# CP-Violating Axions: A Theory Review

Luca Di Luzioa Hector Gisbertb,a Gabriele Levatib,a Paride Paradisib,a Philip
Sørensenb,a aIstituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova,
Via F. Marzolo 8, 35131 Padova, Italy
bDipartimento di Fisica e Astronomia ‘G. Galilei’, Università di Padova,
Via F. Marzolo 8, 35131 Padova, Italy

###### Abstract

We review the physics case for CP-violating axions. In the first part, we
focus on the Quantum Chromodynamics (QCD) axion and argue that new sources of
CP violation beyond QCD misalign the axion solution to the strong CP problem
and can manifest themselves via a tiny scalar axion-nucleon component. We
hence highlight recent advancements in calculating this scalar axion-nucleon
coupling, a parameter that could be probed via axion-mediated force
experiments. In the second part, we focus on axion-like particle (ALP)
interactions entailing the most general sources of CP violation. After
classifying the full set of CP-violating Jarlskog invariants, we report on
recent calculations of ALP contributions to permanent electric dipole moments.
We finally speculate on possible ultraviolet completions of the CP-violating
ALP.

###### keywords:

Axions, CP violation, Axion-mediated forces, Electric dipole moments.

##  1 Introduction

Among light and weakly coupled new physics scenarios beyond the Standard Model
(SM), the Quantum Chromodynamics (QCD) axion stands out as a most compelling
paradigm, by simultaneously providing an elegant solution to the strong CP
problem via the Peccei Quinn (PQ) mechanism 1, 2, 3, 4 and an excellent dark
matter candidate 5, 6, 7. The experimental landscape for axion searches has
considerably broadened in the recent years, see 8, 9 for recent reviews. While
traditional axion searches target the standard axion-photon coupling, novel
detection strategies have been also proposed to probe axion couplings to
nucleons 10, 11, 12, 13, 14 and electrons 15, 16, 17, 18. These are especially
relevant in view of their complementarity in disentangling different
realizations of QCD axion models. Other low-energy signatures of QCD axions
could stem from flavour- and CP-violating processes. Indeed, although
negligible in benchmark axion models 19, 20, 21, 22, sizeable flavour- and CP-
violating axion couplings may be naturally generated in non-minimal scenarios
addressing the SM flavour puzzle 23, 24, 25, 26, 27, 28 and/or accounting for
the observed matter-antimatter asymmetry of the universe.

Standard QCD axion models predict a strict relation between the axion mass and
its decay constant, enforcing an upper bound on the axion mass of about 0.1 eV
after imposing astrophysical and cosmological bounds. Instead, in the more
general framework of axion-like particles (ALPs), such a stringent condition
is relaxed and much larger ALP masses can be conceived. ALP interactions with
SM fermions and gauge bosons are then described via an effective Lagrangian
containing operators up to $d=5$ 29. The opportunity to look for ALPs with
masses above the MeV scale, whose couplings are not tightly constrained by
astrophysical bounds, opens the possibility to probe ALP interactions via
collider searches 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 and flavour-violating
observables, both in the quark 40, 41, 42, 43, 44, 45, 46 and in the lepton
sectors 47, 48, 49.

Conversely, CP-violating observables involving either QCD axions or ALPs have
received relatively less attention. It is hence the purpose of this review to
summarize the physics case for CP-violating axions. In the case of QCD axions,
this is addressed by providing the basic ingredients behind the origin of CP-
violating axion couplings, and by reporting on recent calculations of CP-
violating couplings to nucleons. Instead, in the case of ALPs, we discuss the
most general CP-violating ALP interactions, both for ALP masses above few GeV,
where QCD can be treated perturbatively, and below the GeV scale, where non-
perturbative methods such as chiral perturbation theory ($\chi$pt) are
mandatory. The impact on electric dipole moments (EDMs) is discussed in both
regimes.

The experimental prospects for probing CP-violating axion interactions via
axion-mediated force experiments and future improvements in the measurements
of permanent EDMs of nucleons, nuclei, atoms and molecules will be also
reviewed. The review is structured around two main parts, discussing
respectively the case of CP-violating QCD axions (Sect. 2) and CP-violating
ALPs (Sect. 3).

