Grand Color Axion

Alessandro Valenti$\,{}^{a,b}$, Luca Vecchi$\,{}^{b}$, Ling-Xiao
Xu$\,{}^{a,b}$

a Dipartamento di Fisica e Astronomia “G. Galilei”, Università di Padova,
Italy

b Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova,
Italy

###### Abstract

We present a model that solves the strong CP problem via an axion
parametrically heavier than the standard one. Within this picture the Standard
Model quarks are embedded into a larger non-abelian Grand Color group that at
high scales splits into ordinary QCD and an additional confining dynamics
under which exotic chiral fermions are charged. Crucially, the vacuum
expectation value of the axion is automatically relaxed to zero because the
only renormalizable source of explicit CP violation, beyond those encoded in
the topological angles, is contained in the Standard Model Yukawa couplings,
and is therefore very suppressed. The Grand Color axion potential is
controlled by the scale of the new confining group and is much larger than the
QCD contribution, such that its dynamics is less exposed to the so-called
“axion quality problem”. Potentially observable corrections to the effective
topological angle can also arise, in our model as well as in a large class of
heavy axion scenarios, from non-renormalizable Peccei-Quinn-_conserving_
interactions, which introduce a new “heavy axion quality problem”. Our model
has a very minimal field content, it relies entirely on gauge invariance and
does not require the introduction of additional symmetries beyond the usual
one postulated by Peccei and Quinn. The phenomenology is very rich and can be
tested at colliders as well as via cosmological observations. A particularly
interesting portion of parameter space predicts a visible Grand Color axion of
mass above the GeV and decay constant larger than a few TeV.

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###### Contents

  1. 1 Introduction
  2. 2 A Grand Color group
  3. 3 The axion potential
    1. 3.1 Minimisation at leading order
    2. 3.2 Subleading corrections and heavy axion quality
  4. 4 Phenomenology
  5. 5 Conclusions
  6. A General considerations on the leading order potential

##  1 Introduction

The QCD axion [1, 2, 3] is by far the most popular solution of the strong CP
problem. There are at least three good reasons why this is the case:

  * (i)

It has a very simple setup;

  * (ii)

It allows arbitrary sources of CP violation suppressed by scales much larger
than 1 GeV;

  * (iii)

It can be tested via a wide array of probes.

The QCD axion just requires the existence of an approximate global $U(1)_{\rm
PQ}$, anomalous under QCD, which gets spontaneously broken at some high scale
$f_{a}$. The minimality of these assumptions is envied by all alternative
solutions to the strong CP problem, which instead demand more complicated
structures in the UV and are therefore viewed as less plausible.

The second feature stems from the fact that the solution delivered by the QCD
axion is active at scales of order $\Lambda_{\rm QCD}\sim 1$ GeV. Whatever
fundamental source of CP violation is present at short distances, when we run
down to $\sim\Lambda_{\rm QCD}$ it gets completely encoded in the QCD
topological angle $\bar{\theta}$ and higher-dimensional operators. By relaxing
$\bar{\theta}$ to zero, the QCD axion completely removes the largest source of
CP-violation within the low energy effective field theory. 111The sources of
CP-violation contained in higher-dimensional operators suppressed by the weak
scale and the new physics scale $\Lambda_{\rm UV}$, are easily kept under
control. Operators suppressed by the weak scale lead to very small effects
because of the peculiarity of the SM Yukawa sector. The operators suppressed
by $\Lambda_{\rm UV}$ can be made parametrically smaller by decoupling the new
physics from the QCD scale. Below we will discuss in detail the analogous
effects that arise in our model. This is a truly remarkable property, that
distinguishes the QCD axion from _all_ other solutions to the strong CP
problem. This feature is loved in particular by model-builders because thanks
to it they are liberated from a number of otherwise annoying technical
hurdles: the origin of the flavor hierarchy, the smallness of the weak scale,
baryogenesis, dark matter, etc., are all questions that, in the presence of
the QCD axion, may be addressed without the need to worry about possibly large
CP-odd phases in the new physics. Nevertheless, (ii) is not at all an
essential feature of a viable solution of the strong CP. One may in fact turn
the argument upside down and argue that (ii) represents actually a “drawback”
of the QCD axion, since it indicates that even in the optimistic event of a
confirmation of such a mechanism we will not be able to infer anything new
about the nature of CP violation at short distances.

Leaving any philosophical drift aside, the reason $(iii)$ for its popularity
is a very solid one. Any interesting scenario for physics beyond the SM must
be able to make unambiguous predictions to be confronted with experimental
data. The axion solution of the strong CP problem clearly predicts the
existence of a pseudo-Nambu-Goldstone boson with an irreducible coupling to
gluons

$\displaystyle{\cal L}_{\rm axion}\supset\frac{\bar{a}}{f_{a}}\,\frac{g_{\rm
C}^{2}}{32\pi^{2}}G^{a}_{\mu\nu}\widetilde{G}^{a\,\mu\nu}$ (1.1)

where
$\widetilde{G}^{a\,\mu\nu}=\epsilon^{\mu\nu\alpha\beta}G^{a}_{\alpha\beta}/2$,
that can be tested via collider, astrophysical, as well as cosmological
observations. Current data set limits of the order $f_{a}\gtrsim
10^{8}-10^{9}$ GeV, leaving plenty of room for a solution of the strong CP
problem. What is more is that a Bose condensate of very weakly-coupled axions
may well be the totality or part of the dark matter if $f_{a}\sim 10^{12}$
GeV.

Features (i), (ii), and (iii) are at the origin of the well-deserved
popularity of the QCD axion. And yet, this nice story does not appear fully
convincing to many of us. The reason is also well-known, so much so that it
has been given a name: “axion quality problem” [4, 5, 6]. The potential of the
QCD axion is so small and the current lower bounds on $f_{a}$ so stringent
that tiny perturbations from uncontrollable sources of $U(1)_{\rm
PQ}$-breaking beyond the QCD anomaly, even if suppressed by the Planck scale,
can easily destabilize the solution. In simple terms, the standard axion
solution is apparently a very delicate one.

There are two paths to address this problem. One can either find a mechanism
to suppress the potentially dangerous Planckian perturbations, or find new
corrections to the potential such that the QCD contribution gets effectively
enhanced. In either case, unfortunately, one or both of the attractive
properties (i) and (ii) are lost. Known quantum field theories that address
the quality problem by suggesting mechanisms to suppress quantum gravity
perturbations invoke a number of seemingly ad-hoc gauge or global symmetries
and some end up being rather baroque. This approach to the quality problem
relies on such an intricate structure at short distances that the conceptual
simplicity (i) promised by the QCD axion gets partially if not completely
overshadowed. One may still hope that string theory offers alternative and
more attractive explanations of the axion quality. Unfortunately, our current
understanding of string theory makes it hard to firmly establish the relevance
of this assertion.

Rather than explaining why Planck-scale perturbations are suppressed compared
to the QCD-induced potential, one may alternatively build a model with a heavy
axion, in which the QCD potential is replaced by a much larger and more stable
one. If we follow this path also the IR-effectiveness of the standard QCD
axion, i.e. feature (ii) above, cannot hold anymore. In this paper we
investigate scenarios that pursue this avenue. Here the challenge is to
identify a framework in which the new $U(1)_{\rm PQ}$-breaking effects are
perfectly aligned with the QCD-induced potential so that the overall energy is
still minimized at a value $|\langle\bar{a}\rangle|/f_{a}\lesssim 10^{-10}$,
compatibly with current data. A few such mechanisms have been proposed.

One option is realized in scenarios where the large axion potential still
comes from QCD but, as opposed to the standard mechanism, it is due to new
short-distance effects [7]. If the QCD coupling grows relatively large at some
UV scale, indeed, small instantons may become relevant, and such effects are
naturally aligned with the low energy QCD potential. Unfortunately, a
strongly-coupled UV framework of this type is intrinsically sensitive to the
misaligning effect of whatever new CP-odd phases are present at the UV cutoff
[8, 9]. To firmly establish the viability of this program it is thus necessary
to analyze a concrete realization. To the best of our knowledge the only
explicit and tractable model of this type is the one of Ref. [10]. This work
shows that under reasonable assumptions the strong CP problem may in fact be
solved, though the required setup introduces a few copies of the color gauge
group along with a corresponding axion for each copy, and is therefore not as
minimal as one might have hoped.

Another viable avenue is to postulate a scenario in which $U(1)_{\rm PQ}$ is
anomalous under an additional non-abelian group ${\rm C}^{\prime}$. Provided
the anomaly coefficient is the same as the one of QCD, the axion coupling to
the two sectors reads:

$\displaystyle{\cal L}_{\rm axion}\supset\left(\bar{\theta}_{\rm
C}+\frac{a}{f_{a}}\right)\frac{g_{\rm
C}^{2}}{32\pi^{2}}G^{a}_{\mu\nu}\widetilde{G}^{a\,\mu\nu}+\left(\bar{\theta}_{\rm
C^{\prime}}+\frac{a}{f_{a}}\right)\frac{g_{\rm
C^{\prime}}^{2}}{32\pi^{2}}{G}^{\prime a}_{\mu\nu}{\widetilde{G}}^{\prime
a\,\mu\nu},$ (1.2)

with ${G^{\prime}}^{a}_{\mu\nu}$ indicating the field strength of the ${\rm
C}^{\prime}$ vectors. If it is possible to further identify a structural
condition that ensures

$\displaystyle\bar{\theta}_{\rm C}=\bar{\theta}_{\rm C^{\prime}}$ (1.3)

up to corrections smaller than $10^{-10}$, then a unique axion
$\bar{a}/f_{a}=\bar{\theta}_{\rm C}+a/f_{a}$ can be defined. Its potential may
be naturally dominated by the ${\rm C}^{\prime}$ dynamics and be such that
$\langle\bar{a}\rangle=0$, analogously to QCD [11, 12]. In this framework one
can obtain a sizable axion potential if the new non-abelian sector becomes
strong at scales much larger than $\Lambda_{\rm QCD}$, and the quality problem
is improved. The non-trivial task is explaining (1.3).

We may justify (1.3) by invoking a $Z_{2}$ symmetry [13]. To realize this
program a full copy of the SM is however needed, and in particular the new
confining group must be a mirror copy of QCD, i.e. ${\rm
C}^{\prime}=SU(3)_{\rm C}^{\prime}$. The mirror symmetry must be softly broken
in order to ensure that the mirror sector be sufficiently heavy to have
escaped detection. If the soft breaking is achieved via CP- and flavor-
conserving interactions, any possible correction to (1.3) is controlled by
loops of the SM Yukawas and higher-dimensional operators. It is known [14, 15,
16] that the former corrections, including both threshold as well as RG
effects, are extremely small. The latter can be taken under control as well
provided the soft breaking scale is sufficiently small compared to the UV
cutoff. These mirror models are currently the most studied incarnation of
heavy axion models [17, 18, 19, 20, 21].

