3c{\mathrm{U(1)}}_{B}$,
where ${\mathrm{U(1)}}_{B}$ is the heavy baryon number. We need $p,q$ to be
co-prime and $aq-bp=1$.

At a first glance, the setup by [19] seems to be exactly what we want: a high
quality axion with ${N_{\textsc{DW}}}=1$, since the coefficient of the
$\theta$ coupling to the gluon is $1$. However, again, the problem with the
construction of ${N_{\textsc{DW}}}=1$ is that it’s only kinematic. The fact
that the minimal axion $\theta$ with minimal coupling to gluons exists does
not mean that the correct string where $\theta$ winds from 0 to $2\text{p}$
will form in the early universe. Indeed, from the definition of
$\theta=q\theta_{p}-p\theta_{q}$, we see that $\theta$ winding from 0 to
$2\text{p}$ actually translates to nontrivial windings for the scalar phases
$\theta_{p}$ and $\theta_{q}$. In particular, $\theta_{p}$ and $\theta_{q}$
must have higher winding numbers $a$ and $b$, such that $aq-bp=1$. Because $p$
and $q$ are assumed to be co-prime, such integers $a$ and $b$ always exist,
but the challenge is to ensure that the composite $\Phi_{p}$-$\Phi_{q}$ string
with $\theta_{p}$ winding number $a$ and $\theta_{q}$ winding number $b$ will
actually form in the early universe.

Without loss of generality, consider $v_{p}>v_{q}$. At the time of the first,
$\Phi_{p}$ phase transition, the familiar ${\mathrm{U(1)}}$ gauge strings are
produced. Unless the now massive gauge boson is much heavier than the radial
mode of $\Phi_{p}$, we expect the $\Phi_{p}$ strings to be dominantly winding
number 1.555When the gauge boson is much heavier, we expect string-string
interactions to be attractive independent of relative orientation, which may
enable mergers forming higher winding strings [62, 63, 64]. Therefore, we are
now limited to values of $p$ and $q$ where an integer $b$ exists such that
$q-bp=1$. Later, after $\Phi_{q}$ undergoes its phase transition, the universe
will have strings with either $\Phi_{p}$ or $\Phi_{q}$ windings or both. We
will denote the strings by their $\Phi_{p}$ and $\Phi_{q}$ winding numbers,
$(n,m)$, where $n$ is either $0$ or $\pm 1$, while we leave the possibility of
any integer for $m$ for now. The amount of axion winding around an $(n,m)$
string is $mp-nq$, which is nonzero for generic values of $(n,m)$. This
determines the number of minimal domain walls that can end on a given string,
which we will refer to as the domain wall number of the string and denote
$w=|mp-nq|$ (as opposed to the domain wall number ${N_{\textsc{DW}}}$ of the
theory as a whole). In particular, $(1,m)$ strings will have both a nontrivial
gauge field flux _and_ a nontrivial axion winding, which gives it a
logarithmically divergent tension [65, 66, 67, 68, 60].

In [19], it is argued that at the time of the $\Phi_{q}$ phase transition, in
the presence of a $\Phi_{p}$ string, only $(1,b)$ strings will form since
$m=b$ minimizes the logarithmic divergence proportional to
$|m-\frac{q}{p}|^{2}$. However, this cannot be the case, since formation of
topological defects is an inherently non-local process, where defects emerge
from causally disconnected patches randomly falling into different locations
in the vacuum manifold. Causally disconnected regions should not be able to
communicate with each other to minimize their total energy. Moreover, if there
exists a process that could dynamically change the winding number to minimize
energy at the time of string formation, cosmic strings would not exist at all,
since the true minimal energy state is the vacuum with zero winding
number.666Strings can certainly split or merge to minimize energy, but that
happens strictly after the strings have already formed from the phase
transition.

Many nontrivial questions about the $(1,b)$ string production thus remain: how
many $(1,b)$ strings actually form at the time of the $\Phi_{q}$ phase
transition? Would the abundance of $(1,b)$ string increase later due to
mergers between $(1,m\neq b)$ strings with $(0,\pm 1)$ strings? Compared to
pure ${\mathrm{U(1)}}$ gauge strings where the kinematic constraints for
mergers have been worked out in full detail [69, 70], mergers of strings with
logarithmically divergent tensions are much less understood. Refs. [67, 68]
have conducted simulations in the case of $p=1$ and $q=4$, which showed that
string mergers preferentially generated $(1,4)$ strings that have domain wall
number $w=0$, instead of the desired $(1,3)$ strings with domain wall number
$w=1$. However, for generic values of $p$ and $q$, there is no $(1,m)$ string
with domain wall number $w=0$, and the question remains whether string mergers
preferentially generate the $(1,b)$ string. Lastly, there is also the question
of what is the minimal abundance of $(1,b)$ ($w=1$) strings needed to
annihilate domain walls, relative to other types of strings with $w\neq 1$. As
we will see in the next section, even if all the questions about the string
dynamics are answered and the domain wall problem is resolved, the model
suffers from additional cosmological problems.

In the limit $v_{p}\gg v_{q}$, it is easy to see that the Barr-Seckel model
solution to the axion quality problem reduces to a discrete ${\mathbb{Z}}_{p}$
symmetry. We effectively integrate out the scalar with heavier VEV,
$\Phi_{p}$, and the gauged ${\mathrm{U(1)}}^{\prime}$ reduces to a
${\mathbb{Z}}_{p}$ symmetry in the vacuum. The leftover dynamical scalar
$\Phi_{q}$ has non-minimal charge $q$ mod $p$ under the ${\mathbb{Z}}_{p}$
symmetry. The dynamics of the string formation in this limit is exactly as
discussed in Sec. 2.2. In the simplest case of $q=1$, the first phase
transition at $v_{p}$ exactly produces the correct ${\mathbb{Z}}_{p}$ strings
to annihilate domain walls, where the second phase transition forms strings
where $p$ domain walls join. Again the model suffers from the fact that the
wall-annihilating ${\mathbb{Z}}_{p}$ strings have tensions $\sim v_{p}$ much
higher and uncorrelated with $f_{a}\sim v_{q}$, and the axion abundance from
this model depends on both scales, preventing a precise prediction of the
axion mass. For generic values of $q$ and $p$, as discussed in Sec. 2.2,
whether all minimal domain walls can be destroyed to prevent cosmological
disaster depends on more complicated questions about string dynamics. In this
limit, we would need $p$ to be large to ensure a good quality.

All the discussion above is easy to generalize to the scenario where $p$ and
$q$ have greatest common divisor $d$, so that we write $p={\widetilde{p}}d$
and $q={\widetilde{q}}d$. This is very similar to the discussion in Sec. 2.
The lowest-dimension PQ-violating operator is now
$(\Phi_{p}^{\dagger})^{\widetilde{q}}\Phi_{q}^{\widetilde{p}}$, so we need
${\widetilde{p}}+{\widetilde{q}}$ to be large for a high-quality
${\mathrm{U(1)}}_{\textsc{PQ}}$ symmetry. After $\Phi_{p}$ and $\Phi_{q}$
obtain vacuum expectation values, there is a residual unbroken
${\mathbb{Z}}_{d}\subset{\mathrm{U(1)}}^{\prime}$ gauge symmetry, but it does
not act on the axion and does not affect the domain wall problem. Effectively,
the role of $p$ and $q$ in our discussion above is now played by
${\widetilde{p}}$ and ${\widetilde{q}}$, and the results are unchanged.

###  3.2 Cosmology/quality tension

Assuming that the string dynamics already ensures that ${N_{\textsc{DW}}}=1$
in the Barr-Seckel model, in this subsection we analyze the problems that may
still render the model invalid. These problems include fractionally charged
relics in cosmology, or a more severe quality problem due to Landau poles in
the SM gauge couplings below the UV scale.