##  2 CP-violating QCD axions

The QCD axion originally emerged from the need to “wash out” CP violation from
strong interactions 1, 2, 3, 4. From an effective field theory (EFT)
perspective, the PQ solution to the strong CP problem can be formulated as
follows. The SM Lagrangian is augmented by a spin-0 field $a(x)$ endowed with
a pseudo-shift symmetry, $a\to a+\alpha f_{a}$, that is broken only by the
operator

$\frac{a}{f_{a}}\frac{\alpha_{s}}{8\pi}G\tilde{G}~{},$ (1)

with
$G\tilde{G}\equiv\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}G^{a}_{\mu\nu}G^{a}_{\rho\sigma}$.
After employing the pseudo-shift symmetry to reabsorb the QCD $\theta$ term by
setting $\alpha=-\theta$, one is left in the shifted theory with the operator
in Eq. (1). Hence, the question of CP violation in strong interactions is
traded for a dynamical question about the vacuum expectation value (VEV) of
the axion field

$\theta_{\rm eff}\equiv\frac{\left\langle a\right\rangle}{f_{a}}~{},$ (2)

resulting into an effective $\theta$ parameter, with $|\theta_{\rm
eff}|\lesssim 10^{-10}$ from the non-observation of the neutron EDM (nEDM) 50,
51.

A general result, due to Vafa and Witten 52, ensures that the ground state
energy density of QCD ${\cal E}$ is minimized for $\theta_{\rm eff}=0$, namely
${\cal E}(0)\leq{\cal E}(\theta_{\rm eff})$. The argument is based on an
inequality that exploits the path-integral representation of ${\cal E}$ 53. At
large 4-volume in Euclidean space, one has

$\displaystyle e^{-V_{4}{\cal E}(\theta_{\rm eff})}$
$\displaystyle=\int\mathcal{D}\varphi e^{-S_{0}+i\theta_{\rm eff}{\cal
Q}}=\Big{|}\int\mathcal{D}\varphi e^{-S_{0}+i\theta_{\rm eff}{\cal Q}}\Big{|}$
$\displaystyle\leq\int\mathcal{D}\varphi\Big{|}e^{-S_{0}+i\theta_{\rm
eff}{\cal Q}}\Big{|}=e^{-V_{4}{\cal E}(0)}~{},$ (3)

where $\varphi$ denotes the collection of QCD fields (quarks and gluons),
$S_{0}$ is the QCD Lagrangian in absence of the $\theta$ term and we
introduced the topological charge operator ${\cal Q}\equiv\int
d^{4}x\frac{\alpha_{s}}{8\pi}G\tilde{G}$, which takes integer values in the
background of QCD instantons. A crucial assumption, on which the proof in Eq.
(2) relies, consists in the positive definiteness of the path-integral measure

$\mathcal{D}\varphi\equiv\mathcal{D}A^{a}_{\mu}~{}\mbox{det}\,(\not{D}+M)~{},$
(4)

where the Gaussian path-integral over the fermionic fields has been explicitly
performed. However, while the fermionic determinant is positive definite in a
vector-like theory such as QCD,111Given a non-zero eigenvalue $\lambda$ of the
Euclidean Dirac operator
$i\not{D}=i\sum_{\alpha=1}^{4}\gamma^{\alpha}D_{\alpha}$, that is
$i\not{D}\psi=\lambda\psi$, then also $-\lambda$ is an eigenvalue, since
$i\not{D}(\gamma_{5}\psi)=-\lambda(\gamma_{5}\psi)$. Hence, starting from the
Euclidean Dirac Lagrangian $\overline{\psi}(\not{D}+M)\psi$, the fermionic
determinant reads
$\mbox{det}\,(\not{D}+M)=\Pi_{\lambda}(M-i\lambda)=\Pi_{\lambda>0}(M-i\lambda)(M+i\lambda)=\Pi_{\lambda>0}(M^{2}+\lambda^{2})>0$
(see e.g. 54). this is not the case for a chiral theory like the SM. Hence, we
cannot apply the Vafa-Witten theorem to the SM, although the argument does not
automatically imply that $\theta_{\rm eff}\neq 0$ in the SM. An extra
ingredient of the SM is that CP is explicitly broken in the quark sector by
the Cabibbo-Kobayashi-Maskawa (CKM) phase, which is responsible for a
$\theta_{\rm eff}\neq 0$ term.

