aainstitutetext: Department of Physics, University of California, Santa
Barbara, CA 93106, USAbbinstitutetext: Kavli Institute for Theoretical
Physics, University of California, Santa Barbara, CA 93106, USA

# P not PQ

Nathaniel Craig<EMAIL_ADDRESS>b Isabel Garcia Garcia
<EMAIL_ADDRESS>a Giacomo Koszegi<EMAIL_ADDRESS>a and Amara
McCune<EMAIL_ADDRESS>
###### Abstract

Parity solutions to the strong CP problem are a compelling alternative to
approaches based on Peccei-Quinn symmetry, particularly given the expected
violation of global symmetries in a theory of quantum gravity. The most
natural of these solutions break parity at a low scale, giving rise to a host
of experimentally accessible signals. We assess the status of the simplest
parity-based solution in light of LHC data and flavor constraints,
highlighting the prospects for near-future tests at colliders, tabletop
experiments, and gravitational wave observatories. The origin of parity
breaking and associated gravitational effects play crucial roles, providing
new avenues for discovery through EDMs and gravity waves. These experimental
opportunities underline the promise of generalized parity, rather than Peccei-
Quinn symmetry, as a robust and testable solution to the strong CP problem.

##  1 Introduction

The current upper bound on the size of the neutron electric dipole moment
(EDM) is $|d_{n}|<1.8\cdot 10^{-26}\ e\cdot{\rm cm}$ Abel:2020gbr .111Here we
quote the direct limit; the inferred bound $|d_{n}|<1.6\cdot 10^{-26}\
e\cdot{\rm cm}$ from the 199Hg EDM limit Graner:2016ses  is comparable
assuming no additional contributions to the atomic EDM. In turn, this severely
constrains the size of the QCD vacuum angle, which is required to be

$\bar{\theta}=\theta_{s}+\theta_{q}\lesssim 10^{-10}\ ,$ (1)

where $\theta_{q}$ is the argument of the determinant of the quark mass
matrix, and $\theta_{s}$ the coefficient of the $G\tilde{G}$ operator,

$\mathcal{L}\supset\frac{\theta_{s}\alpha_{s}}{4\pi}{\rm
tr}\left(G^{a}\tilde{G}^{a}\right)\ .$ (2)

In the Standard Model (SM), $\theta_{q}={\rm arg}\ {\rm det}(y_{u}y_{d})$,
with $y_{u,d}$ the Yukawa matrices in the up- and down-quark sectors.
$\bar{\theta}$ provides a physical, basis-independent measurement of
$CP$-violation in the strong sector of the SM.

That $\bar{\theta}$ is constrained to be so tiny is one of the most puzzling
features of the SM, and it is known as the strong CP problem. It stands
alongside the electroweak hierarchy problem and the cosmological constant
problem as one of the three great naturalness puzzles that remain unsolved.
Although numerically the strong CP problem is orders of magnitude less severe
than either of its siblings, it is considerably more robust against anthropic
arguments.222See, however, Kaloper:2017fsa  for arguments to the contrary. As
such, it has drawn renewed attention during an era in which LHC null results
are challenging naturalness-based approaches to the electroweak hierarchy.

Although one could argue that $\theta_{s}=0$ on the basis that QCD
interactions otherwise preserve $CP$, a similar argument cannot be made for a
vanishing $\theta_{q}$. For example, if $CP$ were a good symmetry of the
Yukawa sector then the Yukawa matrices would need to be real. However, real
Yukawas would lead to a vanishing phase in the CKM matrix, in direct conflict
with the $\mathcal{O}(1)$ $CP$-violation observed in the electroweak sector of
the SM. Besides $CP$, a non-zero $\bar{\theta}$ also violates $P$. Again, the
fact that the strong sector preserves parity may allow us to impose
$\theta_{s}=0$. $P$ invariance in the Yukawa sector would require the Yukawa
matrices to be hermitian, in which case $\theta_{q}=0$ too, while still
allowing for a non-zero CKM phase. However, the fact that $P$ is maximally
violated by the electroweak interactions severely weakens this line of
reasoning as an attempt to argue for a small $\bar{\theta}$.

So although $\bar{\theta}$ is a measurement of both $P$ and $CP$ violation by
strong dynamics, the above discussion highlights how the origin of the strong
CP problem in the SM ultimately lies in the features of the _electroweak_
sector. It is the fact that electroweak interactions maximally violate both
$P$ and $CP$ that precludes an understanding of the bound in Eq.(1) based on
the underlying symmetries of the SM.

With this in mind, it is natural to attempt an understanding of the smallness
of $\bar{\theta}$ in the context of theories with an extended electroweak
sector. If either $P$ or $CP$ are good symmetries of the extended theory, then
$\bar{\theta}$ will be forced to vanish. Of course, to account for the $P$ and
$CP$ violation we observe in nature, they must eventually be broken, and a
non-zero $\bar{\theta}$ will be radiatively generated. If the induced
$\bar{\theta}$ is small enough, this class of theories offer a symmetry-based
solution to the strong CP problem. Concrete implementations of this idea based
on spontaneously broken $P$ and $CP$ were first proposed in Babu:1988mw ;
Babu:1989rb ; Barr:1991qx  and Nelson:1983zb ; Barr:1984qx  respectively. It
is the former that will be the focus of this work.333For recent exploration
along these lines, see also Chakdar:2013tca ; DAgnolo:2015uqq ; Hall:2018let .

There is another good reason to consider solutions to strong CP based on the
restoration of spacetime symmetries, namely that these may be realized as
gauge symmetries in the context of string theory Dine:1992ya ; Choi:1992xp .
As such, they can only be broken spontaneously and not explicitly. Depending
on the scale of spontaneous symmetry breaking, the apparent lack of $P$ and
$CP$ violation in the strong sector could therefore be fully, or partially,
explained in this context. Clearly, a resolution to the strong CP problem
along these lines would be especially attractive: it would allow us to
understand the smallness of $\bar{\theta}$ as an accident resulting from the
underlying structure of the UV-completion, as opposed to being the result of a
model-building effort specially designed to address Eq.(1).

From the bottom-up, there are a number of ways the SM can be extended to
accommodate spontaneously broken $P$. However, in order to address the strong
CP problem, a necessary feature of all of them is the presence of an
$SU(2)_{R}$ gauge factor, as well as an extended matter content that mirrors
that of the SM. Crucially, the $SU(3)$ quantum numbers of SM fermions and
their mirror counterparts must be the same, making the presence of additional
colored particles an irreducible feature of these models. With this extended
field content, parity enforces the Yukawa couplings in the two sectors to be
identical. To be phenomenologically acceptable, parity must be broken at some
scale $v^{\prime}$ above the weak scale, with the additional gauge bosons and
mirror quarks being sufficiently heavy to evade experimental constraints.
Naïvely, bounds on the mass of colored particles would seemingly require
$y_{u}v^{\prime}\gtrsim 1\ {\rm TeV}$ Aaboud:2018pii ; Sirunyan:2018omb , in
turn setting a lower bound $v^{\prime}\gtrsim 10^{8}\ {\rm GeV}$. But a
parametric separation of scales between $v$ and $v^{\prime}$ entails an
irreducible amount of fine-tuning $\Delta^{-1}\simeq 2v^{2}/v^{\prime 2}$,
which would become $\Delta^{-1}\lesssim 10^{-12}$ for such a stringent bound
on $v^{\prime}$. Considering that the goal is to naturally explain a number of
$\mathcal{O}(10^{-10})$, parity would hardly seem to remain an attractive
solution to strong CP.

In this paper, we show that the conclusion of the previous paragraph is
premature, and that a parity-breaking scale as low as $18\ {\rm TeV}$ is
consistent with all experimental constraints. This significantly improves the
level of fine-tuning, and leaves an open window for symmetry-based solutions
to strong CP that are based on spontaneously broken parity. The leading
constraint on the low-tuning version of these models comes not from bounds on
colored particles, but from direct searches for $Z^{\prime}$ and $W^{\prime}$
resonances at the LHC Aad:2019fac ; Aad:2019wvl . Future searches for heavy
gauge bosons at current and future colliders are the most promising probes of
this class of theories, with a $100\ {\rm TeV}$ proton collider guaranteed to
make a discovery if the level of fine-tuning is better than $\Delta^{-1}\sim
10^{-5}$ Helsens:2642473 ; Abada:2019lih . Overall, the viability of these
parity-based models makes collider experiments a central testing ground for
solutions to strong CP.

Another attractive feature of this class of solutions to the strong CP problem
is that they are robust against the effects of symmetry-breaking higher
dimensional operators (HDOs) that may arise from short-distance physics
associated with a gravitational UV completion. If parity is a gauge symmetry
of the underlying theory, we are led to consider only those HDOs proportional
to the source of spontaneous symmetry breaking. On the other hand, if parity
were global, the expectation that quantum gravity violates all global
symmetries Zeldovich:1976vq ; Zeldovich:1977be ; Banks:1988yz ;
Giddings:1987cg ; Lee:1988ge ; Abbott:1989jw ; Coleman:1989zu ; Kallosh:1995hi
; Banks:2010zn ; Harlow:2018tng ; Harlow:2018jwu ; Fichet:2019ugl ;
Daus:2020vtf  suggests we should include all HDOs that explicitly violate $P$.
Although the nature of the operators is different in the gauge and global
implementations, the conclusion will be the same: in both cases, the leading
HDOs with $\mathcal{O}(1)$ coefficients may be present without destabilizing
the solution to strong CP.

This stands in stark contrast with the reality of what has traditionally been
the most popular solution to the strong CP problem: the QCD axion
Peccei:1977hh ; Peccei:1977ur ; Wilczek:1977pj ; Weinberg:1977ma ; Kim:1979if
; Shifman:1979if ; Dine:1981rt . In this case, the parameter $\bar{\theta}$ is
promoted to the status of dynamical field, the axion, which is a pseudo-Nambu-
Goldstone boson of a spontaneously broken $U(1)_{PQ}$ _global_ symmetry. A
potential for the axion is induced non-perturbatively by QCD dynamics, and its
vacuum expectation value (vev) adjusts such that $\bar{\theta}=0$, thereby
solving strong CP. To work, the QCD axion potential must dominate to 1 part in
$10^{10}$, overwhelming any other contributions that may arise from additional
degrees of freedom. New dynamics responsible for, say, dark matter,
baryogenesis, or addressing the hierarchy problem, cannot significantly
contribute to the axion potential. Similarly, Planck-suppressed HDOs that
break $U(1)_{PQ}$ must be exceptionally suppressed Barr:1992qq ;
Kamionkowski:1992mf ; Holman:1992us ; Ghigna:1992iv . The mechanism is not
robust. The need for $U(1)_{PQ}$ to be a high quality global symmetry has
become known as the “axion quality problem”. Attempts to turn the QCD axion
into a high quality axion are valuable Chun:1992bn ; Randall:1992ut ;
Cheng:2001ys ; Arvanitaki:2009fg ; Fukuda:2017ylt ; DiLuzio:2017tjx ;
Lillard:2017cwx ; Lillard:2018fdt , but hardly helpful in making a small
$\bar{\theta}$ appear natural.

The goal of this work is to identify the most natural parity-based solution to
the strong CP problem, and highlight its experimental consequences. We do so
by following a strategy that combines the traditional notion of naturalness
with the expectation that gravity violates all global symmetries. The former
singles out a specific implementation of the spectrum of parity-symmetric
models, and underscores the central role of collider experiments in exploring
solutions to strong CP. The latter opens up an entirely new avenue of
exploration for parity solutions to the strong CP problem, ranging from EDM
experiments to gravitational wave observatories, depending on the degree to
which the symmetry remains approximate.

To this end, this article is organized as follows. In section 2 we review the
main features of parity-based solutions to strong CP, and discuss how a low
symmetry breaking scale can be realized while complying with experimental
constraints. We focus on the main phenomenological signatures of these models
that are relevant for collider and flavor experiments in section 3. In section
4, we discuss the size of radiative corrections to both $\bar{\theta}$ and the
EDM of elementary fermions, including charged leptons, depending on the
details of the parity-breaking sector. We explore the effect of Planck-
suppressed HDOs on this class of models in section 5, paying special attention
to a potential gravitational wave signal from the spontaneous breaking of
parity. Section 6 contains our conclusions. Finally, a series of appendices
contain results that have been crucial in our analysis, but may be skipped on
a first reading of the manuscript.

##  2 $P$ to solve strong CP

In this section, we introduce the main features of symmetry-based solutions to
the strong CP problem based on parity. In 2.1 we review the basic idea, as
first introduced in Babu:1988mw ; Babu:1989rb ; Barr:1991qx . We focus on the
scalar potential in 2.2, with an emphasis on the implications for fine-tuning
of the weak scale that arise as a result of the breaking of parity. In 2.3 we
discuss how the scale of additional colored particles can be decoupled from
the parity-breaking scale, in turn minimizing the level of fine-tuning.

###  2.1 Parity as a solution to the strong CP problem

A symmetry-based solution to the strong CP problem based on parity requires
extending the SM both in terms of matter content and gauge interactions. The
minimal implementation of this idea is based on the gauge group

$SU(3)\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{\hat{Y}},$ (3)

as well as a doubling of the matter content of the SM into a ‘mirror’ sector
with identical quantum numbers, except that $SU(2)_{L}$ doublets are now
doublets of $SU(2)_{R}$. Table 1 summarizes the gauge charges in the quark and
Higgs sectors of the theory. (Analogous charge assignments apply in the lepton
sector, which we don’t make explicit.) Crucially, the Higgs sector of the
theory does not introduce additional sources of $CP$-violation — indeed, the
freedom to perform both $SU(2)_{L}$ and $SU(2)_{R}$ gauge transformations
allows us to expand around a vacuum where both vevs are real.

| $Q=\begin{pmatrix}u\\\ d\end{pmatrix}$ | $U^{\dagger}$ | $D^{\dagger}$ | $H$ | $Q^{\prime\dagger}=\begin{pmatrix}u^{\prime\dagger}\\\ d^{\prime\dagger}\end{pmatrix}$ | $U^{\prime}$ | $D^{\prime}$ | $H^{\prime*}$
---|---|---|---|---|---|---|---|---
$SU(3)$ | $\mathbf{3}$ | $\mathbf{3}$ | $\mathbf{3}$ | $\mathbf{\cdot}$ | $\mathbf{3}$ | $\mathbf{3}$ | $\mathbf{3}$ | $\mathbf{\cdot}$
$SU(2)_{L}$ | $\mathbf{2}$ | $\mathbf{\cdot}$ | $\mathbf{\cdot}$ | $\mathbf{2}$ | $\mathbf{\cdot}$ | $\mathbf{\cdot}$ | $\mathbf{\cdot}$ | $\mathbf{\cdot}$
$SU(2)_{R}$ | $\mathbf{\cdot}$ | $\mathbf{\cdot}$ | $\mathbf{\cdot}$ | $\mathbf{\cdot}$ | $\mathbf{2}$ | $\mathbf{\cdot}$ | $\mathbf{\cdot}$ | $\mathbf{2}$
$U(1)_{\hat{Y}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $-\frac{1}{3}$ | $\frac{1}{2}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $-\frac{1}{3}$ | $\frac{1}{2}$
Table 1: Quantum numbers in the quark and Higgs sectors. Mirror sector fields
are distinguished with a prime. We use notation such that all of $Q$, $U$, and
$D$ (as well as their mirror counterparts $Q^{\prime}$, $U^{\prime}$, and
$D^{\prime}$) are left-handed, two-component Weyl fermions, whereas daggered
fields are always right-handed.

With this additional field content, the theory admits an alternative
definition of parity that combines the action of the ‘ordinary’ parity
transformation with an internal symmetry that exchanges the fields of the SM
and mirror sectors. Explicitly, in the gauge, quark, and Higgs sectors:

$\displaystyle{\bf W}^{\mu}_{L}$ $\displaystyle\leftrightarrow{{\bf
W}_{R}}_{\mu},$ (4) $\displaystyle Q,U,D$ $\displaystyle\leftrightarrow
Q^{\prime\dagger},U^{\prime\dagger},D^{\prime\dagger},$ (5) $\displaystyle H$
$\displaystyle\leftrightarrow H^{\prime*},$ (6)

and similarly for leptons. Since $SU(3)$ and $U(1)_{\hat{Y}}$ interactions are
not mirrored, the corresponding gauge fields transform as usual under parity.
Unlike ordinary parity in the SM, this ‘generalized’ parity transformation is
now a good symmetry of the gauge sector of the theory, thanks to the extended
electroweak sector and matter content.

In this context, the strong CP problem is solved as follows. On the one hand,
parity requires that $\theta_{s}=0$, just as one may argue in the SM based on
the properties of the strong sector alone. On the other hand, the presence of
additional colored particles results in an extended quark mass matrix. In
particular, Yukawa terms can be written for both the SM and mirror sectors, of
the form

$\mathcal{L}\supset-\left\\{(y_{u})_{ij}Q_{i}HU_{j}+(y^{\prime}_{u})_{ij}Q^{\prime\dagger}_{i}H^{\prime*}U^{\prime\dagger}_{j}\right\\}+{\rm
h.c.},$ (7)

and similarly for down-type quarks and leptons. As a result, the tree-level
value of $\theta_{q}$ in these models is given by

$\theta_{q}={\rm arg}\ {\rm det}(y_{u}y_{d})+{\rm arg}\ {\rm
det}(y^{\prime*}_{u}y^{\prime*}_{d}).$ (8)

Crucially, demanding that Yukawa interactions preserve parity, which is now a
good symmetry of the extended electroweak sector, enforces the Yukawa
couplings in the two sectors to be identical, i.e.

$y^{\prime}_{f}=y_{f}.$ (9)

In turn, this implies $\theta_{q}=0$, as per Eq.(8), forcing $\bar{\theta}$ to
vanish at tree-level in parity-symmetric models.