Yet, there may be a simpler and more minimal way to justify (1.3), one that
does not require invoking a discrete mirror symmetry. One may in fact embed
color $SU(3)_{\rm C}$ into a larger group at short distances, which we call
_Grand Color_ , and then postulate the latter be broken into the SM times the
new confining group ${\rm C}^{\prime}$. In this setup the structure of eq.
(1.2) emerges at the symmetry-breaking threshold, with (1.3) easily satisfied
at tree-level even when ${\rm C}^{\prime}$ is not an $SU(3)$. This class of
models was first suggested in Ref. [22]. However that paper does not present a
concrete realization. Explicit models have been recently proposed in [23, 24],
but differ qualitatively from ours and the one of [22] due to the presence of
mass terms for the exotic fermions. These latter make it difficult to ensure
(1.3) remains protected against radiative effects. The main results of the
present paper are providing an explicit realization of the Grand Color
scenario that robustly satisfies (1.3) and presenting a careful study of the
vacuum structure of the theory.

In Section 2 we introduce a model with ${\rm C}^{\prime}=Sp(N-3)$ and argue
that in such a framework the condition (1.3) is satisfied up to negligible
radiative effects and higher-dimensional operators, very much like in mirror
models. The axion potential is analyzed in detail and proved to be aligned
with the QCD one in Section 3. This is a highly non-trivial result because in
our scenario the presence of Yukawa interactions prevents from applying the
theorems of [11, 12]. A large confinement scale for ${\rm C}^{\prime}$ implies
a sizable attenuation of the axion quality problem. The phenomenology is
discussed in Section 4 and finally Section 5 presents our conclusions.

##  2 A Grand Color group

The gauge group of our model is $SU(N)_{\text{GC}}\times SU(2)_{\rm L}\times
U(1)_{\rm Y^{\prime}}$ and the entire matter content, SM included, is reported
in table 1. The Grand Color is an $SU(N)_{\text{GC}}$ gauge group and the SM
quarks are in the fundamental and anti-fundamental representations. In order
to cancel gauge anomalies, hypercharge must be partly embedded into the Grand
Color and an abelian factor $U(1)_{Y^{\prime}}$, while to avoid triviality $N$
must be odd [25]. Yet, the leptonic sector remains basically the same as in
the SM, whereas the scalar sector must include at least two additional fields,
$\Phi$ in the adjoint and $\Xi$ in the 2-index anti-symmetric of
$SU(N)_{\text{GC}}$, in order to break Grand Color in a phenomenologically
viable way.

| $SU(N)_{\text{GC}}$ | $SU(2)_{\text{L}}$ | $U(1)_{\text{Y}^{\prime}}$
---|---|---|---
$Q$ | ${\bf N}$ | ${\bf 2}$ | $\frac{1}{2N}$
$U$ | $\overline{\bf N}$ | ${\bf 1}$ | $-\frac{1}{2}-\frac{1}{2N}$
$D$ | $\overline{\bf N}$ | ${\bf 1}$ | $+\frac{1}{2}-\frac{1}{2N}$
$\ell$ | ${\bf 1}$ | ${\bf 2}$ | $-\frac{1}{2}$
$e$ | ${\bf 1}$ | ${\bf 1}$ | $+1$
$H$ | ${\bf 1}$ | ${\bf 2}$ | $+\frac{1}{2}$
$\Phi$ | ${\bf Adj}$ | ${\bf 1}$ | $0$
$\Xi$ | ${\bf N}\otimes_{A}{\bf N}$ | ${\bf 1}$ | $\frac{1}{N}$
Table 1: Minimal field content of the model. The scalars $\Phi,\Xi$ are solely
needed in order to break Grand Color into the SM gauge group.

The most general renormalizable Lagrangian for the fields in table 1 includes
the standard kinetic terms and topological angles, a scalar potential, and a
Yukawa interaction with the Higgs doublet $H$ of the same form as in the SM,

$\displaystyle\mathcal{L}_{\text{Yuk}}=Y_{u}\,QHU+Y_{d}\,Q\widetilde{H}D+Y_{e}\,\ell\widetilde{H}e+{\rm
hc},$ (2.1)

plus the operators $QQ\Xi^{\scalerel*{\dagger}{X}}$ and $UD\Xi$. As explained
in more detail below, though, the presence of the latter interactions would
spoil the key relation (1.3). These couplings can be forbidden in several
ways, for example gauging $\rm B-L$, promoting $\Xi$ to a composite scalar, or
— perhaps less elegantly — invoking a global symmetry. Which of these
mechanisms is actually at work does not concern us. In the following we will
simply assume that (2.1) represent the full set of renormalizable Yukawa
interactions in our model.

The breaking of Grand Color is obtained in two steps:

$\displaystyle SU(N)_{\text{GC}}\times SU(2)_{\text{L}}\times
U(1)_{\text{Y}^{\prime}}$ $\displaystyle\xrightarrow{\braket{\Phi}}$
$\displaystyle SU(3)_{\text{C}}\times SU(N-3)\times SU(2)_{\text{L}}\times
U(1)_{\text{Y}^{\prime}}\times U(1)_{\text{GC}}$ (2.2)
$\displaystyle\xrightarrow{\braket{\Xi}}$ $\displaystyle
SU(3)_{\text{C}}\times Sp(N-3)\times SU(2)_{\text{L}}\times U(1)_{\text{Y}}.$

In the first step the vev of a scalar $\Phi$ breaks $SU(N)_{\rm GC}$ into
$SU(3)_{\text{C}}\times SU(N-3)\times U(1)_{\text{GC}}$. The abelian factor is
normalized such that the fundamental representation of $SU(N)_{\text{GC}}$
decomposes as

$\displaystyle\mathbf{N}\rightarrow(\mathbf{3},\mathbf{1})_{\frac{1}{6}-\frac{1}{2N}}\oplus(\mathbf{1},\mathbf{N-3})_{-\frac{1}{2N}}.$
(2.3)

The second step consists in breaking $SU(N-3)\times U(1)_{\rm GC}\times
U(1)_{\rm Y^{\prime}}\xrightarrow{\braket{\Xi}}Sp(N-3)\times U(1)_{\rm Y}$
through the vev of the $\Xi$ component in the antisymmetric of $SU(N-3)$,
which according to (2.3) carries a $U(1)_{\rm GC}$ charge equal to $-1/N$.
222To avoid any confusion, by $Sp(N-3)$ we denote the group of symplectic
unitary $N-3$ matrices. Consistently, non-triviality of the theory implies
$N-3$ is even. It follows that the unbroken $U(1)_{\rm Y}$ charges are the sum
of $U(1)_{\rm Y^{\prime}}$ and the $U(1)_{\rm GC}$ generators. For simplicity
we take both scalar vevs of order $f_{\rm GC}$. Importantly, because none of
the new scalars $\Phi,\Xi$ has Yukawa couplings one can in principle promote
both of them to composite operators. In that case there would no hidden fine-
tuning in requiring the Grand Color breaking scale be much smaller than the UV
cutoff, i.e. $f_{\rm GC}\ll f_{\rm UV}$. Strictly speaking, the only
naturalness problem that our model necessary suffers from are the usual
hierarchy and cosmological constant problems of the SM.

Below the Grand Color breaking scale $f_{\rm GC}$, the fields $Q,U,D$ split
into the direct sum of the SM quarks plus exotic chiral fermions as shown in
table 2. The exotic fermions $\psi_{q,u,d}$ inherit the Yukawa couplings to
$H$ from (2.1) and are therefore formally the same as the SM ones up to
renormalization effects. Crucially, however, there is no interaction between
the SM fermions and the $\psi$’s apart from higher-dimensional operators
suppressed by $f_{\rm GC}$. This implies that the flavor symmetries of the two
sectors are effectively distinct: loops of the $Sp(N-3)$-charged sector will
never be able to induce flavor-violating processes in the SM.

| $SU(3)_{\text{C}}$ | $Sp(N-3)$ | $SU(2)_{\text{L}}$ | $U(1)_{\text{Y}}$
---|---|---|---|---
$Q=\begin{pmatrix}q\\\ \psi_{q}\end{pmatrix}$ | $\mathbf{3}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\frac{1}{6}$
$\mathbf{1}$ | $\mathbf{N-3}$ | $\mathbf{2}$ | $0$
$U=\begin{pmatrix}u\\\ \psi_{u}\end{pmatrix}$ | $\mathbf{\bar{3}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-\frac{2}{3}$
$\mathbf{1}$ | $\mathbf{N-3}$ | $\mathbf{1}$ | $-\frac{1}{2}$
$D=\begin{pmatrix}d\\\ \psi_{d}\end{pmatrix}$ | $\mathbf{\bar{3}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\frac{1}{3}$
$\mathbf{1}$ | $\mathbf{N-3}$ | $\mathbf{1}$ | $\frac{1}{2}$
Table 2: Decomposition of the quarks below the scale $f_{\rm GC}$. Here
$\psi_{q}=(\psi_{q_{u}},\psi_{q_{d}})$ is an electroweak doublet. The SM
hypercharge $U(1)_{\text{Y}}$ is the sum of $U(1)_{\text{Y}^{\prime}}$ and
$U(1)_{\text{GC}}\subset SU(N)_{\text{GC}}$.

In addition, the field-basis invariant $SU(3)_{\rm C}$ and $Sp(N-3)$
topological angles, inherited by Grand Color as shown in (1.2), at tree-level
satisfy $\bar{\theta}_{\rm C}=\bar{\theta}_{\rm
C^{\prime}}=\theta-\text{arg}\det Y_{u}Y_{d}$, where $\theta$ denotes the
$SU(N)_{\rm GC}$ angle. Radiative effects can spoil this tree-level relation,
and it is mandatory for us to show that the misaligning affects are under
control. There are three different sources of radiative effects that can
potentially invalidate (1.3): the scalar sector, the Yukawa couplings, and
non-renormalizable interactions. The vevs of $\Phi$ and $\Xi$ are the order
parameters of Grand Color breaking and their insertion is necessary to
generate a difference in the two topological angles. Other than that, however,
the scalar sector cannot appreciably contribute to a violation of (1.3) since
the most general renormalizable potential $V(H,\Phi,\Xi)$ is automatically CP-
conserving and its parameters can always be chosen so that CP does not get
broken spontaneously. Furthermore, all radiative corrections due to the Yukawa
sector (2.1) at and below $f_{\rm GC}$ are known to be completely negligible
[14, 15, 16]. Had we allowed the presence of unsuppressed flavor-violating
coefficients for $QQ\Xi^{\scalerel*{\dagger}{X}}$, $UD\Xi$, this nice property
would not have held anymore.333 In this respect our approach differs
qualitatively from [23, 24], where the beyond the SM fermions filling the
Grand Color multiplet are decoupled by giving them large masses. Such a
decoupling may also be achieved in our model, where
$QQ\Xi^{\scalerel*{\dagger}{X}}$, $UD\Xi$ would generate a vector-like mass
matrix $M$ for the $Sp(N-3)$ fermions below $f_{\rm GC}$. Unfortunately,
decoupling would typically violate (1.3) because $M$ introduces a new physical
CP-odd phase that contributes to ${\bar{\theta}}_{\rm C^{\prime}}$ at tree-
level. In order to preserve $|{\bar{\theta}}_{\rm
C^{\prime}}-{\bar{\theta}}_{\rm C}|<10^{-10}$ one would therefore be forced to
demand that $|{\rm Arg}[{\rm det}[M]]|<10^{-10}$. In this paper we avoid this
fine-tuning by forbidding the couplings $QQ\Xi^{\scalerel*{\dagger}{X}}$,
$UD\Xi$. This way the extra fermions remain chiral, like the SM fermions, and
get trapped into the heavy $Sp(N-3)$ hadrons.