New heavy quarks $Q$ introduced in KSVZ-type axion models need to be in
specific representations in order to not breach the current (stringent) bounds
on fractionally (electrically) charged relics [71, 72]. In some
representations, the heavy quarks can only hadronize into fractionally charged
hadrons, which cannot decay into SM particles. The representations with
fractionally charged $Q$-hadrons are thus ruled out directly by constraints on
fractionally charged relics. We are left to deal with only the rest of the
representations in which $Q$ only form integrally charged hadrons and can
decay into SM particles (equivalently, cases where the fields are in
representations of
$[{\mathrm{SU}}(3)_{\textsc{C}}\times{\mathrm{SU}}(2)_{\textsc{L}}\times{\mathrm{U(1)}}_{\textsc{Y}}]/{\mathbb{Z}}_{6}$;
see, e.g., [73, 74]). The authors of [71, 72] conclude that current literature
shows stable $Q$-hadrons are strongly disfavored, given estimates for
annihilation rates of bound states [75, 76]. Combined with constraints of the
$Q$ lifetime, the authors present only 15 representations of heavy quarks
which satisfy the constraints on fractionally charged relics and do not have
Landau poles in the SM gauge couplings at sub-Planckian scales. In particular,
for $Q$ to be in the fundamental of ${\mathrm{SU}}(3)_{\textsc{C}}$ and to
remain a singlet under ${\mathrm{SU}}(2)_{\textsc{L}}$, we need nonzero
${\mathrm{U(1)}}_{\textsc{Y}}$ charge in the form of
$-\frac{1}{3}+\mathbb{Z}$, and for an anti-fundamental like ${\widetilde{Q}}$
we need hypercharge in the form $\frac{1}{3}+\mathbb{Z}$. Therefore we assign
${\mathrm{U(1)}}_{\textsc{Y}}$ charges to the fields
$Q_{0},{\widetilde{Q}}_{p},{\widetilde{Q}}_{-q}$ as the following:

$\displaystyle Y_{Q_{0}}$ $\displaystyle=-\frac{1}{3}+n_{1}=x,$ $\displaystyle
Y_{{\widetilde{Q}}_{p}}$ $\displaystyle=\frac{1}{3}+n_{2}=y,$ $\displaystyle
Y_{{\widetilde{Q}}_{-q}}$ $\displaystyle=\frac{1}{3}+n_{3}=z,$ (28)

where the $n_{i}$’s are integers. One might think that once the heavy quarks
have nonzero ${\mathrm{U(1)}}_{\textsc{Y}}$ charges, the operators that allow
them to decay to SM quarks will also violate ${\mathrm{U(1)}}_{\textsc{PQ}}$
and spoil the axion solution to the strong CP problem. For exmaples, one can
write down mass terms like $mQ_{0}\bar{u}$ and $mQ_{0}\bar{d}$, which have
nonzero PQ charge at dimension 3. However, as is argued in Ref. [77], since
these operators do not acquire a VEV, their modification to the axion
potential is at least as suppressed as the lowest dimensional operator that
involves only fields with VEVs from spontaneous
${\mathrm{U(1)}}_{\textsc{PQ}}$ breaking. The axion solution to the strong CP
problem is not spoiled in this case. However, because of the nonzero
${\mathrm{U(1)}}_{\textsc{Y}}$ charges the model will run into a severe Landau
pole problem, as we sketch out below.

After adding nonzero hypercharges, we now have an additional set of gauge
anomaly cancellation conditions to consider. In the original model in [19],
the ${\mathrm{SU}}(3)_{\textsc{C}}^{3}$,
${\mathrm{SU}}(3)_{\textsc{C}}^{2}{\mathrm{U(1)}}^{\prime}$, and
${\mathrm{SU}}(3)_{\textsc{C}}{\mathrm{U(1)}}^{\prime 2}$ anomalies trivially
cancel. The ${\mathrm{U(1)}}^{\prime 3}$ anomaly can be cancelled with
additional SM singlet fermions without introducing new constraints. With the
nonzero ${\mathrm{U(1)}}_{\textsc{Y}}$ charge assignments in (3.2), we have
the following gauge anomaly cancellation constraints:

$\displaystyle{\mathrm{SU}}(3)_{\textsc{C}}^{2}{\mathrm{U(1)}}_{\textsc{Y}}:\quad$
$\displaystyle(p+q)x+qy+pz=0;$
$\displaystyle{\mathrm{U(1)}}_{\textsc{Y}}^{3}:\quad$
$\displaystyle(p+q)x^{3}+qy^{3}+pz^{3}=0;$
$\displaystyle{\mathrm{U(1)}}_{\textsc{Y}}^{2}{\mathrm{U(1)}}^{\prime}:\quad$
$\displaystyle qy^{2}p+pz^{2}(-q)=0;$
$\displaystyle{\mathrm{U(1)}}_{\textsc{Y}}{\mathrm{U(1)}}^{\prime 2}:\quad$
$\displaystyle qyp^{2}+pzq^{2}=0.$ (29)

The fields $\Phi_{p}$ and $\Phi_{q}$ must have zero hypercharge, because they
obtain large VEVs. Then the structure of the Yukawa couplings in (25) implies
that we must take $x=-y=-z$. All anomaly constraints except the last one (and
including the ${\mathrm{U(1)}}_{\textsc{Y}}$ gravitational anomaly) are
immediately satisfied after this assumption. The last constraint, which
becomes $(qp^{2}+pq^{2})y=0$, cannot be satisfied with the field contents
described above. Instead, we have to enlarge the field content of the theory.
If we simply double the fermion content of Table 1, we would have to give
opposite hypercharges to the two copies. This is inconsistent with the
quantization rule $y=1/3+{\mathbb{Z}}$; it would also double the
${\mathrm{SU}}(3)_{\textsc{C}}^{2}{\mathrm{U(1)}}_{\textsc{PQ}}$ anomaly,
spoiling this model as a solution to the domain wall problem.

The minimal solution is to have three times the original fermion content. To
declutter our notation, we suppress the original copy indices $i,j$ from (25),
and keep only an index $a\in\\{1,2,3\\}$ to label the new copies or
generations of the fields. We take the heavy generations to have hypercharges
$x^{a},y^{a},z^{a}$. All anomaly constraints are then satisfied with
$x^{a}=-y^{a}=-z^{a}$ and $\sum_{a}y^{a}=0$. In order to maintain domain wall
number ${N_{\textsc{DW}}}=1$, we take the third generation to have the
opposite-sign ${\mathrm{U(1)}}^{\prime}$ charge (and, correspondingly, an
opposite-sign ${\mathrm{U(1)}}_{\textsc{PQ}}$ charge) from the first two, so
that it contributes an equal and opposite amount to the
${\mathrm{SU}}(3)_{\textsc{C}}^{2}{\mathrm{U(1)}}_{\textsc{PQ}}$ anomaly. The
Yukawa couplings then take the form

$\displaystyle\mathcal{L}\supset\sum_{a=1,2}\Bigl{(}y_{p}^{a}Q_{0}^{a}{\widetilde{Q}}_{p}^{a}\Phi_{p}^{\dagger}+y_{q}^{a}Q_{0}^{a}{\widetilde{Q}}_{-q}^{a}\Phi_{q}\Bigr{)}+y_{p}^{3}Q_{0}^{3}{\widetilde{Q}}_{p}^{3}\Phi_{p}+y_{q}^{3}Q_{0}^{3}{\widetilde{Q}}_{-q}^{3}\Phi_{q}^{\dagger}+\mathrm{h.c.}$
(30)

