# A Colorful Mirror Solution to the Strong CP Problem

Quentin Bonnefoy<EMAIL_ADDRESS>Berkeley Center for Theoretical
Physics, Department of Physics, University of California, Berkeley, CA 94720,
USA  Theoretical Physics Group, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Lawrence Hall<EMAIL_ADDRESS>Berkeley Center for
Theoretical Physics, Department of Physics, University of California,
Berkeley, CA 94720, USA  Theoretical Physics Group, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Claudio Andrea Manzari<EMAIL_ADDRESS>Berkeley Center for Theoretical Physics, Department of Physics, University of
California, Berkeley, CA 94720, USA  Theoretical Physics Group, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  Christiane Scherb
<EMAIL_ADDRESS>Berkeley Center for Theoretical Physics, Department of
Physics, University of California, Berkeley, CA 94720, USA  Theoretical
Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

###### Abstract

We propose theories of a complete mirror world with parity (P) solving the
strong CP problem. P exchanges the entire Standard Model (SM) with its mirror
copy and two new mass scales arise: $v^{\prime}$ where parity and mirror
electroweak symmetry are spontaneously broken, and $v_{3}$ where the color
groups break to the diagonal strong interactions. The strong CP problem is
solved even if $v_{3}\ll v^{\prime}$. We discuss the energy scales of the
model and the presence of heavy coloured states at the scale $v_{3}$,
accessible at LHC and future colliders. Furthermore, we argue that the
breaking of P introduces negligible contributions to
$\bar{\theta}_{\text{QCD}}$, starting at three-loop order. Finally, we discuss
models which can realize the symmetry breaking dynamically and without
additional hierarchy problems.

##  I Introduction

The resolution of the $U(1)_{A}$ problem by ’t Hooft ’t Hooft (1976a)
established that chiral anomalies, together with the rich structure of the QCD
vacuum, prevent the chiral $U(1)_{A}$ symmetry from being a true quantum
symmetry of QCD. As a byproduct, this picture predicts physical effects from
the CP violating term of the QCD Lagrangian

$\theta_{\text{QCD}}\frac{g_{s}^{2}}{32\pi^{2}}G_{a}^{\mu\nu}\tilde{G}_{a,\mu\nu}\,,$
(1)

where $G_{a}^{\mu\nu}$ is the gluon field strength tensor, $g_{s}$ the strong
coupling constant,
$\tilde{G}_{a}^{\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}G_{a}^{\alpha\beta}$,
and $\theta_{\text{QCD}}$ is a phase parameter whose size determines how badly
CP is broken in strong interactions. Experimental constraints on the electric
dipole moment of the neutron imply $\theta_{\text{QCD}}\lesssim 10^{-10}$
Pendlebury  _et al._ (2015). The lack of understanding of the smallness of
$\theta_{\text{QCD}}$ has been dubbed the strong CP problem. The puzzle is
made even sharper by the presence of weak interactions, which are such that
$\theta_{\text{QCD}}$ receives a contribution through the chiral
transformation needed to diagonalize the quark mass matrix $M$. In the full
theory, the term in Eq. (1) becomes

$\bar{\theta}_{\text{QCD}}\frac{g_{s}^{2}}{32\pi^{2}}G\tilde{G}=(\theta_{\text{QCD}}+\arg\det(M))\frac{g_{s}^{2}}{32\pi^{2}}G\tilde{G}\,.$
(2)

The two contributions to $\bar{\theta}_{\text{QCD}}$ arise from very different
physics and have no reason to cancel in the Standard Model.
Three approaches have therefore been proposed so far in the literature: a
massless quark Kaplan and Manohar (1986); Banks  _et al._ (1994); Hook (2015);
Agrawal and Howe (2018a), spontaneously broken P or CP symmetries Babu and
Mohapatra (1989, 1990); Barr  _et al._ (1991); Nelson (1984); Barr (1984);
Vecchi (2017); Dine and Draper (2015); Hall and Harigaya (2018); Dunsky  _et
al._ (2019); Craig  _et al._ (2021), or a spontaneously broken global chiral
symmetry à la Peccei-Quinn Peccei and Quinn (1977a, b); Weinberg (1978);
Wilczek (1978). (The anthropic principle, often powerful in explaining large
hierarchies, cannot explain the observed level of suppression of
$\bar{\theta}_{\text{QCD}}$ Dine  _et al._ (2018).)
Although it was recognized in the 1970s that parity might solve the strong CP
problem Beg and Tsao (1978); Mohapatra and Senjanovic (1978), early attempts
to construct such theories, based on the Left-Right extension of the
electroweak group to $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$, were
unsuccessful because of CP violating phases in the vacuum expectation values
(vevs) of Higgs bosons transforming as bi-doublets. Babu and Mohapatra
discovered a solution in a simple model having a separate Higgs doublet for
each $SU(2)$ group Babu and Mohapatra (1989). In this model, P forces
$\theta_{\text{QCD}}$ to zero and the fermion Yukawa matrices to be Hermitian,
hence $\bar{\theta}_{\text{QCD}}=0$ at tree-level. Shortly after, the same
authors UV-completed their construction through a see-saw mechanism involving
heavy vector-like fermions Babu and Mohapatra (1990). For a realistic vacuum
to occur at tree-level, P must be softly broken, and the resulting radiative
corrections to $\bar{\theta}_{\text{QCD}}$ occur at 2-loop order. Providing
the soft breaking of parity is limited to operators of dimension 2, these
corrections are plausibly small enough to solve the strong CP problem, while
offering the prospect of an observable neutron electric dipole moment de Vries
_et al._ (2021a). An alternative possibility is that P is exact and a
realistic vacuum for the SM Higgs boson arises radiatively Hall and Harigaya
(2018). In this case, $\bar{\theta}_{\text{QCD}}$ is again unavoidable at
2-loops, and may be probed by future searches for the neutron electric dipole
moment Hall and Harigaya (2018); Hisano  _et al._ (2023).
An alternative model was constructed in Ref. Barr  _et al._ (1991) by doubling
the SM electroweak group to $(SU(2)_{L}\times U(1)_{Y})\times(SU(2)’\times
U(1)’)$. A single SM-like Higgs doublet and three generations of SM-like
fermions were introduced for each of the SM and mirror electroweak sectors.
(Actually, the fermion content is identical to that of Ref. Babu and Mohapatra
(1990), but here the charges prevent the presence of Majorana masses and thus
of a see-saw mechanism.) Both sectors share a common strong interaction. P
forces $\theta_{\text{QCD}}$ to zero, while the Yukawa matrices for SM
fermions are arbitrary but hermitian conjugates of those of mirror fermions,
so that the quark contributions to $\bar{\theta}_{\text{QCD}}$ are cancelled
by the mirror contributions. A hierarchy of Higgs vevs, $v’\gg v$, can again
be obtained either by soft P breaking or by radiative contributions to the
Higgs potential Dunsky  _et al._ (2019); in both cases the contributions to
$\bar{\theta}_{\text{QCD}}$ arise at three loops, as with radiative
contributions in the SM Ellis and Gaillard (1979), and are small. In a final
comment of Ref. Barr  _et al._ (1991), it was suggested that this theory could
be unified, into an $SU(5)\times SU(5)^{\prime}$ or $SO(10)\times
SO(10)^{\prime}$ theory. The resulting $SU(3)\times SU(3)^{\prime}$ group of
strong interactions would be reduced to QCD by breaking to the diagonal
$SU(3)_{\rm QCD}$ combination, with P then fixing the $\bar{\theta}$ parameter
of QCD to zero. Such unified theories have not been constructed, and we are
not aware of any discussions of this mechanism in the literature.
In this paper, we propose the simplest theory of a complete mirror world with
P solving the strong CP problem, and discuss its phenomenology. Parity
exchanges the entire SM for its mirror copy and there are only two relevant
free parameters beyond those of the SM. One is the mass scale $v^{\prime}$,
where parity and mirror electroweak symmetry are spontaneously broken, and the
other is the mass scale $v_{3}$, where $SU(3)\times SU(3)^{\prime}$ breaks to
QCD. Importantly, the strong CP problem is solved even if $v_{3}\ll
v^{\prime}$.
The outline of the paper is as follow: in Sec. II and Sec. III we describe the
mechanism to solve the strong CP problem and its simplest realization. In Sec.
IV we discuss the energy scales of this model, with a focus on the necessary
breaking of parity, and in Sec. V we comment on the loop-induced contributions
to $\bar{\theta}_{\text{QCD}}$ from parity breaking. In Sec. VI we discuss the
alternative dynamical breaking of $SU(3)\times SU(3)^{\prime}$ to QCD by a new
strong force. Finally, we conclude in Sec. VII. Appendices complete the paper:
App. A discusses the option of using CP instead of P, while App. B presents
details on the new heavy colored vector bosons of the model. In App. C we
discuss the breaking of $SU(3)\times SU(3)^{\prime}$ to QCD by fundamental
scalars, and discuss the resulting spectra of heavy colored bosons.