The bottom line is that in our model eq.(1.3) remains satisfied up to the
desired accuracy at the renormalizable level. The most dangerous non-
renormalizable interactions are dimension-5 and dimension-6 operators that
contribute differently to the topological angles once the scalars $\Phi,\Xi$
acquire a vev:

$\displaystyle\frac{\bar{c}_{5}}{f_{\text{UV}}}\frac{g_{\rm
GC}^{2}}{32\pi^{2}}\Phi\,G_{\rm GC}\tilde{G}_{\rm
GC},\;\frac{\bar{c}_{6}}{f_{\text{UV}}^{2}}\frac{g_{\rm
GC}^{2}}{32\pi^{2}}\Phi^{\scalerel*{\dagger}{X}}\Phi\,G_{\rm GC}\tilde{G}_{\rm
GC},\;\frac{\bar{c}_{6}^{\prime}}{f_{\text{UV}}^{2}}\frac{g_{\rm
GC}^{2}}{32\pi^{2}}\Xi^{\scalerel*{\dagger}{X}}\Xi\,G_{\rm GC}\tilde{G}_{\rm
GC}.$ (2.4)

The dominant effect comes from the first interaction, but this can be avoided
by charging $\Phi$ under an additional gauge symmetry, or postulating that
$\Phi$ be the scalar responsible for breaking $U(1)_{\rm PQ}$, in which case
$f_{a}\sim f_{\rm GC}$. The last two operators are more model-independent and
imply $\bar{\theta}_{\rm C}-\bar{\theta}_{\rm C^{\prime}}\sim f_{\rm
GC}^{2}/f_{\rm UV}^{2}$. Taking the Planck scale $f_{\text{UV}}=2.4\times
10^{18}$ GeV as the UV cutoff, satisfying the relation eq.(1.3) up to
corrections of order $10^{-10}$ imposes the constraint $f_{\text{GC}}\lesssim
10^{13}\text{ GeV}$. This bound can be further relaxed if $\Phi,\Xi$ are
composite operators.

Overall, the picture that emerges is qualitatively similar to the
$Z_{2}$-symmetric scenarios: color and the exotic confining dynamics have
basically the same topological angle if no Yukawa couplings are introduced
beyond $Y_{u},Y_{d}$ and the breaking of Grand Color is sufficiently soft.
Under these conditions a unique axion $\bar{a}/f_{a}=\bar{\theta}_{\rm
C}+a/f_{a}$ from the breaking of a $U(1)_{\rm PQ}$ with a Grand Color anomaly
would automatically relax to zero the topological angles of both color and
${\rm C^{\prime}}=Sp(N-3)$. By making the latter confine at a scale $f\gg
f_{\pi}$ much larger than QCD we will see the axion mass can be enhanced and
the axion quality improved while still robustly solving the strong CP problem.
Actually, we will have to require $f$ larger than the weak scale because the
exotic fermions carry electroweak charges (see also Section 4).

The precise origin of the axion is not relevant to our work. What matters is
that its couplings to the $SU(3)_{\rm C}\times Sp(N-3)$ topological terms be
the same. In addition, we will work under the hypothesis that

$\displaystyle f_{a}>f,$ (2.5)

so that the tools of effective field theory can be employed in the next
section to study the axion potential. 444The opposite regime, with $f_{a}<f$,
may nevertheless provide a solution to the strong CP problem but requires a
completely different study. Scenarios in which $f_{a}>f_{\rm GC}$
automatically lead to equal couplings to the $SU(3)_{\rm C}\times Sp(N-3)$
topological terms. It is perhaps worth showing explicitly that the same may
also be true for $f_{a}<f_{\rm GC}$. To see this let us for example UV
complete the axion sector via an interaction $\mathcal{L}_{\text{PQ}}\supset
yFF^{c}\Theta$, with $F$ ($F^{c}$) fermions in the fundamental (anti-
fundamental) of $SU(N)_{\text{GC}}$ carrying $U(1)_{\rm PQ}$ charge $+1$ and
$\Theta$ a scalar of charge $-2$ responsible for breaking $U(1)_{\rm PQ}$
spontaneously at a scale $\sim f_{a}$. In such a model the axion acquires the
very same couplings to the $SU(3)_{\rm C}\times Sp(N-3)$ topological terms
even with $f_{a}<f_{\rm GC}$ because below the Grand Color breaking scale
$F,F^{c}$ split into the direct sum of fermions that are both in the
fundamental representation of $SU(3)_{\rm C}$ and $Sp(N-3)$ and so have the
same Dynkin index. The phase in $y$ does not affect this conclusion. Finally,
we note that for definiteness we decided to work within a KSVZ axion model,
but it should be clear that a DFSZ model would equally do.

##  3 The axion potential

At scales below $f_{\rm GC}$ our model reduces to the SM plus an $Sp(N-3)$
gauge theory with three families of fermions
$\psi_{q}=(\psi_{q_{u}},\psi_{q_{d}})$, $\psi_{u,d}$ charged as shown in table
2, with Yukawa couplings (2.1), and an axion $\bar{a}$ equally coupled to
color $SU(3)_{\rm C}$ and $Sp(N-3)$. All the scalars contained in $\Phi,\Xi$
acquire masses proportional to $f_{\rm GC}$ and decouple. In this section we
discuss the fate of the exotic fermions and the axion potential.

The basic assumption is that $Sp(N-3)$ confines at a scale $f<f_{\rm GC}$
larger than $v\approx 246$ GeV. This hypothesis is certainly realized provided
$N\geq 9$. In order to get rid of an otherwise large mixing between the axion
and the $\eta^{\prime}$ singlet of the $Sp(N-3)$ dynamics we remove the axion
from the topological term via a rotation of the $\psi_{q,u,d}$. This can for
example be achieved via a phase re-definition $\psi_{u}\to
e^{i\bar{a}/{3f_{a}}}\psi_{u}$, where the factor of $3$ denotes the number of
generations, which puts the axion in front of the up-type Yukawas, i.e.
$Y_{u}\to e^{i\bar{a}/{3f_{a}}}Y_{u}$.

The physics at confinement is better described in terms of a strong $Sp(N-3)$
dynamics with an approximate $SU(12)$ global symmetry under which the column
vector $\Psi=(\psi_{q_{u}},\psi_{q_{d}},\psi_{u},\psi_{d})$ transforms as the
fundamental representation. At confinement the chiral condensates
$\langle\psi_{q_{u}}\psi_{q_{d}}\rangle=-\langle\psi_{q_{d}}\psi_{q_{u}}\rangle=\langle\psi_{u}\psi_{d}\rangle\sim
4\pi f^{3}/\sqrt{N}$ break $SU(12)\to Sp(12)$ [26, 27]. To demonstrate this we
first observe that, because all bound states of $Sp(N-3)$ are bosonic, ’t
Hooft anomaly matching implies that $SU(12)$ must be broken. Finally, by the
Vafa-Witten theorem we know that the vectorial subgroup, namely $Sp(12)$,
should remain unbroken [28]. Crucially, the electroweak symmetry is part of
the unbroken group. The choice ${\rm C^{\prime}}=Sp(N-3)$ is essential to
achieve this key property.

The pattern $SU(12)\to Sp(12)$ delivers 65 would-be Nambu-Goldstone bosons
(NGBs) $\Pi$. These are not exact because the weak gauging of $SU(2)_{\rm
L}\times U(1)_{\rm Y}$ and the Yukawa couplings constitute a small explicit
breaking of $SU(12)$. In particular 51 of the would-be NGBs acquire positive
mass squared of order $g^{2}f^{2},g^{\prime 2}f^{2}$ from loops of the
$SU(2)_{\rm L}\times U(1)_{\rm Y}$ vectors. The other 14, denoted by
$\Pi_{0}$, are gauge-neutral and can in principle mix with $\bar{a}$,
similarly to the $\pi_{0}$ in the standard QCD axion. It is the dynamics of
these $\Pi_{0}$ that controls vacuum alignment and in particular the vacuum
expectation value of the axion. The vev of the charged NGBs, instead, vanish
and can be ignored in our discussion. Incidentally, some of the charged NGBs
are electroweak doublets and mix with the fundamental $H$. The heavy linear
combinations are integrated out, whereas we assume that the mass parameter of
$H$ is such that there exists a unique light eigenstate with a small and
negative mass squared. This will play the role of the Higgs doublet of the SM,
$H_{\rm SM}$. The fine-tuning we just invoked is nothing but the usual
hierarchy problem. 555Note that from this observation follows that the true SM
Yukawa couplings in low-energy observables differ compared to $Y_{u,d}$, not
only due to different RG effects, but also because of some mixing angle. We
will neglect these corrections since our results are anyway affected by
uncertainties of ${\cal O}(1)$ from incalculable coefficients.