With the field content outlined in table 1, the theory admits an additional
fermion mass term involving only the $SU(2)$-singlets, of the form

$\mathcal{L}\supset-(\mathcal{M}_{u})_{ij}U_{i}U^{\prime}_{j}+{\rm h.c.}$ (10)

(with analogous terms for down-type quarks and leptons), where invariance
under generalized parity requires that the vector-like mass matrix be
hermitian, i.e. $\mathcal{M}_{f}^{\dagger}=\mathcal{M}_{f}$.444Note that a
non-hermitian mass matrix is compatible with softly broken parity; we will
explore the consequences of such soft breaking in section 4.1. In general, the
expression for $\theta_{q}$ can be conveniently written as

$\theta_{q}={\rm arg}\ {\rm det}(\mathbb{M}_{u}\mathbb{M}_{d}),$ (11)

where $\mathbb{M}_{u}$ and $\mathbb{M}_{d}$ are $6\times 6$ matrices, of the
form

$\mathbb{M}_{f}=\begin{pmatrix}\mathbb{0}&\frac{v^{\prime}}{\sqrt{2}}y^{\prime*}_{f}\\\
\frac{v}{\sqrt{2}}y^{T}_{f}&\mathcal{M}_{f}\end{pmatrix},\qquad{\rm for}\qquad
f=u,d.$ (12)

Due to the zero in the upper-left block of the overall $6\times 6$ mass
matrix, the expression for $\theta_{q}$ remains as in Eq.(8). As we will
discuss in 2.3, the presence of vector-like masses is crucial in implementing
a version of the model with low fine-tuning.555A variation on the model we
have so far discussed entails extending the gauge group in Eq.(3) with an
additional $U(1)$, as first discussed in Barr:1991qx . In this case, SM and
mirror fields are charged under different $U(1)$ factors, which transform into
each other under parity. Although this seems like a minimal modification of
the model presented here, this two-$U(1)$ version does not allow for the
vector-like mass terms of Eq.(10), in turn precluding the implementation of a
low parity-breaking scale.

To obtain a phenomenologically viable model, parity must be broken, with
different vevs in the mirror and SM Higgs sectors. This will induce a non-
vanishing $\bar{\theta}$ beyond tree-level, which must be small enough if the
theory is to remain a bona-fide solution to strong CP. The size of radiative
corrections depends on the details of how parity is spontaneously broken. If
$P$ is broken without breaking $CP$, then the radiatively induced
$\bar{\theta}$ will be no larger than in the SM Barr:1991qx , where
$\bar{\theta}<10^{-19}$ Ellis:1978hq . On the other hand, if $CP$ is also
spontaneously broken (e.g. through the vev of a pseudo-scalar) then a larger
$\bar{\theta}$, as well as a neutron EDM _independent_ of $\bar{\theta}$, may
be radiatively generated. Even in this latter case, we will see that radiative
corrections can be small enough to remain compatible with experimental
constraints. Given that the final size of $\bar{\theta}$ is a somewhat model-
dependent feature of this class of models, we defer a more detailed discussion
of this issue to section 4.

More generally, discussing the leading effect of broken parity on the fine-
tuning of the electroweak sector does not require committing to a specific
implementation of spontaneous symmetry breaking. For this purpose, it will be
enough to focus on the features of the Higgs sector, to which we now turn.

###  2.2 Scalar sector and fine-tuning

For the time being, we will parametrize the necessary breaking of parity
through an explicit soft term in the scalar potential. Of course, such soft
breaking should ultimately be the result of some spontaneous symmetry breaking
dynamics, as we will make more explicit in section 4. In this spirit, the most
general scalar potential involving the $SU(2)_{L}$ and $SU(2)_{R}$ Higgs
doublets takes the form

$V(H,H^{\prime})=-m_{H}^{2}(|H|^{2}+|H^{\prime}|^{2})+\lambda(|H|^{2}+|H^{\prime}|^{2})^{2}+\kappa(|H|^{4}+|H^{\prime}|^{4})+\mu^{2}|H|^{2}.$
(13)

At this level, Eq.(13) is identical to the scalar potential of theories of
Neutral Naturalness, such as Twin Higgs Chacko:2005pe ; Chacko:2005un . The
first two terms respect both parity and a larger accidental $SU(4)$ (really,
$O(8)$) symmetry, while $\kappa$ respects the former but not the latter. The
parameter $\mu^{2}$ softly breaks parity. In the interest of a non-trivial
vacuum structure, we take $m_{H}^{2}>0$. Depending on the relative signs and
sizes of the quartic couplings, the tree-level vacua for $\mu^{2}=0$ either
preserve parity (with $v^{\prime}=v$) or spontaneously break parity (with
$v\neq 0,v^{\prime}=0$ or $v=0,v^{\prime}\neq 0$). A vacuum with
$v^{\prime}\gg v\neq 0$ may be obtained by deforming the theory away from the
parity-symmetric vacuum with nonzero $\mu^{2}$ and $\lambda,\kappa>0$. At
tree-level, the vevs in the SM and mirror Higgs sectors are then given by

$v^{2}=\frac{m_{H}^{2}-\mu^{2}(1+\lambda/\kappa)}{2\lambda+\kappa},\qquad{\rm
and}\qquad v^{\prime
2}=\frac{m_{H}^{2}+\mu^{2}\lambda/\kappa}{2\lambda+\kappa},$ (14)

where $v^{2}\ll v^{\prime 2}$ is necessary in order to obtain a
phenomenologically viable model. After spontaneous symmetry breaking, the
spectrum of the theory contains two scalar fields, $h$ and $h^{\prime}$, with
masses $m_{h}\simeq 2\sqrt{\kappa}v$ and
$m_{h^{\prime}}\simeq\sqrt{2\lambda}v^{\prime}$ respectively, as well as six
Goldstones that become the longitudinal components of the gauge bosons of our
extended electroweak sector. The physical gauge boson spectrum contains
$Z^{\prime}$ and $W^{\prime}$ resonances, which are heavier than their SM
counterparts by a factor of $v^{\prime}/v$. We defer further details of the
scalar and gauge sectors to appendix A.1.

It is clear from Eq.(14) that to obtain a hierarchy of scales between $v$ and
$v^{\prime}$ we need to introduce a tree-level tuning between the parity-
preserving and parity-breaking mass-squared terms. Heuristically, the
necessary fine-tuning is given by

$\Delta^{-1}\equiv\frac{m_{H}^{2}-\mu^{2}(1+\lambda/\kappa)}{m_{H}^{2}}\simeq\frac{(2\lambda+\kappa)v^{2}}{(\lambda+\kappa)v^{\prime
2}}\simeq\frac{2v^{2}}{v^{\prime 2}}.$ (15)

This is an irreducible contribution to the fine-tuning in this class of
models. Insofar as it involves the sensitivity of the weak scale $v$ to
underlying parameters, it may be classified as a tuning associated with the
electroweak hierarchy problem, although it is not necessarily the only such
contribution. For example, a hierarchy of scales $v^{\prime 2}\ll M_{Pl}^{2}$
would constitute an additional source of fine-tuning in the absence of a
stabilizing mechanism. Similarly, additional hierarchies of scales or
couplings in the sector responsible for spontaneously breaking $P$ might
necessitate similar accurate cancellations. However, these are issues that
could, at least _in principle_ , be addressed at some higher scale above
$v^{\prime}$, provided the necessary dynamics do not spoil the smallness of
$\bar{\theta}$ Albaid:2015axa . In contrast, Eq.(15) is forced on us
independently of the UV-completion. Although it is tempting to attach the
tuning in Eq.(15) to the electroweak hierarchy problem and attribute it to
anthropic selection (the perspective advocated in e.g. DAgnolo:2015uqq ;
Hall:2018let ), this necessarily entails some favorable assumptions about the
properties of an anthropic landscape. Here we prefer to render unto strong CP
the things that are strong CP’s, and take the irreducible tuning in Eq.(15) at
face value as a measure of the degree to which a parity model naturally
explains the small value of $\bar{\theta}$ without reintroducing tuning
elsewhere.

With this in mind, in this paper we focus on implementations of parity
solutions to strong CP where the level of fine-tuning, as parametrized in
Eq.(15), is as mild as possible. This will concentrate our attention on a
specific mechanism to generate fermion masses that in turn endows these models
with characteristic phenomenology, as we discuss next.

###  2.3 Fermion masses and a low parity-breaking scale

Fermion mass terms arise from the Yukawa couplings of Eq.(7), as well as from
the vector-like mass involving the $SU(2)$-singlets. In total:

$\mathcal{L}\supset-\left\\{\frac{v}{\sqrt{2}}(y_{u})_{ij}u_{i}U_{j}+\frac{v^{\prime}}{\sqrt{2}}(y_{u})_{ij}^{*}u^{\prime}_{i}U^{\prime}_{j}+(\mathcal{M}_{u})_{ij}U_{i}U^{\prime}_{j}\right\\}+{\rm
h.c.},$ (16)

where we have already set $y^{\prime}_{u}=y_{u}$, as mandated by generalized
parity, and analogous mass terms are present both for down-type quarks and
leptons.

The structure of Eq.(16) allows for two limiting realizations of the fermion
spectrum. If the overall scale of the vector-like mass matrix is $M\ll
v,v^{\prime}$, then fermion masses are generated mainly through the Yukawa
terms, as in the SM. In this case, mirror fermions would be an exact copy of
the SM, just heavier by a factor of $v^{\prime}/v$. Demanding that the
lightest mirror quark is heavy enough to comply with current experimental
constraints requires $m_{u}\times v^{\prime}/v\gtrsim 1\ {\rm TeV}$
Aaboud:2018pii ; Sirunyan:2018omb , in turn setting a lower bound
$v^{\prime}\gtrsim 10^{8}\ {\rm GeV}$. As advertised in the Introduction, this
sets the level of fine-tuning in the electroweak sector to
$\Delta^{-1}\lesssim 10^{-12}$. The phenomenology of parity solutions to
strong CP in this regime was discussed recently in DAgnolo:2015uqq .

On the other hand, the limit $M\gg v,v^{\prime}$, allows for a see-saw
realization of the fermion spectrum, consisting of three light (SM-like)
fermions, and three heavy fermions with mass of order $M$. A sufficiently high
scale for the mass of additional colored particles can now be achieved by
increasing $M$, not $v^{\prime}$. This allows for a much lower parity-breaking
scale, and therefore a much better level of fine-tuning. See-saw
implementations of fermion masses, for both quarks and leptons, are discussed
in Davidson:1987mh ; Davidson:1987tr ; Davidson:1989bx ; Ranfone:1990jf , and
it was in fact in this context that a parity-based solution to the strong CP
problem was first proposed Babu:1988mw ; Babu:1989rb . It is this second
realization of the fermion spectrum that we concentrate on in this work.

The up-quark sector requires special consideration, since the see-saw
mechanism cannot be applied to the top quark while maintaining perturbative
Yukawas. So let us discuss the down-quark and lepton sectors first. (We will
use notation appropriate to the down-quark sector, but emphasize that the same
results apply for leptons.) To leading order in both $v/M$ and $v^{\prime}/M$,
the masses of the light and heavy fermions are obtained by diagonalizing the
$3\times 3$ hermitian matrices

$\frac{vv^{\prime}}{2}y_{d}^{*}\mathcal{M}_{d}^{-1}y_{d}^{T},\qquad\qquad{\rm
and}\qquad\qquad\mathcal{M}_{d},$ (17)

respectively. We make the simplifying assumption that there are no significant
hierarchies in the eigenvalues of $\mathcal{M}_{d}$, and therefore the heavy
quarks appear at a common scale $\sim M$. Parametrically, light quark masses
are then of the form $m_{d_{i}}\sim|y|^{2}vv^{\prime}/M$. The see-saw
mechanism generates fermion masses $m_{d_{i}}\ll v$ while allowing for much
larger Yukawa couplings than in the SM, which is obviously one of the main
attractions of this class of models. Generating the $b$ quark mass through the
see-saw mechanism while maintaining perturbativity sets an upper bound on the
ratio $M/v^{\prime}$, parametrically:

$m_{b}\sim|y|^{2}\frac{vv^{\prime}}{M}\lesssim\frac{vv^{\prime}}{M}\qquad\Rightarrow\qquad\frac{M}{v^{\prime}}\lesssim\frac{v}{m_{b}}\sim
10^{2}.$ (18)

Rotating from the flavor to the mass eigenbasis in the fermion sector can be
conveniently performed step by step at each order in perturbation theory, and
we present a detailed discussion of this procedure in appendix A.2. At zeroth
order in $v^{(\prime)}/M$, it is necessary to perform unitary transformations
acting separately on the $SU(2)$-singlet and doublet fields, of the form:

$d\rightarrow\mathcal{O}^{\dagger}_{d}d\,,\
d^{\prime}\rightarrow\mathcal{O}^{T}_{d}d^{\prime}\ ,\qquad{\rm and}\qquad
D^{\prime}\rightarrow\mathcal{O}^{\dagger}_{D^{\prime}}D^{\prime}\,,\
D\rightarrow\mathcal{O}^{T}_{D^{\prime}}D\,.$ (19)

$\mathcal{O}_{d}$ and $\mathcal{O}_{D^{\prime}}$ are $3\times 3$ unitary
matrices acting on flavor space that diagonalize the first and second matrices
of Eq.(17), respectively. At first order in $v^{(\prime)}/M$, a further
rotation is required that mixes the $SU(2)$-singlet and doublet fields as
follows:

$\begin{pmatrix}d\\\
D^{\prime}\end{pmatrix}\rightarrow\begin{pmatrix}\mathbb{1}_{3}&\epsilon_{d}^{\dagger}\\\
-\epsilon_{d}&\mathbb{1}_{3}\end{pmatrix}\begin{pmatrix}d\\\
D^{\prime}\end{pmatrix},\qquad{\rm and}\qquad\begin{pmatrix}d^{\prime}\\\
D\end{pmatrix}\rightarrow\begin{pmatrix}\mathbb{1}_{3}&\epsilon^{\prime\dagger}_{d}\\\
-\epsilon^{\prime}_{d}&\mathbb{1}_{3}\end{pmatrix}\begin{pmatrix}d^{\prime}\\\
D\end{pmatrix},$ (20)

where $\epsilon_{d}$ and $\epsilon^{\prime}_{d}$ are $3\times 3$ matrices with
entries of $\mathcal{O}(v/M)$ and $\mathcal{O}(v^{\prime}/M)$ respectively,
and whose explicit expressions are given in Eq.(95).

Using Dirac notation, the left- and right-handed components of the light and
heavy mass eigenstates are then given by

${d_{i}}_{L}=\begin{pmatrix}d_{i}\\\ 0\end{pmatrix},\
{d_{i}}_{R}=\begin{pmatrix}0\\\ d^{\prime\dagger}_{i}\end{pmatrix},\qquad{\rm
and}\qquad{D_{i}}_{L}=\begin{pmatrix}D^{\prime}_{i}\\\ 0\end{pmatrix},\
{D_{i}}_{R}=\begin{pmatrix}0\\\ D^{\dagger}_{i}\end{pmatrix}.$ (21)

In particular, notice that the right-handed components of the light (SM-like)
fermions consist of the corresponding component of the $SU(2)_{R}$-doublets,
up to corrections of $\mathcal{O}(v^{\prime}/M)$. This feature plays a crucial
role in the phenomenology of these models. In particular, it leads to
unsuppressed couplings between $SU(2)_{R}$ gauge bosons, and the right-handed
currents of the SM-like fermions. As we will discuss in 3.1, this leads to the
most stringent bound on the parity-breaking scale.

As far as the up-quark sector is concerned, the see-saw mechanism can be
implemented for the $u$ and $c$ quarks, with the corresponding heavy partners
appearing at the scale $\sim M$. The mass eigenstates for the first two
generations are as in Eq.(21). The top sector, on the other hand, cannot be
significantly “see-sawed”. Instead, it consists of light and heavy top
partners with tree-level masses $m_{t}\simeq y_{t}v/\sqrt{2}$ and
$m_{t^{\prime}}\simeq m_{t}\times v^{\prime}/v$, respectively. In Dirac
notation, and at zeroth order in $v^{(\prime)}/M$, the mass eigenstates are
now purely made of SM and mirror sector fields, i.e.

$t_{L}=\begin{pmatrix}u_{3}\\\ 0\end{pmatrix},\ t_{R}=\begin{pmatrix}0\\\
U_{3}^{\dagger}\end{pmatrix},\qquad{\rm and}\qquad
t^{\prime}_{L}=\begin{pmatrix}U_{3}^{\prime}\\\ 0\end{pmatrix},\
t^{\prime}_{R}=\begin{pmatrix}0\\\ u_{3}^{\prime\dagger}\end{pmatrix}.$ (22)

As usual, rotation matrices in the quark sector are constrained by the
requirement that the CKM matrix is reproduced appropriately, which in this
case implies $V=\mathcal{O}_{u}\mathcal{O}^{\dagger}_{d}$, up to corrections
of $\mathcal{O}(v^{2}/M^{2})$. Further details concerning the mass
diagonalization procedure in the fermion sector can be found in appendix A.2.

This finalizes our discussion of the main characteristics of parity solutions
to strong CP that feature low fine-tuning in the electroweak sector. Before
moving on, we include in figure 1 a schematic representation of the typical
spectrum of these models. Amusingly, the combination of parity and the see-saw
mechanism leads to a spectrum of partner particles strikingly reminiscent of a
“natural” left-right Twin Higgs model Chacko:2005un ; Goh:2006wj  with light
top and $W/Z$ partners.

Figure 1: Schematic illustration of the particle spectrum of parity solutions
to the strong CP problem in their least tuned version. The lightest exotic
particles are $W^{\prime}$ and $Z^{\prime}$ resonances, followed by a top
partner appearing at a scale of order $v^{\prime}$. A mirror Higgs is also
part of the low-lying spectrum, with
$m_{h^{\prime}}\simeq\sqrt{2\lambda}v^{\prime}$. (For illustration, we have
chosen $\lambda=\mathcal{O}(1)$, but note that $h^{\prime}$ could be much
lighter if $\lambda\ll 1$.) Additional mirror quarks have masses of order the
see-saw scale, $M\gg v^{\prime}$. The lepton sector must also be “see-sawed”,
with mirror leptons similarly appearing well above $v^{\prime}$ (although we
emphasize that the see-saw scales in the quark and lepton sectors need not
coincide).

##  3 Dial $P$ for Phenomenology

We now turn to the phenomenology of natural parity solutions to strong CP,
beginning with direct bounds from the LHC in section 3.1 before turning to
indirect flavor constraints in 3.2. The collider and flavor phenomenology of
similar left-right models has been the topic of previous work Babu:1988mw ;
Babu:1989rb ; Ranfone:1990jf ; Kiyo:1998zm ; Goh:2006wj , and our focus here
will be on those “irreducible” signatures that are mandated by the structure
of the theory in its capacity as a solution to strong CP. A more in depth
study of collider and flavor signatures in light of forthcoming data can
illuminate the additional structure of these models, and it is a worthwhile
direction for continued study.

###  3.1 Collider bounds

The doubling of the electroweak sector gives rise to a plethora of
experimental signatures at colliders, ranging from additional vector bosons
(the $W^{\prime}$ and $Z^{\prime}$ of spontaneously broken $SU(2)_{R}$) to
vector-like quarks (the $SU(2)$-singlet fermions) to additional Higgs bosons.
Ultimately, given that the $Z^{\prime}$ and $W^{\prime}$ gauge bosons acquire
masses exclusively from $SU(2)_{R}$ breaking, and inherit couplings to the SM-
like quarks and leptons, collider searches for these additional vectors place
the most solid and strongest direct bounds on the models under consideration.