###  2.1 The QCD axion ground state in the presence of CP violation

Given a short-distance CP-violating local operator $\mathcal{O}_{\rm CPV}(x)$,
we can estimate the value of $\theta_{\rm eff}$ by expanding the axion
potential as

$V(\theta_{\rm eff})=K^{\prime}\theta_{\rm eff}+\frac{1}{2}K\theta_{\rm
eff}^{2}+\mathcal{O}(\theta^{3}_{\rm eff})~{},$ (5)

and focus on the $\theta_{\rm eff}\ll 1$ regime, since we know that
$|\theta_{\rm eff}|\lesssim 10^{-10}$. Here, $K$ is a 2-point function also
known as topological susceptibility 20

$K=i\int d^{4}x\langle
0|\mathcal{T}\frac{\alpha_{s}}{8\pi}G\tilde{G}(x)\frac{\alpha_{s}}{8\pi}G\tilde{G}(0)|0\rangle~{},$
(6)

where $\mathcal{T}$ denotes the time-ordered product. This quantity is related
to the axion mass via $K=m_{a}^{2}f_{a}^{2}$, while $K^{\prime}$ is a 1-point
function given by 55

$K^{\prime}=i\int d^{4}x\langle
0|\mathcal{T}\frac{\alpha_{s}}{8\pi}G\tilde{G}(x)\mathcal{O}_{{\rm
CPV}}(0)|0\rangle~{}.$ (7)

Note that $K^{\prime}\neq 0$ because $G\tilde{G}$ and $\mathcal{O}_{{\rm
CPV}}$ are both CP-odd, and CP-violating effects can be safely neglected for
the evaluation of the QCD matrix element. The induced $\theta_{\rm eff}$ is
hence obtained via the direct minimization of $V(\theta_{\rm eff})$ in Eq.
(5), yielding

$\theta_{\rm eff}\simeq-\frac{K^{\prime}}{K}~{}.$ (8)

The value of $\theta_{\rm eff}$ in the SM was estimated by Georgi and Randall
in Ref. 56. At energies below $\Lambda_{\chi}=4\pi f_{\pi}$ with
$f_{\pi}\simeq 92$ MeV one can write a flavour conserving, CP-violating
operator

$\mathcal{O}^{\rm SM}_{\rm CPV}=\frac{G_{F}^{2}}{m_{c}^{2}}J_{\rm
CKM}[\overline{u}\gamma^{\mu}(1-\gamma_{5})d\cdot\overline{d}\gamma_{\mu}]\not{D}[\gamma^{\nu}(1-\gamma_{5})s\cdot\overline{s}\gamma_{\nu}(1-\gamma_{5})u]~{},$
(9)

which is obtained in the SM after integrating out the $W$ boson and the charm
quark (see diagram in Fig. 1 of 56), while $J_{\rm
CKM}=\mbox{Im}\,V_{ud}V^{*}_{cd}V_{cs}V^{*}_{us}\simeq 3\times 10^{-5}$ is the
reduced Jarlskog invariant 57. A proper evaluation of the $K^{\prime}$ matrix
element in the presence of $\mathcal{O}^{\rm SM}_{\rm CPV}$ is a non-trivial
task. However, by the rules of naive dimensional analysis (NDA) 58 one expects
$K^{\prime}\sim f_{\pi}^{4}\frac{G_{F}^{2}}{m_{c}^{2}}J_{\rm
CKM}f_{\pi}^{4}\Lambda^{2}_{\chi}$ 56.222The extension of NDA power counting
arguments to CP-violating operators beyond the SM can be found in Ref. 59.
Hence, taking also $K\sim f_{\pi}^{4}$ (indeed, $K\simeq(76\ \text{MeV})^{4}$
– see 60, 61), one obtains

$\theta^{\rm SM}_{\rm eff}\sim\frac{G_{F}^{2}}{m_{c}^{2}}J_{\rm
CKM}f_{\pi}^{4}\Lambda^{2}_{\chi}\sim 10^{-19}~{},$ (10)

which should be taken as an indicative estimate of the CKM contribution.