This argument can be understood by thinking about the chiral rotation of the
$Q_{0}{\widetilde{Q}}_{p}$ combination in the interaction term
$Q_{0}{\widetilde{Q}}_{p}\Phi_{p}^{\dagger}$. In order to cancel the
${\mathrm{SU}}(3)_{\textsc{C}}^{2}{\mathrm{U(1)}}_{\textsc{PQ}}$ anomaly
contribution from rephasing $\Phi_{p}$ between two generations, we need one
generation to couple to $\Phi_{p}^{\dagger}$ and the other couple to
$\Phi_{p}$. This coupling is determined by the ${\mathrm{U(1)}}^{\prime}$
charges of the $Q_{0}$ and ${\widetilde{Q}}_{p}$ fields. The same logic
follows for the coupling to $\Phi_{q}$. Since the axion is a linear
combination of the rephasing of $\Phi_{p}$ and $\Phi_{q}$, the above treatment
cancels the ${\mathrm{SU}}(3)_{\textsc{C}}^{2}{\mathrm{U(1)}}_{\textsc{PQ}}$
anomaly between two generations, and the remaining generation contributes
${N_{\textsc{DW}}}=1$.

We choose the smallest possible hypercharge assignments to make the Landau
pole constraint as mild as possible, choosing $n_{i}=0$ in (3.2) for the first
two generations and $n_{i}=1$ for the third generation to achieve
$\sum_{a}y^{a}=0$. The additional matter field content with the fields’
representations is displayed in Table 2 for the readers’ convenience.

$\begin{array}[]{ c| c c c c
}&{\mathrm{SU}}(3)_{\textsc{C}}&{\mathrm{U(1)}}_{\textsc{Y}}&{\mathrm{U(1)}}^{\prime}&\text{copies}\\\
\hline\cr
Q_{0}^{i}&\bm{3}&x^{i}=(-\frac{1}{3},-\frac{1}{3},\frac{2}{3})&(0,0,0)&(p+q)\times
3\\\
\hline\cr{\widetilde{Q}}^{i}_{p}&\bm{\overline{3}}&y^{i}=(\frac{1}{3},\frac{1}{3},-\frac{2}{3})&(p,p,-p)&q\times
3\\\
{\widetilde{Q}}^{i}_{-q}&\bm{\overline{3}}&z^{i}=(\frac{1}{3},\frac{1}{3},-\frac{2}{3})&(-q,-q,q)&p\times
3\\\ \hline\cr\end{array}$ Table 2: We list the modified fermion contents of
our adaptation of the model in Ref. [19]. The fields are still singlets under
${\mathrm{SU}}(2)_{\textsc{L}}$, but they now acquire nontrivial
${\mathrm{U(1)}}_{\textsc{Y}}$ charge. The three generations of fields are
required to cancel gauge anomalies as well as to maintain
${N_{\textsc{DW}}}=1$. The ${\mathrm{U(1)}}_{\textsc{PQ}}$ charge closely
resembles that in Table 1. The quark combination
$Q_{0}^{3}{\widetilde{Q}}_{p}^{3}$ in the third generation has opposite
${\mathrm{U(1)}}_{\textsc{PQ}}$ charge from the first two, and the interaction
with the scalars is correspondingly modified. All the fields in this table are
left-handed Weyl fermions.

It is straightforward to calculate the one-loop $\beta$-functions of
${\mathrm{SU}}(3)_{\textsc{C}}$ and ${\mathrm{U(1)}}_{\textsc{Y}}$ in order to
determine the location of the Landau poles. Consider the
${\mathrm{SU}}(3)_{\textsc{C}}$ running coupling at one-loop:

$\displaystyle\frac{1}{\alpha_{s}(\mu)}=\frac{1}{\alpha_{s}(\mu_{0})}-\frac{b_{0}}{2\text{p}}\ln{\frac{\mu}{\mu_{0}}}\,,$
(31)

where $b_{0}=-11+\frac{2}{3}n_{f}$. In the following we assume that the Yukawa
couplings of the quarks to the scalars are 1. Decreasing the Yukawa couplings
and thus the heavy quark masses will make the Landau pole problem worse since
the heavy quarks start affecting the running of the coupling at a lower scale.
We have checked that increasing the Yukawa coupling to the maximum possible
value of $4\text{p}$ will not change the conclusion of our analysis either. We
run the coupling from mass of the top quark $m_{t}$ to the smaller VEV of the
two scalars, $v_{q}$, with SM fields only. Above $v_{q}$ and below $v_{p}$ we
have $3\times p$ additional _Dirac_ pairs of $({\widetilde{Q}}_{-q},Q_{0})$
fields, and above $v_{p}$ we have another $3\times q$ additional
$({\widetilde{Q}}_{p},Q_{0})$ fields. Requiring the
${\mathrm{SU}}(3)_{\textsc{C}}$ Landau pole to be below the new physics scale
$\Lambda$, we need

$\displaystyle-7\ln\frac{m_{t}}{\Lambda}+2\Bigl{(}q\ln\frac{v_{p}}{\Lambda}+p\ln\frac{v_{q}}{\Lambda}\Bigr{)}+\frac{2\text{p}}{\alpha_{s}(m_{t})}>0\,.$
(32)

Similarly, we can derive the constraints for the
${\mathrm{U(1)}}_{\textsc{Y}}$ Landau pole:

$\displaystyle\frac{41}{6}\ln\frac{m_{t}}{\Lambda}+4\sum_{i}Q_{i,Y}^{2}\Bigl{(}q\ln\frac{v_{p}}{\Lambda}+p\ln\frac{v_{q}}{\Lambda}\Bigr{)}+\frac{2\text{p}}{\alpha_{\textsc{Y}}(m_{t})}>0\,,$
(33)

where
$\sum_{i}Q_{i,Y}^{2}=\sum_{i}x_{i}^{2}=\sum_{i}y_{i}^{2}=\sum_{i}z_{i}^{2}=\frac{2}{3}$
is the sum over squares of ${\mathrm{U(1)}}_{\textsc{Y}}$ charges for each
quark field for all 3 generations. Finally, the quality problem constraint is
that the correction to axion potential from operators of the form
$g(\Phi_{p}^{\dagger})^{q}(\Phi_{q})^{p}/\Lambda^{p+q-4}$ is smaller than
$10^{-10}\Lambda_{\textsc{IR}}^{4}$, where
$\Lambda_{\textsc{IR}}^{4}=\frac{m_{u}m_{d}}{(m_{u}+m_{d})^{2}}m_{\pi}^{2}f_{\pi}^{2}$
is the size of the QCD-generated contribution to the axion potential. We then
require when $\Phi_{p}$ and $\Phi_{q}$ get VEVs,

$\displaystyle\ln
g+\Bigl{(}q\ln\frac{v_{p}}{\Lambda}+p\ln\frac{v_{q}}{\Lambda}\Bigr{)}-4\ln\frac{\Lambda_{\textsc{IR}}}{\Lambda}<\ln
10^{-10}\,.$ (34)

Figure 3: We show that the three constraints from Standard Model Landau poles
and axion quality exclude the parameter space of the model in Sec. 3.2
completely. The shaded region for each color indicates the excluded region by
imposing the given constraint, i.e. the blue (orange) region indicates where
the ${\mathrm{SU}}(3)_{\textsc{C}}$ (${\mathrm{U(1)}}_{\textsc{Y}}$) Landau
pole is below the UV scale $\Lambda$, and the green region is where the
correction to the axion potential is larger than $10^{-10}$. In the plot the
Yukawa couplings and the coupling strength of the lowest-dimensional PQ
breaking operator $g$ are set to 1, but varying the couplings does not change
the conclusion unless $g$ is taken extremely small. We do not assume the value
any other parameters in the model, as the dependence can be fully captured by
the combination on the $y$-axis, which is
$q\log_{10}\frac{v_{p}}{\Lambda}+p\log_{10}\frac{v_{q}}{\Lambda}$ for the
model in Sec. 3.2. We later discuss a different model in Sec. 4, and find
similar conclusions. The $y$-axis dependence in that case becomes
$n\log_{10}\frac{m_{Q}}{\Lambda}$. See the text for more detail.