The dynamics of the neutral NGBs can be effectively described observing that
the electroweak symmetry leaves intact a smaller $SU(3)_{q}\times
SU(3)_{u}\times SU(3)_{d}\times U(1)_{\text{B}}$ global subgroup of $SU(12)$,
associated to the independent flavor rotations of $\psi_{q},\psi_{u},\psi_{d}$
as well as the $Sp(N-3)$ baryon number under which $\psi_{u},\psi_{d}$ have
charge opposite to $\psi_{q}$. The vacuum condensates break this symmetry down
to $SO(3)_{q}\times SU(3)_{u-d}$. As a result the 14 neutral NGBs can be
effectively parametrized in terms of three matrices: a special, unitary and
symmetric matrix $\Sigma_{L}\in SU(3)_{q}/SO(3)_{q}$, a special, unitary
matrix $\Sigma_{R}\in SU(3)_{u}\times SU(3)_{d}/SU(3)_{u-d}$, and finally
$\eta_{\rm B}$, the NGB of $U(1)_{\rm B}$. The boson $\eta_{\rm B}$ remains an
exactly massless state because the baryon number is not explicitly broken. We
will discuss its phenomenology in Section 4. The remaining 13 neutral scalars,
along with the axion, acquire a potential from the Yukawa interactions of
$\psi_{q,u,d}$. We stress that these couplings are the same as those of the SM
quarks at the threshold $f_{\rm GC}$, though below that scale they renormalize
differently. At the scale $f$ relevant for the present discussion the
$\psi_{q,u,d}$ couplings, which to avoid over-complicating our notation will
still be denoted by $Y_{u,d}$, are expected to be somewhat larger than the SM
Yukawas by a flavor-universal factor due to loops of the $Sp(N-3)$ dynamics.
Expanding in powers of $Y_{u,d}$ the most general potential reads:

$\displaystyle
V_{\text{neutral}}=\frac{c_{ud}}{N}f^{4}\,\text{tr}\left[Y_{u}\Sigma_{R}Y_{d}^{t}\Sigma_{L}\right]e^{i\frac{\bar{a}}{N_{g}f_{a}}}+{\rm
hc}+{\cal O}(Y^{4},v^{2}/f^{2}),$ (3.1)

with $N_{g}=3$ the number of generations. The dominant contribution arises
from a loop of $H$ and the $Sp(N-3)$ dynamics. The factor of $N$ has been
identified using a large $N$ scaling and recalling that $f^{2}\propto N$. The
parameter $c_{ud}$ is a real incalculable quantity. Subleading corrections
contain $|H_{\rm SM}|^{2}$ and/or higher order insertions of the Yukawa
couplings. The former cannot affect qualitatively the potential; such
corrections are necessarily small because we are interested in the chiral
regime $v\lesssim f$ (see also Section 4). The latter will be argued to be
negligible in Section 3.2.

Contrary to the standard QCD axion, the theory under consideration contains a
light fundamental scalar with Yukawa couplings to the fermions $\Psi$ and it
is not possible to directly apply the results of [12] in order to argue that
$\langle\bar{a}\rangle=0$. The minimization problem is therefore conceptually
different from QCD. In particular, in QCD [12] imply that the vev of the pions
must vanish and the low energy dynamics contains no CP violation other than
the one encoded in $\bar{\theta}$. In our scenario, on the other hand, the
axion effective potential can depend non-trivially on the vacuum configuration
of the $\Pi_{0}$’s. We will have to prove $\langle\bar{a}\rangle=0$ by brute
force. This is what we will do in the next section.

Before turning to the minimization of the potential, though, we stress that
(3.1) possesses a $Z_{N_{g}}\subset SU(4N_{g})$ symmetry under which
$\Sigma_{R,L}\rightarrow e^{\pm i2\pi n/N_{g}}\Sigma_{R,L}$. This discrete
symmetry signals the presence of a set of inequivalent vacua sharing the same
perturbative mass spectrum and axion vev, which may indicate a cosmological
domain-wall problem if the temperature of the Universe ever exceeded $f$.
666These domain-walls are stable because $Z_{3}\subset U(1)_{\rm B}$, and
explicit breaking of $U(1)_{\rm B}$ occurs via effective operators of an
extremely high dimensionality, since the baryon number is very well protected
by our gauge symmetries. This issue adds to the more familiar domain-wall
problem of axion models, which takes place at the scale $f_{a}>f$.

###  3.1 Minimisation at leading order

The potential (3.1) involves 14 fields (13 neutral NGBs $\Pi_{0}$ and the
axion $\bar{a}$), and its minimization is highly non-trivial. To perform this
task we find it convenient to first discuss the properties of the more general
structure

$\displaystyle V_{\text{neutral}}^{\rm
LO}=V_{0}\left(\Pi_{0}/f\right)e^{i\bar{a}/f_{a}N_{g}}+{\rm hc},$ (3.2)

where the number of light fermion generations $N_{g}$ as well as the explicit
expression of $V_{0}$ are left arbitrary. Interestingly, both the potentials
of our model and that of the standard QCD axion have precisely this form.
Therefore some of the results discussed here have a rather general validity.
In particular, in appendix A we demonstrate that the absolute minimum of (3.2)
is found by maximizing $|V_{0}|$, whereas the axion vacuum is determined by
$\braket{\bar{a}}/f_{a}=N_{g}(\pi-\phi)\text{ mod }2\pi$, where $\phi={\rm
arg}[V_{0}]$ at the extremum.

In the case at hand $V_{0}$ is given in equation (3.1), and in the basis in
which $Y_{u}=\widehat{Y}_{u}$ is diagonal can be written as

$\displaystyle V_{0}$ $\displaystyle=$
$\displaystyle\frac{c_{ud}}{N}f^{4}\,\text{tr}[\widehat{Y}_{u}\Sigma_{R}\widehat{Y}_{d}V_{\text{CKM}}^{\scalerel*{\dagger}{X}}\Sigma_{L}]$
$\displaystyle=$
$\displaystyle\frac{c_{ud}}{N}f^{4}\,[\widehat{Y}_{u}]_{i}[A]_{ii},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}A=\Sigma_{R}\widehat{Y}_{d}V_{\rm
CKM}^{\scalerel*{\dagger}{X}}\Sigma_{L}.$

$|V_{0}|$ is maximized when $A$ is aligned as much as possible along
$\widehat{Y}_{u}$, with the corresponding entries satisfying
$|[A]_{33}|>|[A]_{22}|>|[A]_{11}|$. Suppose for the time being that it is
possible to find a configuration $\Sigma_{L,R}$ that fully diagonalizes $A$,
so that the diagonal entries read $[A]_{ii}=|[A]_{ii}|e^{i\phi_{i}}$, where
$\phi_{i}$ are phases subject to $\phi_{1}+\phi_{2}+\phi_{3}=2\pi n$ because
of the constraint $\det[A]=\det[\widehat{Y}_{d}]\in\mathds{R}$. Under this
hypothesis $|V_{0}|$ would be maximized when $\phi_{i}=\phi_{j}$ is common to
all entries, such that the trace becomes a coherent sum of terms, and the
minimum configuration would read $\phi_{i}=2\pi n/N_{g}$. The phase of $V_{0}$
at the minimum would finally be $\phi=\text{arg}\,c_{ud}+2\pi n/N_{g}$, and
from eq. (A.7) we would infer that
$\braket{\bar{a}}/f_{a}=N_{g}(\pi-(\text{arg}\,c_{ud}+2\pi
n/N_{g}))=N_{g}(\pi-\text{arg}\,c_{ud})$, or

$\displaystyle\frac{\braket{\bar{a}}}{f_{a}}=\begin{cases}0\text{ mod
}2\pi&{\rm if}~{}N_{g}={\rm even}\\\ 0\text{ mod }2\pi&{\rm if}~{}N_{g}={\rm
odd~{}and}~{}c_{ud}<0\\\ \pi\text{ mod }2\pi&{\rm if}~{}N_{g}={\rm
odd~{}and}~{}c_{ud}>0\end{cases}.$ (3.4)

This shows that, as long as $A$ can be diagonalized, the system has a natural
tendency to relax the axion to a CP-conserving vev. Thus the axion vev is
usually vanishing, though for odd $N_{g}$ and positive $c_{ud}$ we get
${\braket{\bar{a}}}/{f_{a}}=\pi$. Despite being CP-conserving, the latter
option is not phenomenologically acceptable because incompatible with the
Gell-Mann-Okubo relations [29]. In the standard QCD axion the result of [12]
ensures that ${\braket{\bar{a}}}/{f_{a}}=0$, which implies that $c_{ud}$ must
be negative. In our model later on we will offer some argument indicating that
$c_{ud}$ should be negative.

Unfortunately, it is possible to prove that as soon as $N_{g}\geq 3$ the
matrix $A$ cannot be exactly diagonalized because $\Sigma_{L}$, being unitary-
symmetric, does not contain enough degrees of freedom to diagonalize
$A^{\dagger}A$. In scenarios with $N_{g}\geq 3$ the logic leading to (3.4) can
thus at most be approximate. And yet, we find (at least for the physically
relevant case $N_{g}=3$) that (3.4) remains valid. Despite the impossibility
of diagonalizing $A$, in fact, the relation ${\rm
det}[A]=A_{11}A_{22}A_{33}+\Delta$ holds up to a very small perturbation
$\Delta$. Eq. (3.4) then applies because in a perturbative expansion for small
off-diagonal elements the dynamical phases of the three diagonal elements of
$A$ are determined at leading order to be $2\pi n/N_{g}$. That is, the
corresponding fluctuations fall into a deep potential well, which cannot be
destabilized by the next to leading corrections due to $\Delta$. As a result
the overall phase of $V_{0}$ is still determined by
$\phi=\text{arg}\,c_{ud}+2\pi n/N_{g}$ and the axion vev by (3.4), as if $A$
could be exactly diagonalized.

Even though the above arguments seem rather convincing to us, an explicit
calculation would help lifting any doubt on (3.4). Furthermore, an explicit
analysis is necessary to compute the masses of the NGBs and the axion. In the
following we will thus verify (3.4) and calculate the axion mass for
$N_{g}=1$, where in fact $A$ is trivially diagonalized, as well as for
$N_{g}=2$, where it can be fully diagonalized by the NGB matrices.
Subsequently we will consider the phenomenologically relevant case $N_{g}=3$.
Along the way we will argue in favor of $c_{ud}<0$.

#### Warming up with $N_{g}=1$ and $N_{g}=2$

The $N_{g}=1$ case is almost trivial, since the spectrum of NGBs is composed
of a charged composite Higgs, that is not relevant to vacuum alignment, and
the exact flat direction $\eta_{\rm B}$. The potential simply reduces to a
potential for the axion:

$\displaystyle V_{\text{neutral}}^{\rm
LO}=2\,\frac{c_{ud}}{N}f^{4}\,y_{u}y_{d}\cos\left(\frac{\bar{a}}{f_{a}}\right)\qquad\qquad\qquad(N_{g}=1)$
(3.5)

where $y_{u},y_{d}$ are the up and down quark Yukawas. This potential is
minimised at $\braket{\bar{a}}/f_{a}=0\text{ mod }2\pi$ if $c_{ud}<0$ or
$\braket{\bar{a}}/f_{a}=\pi\text{ mod }2\pi$ if $c_{ud}>0$, as expected from
(3.4). The axion mass is given by

$\displaystyle
m_{a}^{2}=2\,\frac{|c_{ud}|}{N}\,y_{u}y_{d}\frac{f^{4}}{f_{a}^{2}}\qquad(N_{g}=1).$
(3.6)

It is possible to show that for $N_{g}=1$ the parameter $c_{ud}$ must be
negative. The argument is a bit involved and will only be sketched here.