It is interesting to note that the PQ mechanism would have been ineffective in
a modified SM with the QCD and the Fermi scales closer to each other and/or in
the presence of a trivial flavour structure such that $J_{\rm CKM}\sim 1$.
Also, the PQ mechanism is not generically going to work in a low-scale theory
beyond the SM with generic CP violation. To see this, assume a $d=6$ CP-
violating operator $\mathcal{O}_{\rm CPV}/\Lambda_{\rm BSM}^{2}$, coupled to
QCD. Following an estimate similar to that in Eq. (10) one obtains

$\theta^{\rm BSM}_{\rm eff}\sim\left(\frac{\Lambda_{\chi}}{\Lambda_{\rm
BSM}}\right)^{2}\sim 10^{-10}\left(\frac{100\ \text{TeV}}{\Lambda_{\rm
BSM}}\right)^{2}~{},$ (11)

where in the second step we have normalized $\theta^{\rm BSM}_{\rm eff}$ to
the bound set by the nEDM. This expression suggests that the QCD axion could
act as a low-energy probe of high-energy sources of CP-violation, by testing
generic CP-violating new physics up to the scale $\Lambda_{\rm BSM}\lesssim
100$ TeV.

An axion VEV could also be generated from an ultraviolet (UV) source of PQ
breaking that is not aligned to the QCD anomaly. This is motivated by the fact
that the U(1)PQ, being a global symmetry, does not need to be exact and it is
expected to be broken by quantum gravity effects (see e.g. 62). Hence, the
axion VEV could be displaced from zero, leading to the so-called PQ quality
problem 63, 64, 65, that is the need to explain why UV physics does not
generically spoil the QCD axion solution to the strong CP problem. There
actually exist UV constructions, based e.g. on extra gauge symmetries (see
e.g. 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76), which ensure a controlled
solution to the PQ quality problem. In the following, we will assume as a
canonical example a PQ-breaking effective operator of the type

$V_{\text{PQ-break}}=e^{i\delta_{\theta}}\frac{\Phi^{n}}{\Lambda_{\rm
UV}^{n-4}}+\text{h.c.}~{},$ (12)

with $n>4$. Here, $\delta_{\theta}$ denotes a generic phase, $\Lambda_{\rm
UV}$ is some UV scale like the Planck mass and
$\Phi=\frac{f_{a}}{\sqrt{2}}e^{i\theta}$ is a complex scalar, with the radial
mode integrated out and the angular mode $\theta=a/f_{a}$ given by the axion.
In terms of $\theta$, the PQ-breaking potential reads

$\displaystyle V_{\text{PQ-
break}}=\frac{1}{2^{n/2-1}}\frac{f_{a}^{n}}{\Lambda_{\rm
UV}^{n-4}}\cos(n\theta+\delta_{\theta})~{}.$ (13)

In general, the phase of this potential, parameterized by $\delta_{\theta}$,
needs not be aligned with the CP-conserving minimum of the QCD potential at
$\theta=0$. Therefore, we generically expect $\delta_{\theta}$ to be
$\mathcal{O}(1)$. The balance between $V_{\rm PQ-break}$ and the standard QCD
axion potential, $V_{\rm
QCD}=\frac{1}{2}f_{a}^{2}m_{a}^{2}\theta^{2}+\mathcal{O}(\theta^{4})$, yields
the axion VEV (approximated for $\theta\ll 1$)

$\displaystyle\theta_{\rm eff}\simeq-\frac{n\Lambda_{\rm
UV}^{4-n}f_{a}^{n}\sin\delta_{\theta}}{2^{\frac{n}{2}-1}f_{a}^{2}m_{a}^{2}}~{},$
(14)

which effectively also breaks CP. Taking for instance $\sin\delta_{\theta}=1$,
$\Lambda_{\rm UV}=M_{\rm Pl}$ and $f_{a}=10^{8}$ GeV, the nEDM bound
$|\theta_{\rm eff}|\lesssim 10^{-10}$ translates into $n\geq 9$.

While the nEDM directly probes $\theta_{\rm eff}$, the presence of a
propagating light axion field could offer additional ways to test the axion
ground state, as discussed in the following.

###  2.2 CP-violating axion-nucleon couplings

A remarkable consequence of $\theta_{\rm eff}\neq 0$ is the generation of a
_scalar_ axion coupling to nucleons, $g^{S}_{aN}a\overline{N}N$ (with
$N=p,n$), which can be searched for in axion-mediated force experiments as
suggested by Moody and Wilczek 77. The phenomenological relevance of these
observables will be discussed in Sect. 2.3. To understand the origin of
$g^{S}_{aN}$, let us define the canonical axion field as the excitation over
its VEV, $a\to\theta_{\rm eff}f_{a}+a$, and consider the two-flavour QCD axion
Lagrangian