#### Neutral currents

The $Z^{\prime}$ resonance inherits couplings to both the left- and right-
handed currents of SM-like fermions. In the down-type quark and lepton
sectors, these are both flavor diagonal and generation universal. After
rotating to the mass eigenbasis in both the gauge and fermion sectors, as
outlined in appendix A, we find:

$\mathcal{L}\supset
gZ^{\prime}_{\mu}\sum_{i=1}^{3}\left(z^{\prime}_{d_{R}}\bar{d_{i}}_{R}\gamma^{\mu}{d_{i}}_{R}+z^{\prime}_{e_{R}}\bar{e_{i}}_{R}\gamma^{\mu}{e_{i}}_{R}+z^{\prime}_{\nu_{R}}\bar{\nu_{i}}_{R}\gamma^{\mu}{\nu_{i}}_{R}\right)+\\{R\rightarrow
L\\}\ .$ (23)

As discussed in section 2.3, the see-saw implementation of fermion masses
leads to unsuppressed couplings between the $SU(2)_{R}$ gauge bosons and
right-handed fermions. Up to corrections of $\mathcal{O}(\sin^{2}\theta_{w})$,
these are identical to the couplings between the SM $Z$ and left-handed
currents. Specifically:

$z^{\prime}_{d_{R}}\simeq-\frac{g}{2}\left(1+\mathcal{O}({\rm
s}_{w}^{2})\right),\quad
z^{\prime}_{e_{R}}\simeq-\frac{g}{2}\left(1+\mathcal{O}({\rm
s}_{w}^{2})\right),\quad{\rm and}\quad
z^{\prime}_{\nu_{R}}\simeq\frac{g}{2}\left(1+\mathcal{O}({\rm
s}_{w}^{2})\right),$ (24)

where ${\rm s}_{w}\equiv\sin\theta_{w}$, and we have ignored corrections of
$\mathcal{O}(v^{2}/v^{\prime 2})$. On the other hand, couplings of the
$Z^{\prime}$ to left-handed currents are now suppressed:

$z^{\prime}_{d_{L}}\simeq-\frac{g}{6}\frac{{\rm s}_{w}{\rm t}_{w}}{\sqrt{\cos
2\theta_{w}}}=\mathcal{O}({\rm s}_{w}^{2}),\quad{\rm and}\quad
z^{\prime}_{e_{L}}=z^{\prime}_{\nu_{L}}=-3z^{\prime}_{d_{L}}.$ (25)

The situation in the up-quark sector is somewhat different. Now, couplings
between the $Z^{\prime}$ and the right-handed fermion currents are no longer
universal. Instead, we have:

$\mathcal{L}\supset
gZ^{\prime}_{\mu}\left(\sum_{i=1}^{3}z^{\prime}_{u_{L}}\bar{u_{i}}_{L}\gamma^{\mu}{u_{i}}_{L}+\sum_{i=1}^{2}z^{\prime}_{u_{R}}\bar{u_{i}}_{R}\gamma^{\mu}{u_{i}}_{R}+z^{\prime}_{t_{R}}\bar{t}_{R}\gamma^{\mu}t_{R}\right).$
(26)

As before, $Z^{\prime}$ couplings to first and second generation right-handed
currents are unsuppressed, and are given by

$z^{\prime}_{u_{R}}\simeq\frac{g}{2}\left(1+\mathcal{O}({\rm
s}_{w}^{2})\right)\ ,$ (27)

whereas those to left-handed fermions, as well as to the right-handed top, now
read

$z^{\prime}_{u_{L}}=\frac{z^{\prime}_{t_{R}}}{4}\simeq-\frac{g}{6}\frac{{\rm
s}_{w}{\rm t}_{w}}{\sqrt{\cos 2\theta_{w}}}=\mathcal{O}({\rm s}_{w}^{2})\ .$
(28)

Bounds on the $Z^{\prime}$ mass from its production at the LHC will therefore
be similar to those found in the so-called Sequential Standard Model, which
features a $Z^{\prime}$ resonance that is just a heavy copy of the SM $Z$. In
the present model, couplings of the $Z^{\prime}$ to SM fermions are similar to
those of the $Z$ after the replacement $L\leftrightarrow R$ — a replacement
that does not affect either the production cross section or the decay rates
into light fermions. The most constraining limits thus come from Aad:2019fac ,
where a search focused on leptonic final states sets a lower bound
$m_{Z^{\prime}}\gtrsim 5$ TeV. In turn, this translates into a lower limit on
the scale of parity breaking of order $v^{\prime}\gtrsim 13$ TeV.

#### Charged currents

$W^{\prime}$ gauge bosons interact with the right-handed SM fermions in a way
that mirrors the interactions between their left-handed counterparts and SM
$W$. In the lepton sector:

$\mathcal{L}\supset\frac{g}{\sqrt{2}}\sum_{i,j=1}^{3}\left(B_{ij}W^{+}_{\mu}\bar{\nu_{i}}_{L}\gamma^{\mu}{e_{j}}_{L}+B^{\prime}_{ij}W^{\prime+}_{\mu}\bar{\nu_{i}}_{R}\gamma^{\mu}{e_{j}}_{R}\right)+{\rm
h.c.},$ (29)

where $B=B^{\prime}=\mathcal{O}_{\nu}\mathcal{O}_{e}^{\dagger}$, up to
corrections of order ${v^{(\prime)}}^{2}/M^{2}$. As far as the quark sector is
concerned, the up-type sector again requires special consideration. We find:

$\mathcal{L}\supset\frac{g}{\sqrt{2}}\sum_{i,j=1}^{3}W^{+}_{\mu}V_{ij}\bar{u_{i}}_{L}\gamma^{\mu}{d_{j}}_{L}+{\rm
h.c.},$ (30)

with $V=\mathcal{O}_{u}\mathcal{O}_{d}^{\dagger}+\mathcal{O}(v^{2}/M^{2})$,
whereas

$\mathcal{L}\supset\frac{g}{\sqrt{2}}W^{\prime+}_{\mu}\sum_{j=1}^{3}\left(\sum_{i=1}^{2}V^{\prime}_{ij}\bar{u_{i}}_{R}\gamma^{\mu}{d_{j}}_{R}+\Delta
V^{\prime}_{3j}\bar{t}_{R}\gamma^{\mu}{d_{j}}_{R}\right)+{\rm h.c.}$ (31)

Up to corrections of $\mathcal{O}(v^{\prime 2}/M^{2})$, we have
$V^{\prime}=V$, and $\Delta V^{\prime}_{3j}=(\epsilon^{\prime*}_{u}V)_{3j}$.
The $3\times 3$ matrix $\epsilon^{\prime}_{u}$, whose entries are suppressed
by a factor of $v^{\prime}/M\ll 1$, is given explicitly in Eq.(101).

As with the $Z^{\prime}$, we expect bounds on the $W^{\prime}$ to be
comparable to those in the Sequential Standard Model. Current direct searches
set stringent constraints on such $W^{\prime}$ resonances, of order
$m_{W^{\prime}}\gtrsim 6$ TeV Aad:2019wvl . In turn, this sets the strongest
limit on the scale of parity breaking: $v^{\prime}\gtrsim 18$ TeV. Although
direct searches for vector-like quarks and additional Higgs bosons are also
germane, they lead to significantly weaker bounds on the scale of parity
breaking compared to $W^{\prime}$ and $Z^{\prime}$ searches. For example, null
results in searches for vector-like top partners Aaboud:2018pii ;
Sirunyan:2018omb  lead to $v^{\prime}\gtrsim 2$ TeV, with comparable bounds
coming from searches for SM-singlet scalars.

Looking to the future, a 100 TeV $pp$ collider such as the proposed FCC-hh
should be sensitive to $W^{\prime}$ and $Z^{\prime}$ bosons as heavy as $\sim
40$ TeV Helsens:2642473 ; Abada:2019lih , corresponding to $v^{\prime}\gtrsim
120$ TeV. This would comprehensively cover the most natural parameter space
consistent with current data, and the non-observation of heavy vectors at such
a collider would suggest that parity solutions are tuned at the
$\Delta^{-1}\sim 10^{-5}$ level. In this respect, future colliders provide a
decisive test of parity solutions to the strong CP problem.

###  3.2 Flavor constraints

In the SM, flavor-changing neutral currents (FCNCs) are absent at tree-level,
appearing only at one-loop, and being additionally suppressed by the GIM
mechanism. As a result, precision measurements of flavor-violating processes
often imply stringent constraints on extensions of the SM. In the class of
models under consideration, FCNCs arise already at tree-level, mediated by the
$Z$ and $Z^{\prime}$ gauge bosons, as well as the scalars $h$ and
$h^{\prime}$. However, their size is suppressed by factors of the Yukawa
couplings of the relevant fermions, making their effect negligible. At one-
loop, FCNCs proceeding via box diagrams involving $W^{\prime}$ gauge bosons
and mirror up-type quarks can lead to deviations in kaon properties, in turn
setting the leading constraints on the flavor structure of these models.

#### Tree-level FCNCs

Rotating from the gauge to the mass eigenbasis in the fermion and gauge boson
sectors, as specified in appendix A, leads to the presence of flavor-changing
interactions between the $Z$ and the SM-like fermions. For example, in the
down-quark sector there are new interactions of the form

$\mathcal{L}\supset\frac{g}{2{\rm
c}_{w}}(\epsilon_{d}^{\dagger}\epsilon_{d})_{ij}Z_{\mu}\bar{d_{i}}_{L}\gamma^{\mu}{d_{j}}_{L},$
(32)

where $\epsilon_{d}$ is a $3\times 3$ matrix acting on flavor space whose
explicit form is given in Eq.(95). Integrating out the $Z$, the effective
hamiltonian relevant to describe $|\Delta F|=1$ processes, such as the
leptonic decay of $B$ mesons, now contains additional terms, of the form

$\Delta\mathcal{H}_{\rm
eff}\simeq-\sqrt{2}G_{F}(\epsilon_{d}^{\dagger}\epsilon_{d})_{32}\cos(2\theta_{w})(\bar{b}_{L}\gamma^{\mu}s_{L})(\bar{\mu}_{L}\gamma_{\mu}\mu_{L})+{\rm
h.c.}$ (33)

(An analogous term involving right-handed muons is also present, but
suppressed by a factor of ${\rm s}_{w}^{2}$, so we neglect it in the
subsequent discussion.)

The deviation with respect to the SM prediction for the branching fraction of
the process $B_{s}^{0}\rightarrow\mu^{+}\mu^{-}$ as a result of the operator
in Eq.(33) can be written as

$r_{\mu\mu}\equiv\frac{{\rm BR}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})_{\rm
BSM}}{{\rm BR}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})_{\rm
SM}}-1\simeq\frac{|C^{\rm(SM)}_{10}+C^{\rm(BSM)}_{10}|^{2}}{|C^{\rm(SM)}_{10}|^{2}}-1,$
(34)

where $C^{\rm(SM)}_{10}$ and $C^{\rm(BSM)}_{10}$ are the SM and BSM
contributions to the Wilson coefficient of the four-fermion operator
$(\bar{b}_{L}\gamma^{\mu}s_{L})(\bar{\mu}\gamma_{\mu}\gamma^{5}\mu)$. In the
SM

$C^{\rm(SM)}_{10}=\frac{G_{F}}{2\sqrt{2}}\frac{\alpha}{4\pi}(V_{tb}^{*}V_{ts})\tilde{C}^{\rm(SM)}_{10},$
(35)

with $\tilde{C}^{\rm(SM)}_{10}\simeq 4.41$ Khodjamirian:2010vf , whereas from
Eq.(33) we have

$C^{\rm(BSM)}_{10}\simeq\frac{G_{F}}{\sqrt{2}}\cos(2\theta_{w})(\epsilon_{d}^{\dagger}\epsilon_{d})_{32}.$
(36)

A stringent upper bound on the size of the
$(\epsilon_{d}^{\dagger}\epsilon_{d})_{32}$ coefficient arises from the
requirement that the masses of the down-type quarks are correctly reproduced
in this model. From Eq.(95), we have

$(\epsilon_{d}^{\dagger}\epsilon_{d})_{32}=\frac{v^{2}}{2}\sum_{i}\frac{(\tilde{y}_{d})_{3i}(\tilde{y}_{d})_{2i}^{*}}{m_{D_{i}}^{2}}\sim\frac{v^{2}}{M^{2}}\sum_{i}(\tilde{y}_{d})_{3i}(\tilde{y}_{d})_{2i}^{*}\lesssim\frac{v}{M}\frac{\sqrt{m_{b}m_{s}}}{v^{\prime}}\ll
1,$ (37)

where in the last step we have made use of the upper bound in Eq.(94). We then
have, parametrically

$r_{\mu\mu}\sim\frac{2|C^{\rm(BSM)}_{10}|}{|C^{\rm(SM)}_{10}|}\lesssim
10^{-3}\left(\frac{18\ {\rm
TeV}}{v^{\prime}}\right)^{2}\left(\frac{v^{\prime}}{M}\right).$ (38)

This effect is much smaller than the theoretical and experimental errors on
${\rm BR}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$, which are both on the order of
$10\%$ Zyla:2020zbs .666The effective operator
$(\bar{b}_{R}\gamma^{\mu}s_{R})(\bar{\mu}\gamma_{\mu}\gamma^{5}\mu)$ is also
generated after integrating out the $Z$, with a Wilson coefficient
$C^{\prime\rm(BSM)}_{10}\sim C^{\rm(BSM)}_{10}$ that enters into Eq.(34) in a
similar manner. The presence of this operator does not quantitatively affect
our analysis.

The effects of $Z$-mediated FCNCs on other processes are even more suppressed.
For example, $|\Delta F|=2$ processes such as kaon mixing require two
insertions of the (tiny) flavor-violating coefficient. In the lepton sector,
even $|\Delta F|=1$ decays are virtually unobservable, as the effect is now
suppressed by the masses of the relevant leptons. FCNCs mediated by the SM
Higgs are similarly negligible, since the corresponding Wilson coefficients
feature the same suppression as those from $Z$ exchange, on top the smaller
coupling between the Higgs and light fermions. Flavor-changing interactions
mediated by the $Z^{\prime}$ and $h^{\prime}$ are further suppressed by an
additional factor of $m_{Z}^{2}/m_{Z^{\prime}}^{2}$ and
$m_{h}^{2}/m_{h^{\prime}}^{2}$ respectively, making them irrelevant. Overall,
the strong suppression of the tree-level FCNCs that occurs naturally in these
models makes their effects negligible.

#### One-loop FCNCs

Another source of FCNCs beyond those present in the SM arises at one loop. In
these models, the familiar box diagram that describes meson mixing in the SM
is now accompanied by similar diagrams that include $W^{\prime}$s as well as
the additional (heavy) up-type quarks running inside the loop, as we show in
figure 2.

Figure 2: Additional box diagrams contributing to kaon mixing in the models
under consideration include any of the up-type quarks propagating inside the
loop, as well as (left) two $W$ bosons, (center) one $W$ and one $W^{\prime}$,
and (right) two $W^{\prime}$s. Quantitatively, it is diagrams with one $W$ and
one $W^{\prime}$ that are the most relevant.

Given the level of experimental precision in measurements of kaon mixing
parameters, even a modification to this process at the loop level can be a
significant source of constraints.

The relevant interactions are those involving the SM-like down-type quarks and
both the $W$ and $W^{\prime}$ gauge bosons. They are given by

$\mathcal{L}\supset\frac{g}{\sqrt{2}}W^{+}_{\mu}\sum_{j,i=1}^{3}\left(V_{ij}\bar{u_{i}}_{L}\gamma^{\mu}{d_{j}}_{L}+\Delta
V_{ij}\bar{U_{i}}_{L}\gamma^{\mu}{d_{j}}_{L}\right)+{\rm h.c.},$ (39)

where ${U_{3}}_{L}=t^{\prime}_{L}$ here and $\Delta V=\epsilon_{u}V$, as well
as

$\begin{split}\mathcal{L}\supset\frac{g}{\sqrt{2}}W^{\prime+}_{\mu}\sum_{j=1}^{3}&\left(\sum_{i=1}^{2}V_{ij}\bar{u_{i}}_{R}\gamma^{\mu}{d_{j}}_{R}+V_{3j}\bar{t}^{\prime}_{R}\gamma^{\mu}{d_{j}}_{R}\right.\\\
&\left.+\sum_{i=1}^{2}\Delta
V^{\prime}_{ij}\bar{U_{i}}_{R}\gamma^{\mu}{d_{j}}_{R}+\Delta
V^{\prime}_{3j}\bar{t}_{R}\gamma^{\mu}{d_{j}}_{R}\right),\end{split}$ (40)

with $\Delta V^{\prime}=\epsilon^{\prime*}_{u}V$. The entries of the $3\times
3$ matrices $\epsilon_{u}$ and $\epsilon^{\prime}_{u}$ are $\mathcal{O}(v/M)$
and $\mathcal{O}(v^{\prime}/M)$ respectively, and explicit expressions can be
found in Eq.(101).

The detailed expressions, including loop functions, relevant to estimate the
contributions to the kaon mixing parameters $\Delta m_{K}$ and
$|\epsilon_{K}|$ can be found in appendix C. Additional box diagrams including
two $W$s or two $W^{\prime}$s always lead to a contribution which is much
smaller than that of the SM, and can therefore be neglected. The leading
contribution arises from diagrams including one $W$ and one $W^{\prime}$. In
this case, there is an “irreducible” contribution to both parameters
(irreducible in the sense that it can only be “turned-off” by increasing
$v^{\prime}$), which comes from the $u$ and $c$ quarks, whose couplings to the
$W^{\prime}$ gauge boson are set to be equal to those of the CKM matrix as a
result of generalized parity. The size of this correction reads

$(\Delta m_{K})_{u,c}\approx-6\cdot 10^{-16}\ {\rm GeV}\left(\frac{6\ {\rm
TeV}}{m_{W^{\prime}}}\right)^{2},\qquad{\rm
and}\qquad|\epsilon_{K}|_{u,c}\approx 7\cdot 10^{-5}\left(\frac{6\ {\rm
TeV}}{m_{W^{\prime}}}\right)^{2},$ (41)

which in both cases is an order of magnitude below the theoretical error in
the corresponding SM prediction, for values of $m_{W^{\prime}}$ consistent
with the direct bounds discussed in 3.1.