Our argument starts by considering a modified $N_{g}=1$ scenario in which
$Y_{u}=Y_{d}$ and only the neutral component of the fundamental Higgs is
dynamical. This is certainly not our model, but its effective potential is
just a simple generalization of ours because the UV diagrams that generate it,
in terms of fundamental fermions and $H$, are virtually identical to those in
our model modulo corrections of order $v^{2}/f^{2}$. In particular, the sign
of the overall coefficient $c_{ud}$ is exactly the same in the two scenarios
because determined by equal correlators in the unperturbed $Sp(N-3)$ theory.
The conclusion that the axion vev vanishes only for $c_{ud}<0$ remains valid.
But crucially, in the modified model the fermionic determinant arising from
the integration of $\Psi$ is real and positive definite because the fermionic
spectrum is effectively doubled, see [11]. Therefore the result of [12] can be
used to argue that the axion must be minimized at zero, and hence indirectly
that $c_{ud}<0$. This for us is proof that the coefficient $c_{ud}$ in (3.5)
is negative.

The minimization of the $N_{g}=2$ case is more interesting. In this case the
potential (3.1) depends on a CKM matrix that can be written in terms of the
Cabibbo angle:

$\displaystyle V_{\text{CKM}}=\begin{pmatrix}\cos\theta_{c}&\sin\theta_{c}\\\
-\sin\theta_{c}&\cos\theta_{c}\end{pmatrix}\qquad(N_{g}=2).$ (3.7)

As anticipated at the end of Section 3, we find two inequivalent vacua related
by a $Z_{2}$ symmetry. These are given by

$\displaystyle
c_{ud}>0:~{}~{}~{}\langle\Sigma_{L}\rangle=(\pm)\left(\begin{matrix}i\cos\theta_{c}&i\sin\theta_{c}\\\
i\sin\theta_{c}&-i\cos\theta_{c}\end{matrix}\right),~{}~{}~{}~{}~{}~{}~{}~{}\langle\Sigma_{R}\rangle=(\pm)\left(\begin{matrix}i&0\\\
0&-i\end{matrix}\right),~{}~{}~{}~{}~{}~{}\braket{\bar{a}}=0,$ (3.8)
$\displaystyle
c_{ud}<0:~{}~{}~{}\langle\Sigma_{L}\rangle=(\pm)\left(\begin{matrix}i\cos\theta_{c}&i\sin\theta_{c}\\\
i\sin\theta_{c}&-i\cos\theta_{c}\end{matrix}\right),~{}~{}~{}~{}~{}~{}~{}~{}\langle\Sigma_{R}\rangle=(\mp)\left(\begin{matrix}i&0\\\
0&-i\end{matrix}\right),~{}~{}~{}~{}~{}~{}\braket{\bar{a}}=0.$ (3.9)

With two generations the vacuum configurations precisely diagonalise
$A=\Sigma_{R}\widehat{Y}_{d}V_{\text{CKM}}^{\scalerel*{\dagger}{X}}\Sigma_{L}$,
and in both cases $\braket{\bar{a}}/f_{a}=0$ consistently with (3.4). In this
vacuum all scalar excitations (except for the exact flat direction $\eta_{\rm
B}$) are massive.

The $N_{g}=2$ case is so simple to handle analytically that we were able to
find an explicit expression for the effective axion potential. This is
obtained by solving the equation of motion for the neutral NGBs $\Pi_{0}$ and
plugging it back into $V_{\rm neutral}$. It is a reliable approximation of the
axion self-couplings in the limit $f\ll f_{a}$ in which the neutral NGBs are
much heavier than the axion. We find

$\displaystyle
V_{\text{eff}}\,\left(\frac{\bar{a}}{f_{a}}\right)=-2\,\frac{|c_{ud}|}{N}f^{4}\,\text{tr}[\widehat{Y}_{u}\widehat{Y}_{d}]\sqrt{1-4\frac{\det[\widehat{Y}_{u}\widehat{Y}_{d}]}{\left(\text{tr}[\widehat{Y}_{u}\widehat{Y}_{d}]\right)^{2}}\sin^{2}\left(\frac{\bar{a}}{2f_{a}}\right)}\qquad(N_{g}=2),$
(3.10)

which is consistently minimized at $\langle\bar{a}\rangle/f_{a}=0$ mod $2\pi$.
This result is reminiscent of the potential of the QCD axion in 2-flavor QCD.
The axion mass immediately follows:

$\displaystyle
m_{a}^{2}=2\frac{|c_{ud}|}{N}\,\frac{\det[\widehat{Y}_{u}\widehat{Y}_{d}]}{\text{tr}[\widehat{Y}_{u}\widehat{Y}_{d}]}\,\frac{f^{4}}{f_{a}^{2}}\qquad(N_{g}=2).$
(3.11)

Eq. (3.10) is very valuable to us because we will not be able to obtain an
explicit expression for $N_{g}=3$. It is therefore useful to extract as much
information as possible from it. First, we observe that in the limit of a
heavy second generation eqs (3.10), (3.11) reduce to eqs (3.5), (3.6). This is
a highly non-trivial check of the consistency of our results. It is a
consequence of the fact that a large Yukawa coupling for the second generation
implies that a number of NGBs becomes much heavier than those associated to
the light first generation. Up to corrections suppressed by the heavy NGB
mass, therefore, the potential should reduce to the $N_{g}=1$ case, which is
what we see here explicitly. The very same logic constrains the structure of
the $N_{g}=3$ potential, as we will verify numerically.

A second important lesson we can draw from (3.10), and more readily (3.6), is
that the non-trivial dependence of the axion potential should be controlled by
$\det[\widehat{Y}_{u}\widehat{Y}_{d}]$. This is also a very general result,
independent of $N_{g}$. Indeed, if any of the eigenvalues of $Y_{u}$ or
$Y_{d}$ were to vanish the UV Lagrangian would be invariant under an
additional anomalous axial symmetry which could be combined with $U(1)_{\rm
PQ}$ to obtain an exact unbroken one. In that situation the axion would become
an exact flat direction. Hence the axion potential must be proportional to at
least a power of all the eigenvalues. The quantity
$\det[\widehat{Y}_{u}\widehat{Y}_{d}]$ is the simplest object with this
property. The axion mass squared cannot be simply proportional to the
determinant unless $N_{g}=1$, however. By counting the units of $\hbar$ we
need at least $2N_{g}-2$ additional coupling constants in the denominator, so
dimensional analysis forces $m_{a}^{2}$ to be inversely proportional to an
appropriate combination of $Y_{u},Y_{d}$ as found in (3.11). The significant
hierarchy in the SM fermion masses and the decoupling properties mentioned in
the previous paragraph, together indicate that the axion mass in our model is
always numerically close to (3.6).

#### The real world: $N_{g}=3$

Having checked the simplified scenarios $N_{g}=1,2$, we can now turn to the
realistic case $N_{g}=3$. The previous calculations support the correctness of
our qualitative argument of section 3.1, and as such we expect the leading
order potential (3.1) to be minimised at $\braket{\bar{a}}/f_{a}=0\,(\pi)$ for
$c_{ud}<0\,(>0)$. Unfortunately, in the case $N_{g}=3$ the potential involves
14 fields and it is not possible to approach the problem analytically. For
this reason we employ a customised MATHEMATICA algorithm which enables us to
numerically find the minimum of the potential up to a very high accuracy. The
minimisation procedure is repeated many times in order to statistically
validate the result. The Yukawa couplings that appear in the potential are
renormalized at the scale $\sim 4\pi f/\sqrt{N}$ by the $Sp(N-3)$ dynamics. As
a benchmark we employ the PDG data for $V_{\text{CKM}}$ and the numerical
values of $\widehat{Y}_{u},\widehat{Y}_{d}$ that correspond to the SM quark
Yukawas evaluated at the TeV scale. Changing the numerical value of these
couplings does not affect our results qualitatively. What we find is exactly
what anticipated in section 3.1: the distinct vacua configurations are related
by a $Z_{3}$ symmetry $\Sigma_{R,L}\rightarrow e^{\pm i2\pi n/3}\Sigma_{R,L}$;
all vacua give rise to the same $A$, which is diagonal up to small off-
diagonal elements; the axion is minimised at $\braket{\bar{a}}/f_{a}=0$ or
$\pi$ depending on the sign of $c_{ud}$.

In the study of the $N_{g}=1$ toy model we gave an argument supporting the
claim that $c_{ud}$ is negative. At sufficiently large $N$ there is no
distinction between the coefficients $c_{ud}$ for the $N_{g}=1$ and $N_{g}=3$
scenarios. We are therefore motivated to conjecture that $c_{ud}<0$ also for
$N_{g}=3$, and from now on work under this hypothesis. Given the central role
played by this hypothesis, it would be interesting to find an independent
proof, for instance using lattice QCD techniques.

We are now interested in studying the spectrum of the NGBs and the axion. To
read off the masses we first canonically normalise the kinetic term $f^{2}{\rm
tr}[\partial_{\mu}\Sigma_{L,R}\partial^{\mu}\Sigma_{L,R}]$, altered by the vev
of the NGBs, and then perform an $SO(14)$ rotation to diagonalise the Hessian
of the potential. As a result of this operation, the masses of the 13 neutral
NGBs are found to be

$\displaystyle m_{\Pi_{0}}^{2}\simeq\left(\begin{matrix}6.4\times 10^{-2}\\\
2.5\times 10^{-2}\\\ 2.4\times 10^{-2}\\\ 2.4\times 10^{-2}\\\ 2.4\times
10^{-2}\\\ 2.4\times 10^{-2}\\\ 2.3\times 10^{-2}\\\ 2.7\times 10^{-5}\\\
1.2\times 10^{-5}\\\ 2.5\times 10^{-6}\\\ 1.6\times 10^{-6}\\\ 1.6\times
10^{-6}\\\ 4.0\times 10^{-8}\end{matrix}\right)\times\frac{|c_{ud}|}{N}f^{2}$
(3.12)

for all the three $Z_{3}$-symmetric vacua configurations. The eigenstate
corresponding to the axion is the lightest one for any $f<f_{a}$. The mass
that we extract numerically respects the scaling suggested in the previous
section, namely

$\displaystyle m_{a}^{2}\simeq
2\frac{|c_{ud}|}{N}y_{u}y_{d}\,\frac{f^{4}}{f_{a}^{2}}\simeq 1.8\times
10^{-10}\times\frac{|c_{ud}|}{N}\,\frac{f^{4}}{f_{a}^{2}}\qquad(N_{g}=3).$
(3.13)

The NGBs’ masses are substantially unaffected by the mixing with the axion as
long as $f/f_{a}\ll 1$ (even though the mixing will turn out to be important
for phenomenology, see section 4). In the extreme limit $f/f_{a}\to 1$ the
mixing impacts the NGBs masses by a few $10\%$, with the lightest being
affected the most.

We conclude this section emphasizing that the axion mass (3.13) is
parametrically enhanced with respect to the standard QCD axion as long as
$f\gtrsim 10^{2}f_{\pi}$. The goal outlined in Section 1 is fulfilled.

###  3.2 Subleading corrections and heavy axion quality

The analysis carried out so far demonstrates that our axion has a large mass
and a leading order potential minimized at $\braket{\bar{a}}=0$. Higher order
corrections cannot affect this result unless they introduce new sizable
sources of CP-violation or flavor-violation. We will argue next that
subleading effects due to renormalizable interactions do not spoil our
solution of the strong CP problem and that the effect of non-renormalizable
operators can be taken under control.