$\mathcal{L}_{a}=\frac{\alpha_{s}}{8\pi}\left(\theta_{\rm
eff}+\frac{a}{f_{a}}\right)G\tilde{G}-\overline{q}_{L}M_{q}q_{R}+\text{h.c.}~{},$
(15)

where $M_{q}=\mbox{diag}\,(m_{u},m_{d})$. Upon the quark field redefinition

$q\to e^{\frac{i}{2}\gamma_{5}\left(\theta_{\rm
eff}+\frac{a}{f_{a}}\right)Q_{a}}q~{},$ (16)

with
$Q_{a}=M_{q}^{-1}/\mbox{Tr}\,M_{q}^{-1}=\mbox{diag}\,(\frac{m_{d}}{m_{u}+m_{d}},\frac{m_{u}}{m_{u}+m_{d}})$,
the $G\tilde{G}$ term in Eq. (15) gets rotated away, while the quark mass
matrix becomes

$M_{q}\to M_{a}=e^{i\gamma_{5}\left(\theta_{\rm
eff}+\frac{a}{f_{a}}\right)Q_{a}}M_{q}~{},$ (17)

where we used $[M_{q},Q_{a}]=0$. Hence, in the new basis we have

$\displaystyle\mathcal{L}_{a}$
$\displaystyle=-\overline{q}\left[\frac{M_{a}+M^{\dagger}_{a}}{2}+\gamma_{5}\frac{M_{a}-M^{\dagger}_{a}}{2}\right]q$
$\displaystyle=-\overline{q}\cos\left(\left(\theta_{\rm
eff}+\frac{a}{f_{a}}\right)Q_{a}\right)M_{q}q-\overline{q}i\gamma_{5}\sin\left(\left(\theta_{\rm
eff}+\frac{a}{f_{a}}\right)Q_{a}\right)M_{q}q~{}.$ (18)

Focusing on the scalar component (i.e. the term without $\gamma_{5}$)

$\displaystyle\mathcal{L}_{a}$
$\displaystyle\supset-\overline{q}\cos\left(\theta_{\rm
eff}Q_{a}\right)\cos\left(\frac{a}{f_{a}}Q_{a}\right)M_{q}q+\overline{q}\sin\left(\theta_{\rm
eff}Q_{a}\right)\sin\left(\frac{a}{f_{a}}Q_{a}\right)M_{q}q$
$\displaystyle\simeq\theta_{\rm
eff}\frac{a}{f_{a}}\overline{q}~{}Q_{a}^{2}M_{q}q=\frac{\theta_{\rm
eff}}{f_{a}}\frac{m_{u}m_{d}}{m_{u}+m_{d}}a\left(\frac{m_{d}}{m_{u}+m_{d}}\overline{u}u+\frac{m_{u}}{m_{u}+m_{d}}\overline{d}d\right)~{},$
(19)

where in the second step we kept only a term linear in $a/f_{a}$ and expanded
for small $\theta_{\rm eff}$, while in the last step we employed the explicit
representation of $Q_{a}$ given below Eq. (16). Hence, the scalar axion
coupling to nucleons is

$\displaystyle g^{S}_{aN}$ $\displaystyle=\frac{\theta_{\rm
eff}}{f_{a}}\frac{m_{u}m_{d}}{m_{u}+m_{d}}\left(\frac{m_{d}}{m_{u}+m_{d}}\langle
N|\overline{u}u|N\rangle+\frac{m_{u}}{m_{u}+m_{d}}\langle
N|\overline{d}d|N\rangle\right)$ (20) $\displaystyle=\frac{\theta_{\rm
eff}}{f_{a}}\frac{m_{u}m_{d}}{m_{u}+m_{d}}\left[\frac{\langle
N|\overline{u}u+\overline{d}d|N\rangle}{2}+\left(\frac{m_{d}-m_{u}}{m_{u}+m_{d}}\right)\frac{\langle
N|\overline{u}u-\overline{d}d|N\rangle}{2}\right]~{},$

where in the last step we isolated the iso-spin singlet component,
$\overline{u}u+\overline{d}d$. Focusing on the leading iso-spin singlet term
(as in 77), one has

$g^{S}_{aN}\simeq\frac{\theta_{\rm
eff}}{f_{a}}\frac{m_{u}m_{d}}{m_{u}+m_{d}}\frac{\langle
N|\overline{u}u+\overline{d}d|N\rangle}{2}\simeq 1.3\cdot
10^{-12}~{}\theta_{\rm eff}\left(\frac{10^{10}\ \text{GeV}}{f_{a}}\right)~{}.$
(21)

For the numerical evaluation, we employed the value of the pion-nucleon sigma
term $\sigma_{\pi N}=\langle
N|\overline{u}u+\overline{d}d|N\rangle(m_{u}+m_{d})/2=59.1\pm 3.5$ MeV 78. As
pointed out in Ref. 79, a factor $1/2$ was missed in the original derivation
of Moody and Wilczek 77.