Contributions from box diagrams that involve additional members of the up-
quark sector additionally depend on the see-saw scale $M$, as well as the size
of both diagonal and off-diagonal entries in the up-type Yukawa matrices. As
far as $\Delta m_{K}$ is concerned, the leading contribution comes from
diagrams where the $u$ and $c$ quarks propagate inside the loop, and so it is
roughly equal to the result in Eq.(41), even for a see-saw scale $M$ that sits
only slightly above $v^{\prime}$. In contrast, the contribution to
$|\epsilon_{K}|$ can be large, and it is dominated by diagrams where the $t$
quark propagates inside the loop. Choosing the individual entries in the
Yukawa couplings to saturate the upper bound given in Eq.(100),
$|\epsilon_{K}|$ sets a lower bound on $M$ than can range between $750$ TeV
and $1000$ TeV (depending on whether the leading contribution interferes
destructively or constructively with the SM result) for $v^{\prime}\sim 18$
TeV. This value of $M$ sits comfortably within the upper bound $M\lesssim
10^{2}v^{\prime}$, which follows from the requirement of perturbative Yukawas,
as discussed around Eq.(18). Alternatively, even for $v^{\prime}=18$ TeV and
$M=40$ TeV, an additional suppression by a factor of $\mathcal{O}(0.1)$ in the
off-diagonal elements of the up-type Yukawas with respect to their upper bound
is enough to bring the predicted value of $|\epsilon_{K}|$ within the allowed
range.

Overall, the class of parity solutions to the strong CP problem that we focus
on in this work can comfortably satisfy existing constraints from flavor
physics. Flavor-changing processes are, nevertheless, an interesting probe of
the structure of these models, and a more in-depth investigation is a
promising avenue for future work.

##  4 Broken parity and the neutron EDM

As we discussed in section 2, parity-symmetric theories predict a vanishing
$\bar{\theta}$, therefore offering a potential solution to the strong CP
problem. However, the breaking of parity that is necessary for
phenomenological reasons implies that, although zero at tree-level, a non-zero
$\bar{\theta}$ may be generated radiatively. In this section, we investigate
in detail the size of radiative corrections to both $\bar{\theta}$, and the
EDM of elementary fermions. We focus on the effect of non-gravitational
interactions, and leave gravitational considerations to section 5.

The size of radiative corrections to the $\bar{\theta}$ parameter is a
somewhat model-dependent question, as it depends on the details of how parity
is broken. For instance, we could regard generalized parity to be a global
symmetry that is only broken softly by dimensionful parameters, as in Eq.(13).
More realistically, we might expect that the breaking of parity is
spontaneous, and not explicit. This must certainly be the case if, for
example, parity were a gauge symmetry of the UV theory. Even in this case,
there are two qualitatively different options: either parity is broken without
breaking $CP$ (e.g. through a symmetry-breaking sector with two scalar fields
that obtain asymmetric vevs); or both parity and $CP$ are broken
simultaneously (e.g. through the vev of a pseudo-scalar). The former situation
is quantitatively similar to the global case. In the latter, however, the
symmetry-breaking sector can introduce an additional source of $CP$-violation
beyond that present in the SM, and a non-vanishing $\bar{\theta}$ can arise
already at one loop.

In the remainder of this section, we discuss the three qualitatively different
possibilities for the breaking of parity, with a focus on the implications for
the size of radiative corrections to the neutron EDM.

###  4.1 Softly broken parity

We will first discuss the possibility of parity being broken softly, only as a
result of dimensionful parameters. Performing this analysis will give us an
understanding of the irreducible effects that will be present in any theory
where the breaking of parity happens dynamically.

There are two potential sources of soft breaking. One corresponds to the
$\mu^{2}$ term in the scalar potential of Eq.(13), which splits the Higgs vevs
in the SM and mirror sectors. As anticipated in the introduction, if this was
the only source of parity-violation, radiative corrections to $\bar{\theta}$
would be no larger than in the SM Barr:1991qx . Another potential source of
soft breaking are the vector-like mass matrices of Eq.(10). Relaxing the
requirement that these be hermitian introduces a soft breaking of both
generalized parity and $CP$. In this case, a correction to the EDMs of
elementary charged fermions (both quarks and leptons) arises already at one
loop, whereas $\bar{\theta}$ remains zero both at the tree- and one-loop
levels. In turn, this translates into a contribution to the neutron EDM
_independent_ of $\bar{\theta}$.

Taking the vector-like mass matrices of the $SU(2)$-singlets to be general
complex matrices, WLOG we may write them as

$\mathcal{M}^{\prime}_{f}=\mathcal{M}_{f}+i\Delta\mathcal{M}_{f},$ (42)

where both $\mathcal{M}^{\dagger}_{f}=\mathcal{M}_{f}$ and
$\Delta\mathcal{M}^{\dagger}_{f}=\Delta\mathcal{M}_{f}$. If
$\Delta\mathcal{M}_{f}$ is non-vanishing, $\mathcal{M}^{\prime}_{f}$ is no
longer hermitian, therefore (softly) breaking both generalized parity and
$CP$. At one-loop, a non-zero $\Delta\mathcal{M}_{f}$ leads to a non-vanishing
contribution to the EDM of elementary fermions, with the relevant diagrams
depicted in figure 3.

Figure 3: One-loop diagrams generating a non-zero EDM for the $d$ quark in the
presence of soft breaking of generalized parity through non-hermitian vector-
like mass matrices for the $SU(2)$-singlets. The leading contribution arises
from the diagram where $h^{\prime}$, and the heavy mirror quarks, $D_{j}$,
propagate inside the loop. Analogous diagrams are present in both the up-quark
and lepton sectors.

The result is dominated by diagrams where the mirror Higgs, $h^{\prime}$, and
the heavy mirror fermions propagate inside the loop. We present a detailed
calculation in appendix B.1. For any of the light SM fermions, we find

$\frac{d_{f}}{e}\simeq\frac{n_{f}Q_{f}m_{f}}{32\pi^{2}M^{2}}\times\mathcal{O}\left(\frac{|\Delta\mathcal{M}|}{M}\right).$
(43)

where $n_{f}$ is the number of mirror fermions appearing at the see-saw scale
in each fermion sector (i.e. $n_{d}=n_{e}=3$, and $n_{u}=2$), and
$|\Delta\mathcal{M}|$ refers to the typical size of the entries in the
$\Delta\mathcal{M}$ matrix.

Taking the soft-breaking through $\Delta\mathcal{M}$ to be $\mathcal{O}(1)$,
we find, parametrically,

$d_{u},d_{d}\sim 10^{-28}\left(\frac{40\ {\rm TeV}}{M}\right)^{2}e\cdot{\rm
cm}.$ (44)

In turn, this will translate into an EDM for the neutron of approximately the
same size. For illustration, we have normalized the above expression to a
value of $M$ that is roughly a factor of two larger than the current lower
bound on $v^{\prime}$. The corresponding result lies two orders of magnitude
below the current experimental bound on $d_{n}$, and could fall within reach
of future experiments depending on the value of the see-saw scale (see e.g.
Hutzler:2020lmj  for a survey of prospective molecule-based searches promising
orders-of-magnitude improvement in sensitivity to hadronic CPV).

The see-saw mechanism must also be implemented in the charged lepton sector.
If it were not, the mirror partner of the SM electron would appear at a scale
$m_{e^{\prime}}\simeq m_{e}\times v^{\prime}/v$, which would be as low as
$\sim 40$ MeV for the least fine-tuned version of the model where
$v^{\prime}\simeq 18$ TeV. Since mirror fermions carry the same
electromagnetic charge as their SM counterparts, this possibility is obviously
ruled out. As a result, a non-zero electron EDM is also a generic prediction
of this class of theories. Parametrically

$d_{e}\sim 10^{-29}\left(\frac{90\ {\rm TeV}}{M}\right)^{2}e\cdot{\rm cm},$
(45)

where we have chosen the see-saw scale in the lepton sector so as to saturate
the current upper bound on the electron EDM, which is $|d_{e}|<1.1\cdot
10^{-29}\ e\cdot{\rm cm}$ Andreev:2018ayy .

Although an EDM is generated at one loop for the various elementary fermions,
$\bar{\theta}$ remains zero at this order. At tree-level, it is easy to see
that $\bar{\theta}=0$, even in the presence of non-hermitian vector-like mass
matrices. Working in the flavor basis, the full $6\times 6$ mass matrices in
both the up- and down-quark sectors are only modified with respect to Eq.(12)
by replacing $\mathcal{M}_{f}$ with $\mathcal{M}^{\prime}_{f}$ in the bottom-
right block. We then have

$\det\mathbb{M}_{f}=\det\begin{pmatrix}\mathbb{0}_{3}&\frac{v^{\prime}}{\sqrt{2}}y^{\prime*}_{f}\\\
\frac{v}{\sqrt{2}}y^{T}_{f}&\mathcal{M}^{\prime}_{f}\end{pmatrix}\propto\det\left(y^{\prime*}_{f}y^{T}_{f}\right)\qquad{\rm
for}\qquad f=u,d,$ (46)

which is real regardless of $\mathcal{M}^{\prime}_{f}$, since
$y^{\prime}_{f}=y_{f}$. This is clearly an accidental consequence of the zero
appearing in the upper-left corner of the quark mass matrix — the gauge
structure of the theory does not allow for relevant operators with the
appropriate quantum numbers to fill that block. The vanishing of
$\bar{\theta}$ at one-loop is less immediately obvious. The relevant
calculation was performed in Babu:1989rb , and it is also apparent as a
byproduct of our EDM calculation in appendix B.1. As already emphasized in
Babu:1989rb , a non-zero correction to $\bar{\theta}$ could appear at the two-
loop order, and would lead to an additional contribution to the neutron EDM
that could be comparable in size to the one discussed here.

###  4.2 Spontaneously broken parity and $CP$

Perhaps more compellingly — and necessarily, if parity is a gauge symmetry —
the breaking of parity can be accomplished through the vacuum expectation
value of an additional field. The most minimal realization actually entails
the breaking of both parity and $CP$ through the vev of a pseudo-scalar field
$\phi$. The soft term in Eq.(13) is generated by pseudo-scalar couplings to
the Higgs sector, of the form

$V\supset\mu_{\phi}\phi(|H|^{2}-|H^{\prime}|^{2})+\lambda_{\phi}\phi^{2}(|H|^{2}+|H^{\prime}|^{2})\
.$ (47)

The first term above splits the two vevs, and to obtain $v^{\prime}\gg v$
entails

$\mu_{\phi}v_{\phi}\sim\kappa v^{\prime 2}\ .$ (48)

A natural possibility is to take $v_{\phi}\sim\mu_{\phi}\sim v^{\prime}$, with
$\kappa=\mathcal{O}(1)$. However, $\kappa\ll 1$ is also possible, especially
since this coupling breaks the otherwise accidental $SU(4)$ global symmetry of
the scalar potential in Eq.(13). Indeed, the quartic coupling of the SM-like
Higgs is $\lambda_{SM}\sim 2\kappa,$ suggesting $\kappa\lesssim 0.1$ and thus
a pseudo-scalar vev $v_{\phi}$ that is numerically somewhat smaller than
$v^{\prime}$.

Crucially, there is an additional operator consistent with all symmetries that
involves $\phi$ and the $SU(2)$-singlet fermions, of the form Albaid:2015axa :

$\mathcal{L}\supset i(\bar{y}_{d})_{ij}\phi D_{i}D_{j}^{\prime}+{\rm h.c.},$
(49)

and similarly for up-type quarks and leptons. The $\bar{y}_{f}$ matrices must
be hermitian in order to respect generalized parity. When $\phi$ gets a vev,
this term breaks both parity _and_ $CP$. In the notation of section 4.1, a
non-hermitian contribution to the vector-like masses in the fermion sector is
generated, of the form $\Delta\mathcal{M}_{f}=\bar{y}_{f}v_{\phi}$. More
importantly, new interactions involving the pseudo-scalar lead to a non-zero
contribution to $\bar{\theta}$ already at one loop, which sets stringent
constraints on the size of these couplings. The relevant diagrams are those on
the left of figure 3, minus the external photon line, and allowing for $\phi$
to propagate inside the loop. In appendix B.2, we present a detailed
calculation, performed in the mass eigenbasis, of the one-loop correction to
the quark mass matrix, and the corresponding correction to $\bar{\theta}$, in
the context of left-right models with a see-saw fermion structure. In the
remainder of this section, we will reproduce the parametric contribution to
$\bar{\theta}$ from the down-quark sector using a spurion analysis that the
reader might find more instructive.

If we parametrize the one-loop correction to the $6\times 6$ mass matrix in
the down-quark sector in terms of $3\times 3$ blocks, as follows

$\Delta\mathbb{M}_{d}\equiv\begin{pmatrix}\Delta_{d^{\prime}d}&\Delta_{d^{\prime}D^{\prime}}\\\
\Delta_{Dd}&\Delta_{DD^{\prime}}\end{pmatrix},$ (50)

then the corresponding contribution to $\bar{\theta}$ from the down-quark
sector can be written as

$\begin{split}\theta_{d}&\equiv{\rm arg}\
\det\left(\mathbb{M}_{d}+\Delta\mathbb{M}_{d}\right)\\\
&\simeq\operatorname{Im}\operatorname{tr}\left(\mathbb{M}^{-1}_{d}\Delta\mathbb{M}_{d}\right)\\\
&=\operatorname{Im}\operatorname{tr}\left\\{-\left(\frac{vv^{\prime}}{2}y^{\prime*}_{d}\mathcal{M}^{-1}_{d}y^{T}_{d}\right)^{-1}\Delta_{d^{\prime}d}+\left(\frac{v}{\sqrt{2}}y^{T}_{d}\right)^{-1}\Delta_{Dd}+\left(\frac{v^{\prime}}{\sqrt{2}}y^{\prime*}_{d}\right)^{-1}\Delta_{d^{\prime}D^{\prime}}\right\\}.\end{split}$
(51)

Notice that the $\Delta_{DD^{\prime}}$ block does not contribute to
$\theta_{d}$ at this order, which again is a consequence of the zero in the
upper-left corner of $\mathbb{M}_{d}$.

We will now estimate the size of the $\Delta$ matrices appearing in Eq.(51)
through a spurion analysis, as follows. The lagrangian will remain invariant
under $SU(3)$ flavor transformations on the various quark fields, of the form

$d\rightarrow\mathcal{R}_{Q}d,\qquad
d^{\prime}\rightarrow\mathcal{R}_{Q^{\prime}}d^{\prime},\qquad{\rm and}\qquad
D\rightarrow\mathcal{R}_{D}D,\qquad
D^{\prime}\rightarrow\mathcal{R}_{D^{\prime}}D^{\prime},$ (52)

provided the various Yukawa couplings, as well as the vector-like mass matrix,
similarly transform in an appropriate manner. The correct transformation rules
for these objects are given by

$\displaystyle
y^{T}_{d}\rightarrow\mathcal{R}_{D}^{*}y^{T}_{d}\mathcal{R}_{Q}^{\dagger}\
,\qquad\qquad\qquad$ $\displaystyle
y^{\prime*}_{d}\rightarrow\mathcal{R}_{Q^{\prime}}^{*}y^{\prime*}_{d}\mathcal{R}_{D^{\prime}}^{\dagger}\
,$ (53)
$\displaystyle\mathcal{M}_{d}\rightarrow\mathcal{R}_{D}^{*}\mathcal{M}_{d}\mathcal{R}_{D^{\prime}}^{\dagger}\
,\qquad\qquad\qquad$
$\displaystyle\bar{y}_{d}\rightarrow\mathcal{R}_{D}^{*}\bar{y}_{d}\mathcal{R}_{D^{\prime}}^{\dagger}\
.$

On the other hand, the $\Delta$ matrices of Eq.(51) must similarly transform
as follows:

$\Delta_{d^{\prime}d}\rightarrow\mathcal{R}_{Q^{\prime}}^{*}\Delta_{d^{\prime}d}\mathcal{R}_{Q}^{\dagger}\
,\quad\Delta_{d^{\prime}D^{\prime}}\rightarrow\mathcal{R}_{Q^{\prime}}^{*}\Delta_{d^{\prime}D^{\prime}}\mathcal{R}_{D^{\prime}}^{\dagger}\
,\quad{\rm
and}\quad\Delta_{Dd}\rightarrow\mathcal{R}_{D}^{*}\Delta_{Dd}\mathcal{R}_{Q}^{\dagger}\
.$ (54)

It is now straightforward to identify the leading objects that transform as in
Eq.(54) and contain a single insertion of $\bar{y}_{d}$. These are of the form

$\Delta_{d^{\prime}d}\sim\frac{vv^{\prime}v_{\phi}}{16\pi^{2}}\left(y^{\prime*}_{d}\mathcal{M}^{-1}_{d}\bar{y}_{d}\mathcal{M}^{-1}_{d}y^{T}_{d}\right),$
(55)

whereas

$\Delta_{d^{\prime}D^{\prime}}\sim\frac{v^{\prime}v_{\phi}}{16\pi^{2}}\left(y^{\prime*}_{d}\mathcal{M}^{-1}_{d}\bar{y}_{d}\right),\qquad{\rm
and}\qquad\Delta_{Dd}\sim\frac{vv_{\phi}}{16\pi^{2}}\left(\bar{y}_{d}\mathcal{M}^{-1}_{d}y^{T}_{d}\right).$
(56)

We have also included numerical factors to account for the loop suppression,
as well as to take into account that the contribution to $\bar{\theta}$ must
not diverge in the limits where either $v$ or $v^{\prime}$ vanish. Plugging
this back into Eq.(51), we find that all three terms give a contribution of
the same size. Parametrically:

$\theta_{d}\sim\frac{v_{\phi}}{16\pi^{2}}\operatorname{tr}\left(\mathcal{M}^{-1}_{d}\bar{y}_{d}\right)\sim\frac{|\bar{y}_{d}|v_{\phi}}{16\pi^{2}M}.$
(57)

This result is consistent with the more detailed calculation of the
contribution to $\bar{\theta}$ from the quark sector presented in appendix
B.2.

Requiring that $\bar{\theta}\lesssim 10^{-10}$ sets an upper bound on the
typical size of the entries of the $\bar{y}$ matrices in the quark sector, of
the form

$\bar{y}\lesssim 10^{-8}\frac{M}{v_{\phi}}\lesssim 10^{-6},$ (58)

where in the last step we have assumed that $v_{\phi}\sim v^{\prime}$, and
have taken into account the upper bound on the see-saw scale $M$ as given in
Eq.(18).

This result bring us to the following conclusion: if the spontaneous breaking
of parity also implies breaking $CP$, then any interaction between the quark
and symmetry breaking sectors must be extremely weak. Fortunately, if
$\bar{y}=0$ at tree-level, a non-zero value of $\bar{y}$ will not be generated
radiatively. Indeed, $\bar{y}$ and the vector-like mass matrices $\mathcal{M}$
are the only two parameters that violate the $\mathbb{Z}_{2}$ symmetry acting
on the matter fields of the mirror sector. The breaking through $\mathcal{M}$,
however, is soft, and therefore will not translate into a non-zero $\bar{y}$
at the loop order. In this sense, a vanishing $\bar{y}$ is technically
natural.