#### Renormalizable interactions

In the renormalizable version of our scenario the effective axion potential
$V_{\rm eff}(\bar{a}/f_{a})$ depends on flavor-invariant combinations of the
parameters $Y_{u,d}$, $\langle\Sigma_{L,R}\rangle$. The Yukawas parametrize
_explicit_ CP violation, whereas the NGB vacuum potentially represents an
independent source of _spontaneous_ CP violation. A non-vanishing vev for the
axion is induced by CP-odd, flavor-conserving combinations of these
parameters.

Crucially, in our model all CP-odd invariants must necessarily be proportional
to explicit CP-violation. This follows from the fact that spontaneous CP
violation does not take place. As a first simple check of this statement, let
us inspect the $N_{g}=2$ toy model, where we have an explicit analytic
solution. Here the CKM matrix is real, i.e. there is no explicit CP violation,
and it is readily seen that eqs (3.8) and (3.9) preserve the generalised CP
transformation $\Sigma_{R,L}\to-\Sigma_{R,L}^{*}$. A far less trivial check is
obtained for $N_{g}=3$. In that case we verified that, when we switch off the
CKM phase, the explicit numerical solution of the leading order potential in
(3.1) also satisfies the relation $\braket{\Sigma_{R,L}}=e^{\pm i2\pi
n/3}\braket{\Sigma_{R,L}}^{*}$. In other words, in the absence of explicit
violation, CP is not spontaneously broken.

An important consequence of what we just demonstrated is that any CP-odd
flavor invariant in $V_{\rm eff}$ must be proportional to the explicit CP-
violation in $Y_{u,d}$. A key property of our theory, inherited from the SM,
is that explicit CP-violation should disappear whenever two of the eigenvalues
of the SM quark mass matrix squared are degenerate, or any mixing angle goes
to zero, or when the CKM phase vanishes. This is very important. We have
already seen that the non-trivial part of the effective axion potential must
be proportional to ${\rm det}[Y_{u}Y_{d}]$. Here we find that any explicit CP-
violating interaction of the axion must contain a further suppression that
disappears in the above limits. Such a suppression is so significant that
explicit CP-violation in $V_{\rm eff}$ becomes effectively innocuos: CP-odd
flavor-invariant combinations of the Yukawas and $\langle\Sigma_{L,R}\rangle$
may arise in $V_{\rm eff}$ only at very high order in an expansion in
$Y_{u,d}$ and are numerically extremely small. It is therefore not surprising
that, even including the CKM phase, our ${\cal O}(Y^{2})$ potential does not
induce an axion vev. In fact, at ${\cal O}(Y^{2})$ the effect of explicit CP
violation cannot be visible, the only flavor-invariant candidate
$\text{tr}\left[Y_{u}\braket{\Sigma_{R}}Y_{d}^{t}\braket{\Sigma_{L}}\right]$
is real and hence there is nothing that can be on the right-hand side of
$\braket{\bar{a}}=0$.

In summary, subleading corrections to the effective axion potential are either
even in $\bar{a}$ or odd, the latter being proportional to the explicit CP-
violation. Terms even in $\bar{a}$ cannot destabilize our solution because the
leading order theory has no flat directions. The terms odd in $\bar{a}$ are
however dangerous if they include a tadpole. In that case the axion vev is
shifted from the origin. Still, the shift must be proportional to the tiny
explicit CP-violating phase that controls the tadpole and the vacuum
expectation value would thus be safely below
$|\langle{\bar{a}}\rangle|/f_{a}\lesssim 10^{-10}$. Our axion dynamically
solves the strong CP problem like in the standard QCD scenario. Higher
dimensional operators with new CP-violating or flavor-violating couplings can
however introduce new CP-odd flavor invariants which can be numerically more
relevant that those of the renormalizable theory. These effects are discussed
next.

#### Higher-dimensional operators

At the root of the axion quality problem is the fact that a huge $f_{a}$ makes
the axion potential extremely sensitive to cutoff-suppressed $U(1)_{\rm
PQ}$-breaking interactions. As reviewed in Section 1 this sensitivity may be
alleviated by increasing the axion mass. However, there is no free lunch. To
enhance the axion mass, the confinement scale $f$ has to be rather large as
well. As a result, cutoff-suppressed $U(1)_{\rm PQ}$-_conserving_
contributions to the axion potential, as long as they are CP- or flavor-
violating, may become important and in principle spoil the solution of the
strong CP problem.

More precisely, consider the flavor-violating but $U(1)_{\rm PQ}$-preserving
operator

$\displaystyle\frac{\bar{c}_{ijkl}}{f_{\text{UV}}^{2}}Q_{i}Q_{j}U_{k}D_{l},$
(3.14)

where $i,j,k,l$ are flavor indices. This has precisely the same axial
$U(1)_{A}$ charges as the leading order potential in (3.1) and thus represents
a modification $\delta V_{0}\sim\bar{c}\,16\pi^{2}f^{6}/(N^{2}f_{\rm UV}^{2})$
of the quantity $V_{0}\sim{\rm Tr}[Y_{u}Y_{d}]f^{4}/N$ defined in (3.1). This
is not aligned with (3.1) for generic ${\bar{c}}_{ijkl}$, and must therefore
be small. According to eq.(A.2) the axion vev is of order
$\langle\bar{a}\rangle/f_{a}\sim{\rm Im}[\delta V_{0}]/|V_{0}|$, where the
imaginary part of the flavor invariant $\delta V_{0}$ may come either from
${\bar{c}}_{ijkl}$ directly or from phases of the leading NGB vev, which
become physical when contracted with a flavor-violating ${\bar{c}}_{ijkl}$.
The requirement that the effective topological angle be less than $10^{-10}$
becomes

$\displaystyle{f}\lesssim 10^{-7}f_{\rm UV},$ (3.15)

which for a maximal UV cutoff of order $f_{\rm UV}=2.4\times 10^{18}~{}{\rm
GeV}$ reads ${f}\lesssim 10^{11}$ GeV. Somewhat similar considerations apply
to operators that do not violate the axial $Sp(N-3)$ symmetry $U(1)_{A}$ but
still violate flavor, like

$\displaystyle\frac{\bar{c}_{ijkl}}{f_{\rm
UV}^{2}}(\Psi_{i}\Psi_{j})(\Psi_{k}\Psi_{l})^{\dagger}.$ (3.16)

This operator modifies the NGB vev and in turn shifts the axion minimum. As a
conservative bound we impose (3.15). Moreover, we could have operators that do
not violate flavor, but contribute new CP-odd flavor-conserving phases to the
axion effective potential. A typical example is the Weinberg operator

$\displaystyle\frac{\bar{c}_{W}}{M_{\rm UV}^{2}}\frac{g^{3}_{\rm
GC}}{16\pi^{2}}G_{\rm GC}G_{\rm GC}\widetilde{G}_{\rm GC}.$ (3.17)

The new CP-odd phase should be smaller than $10^{-10}$ to guarantee a solution
of the strong CP problem. From this requirement the weaker upper bound
$\bar{c}_{W}f^{2}/f^{2}_{\rm UV}\lesssim 10^{-10}$ follows.

Finally, let us come back to the original motivation of our work: the axion
quality problem. As a crude estimate, imposing that a $U(1)_{\rm
PQ}$-violating operator of dimension $d$ does not significantly alter the
axion vev implies

$\displaystyle g_{\rm
UV}^{2}f_{a}^{4}\left(\frac{f_{a}}{f_{\text{UV}}}\right)^{d-4}\lesssim
10^{-10}m_{a}^{2}f_{a}^{2},$ (3.18)

where $g_{\rm UV}$ is a coupling of the UV dynamics. For a given value of
$f_{a}$ this represents a lower bound on $d-4$. Comparing to the lower bound
in the standard QCD axion, i.e. $(d-4)_{\text{std}}$, this reads

$\displaystyle(d-4)=(d-4)_{\text{std}}-\frac{\ln{m_{a}^{2}}/{m_{a,{\rm
std}}^{2}}}{\ln{f_{\rm UV}}/{f_{a}}}.$ (3.19)

The quality problem is logarithmically sensitive to the mass ratio
${m_{a}^{2}}/{m_{a,{\rm std}}^{2}}$ and becomes more and more sensitive to
this quantity as $f_{a}$ gets larger. In our model we find an appreciable
improvement as long as $f\gg 10^{2}f_{\pi}$, see (3.13). As a numerical
example, for $f_{a}=10^{10}$ GeV and $f=10^{8}$ GeV we get an axion mass of a
few GeV. This corresponds to $d\gtrsim 7$, which is a significant improvement
compared to $d_{\text{std}}\gtrsim 10$.

##  4 Phenomenology

The phenomenology of our model is extremely rich and cannot be investigated in
depth here. In this section we present a first qualitative assessment.

The $Sp(N-3)$ dynamics generates many massive hadrons, all of which are
unstable because there is no unbroken flavor symmetry that protects them.
Heavy hadrons of mass $\propto 4\pi f/\sqrt{N}$ as well as baryons quickly
decay into NGBs. The electroweak-charged NGBs decay into the SM Higgs boson,
$W^{\pm},Z^{0}$, and neutral $\Pi_{0}$’s. The latter are much more long-lived,
and decay dominantly into QCD hadrons and/or photons via the mixing with the
axion and the $\eta^{\prime}$ of the $Sp(N-3)$ dynamics. Less relevant decay
channels for $\Pi_{0}$’s are into SM fermions via non-renormalizable
interactions generated at the scale $f_{\rm GC}$. Very likely, yet, only the
lightest hadrons were significantly produced in the early Universe because in
order to robustly avoid a domain-wall problem associated to the
$Z_{3}$-degeneracy of the NGB potential reheating must probably have occurred
after $Sp(N-3)$ confinement (see discussion below eq. (3.1)).