We observe that the combination
$q\ln\frac{v_{p}}{\Lambda}+p\ln\frac{v_{q}}{\Lambda}$ is the defining
parameter of this family of models. Combining the three constraints into one
exclusion plot shown in Fig. 3, where we take $g=1$, we see that the parameter
space is completely ruled out.

One might ask how small $g$ would need to be to avoid this conclusion.
Combining (34) and (32), we derive

$g<10^{-10}\left(\frac{\Lambda_{\textsc{IR}}}{m_{t}}\right)^{7/2}\sqrt{\frac{\Lambda_{\textsc{IR}}}{\Lambda}}{\mathrm{e}}^{\pi/\alpha_{s}(m_{t})}\approx
10^{-17}\sqrt{\frac{10^{12}\,\mathrm{GeV}}{\Lambda}}.$ (35)

This shows that, for any reasonable cutoff on a QCD axion model, solving the
quality problem while avoiding a Landau pole for QCD requires that the
coefficient of the lowest-dimension PQ-violating operator be tuned much
smaller than ${\overline{\theta}}$ itself. Thus, the Strong CP problem is
simply not solved, but (at best) transformed into a similar problem of why $g$
is so small.

##  4 An ${\mathrm{SU}}(n)\times{\mathrm{SU}}(n)$ symmetry for quality

In this section we will discuss a model of a high quality axion protected by a
nonabelian gauge symmetry, following [56]. Many of the considerations are
closely parallel to those in the previous section, so our discussion will be
relatively brief.

###  4.1 Scalar potential and axion strings

The theory has a gauge symmetry
$G={\mathrm{SU}(n)_{\textsc{L}}}\times{\mathrm{SU}(n)_{\textsc{R}}}$, with a
field $Y$ in the bifundamental
$(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}},\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}})$
representation. The renormalizable potential $V(Y)$ contains a tachyonic
$\mathrm{tr}(YY^{\dagger})$ term and quartic terms which are a linear
combination of $\mathrm{tr}(YY^{\dagger}YY^{\dagger})$ and
$\mathrm{tr}(YY^{\dagger})^{2}$, which lead to a vacuum expectation value
$\langle Y\rangle\propto\mathds{1}$ breaking
${\mathrm{SU}(n)_{\textsc{L}}}\times{\mathrm{SU}(n)_{\textsc{R}}}$ to the
diagonal ${\mathrm{SU}(n)_{\textsc{V}}}$. This symmetry breaking pattern is
familiar from the flavor symmetry of QCD, and the structure of the potential
for $Y$ mimics that of the linear sigma model. The renormalizable potential
also has an accidental ${\mathrm{U(1)}}$ global symmetry rephasing $Y$, which
is of high quality; the lowest-dimension operator breaking it is $\det Y$.
This is the Peccei-Quinn symmetry that is spontaneously broken to provide an
axion,

$\theta(x)=\arg(\det Y(x)).$ (36)

To understand the formation of axion strings, consider the limit in which the
gauge couplings are turned off and we focus on the dynamics induced by the
scalar potential. The renormalizable potential has a global symmetry

$G_{V}=[{\mathrm{SU}(n)_{\textsc{L}}}\times{\mathrm{SU}(n)_{\textsc{R}}}\times{\mathrm{U(1)}}]/({\mathbb{Z}}_{n}\times{\mathbb{Z}}_{n}).$
(37)

Here ${\mathrm{SU}(n)_{\textsc{L}}}:Y\mapsto LY$,
${\mathrm{SU}(n)_{\textsc{R}}}:Y\mapsto YR^{\dagger}$, and
${\mathrm{U(1)}}:Y\mapsto{\mathrm{e}}^{{\mathrm{i}}\alpha}Y$. The
${\mathbb{Z}}_{n}$ group generated by
$L=R=\exp(2\text{p}{\mathrm{i}}/n)\mathds{1}$ and $\alpha=0$ acts trivially on
$Y$, as does the ${\mathbb{Z}}_{n}$ group generated by
$L=\exp(2\text{p}{\mathrm{i}}/n)\mathds{1},R=\mathds{1},\alpha=-2\text{p}/n$,
so we have taken the quotient by the product of these two ${\mathbb{Z}}_{n}$
groups to obtain the group $G_{V}$ acting faithfully on $Y$.

The vacuum expectation value $\langle Y\rangle=v\mathds{1}$ spontaneously
breaks the symmetry to

$H_{V}=\mathrm{PSU}(n)_{\textsc{V}}\equiv{\mathrm{SU}(n)_{\textsc{V}}}/{\mathbb{Z}}_{n}.$
(38)

That is, pairs $L=R=V\in{\mathrm{SU}(n)}$ acting as $Y\mapsto VYV^{\dagger}$
leave $\langle Y\rangle$ invariant, but the cases where $V$ is in the center
of ${\mathrm{SU}(n)}$ acted trivially on any $Y$ and were quotiented out in
$G_{V}$, so we must take the ${\mathbb{Z}}_{n}$ quotient here as well.

The phase transition in which $Y$ acquires a vacuum expectation value will
lead, by the Kibble-Zurek mechanism, to the production of strings
characterized by the fundamental group $\pi_{1}(G_{V}/H_{V})$. We claim that
$\pi_{1}(G_{V}/H_{V})\cong{\mathbb{Z}}$. To see this, we can realize $G_{V}$
as a quotient of the simply connected space
${\mathrm{SU}(n)_{\textsc{L}}}\times{\mathrm{SU}(n)_{\textsc{R}}}\times{\mathbb{R}}$
by the freely-acting group ${\mathbb{Z}}_{n}\times{\mathbb{Z}}$, where
${\mathbb{Z}}_{n}$ is generated by
$(L,R,\alpha)\mapsto({\mathrm{e}}^{2\text{p}{\mathrm{i}}/n}L,{\mathrm{e}}^{2\text{p}{\mathrm{i}}/n}R,\alpha)$
and ${\mathbb{Z}}$ is generated by
$(L,R,\alpha)\mapsto({\mathrm{e}}^{2\text{p}{\mathrm{i}}/n}L,R,\alpha-2\text{p}/n)$.
Hence $\pi_{1}(G_{V})\cong{\mathbb{Z}}_{n}\times{\mathbb{Z}}$. We also have
$\pi_{1}(H_{V})\cong{\mathbb{Z}}_{n}$. The inclusion $H_{V}\hookrightarrow
G_{V}$ induces a map identifying $\pi_{1}(H_{V})$ with the ${\mathbb{Z}}_{n}$
factor in $\pi_{1}(G_{V})$. We use the homotopy exact sequence
$\ldots\to\pi_{1}(H_{V})\to\pi_{1}(G_{V})\to\pi_{1}(G_{V}/H_{V})\to\pi_{0}(H_{V})\cong
0$ to conclude that $\pi_{1}(G_{V}/H_{V})\cong{\mathbb{Z}}$. This fundamental
group is generated by the image of the path
$[0,1]\to{\mathrm{SU}(n)_{\textsc{L}}}\times{\mathrm{SU}(n)_{\textsc{R}}}\times{\mathrm{U(1)}}$
given by