###  4.3 Spontaneously broken parity alone

A less minimal possibility is to spontaneously break parity while preserving
$CP$ through the addition of two scalar fields, $\sigma$ and
$\sigma^{\prime}$, whose vevs differ. This can be achieved if this symmetry
breaking sector has a scalar potential of the form

$V_{\sigma}=-\frac{m_{\sigma}^{2}}{2}(\sigma^{2}+\sigma^{\prime
2})+\frac{\lambda_{1}}{4}(\sigma^{2}+\sigma^{\prime
2})^{2}+\frac{\lambda_{2}}{4}\sigma^{2}\sigma^{\prime 2}\ .$ (59)

where for simplicity we have forbidden cubic terms by imposing an additional
$\mathbb{Z}_{2}$ symmetry. If $\lambda_{2}>0$, the vacua lie at
$\langle\sigma\rangle=0,$
$\langle\sigma^{\prime}\rangle=\pm\sqrt{m_{\sigma}^{2}/\lambda_{1}}$ and
viceversa. This option is not viable for the Higgs potential itself, which
requires both $v$ and $v^{\prime}$ to be nonzero, but is perfectly adequate
for an additional scalar sector.

Parity breaking can then be translated into the Higgs sector by writing
appropriate couplings of the form

$V\supset\lambda_{\sigma}\left(\sigma^{2}|H|^{2}+\sigma^{\prime
2}|H^{\prime}|^{2}\right)+\lambda^{\prime}_{\sigma}(\sigma^{\prime
2}|H|^{2}+\sigma^{2}|H^{\prime}|^{2})\ .$ (60)

These terms are compatible with the generalized parity introduced in section
2, acting additionally as $\sigma\leftrightarrow\sigma^{\prime}$. Provided
$\lambda_{\sigma}\neq\lambda^{\prime}_{\sigma}$, this will generate the soft
term in Eq.(13) proportional to $\lambda_{\sigma}-\lambda^{\prime}_{\sigma}$.
For example, in the vacuum with $\langle\sigma^{\prime}\rangle\neq 0$,
$v^{\prime}\gg v$ corresponds to

$(\lambda^{\prime}_{\sigma}-\lambda_{\sigma})\langle\sigma^{\prime}\rangle^{2}\sim\kappa
v^{\prime 2}.$ (61)

As this scenario breaks $P$ without breaking $CP$ (and the additional
$\mathbb{Z}_{2}$ symmetry acting on the $\sigma$s forbids marginal couplings
between $\sigma,\sigma^{\prime}$ and fermion bilinears), there are no
significant additional contributions to the neutron EDM. There is, of course,
the possibility of collider signatures coming from the Higgs portal coupling
in Eq.(60), most notably mixing between the Higgs and the scalar that acquires
a vev, as well as invisible decays of the Higgs if kinematically allowed. The
two scalars acquire masses of order
$\sqrt{2\lambda_{1}}\langle\sigma^{\prime}\rangle\sim\sqrt{\lambda_{1}}v^{\prime}$
and
$\sqrt{\lambda_{2}/2}\langle\sigma^{\prime}\rangle\sim\sqrt{\lambda_{2}}v^{\prime}$,
respectively, and it is certainly possible for one to be lighter than half the
Higgs mass depending on the values of $\lambda_{1,2}$.

##  5 Strong CP and quantum gravity

As we discussed in the Introduction, the strong CP problem arises out of the
difficulty of reconciling the smallness of $\bar{\theta}$ with the
$\mathcal{O}(1)$ violation of both parity and $CP$ by the electroweak sector.
In turn, all attempts to address this puzzle are themselves based on the
introduction of an additional symmetry beyond those of the SM. However, there
is strong evidence that within a theory of quantum gravity, global symmetries
cannot be exact — they must be either broken, or gauged. The origin of this
statement goes back a long way Zeldovich:1976vq ; Zeldovich:1977be ;
Banks:1988yz ; Giddings:1987cg ; Lee:1988ge ; Abbott:1989jw ; Coleman:1989zu ;
Kallosh:1995hi ; Banks:2010zn , and to some extent it has recently been
established Harlow:2018tng ; Harlow:2018jwu . Of course, the single most
pressing issue for phenomenology is to establish a lower bound on the amount
of global symmetry violation that must be present in the IR. Attempts at
finding such a “universal” lower bound have been made Fichet:2019ugl ;
Daus:2020vtf , but a fully satisfactory answer remains elusive. Absent a full
understanding of how quantum gravity affects global symmetries at low
energies, we can at least attempt to assess the robustness of an EFT against
global symmetry violation by considering the impact of HDOs suppressed by the
appropriate power of $M_{Pl}$. This both constrains the viable parameter space
of parity solutions to strong CP and illustrates the sense in which $P$,
rather than $U(1)_{PQ}$, provides a solution to the strong CP problem that is
robust against the expected intrusion of quantum gravity. Beyond imposing
constraints, these higher-dimensional operators also lead to new experimental
signatures associated with the spontaneous breaking of parity, which we
explore in section 5.2.

###  5.1 Constraints from Planck-suppressed operators

The observation that the breaking of global symmetries by quantum gravity can
have a profound impact on the validity of the QCD axion solution to strong CP
was first made in Barr:1992qq ; Kamionkowski:1992mf ; Holman:1992us ;
Ghigna:1992iv . Planck-suppressed HDOs that violate the $U(1)_{PQ}$ symmetry
carried by the field $\Phi$, the phase of which is the axion, are of the form

$\mathcal{L}\supset\frac{\eta}{M_{Pl}^{d-4}}|\Phi|^{d-n}\Phi^{n}+{\rm h.c.}$
(62)

Here, $d$ is the operator dimension, $n$ its units of $U(1)_{PQ}$ charge (so
$n\geq 1$ in order to break the symmetry), and $\eta$ a coupling that will in
general feature arbitrary real and imaginary parts. HDOs of this form
contribute to the axion potential, and, in general, will displace the axion
vev away from the value leading to a small $\bar{\theta}$. Following
Kamionkowski:1992mf , requiring that the shift in the axion vev is small
enough so as not to spoil the solution to strong CP translates into the
following upper bound

$|\eta|\left(\frac{f_{a}}{\sqrt{2}M_{Pl}}\right)^{d}\lesssim
10^{-81}\bar{\theta}\lesssim 10^{-91},$ (63)

where $f_{a}$ is the scale of $U(1)_{PQ}$ spontaneous symmetry breaking
(alternatively, the axion decay constant), which is experimentally constrained
to be between $10^{8}$ and $10^{17}$ GeV Zyla:2020zbs . Focusing on operators
of dimension $d=5$, this translates into an upper bound on the size of $\eta$,
of the form

$|\eta|\lesssim 10^{-55}\left(\frac{10^{12}{\rm
GeV}}{f_{a}}\right)^{5}\left(\frac{\bar{\theta}}{10^{-10}}\right)\ .$ (64)

In other words, for all experimentally allowed values of the axion decay
constant, the $U(1)_{PQ}$ symmetry must remain an approximate global symmetry
to an exceptional degree. This is clearly one of the most significant
drawbacks of the axion solution to strong CP.

In the remainder of this section we study the effect of Planck-suppressed HDOs
on parity solutions to the strong CP problem. We consider separately the cases
where parity is global or gauged. The nature of the HDOs under consideration
will be different, but in both cases we will see that even $\mathcal{O}(1)$
coefficients are compatible with solving strong CP.

#### Parity as a global symmetry

If we regard parity as a global symmetry, then we must consider the effect of
HDOs that explicitly violate $P$. The relevant dimension-5 HDOs were already
identified in Berezhiani:1992pq , and they are of the form

$\mathcal{L}\supset\frac{1}{M_{Pl}}\left[(\alpha_{u})_{ij}(H^{\prime}Q_{i}^{\prime})(HQ_{j})+(\alpha_{d})_{ij}(H^{\prime\dagger}Q_{i}^{\prime})(H^{\dagger}Q_{j})\right]+\textrm{h.c.}$
(65)

Notice that if $\alpha_{f}=\alpha_{f}^{\dagger}$ then the above terms would be
parity-symmetric. In general, however, the $\alpha_{f}$’s will _not_ be
hermitian, and it is under this assumption that we proceed.

Setting the Higgs to their vevs, Eq.(65) leads to a correction to the quark
mass matrix that, for arbitrary $\alpha_{f}$’s, does not respect generalized
parity. The leading contribution to $\bar{\theta}$ will come from the
contributions to the up- and down-quark masses, which are of the form

$\delta m_{u}\simeq\frac{vv^{\prime}(\alpha_{u})_{11}}{2M_{Pl}},\qquad{\rm
and}\qquad\delta m_{d}\simeq\frac{vv^{\prime}(\alpha_{d})_{11}}{2M_{Pl}}.$
(66)

In turn,

$\theta_{q}\simeq\frac{\operatorname{Im}(\delta
m_{u})}{m_{u}}+\frac{\operatorname{Im}(\delta m_{d})}{m_{d}}\sim
10^{5}\frac{|\alpha|v^{\prime}}{2M_{Pl}},$ (67)

where in the last step we have used $m_{u}/v\sim m_{d}/v\sim 10^{-5}$.
Requiring that the above contribution is smaller than the current bound on
$\bar{\theta}$ translates into an upper bound on the parity breaking scale:

$v^{\prime}\lesssim\frac{20\ {\rm
TeV}}{|\alpha|}\left(\frac{\bar{\theta}}{10^{-10}}\right).$ (68)

Notice this upper bound is (just) compatible with the lower bound
$v^{\prime}\gtrsim 18$ TeV from direct searches of $W^{\prime}$ gauge bosons,
as discussed in section 3.1. As a result, if _global_ generalized parity is
responsible for solving strong CP, an $\mathcal{O}(1)$ violation of the
symmetry due to gravitational effects would imply a contribution to
$\bar{\theta}$ accessible in near-future experiments.

#### Parity as a gauge symmetry

If parity is instead a gauge symmetry of the underlying theory, HDOs that
explicitly violate $P$ are therefore not allowed. Planck-suppressed operators
such as those in Eq.(65) might still be generated, but only with
$\alpha_{f}=\alpha_{f}^{\dagger}$, and therefore will not contribute to
$\bar{\theta}$. Instead, the operators of interest must be proportional to the
source of spontaneous symmetry breaking. If the latter takes place via the vev
of a pseudo-scalar, as discussed in section 4.2, then there are two
dimension-5 HDOs that satisfy this requirement, namely:

$\mathcal{L}\supset\eta_{s}\frac{\phi\alpha_{s}}{4\pi
M_{Pl}}\operatorname{tr}\left(G^{a}\tilde{G}^{a}\right),$ (69)

and

$\mathcal{L}\supset\frac{i\phi}{M_{Pl}}\left\\{(\zeta_{u})_{ij}Q_{i}HU_{j}+(\zeta^{\prime}_{u})_{ij}Q^{\prime}_{i}H^{\prime}U^{\prime}_{j}+(\zeta_{d})_{ij}H^{\dagger}Q_{i}D_{j}+(\zeta^{\prime}_{d})_{ij}H^{\prime\dagger}Q^{\prime}_{i}U^{\prime}_{j}\right\\}+{\rm
h.c.},$ (70)

with $\eta_{s}\in\mathbb{R}$, and $\zeta^{\prime}_{f}=\zeta_{f}^{*}$ so as to
satisfy generalized parity.

The operator of Eq.(69) will generate a contribution to $\theta_{s}$ after
spontaneous symmetry breaking, of the form

$\theta_{s}\simeq\frac{\eta_{s}v_{\phi}}{M_{Pl}}.$ (71)

Assuming that $v_{\phi}\sim v^{\prime}$, demanding that this contribution is
smaller than the current bound on $\bar{\theta}$ leads to an upper bound on
the parity breaking scale

$v_{\phi}\sim v^{\prime}\lesssim\frac{10^{9}\ {\rm
GeV}}{\eta_{s}}\left(\frac{\bar{\theta}}{10^{-10}}\right),$ (72)

which is clearly well above current bounds on $v^{\prime}$.

At the same time, once $\phi$ gets its vev, the operator of Eq.(70) leads to
an extra contribution to the Yukawa couplings of the up- and down-type quarks
in the SM and mirror sectors that are _not_ parity-symmetric. In turn, this
will lead to an additional contribution to the mass eigenvalues of the light
quarks which will in general contain an imaginary component. For example, in
the down quark sector

$\operatorname{Im}(\delta m_{d_{i}})\simeq
vv^{\prime}\operatorname{Im}\left\\{\frac{iv_{\phi}}{M_{Pl}}\sum_{j}\frac{(y_{d})^{*}_{ij}(\zeta_{d})_{ij}}{m_{D_{j}}}\right\\}\sim\frac{vv^{\prime}}{M}\frac{|\zeta_{d}|v_{\phi}}{M_{Pl}}|(y_{d})_{i\star}|,$
(73)

where $|(y_{d})_{i\star}|$ refers to the typical size of the entries in the
$i$-th row of the $y_{d}$ matrix. The leading contribution to $\theta_{q}$
will come from the up and down quarks. In total:

$\theta_{q}\simeq\frac{\operatorname{Im}(\delta
m_{u})}{m_{u}}+\frac{\operatorname{Im}(\delta
m_{d})}{m_{d}}\simeq\frac{v_{\phi}}{M_{Pl}}\frac{vv^{\prime}}{M}\left\\{\frac{|\zeta_{u}|(y_{u})_{1\star}}{m_{u}}+\frac{|\zeta_{d}|(y_{d})_{1\star}}{m_{d}}\right\\}.$
(74)

Taking into account the upper bound on the entries of the Yukawa couplings
necessary to reproduce the light quark masses (see Eq.(94)), as well as the
requirement that $v^{\prime}\lesssim M$ in order to implement the see-saw
mechanism, the previous equation implies

$\theta_{q}\lesssim 10^{2}\frac{|\zeta|v_{\phi}}{M_{Pl}}\ ,$ (75)

where we have set $m_{u}/v\sim m_{d}/v\sim 10^{-5}$, and have assumed that
$|\zeta_{u}|\sim|\zeta_{d}|$. In turn, taking $v_{\phi}\sim v^{\prime}$, this
sets an upper bound on the scale of spontaneous symmetry breaking:

$v_{\phi}\sim v^{\prime}\lesssim\frac{10^{7}\ {\rm
GeV}}{|\zeta|}\left(\frac{\bar{\theta}}{10^{-10}}\right).$ (76)

As before, this is fully compatible with current experimental bounds on the
parity-breaking scale, even for $\mathcal{O}(1)$ coefficients of the
corresponding HDOs.

###  5.2 Gravitational waves from the spontaneous breaking of parity

Beyond providing additional constraints on the parameter space of parity
solutions to strong CP, the expected effects of gravity also introduce new
experimental signatures. Here we highlight one possibility, namely the impact
of HDOs when parity is a spontaneously broken global symmetry. The spontaneous
breaking of discrete symmetries can lead to the formation of a network of
domain walls in the early universe, provided the reheating temperature after
inflation is above the scale of spontaneous symmetry breaking Kibble:1976sj .
If the spontaneously broken symmetry is global, but otherwise exact, a domain
wall configuration interpolates between two distinct vacua that are
degenerate, making these defects topologically stable objects. The formation
of such networks can be fatal on two grounds. On the one hand, the energy
density in domain walls redshifts slower than that of matter or radiation, and
would eventually dominate the universe’s energy budget. If this happened
before the current epoch, the rapid expansion of the subsequent domain-wall-
dominated era would be at odds with observation. On the other hand, even if
only a subdominant component of the total energy density was in the form of
domain walls today, their effect on large-scale density fluctuations rules out
defects with characteristic scales above $\sim 1\ {\rm MeV}$ Zeldovich:1974uw
. These considerations are often referred to as the “domain wall problem” of
theories with spontaneously broken discrete symmetries.

These problems are largely solved when we take into consideration that, within
a theory of quantum gravity, we expect all symmetries to be either broken or
gauged Zeldovich:1976vq ; Zeldovich:1977be ; Banks:1988yz ; Giddings:1987cg ;
Lee:1988ge ; Abbott:1989jw ; Coleman:1989zu ; Kallosh:1995hi ; Banks:2010zn ;
Harlow:2018tng ; Harlow:2018jwu  — an expectation that includes spacetime
symmetries Harlow:2018tng ; Harlow:2018jwu . In this context, the domain wall
network is unstable, rendering its earlier formation largely unproblematic.
Moreover, the significant amount of gravitational radiation emitted in the
process results in a stochastic gravitational wave background that may be
within reach of current and future observatories. We discuss this possibility
in the remainder of this section.

We will focus first on the scenario where parity is a global symmetry that is
only explicitly broken by gravitational effects. At low energies, the
symmetry-breaking dynamics will enter the effective potential for $\phi$
through HDOs that violate parity. One such operator is of the form

$V\supset\epsilon\frac{\phi^{5}}{M_{Pl}}\ .$ (77)

This breaks the degeneracy between the two previously degenerate vacua,
corresponding to $\langle\phi\rangle=\pm v_{\phi}$. Parametrically, the energy
difference now reads

$\delta V\sim\frac{\epsilon v_{\phi}^{5}}{M_{Pl}}\ .$ (78)

If the reheating temperature is above the scale of spontaneous symmetry
breaking, then we expect that a network of domain walls will be formed once
the temperature of the universe drops bellow $T\sim v_{\phi}$ Kibble:1976sj .
Numerical Press:1989yh ; Garagounis:2002kt ; Oliveira:2004he ; Avelino:2005kn
; Leite:2011sc  and analytical Hindmarsh:1996xv ; Hindmarsh:2002bq  studies
suggest that, shortly after formation, the network evolves according to a
scaling solution, with $\rho_{\rm DW}(t)\simeq\sigma/t$, and typical domain
wall size comparable to the Hubble scale $H(t)^{-1}$. $\sigma$ corresponds to
the tension of the domain walls, which in our model is of the form
$\sigma\sim\sqrt{\kappa_{\phi}}v_{\phi}^{3}$, where $\kappa_{\phi}$ refers to
the quartic coupling in the $\phi$ potential. Two competing effects determine
the network’s subsequent evolution. On the one hand, the pressure difference
between the two vacua exerts a force per unit area of order $\sim\delta V$. On
the other, the tension per unit area acting on a wall with curvature radius
$R$ is $\sim\sigma/R$. In the scaling regime, $R(t)\sim H(t)^{-1}\sim t$
(assuming the universe is radiation dominated), and therefore the effect of
tension decreases with time. Eventually, the pressure difference between the
two vacua dominates, causing the network to collapse at a time of order

$t_{\rm coll.}\sim\frac{\sigma}{\delta
V}\sim\frac{\sqrt{\kappa_{\phi}}M_{Pl}}{\epsilon v_{\phi}^{2}}\ .$ (79)

Clearly, the domain wall network could be very long-lived if $\epsilon\lll 1$.
The requirement that collapse takes place either before the universe becomes
domain wall dominated, or before the start of BBN (so as to avoid energy
injection into the SM plasma that would disrupt light element formation), sets
a lower bound on $\epsilon$ as a function of the spontaneous symmetry breaking
scale. This is depicted in figure 4, where the BBN and domain-wall-domination
restrictions dominate for values of $v_{\phi}$ below and above $\sim 73$ TeV
respectively.