##  II Solution to the Strong CP Problem

We mirror the full SM gauge group and therefore start with $SU(3)\times
SU(2)_{L}\times U(1)_{Y}\times SU(3)^{\prime}\times SU(2)^{\prime}\times
U(1)^{\prime}$. In Left-Right theories, one leaves the fermion content of the
SM unchanged, and the right-handed (RH) fermions are grouped into prime
electroweak doublets. Here, however, we cannot simply assign prime color to
such a doublet, as that would generate $SU(3)^{(\prime)}{}^{3}$ gauge
anomalies. Therefore, we are led to a complete mirror theory, where one also
doubles the fermionic spectrum, as shown in Tab. 1. We also have two Higgs
bosons, each charged under only one of the two worlds and responsible for the
breaking of its electroweak sector. Our mirror world lagrangian thus reads

${\cal L}={\cal L}_{\text{SM}}+{\cal
L}_{\text{SM’}}+\tilde{\lambda}\absolutevalue{H}^{2}\absolutevalue{H’}^{2}\ ,$
(3)

where ${\cal L}_{\text{SM}}^{\prime}$ refers to the Lagrangian of the Standard
model, but where all fields and couplings are primed. For simplicity, the
kinetic mixing and dimension-5 neutrino mass operators are not shown, as we do
not need them for our analysis.

| $SU(3)$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $SU(3)^{\prime}$ | $SU(2)^{\prime}$ | $U(1)^{\prime}$
---|---|---|---|---|---|---
$Q$ | $\mathbf{3}$ | $\mathbf{2}$ | 1/6 | $\mathbf{1}$ | $\mathbf{1}$ | 0
$u^{c}$ | $\overline{\mathbf{3}}$ | $\mathbf{1}$ | -2/3 | $\mathbf{1}$ | $\mathbf{1}$ | 0
$d^{c}$ | $\overline{\mathbf{3}}$ | $\mathbf{1}$ | 1/3 | $\mathbf{1}$ | $\mathbf{1}$ | 0
$L$ | $\mathbf{1}$ | $\mathbf{2}$ | -1/2 | $\mathbf{1}$ | $\mathbf{1}$ | 0
$e^{c}$ | $\mathbf{1}$ | $\mathbf{1}$ | -1 | $\mathbf{1}$ | $\mathbf{1}$ | 0
$H$ | $\mathbf{1}$ | $\mathbf{2}$ | 1/2 | $\mathbf{1}$ | $\mathbf{1}$ | 0
$Q^{\prime}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $\overline{\mathbf{3}}$ | $\mathbf{2}$ | -$1/6$
$u^{\prime c}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $\mathbf{3}$ | $\mathbf{1}$ | $2/3$
$d^{\prime c}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $\mathbf{3}$ | $\mathbf{1}$ | -$1/3$
$L^{\prime}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $\mathbf{1}$ | $\mathbf{2}$ | $1/2$
$e^{\prime c}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $\mathbf{1}$ | $\mathbf{1}$ | $1$
$H^{\prime}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $\mathbf{1}$ | $\mathbf{2}$ | -$1/2$
Table 1: Left-handed (LH) fields of the model, related by parity as shown in
Eq. (5).

The gauge and field content of the model is now such that one can pair the
fields via spacetime parity. More precisely, we compose the usual action of P
Branco  _et al._ (1999) with a $\mathbb{Z}_{2}$ symmetry which exchanges the
SM and mirror fields. At the level of the gauge bosons, one has

$A^{a}_{\mu}(t,\vec{r})\xrightarrow[]{\mathbb{Z}_{2}}A^{\prime
a}_{\mu}(t,\vec{r})\xrightarrow[]{P}A^{\prime a}{}^{\mu}(t,-\vec{r})\ .$ (4)

(Due to the independent C invariance of each Yang-Mills theory, even with
$\theta$ terms, it is actually equivalent to impose either P or CP. For
simplicity in the following we impose P, but the discussion proceeds
analogously imposing CP, as discussed in App. A.) Parity interchanges LH and
RH fermions, and hence sends

$Q(t,\vec{r})\xrightarrow[]{P\circ\mathbb{Z}_{2}}\gamma^{0}Q^{\prime\,c}(t,-\vec{r})\,,$
(5)

with $Q^{\prime}$ in the $\overline{\mathbf{3}}$ of $SU(3)^{\prime}$, and
similarly for the other fermions. It also exchanges
$H\leftrightarrow\tilde{H^{\prime}}$. The $\theta$ angles are odd under P,
hence $\theta=-\theta^{\prime}$ and ${\cal L}$ only contains

$\begin{split}\frac{\theta}{16\pi^{2}}\left[g_{3}^{2}\Tr\left(G\tilde{G}\right)-g_{3}^{\prime
2}\Tr\left(G^{\prime}\tilde{G}^{\prime}\right)\right]\,.\end{split}$ (6)

Finally, parity imposes that the SM and mirror gauge couplings are equal, in
particular $g_{3}=g_{3}^{\prime}$, but we keep the two couplings explicit in
(6) since they will run differently below the scale of P breaking. From (6),
it appears clearly that breaking $SU(3)\times SU(3)^{\prime}$ to its diagonal
subgroup (then identified with $SU(3)_{\rm QCD}$) provides a perfect
cancellation of $\bar{\theta}_{\rm QCD}$, as long as no new phases are
introduced by the sector which generates this breaking. This discussion
accounts also for the electroweak contributions: P sends $\bar{Q}u\tilde{H}$
to $\left(\bar{Q}^{\prime}u^{\prime}\tilde{H^{\prime}}\right){}^{\dagger}$,
hence the Yukawa matrices in the SM and mirror sectors are hermitian
conjugates of one another. Thus, the appropriate quark rephasings contribute

$\begin{split}\delta\theta&=\arg\det Y_{u}Y_{d}\ ,\\\
\delta\theta^{\prime}&=\arg\det Y_{u}^{\prime}Y_{d}^{\prime}=-\arg\det
Y_{u}Y_{d}=-\delta\theta\,,\end{split}$ (7)

respecting Eq. (6). We therefore extend the Lagrangian of our model in order
to accommodate the color breaking:

$\mathcal{L}\to\mathcal{L}+\mathcal{L}_{\text{\rm breaking}}\ .$ (8)

We discuss $\mathcal{L}_{\text{\rm breaking}}$ in more details below.

##  III $SU(3)\times SU(3)^{\prime}$ breaking

The mechanism presented in Sec II does not depend on a specific symmetry
breaking sector, $\mathcal{L}_{\text{\rm breaking}}$, as long as it provides
the breaking pattern: $SU(3)\times SU(3)^{\prime}\to SU(3)_{\rm QCD}$. We
present here a simple realization of this symmetry breaking by the vev of a
bi-fundamental scalar field $\Sigma_{ii^{\prime}}$. We nevertheless stress
that this realization is by no means unique. For instance, models that
dynamically break the color groups to their diagonal subgroup are discussed in
Sec. VI.
Modulo conjugation, we can consider two cases for the charges of $\Sigma$:
$(\mathbf{3},\mathbf{3})$ or $(\mathbf{3},\overline{\mathbf{3}})$. Calling
$U,U^{\prime}$ the unitary group actions in the $\mathbf{3}$ representation,
one has

$\Sigma\left\\{\begin{matrix}\xrightarrow[]{(\mathbf{3},\mathbf{3})}U\Sigma
U^{\prime T}\\\ \xrightarrow[]{(\mathbf{3},\overline{\mathbf{3}})}U\Sigma
U^{\prime\dagger}\end{matrix}\right.\,.$ (9)

When $\Sigma$ acquires a vev proportional to the identity matrix,
$\langle\Sigma\rangle=\frac{v_{3}}{\sqrt{6}}\mathbf{1}$, the vacuum preserves
the diagonal $SU(3)_{\text{QCD}}$, which acts as

$\langle\Sigma\rangle\to V\langle\Sigma\rangle
V^{\dagger}=\langle\Sigma\rangle\ .$ (10)

Projecting onto the massless gluons, one finds

$g_{3}^{2}G\tilde{G}=g_{3}^{\prime 2}G^{\prime}\tilde{G}^{\prime}\ ,$ (11)

confirming the cancellation of $\theta_{\text{QCD}}$ from Eq. (6) (see App. B
for details). Anticipating the discussion in Sec. IV, we stress that this
cancellation holds even when $g_{3}\neq g_{3}^{\prime}$, as long as
$\theta=-\theta^{\prime}$.
Such a vev is easy to achieve via the most general potential of $\Sigma$,
which reads