$t\mapsto\left(\exp\left[{\mathrm{i}}t\frac{2\text{p}}{n}\begin{pmatrix}1&&&&\\\
&1&&&\\\ &&\ddots&&\\\ &&&1&\\\
&&&&-(n-1)\end{pmatrix}\right],\mathds{1},\exp(-2\text{p}{\mathrm{i}}t/n)\right).$
(39)

In the covering space, this path winds from the origin to a central element in
${\mathrm{SU}(n)_{\textsc{L}}}$ and winds $1/n$ of the way around
${\mathrm{U(1)}}$; this endpoint is identified with the origin by the second
${\mathbb{Z}}_{n}$ factor in (37). The Kibble-Zurek mechanism leads to the
production of cosmic strings associated with this path, which correspond to
strings of minimal winding for the axion $\theta$. We do not expect that
gauging ${\mathrm{SU}(n)_{\textsc{L}}}\times{\mathrm{SU}(n)_{\textsc{R}}}$ and
adding Yukawa interactions will alter this conclusion. Unlike for the models
of Sec. 2.3 and Sec. 3, then, there is no ambiguity about string formation
dynamics in this model: the domain wall problem can be solved, if the model is
free of other cosmological difficulties.

###  4.2 Standard Model couplings and Landau pole constraint

The model of [56] adds quark fields charged under
${\mathrm{SU}(n)_{\textsc{L}}}\times{\mathrm{SU}}(3)_{\textsc{C}}$ and
${\mathrm{SU}(n)_{\textsc{R}}}\times{\mathrm{SU}}(3)_{\textsc{C}}$, which give
rise through triangle diagrams to the couplings of the axion to gluons. These
fields were chosen to be electroweak singlets, and so [56] assumes a pre-
inflation scenario to avoid the cosmological disaster of fractionally charged
particles. However, as in Sec. 3.2, we can consider a model in the post-
inflation scenario with additional fields carrying hypercharge to eliminate
this problem.

$\begin{array}[]{ c|c c c c c c | c }&\text{spin}&{\mathrm{SU}(n)_{\textsc{L}}}&{\mathrm{SU}(n)_{\textsc{R}}}&{\mathrm{SU}}(3)_{\textsc{C}}&{\mathrm{U(1)}}_{\textsc{Y}}&\text{copies}&{\mathrm{U(1)}}_{\textsc{PQ}}\\\ \hline\cr Y&0&\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}&\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}&\bm{1}&0&1&1/n\\\ \hline\cr D^{(i)}&1/2&\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}&\bm{1}&\bm{3}&-1/3&2&1/n\\\ \widetilde{D}^{(i)}&1/2&\bm{1}&\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}&\bm{\overline{3}}&+1/3&2&0\\\ \hline\cr U&1/2&\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}&\bm{1}&\bm{3}&+2/3&1&-1/n\\\ \widetilde{U}&1/2&\bm{1}&\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}&\bm{\overline{3}}&-2/3&1&0\\\ \hline\cr\Psi^{(j)}&1/2&\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}&\bm{1}&\bm{1}&0&3&-1/n\\\ \widetilde{\Psi}^{(j)}&1/2&\bm{1}&\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}&\bm{1}&0&3&0\\\ \hline\cr\end{array}$ Table 3: Matter content of a model with an ${\mathrm{SU}(n)}\times{\mathrm{SU}(n)}$ gauge symmetry to address the quality problem. This closely follows [56], but we have added nontrivial hypercharge assignments to avoid fractionally charged particles. This then requires a more elaborate field content, chosen to cancel gauge anomalies and to ensure domain wall number 1. All spin-$1/2$ fields in table are left-handed Weyl fermions.

Specifically, we consider the matter content shown in Table 3, together with
the Yukawa couplings

$(y_{D})_{ik}D^{(i)}Y^{\dagger}\widetilde{D}^{(k)}+y_{U}UY\widetilde{U}+(y_{\Psi})_{jl}\Psi^{(j)}Y\widetilde{\Psi}^{(l)}+\mathrm{h.c.}$
(40)

The symmetry ${\mathrm{U(1)}}_{\textsc{PQ}}$ is an approximate accidental
global symmetry; we have normalized the charges to be fractional to signal
that the $2\text{p}$-periodic axion field $\theta$ is the phase of $\det Y$,
rather than $Y$ itself. This gives rise to the correctly normalized domain
wall number from the
${\mathrm{SU}}(3)_{\textsc{C}}^{2}\text{-}{\mathrm{U(1)}}_{\textsc{PQ}}$
anomaly coefficient: $2\times n\times 1/n+n\times(-1/n)=1$. Importantly, the
${\mathrm{SU}(n)}_{\textsc{L,R}}^{2}\text{-}{\mathrm{U(1)}}_{\textsc{PQ}}$
mixed anomalies cancel, ensuring that the new nonabelian dynamics does not
spoil the solution to the Strong CP problem. Aside from
${\mathrm{U(1)}}_{\text{PQ}}$, all other symmetries in the table are gauged,
and the charges are chosen so that anomalies cancel. For example, the mixed
${\mathrm{SU}(n)_{\textsc{L}}}^{2}\text{-}{\mathrm{U(1)}}_{\textsc{Y}}$
anomaly cancels because we have two copies of the $D$ fields with hypercharge
$-1/3$, but only one copy of the $U$ field with hypercharge $+2/3$. The
${\mathrm{SU}(n)_{\textsc{L}}}^{3}$ anomaly cancels because we have 2 copies
of a color triplet in the  representation and one color triplet and three
copies of a singlet in the
$\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$
representation, so altogether there are six fundamentals and six
antifundamentals. Similarly, one can check that all of the other gauge anomaly
cancellation conditions hold.

Much as in Sec. 3.2, the large number of added fields poses a Landau pole
problem. We assume that all the additional fermion fields get mass $m_{Q}\sim
v/\sqrt{2}$. This is valid if all Yukawa couplings are of the same order. The
masses are already naturally degenerate within each
${\mathrm{SU}}(3)_{\textsc{C}}$ and ${\mathrm{SU}}(n)_{\textsc{L+R}}$
multiplet. Following a similar analysis as before, in which the fermions alter
the beta functions above the scale $m_{Q}$, we arrive at the Landau pole
constraints for ${\mathrm{SU}}(3)_{\textsc{C}}$ and
${\mathrm{U(1)}}_{\textsc{Y}}$:

$\displaystyle-7\ln\frac{m_{t}}{\Lambda}+2n\ln\frac{m_{Q}}{\Lambda}+\frac{2\text{p}}{\alpha_{s}(m_{t})}>0\,,$
(41) $\displaystyle\frac{41}{6}\ln\frac{m_{t}}{\Lambda}+4n\sum
Q_{i,Y}^{2}\ln\frac{m_{Q}}{\Lambda}+\frac{2\text{p}}{\alpha_{\textsc{Y}}(m_{t})}>0\,.$
(42)

Here $\sum Q_{i,Y}^{2}=\frac{2}{3}$ sums over $D^{(i)},U,\Psi^{(j)}$, or
equivalently $\widetilde{D}^{(i)},\widetilde{U},\widetilde{\Psi}^{(j)}$.

The quality problem constraint comes from requiring the correction to the
axion potential [56]

$\displaystyle\Delta
V=\Big{(}\frac{v}{\sqrt{2}}\Big{)}^{n}\frac{\kappa}{\Lambda^{n-4}}$ (43)

to satisfy $\Delta V/\Lambda_{\textsc{IR}}^{4}<10^{-10}$. Here $\kappa$ is the
coupling strength of the lowest-dimensional PQ breaking operator $\det Y$, and
we take $\kappa$ to be $\mathcal{O}(1)$. This translates to

$\displaystyle\ln\kappa+n\ln\frac{m_{Q}}{\Lambda}-4\ln\frac{\Lambda_{\textsc{IR}}}{\Lambda}<\ln
10^{-10}\,,$ (44)

where we have used $m_{Q}\sim v/\sqrt{2}$.