The hadrons can be directly produced at the LHC and future colliders. In
addition, the $Sp(N-3)$ dynamics can be indirectly probed via precision
measurements. As a rough measure of the current impact of these constraints we
impose the qualitative bound $f\gtrsim$ TeV. In this regime the low energy
signatures are mainly controlled by $\bar{a}$ and $\Pi_{0}$ (and, if present,
$\eta_{B}$; see below). The effective field theory is governed by the
couplings to the topological terms of the gluon and the photon

$\displaystyle\mathcal{L}_{\rm EFT}$ $\displaystyle\supset$
$\displaystyle\frac{1}{2}(\partial\bar{a})^{2}-\frac{m_{a}^{2}}{2}\bar{a}^{2}+\frac{g_{\rm
C}^{2}}{32\pi^{2}}\frac{\bar{a}}{f_{a}}G\widetilde{G}+\bar{c}_{a\gamma\gamma}\frac{e^{2}}{32\pi^{2}}\frac{\bar{a}}{f_{a}}F\widetilde{F}$
$\displaystyle+$
$\displaystyle\frac{1}{2}(\partial\Pi_{0,i})^{2}-\frac{m_{\Pi_{0},i}^{2}}{2}\Pi_{0,i}^{2}+\frac{g_{\rm
C}^{2}}{32\pi^{2}}\bar{c}_{\Pi_{0}gg}^{i}\frac{\Pi_{0,i}}{f}G\widetilde{G}+\bar{c}^{i}_{\Pi_{0}\gamma\gamma}\frac{e^{2}}{32\pi^{2}}\frac{\Pi_{0,i}}{f}F\widetilde{F}.$

The effective couplings to the $Z^{0}$ and $W^{\pm}$ bosons are
phenomenologically less relevant. Assuming that the $U(1)_{\rm PQ}$ has no
electroweak anomaly, and momentarily ignoring the mixing with $\Pi_{0}$, the
coefficient $\bar{c}_{a\gamma\gamma}$ can be computed by moving the axion from
the $SU(N)_{\text{GC}}$ topological term to the Yukawas with an anomalous
chiral rotation, and re-placing it only in front of the QCD topological term
below the Grand Color breaking. Recalling that the SM hypercharge is given by
a combination of $U(1)_{\text{Y}^{\prime}}$ and $U(1)_{\text{GC}}$, we get

$\displaystyle\bar{c}_{a\gamma\gamma}=-\frac{1}{2}\left(N-3\right)\left(1-\frac{1}{3N}\right).$
(4.2)

Consistently with expectations, this expression vanishes for $N=3$, when our
model reduces to a standard KSVZ scenario. We will consider $N=13$ for
definiteness, noting that a number of colors $>17$ would typically induce a
Landau pole for $SU(2)_{\rm L}$ below the Planck scale whereas for $N<9$ the
condition $f\gtrsim$ TeV would not be attained. We verified that with these
parameters the condition $f<f_{\rm GC}\lesssim 10^{13}\text{ GeV}$ is also
satisfied.

The neutral NGBs have no bare coupling to the SM vectors. However, they
acquire them from the mixing with $\bar{a}$ and the heavy $\eta^{\prime}$ of
$Sp(N-3)$. These contributions are parametrically of order
$\bar{c}_{\Pi_{0}gg}\sim f/f_{a}$ and $\bar{c}_{\Pi_{0}\gamma\gamma}\sim{\rm
max}[m_{\Pi_{0}}^{2}/m_{\eta^{\prime}}^{2},f/f_{a}]$. As a result the decay
rate into gluons, controlled by the $\bar{a}-\Pi_{0}$ mixing, is of order

$\displaystyle\Gamma_{\Pi_{0}\to gg}\gtrsim\Gamma_{\bar{a}\to
gg}\frac{f_{a}}{f},$ (4.3)

where we took into account the different scaling of the masses with $f,f_{a}$
(see (3.13) and (3.12)). Importantly, the rate is always greater than
$\Gamma_{\bar{a}\to gg}$ in the regime (2.5). This parametric estimate is
confirmed by an accurate numerical analysis, which also reveals that some
$\Pi_{0}$ can have rates several orders of magnitude larger than shown in
(4.3). Note also that the NGB mixing with the axion does not appreciably
modify (4.2). As long as $f\ll f_{a}$ the effect is parametrically suppressed,
and we numerically verified that as $f$ approaches $f_{a}$ the change in the
axion coupling to photons is still at most ${\cal O}(10\%)$.

The axion mass is given in eq. (3.13) and in the allowed regime $f\gtrsim$ TeV
is always larger than the standard one. Furthermore, as we saw around (3.15),
a conservative condition for the strong CP problem to be solved is $f\lesssim
10^{11}$ GeV, where we identified the UV cutoff with the Planck scale.
Combining the two bounds we see that our scenario populates the light-pink
bend in the $m_{a}-f_{a}$ plot of Fig. 1 labeled by “Grand Color axion”,
defined by the implicit relation $10^{3}~{}{\rm GeV}\leq f\leq 10^{11}$ GeV —
where $f=f(m_{a},f_{a})$ is given by (3.13). We included a hard cut at
$f<f_{a}$ to indicate the regime of validity of the effective field theory
approach adopted in this paper, see eq. (2.5). In the grey region $f>f_{a}$
our results do not necessarily apply, though without a detailed analysis this
region cannot be excluded. The dotted grey lines in Fig. 1 show contour
regions of eq. (3.19) with $f_{\rm UV}=2.4\times 10^{18}$ GeV. The axion
quality problem is progressively more alleviated as we move towards the upper-
right corner. The regime with high quality is the one in which, for a fixed
$f_{a}$, the axion mass is maximal, i.e. the confinement scale reaches the
extreme value $f\to f_{a}$ compatible with (2.5).

Figure 1: Collection of the main bounds on $f_{a}$ vs $m_{a}$, as discussed in
section 4. For definiteness we assumed $N=13$ and $|c_{ud}|=1$. The light-pink
bend denoted by “Grand Color axion” identifies the region populated by our
scenario.

The “Grand Color axion” bend is mostly probed by cosmological observations
from BBN and CMB physics (blue), astrophysics (green), and collider
experiments (brown). In particular, the left boundary of the blue region is
taken from the collection of bounds in [30]. The rightmost part of the
cosmology bound is due mainly to the $N_{\rm eff}$ bounds from [31] and the
requirement that the total axion decay rate satisfies $\Gamma_{\rm tot}\geq
3H(T_{\rm BBN})$, where $H(T_{\rm BBN})$ is the Hubble rate when the Universe
reached temperatures of order $T_{\rm BBN}=4$ MeV [32], in order not to
interfere with Big-Bang-Nucleosynthesis. The hadronic decay rate has been
calculated adapting the results of [33] to our model. The axion decay rate
takes into account (4.2), includes also the mixing with the QCD mesons, and is
modified with respect to the standard case because of the much larger value of
$m_{a}$. For the astrophysics bounds we refer to [30]. The collider bounds on
top of Fig. 1 are taken from [34] and [35]. The white regions are currently
viable.

In the far bottom-left of the “Grand Color axion” bend one should make sure
that the heavier NGBs are sufficiently long-lived to avoid disrupting the
primordial abundance of light elements. However, because of their large
masses, vacuum misalignment typically over-produces them in the form of a Bose
condensate of $\bar{a},\Pi_{0}$ unless the initial misalignment angles are
extremely small. Moreover, in that region the axion quality problem is not
ameliorated compared to the standard scenario. For these reasons we believe
the upper-right region is more motivated.

The upper-right corner of the “Grand Color axion” bend is also more
interesting because in that regime ongoing and future accelerators as well as
future CMB observations are able to explore our model. In particular, because
$f_{a}$ can approach the TeV scale this scenario can be probed for example at
HL-LHC [21], Kaon and Hyperon factories [34], DUNE [35], NA62 [36], Belle II
[37, 38] and MATHUSLA [39]. The phenomenological signatures are much richer
than the standard QCD axion because of a variety of axion-like-particles
$\Pi_{0}$ with anomalous couplings, see (4). Future CMB surveys will also
significantly improve measurements of $N_{\rm eff}$ and will be able to
constrain a wider region of parameter space on the right of the blue area. In
contrast to the accelerator searches mentioned above, however, here the
neutral NGBs are not expected to play any relevant role because are much
heavier and decay faster than the axion, see (4.3). Yet, they might still lead
to non-standard cosmological signatures, albeit quite indirect. The energy
stored in a Bose condensate of neutral NGBs via vacuum misalignment may be
estimated as
$\rho_{\Pi_{0}}(T)\sim\theta_{r}^{2}m_{\Pi_{0}}^{2}f^{2}({T}/{T_{r}})^{3}$,
with $\theta_{r}$ denoting the misalignment angle at the end of inflation and
$T_{r}$ the reheating temperature. For certain values of masses and decay
rates the temperature $T_{\rm m}$ at which $\rho_{\Pi_{0}}(T_{\rm m})$
dominates over radiation is actually larger than the decay temperature $T_{\rm
d}\sim\sqrt{\Gamma_{\Pi_{0}}M_{\rm Pl}}$. When this happens the Universe
undergoes an early period of matter domination which might result in a
depletion of the primordial densities of visible and dark matter.

One last comment should be added before closing. In addition to
$\Pi_{0},\bar{a}$, our model generically features a virtually massless
photophobic axion-like particle $\eta_{B}$ from the breaking of $U(1)_{\rm
B}\subset SU(12)$. It acquires no potential from the Yukawa interactions,
since these are $U(1)_{\rm B}$-symmetric, and only has electroweak anomalous
couplings. Its effective Lagrangian reduces to

$\displaystyle\mathcal{L}_{\rm EFT}$ $\displaystyle\supset$
$\displaystyle\frac{1}{2}(\partial\eta_{\rm B})^{2}+\frac{\eta_{\rm
B}}{f_{B}}\left(\frac{g_{\rm L}^{2}}{32\pi^{2}}W\widetilde{W}-\frac{g_{\rm
Y}^{2}}{32\pi^{2}}Y\widetilde{Y}\right),$ (4.4)

where $f_{B}=(2/\sqrt{3})f/N$, plus derivative interactions with itself and
the other NGBs. While its coupling to photons vanish, 1-loop generated
interactions to the SM fermions lead to a constraint of order $f\gtrsim
300\times N$ TeV [40] whereas, under the hypothesis $T_{f}<f$, the impact on
$N_{\rm eff}$ is minimal. The bound on $f$ just quoted is much stronger than
those considered before. To obliterate the problem we may get rid of
$\eta_{B}$ by gauging $U(1)_{\rm B-L}$, as suggested for other reasons in
Section 2. In order for $\eta_{\rm{B}}$ to be gauged away, all the fundamental
scalars should be neutral and right-handed neutrinos should be added in order
to ensure gauge anomalies cancellation.

##  5 Conclusions

We constructed a concrete model for a heavy axion arising from an enlarged
color sector, which is conceptually and structurally very simple, and has an
extremely minimal field content. Including all SM fields as well, the latter
entirely fits in Table 1. No mirror copy of the SM, nor of color, is required.
In addition, there is no need for additional global symmetries besides the
familiar $U(1)_{\rm PQ}$. The necessary coupling structure may naturally
result as a consequence of gauge invariance and the field content.

The unique sources of CP-violation in the renormalizable part of the theory
are included in the Yukawa couplings $Y_{u,d}$ and the topological angles,
exactly as in the SM. Hence, radiative corrections cannot spoil the equality
of the couplings of the axion to QCD and the new confining group ${\rm
C^{\prime}}=Sp(N-3)$. From the very same reason follows that the effective
Grand Color axion potential is automatically aligned with the QCD one: our
scenario introduces no exotic flavor-conserving CP-odd phases, and is
therefore approximately CP-invariant up to a very high accuracy, like the QCD
potential.