$\begin{split}V(\Sigma)=&-m^{2}\Tr(\Sigma\Sigma^{\dagger})+c\Tr^{2}(\Sigma\Sigma^{\dagger})\\\
&+\tilde{c}\Tr(\Sigma\Sigma^{\dagger})^{2}+\left(\tilde{m}\det(\Sigma)+h.c.\right)\
.\end{split}$ (12)

In addition to this potential, ${\cal L}_{\rm breaking}$ contains mixing terms
between $\Sigma$ and the Higgs fields. Since $v_{3}\gg v$, the couplings to
$H$ are irrelevant for our discussion, while the vev of $H^{\prime}$ simply
shifts the couplings shown in Eq. (12). The vacua of this model have been
thoroughly studied in Ref. Bai and Dobrescu (2018a): there are parameter
ranges where the unbroken gauge symmetry in the global minimum is $U(1)^{2}$,
$SU(2)^{2}\times U(1)$ or $SU(3)$. Only the latter is of interest for us,
which is for instance the only (global and local) minimum when $m^{2}\geq 0$
and $c,\tilde{c}\geq 0$ (see Ref. Bai and Dobrescu (2018a) for the complete
set of conditions), but we also note that the rich structure of vacua is
interesting from a cosmological perspective. Finally, we stress that with our
choice of charges, $\Sigma$ does not couple to fermions at the renormalizable
level and its vev does not reintroduce CP phases in their mass matrices.
We discuss the need of parity breaking in the next section, however here we
note that if $v_{3}$ is larger than the parity breaking scale, the low energy
description of our model coincides with models where only electroweak forces
are mirrored Babu and Mohapatra (1989, 1990); Barr  _et al._ (1991); Hall and
Harigaya (2018); Dunsky  _et al._ (2019); Craig  _et al._ (2021). This is
clear when $\Sigma$ transforms in the $(\mathbf{3},\overline{\mathbf{3}})$: in
that case, $Q^{\prime}$ is a $\overline{\mathbf{3}}$ of $SU(3)_{\text{QCD}}$
and one retains a P transformation in the IR. Indeed, (10) implies that
$SU(3)_{\rm QCD}$ acts as

$Q\to VQ\ ,\quad
Q^{\prime}\left\\{\begin{matrix}\xrightarrow[]{\Sigma\in(\mathbf{3},\mathbf{3})}VQ^{\prime}\\\
\xrightarrow[]{\Sigma\in(\mathbf{3},\overline{\mathbf{3}})}V^{*}Q^{\prime}\end{matrix}\right.\
.$ (13)

On the other hand, experimental bounds are much stricter on the scale of
parity breaking than they are on $v_{3}$, hence the scenario where $v_{3}$ is
at the lowest possible scale is the most phenomenologically interesting and
novel. However, this choice of a low vev for $\Sigma$ suffers from the usual
scalar hierarchy problem, which arises here at tree-level due to the couplings
of $\Sigma$ to $H^{\prime}$. The dynamical breaking models of Sec. VI address
this issue.
Further details on the associated spectrum of physical scalars in the IR are
given in App. C.

##  IV Parity Breaking and
Energy Scales

We showed in the previous sections how to obtain a perfect cancellation of
$\theta_{\text{QCD}}$ when P connects the SM to its mirror copy. However, if P
is unbroken, this possibility is ruled out by experiment. For instance, for
exact P the two Higgs vevs are the same, doubling the number of colored
fermions at low energies. More generally, collider and cosmological probes
require the mirror sector to decouple at low energies, and therefore P must be
broken at some high energy scale. This is most easily achieved by making the
vev of the mirror Higgs much larger than that of its SM companion:
$v^{\prime}\gg v$. Such a hierarchical vacuum can be obtained at tree-level,
via explicit soft breaking of parity Babu and Mohapatra (1989, 1990); Craig
_et al._ (2021), or through loop-induced corrections, as in the Higgs Parity
mechanism Dunsky  _et al._ (2019). These mechanisms for spontaneous breaking
of parity in the electroweak sector can occur even when the Higgs doublets
have quartic couplings to the colored $\Sigma$ field, whether $v_{3}$ is
larger or smaller than $v^{\prime}$. In any case, since the parity breaking is
spontaneous or soft, the mirror Yukawa matrices remain the hermitian
conjugates of those of the SM at the scale $v^{\prime}$, and thus do not
introduce new parameters. The present solution to the strong CP problem is
therefore completely defined by two energy scales: $v^{\prime}$ and $v_{3}$.
Nevertheless, there may be additional parameters associated with the
particular $SU(3)\times SU(3)^{\prime}$ breaking mechanism, as seen for
instance in Sec. III. In general, these parameters are not relevant for the
strong CP problem as long as they provide the right breaking pattern
$SU(3)_{C}\times SU(3)_{C}^{\prime}\to SU(3)_{\rm QCD}$, but they are relevant
for studying the phenomenology of any precise model.
Independently of the breaking mechanism, there is the requirement that the
model solves the strong CP problem, despite parity being broken. Contributions
to $\bar{\theta}_{\text{QCD}}$ beyond those of Section II, which cancel, can
be classical or quantum. Quantum contributions to $\bar{\theta}_{\text{QCD}}$
are discussed in the next section. Classical ones yield an upper bound on
$v^{\prime}$ due to expected new physics at most at the Planck scale. More
precisely, there are dangerous dimension-6 operators of the form

$\frac{g_{3}^{2}\Tr
G\tilde{G}}{16\pi^{2}M_{P}^{2}}\left(\lambda\absolutevalue{H}^{2}+\lambda^{\prime}\absolutevalue{H^{\prime}}^{2}\right)-\left(g_{3},G,H\leftrightarrow
g_{3}^{\prime},G^{\prime},H^{\prime}\right)\ ,$ (14)

where the pattern of couplings is chosen so as to respect parity. (Planck-
suppressed operators could also explicitly break parity if it is global
Berezhiani  _et al._ (1993), but not if it is gauged Choi  _et al._ (1993);
Dine  _et al._ (1992).) When the two Higgses receive different vevs, such
operators reintroduce
$\bar{\theta}_{\text{QCD}}=(\lambda-\lambda^{\prime})(v^{2}-v^{\prime}{}^{2})/M_{P}^{2}$.
For order one Wilson coefficients $\lambda^{(\prime)}$ this imposes
$v^{\prime}\lesssim 10^{14}\,{\rm GeV}$, as shown in Fig. 1. There can also be
corrections to the Yukawa couplings of the form

$\bar{Q}\frac{Y_{u,1}\absolutevalue{H}^{2}+Y_{u,2}\absolutevalue{H^{\prime}}^{2}}{M_{P}^{2}}u\tilde{H}+\left(\begin{matrix}Y_{u,i},Q,u,H\\\
\leftrightarrow
Y_{u,i}^{\dagger},Q^{\prime},u^{\prime},H^{\prime}\end{matrix}\right)\ ,$ (15)