Figure 4: Constraints on the size of the coefficient of the Planck-suppressed
HDO of Eq.(77), as a function of the scale of spontaneous symmetry breaking,
$v_{\phi}$. The region $v_{\phi}\lesssim 18$ TeV (green) is in conflict with
direct bounds on the mass of $W^{\prime}$ and $Z^{\prime}$ resonances, under
the assumption that $v_{\phi}\sim v^{\prime}$, as discussed in section 3.1.
Values of $\epsilon$ that are too small (blue) do not destabilize the domain
wall network early enough to either avoid a domain-wall-dominated era, or to
ensure collapse before the onset of BBN, and are therefore ruled out. The
region of parameter space in pink is experimentally allowed, but the level of
fine-tuning in the electroweak sector worsens as $v_{\phi}$ is increased
(corresponding to a darker shade). The dashed line at $v_{\phi}\simeq 3.5\cdot
10^{4}$ TeV corresponds to a fine-tuning of $\mathcal{O}(10^{-10})$ in the
electroweak sector. (For illustration, we have set the quartic coupling in the
pseudo-scalar potential to be $\kappa_{\phi}=1$ in this plot.)

As can be appreciated in the figure, in the region of parameter space where
the fine-tuning is better than $10^{-10}$ (that is, $v_{\phi}\sim
v^{\prime}\lesssim 3.5\cdot 10^{4}$ TeV), $\epsilon$ may be as small as
$\mathcal{O}(10^{-13})$.

The collapse of a domain wall network leads to the production of gravitational
waves Vilenkin:1981zs ; Vachaspati:1984yi . On dimensional grounds, one would
expect the energy density in gravitational radiation to be of the form
$\rho_{\rm gw}\sim G_{N}\sigma^{2}$ (the mandatory power of $G_{N}$ times the
necessary factors of $\sigma$ to make up dimensions), an expectation that is
largely upheld by numerical analysis Gleiser:1998na ; Dufaux:2007pt ;
Hiramatsu:2010yz ; Kawasaki:2011vv . The resulting gravitational wave spectrum
has an extended shape, peaking at a frequency corresponding to the Hubble size
at the time of collapse (corresponding to the typical size of the domain
walls), and falling off as $1/f$ for larger frequencies. At the present epoch,
the peak frequency of the gravitational wave signal is given by

$\displaystyle f_{*}$ $\displaystyle\simeq 10^{-9}\ {\rm Hz}\left(\frac{T_{\rm
coll.}}{10^{-2}\ {\rm GeV}}\right)\left(\frac{g_{*}(T_{\rm
coll.})}{10}\right)^{1/6}$ (80) $\displaystyle\sim 10^{-9}\ {\rm
Hz}\left(\frac{v_{\phi}}{18\ {\rm
TeV}}\right)\left(\frac{\epsilon}{10^{-12}}\right)^{1/2}\left(\frac{1}{\kappa_{\phi}}\right)^{1/4},$

and the energy density in gravitational radiation at frequency peak reads
Saikawa:2017hiv

$\displaystyle\Omega_{\rm gw}h^{2}(f_{*})$ $\displaystyle\simeq 2\cdot
10^{-10}\left(\frac{\sigma}{(20\ {\rm
TeV})^{3}}\right)^{2}\left(\frac{10^{-2}\ {\rm GeV}}{T_{\rm
coll.}}\right)^{4}\left(\frac{10}{g_{*}(T_{\rm coll.})}\right)^{4/3}$ (81)
$\displaystyle\sim 10^{-10}\left(\frac{v_{\phi}}{18\ {\rm
TeV}}\right)^{2}\left(\frac{10^{-12}}{\epsilon}\right)^{2}\left(\frac{\kappa_{\phi}}{1}\right)^{2},$

where $T_{\rm coll.}$ refers to the temperature of the SM plasma at a time
$t_{\rm coll.}$, and we have assumed that network collapse takes place during
radiation domination.777 In the second steps of Eq.(80) and (81), we have
substituted $T_{\rm coll.}$ by the corresponding expression in terms of the
model’s fundamental parameters, while ignoring a weak dependence on $g_{*}$.

Figure 5 shows the region that can be spanned by the peak of the stochastic
gravitational wave background in the $f_{*}$ vs. $\Omega_{\rm gw}h^{2}(f_{*})$
plane, together with the sensitivity curves of a number of gravitational wave
experiments. The lower bound on $\epsilon$ depicted in figure 4 translates
into a lower bound on $f_{*}$ for each value of the symmetry breaking scale
(e.g. $f_{*}\gtrsim 10^{-9}$ Hz for $v_{\phi}\simeq 18$ TeV). As can be seen
in figure 5, a region of parameter space with low $v^{\prime}$ falls within
reach of gravitational wave observatories probing the low frequency regime
such as SKA Janssen:2014dka , NANOGrav Arzoumanian:2018saf , and the EPTA
Lentati:2015qwp . As $\epsilon$ is increased, the collapse of the domain wall
network occurs earlier, further suppressing the current value of the energy
density in gravitational radiation by the corresponding redshift factor.

Figure 5: In blue, region of parameter space where the frequency peak of the
stochastic gravitational wave signal, $f_{*}$, as well as the corresponding
energy density, $\Omega_{\rm gw}h^{2}(f_{*})$, can fall in light of the
experimental constraints on the various model parameters summarized in figure
4. Dotted lines correspond to constant $\epsilon$. The region to the right of
the dashed line corresponding to $v_{\phi}=3.5\cdot 10^{4}$ TeV features a
level of fine-tuning worse than $1$ part in $10^{10}$, and it is therefore
less attractive. Sensitivity curves for a variety of gravitational wave
experiments are shown, including the pulsar timing arrays EPTA Lentati:2015qwp
, NANOGrav Arzoumanian:2018saf , SKA Janssen:2014dka  (observation time of 5,
10 and 20 years as indicated), as well as the space-based interferometers LISA
Audley:2017drz , and DECIGO Seto:2001qf . (For illustration, we have set the
quartic coupling in the pseudo-scalar potential to be $\kappa_{\phi}=1$ in
this plot.)

Our discussion so far applies in the context of global discrete symmetries
provided that they either do not descend from a continuous symmetry, or that,
if they do, the symmetry breaking scale of the continuous factor is above the
reheating temperature, so that a network of cosmic strings is not formed in
the early universe. On the other hand, if the reheating temperature is larger
than the scale set by the tension of the strings, $\mu$, then a string network
will be formed first, with the strings later joined by domain walls. The
entire string-wall network now evolves together, and the problem features an
additional time scale, given by

$t_{*}\sim\frac{\mu}{\sigma}\ .$ (82)

At $t\sim t_{*}$, the force per unit length on a string of radius $R(t)\sim
t$, given by $\sim\mu/t$, becomes comparable to the wall tension. The system
then becomes dominated by the tension of the domain walls, causing the network
to shrink, and break down into pieces that will further decay into
gravitational waves (or, potentially, also massive particles, depending on
their size and the relevant particle spectrum) Vilenkin:1981zs ;
Vachaspati:1984yi . If this timescale is shorter than $t_{\rm coll.}$, the
earlier destruction of the network of defects could move any potential
gravitational wave signal into an unobservable regime.

The discussion of the previous paragraph is especially relevant if parity is
instead realized as a gauge symmetry, for which explicit breaking is no longer
allowed. Naïvely, one would hope that the gauge case would be cosmologically
more benign: the gauge equivalence of the two vacua makes them no longer
distinct, eliminating the topological stability of the domain walls. Indeed,
domain walls can be destroyed by a process in which a string loop is nucleated
on the wall, further growing to destroy the entire defect. However, the
corresponding nucleation probability is proportional to
$e^{-\mu^{3}/\sigma^{2}}$ Preskill:1992ck , which will be exceedingly small
for any reasonable separation of scales between the string and wall tensions,
therefore rendering gauge domain walls effectively stable. It is therefore
crucial that the reheating temperature is above the string tension scale, so
that a string network is formed that can later result in the entire collapse
of the subsequent string-wall network. The cosmological implications, as well
as potential gravitational wave signatures, of a discrete parity symmetry that
is gauged will be further explored in future work.

##  6 Conclusions

The strong CP problem remains one of the great naturalness problems of the
Standard Model, and is perhaps the most compelling in light of its resistance
to straightforward anthropic explanations. Fully satisfying solutions to the
problem remain elusive given the expected violation of global symmetries in a
theory of quantum gravity, which demands extensive effort to protect the
Peccei-Quinn symmetry underlying axion-based approaches. In this work we have
pursued a possibility that is more transparently robust against the effects of
quantum gravity, revisiting parity-based solutions to the strong CP problem.
Our approach highlights the experimental signatures associated with the most
natural regions of parameter space in these models, as well as ancillary
signatures that are dependent upon the detailed mechanism of parity breaking.

The notion of naturalness within this parameter space is governed by the
tuning associated with the separation of scales of $SU(2)_{L}$ and $SU(2)_{R}$
breaking, which are related by generalized parity. Given this tuning, “see-
saw” vector-like masses for the $SU(2)$-singlet fermions play a key role in
allowing the scale of $SU(2)_{R}$ breaking to be lowered toward its most
natural value consistent with experimental constraints. Within this framework,
the LHC provides the strongest test of natural parity-based solutions to the
strong CP problem, probing the scale of $SU(2)_{R}$ breaking through searches
for $W^{\prime}$ and $Z^{\prime}$ vector bosons as well as vector-like quarks
and additional Higgs bosons. This leaves parity solutions tuned at the $\sim
10^{-3}$ level, which while not fully natural remains a significant
improvement in explaining the observed $\bar{\theta}\lesssim 10^{-10}$. The
extended reach for heavy resonances at future colliders such as FCC-hh will
decisively test these parity solutions at the level of $\sim 10^{-5}$ tuning.
Constraints on new sources of flavor violation play a complementary role, with
additional sensitivity to the scale of vector-like fermions and the underlying
model of flavor.

The detailed mechanism of parity breaking gives rise to additional signatures
within reach of near-future tabletop experiments and gravitational wave
observatories. Soft parity and $CP$-violating terms give rise to EDMs for
elementary fermions at one-loop, both quarks and charged leptons, which
provide a pathway to discovery in precision searches for $CP$-violation in
molecular systems. Spontaneous violation of parity and $CP$ through the vev of
a pseudo-scalar gives rise to additional one-loop contributions to
$\bar{\theta}$, which provides an additional pathway to discovery and already
requires the source of parity violation to be sequestered from the quark
sector (albeit in a technically natural way). The expected violation of global
symmetries in a theory of quantum gravity further shapes the viable parameter
space and potential experimental signatures through the impact of various
Planck-suppressed operators whose form depends on the underlying parity-
breaking mechanism. If parity is a global symmetry that is broken both
spontaneously (by a pseudo-scalar vev) and explicitly (by gravitational
effects), collapse of the domain wall network associated with the spontaneous
breaking of parity can generate a gravity wave signal accessible at low-
frequency gravitational wave observatories. In this respect, the violation of
global symmetries by gravitational effects is a feature of parity-based
solutions to the strong CP problem, rather than a bug. Taken together, these
experimental opportunities warrant further exploration of generalized parity
as a solution to strong CP.

## Acknowledgments

We thank P. Draper and A. Jayich for useful conversations. The research of IGG
is funded by the Gordon and Betty Moore Foundation through Grant GBMF7392, and
in part by the National Science Foundation under Grant No. NSF PHY-1748958.
The research of NC, GK, and AM is supported in part by the Department of
Energy under the grant DE-SC0011702 and the Cottrell Scholar Program through
the Research Corporation for Science Advancement.

##  Appendix A Mass eigenstates

###  A.1 Gauge and Higgs sectors

With the gauge group of Eq.(3), and the Higgs sector specified in table 1,
spontaneous symmetry breaking takes place in two steps, as follows

$SU(2)_{L}\times SU(2)_{R}\times U(1)_{\hat{Y}}\xrightarrow{v^{\prime}\neq
0}SU(2)_{L}\times U(1)_{Y}\xrightarrow{v\neq 0}U(1)_{EM}.$ (83)

The physical spectrum contains SM-like $Z$, $W^{\pm}$, and $\gamma$ gauge
bosons, as well as exotic $Z^{\prime}$ and $W^{\prime\pm}$ excitations. At
tree-level, no mixing occurs in the charged gauge boson sector, and the mass
eigenstates are given in terms of the gauge eigenbasis by the usual
expression:

$W^{\pm}=\frac{1}{\sqrt{2}}(W^{1}\mp iW^{2})\ ,$ (84)

and similarly in the $W^{\prime}$ sector. Tree-level masses are of the form
$m_{W}=gv/2$ and $m_{W^{\prime}}=gv^{\prime}/2$, where we have assumed that
$g^{\prime}=g$, as mandated by generalized parity. By contrast, in the neutral
gauge boson sector mixing between SM and mirror fields takes place already at
tree-level. At zeroth order in a $v/v^{\prime}$ expansion, the gauge
eigenstates can be written in the mass eigenbasis as follows

$\begin{pmatrix}W^{\prime 3}_{\mu}\\\ W^{3}_{\mu}\\\
\hat{B}_{\mu}\end{pmatrix}=\begin{pmatrix}\frac{\sqrt{\cos
2\theta_{w}}}{\cos\theta_{w}}&&&-\sin\theta_{w}\tan\theta_{w}&&&\sin\theta_{w}\\\
0&&&\cos\theta_{w}&&&\sin\theta_{w}\\\
-\tan\theta_{w}&&&-\tan\theta_{w}\sqrt{\cos 2\theta_{w}}&&&\sqrt{\cos
2\theta_{w}}\end{pmatrix}\begin{pmatrix}Z^{\prime}_{\mu}\\\ Z_{\mu}\\\
A_{\mu}\end{pmatrix},$ (85)

where $\sin^{2}\theta_{w}\simeq 0.231$ as usual. Corrections to the above
expression arise at $\mathcal{O}(v^{2}/v^{\prime 2})$. Masses for the SM-like
$Z$ and mirror $Z^{\prime}$ are given by

$m_{Z}=\frac{gv}{2\cos\theta_{w}}+\mathcal{O}\left(\frac{v^{2}}{v^{\prime
2}}\right),\qquad{\rm and}\qquad
m_{Z^{\prime}}=\frac{gv^{\prime}\cos\theta_{w}}{2\sqrt{\cos
2\theta_{w}}}+\mathcal{O}\left(\frac{v^{2}}{v^{\prime 2}}\right).$ (86)

After electroweak symmetry breaking, the Higgs sector consists of two real
scalar fields, $h$ and $h^{\prime}$, with masses given by $m_{h}\simeq
2\sqrt{\kappa}v$ and $m_{h^{\prime}}\simeq\sqrt{2\lambda}v^{\prime}$. Rotating
from the gauge to the mass eigenbasis can be performed as follows

$\begin{pmatrix}h\\\
h^{\prime}\end{pmatrix}\rightarrow\begin{pmatrix}\cos\alpha&&\sin\alpha\\\
-\sin\alpha&&\cos\alpha\end{pmatrix}\begin{pmatrix}h\\\
h^{\prime}\end{pmatrix}\ ,$ (87)

with mixing angle $\alpha\sim v/v^{\prime}$.

###  A.2 Fermion sector

Rotating from the flavor to the mass eigenbasis in the fermion sector requires
solving the eigenvalue problem for the $6\times 6$ matrices
$\mathbb{M}_{f}^{\dagger}\mathbb{M}_{f}$, and
$\mathbb{M}_{f}\mathbb{M}_{f}^{\dagger}$, with $\mathbb{M}_{f}$ as given in
Eq.(12). This can be conveniently done as a perturbation expansion in
$v/M,v^{\prime}/M\ll 1$. In this section, we summarize the relevant results of
this procedure. We focus first on the down-quark and lepton sectors (although
we will use notation appropriate to the down-quark sector, we emphasize that
identical results apply for leptons). The singularities of the up sector as
related to the top quark merit a separate discussion that we present later.

#### Down-type quarks and leptons

The mass eigenvalues in the down-quark sector can be found by diagonalizing
the two $3\times 3$ matrices

$\frac{v^{\prime}v}{2}y^{\prime*}_{d}\mathcal{M}^{-1}_{d}y^{T}_{d},\qquad{\rm
and}\qquad\mathcal{M}_{d}.$ (88)

In full generality, i.e. without yet imposing generalized parity, the above
matrices are not necessarily hermitian, and two unitary matrices are needed in
order to bring them into real diagonal form. This corresponds to the unitary
transformations

$d\rightarrow\mathcal{O}^{\dagger}_{d}d,\
d^{\prime}\rightarrow\mathcal{O}^{\dagger}_{d^{\prime}}d^{\prime},\qquad{\rm
and}\qquad D\rightarrow\mathcal{O}^{\dagger}_{D}D,\
D^{\prime}\rightarrow\mathcal{O}^{\dagger}_{D^{\prime}}D^{\prime}.$ (89)

By definition, the rotation matrices are such that

$\mathbb{m}_{d}\equiv\mathcal{O}^{*}_{d^{\prime}}\left(\frac{v^{\prime}v}{2}y^{\prime*}_{d}\mathcal{M}_{d}^{-1}y_{d}^{T}\right)\mathcal{O}^{\dagger}_{d}={\rm
diag}(m_{d_{i}})\ ,$ (90)

and

$\mathbb{m}_{D}\equiv\mathcal{O}^{*}_{D}\mathcal{M}_{d}\mathcal{O}^{\dagger}_{D^{\prime}}={\rm
diag}(m_{D_{i}})\ ,$ (91)

where $m_{d_{i}}$ and $m_{D_{i}}$ are the masses of the SM and exotic heavy
quarks respectively. As advertised in section 2.3, we will make the
simplifying assumption that all three mirror quarks appear at a common scale
$m_{D_{i}}\sim M\gg v,v^{\prime}$. Imposing generalized parity makes both
matrices in Eq.(88) hermitian. In this case, a single unitary matrix suffices
to make them diagonal, and we have
$\mathcal{O}_{d^{\prime}}=\mathcal{O}^{*}_{d}$ and
$\mathcal{O}_{D}=\mathcal{O}^{*}_{D^{\prime}}$.