The exclusion plot for this model looks the same as Fig. 3 from the model in
Sec. 3.2, except the defining parameter on the $y$-axis is now
$n\ln\frac{m_{Q}}{\Lambda}$. This is clear to see by comparing the constraint
equations themselves. We conclude that this class of models with nonabelian
gauge symmetry also presents tensions between the quality problem constraint
and standard post-inflation cosmology. We reiterate that the domain wall
problem is indeed solved here, as the string formation dynamics is
unambiguous, unlike the models considered in Sec. 2 and Sec. 3, where it
remains unclear without detailed simulation whether the correct strings can
form to annihilate domain walls. Nonetheless, the model fails either because
of fractionally charged relics or because it does not solve the quality
problem (and hence, does not solve the Strong CP problem).

##  5 Comments on composite axion models

The models that we have discussed above have elementary scalar fields, which
introduce a fine tuning problem of their own. This scalar hierarchy problem
could be solved by supersymmetry or compositeness. It has long been
appreciated that a composite axion model could naturally explain the axion’s
origin as a pseudo-Nambu-Goldstone boson, much like the pions of QCD [78]. The
minimal such models are free of a scalar hierarchy problem, but do not solve
the quality problem because they admit low-dimension PQ-violating operators.
Over the years, a number of examples have been constructed in which additional
gauge symmetries forbid such operators and solve the quality problem; see,
e.g., [21, 77, 79, 80, 81, 82]. These ensure that the lowest-dimensional PQ-
violating operator has a large dimension by realizing it as a baryonic
operator of some ${\mathrm{SU}}(m)$ gauge theory (e.g., [21]) or as a product
across many links in a moose diagram (e.g., [79]).

All of these models feature a confining ${\mathrm{SU}}(n)$ gauge theory, and
all of them predict a domain wall number that is a multiple of $n$. Thus, none
of them is a good candidate for a post-inflation axion realizing the minimal
solution to the domain wall problem.

The reason that the domain wall number is a multiple of $n$ in these models is
that the heavy fields carrying PQ charge are fundamentals of both
${\mathrm{SU}}(n)$ and ${\mathrm{SU}}(3)_{\textsc{C}}$. Integrating out fields
carrying PQ charge $p_{i}$, ${\mathrm{SU}}(n)$ representation $r_{i}$, and
${\mathrm{SU}}(3)_{\textsc{C}}$ representation $s_{i}$ gives

$k_{G}=\sum_{i}p_{i}\dim(r_{i})2I(s_{i}),$ (45)

with $I(s_{i})$ the Dynkin index normalized to $1/2$ for the fundamental
representation. For the moment, we assume that $p_{i}\in{\mathbb{Z}}$ (an
assumption that we will revisit below). We have $2I(s_{i})\in{\mathbb{Z}}$,
and for the fundamental representation  of ${\mathrm{SU}}(n)$,
$\dim(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})=n$.
From this expression, we see that in order to modify a composite axion model
to have $|k_{G}|=1$, we must necessarily have fields contributing to $k_{G}$
that transform in different representations of ${\mathrm{SU}}(n)$.

Let us consider a composite model with the following assumptions:

  * •

It is based on strong dynamics arising from an asymptotically free
${\mathrm{SU}}(n)$ gauge theory.

  * •

All of the PQ-charged fields contributing to $k_{G}$ transform nontrivially
under ${\mathrm{SU}}(n)$.

  * •

The PQ symmetry has no mixed anomaly with ${\mathrm{SU}}(n)$; otherwise, the
would-be axion would, like the $\eta^{\prime}$ in QCD, acquire a large mass.

We will see that, even before considering dynamics or how to arrange for a
high-quality accidental PQ symmetry, there are some quite general difficulties
with achieving domain wall number 1 from this starting point.

Asymptotic freedom of ${\mathrm{SU}}(n)$ is a strong restriction on the matter
content. At one loop, it requires that

$\frac{11}{3}n-\frac{1}{3}\sum_{i}2I(r_{i})\dim(s_{i})>0,$ (46)

where the sum is over left-handed Weyl fermions $\psi_{i}$ transforming in the
$r_{i}$ representation of ${\mathrm{SU}}(n)$ and the $s_{i}$ representation of
${\mathrm{SU}}(3)_{\textsc{C}}$. For a nontrivial representation of
${\mathrm{SU}}(3)_{\textsc{C}}$ we have $\dim(s_{i})\geq 3$. Hence, only
representations satisfying

$2I(r_{i})<\frac{11}{3}n$ (47)

can preserve asymptotic freedom. This eliminates large representations from
consideration. For example, the symmetric three-index tensor has
$2I(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})=(n+2)(n+3)/2$,
which exceeds $11n/3$ for all ${\mathrm{SU}}(n)$. The fundamental, two-index
tensors, and adjoint always obey the inequality. Other tensors do in special
cases, e.g., the fully antisymmetric three-index tensor
($2I(\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{9.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})=(n-3)(n-2)/2$)
when $n<12$, the mixed three-index tensor
($2I(\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})=n^{2}-3$)
when $n=4$, and even the four-index fully antisymmetric tensor when $n=8$.

Simply having multiple representations of ${\mathrm{SU}}(n)$ available is no
guarantee that we can find a viable model with domain wall number 1. For
example, any model with only fundamentals and symmetric or antisymmetric two-
index tensor representations will have non-minimal domain wall number. This is
because, for each of these representations $r$, we have $n\mid\dim(r)$ when
$n$ is odd, and $\frac{n}{2}\mid\dim(r)$ when $n$ is even. In fact, we can
make a stronger statement. A representation $r$ of ${\mathrm{SU}}(n)$ has an
associated $n$-ality $z_{n}(r)$, which is the representation’s charge under
the ${\mathbb{Z}}_{n}$ center of ${\mathrm{SU}}(n)$, or equivalently the
number of boxes in the Young tableau modulo $n$. Any representation has the
property that

$z_{n}(r)\dim(r)\cong 0\pmod{n}.$ (48)

This follows from the fact that for any ${\mathrm{SU}}(n)$ representation
$\rho$, $\det\rho(g)=1$ for any $g\in{\mathrm{SU}}(n)$ (and in particular, for
$g$ in the center); this is because $\det\circ\rho$ gives a one-dimensional
representation of ${\mathrm{SU}}(n)$ and all such representations are trivial.
Using (48), we can exclude many possible models in which all of the $\dim(r)$
terms in (45) have certain factors of $n$ in common. For example, if $n$ is a
prime power $p^{k}$, then the dimension of every representation of nonzero
$n$-ality is divisible by $p$, and achieving domain wall number 1 requires the
use of a representation of zero $n$-ality (like the adjoint).