and similarly in the down sector. Those reintroduce
$\bar{\theta}_{\text{QCD}}=\imaginary\Tr\left(Y_{u}^{-1}\left[Y_{u,1}+Y_{u,2}\right]\right)(v^{2}-v^{\prime
2})/M_{P}^{2}$, which is enhanced by the inverse of the small quark Yukawas.
Thus, for arbitrary $Y_{u,i}$ the bound shown in Fig. 1 is strengthened by
roughly $2.5$ orders of magnitude. However, that strongly depends on the
flavor structure of the matrices $Y_{u,i}$, which may be similar to that of
$Y_{u}$ in a full model of flavor. In that case, the bound is unchanged.
A second set of model-independent constraints come from collider bounds on the
mirror quarks, which become charged under QCD below $v_{3}$. The lightest
mirror-quarks are stable and, once they are pair-produced in p-p collisions,
they form colorless bound states with quarks or gluons produced by their own
color field Fairbairn  _et al._ (2007). Therefore, the best constraints come
from LHC searches for heavy stable electrically charged particles CMS (2021);
ATL (2022). Our dark quarks form bound states of charge $\pm 2/3,\,\pm
1/3,\,\pm 4/3$, whose masses can be assumed to be very close to the mass of
its mirror constituent, the lightest of which is the mirror up-quark,
$u^{\prime}$. We recasted the ATLAS search for stable gluinos and charginos of
Ref. ATL (2022), finding a lower bound of $m_{u^{\prime}}\gtrsim 1.3\;\rm
TeV$. The fermion Yukawas of the two worlds being related by parity, this
bound translates into $v^{\prime}\approx\sqrt{2}m_{u^{\prime}}/Y_{u}\gtrsim
1.5\times 10^{8}\;\rm GeV$, as shown in Fig. 1.
Further constraints come from the different running of the gauge couplings and
the presence of additional particle fields. While the former are qualitatively
independent of the $SU(3)\times SU(3)^{\prime}$ breaking mechanism, the latter
are strongly model dependent. For concreteness, we now consider the
spontaneously symmetry breaking mechanism provided by the bi-fundamental
scalar discussed in Sec. III, and show the bounds in Fig. 1. The running of
$g_{3}$ differs from that of $g_{3}^{\prime}$ below $v^{\prime}$, since the
quarks are much heavier in the mirror sector111There are also parity-invariant
terms which lift the degeneracy between the $SU(3)^{(\prime)}$ gauge couplings
below $v^{\prime}$ at the classical level,  $\frac{\Tr
G^{2}}{M_{P}^{2}}\left(c\absolutevalue{H}^{2}+c^{\prime}\absolutevalue{H^{\prime}}^{2}\right)+\left(G,H,g_{3}\leftrightarrow
G^{\prime},H^{\prime},g_{3}^{\prime}\right)\ .$ Due to the small
$v^{\prime}/M_{P}$ ratio, such effects are negligible with respect to those
from RG running.. As we anticipated in Section III, this does not affect the
cancellation of $\bar{\theta}_{\text{QCD}}$. Nevertheless, we ought to require
that $SU(3)^{\prime}$ does not confine before $v_{3}$, as it would modify the
potential of $\Sigma$ and spoil the required breaking pattern of the strong
groups. We then obtain another upper bound for $v^{\prime}$ as a function of
$v_{3}$, as shown in Fig. 1. Note that we can display the bound in such a two-
dimensional plot due to the fact that, in our setup, for a given field content
and for given values of $v_{3}$ and $v^{\prime}$, the RG running of the gauge
couplings is fully fixed by the requirement that they should match to the QCD
coupling constant at $v_{3}$ and that they should be equal at $v^{\prime}$.
At the scale $v_{3}$ a massive vector octet coupled to (mirror) quarks and QCD
gluons appears, as well as various scalar states, some of which are charged
under $SU(3)_{\text{QCD}}$, as discussed in App. B and C, respectively. Bounds
from collider searches of the former have been discussed extensively in Refs.
Dobrescu  _et al._ (2008); Bai and Dobrescu (2018b) and are shown in Fig. 1 in
the $v^{\prime}-v_{3}$ plane. They forbid the region where both parameters are
small. In particular, taking into account the requirement that
$SU(3)^{\prime}$ does not confine, we find $v_{3}\gtrsim 0.85\,\text{\rm
TeV}$. For these values of $v_{3}$, the collider bounds from coloured scalars
can always be avoided by an appropriate choice of the parameters in the
$\Sigma$ potential Bai and Dobrescu (2018b), so we do not discuss them
further.
As we will see below, in other realizations of the color breaking mechanism
there may be scalars or fermions significantly lighter than the heavy gluons,
resulting in the blue region extending towards the right side of the plot.

Figure 1: Allowed parameter space in the $v_{3}-v^{\prime}$ plane. The
excluded region by collider searches of heavy gluons (blue) and heavy quarks
(gray), the bounds from the requirement that $SU(3)^{\prime}$ does not confine
before $v_{3}$ (solid red line) and that the higher dimensional operators do
not reintroduce a sizable $\theta_{\rm QCD}$ (dashed orange line) are shown.

##  V Quantum corrections to $\bar{\theta}$

The fact that all classical sources of strong CP violation cancel to
sufficient accuracy does not yet ensure that the model solves the strong CP
problem: one also needs to check that the spontaneous breaking of parity does
not contribute through large loop corrections to $\bar{\theta}_{\text{QCD}}$
Ellis and Gaillard (1979); de Vries  _et al._ (2019); Valenti and Vecchi
(2021); de Vries  _et al._ (2021b, a). We argue that, for our model, no
contributions to $\bar{\theta}_{\text{QCD}}$ exist before three-loop order.
In order to do so, it is quite useful to keep in mind that one needs both P
and CP violation to generate a non-zero $\bar{\theta}_{\text{QCD}}$, and to
look for sources of CP violation in the present model. To begin with, the vev
of $\Sigma$ does not spontaneously break (C)P, thus it does not introduce any
new CP phase. Indeed, for real $\tilde{m}$ (which can always be achieved upon
rephasing $\Sigma$), Ref. Bai and Dobrescu (2018a) showed that
$\langle\Sigma\rangle$ can be chosen to be diagonal and real. Therefore, the
gauge and self interactions of $\Sigma$, as well as its vev, respect the
various discrete symmetries which act as follows when $\Sigma$ transforms as a
$\left(\mathbf{3},\mathbf{\bar{3}}\right)$ (see App. A),

$\Sigma\xrightarrow[]{P\circ\mathbb{Z}_{2}}\Sigma^{\dagger}\xrightarrow[]{C}\Sigma^{T}\
.$ (16)

In particular, a real $\tilde{m}$ is compatible with P. Similarly, the vevs of
the two Higgses can both be chosen real via gauge transformations, hence
neither the scalar potential nor the scalar vevs break CP. Therefore, the
physical sources of CP violation in the present model are fully contained in
the Yukawa matrices, and are therefore constrained by the large $U(3)^{6}$
quark flavor symmetry. Consequently, they reduce to two copies of the Jarlskog
invariant of the SM Jarlskog (1985a, b). At energies larger than $v^{\prime}$,
parity equates them and a single CKM-like phase remains. Below $v^{\prime}$,
parity is broken and the mass matrices run differently.
As a consequence, there are no diagrams contributing to
$\bar{\theta}_{\text{QCD}}$ before the three-loop order. The argument goes as
follows: corrections to $\bar{\theta}_{\text{QCD}}$ are associated to loop
corrections to the two-point functions of fermions, while Jarlskog-like
structure only arise in diagrams with at least four CKM insertions. It is
quite simple to see that the simplest diagrams arise at two-loops, with two W
boson propagators closing onto a single quark line. (Two-loop diagrams which
contain a closed quark loop effectively involve two quarks generations, and
cannot violate CP.) However, those involve a single SM copy at a time and it
has been shown in Ref. Ellis and Gaillard (1979) that they vanish.222It has
been shown that minimal models of spontaneously broken parity generate two-
loop contributions to $\bar{\theta}$ de Vries  _et al._ (2021a). However,
those are associated to the presence of Majorana masses for the weak singlets,
introducing extra physical sources of CP violation. The gauge charges of our
fermions are such that such Majorana masses are forbidden.
In our model, it also turns out that there are very few three-loop diagrams
beyond those already considered by Ref. Ellis and Gaillard (1979). New
diagrams would either mix the two SM copies, or involve the new boson
$\Sigma$. Above the scale $v_{3}$, both kind of diagrams require four W
bosons, and at least two gluons lines in addition to a line of either $\Sigma$
or a mirror fermion, or a vertex mixing $H,H^{\prime}$. They are all seen to
be at least four-loop suppressed. Below $v_{3}$, a single kind of new three-
loop diagrams exists, namely the exact copies of the leading diagrams
considered in Ref. Ellis and Gaillard (1979), upon replacing massless gluon
propagators by massive ones and $g_{s}\to(g_{3}/g_{3}^{\prime})^{\pm 1}g_{s}$
for each of the two copies. Since our massive gluons are heavy and $g_{3}\sim
g_{3}^{\prime}$, their contributions is at most comparable to that of massless
gluons. We therefore conclude that loop contributions to $\theta_{\text{QCD}}$
in our model are comparable to those in the SM, and totally negligible.
Let us stress that, although we explicitly referred to $\Sigma$ in the
previous discussion, the conclusion also holds for composite models, which are
the subject of the next section. Indeed, they cannot generate a three-loop
diagram connecting the two worlds either, and, as we will argue, they do not
generate new phases that would shift $\theta_{\text{QCD}}$.
Finally, there are non-perturbative contributions to $\theta_{\text{QCD}}$
which are sensitive to the non-zero CP-odd $\theta$ angles above $v_{3}$.
Despite being non-perturbative, those effects can be sizeable due to the fact
that the gauge couplings of $SU(3)\times SU(3)^{\prime}$ can be much larger
than that of $SU(3)_{\text{QCD}}$ at $v_{3}$ Affleck (1981); Csaki and
Murayama (1998); Agrawal and Howe (2018a, b); Csáki  _et al._ (2020). In
particular, the small instantons generate fermionic ’t Hooft determinants ’t
Hooft (1976b) which lead to corrections to the fermion masses. As said above,
the absence of free parameters beyond $v_{3}$ and $v^{\prime}$ in our model
allows us to compute those terms unambiguously. Those involving the SM
fermions are sufficiently suppressed by the product of all Yukawa couplings.
While those involving the mirror fermions are suppressed by the small gauge
couplings when $v^{\prime}$ is large, they can become sizeable when the mirror
up quark mass is close to $v_{3}$. Nevertheless, such situations also
correspond to gauge couplings which remain small at $v_{3}$, as can be seen
from Fig. 1. We have checked that the induced shift of $\theta_{\text{QCD}}$
is compatible with the current bounds, for all available parts of our
parameter space. In particular, it does not exclude more regions of Fig. 1.