Note that Eq. (21) is formally correct only in the case in which the origin of
the axion VEV is due to a UV source of PQ breaking, as e.g. in Eq. (12). On
the other hand, in the more general case of CP-violating sources related to
QCD, i.e. a short distance operator made of quarks and gluons, Eq. (21) turns
out to be conceptually unsatisfactory when we focus on the relevant question:
_how to properly correlate the scalar axion-nucleon coupling with the nEDM
bound?_ The reason being the Eq. (21) misses extra contributions due to meson
tadpoles ($\pi^{0}$, $\eta$, $\eta^{\prime}$), which are generated by the same
UV source of CP violation responsible for $\theta_{\rm eff}\neq 0$ and are of
the same size of the term proportional to $\theta_{\rm eff}$. This improvement
was recently taken into account in Ref. 79 which, including as well iso-spin
breaking effects from the leading order axion-baryon-meson chiral Lagrangian,
found the master formula

$\displaystyle{g}^{S}_{an,\,p}$
$\displaystyle\simeq\frac{4B_{0}\,m_{u}m_{d}}{f_{a}(m_{u}+m_{d})}\bigg{[}\pm(b_{D}+b_{F})\frac{\left\langle\pi^{0}\right\rangle}{F_{\pi}}+\frac{b_{D}-3b_{F}}{\sqrt{3}}\frac{\left\langle\eta_{8}\right\rangle}{F_{\pi}}-\sqrt{\frac{2}{3}}(3b_{0}+2b_{D})\frac{\left\langle\eta_{0}\right\rangle}{F_{\pi}}$
$\displaystyle-\left(b_{0}+(b_{D}+b_{F})\frac{m_{u,d}}{m_{d}+m_{u}}\right)\theta_{\rm
eff}\bigg{]}~{},$ (22)

where for clarity we neglected $m_{u,d}/m_{s}$ terms. Here,
$B_{0}=m_{\pi}^{2}/(m_{d}+m_{u})$ while the hadronic Lagrangian parameters
$b_{D,F}$ are determined from the baryon octet mass splittings, $b_{D}\simeq
0.07\,\rm GeV^{-1}$, $b_{F}\simeq-0.21\,\rm GeV^{-1}$ at LO 80. The value of
$b_{0}$ is determined from the pion-nucleon sigma term as
$b_{0}\simeq-\sigma_{\pi N}/4m_{\pi}^{2}$. From the determination in 78 one
obtains $b_{0}\simeq-0.76\pm 0.04\,\rm GeV^{-1}$. Given $\sigma_{\pi
N}\equiv\langle N|\overline{u}u+\overline{d}d|N\rangle\,(m_{u}+m_{d})/2$, the
isospin symmetric $b_{0}\theta_{\rm eff}$ term reproduces exactly Eq. (21).

In general, $g^{S}_{aN}$ and $d_{n}$ are _not_ proportional, as it would
follow instead from Eq. (21). For instance, exact cancellations among the VEVs
can happen for $d_{n}$ 81, 82 which have no counterpart in $g^{S}_{aN}$. Note
that $\theta_{\rm eff}$ and the meson VEVs in Eq. (2.2) are meant to be
computed from high-energy sources of CP violation, represented by an effective
operator $\mathcal{O}_{\rm CPV}$. In Ref. 79 an explicit example was worked
out in the context of 4-quark operators of the type $\mathcal{O}_{\rm
CPV}=(\overline{q}q)(\overline{q}^{\prime}i\gamma_{5}q^{\prime})$ with
$q,q^{\prime}=u,d,s$, arising e.g. by integrating out the heavy $W_{R}$ boson
in left-right symmetric models. Building on the detailed analysis of Ref. 82,
both $g^{S}_{aN}$ and $d_{n}$ were computed in the minimal left-right
symmetric model with $\mathcal{P}$-parity 83, 84, showing a non-trivial
interplay which deviates sizeably from the naive scaling implied by Eq. (21),
that is $g^{S}_{aN}\propto d_{n}\propto\theta_{\rm eff}$.