For an example where $n$ is not a prime power, consider the case of
${\mathrm{SU}}(6)$. We have
$\dim(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})=6$,
$\dim(\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})=15$,
and
$\dim(\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{9.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})=20$.
Any model that exploits only two of these representations has non-minimal
domain wall number (divisible by 3 using  and  , by 2 using  and  , or by 5
using  and  ). A model that exploits all three representations (or their
conjugates) has a chance. However, such a model has a very large amount of
matter! If we add fields in the
$(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}},\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})$,
$(\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}},\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})$,
and
$(\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{9.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}},\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})$
of ${\mathrm{SU}}(6)\times{\mathrm{SU}}(3)_{\textsc{C}}$ (possibly with
conjugates on some of these labels), we have added $3(6+15+20)=123$ new fields
that must carry hypercharge (to avoid fractional charges, as usual). If we
assume that all of these fields have the minimal hypercharge $\pm 1/3$
compatible with their ${\mathrm{SU}}(3)_{\textsc{C}}$ representation, then
they are sufficient to drive ${\mathrm{U(1)}}_{\textsc{Y}}$ to a Landau pole
below the Planck scale unless $f_{a}\gtrsim 4\times 10^{15}\,\mathrm{GeV}$,
which would lead to an overabundance of dark matter in the post-inflation
scenario. In fact, this field content is insufficient: we must cancel the
${\mathrm{U(1)}}_{\textsc{Y}}\text{-}{\mathrm{SU}}(6)^{2}$ anomaly, among
others, so the situation can only become worse. And so far we have only
discussed field content; we must also ask whether the model admits an
approximate Peccei-Quinn symmetry of high quality, what the PQ charge
assignments are, and whether the model has the appropriate dynamics to
spontaneously break PQ without unwanted cosmological relics! It seems likely
that, even if we lower the cutoff below the Planck scale, no model along these
lines could be viable.

Next, let us consider the case of the adjoint representation. We have
$2I(\mathrm{Adj})=2n$, so we can add a field $\lambda$ that is an
$(\mathrm{Adj},\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})$
of ${\mathrm{SU}}(n)\times{\mathrm{SU}}(3)_{\textsc{C}}$ without spoiling the
asymptotic freedom of ${\mathrm{SU}}(n)$, but we cannot add two of them. We
could also add
$(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}},\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})$
or
$(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}},\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}})$
representations (at least one, possibly more, depending on Peccei-Quinn charge
assignments) to achieve domain wall number 1. Along the lines of the previous
paragraph, we have added at least $3(n^{2}+n-1)$ new fields that must carry
hypercharge at least $\pm 1/3$, so $n$ cannot be too large. For example,
demanding a Landau pole below the Planck scale for
$f_{a}=10^{11}\,\mathrm{GeV}$, we must have $n<7$. One could further consider
anomaly cancelation and other constraints on the model, but there is another
problem to consider that is specific to this case. The adjoint is a real
representation, and there is no reason to expect that
$\langle\mathrm{tr}(\lambda\lambda)\rangle$ should be zero. This expectation
value would spontaneously break ${\mathrm{SU}}(3)_{\textsc{C}}$ and
${\mathrm{U(1)}}_{\textsc{Y}}$ at the scale $f_{a}$, in obvious contradiction
to the world around us.

One of the assumptions that we made above, integer PQ charges
$p_{i}\in{\mathbb{Z}}$, can be modified in cases where a discrete subgroup of
${\mathrm{U(1)}}_{\textsc{PQ}}$ is gauged. As we emphasized at the end of Sec.
2.2, in such cases, the formula (45) that we have used to compute $k_{G}$ may
overcount the true number of gauge-inequivalent vacua. Instead, we should
divide by the number of vacua connected by discrete gauge transformations
acting on the axion. In other words, the Lazarides-Shafi mechanism [48] gives
a potential loophole in the above argument, just as it did in our examples in
previous sections. A gauged discrete subgroup of
${\mathrm{U(1)}}_{\textsc{PQ}}$ cannot be embedded in the confining
${\mathrm{SU}}(n)$ group itself, because we have assumed that the axion is a
meson of the confining sector (and hence that it is invariant under
${\mathrm{SU}}(n)$ transformations). Thus, to exploit the loophole, we must
embed a discrete subgroup of ${\mathrm{U(1)}}_{\textsc{PQ}}$ in a different
gauge symmetry, i.e., a gauged flavor symmetry, from the ${\mathrm{SU}}(n)$
viewpoint. In doing so, we will inevitably encounter difficulties that are
closely analogous to those we saw in examples in Sec. 2.3, Sec. 3, and Sec. 4,
and often worse. For example, in the model of [21] (further analyzed in [77]),
there is an ${\mathrm{SU}}(n)$ confining group and a further
${\mathrm{SU}}(m)$ gauge group that is a flavor symmetry from the
${\mathrm{SU}}(n)$ viewpoint. The naive domain wall number is $nm$, but the
lowest dimension gauge invariant operator is baryonic from the
${\mathrm{SU}}(m)$ viewpoint; effectively, the ${\mathbb{Z}}_{m}$ center of
${\mathrm{SU}}(m)$ coincides with a subgroup of
${\mathrm{U(1)}}_{\textsc{PQ}}$ and is gauged. This reduces the domain wall
number from $nm$ to $n$, but does not solve the problem.

While the considerations in this section do not constitute a rigorous no-go
theorem, they show that severe difficulties arise in attempts to construct a
composite axion model with minimal domain wall number.

##  6 Conclusions

We have seen that a wide variety of post-inflation axion models exhibit a
fundamental tension: modifying the model to solve the quality problem
introduces cosmological problems; modifying the model to solve cosmological
problems resurrects the quality problem. The details have varied from model to
model, but there are common features. Models that solve the quality problem
generally rely on additional gauge symmetries under which the axion
transforms. These gauge symmetries can relate different, apparently distinct
vacua that are separated by domain walls at the time of the phase transition.
The gauge equivalence implies that, in principle, such domain walls can always
be destroyed by axion strings. Whether such strings form in the early
universe, however, is a detailed dynamical question. A further complication
arises from strong experimental constraints on the existence of fractionally
charged particles, which require that particles with
${\mathrm{SU}}(3)_{\textsc{C}}$ center charge also carry nonzero hypercharge.
Adding matter charged under the new gauge symmetries invoked to solve the
quality problem, and canceling all gauge anomalies including those involving
hypercharge, often requires the introduction of many fields charged under the
Standard Model gauge group. These can drive the couplings strong at a low
scale $\Lambda$, which appears in the denominator of higher-dimension
operators, exacerbating the quality problem.

###  6.1 Brief discussion of other models

Our observations extend to a range of other models incorporating additional
physical mechanisms. For example, supersymmetric axion models offer compelling
possibilities to solve both the electroweak hierarchy problem and the axion
hierarchy problem (i.e., the question of why $f_{a}\ll M_{\mathrm{Pl}}$).
These problems can be linked, for example through the Kim-Nilles [83]
mechanism, where a Peccei-Quinn global symmetry forbids a simple $\mu
H_{u}H_{d}$ term in the superpotential but allows a term of the form
$\frac{1}{\Lambda}S^{2}H_{u}H_{d}$. The phase of $S$ can play the role of the
axion, predicting that $\mu\Lambda\sim f_{a}^{2}$. To protect the axion
quality, a discrete ${\mathbb{Z}}_{p}$ gauge symmetry [84] or
${\mathbb{Z}}_{24}$ gauge R-symmetry [85, 86, 87, 88] has been proposed. The
arguments of Sec. 2 carry over directly: such models have a domain wall
problem in the post-inflation scenario, unless they are extended in some way
to explain the cosmological origin of ${\mathbb{Z}}_{p}$ strings. Any
supersymmetric model implementing the Lazarides-Shafi mechanism will differ in
two respects from examples we have discussed. On the one hand, holomorphy
restricts the set of allowed superpotential terms, potentially ameliorating
the quality problem. On the other, supersymmetry increases the number of
fields in the theory and thus tends to push Landau poles to lower scales.
Perhaps there is a case where the effect of holomorphy is more important,
allowing the model to succeed where the models of Sec. 3 and Sec. 4 failed. A
search for such models is beyond the scope of this paper.