##  VI Dynamical Color Breaking

In Sec. III, we considered a small $\langle\Sigma\rangle/v^{\prime}$ ratio,
which allows us to make the model as testable as possible. Nevertheless, from
a theory perspective this may call for an explanation, as it suffers from a
hierarchy problem. Here, we explore dynamical color breaking as an
alternative. In addition to evading hierarchy problems, such scenarios can be
characterized by a single additional parameter, while the $\Sigma$ potential
has four.
A tentative model which replaces the $\Sigma$ dynamics has the field and gauge
content shown in Table 2.

| $SU(N)$ | $SU(3)$ | $SU(3)^{\prime}$
---|---|---|---
$\psi_{L}$ | $\mathbf{N}$ | $\mathbf{3}$ | $\mathbf{1}$
$\psi_{R}$ | $\mathbf{N}$ | $\mathbf{1}$ | $\mathbf{3^{\prime}}$
Table 2: Field and gauge content leading to a dynamical generation of $v_{3}$.
Subscripts indicate chirality. $\mathbf{3^{\prime}}$ refers to $\mathbf{3}$ or
$\mathbf{\bar{3}}$. Spectator fermions cancel the anomalies.

The representation $\mathbf{3^{\prime}}$ can be taken to be $\mathbf{3}$ or
$\mathbf{\bar{3}}$, mapping to $\Sigma$ in the $\left(3,\bar{3}\right)$ or
$\left(3,3\right)$, respectively. There are $SU(3)^{(\prime)3}$ anomalies,
which must be cancelled by spectator fermions, to be discussed below. When the
$SU(N)$ gauge group becomes strong and confines (which happens for $N\geq 2$,
as there are three flavors), one expects a quark condensate to form and break
the axial part of the flavor group Coleman and Witten (1980). The full flavor
group being gauged, all vacua are equivalent to the one such that

$\langle\bar{\psi}_{L}\psi_{R}\rangle=v_{3}^{3}\mathbf{1}_{3\times 3}\ ,$ (17)

showing that the breaking pattern is the one announced, namely to the diagonal
vectorlike $SU(3)_{\text{QCD}}$ subgroup. The eight NGBs of the broken
$SU(3)_{A}$ flavor group are all eaten up by the heavy gluons, while the
would-be NGB of the axial $U(1)_{A}$ receives a large mass from the mixed
anomaly with $SU(N)$. A remark on the unbroken $U(1)_{V}$ is also in order: it
has a mixed anomaly with both $SU(3)$ and $SU(3)^{\prime}$, which shifts
$\theta-\theta^{\prime}$ but not $\theta+\theta^{\prime}$. Therefore, parity
in the UV is still the ingredient that solves the strong CP problem. How it
acts depends on the precise structure of the model and will be discussed
below.
The spectator fermions that cancel the gauge anomalies are colored, hence they
must be made massive. We discuss different options below, which we classify
according to whether they lead to an high or low energy scale for $v_{3}$.

###  VI.1 Models with high $v_{3}$

The colored spectators could be given a mass through a 4-Fermi interaction
with the fermions participating in the above condensate. Focusing on
fundamentals of $SU(3)^{(\prime)}$, we consider $\psi^{\prime}_{L/R,i=1...N}$,
where $i$ is a flavor index, in the conjugate representation of
$SU(3)^{(\prime)}$ with respect to $\psi_{L/R}$, respectively. We can then add
to the lagrangian

${\cal
L}\supset\frac{c_{ij}}{M_{\text{UV}}^{2}}\overline{\psi_{L}^{a}}\psi_{R}^{a^{\prime}}\overline{\psi_{R,ia^{\prime}}^{\prime}}\psi_{L,ja}^{\prime}+h.c.\
,$ (18)

where we made the contracted $SU(3)^{(\prime)}$ indices explicit. Due to the
gauge structure of that model, parity must exchange
$\psi_{L}\leftrightarrow\gamma^{0}\psi_{R}$ (hence we must choose
$\mathbf{3^{\prime}}=\mathbf{3}$),
$\psi_{L}^{\prime}\leftrightarrow\gamma^{0}\psi_{R}^{\prime}$ and leave the
gauge factor $SU(N)$ invariant. Therefore, $c=c^{\dagger}$ and
$\bar{\theta}_{\text{QCD}}$ is not affected by the mass diagonalization of the
spectators. In addition, the condensate, on which parity acts as hermitian
conjugation, does not break parity. Finally, the theta angle $\theta_{N}$ of
$SU(N)$ must vanish. Therefore, the would-be NGB of $U(1)_{A}$, which relaxes
$\theta_{N}$ to zero and couples like $\theta+\theta^{\prime}$, does not shift
the latter as it does not acquire a vev.
By consistency, the cutoff $M_{\text{UV}}$ ought to be at least $v^{\prime}$.
Consequently, imposing that the new colored particles are heavier than a TeV
leads to the following bound on $v_{3}$:

$v_{3}\gtrsim 10^{7}\text{ GeV
}\left(\frac{M_{\text{UV}}}{v^{\prime}}\right)^{2\over
3}\left(\frac{v^{\prime}}{10^{9}\text{ GeV}}\right)^{2\over 3}\ ,$ (19)

which is such that the heavy gluons cannot be seen at the LHC. Here, the
lightest BSM particles, which can be searched for, are colored fermions (found
both in the mirror and spectator sector).
Another option, free of any explicit cutoff, instead consists in charging the
spectator fermions under an additional confining gauge theory. This leads us
to the gauge and field content of Table 3.

| $SU(N)$ | $SU(N)^{\prime}$ | $SU(3)$ | $SU(3)^{\prime}$
---|---|---|---|---
$\psi_{L}$ | $\mathbf{N}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{1}$
$\psi_{R}$ | $\mathbf{N}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3^{\prime}}$
$\psi_{L}^{\prime}$ | $\mathbf{1}$ | $\mathbf{N}$ | $\mathbf{\bar{3}}$ | $\mathbf{1}$
$\psi_{R}^{\prime}$ | $\mathbf{1}$ | $\mathbf{N}$ | $\mathbf{1}$ | $\mathbf{\bar{3}^{\prime}}$
Table 3: Field and gauge content extending that of Table 2 and leading to a
dynamical generation of $v_{3}$. Subscripts indicate chirality.
$\mathbf{3^{\prime}}$ refers to $\mathbf{3}$ or $\mathbf{\bar{3}}$.

The model is easily seen not to generate any gauge anomalies, as it is
vectorlike with respect to each gauge factor independently. (It is chiral at
the level of the whole gauge group, but all mixed anomalies are trivially
zero.)
Two scenarios can be considered: the two $SU(N)$ groups confine at different
or similar scales, depending on the charge assignments that determine the
action of parity. When $\mathbf{3^{\prime}}=\mathbf{3}$, parity must exchange
$\psi_{L}^{(\prime)}\leftrightarrow\gamma^{0}\psi_{R}^{(\prime)}$ and leave
the $SU(N)$ factors unchanged, thereby making the gauge couplings and the
confining scales $\Lambda$ and $\Lambda^{\prime}$ independent. When
$\mathbf{3^{\prime}}=\mathbf{\bar{3}}$, parity must be chosen to exchange the
two $SU(N)$, as well as
$\psi_{L}\leftrightarrow\gamma^{0}\psi_{R}^{\prime},\psi_{L}^{\prime}\leftrightarrow\gamma^{0}\psi_{R}$.
This equates the two gauge couplings in the UV and relates the two confining
scales, $\Lambda\approx\Lambda^{\prime}$. (The equality may not hold exactly,
due to the remark made in footnote 1.) This second option is consistent with a
low $v_{3}$ and will be discussed in the next section. In the first case, we
can pick $\Lambda\geq\Lambda^{\prime}$ without loss of generality, and the
only possible condensates are