Our scalar potential has a characteristic ${\cal O}(Y^{2})$ form but also
shares some similarity with the standard one due to the presence of a mixing
between the axion and the heavy neutral Nambu-Goldstone modes, somewhat
analogous to the $\bar{a}-\pi_{0}$ mixing in the standard scenario. The
resulting Grand Color axion mass is very distinctive and scales as:

$\displaystyle m_{a}^{2}\sim\frac{y_{u}y_{d}}{N}\frac{f^{4}}{f_{a}^{2}},$
(5.1)

where $y_{u,d}$ are the up- and down-quark Yukawas renormalized by the
$Sp(N-3)$ dynamics at $\sim 4\pi f/\sqrt{N}$. By construction the axion
potential in our model is a 2-loop-sized effect proportional to two powers of
Yukawas, and eq. (5.1) follows from dimensional analysis and simple physical
considerations. This expression differs from the one predicted by existing
heavy axion models and its magnitude falls somewhat in between mirror and UV-
instanton models. It is significantly enhanced compared to the one predicted
by potentials dominated by small instantons. As a result, an improvement in
axion quality is achieved with a significantly smaller $f$, and hence a
reduced sensitivity to physics at the cutoff scale. The scaling in (5.1) is
however suppressed compared to what is found in $Z_{2}$-symmetric models, and
so for a similar axion mass our $f$ needs to be larger.

While corrections to the effective topological angle
$\langle\bar{a}\rangle/f_{a}$ from renormalizable couplings are virtually
negligible, the effect of $U(1)_{\rm PQ}$-violating as well as $U(1)_{\rm
PQ}$-conserving higher-dimensional operators can in principle alter the axion
vev in a significant way. The sensitivity to Peccei-Quinn-violating
interactions suppressed by the UV cutoff is reduced compared to the standard
scenario, and the quality is certainly improved because of the larger axion
mass. Yet, we found that heavy axion models develop a novel sensitivity to
_Peccei-Quinn-preserving_ deformations. 777During the completion of this work,
ref. [41] appeared stressing this very same point. Their numerical estimates
are based on 1-instanton calculations and therefore not reliable for our model
nor for mirror-symmetric scenarios. Yet their conclusions are qualitatively
general and agree with ours. This “heavy axion quality problem” is generically
shared by all models that attempt to increase the axion mass via a new strong
coupling at $f\gg f_{\pi}$, and ours is no exception. Future measurements of
the dipole moments of the neutron, as well as of atoms and molecules will
potentially be able to set upper bounds on the axion mass of these scenarios.

Finally, we should point out that in this paper we presented just one of a
larger set of interesting Grand Color scenarios. Our choice ${\rm
C}^{\prime}=Sp(N-3)$ was for example motivated by the need to avoid
electroweak symmetry breaking at the scale $f$, but other options are
possible. It would be interesting to explore alternative Grand Color
scenarios, for example models with different ${\rm C}^{\prime}$, scenarios
compatible with a complete grand unification into a simple gauge group, or
scenarios with composite $\Phi,\Xi$, and construct explicit models with
$f_{a}\sim f$. Besides robustly addressing the strong CP problem with an
alleviated quality problem, these models feature a number of distinctive
phenomenological signatures, which in this paper we just began to explore.

## Acknowledgments

We thank F. D’Eramo and L. Di Luzio for discussions on the cosmological and
collider bounds. This research was partly supported by the Italian MIUR under
contract 2017FMJFMW (PRIN2017), the “iniziativa specifica” Physics at the
Energy, Intensity, and Astroparticle Frontiers (APINE) of Istituto Nazionale
di Fisica Nucleare (INFN), and the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
860881-HIDDeN.

##  Appendix A General considerations on the leading order potential

In this appendix we derive the minimization condition for the leading order
potential of the neutral scalars. This can be compactly written as in eq.
(3.2). The extremality conditions read

$\displaystyle\begin{cases}\frac{\delta
V_{0}}{\delta\Pi_{m}}e^{i\braket{\bar{a}}/f_{a}N_{g}}+\frac{\delta
V^{*}_{0}}{\delta\Pi_{m}}e^{-i\braket{\bar{a}}/f_{a}N_{g}}=0\\\
V_{0}e^{i\braket{\bar{a}}/f_{a}N_{g}}-V_{0}^{*}e^{-i\braket{\bar{a}}/f_{a}N_{g}}=0.\end{cases}$
(A.1)

Combining these equations (we will show below that $|V_{0}|\neq 0$) we get
$e^{i\braket{\bar{a}}/f_{a}N_{g}}=\pm V_{0}^{*}/|V_{0}|$, so that the above
system can be rewritten as

$\displaystyle\begin{cases}\frac{\delta|V_{0}|}{\delta\Pi_{m}}=0\\\
\sin\left(\frac{\braket{\bar{a}}}{N_{g}f_{a}}\right)=\mp\frac{{\rm
Im}[V_{0}]}{|V_{0}|}\end{cases}$ (A.2)

from which it is clear that the vacuum is obtained extremizing $|V_{0}|$, and
a necessary condition for the strong CP problem to be solved is that $V_{0}$
is real at the minimum.

To see whether the vacuum is actually at the minimum or at the maximum of
$|V_{0}|$ we must study the Hessian, which reads

$\displaystyle{\cal H}=\left(\begin{matrix}M_{\Pi\Pi}^{2}&M^{2}_{a\Pi}\\\
M^{2}_{a\Pi}&M_{aa}^{2}\end{matrix}\right)$ (A.3)

with

$\displaystyle[M_{\Pi\Pi}^{2}]_{mn}$ $\displaystyle=$
$\displaystyle\frac{1}{f^{2}}\frac{\delta^{2}V_{0}}{\delta\Pi_{m}\delta\Pi_{n}}e^{i\braket{\bar{a}}/f_{a}N_{g}}+\frac{1}{f^{2}}\frac{\delta^{2}V^{*}_{0}}{\delta\Pi_{m}\delta\Pi_{n}}e^{-i\braket{\bar{a}}/f_{a}N_{g}}$
(A.4) $\displaystyle\left[M_{a\Pi}^{2}\right]_{m}$ $\displaystyle=$
$\displaystyle\frac{i}{ff_{a}N_{g}}\frac{\delta
V_{0}}{\delta\Pi_{m}}e^{i\braket{\bar{a}}/f_{a}N_{g}}-\frac{i}{ff_{a}N_{g}}\frac{\delta
V^{*}_{0}}{\delta\Pi_{m}}e^{-i\braket{\bar{a}}/f_{a}N_{g}}$ (A.5)
$\displaystyle M^{2}_{aa}$ $\displaystyle=$
$\displaystyle-\frac{1}{f_{a}^{2}N^{2}_{g}}V_{0}e^{i\braket{\bar{a}}/f_{a}N_{g}}-\frac{1}{f_{a}^{2}N^{2}_{g}}V^{*}_{0}e^{-i\braket{\bar{a}}/f_{a}N_{g}}.$

The NGB-axion mixing and the pure axion term are order $f/f_{a}$ and
$f^{2}/f_{a}^{2}$. For simplicity we will work in the limit $f\ll f_{a}$, so
we can treat them as perturbations, but our conclusions will apply in general.
In this way, if $M^{2}_{\Pi\Pi}$ has no flat directions the lightest
eigenvalue approximately reads

$\displaystyle
M_{a}^{2}=M_{aa}^{2}-M^{2}_{a\Pi}[M^{2}_{\Pi\Pi}]^{-1}M^{2}_{a\Pi}+{\cal
O}(f^{4}/f_{a}^{4})\leq M^{2}_{aa}.$ (A.6)

Thus a necessary condition for stability is $M_{aa}^{2}\geq M_{a}^{2}\geq 0$,
which translates into

$\displaystyle V_{0}e^{i\braket{\bar{a}}/N_{g}f_{a}}=-|V_{0}|.$ (A.7)

This tells that the absolute minimum of the potential is reached when
$V_{\text{neutral}}^{\rm
LO}=2V_{0}e^{i\braket{\bar{a}}/N_{g}f_{a}}=-2|V_{0}|$, and hence the vacuum is
obtained by maximizing $|V_{0}|$. This justifies the earlier assumption
$V_{0}\neq 0$, for $V_{0}=0$ would be energetically disfavored. The conclusion
just derived has been obtained for $f\ll f_{a}$ but in fact has general
validity because a mass mixing always pushes the lightest eigenstate to lower
values, so the condition $M_{aa}^{2}\geq 0$ is anyway necessary. The presence
of flat directions in $M^{2}_{\Pi\Pi}$ would not alter the conclusion either.
Indeed, for the same reason we just explained these directions cannot mix with
the axion otherwise they would turn tachyonic after the mixing is removed. It
follows that $\Pi$ flat directions cannot affect the axion mass nor the
argument leading to (A.7).

In summary, we demonstrated in complete generality that the minimum of $V^{\rm
LO}_{\text{neutral}}$ is obtained by maximizing $|V_{0}|$ with respect to the
goldstone fields. The axion vev follows. Indeed, writing
$V_{0}=|V_{0}(\braket{\Pi})|e^{i\phi(\braket{\Pi}/f)}$, eq. (A.7) indicates
that the value of the axion at the minimum is determined by
$\braket{\bar{a}}/N_{g}f_{a}+\phi=\pi\text{ mod }2\pi$.

The solution of the strong CP problem requires $N_{g}(\pi-\phi)$ be a multiple
of $2\pi$, and this cannot be assessed unless an explicit form of $V_{0}$ is
given. In Section 3.1 we analyzed in detail the potential of our scenario.
Here, as a quick check of our results, we study the leading order potential
for the standard axion in 2-flavor QCD. Note that the same structure (3.1)
applies to the standard QCD axion provided $\widehat{Y}_{u}$ is interpreted as
the quark mass matrix and $A$ the pion matrix. In the QCD case the axion is
rotated such that the quark masses are positive, i.e. $m_{u,d}>0$, and the
axion field appears as in (3.2) with $N_{g}=2$. Switching off the charged pion
components we have
$V_{0}=C[m_{u}e^{i\pi_{0}/f_{\pi}}+m_{d}e^{-i\pi_{0}/f_{\pi}}]$, with $C$ some
constant. For $m_{u}\neq 0$ we find that $|V_{0}|$ has two extrema, one at
$\langle\pi_{0}\rangle=0$ and the other at $\langle\pi_{0}\rangle=\pi/2$. At
the two extrema the function $|V_{0}|$ is respectively given by $m_{u}+m_{d}$
and $|m_{u}-m_{d}|$. Hence the absolute maximum of $|V_{0}|$ is attained when
$\langle\pi_{0}\rangle=0$ and the strong CP problem, as well-known, is solved,
i.e. $\braket{\bar{a}}/f_{a}=2\pi\sim 0$. When $m_{u}=0$ the function
$|V_{0}|=|C|m_{d}$ is constant and $\phi={\rm arg}[V_{0}]$ arbitrary. The
argument above tells us that the axion is now a flat direction, as expected.

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