Recently a novel axion scenario has received some attention, in which a
${\mathbb{Z}}_{p}$ symmetry acts to permute $p$ copies of the Standard Model
that all couple to a common axion [89]. This model predicts an exponentially
suppressed axion mass, for a given $f_{a}$, compared to the standard QCD
axion, and thus it necessarily predicts a different cosmological history and
dark matter abundance than the standard scenario. Nonetheless, one could ask
whether such a model admits a viable cosmology with a post-inflation Peccei-
Quinn phase transition. This question can only be answered in specific UV
completions. A KSVZ-like UV completion of this model, with a complex scalar
$S$ charged under ${\mathbb{Z}}_{p}$ obtaining a VEV and coupling to all $p$
sectors, was argued to solve the axion quality problem [90]. Such a model does
not have a viable post-inflation cosmology, because $S$ would be in thermal
contact with all $p$ sectors, producing an overwhelming dark radiation
problem. A different, composite UV completion discussed in [90] does not solve
the quality problem at all.

Our assessment of the quality problem has assumed that operators appear
suppressed by a UV cutoff $\Lambda$, below $M_{\mathrm{Pl}}$ and any Landau
poles, with $O(1)$ coefficients. A model like that of Sec. 4, which solves the
domain wall problem and has no fractionally charged relics, could give rise to
a viable post-inflation QCD axion if our analysis of the quality problem is
flawed. This raises the question: are there axion models in which the
coefficients of PQ-violating operators are much smaller than $O(1)$? Within
effective field theory, one could simply postulate that there is a Peccei-
Quinn symmetry broken only by a very small spurion, which suppresses the
coefficients of PQ-violating operators. From our viewpoint, such a model does
not solve the Strong CP problem at all; it simply shifts the question of why
$|{\overline{\theta}}|$ is small to the question of why the spurion is small.
On the other hand, a UV completion that successfully accounts for a small
spurion could solve the quality problem. One approach would be to search for
an even larger gauge group that is spontaneously broken at higher energies,
but this doesn’t add a qualitatively new ingredient compared to our examples
and seems unlikely to help. A different physical mechanism is needed. Locality
in extra dimensions can lead to exponentially small coefficients of higher-
dimension operators, if the fields involved are located at sites a distance
$L$ apart in extra dimensions and interactions between them are mediated by
bulk fields with mass $m\gg 1/L$. This is not immediately useful for a model
like that of Sec. 4, since the PQ-violating operator $\det(Y)$ involves a
self-interaction rather than an interaction among fields that can be spatially
separated, but there may be models where it succeeds. As already mentioned in
Sec. 1, a large class of models that achieves exponential suppression of
corrections to the axion potential relies on the axion as a zero mode of an
extra-dimensional gauge field [26], but such models have no 4d Peccei-Quinn
phase transition and so do not have a post-inflation cosmology. An interesting
alternative is the case where a ${\mathrm{U(1)}}$ gauge field obtains a
Stueckelberg mass by the 4d Green-Schwarz mechanism [91, 92]. In this case,
the would-be axion is eaten by an anomalous gauge field, but can leave behind
an exponentially good approximate global symmetry.777Ordinarily, when we higgs
a ${\mathrm{U(1)}}$ gauge symmetry, there is no good approximate global
symmetry left behind. We can insert factors of $\langle\phi\rangle$ to violate
charge, and the mass of the gauge field is proportional to
$\langle\phi\rangle$, so symmetry breaking effects are large at the higgsing
scale. In the Green-Schwarz case, however, charge breaking effects come from
factors of $\langle{\mathrm{e}}^{-T}\rangle$ where the axion $\mathrm{Im}\,T$
shifts under ${\mathrm{U(1)}}$ transformations. The gauge field mass, on the
other hand, depends on the second derivative of the kinetic term for $T$ (and
is often, parametrically, a UV scale multiplied by a power of $T$). Thus,
global symmetry breaking effects can be exponentially small compared to the
Stueckelberg mass scale. The phase of another field charged under the
anomalous gauge symmetry can then play the role of the axion; because it is
the phase of a 4d field, such models can have an ordinary 4d Peccei-Quinn
phase transition. Models with this structure arise in string theory and are
often referred to as “open string axion models,” although the original example
was in the context of the heterotic (closed) string (see, e.g., [93, 94, 28,
95, 96, 97, 98, 99], or [100] for similar structure in a phenomenological
extra dimensional model). These models potentially offer a compelling way out
of the problems we have discussed. On the other hand, in the minimal
incarnation of this mechanism, the $D$-term potential ensures that the PQ
breaking scale is close to the string scale, which would make it difficult to
find a model of inflation happening at yet higher energies. It would be
interesting to more fully explore the structure of open string axion models
that naturally have a separation of scales and the extent to which they can
achieve a viable post-inflation axion cosmology.

###  6.2 Concluding questions

We conclude by listing a set of questions raised by this work. To answer some
of these questions, we expect that numerical simulations of non-minimal axion
models could be useful. Others may be amenable to model-building.

  * •

What is the string formation dynamics in the model of [55] with
${\mathrm{SU}}(N)$ broken to ${\mathrm{SO}}(N)$ by a symmetric tensor (for odd
$N$)? From our discussion in Sec. 2.3, we expect that the strings that form
will not be suitable for solving the domain wall problem, but a classical
lattice simulation could give a definitive answer. An analytic study of
semiclassical string solutions, especially of the relative tension for strings
of different winding, could also be instructive.

  * •

In a scenario with multiple types of strings, such as the model discussed in
Sec. 3, what is the dynamics of string formation? In particular, is the
minimal string (on which a single domain wall ends) formed, and in what
abundance relative to other strings? How does the answer to this question
depend on parameters in the theory (e.g., the relative size of gauge couplings
and quartic couplings, or the ratios of different VEVs)?

  * •

If a string network forms that contains a mix of strings on which single
domain walls end and strings on which multiple domain walls end, what happens
at the QCD phase transition when the domain wall network forms? What
population of minimal strings is sufficient to solve the domain wall problem?

  * •

Are there viable composite axion models that have domain wall number 1
(possibly invoking the Lazarides-Shafi mechanism)?

  * •

Are there post-inflation axion models that solve the axion quality problem
(and hence the Strong CP problem), have domain wall number 1, and are free of
cosmological difficulties?

If the answer to the last question is “yes,” then within such models one may
be able to predict a target axion dark matter mass, along the lines of recent
simulations of axion strings in the post-inflation scenario. However, it may
be that these models have non-minimal cosmological dynamics, axion strings of
higher tension, or other novelties. Independent of the answer to that
question, we believe that the difficulties we have encountered constructing a
post-inflation axion scenario that solves the Strong CP problem with a viable
cosmology provide a reason to be wary of targeting any specific model or
scenario. The QCD axion already provides a well-defined target parameter space
across several orders of axion mass. The entire suite of experiments aiming to
“delve deep and search wide” [101] is crucial in the search for the axion.

## Acknowledgments

We have used the LaTeX source of [102] to draw Young tableaux (and also found
it a convenient reference for group theory factors). We would like to thank
Nima Arkani-Hamed, Roberto Contino, Ben Heidenreich, Anson Hook, Joshua Lin,
Maxim Perelstein, Alessandro Podo, Raman Sundrum, and Neal Weiner for helpful
discussions. QL is supported by the DOE grant DE-SC0013607, the NSF grant
PHY-2210498 and PHY-2207584, and the Simons Foundation. MR is supported in
part by the DOE Grant DE-SC0013607. ZS is supported by a fellowship from the
MIT Department of Physics. This work was performed in part at the Aspen Center
for Physics, which is supported by National Science Foundation grant
PHY-2210452.

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