$\langle\overline{\psi_{L}}\psi_{R}\rangle={\cal V}_{3}^{3}\mathbf{1}_{3\times
3}\ ,\ \langle\overline{\psi^{\prime}_{L}}\psi_{R}^{\prime}\rangle={\cal
V}_{3}^{\prime 3}\mathbf{1}_{3\times 3}\ ,$ (20)

where $v_{3}\approx{\cal V}_{3}\approx\Lambda/(4\pi)\geq{\cal
V}^{\prime}_{3}\approx\Lambda^{\prime}/(4\pi)$. As said above, the first
condensate can always be brought to that form through gauge transformations,
while the second is selected by vacuum alignment Weinberg (1976); Susskind
(1979); Peskin (1980); Preskill (1981) from the vectorlike QCD gauge group
below $v_{3}$. The condensates do not break parity but break the (non-
anomalous) $SU(3)_{A}^{2}$ flavor symmetry, whose $16$ NGBs decompose into the
eight longitudinal components of the heavy gluons, and a QCD colored octet of
physical scalars $\pi^{a}$, of mass

$m_{\pi^{a}}^{2}\sim\frac{\alpha_{s}}{4\pi}\Lambda^{\prime}{}^{2}\ ,$ (21)

and phenomenology identical to that of the octet of scalars in App. C. Since
there are collider bounds on such colored scalar octets (which, as seen in
Appendix C, can decay to gluons), $\Lambda^{\prime}$ is bounded from below.
When the $SU(N)^{(\prime)}$ coupling constants are not tuned to be very
similar at the scale $v^{\prime}$, we expect that $\Lambda^{\prime}\ll v_{3}$.
Therefore, as above, the heavy gluons are not accessible at the LHC, but the
light NGBs can be looked for. As before, the unbroken anomalous $U(1)_{V}$
symmetries do not affect $\theta+\theta^{\prime}$, while the two would-be
NGBs, which relax to zero the already-vanishing $\theta_{N^{(\prime)}}$, do
not get a vev and do not shift $\theta+\theta^{\prime}$ either.

###  VI.2 Models with low $v_{3}$

In the second case where $\mathbf{3^{\prime}}=\mathbf{\bar{3}}$ and
$\Lambda\approx\Lambda^{\prime}$, $v_{3}$ can be chosen to saturate the
current experimental bounds, such that the heavy gluons can be in principle
searched for. In that case, we note that the leading-order techniques of Refs.
Weinberg (1976); Susskind (1979); Peskin (1980); Preskill (1981) do not allow
us to determine the precise vacuum and cannot rule out the possibility that
$SU(3)_{\text{QCD}}$ is also partly broken by the vacuum. (In that case,
parity would also be spontaneously broken. Due to the gauge invariance, the
only relevant cases are unbroken $SU(2)\times U(1)$ and $U(1)^{2}$.) A
definite answer is found upon analyzing the vacuum beyond leading order, as
done e.g. in Refs. Peskin (1980); Chadha and Peskin (1981a, b). We find
instabilities for all cases but the one where $SU(3)_{\text{QCD}}$ is
conserved, for which the vacuum is again that of Eq. (20). (Due to the
identical confinement scales, it respects parity.) The physical spectrum is
then again made of an octet of heavy gluons and an octet of NGBs. However, the
masses that the latter receive from QCD are not given by Eq. (21), but arise
at higher order, namely only at order ${\cal O}(\alpha_{s}^{2})$ (see e.g.
Peskin (1980); Chadha and Peskin (1981a, b) for an explanation based on
symmetries). More precisely, we find that the scalars receive a mass

$m_{\pi^{a}}^{2}\sim\frac{\alpha_{s}m_{G}^{2}}{4\pi}\log\left(\frac{\Lambda^{2}}{m_{G}^{2}}\right)\
,$ (22)

where the heavy gluon mass $m_{G}^{2}\sim\alpha_{s}v_{3}^{2}$. The NGBs are
thus lighter (by roughly an order of magnitude) than the gauge bosons, and
would constitute the first collider signals. Given current constraints on the
NGBs, $v_{3}\gtrsim\,\text{30 TeV}$ and the heavy gluons could be within the
reach of future colliders, unlike in the previous section. Something non-
trivial happens at the level of the would-be NGBs of the two $U(1)_{A}$: they
now can relax the non-vanishing $\theta_{N^{(\prime)}}$, but those are
opposite due to parity, hence the vevs of the two would-be NGBs are opposite.
Since they couple identically to $G^{(\prime)}\tilde{G}^{(\prime)}$, they do
not shift $\theta+\theta^{\prime}$.
In the case of a single $SU(N)$, we added $SU(N)$-singlet spectator fermions
to cancel the color anomalies. Another possibility is to double the spectrum
of strongly interacting fermions, so that the color anomalies are cancelled
without spectators, as in the field content of Table 4.

| $SU(N)$ | $SU(3)$ | $SU(3)^{\prime}$
---|---|---|---
$\psi_{L}$ | $\mathbf{N}$ | $\mathbf{3}$ | $\mathbf{1}$
$\tilde{\psi}_{L}$ | $\mathbf{N}$ | $\mathbf{\bar{3}}$ | $\mathbf{1}$
$\psi_{R}^{\prime}$ | $\mathbf{N}$ | $\mathbf{1}$ | $\mathbf{3}$
$\tilde{\psi}_{R}^{\prime}$ | $\mathbf{N}$ | $\mathbf{1}$ | $\mathbf{\bar{3}}$
Table 4: Field and gauge content extending that of Table 2 and leading to a
dynamical generation of $v_{3}$.

Here, parity leaves the $SU(N)$ group unaffected and exchanges
$\psi_{L}\leftrightarrow\gamma^{0}\psi_{R}^{\prime}$, and similarly for
$\tilde{\psi}^{(\prime)}$.
There are six flavors of fundamentals of $SU(N)$; thus, for $N\geq 2$, $SU(N)$
confines. The quark condensate forms and spontaneously breaks the $SU(6)_{A}$
flavor symmetry Coleman and Witten (1980), part of which is weakly gauged. The
largest possible unbroken piece of $SU(3)\times SU(3)^{\prime}$ is
$SU(3)_{\text{QCD}}$. As above, the leading-order techniques of Refs.Weinberg
(1976); Susskind (1979); Peskin (1980); Preskill (1981) do not determine the
precise vacuum and cannot rule out the possibility that $SU(3)_{\text{QCD}}$
is also partly broken by the vacuum. A next-to-leading-order analysis sheds
light on the actual vacuum: we found instabilities for selected situations
where the gauge group breaks to $SU(2)\times U(1)$ or $U(1)^{2}$. Scanning
over all possible vacuum alignments goes beyond the scope of this paper and is
left for future work. Assuming $SU(3)_{\text{QCD}}$ does not break, two
equivalent fermion condensates can form:

$\langle\overline{\psi_{L}}\psi_{R}^{\prime}\rangle=\langle\overline{\tilde{\psi}_{L}}\tilde{\psi}_{R}^{\prime}\rangle=v_{3}^{3}\;\mathbf{1}_{3\times
3}$ (23)

with all other condensates vanishing, or

$\langle\overline{\psi_{L}}\tilde{\psi}_{R}^{\prime}\rangle=\langle\overline{\tilde{\psi}_{L}}\psi_{R}^{\prime}\rangle=v_{3}^{3}\;\mathbf{1}_{3\times
3}$ (24)

with all other condensates vanishing. These realize the same breaking pattern
as $\Sigma$ in the $\left(3,\bar{3}\right)$ and $\left(3,3\right)$,
respectively, leading to $35$ NGBs, $\Pi^{A}$, decomposing into
$\mathbf{1}+\mathbf{3}+\mathbf{\bar{3}}+\mathbf{6}+\mathbf{\bar{6}}+\mathbf{8}+\mathbf{8}$
under the unbroken $SU(3)_{\text{QCD}}$. One octet is eaten by the gauge boson
of $SU(3)_{A}$, and the would-be NGB of $U(1)_{A}$ gets a large mass $\sim
v_{3}$, due to its mixed anomaly with $SU(N)$. As above, it does not get a vev
and therefore it does not shift $\theta+\theta^{\prime}$. The singlet NGB
remains massless, protected by the broken generator
$\text{diag}\left(1,1,1,-1,-1,-1\right)$ of $SU(6)_{A}$, which has no mixed
anomalies with $SU(N)$ nor with $SU(3)^{(\prime)}$. As above, all the colored
NGB only receive a mass from QCD at order ${\cal O}(\alpha_{s}^{2})$.