More recent works have readdressed the calculation of $g^{S}_{aN}$ in terms of
short-distance sources of CP violation. Ref. 85 considered the contribution of
EDMs and color EDMs of quarks, and revisited as well the SM contribution
arising from the CKM phase. Ref. 86 provided instead a systematic calculation
of CP-violating axion couplings, starting from the CP-violating basis of low-
energy EFT (LEFT) operators of quarks and gluons, and employing at low-energy
a chiral Lagrangian approach, similar to the one of Eq. (2.2).

###  2.3 Axion-mediated forces vs nEDM

Including both scalar and pseudo-scalar couplings to matter fields, the axion
interaction Lagrangian can be written as

$\mathcal{L}^{\rm int}_{a}\supset
g^{S}_{aN}a\overline{N}N+g^{P}_{af}a\overline{f}i\gamma_{5}f~{},$ (23)

where $f=N,e$, and $g^{P}_{af}=C_{f}m_{f}/f_{a}$ denoting the usual pseudo-
scalar axion coupling, with $C_{f}\sim\mathcal{O}(1)$ in benchmark axion
models 87.

By taking the non-relativistic limit of $\mathcal{L}^{\rm int}_{a}$ one
obtains different kinds of static potentials, which can manifest themselves as
new axion-mediated macroscopic forces 77. The latter can be tested in
laboratory experiments, and hence do not rely on model-dependent axion
production mechanisms, as in the case of dark matter axions (haloscopes) or to
a lesser extent solar axions (helioscopes). An updated review of axion-
mediated force experiments and relevant limits can be found in Ref. 88 (see
also 89, 8, 9).

Axion-induced potentials can be of three types, depending on the combination
of couplings involved, i.e. $g^{S}_{aN}g^{S}_{af}$ (monopole-monopole),
$g^{S}_{aN}g^{P}_{af}$ (monopole-dipole) and $g^{P}_{af}g^{P}_{af}$ (dipole-
dipole). The idea of searching for dipole-dipole axion interactions in atomic
physics is as old as the axion itself 3. However, dipole-dipole forces turn
out to be spin suppressed in the non-relativistic limit and suffer from large
backgrounds from ordinary magnetic forces. Searches based on monopole-monopole
interactions, like tests of gravity on macroscopic scales, are in principle
much more powerful. However, under the theoretical prejudice that we are after
the QCD axion, the $\theta_{\rm eff}^{2}$ suppression (given the nEDM bound)
is such that current experiments are still some orders of magnitude away from
testing the QCD axion.

Figure 1: Parameter space of axion-mediated monopole-dipole forces. Figure
adapted from Ref. 88.

The sweet spot is given by monopole-dipole searches which, as shown in Fig. 1,
will enter into the QCD axion region. In this case, the axion-mediated
potential between nucleons and a spin sample of fermions of type $f$ is given
by 77

$V(r)=g^{S}_{aN}g^{P}_{af}\frac{\hat{\sigma}\cdot\hat{r}}{8\pi
M_{f}}\left(\frac{m_{a}}{r}+\frac{1}{r^{2}}\right)e^{-m_{a}r}~{},$ (24)

which implies that for $m_{a}r\lesssim 1$, so that the potential is not
exponentially suppressed, $V(r)$ falls like $1/r^{2}$. A new detection concept
by the ARIADNE collaboration 12, 13 plans to use nuclear magnetic resonance
techniques to probe the axion field sourced by an unpolarized material via a
sample of nucleon spins.333A similar approach is pursued by the
QUAX-$g_{p}g_{s}$ collaboration 16, 17 using instead electron spins. Note that
the yellow QCD axion band in Fig. 1 is obtained by employing Eq. (21) for the
scalar axion-nucleon coupling, with the value of $\theta_{\rm eff}$ spanning
from the SM estimate $\theta_{\rm eff}\sim 10^{-19}$ (lower side) to the limit
imposed by the nEDM $\theta_{\rm eff}\simeq 10^{-10}$ (upper side). However,
as argued in Sect. 2.2, the relation between $g^{S}_{aN}$ and $d_{n}$ is model
dependent and hence, given a specific source of CP-violation, it should be
assessed case by case in order to properly determine the parameter space
region that is allowed by the nEDM bound (for a comprehensive analysis, see
Ref. 86).