It is convenient to define two new matrices corresponding to the Yukawa
couplings in this new basis

$\tilde{y}_{d}\equiv\mathcal{O}^{*}_{d}y_{d}\mathcal{O}^{\dagger}_{D},\qquad{\rm
and}\qquad\tilde{y}^{\prime}_{d}\equiv\mathcal{O}_{d^{\prime}}y^{\prime}_{d}\mathcal{O}^{T}_{D^{\prime}}\
.$ (92)

With this definition, the tree-level masses of the SM-like fermions read

$m_{d_{i}}=\frac{vv^{\prime}}{2}\sum_{j}\frac{(\tilde{y}^{\prime*}_{d})_{ij}(\tilde{y}_{d})_{ij}}{m_{D_{j}}}=\frac{vv^{\prime}}{2}\sum_{j}\frac{|(\tilde{y}_{d})_{ij}|^{2}}{m_{D_{j}}}\
,$ (93)

where the last step holds provided we impose generalized parity. From this
expression, we can find an upper bound on the individual entries in the Yukawa
matrix, of the form

$|(\tilde{y}_{d})_{ij}|\lesssim\left(\frac{2m_{d_{i}}m_{D_{j}}}{vv^{\prime}}\right)^{1/2}\sim\left(\frac{m_{d_{i}}M}{vv^{\prime}}\right)^{1/2}\
.$ (94)

As advertised in section 2.3, bringing the full $6\times 6$ matrix of Eq.(12)
into diagonal form requires a further transformation that mixes the
$SU(2)$-doublet and singlet fields, as specified in Eq.(20). In terms of the
$\tilde{y}_{d}$ and $\tilde{y}^{\prime}_{d}$ couplings defined earlier, the
$3\times 3$ blocks appearing in Eq.(20) can be written as

$\epsilon_{d}=\frac{v}{\sqrt{2}}\mathbb{m}_{D}^{-1}\tilde{y}_{d}^{T},\qquad\qquad{\rm
and}\qquad\qquad\epsilon^{\prime}_{d}=\frac{v^{\prime}}{\sqrt{2}}\mathbb{m}_{D}^{-1}\tilde{y}^{\prime\dagger}_{d},$
(95)

whose entries are of $\mathcal{O}(v/M)$ and $\mathcal{O}(v^{\prime}/M)$
respectively. When generalized parity is only broken by the different vev’s in
the SM and mirror sectors, we have
$\epsilon^{\prime}_{d}=(v^{\prime}/v)\epsilon_{d}^{*}$.

#### Up-type quarks

The diagonalization procedure in the up-quark sector is analogous to that for
down-type quarks and leptons, although this time accommodating for the
singularities of the third generation for which the see-saw mechanism cannot
be implemented.

As before, at zeroth order in $v^{(\prime)}/M$, we perform transformations of
the form

$u\rightarrow\mathcal{O}^{\dagger}_{u}u,\
u^{\prime}\rightarrow\mathcal{O}^{\dagger}_{u^{\prime}}u^{\prime},\qquad{\rm
and}\qquad U\rightarrow\mathcal{O}^{\dagger}_{U}U,\
U^{\prime}\rightarrow\mathcal{O}^{\dagger}_{U^{\prime}}U^{\prime}.$ (96)

On the one hand, the matrices $\mathcal{O}_{U}$ and $\mathcal{O}_{U^{\prime}}$
must be chosen such that the vector-like mass matrix $\mathcal{M}_{u}$ is
brought into diagonal form. In this case, we make the assumption that two of
the eigenvalues of $\mathcal{M}_{u}$ are $m_{U_{1}},m_{U_{2}}\sim M$, whereas
the third one is much smaller, and for simplicity we will take it to vanish in
what follows. On the other hand, the matrices $\mathcal{O}_{u}$ and
$\mathcal{O}_{u^{\prime}}$ must now be such that

$(\tilde{y}^{\prime*}_{u}\hat{\mathbb{m}}_{U}^{-1}\tilde{y}^{T}_{u})_{ij}=\delta_{ij}(\tilde{y}^{\prime*}_{u}\hat{\mathbb{m}}_{U}^{-1}\tilde{y}^{T}_{u})_{ii},\qquad\quad{\rm
and}\quad\qquad(\tilde{y}_{u})_{i3}=(\tilde{y}^{\prime}_{u})_{i3}=0$ (97)

for $i,j=1,2$, and where $\hat{\mathbb{m}}_{U}^{-1}\equiv{\rm
diag}(m_{U_{1}}^{-1},m_{U_{2}}^{-1},0)$, and the $\tilde{y}_{u}$ and
$\tilde{y}^{\prime}_{u}$ matrices are defined as in Eq.(92). Moreover, we
define $y_{t}\equiv\tilde{y}_{33}$, and
$y_{t^{\prime}}\equiv\tilde{y}^{\prime*}_{33}$, which we may choose to be real
and positive.

With this preliminaries, the tree-level mass eigenvalues in the top sector
read

$m_{t}=\frac{y_{t}}{\sqrt{2}}v\ ,\qquad{\rm and}\qquad
m_{t^{\prime}}=\frac{y_{t^{\prime}}}{\sqrt{2}}v^{\prime},$ (98)

with $y_{t^{\prime}}=y_{t}$ if we impose generalized parity. For the first and
second generation, we have instead

$m_{u_{i}}=\frac{vv^{\prime}}{2}\sum_{j=1}^{2}\frac{(\tilde{y}^{\prime*}_{u})_{ij}(\tilde{y}_{u})_{ij}}{m_{U_{j}}}=\frac{vv^{\prime}}{2}\sum_{j=1}^{2}\frac{|(\tilde{y}_{u})_{ij}|^{2}}{m_{U_{j}}},$
(99)

where the last step holds provided we impose generalized parity. As before, we
can now obtain an upper bound on the individual Yukawa entries, of the form

$|(\tilde{y}_{u})_{ij}|\lesssim\left(\frac{2m_{u_{i}}m_{U_{j}}}{vv^{\prime}}\right)^{1/2}\sim\left(\frac{m_{u_{i}}M}{vv^{\prime}}\right)^{1/2}\qquad{\rm
for}\qquad i,j=1,2.$ (100)

A further transformation mixing the $SU(2)$-singlet and doublet components is
again necessary in order to diagonalize the full $6\times 6$ mass matrix,
which can be written as in Eq.(20). The corresponding $\epsilon_{u}$ and
$\epsilon^{\prime}_{u}$ blocks are now given by

$\displaystyle(\epsilon_{u})_{ij}$
$\displaystyle=\frac{v}{\sqrt{2}}(\hat{\mathbb{m}}_{U}^{-1}\tilde{y}^{T})_{ij}\
,\qquad$ $\displaystyle(\epsilon_{u})_{3j}$
$\displaystyle=-\frac{vv^{\prime}}{2}\frac{m_{t^{\prime}}(\tilde{y}^{\prime*}\hat{\mathbb{m}}_{U}^{-1}\tilde{y}^{T})_{3j}}{m_{t^{\prime}}^{2}-\delta_{j3}m_{t}^{2}}\
,$ (101) $\displaystyle(\epsilon^{\prime}_{u})_{ij}$
$\displaystyle=\frac{v^{\prime}}{\sqrt{2}}(\hat{\mathbb{m}}_{U}^{-1}\tilde{y}^{\prime
T})^{*}_{ij}\ ,\qquad$ $\displaystyle(\epsilon^{\prime}_{u})_{3j}$
$\displaystyle=-\frac{vv^{\prime}}{2}\frac{m_{t}(\tilde{y}^{*}\hat{\mathbb{m}}_{U}^{-1}\tilde{y}^{\prime
T})^{*}_{3j}}{m_{t}^{2}-\delta_{j3}m_{t^{\prime}}^{2}}\ .$

for $i=1,2$ and $j=1,2,3$. Just as in the down-quark sector, if generalized
parity is only broken by the difference between $v$ and $v^{\prime}$, we have
$\epsilon^{\prime}_{u}=(v^{\prime}/v)\epsilon^{*}_{u}$.

##  Appendix B Radiatively induced EDM

###  B.1 One-loop EDM

We will now present a calculation of the one-loop correction to the EDM of
elementary charged fermions that arises under the assumption that parity is
only broken softly, both in the scalar potential and through the presence of
non-hermitian vector-like masses for the $SU(2)$-singlets. The relevant
diagrams are those featured in figure 3. We will concentrate first on diagrams
where either $h$ or $h^{\prime}$ propagate inside the loop.

In full generality, a Dirac fermion $f$ that interacts with another fermion
$\psi$, and a neutral scalar $\phi$ through Yukawa couplings of the form

$\mathcal{L}\supset L(\bar{f}_{R}\psi_{L})\phi+R(\bar{f}_{L}\psi_{R})\phi+{\rm
h.c.},$ (102)

will receive a one-loop EDM given by

$\frac{d_{f}}{e}=\frac{Q_{\psi}}{16\pi^{2}}\frac{m_{\psi}}{m^{2}_{\phi}}A(r)\operatorname{Im}\left(LR^{*}\right),$
(103)

where $r=m_{\psi}^{2}/m_{\phi}^{2}$, and the loop function $A$ is given by

$A(r)=\frac{1}{2(1-r)^{2}}\left(3-r+\frac{2\log r}{1-r}\right).$ (104)

In the model we are considering, the Yukawa interactions involving both light
and heavy fermions can be written as

$\mathcal{L}\supset-\sum_{s=h,h^{\prime}}s\left(\bar{d}_{R}\hat{\eta}^{s}d_{L}+\bar{d}_{R}\hat{\beta}^{s}D_{L}+\bar{D}_{R}\hat{\gamma}^{s}d_{L}+\bar{D}_{R}\hat{\delta}^{s}D_{L}\right)+\rm{h.c.},$
(105)

where we are using notation specific to the down-quark sector, but analogous
expressions apply for up-quarks and leptons (although the specific form of the
Yukawa matrices will differ). It will be convenient to write the above
matrices as $\hat{\omega}^{s}=R_{1s}\omega^{h}+R_{2s}\omega^{h^{\prime}}$ (for
$\omega=\eta,\beta,\gamma,\delta$), where $R_{11}=R_{22}=\cos\alpha$ and
$R_{12}=-R_{21}=\sin\alpha$, with $\alpha\sim v/v^{\prime}$ the mixing angle
the Higgs sector. For the down-quark sector, the $\omega$ matrices can be
conveniently written as follows

$\eta^{h}=-\frac{\mathbb{m}_{d}}{v},\qquad\beta^{h}=-\frac{1}{\sqrt{2}}\left(\mathbb{m}_{d}\tilde{y}^{*}\mathbb{m}_{D}^{-1}\right),\qquad\gamma^{h}=\frac{\tilde{y}^{T}}{\sqrt{2}},\qquad\delta^{h}=\frac{v}{2}\left(\tilde{y}^{T}\tilde{y}^{*}\mathbb{m}_{D}^{-1}\right),$
(106)

and

$\eta^{h^{\prime}}=-\frac{\mathbb{m}_{d}}{v^{\prime}},\qquad\beta^{h^{\prime}}=\frac{\tilde{y}^{\prime*}}{\sqrt{2}},\qquad\gamma^{h^{\prime}}=-\frac{1}{\sqrt{2}}\mathbb{m}_{D}^{-1}\tilde{y}^{\prime
T}\mathbb{m}_{d},\qquad\delta^{h^{\prime}}=\frac{v^{\prime}}{2}\left(\mathbb{m}_{D}^{-1}\tilde{y}^{\prime
T}\tilde{y}^{\prime*}\right).$ (107)

The one-loop correction to the EDM of one of the SM-like quarks, $d_{i}$, is
dominated by diagrams where the heavy mirror quarks propagate inside the loop.
Since $m_{D_{j}}\sim M\gg m_{h},m_{h^{\prime}}$, we can expand the loop
function $A(r)$ in the limit $r\gg 1$. Keeping the first two terms, we find

$\frac{d_{d_{i}}}{e}\simeq-\sum_{j,s}\frac{Q_{d}}{32\pi^{2}m_{D_{j}}}\left(1-\frac{m_{s}^{2}}{m_{D_{j}}^{2}}\right)\operatorname{Im}\left(\hat{\beta}^{s}_{ij}\hat{\gamma}^{s}_{ji}\right).$
(108)

The last factor in the previous expression can be written as

$\operatorname{Im}(\hat{\beta}^{s}_{ij}\hat{\gamma}^{s}_{ji})=R_{1s}R_{2s}\operatorname{Im}(\beta^{h}_{ij}\gamma^{h^{\prime}}_{ji}+\beta^{h^{\prime}}_{ij}\gamma^{h}_{ji}),$
(109)

where we have taken into account that
$\beta^{h}_{ij}\gamma^{h}_{ji},\beta^{h^{\prime}}_{ij}\gamma^{h^{\prime}}_{ji}\in\mathbb{R}$,
so those combinations don’t appear on the right-hand-side. When summing over
$s$ in Eq.(108), the contribution from the leading term in the
$m_{s}^{2}/m_{D_{j}}^{2}\ll 1$ expansion vanishes since
$\sum_{s}R_{1s}R_{2s}=0$. The leading contribution to $d_{d_{i}}$ then reads

$\begin{split}\frac{d_{d_{i}}}{e}&\simeq\sum_{j,s}\frac{Q_{d}}{32\pi^{2}}\frac{m_{s}^{2}}{m_{D_{j}}^{3}}R_{1s}R_{2s}\operatorname{Im}\left(\beta^{h}_{ij}\gamma^{h^{\prime}}_{ji}+\beta^{h^{\prime}}_{ij}\gamma^{h}_{ji}\right)\\\
&\simeq\sum_{j}\frac{Q_{d}}{32\pi^{2}}\frac{m_{h^{\prime}}^{2}}{m_{D_{j}}^{3}}\sin\alpha\operatorname{Im}\left(\beta^{h^{\prime}}_{ij}\gamma^{h}_{ji}\right),\end{split}$
(110)

where in the last step we have neglected the first term in parenthesis since
it is suppressed by a factor of $\mathcal{O}((vv^{\prime}/M^{2})^{2})$ with
respect to the second, and we have only kept the contribution from
$h^{\prime}$, since the contribution from $h$ is suppressed by an additional
factor of $m_{h}^{2}/m_{h^{\prime}}^{2}$.

From Eq.(106) and (107), we find

$\beta^{h^{\prime}}_{ij}\gamma^{h}_{ji}=\frac{1}{2}\tilde{y}^{\prime*}_{ij}\tilde{y}_{ij}.$
(111)

When generalized parity is a good symmetry, we have
$\tilde{y}^{\prime}=\tilde{y}$, and therefore the above term is real, in turn
leading to a vanishing EDM. In the presence of soft breaking through non-
hermitian vector-like masses, the relationship $\tilde{y}^{\prime}=\tilde{y}$
no longer holds, even if the Yukawa couplings in the flavor basis remain
identical since the breaking is soft. To see that the equality of the Yukawa
couplings in the SM and mirror sectors no longer holds in the mass basis, it
is useful to remind ourselves of the fermion mass diagonalization procedure
discussed in appendix A.2. When generalized parity remains unbroken, the
unitary matrices of Eq.(89) are such that
$\mathcal{O}_{F}=\mathcal{O}_{F^{\prime}}^{*}$ and
$\mathcal{O}_{f^{\prime}}=\mathcal{O}_{f}^{*}$. However, the non-hermiticity
of the vector-like mass matrix means the unitary matrices needed to bring the
$3\times 3$ matrices of Eq.(88) into real diagonal form will no longer satisfy
this simple relation. Instead, writing the new vector-like mass matrices as
$\mathcal{M}_{f}+i\Delta\mathcal{M}_{f}$, with both $\mathcal{M}_{f}$ and
$\Delta\mathcal{M}_{f}$ hermitian, the new unitary matrices are modified as
follows

$\displaystyle\mathcal{O}_{F^{\prime}}$
$\displaystyle\rightarrow\tilde{\mathcal{O}}_{F^{\prime}}=\mathcal{O}_{F^{\prime}}+\Delta_{F^{\prime}},\qquad\qquad$
$\displaystyle\mathcal{O}_{F}\rightarrow\tilde{\mathcal{O}}_{F}=\mathcal{O}^{*}_{F^{\prime}}-\Delta^{*}_{F^{\prime}},$
(112) $\displaystyle\mathcal{O}_{f}$
$\displaystyle\rightarrow\tilde{\mathcal{O}}_{f}=\mathcal{O}_{f}+\Delta_{f},\qquad\qquad$
$\displaystyle\mathcal{O}_{f^{\prime}}\rightarrow\tilde{\mathcal{O}}_{f^{\prime}}=\mathcal{O}^{*}_{f}-\Delta^{*}_{f}.$

The $\Delta$ matrices arise at
$\mathcal{O}\left(|\Delta\mathcal{M}|/M\right)$, and are given by

$\left(\Delta_{F^{\prime}}\right)_{ij}=i\sum_{k\neq
i}\frac{[\Delta\tilde{\mathcal{M}},\mathbb{m}_{F}]_{ik}}{m^{2}_{F_{i}}-m^{2}_{F_{k}}}\left(\mathcal{O}_{F^{\prime}}\right)_{kj},\quad{\rm
and}\quad\left(\Delta_{f}\right)_{ij}=i\sum_{k\neq
i}\frac{[\Delta\tilde{m},\mathbb{m}_{f}]_{ik}}{m^{2}_{f_{i}}-m^{2}_{f_{k}}}\left(\mathcal{O}_{f}\right)_{kj},$
(113)

where

$\Delta\tilde{\mathcal{M}}\equiv\mathcal{O}_{F^{\prime}}\Delta\mathcal{M}\mathcal{O}_{F^{\prime}}^{\dagger}\qquad{\rm
and}\qquad\Delta\tilde{m}\equiv\frac{vv^{\prime}}{2}\tilde{y}^{*}\mathbb{m}_{F}^{-1}\Delta\tilde{\mathcal{M}}\mathbb{m}_{F}^{-1}\tilde{y}^{T}.$
(114)

Using the above expressions in the definition of $\tilde{y}$, it is possible
to write

$\tilde{y}^{\prime*}=\tilde{y}^{*}(\mathbb{1}+\xi),$ (115)

where $\xi$ is a matrix with entries of $\mathcal{O}(|\Delta\mathcal{M}|/M)$.
Explicitly, after some massaging,

$\tilde{y}^{\prime*}_{ij}=\sum_{l}\tilde{y}^{*}_{il}\left(\delta_{lj}+i\frac{\Delta\tilde{\mathcal{M}}_{lj}}{m_{F_{l}}+m_{F_{j}}}(1-\delta_{lj})+i\frac{vv^{\prime}}{2}\sum_{n,k\neq
i}\frac{\tilde{y}^{*}_{kj}\tilde{y}_{kn}}{m_{f_{i}}+m_{f_{k}}}\frac{\Delta\tilde{\mathcal{M}}_{ln}}{m_{F_{l}}m_{F_{n}}}\right).$
(116)

In total:

$\operatorname{Im}(\tilde{y}^{\prime*}_{ij}\tilde{y}_{ij})=|\tilde{y}_{ij}|^{2}\times\mathcal{O}\left(\frac{\Delta\mathcal{M}}{M}\right).$
(117)

Plugging this back into Eq.(110), we have

$\frac{d_{d_{i}}}{e}\simeq\sum_{j}\frac{Q_{d}}{32\pi^{2}}\frac{m_{h^{\prime}}^{2}}{m_{D_{j}}^{3}}\sin\alpha\frac{1}{2}\operatorname{Im}(\tilde{y}^{\prime*}_{ij}\tilde{y}_{ij})\simeq\frac{n_{d}Q_{d}}{32\pi^{2}}\frac{m_{d_{i}}}{M^{2}}\times\mathcal{O}\left(\frac{\Delta\mathcal{M}}{M}\right),$
(118)

with $n_{d}=3$ the number of mirror fermions appearing at the see-saw scale in
the down-quark sector. The above expression also applies to the lepton sector,
after making the obvious substitutions. In the up-quark sector, the
expressions for the Yukawa couplings are somewhat different to those in
Eq.(106) and (107), but can be similarly found by following the flavor-to-
mass-basis rotation procedure outlined in section A.2. In the end, diagrams
where $h^{\prime}$ and the mirror partners of the $u$ and $c$ quarks propagate
inside the loop give the leading contribution to the one-loop EDM. Thus, the
above expression also applies for the up-quark sector, this time with
$n_{u}=2$ instead.