##  VII Conclusion

We have proposed simple theories of a complete mirror world where parity
composed with the mirror exchange symmetry solves the strong CP problem. The
new feature of our construction is the presence of a mirror strong
interaction, and therefore of two non-vanishing $\theta$ angles, which cancel
each other once the color groups are spontaneously broken to their diagonal
subgroup. The solution to the strong CP problem and the experimental viability
of the model rely on two symmetry breakings: the diagonal breaking of the
color groups at the scale $v_{3}$ makes the effective low-energy
$\bar{\theta}$ angle vanish through destructive interference, while parity is
broken by a large mirror electroweak scale $v^{\prime}\gg v$. We focus on the
scenario where $v_{3}\ll v^{\prime}$; if $v_{3}\geq v^{\prime}$, the effective
theory below $v_{3}$ is the model of Ref. Barr  _et al._ (1991); Dunsky  _et
al._ (2019). In addition, saturating the experimental constraints allows
$v_{3}$ to be much below $v^{\prime}$, leading to the richest phenomenology;
new colored states may be accessible at colliders. Due to the symmetry
structure of the model, the loop corrections to the QCD $\bar{\theta}$ angle
are shown to be under control everywhere in parameter space. We stress the
high predictive power of this mirror world, with only two scales
characterizing its qualitative features: $v_{3}$ and $v^{\prime}$. All the
other scales in the mirror world are related by parity to those in the SM. It
is also worth noting the rich cosmology of our models due to the presence of
heavy fermions, scalars and vectors at different energy scales, as well as
various phase transitions. These topics are currently under investigation.

## Acknowledgments

We thank the members of the Berkeley Center for Theoretical Physics for useful
discussions. We also thank Simon Knapen, Hitoshi Murayama, Michael Peskin and
Pablo Quílez for discussions on composite models, Simon Knapen and Dean
Robinson for discussions on collider bounds, and Pablo Quílez for suggesting
that we check the contributions of small instantons. We are deeply grateful to
Simone Pagan Griso for guiding us through the appropriate LHC searches. This
work is supported by the Office of High Energy Physics of the U.S. Department
of Energy under contract DE-AC02-05CH11231 and by the NSF grant PHY-2210390.
CS acknowledges additional support through the Alexander von Humboldt
Foundation.

##  Appendix A P or CP

As anticipated in Sec. II, due to the independent C invariance of each Yang-
Mills theory (even with $\theta$ terms), the strong CP problem can be solved
imposing either P or CP. In fact, with one Weyl fermion, CP is the only well-
defined transformation, while the difference between P and CP is blurred when
one considers two mirror Weyl fermions and insists on exchanging them. In the
main text we made the choice of imposing P, while here we discuss CP.
The action of (C)P on the gauge fields $A_{\mu}^{a}$, when composed with a
$\mathbb{Z}_{2}$ symmetry which exchanges the SM and mirror fields, would
yield Branco  _et al._ (1999)

$\begin{split}A^{a}_{\mu}(t,\vec{r})\xrightarrow[]{\mathbb{Z}_{2}}A^{\prime
a}_{\mu}(t,\vec{r})&\xrightarrow[]{P}A^{\prime a}{}^{\mu}(t,-\vec{r})\\\
&\xrightarrow[]{C}-s^{a}A^{\prime a}{}^{\mu}(t,-\vec{r})\ ,\end{split}$ (25)

where $T^{a}{}^{*}=s^{a}T^{a}$ for $s^{a}=\pm 1$ and $T^{a}$ is a generator of
the gauge algebra. It is clear that one is free to relabel $-s^{a}A^{\prime
a}{}^{\mu}\to A^{\prime a}_{\mu}$, (for each gauge factor independently)
without affecting $A^{a}_{\mu}$, which would turn the above P into CP, and
reciprocally. This relabelling changes a given representation into its
conjugate and flips the hypercharge, as it should for the action of charge
conjugation. There are no physical consequences of either choice.
As for P, CP needs to send a RH fermion to a LH fermion and reciprocally.
Therefore, a quark $Q$ must be sent to $Q^{\prime\,c}$,

$Q\xrightarrow[]{CP\circ\mathbb{Z}_{2}}\gamma^{0}Q^{\prime\,c}\,,$ (26)

with the spacetime dependence left implicit. Imposing parity demands that
$Q^{\prime}$ is in the $\overline{\mathbf{3}}$ of $SU(3)^{\prime}$, and in the
$\mathbf{3}$ when CP is imposed. Indeed,

$\begin{split}\bar{Q}\gamma^{\mu}T^{a}QG_{\mu}^{a}\xrightarrow[]{P\circ\mathbb{Z}_{2}}&-\bar{Q}^{\prime}\gamma_{\mu}T^{a}{}^{T}Q^{\prime}G^{\mu}{}^{a}\\\
&=\bar{Q}’\gamma_{\mu}\left(-T^{a}{}^{*}\right)Q’G^{\mu}{}^{a}\ ,\\\
\bar{Q}\gamma^{\mu}T^{a}QG_{\mu}^{a}\xrightarrow[]{CP\circ\mathbb{Z}_{2}}&\
\bar{Q}’\gamma_{\mu}s^{a}T^{a}{}^{T}Q’G^{\mu}{}^{a}\\\
&=\bar{Q}’\gamma_{\mu}T^{a}Q’G^{\mu}{}^{a}\,\end{split}$ (27)

(for $T^{a}=\lambda^{a}/2$, with $\lambda^{a}$ a Gell-Mann matrix), where one
recognizes the generators in the $\overline{\mathbf{3}}$ as the opposite of
the complex conjugates of those in the $\mathbf{3}$. More generally, the
fermions charges for CP would be given by Tab. 1 where all prime
representations are conjugated. The two transformations in Eq. (27) can be
exchanged upon renaming the gluons appropriately. The $\bar{\theta}$ angles
are odd under CP as under P, and the condition on the Yukawa matrices are
independent of the choice of P or CP, hence
$\bar{\theta}=-\bar{\theta}^{\prime}$ and the discussion proceeds exactly as
in the main text.
At the level of $\Sigma$, which as we argued can be chosen to transform as a
$(\mathbf{3},\mathbf{3})$ or a $(\mathbf{3},\overline{\mathbf{3}})$, the (C)P
transformation can be found by demanding that the gauge interactions are
invariant. Those are found in the kinetic term of $\Sigma$, which reads
$\Tr\left(\left(D_{\mu}\Sigma\right]^{\dagger}D_{\mu}\Sigma\right)$, with

$\begin{split}(\mathbf{3},\overline{\mathbf{3}})\quad&D_{\mu}\Sigma=\partial_{\mu}\Sigma-
ig_{3}T^{a}G^{a}_{\mu}\Sigma+ig_{3}^{\prime}\Sigma T^{a}G^{\prime
a}_{\mu}\,,\\\
(\mathbf{3},\mathbf{3})\quad&D_{\mu}\Sigma=\partial_{\mu}\Sigma-
ig_{3}T^{a}G^{a}_{\mu}\Sigma-ig_{3}^{\prime}\Sigma T^{a}{}^{T}G^{\prime
a}_{\mu}\,.\end{split}$ (28)

As anticipated in Eq. (16), this imposes the following choice

$\begin{split}(\mathbf{3},\overline{\mathbf{3}})\quad&\Sigma\xrightarrow[]{P\circ\mathbb{Z}_{2}}\Sigma^{\dagger}\xrightarrow[]{C}\Sigma^{T}\,,\\\
(\mathbf{3},\mathbf{3})\quad&\Sigma\xrightarrow[]{P\circ\mathbb{Z}_{2}}\Sigma^{T}\xrightarrow[]{C}\Sigma^{\dagger}\,.\end{split}$
(29)

These transformations restrict a class of higher-dimensional operators which
could generate a non-zero $\theta_{\text{QCD}}$, $\Tr\left(\Sigma
G^{\prime}\Sigma^{\dagger}\tilde{G}\right)$ where $G^{(\prime)}\equiv
G^{(\prime)a}T^{a}$. Such an operator is real, and it is forbidden by parity
if $\Sigma$ transforms as a $(\mathbf{3},\overline{\mathbf{3}})$. When
$\Sigma$ transforms as a $(\mathbf{3},\mathbf{3})$, one can write
$\Tr\left(\Sigma
G^{\prime}\Sigma^{\dagger}\tilde{G}\right)-\left(G^{(\prime)}\to
G^{(\prime)*}\right)$, which cancels when $\Sigma$ is replaced by its vev.
This is consistent with the fact that $\langle\Sigma\rangle$ does not break P
nor CP.

##  Appendix B Heavy Gluons

The breaking of $SU(3)\times SU(3)^{\prime}\to SU(3)_{QCD}$ results in 8
massless gluons and one heavy vector octet. The breaking being diagonal,
implies that, up to a gauge transformation,

$g^{\prime}_{3}G_{\mu}^{a}=\pm g^{\prime}_{3}G_{\mu}^{\prime a}$ (30)

once one projects onto the massless gluons, where the sign may depend on $a$.
For instance, when one introduces the vev
$\langle\Sigma\rangle=\frac{v_{3}}{\sqrt{6}}\mathbf{1}$ of the bi-fundamental
scalar $\Sigma$, its kinetic term contains

$\Tr\left(\left(D_{\mu}\Sigma\right]^{\dagger}D_{\mu}\Sigma\right)\supset\frac{v_{3}^{2}}{12}\left\\{\begin{array}[]{l}(\mathbf{3},\mathbf{3}):(g_{3}G^{a}_{\mu}+s^{a}g_{3}^{\prime}G^{\prime
a}_{\mu})^{2}\\\
(\mathbf{3},\overline{\mathbf{3}}):(g_{3}G^{a}_{\mu}-g_{3}^{\prime}G^{\prime
a}_{\mu})^{2}\end{array}\right.\ .$ (31)

Thus, the projection onto the massless gluons yields

$g_{3}^{2}G\tilde{G}=g_{3}^{\prime 2}G^{\prime}\tilde{G}^{\prime}\ ,$ (32)

which shows that the $\bar{\theta}$ term vanishes in the IR (see Eq. (6)).
Choosing the realization where
$\Sigma\in\left(\mathbf{3},\mathbf{\bar{3}}\right)$ for definiteness, we can
diagonalize the gluon matrix in Eq. (31) to obtain the mass eigenstates
$G_{SM}$ and $G_{H}$:

$\displaystyle G^{\mu\,a}_{SM}=$
$\displaystyle\frac{g_{3}^{\prime}G^{\mu\,a}+g_{3}G^{\prime\,\mu\,a}}{\sqrt{g_{3}^{\prime\,2}+g_{3}^{2}}}\,,$
(33) $\displaystyle G^{\mu\,a}_{H}=$
$\displaystyle\frac{g_{3}G^{\mu\,a}-g_{3}^{\prime}G^{\prime\,\mu\,a}}{\sqrt{g_{3}^{\prime\,2}+g_{3}^{2}}}\,.$
(34)

Below $v_{3}$ one recovers the SM group of strong interactions, $SU(3)_{\rm
QCD}$, plus a massive spin-one octet. The gauge-fermion interactions in the
mass basis now reads:

$\displaystyle\mathcal{L}\supset$ $\displaystyle-g_{\rm
QCD}\bar{q}\gamma_{\mu}qT^{a}\left(-\tan\theta
G^{\mu\,a}_{H}+G^{\mu\,a}_{SM}\right)$ (35) $\displaystyle-g_{\rm
QCD}\bar{q}^{\prime}\gamma_{\mu}q^{\prime}T^{a}\left(\tan^{-1}\theta
G^{\mu\,a}_{H}+g_{s}G^{\mu\,a}_{SM}\right)\,,$

where we identified the QCD coupling constant $g_{\rm
QCD}\equiv\frac{g_{3}g_{3}^{\prime}}{\sqrt{g_{3}^{2}+g_{3}^{\prime\,2}}}$ and
defined $\tan\theta\equiv g_{3}/g_{3}^{\prime}$. Eq. 35 shows that a heavy
gluon can decay to two light quarks.

##  Appendix C Physical Colored Scalars

The physical scalars from $\Sigma$ decompose into two real singlets of
$SU(3)_{C}$ and an octet, and have been thoroughly studied in Ref. Bai and
Dobrescu (2018b). In unitary gauge,

$\Sigma=\frac{v_{3}+\sigma+i\tilde{\sigma}}{\sqrt{6}}\mathbf{1}+\sigma^{a}T^{a}\
.$ (36)

From the potential of $\Sigma$, in Eq. (12), one finds

$v_{3}=\frac{\sqrt{3}\left(\sqrt{8(3c+\tilde{c})m^{2}+\tilde{m}^{2}}-\tilde{m}\right)}{2\sqrt{2}(3c+\tilde{c})}$
(37)

and the interactions read, in the mass basis,

$\displaystyle\mathcal{L}=$
$\displaystyle\frac{m_{0^{+}}^{2}}{2}\sigma^{2}+\frac{m_{0^{-}}^{2}}{2}\tilde{\sigma}^{2}+\frac{m_{8}^{2}}{2}\sigma^{a}{}^{2}$
(38)
$\displaystyle+\left(g_{8}+\lambda^{\prime}_{0^{+}8}\sigma\right)d^{abc}\sigma^{a}\sigma^{b}\sigma^{c}+g_{0^{+}}\sigma^{3}$
$\displaystyle+\lambda_{0^{+}}\sigma^{4}+\lambda_{0^{-}}\tilde{\sigma}^{4}+g_{0^{+}0^{-}}\tilde{\sigma}^{2}\sigma+\lambda_{0^{+}0^{-}}\tilde{\sigma}^{2}\sigma^{2}$
$\displaystyle+\lambda_{8}\left(\sigma^{a}{}^{2}\right)^{2}+\left(g_{0^{+}8}\sigma+\lambda_{0^{+}8}\sigma^{2}+\lambda_{0^{-}8}\tilde{\sigma}^{2}\right)\sigma^{a}{}^{2}\
,$

where $d^{abc}\equiv\frac{1}{2}\Tr\left(T^{a}\left[T^{b},T^{c}\right]\right)$.
The masses are

$\begin{split}m_{0^{+}}^{2}&=v_{3}^{2}\tilde{c}+\frac{2v_{3}\tilde{m}}{\sqrt{6}}+3\,c\,v_{3}^{2}-m^{2}\,,\\\
m_{0^{-}}^{2}&=\frac{v_{3}^{2}\tilde{c}}{3}-\frac{2v_{3}\tilde{m}}{\sqrt{6}}+c\,v_{3}^{2}-m^{2}\,,\\\
m_{8}^{2}&=v_{3}^{2}\tilde{c}-\frac{v_{3}\tilde{m}}{\sqrt{6}}+c\,v_{3}^{2}-m^{2}\,.\end{split}$
(39)

$\sigma^{a},\tilde{\sigma}$ and the 8 longitudinal polarizations of $G_{H}$
are the 17 Nambu-Goldstone bosons (NGBs) of the $O(18)$ symmetry of the
$\Sigma$ kinetic term, spontaneously broken to $O(17)$ by
$\langle\Sigma\rangle$. It is partly gauged and partly explicitly broken,
which can be seen from the tree level masses above, once $v_{3}$ is fixed to
its value. In the potential, only $\tilde{m}$ breaks the $U(1)$ symmetry of
which $\tilde{\sigma}$ is the NGB, and its mass can consistently be seen to
vanish in the $\tilde{m}\to 0$ limit. In this limit, the mass of $\sigma^{a}$
is found to be proportional to $\tilde{c}$, the only coupling left in the
potential which breaks $O(18)$.
The trilinear couplings read

$\begin{split}g_{0^{+}}&=\frac{v_{3}\tilde{c}}{3}+\frac{\tilde{m}}{3\sqrt{6}}+v_{3}\,c\,,\\\
g_{8}&=\frac{2\sqrt{2}v_{3}\tilde{c}}{\sqrt{3}}+\frac{2\tilde{m}}{3}\,,\\\
g_{0^{+}0^{-}}&=\frac{v_{3}\tilde{c}}{3}-\frac{\tilde{m}}{\sqrt{6}}+v_{3}\,c\,,\\\
g_{0^{+}8}&=v_{3}\tilde{c}-\frac{\tilde{m}}{2\sqrt{6}}+v_{3}\,c\,,\\\
\end{split}$ (40)

and the quartic couplings are

$\begin{split}\lambda_{0^{+}}&=\lambda_{0^{-}}=\frac{\tilde{c}}{12}+\frac{c}{4}\,,\\\
\lambda_{8}&=\frac{\tilde{c}}{8}+\frac{c}{4}\,,\quad\lambda_{0^{+}8}=\frac{\tilde{c}}{2}+\frac{c}{2}\,,\\\
\lambda_{0^{+}0^{-}}&=\frac{\tilde{c}}{6}+\frac{c}{2}\,,\quad\lambda^{\prime}_{0^{+}8}=\frac{4\tilde{c}}{\sqrt{6}}\,.\end{split}$
(41)

From these couplings we see that $\sigma$ and $\sigma^{a}$ can decay to gluons
via a loop of $\sigma^{a}$. (The new scalar bosons could also decay to other
new bosons, depending on their masses.) Through the scalar potential, the
P-odd $\tilde{\sigma}$ is stable, but it can decay to gluons and quarks via an
intermediate $\sigma^{a}$ and $G_{H}$. This happens via the $\Sigma$ kinetic
term which contains the interaction

$\frac{\left(1+\tan^{2}\theta\right)g_{s}}{\sqrt{6}\tan\theta}G_{H\mu}^{a}\tilde{\sigma}\overset{\leftrightarrow}{\partial^{\mu}}\sigma^{a}\
.$ (42)

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