Additional contributions arise from diagrams where $Z$ and $Z^{\prime}$
propagate inside the loop (see figure 3). In this case, the leading
contribution arises from diagrams involving $Z^{\prime}$ as well as heavy
mirror fermions. In total, the final result is parametrically the same as that
in Eq.(118), except for an additional suppression by a factor of
$g^{2}\sin^{2}\theta_{w}$.

Although the potential one-loop correction to $\bar{\theta}$ that could arise
as a result of the soft breaking through non-hermitian vector-like masses was
already shown to vanish in Babu:1989rb , this can also be seen from the
calculation we have just performed. The relevant diagrams contributing to the
quark mass matrix, and therefore to $\bar{\theta}$, are those of figure 3,
minus the external photon line. So although the appropriate loop function will
be different, the overall correction will be similarly proportional to
Eq.(111). Using Eq.(90) to rewrite $\tilde{y}^{\prime}$ in terms of
$\tilde{y}$, and the diagonal mass matrices, we find

$\operatorname{Im}(\delta
m_{d_{i}})\propto\operatorname{Im}(\tilde{y}^{\prime*}_{ij}\tilde{y}_{ij})\propto
m_{d_{i}}\operatorname{Im}((\tilde{y}^{-1})_{ji}\tilde{y}_{ij}).$ (119)

As a result, the corresponding contribution to $\bar{\theta}$ from the down-
quark sector reads

$\sum_{i}\frac{\operatorname{Im}(\delta
m_{d_{i}})}{m_{d_{i}}}\propto\operatorname{Im}\left(\sum_{i}(\tilde{y}^{-1})_{ji}\tilde{y}_{ij}\right)=0.$
(120)

Notice the sum over quark flavors is crucial in the above cancellation.

###  B.2 One-loop $\bar{\theta}$

The calculation of the one-loop correction to the quark mass matrix, and, in
turn, to $\bar{\theta}$, proceeds along similar lines to the EDM calculation
we have just discussed. The leading contribution to $\bar{\theta}$ comes from
corrections to the light quark masses, and it is due to diagrams where either
$h^{\prime}$ or $\phi$ propagate inside the loop.

Rotating from the gauge to the mass basis in the scalar sector requires
performing a transformation $s_{i}\rightarrow R_{ij}s_{j}$, with
$s_{i}=\\{h,h^{\prime},\phi\\}$, and $R$ is a $3\times 3$ orthogonal matrix
that we parametrize in terms of the various mixing angles as

$R=\begin{pmatrix}{\rm c}_{\alpha}{\rm c}_{\beta}&&&{\rm c}_{\alpha}{\rm
s}_{\beta}{\rm s}_{\gamma}+{\rm s}_{\alpha}{\rm c}_{\gamma}&&&-{\rm
c}_{\alpha}{\rm s}_{\beta}{\rm c}_{\gamma}+{\rm s}_{\alpha}{\rm s}_{\gamma}\\\
-{\rm s}_{\alpha}{\rm c}_{\beta}&&&-{\rm s}_{\alpha}{\rm s}_{\beta}{\rm
s}_{\gamma}+{\rm c}_{\alpha}{\rm c}_{\gamma}&&&{\rm s}_{\alpha}{\rm
s}_{\beta}{\rm c}_{\gamma}+{\rm c}_{\alpha}{\rm s}_{\gamma}\\\ {\rm
s}_{\beta}&&&-{\rm c}_{\beta}{\rm s}_{\gamma}&&&{\rm c}_{\beta}{\rm
c}_{\gamma}\end{pmatrix}$ (121)

where ${\rm c}_{\alpha}=\cos\alpha$, ${\rm s}_{\alpha}=\sin\alpha$, etc.
Parametrically, we expect ${\rm c}_{\alpha}\sim{\rm c}_{\beta}\sim{\rm
c}_{\gamma}=\mathcal{O}(1)$, whereas ${\rm s}_{\alpha}\sim v/v^{\prime}$,
${\rm s}_{\beta}\sim v/v_{\phi}$, and ${\rm s}_{\gamma}\sim
v_{\phi}/v^{\prime}$. In the down-quark sector, the Yukawa interactions of
Eq.(105) need to be extended to include $\phi$ in the sum, and the
$\hat{\omega}^{s}$ matrices are now given by

$\hat{\omega}^{s}=R_{1s}\omega^{h}+R_{2s}\omega^{h^{\prime}}+R_{3s}\omega^{\phi}\qquad{\rm
for}\qquad\omega=\eta,\beta,\gamma,\delta.$ (122)

The expressions for $\omega^{h}$ and $\omega^{h^{\prime}}$ are as in Eq.(106)
and (107), whereas for $\phi$ we have

$\displaystyle\eta^{\phi}$
$\displaystyle=-i\frac{vv^{\prime}}{2}\left(\tilde{y}^{\prime*}\mathbb{m}_{D}^{-1}\tilde{\bar{y}}\mathbb{m}_{D}^{-1}\tilde{y}^{T}\right),\qquad$
$\displaystyle\beta^{\phi}$
$\displaystyle=\frac{iv^{\prime}}{\sqrt{2}}\left(\tilde{y}^{\prime*}\mathbb{m}_{D}^{-1}\tilde{\bar{y}}\right),$
(123) $\displaystyle\gamma^{\phi}$
$\displaystyle=\frac{iv}{\sqrt{2}}\left(\tilde{\bar{y}}\mathbb{m}_{D}^{-1}\tilde{y}^{T}\right),\qquad$
$\displaystyle\delta_{\phi}$ $\displaystyle=-i\tilde{\bar{y}},$

where
$\tilde{\bar{y}}\equiv\mathcal{O}_{F^{\prime}}\bar{y}\mathcal{O}_{F^{\prime}}^{\dagger}$,
as usual.

In the notation of Eq.(102), the one-loop correction to
$\operatorname{Im}(\delta m_{f})$ is given by

$\operatorname{Im}(\delta
m_{f})=\frac{m_{\psi}}{16\pi^{2}}\mathcal{F}(m_{\psi},m_{\phi})\operatorname{Im}(LR^{*}),$
(124)

where the loop function $\mathcal{F}$ now reads

$\mathcal{F}(m_{\psi},m_{\phi})=\frac{1}{m_{\psi}^{2}-m_{\phi}^{2}}\left[m_{\psi}^{2}\left(\log\frac{m_{\psi}^{2}}{\mu^{2}}-1\right)-m_{\phi}^{2}\left(\log\frac{m_{\phi}^{2}}{\mu^{2}}-1\right)\right].$
(125)

For the case at hand, the leading one-loop correction to the mass of the SM-
like fermions involves diagrams where the heavy mirror partners appearing at
scale $M$ propagate inside the loop. Specifically, in the down-quark sector,
we have

$\operatorname{Im}(\delta
m_{d_{i}})=\sum_{s,j}\frac{m_{D_{j}}}{16\pi^{2}}\mathcal{F}(m_{D_{j}},m_{s})\operatorname{Im}(\hat{\beta}^{s}_{ij}\hat{\gamma}^{s}_{ji}).$
(126)

In analogy to the discussion in the previous section, the leading term in
$\mathcal{F}$ in the limit $m_{D_{j}}\gg m_{s}$ is independent of $m_{s}$, and
its contribution to $\operatorname{Im}(\delta m_{f})$ vanishes as a result of
the orthogonality of the mixing matrix in the scalar sector. The leading
correction to $\operatorname{Im}(\delta m_{d_{i}})$ then reads

$\begin{split}\operatorname{Im}(\delta
m_{d_{i}})&\simeq\sum_{j,s}\frac{m_{s}^{2}}{16\pi^{2}m_{D_{j}}}\log\frac{m_{D_{j}}^{2}}{m_{s}^{2}}\operatorname{Im}(\hat{\beta}^{s}_{ij}\hat{\gamma}^{s}_{ji})\\\
&\simeq\sum_{j,s=h^{\prime},\phi}\frac{m_{s}^{2}}{16\pi^{2}m_{D_{j}}}\log\frac{m_{D_{j}}^{2}}{m_{s}^{2}}\operatorname{Im}({\rm
s}_{\gamma}\beta^{h^{\prime}}_{ij}\gamma^{\phi}_{ji}+{\rm s}_{\gamma}{\rm
s}_{\alpha}\beta^{\phi}_{ij}\gamma^{h}_{ji}),\end{split}$ (127)

where in the last step we have neglected the contribution from $h$, which is
suppressed by a factor of $m_{h}^{2}/m_{s}^{2}$ compared to that from
$h^{\prime}$ and $\phi$. The two terms inside the parenthesis are given by

${\rm
s}_{\gamma}\operatorname{Im}(\beta^{h^{\prime}}_{ij}\gamma^{\phi}_{ji})={\rm
s}_{\gamma}\frac{v}{2}\frac{\tilde{y}^{\prime*}_{ij}\tilde{\bar{y}}_{jk}\tilde{y}_{ik}}{m_{D_{k}}}\sim{\rm
s}_{\gamma}\frac{m_{d_{i}}\bar{y}}{v^{\prime}}\sim\frac{v_{\phi}m_{d_{i}}\bar{y}}{v^{\prime
2}},$ (128)

where in the last step we have substituted ${\rm s}_{\gamma}\sim
v_{\phi}/v^{\prime}$, as we expect when $v_{\phi}\lesssim v^{\prime}$, and

${\rm s}_{\gamma}{\rm
s}_{\alpha}\operatorname{Im}(\beta^{\phi}_{ij}\gamma^{h}_{ji})={\rm
s}_{\gamma}{\rm
s}_{\alpha}\frac{v^{\prime}}{2}\frac{\tilde{y}^{\prime*}_{ik}\tilde{\bar{y}}_{kj}\tilde{y}_{ij}}{m_{D_{k}}}\sim{\rm
s}_{\gamma}{\rm
s}_{\alpha}\frac{m_{d_{i}}\bar{y}}{v}\sim\frac{v_{\phi}m_{d_{i}}\bar{y}}{v^{\prime
2}}.$ (129)

Both terms are therefore of the same order. When $v_{\phi}\lesssim
v^{\prime}$, the contribution from $\phi$ to $\operatorname{Im}(\delta
m_{d_{i}})$ is subleading to that from $h^{\prime}$. Setting
$m_{h^{\prime}}\simeq\sqrt{2\lambda}v^{\prime}\sim v^{\prime}$, we then have

$\operatorname{Im}(\delta
m_{d_{i}})\sim\frac{m_{d_{i}}}{16\pi^{2}}\frac{\bar{y}v_{\phi}}{M}\log\frac{M^{2}}{m_{h^{\prime}}^{2}},$
(130)

and the contribution to $\bar{\theta}$ from the down-quark sector reads

$\bar{\theta}\simeq\sum_{i}\frac{\operatorname{Im}(\delta
m_{d_{i}})}{m_{d_{i}}}\sim\frac{1}{16\pi^{2}}\frac{\bar{y}v_{\phi}}{M}\log\frac{M^{2}}{m_{h^{\prime}}^{2}}.$
(131)

The above expression agrees with the parametric estimate presented in section
4.2, except for the log factor that is not captured in our spurion analysis.

##  Appendix C Kaon mixing

The $\Delta m_{K}$ and $|\epsilon_{K}|$ parameters characterizing the kaon
sector can be written as

$\Delta m_{K}=2{\rm Re}(m^{K}_{12}),\qquad{\rm
and}\qquad|\epsilon_{K}|=\frac{\kappa_{\epsilon}|\operatorname{Im}(m^{K}_{12})|}{\sqrt{2}\Delta
m_{K}},$ (132)

where $m^{K}_{12}\equiv\frac{1}{2m_{K}}\langle K^{0}|\mathcal{H}_{\rm
eff}|\bar{K}^{0}\rangle$, and $\mathcal{H}_{\rm eff}$ refers to the effective
hamiltonian appropriate to describe kaon mixing.

In the SM, $\mathcal{H}_{\rm eff}$ is generated at one-loop through box
diagrams involving two $W$ gauge bosons. The corresponding contribution reads

$\mathcal{H}_{\rm
eff}\supset-\frac{G_{F}^{2}m_{W}^{2}}{4\pi^{2}}(\bar{d}_{L}\gamma_{\mu}s_{L})(\bar{d}_{L}\gamma^{\mu}s_{L})\sum_{\alpha,\beta}\lambda_{\alpha}\lambda_{\beta}F(x_{\alpha},x_{\beta})+{\rm
h.c.},$ (133)

where the loop function $F$ is given by Inami:1980fz

$\displaystyle F(x_{\alpha},x_{\beta})$
$\displaystyle=\frac{x_{\alpha}^{2}\log
x_{\alpha}}{(x_{\beta}-x_{\alpha})(1-x_{\alpha})^{2}}\left(1-2x_{\beta}+\frac{x_{\alpha}x_{\beta}}{4}\right)+\\{x_{\alpha}\leftrightarrow
x_{\beta}\\}$ (134)
$\displaystyle+\frac{1}{(1-x_{\alpha})(1-x_{\beta})}\left(\frac{7x_{\alpha}x_{\beta}}{4}-1\right),$

and $\lambda_{\alpha}=V_{\alpha d}^{*}V_{\alpha s}$ for $\alpha=u,c,t$. In the
present model, the sum over $\alpha$ and $\beta$ in Eq.(133) must be extended
to include the additional members of the up-quark sector. The corresponding
couplings can be read off from Eq.(39), and are given by

$\lambda_{\alpha}=\Delta V_{\alpha d}^{*}\Delta V_{\alpha s}\qquad{\rm
for}\qquad\alpha=U,C,T.$ (135)

An additional contribution to $\mathcal{H}_{\rm eff}$ arises from diagrams
involving one $W$ and one $W^{\prime}$. In this case:

$\mathcal{H}_{\rm
eff}\supset-\frac{G_{F}^{2}m_{W}^{2}}{4\pi^{2}}\beta(\bar{d}_{R}s_{L})(\bar{d}_{L}s_{R})\sum_{\alpha,\beta}\lambda^{LR}_{\alpha}\lambda^{RL}_{\beta}\tilde{F}(\beta,x_{\alpha},x_{\beta})+{\rm
h.c.},$ (136)

where $\beta\equiv m_{W}^{2}/m_{W^{\prime}}^{2}=v^{2}/v^{\prime 2}$, and the
loop function now reads Ecker:1985vv

$\tilde{F}(\beta,x_{\alpha},x_{\beta})=\sqrt{x_{\alpha}x_{\beta}}\left\\{(1+\beta)I_{2}(x_{\alpha},x_{\beta},\beta)-(4+\beta
x_{\alpha}x_{\beta})I_{1}(x_{\alpha},x_{\beta},\beta)\right\\},$ (137)

with

$\displaystyle I_{1}(\beta,x_{\alpha},x_{\beta})$
$\displaystyle=\frac{x_{\alpha}\log x_{\alpha}}{(1-x_{\alpha})(1-\beta
x_{\alpha})(x_{\alpha}-x_{\beta})}+\\{x_{\alpha}\leftrightarrow
x_{\beta}\\}-\frac{\beta\log\beta}{(1-\beta)(1-\beta x_{\alpha})(1-\beta
x_{\beta})},$ (138) $\displaystyle I_{2}(\beta,x_{\alpha},x_{\beta})$
$\displaystyle=\frac{x^{2}_{\alpha}\log x_{\alpha}}{(1-x_{\alpha})(1-\beta
x_{\alpha})(x_{\alpha}-x_{\beta})}+\\{x_{\alpha}\leftrightarrow
x_{\beta}\\}-\frac{\log\beta}{(1-\beta)(1-\beta x_{\alpha})(1-\beta
x_{\beta})}.$

The relevant couplings follow from the interactions in Eq.(39) and (40). For
the $u$ and $c$ quarks, we have
$\lambda^{LR}_{\alpha}=\lambda^{RL}_{\alpha}=\lambda_{\alpha}$, whereas for
their heavy partners

$\lambda^{LR}_{\alpha}=\Delta V^{*}_{\alpha d}\Delta V^{\prime}_{\alpha
s},\qquad{\rm and}\qquad\lambda^{RL}_{\alpha}=\Delta V^{\prime*}_{\alpha
d}\Delta V_{\alpha s},\qquad{\rm for}\qquad\alpha=U,C.$ (139)

In the top sector, on the other hand, we have

$\lambda^{LR}_{t}=V^{*}_{td}\Delta V^{\prime}_{3s},\ \lambda^{RL}_{t}=\Delta
V^{\prime*}_{3d}V_{ts},\qquad{\rm and}\qquad\lambda^{LR}_{T}=\Delta
V^{*}_{3d}V_{ts},\ \lambda^{RL}_{T}=V^{*}_{td}\Delta V_{3s}.$ (140)

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