In order to have a testable signal in monopole-dipole axion searches a
sizeable source of CP violation beyond the SM is hence required. As a
reference value in terms of $\theta_{\rm eff}$ (using Eq. (21) for
$g^{S}_{aN}$) one would need at least $\theta_{\rm eff}\gtrsim 10^{-13}$, in
order for this effect to be testable at the ultimate phase of ARIADNE (cf.
Fig. 1). While the SM prediction, $\theta_{\rm eff}\sim 10^{-19}$, is far from
being testable, new sources of CP violation beyond the CKM phase are needed to
explain the matter-antimatter asymmetry of the universe. Even if they are
decoupled at very high-energy scales, these CP-violating sources might
contribute sizeably to $\theta_{\rm eff}$, as shown by the estimate in Eq.
(11), and could become accessible in axion-mediated force experiments.

Future improvements of EDM limits will also play a crucial role in
disentangling different sources of CP violation coupled to QCD, also in the
presence of a light axion field (see e.g. 90, 91). From this perspective, the
interplay of low-energy CP-violating phenomena involving the axion field, such
as axion-mediated forces and EDMs, offers a new window for testing high-energy
sources of CP-violation.

###  2.4 Cosmological signatures of CP-violating axions

Today, CP-violating axion potentials are significantly limited by the nEDM
constraint, making them strongly subdominant compared to the axion potential
from QCD. However, the potential generated by QCD in the form of the
topological susceptibility $K(T)$, defined in Eq. (6), becomes strongly
suppressed at temperatures above the QCD crossover. Lattice QCD determinations
of $K(T)$, such as 61, can be approximately fitted with the analytical
expression

$\displaystyle\frac{K(T)}{f_{a}^{2}}=m_{a}^{2}(T)=m_{a}^{2}\,\max\left[\left(\frac{T}{T_{c}}\right)^{-2b},\
1\right]~{},$ (25)

where $T_{c}=150$ MeV and $b=3.92$. At early times, the QCD axion potential is
subdominant to Hubble friction and the field remains frozen at a constant
value. Later, once the temperature $T_{\rm osc}$ is reached where the
condition $m_{a}(T_{\rm osc})\approx 3H(T_{\rm osc})$ is satisfied, the field
starts oscillating and the evolution can be described in the WKB approximation
as $a\approx m_{a}^{-1/2}(T)\ R^{-3/2}$, where $R$ denotes the cosmological
scale factor. For an axion with no other significant potential, this leads to
a dark matter relic density of the form

$\displaystyle\rho_{a,\,\rm
today}\simeq\frac{1}{2}m_{a}^{2}f_{a}^{2}\delta^{2}\frac{m_{a}(T_{\rm
osc})}{m_{a}}\frac{g_{*s}(T_{\rm today})}{g_{*s}(T_{\rm
osc})}\left(\frac{T_{\rm today}}{T_{\rm osc}}\right)^{3}~{},$ (26)

where we neglected any anharmonic effects, which contribute an
$\mathcal{O}(1)$ factor unless $\delta$ is tuned to $\pi$, and made the
simplifying assumption that the field can be immediately described by the WKB
approximation once $m_{a}(T)\approx 3H(T)$ is satisfied. Here,
$\delta\in[-\pi,\pi]$ is the initial misalignment angle, which we generically
expect to be $\mathcal{O}(1).$ To reproduce a fraction $r\equiv\rho_{a,\,\rm
today}/\rho_{\rm DM,\,today}$, this corresponds to an oscillation temperature
of

$\displaystyle T_{\rm osc}\simeq 900\text{ MeV }\times
r^{-1/7}\delta^{2/7}~{},$ (27)

where we approximated $b=4$ to simplify the functional form. This implies that
the CP-preserving potential is suppressed relative to the present-day value by

$\displaystyle\frac{K(T_{\rm osc})}{K}\simeq 5\times 10^{-7}\times
r^{8/7}\delta^{-16/7}~{}.$ (28)

Assuming an $\mathcal{O}(1)$ misalignment angle and that the axion makes up
the whole dark matter density thus implies that the CP-preserving potential
$\propto K(T)$ is suppressed by a factor of more than $10^{6}$ relative to the
present day value. Therefore, although any CP-violating potential must be
subdominant today, it is conceivable that it played a relevant role at
temperatures at or above the QCD crossover.

The possibility for a CP-violating potential to affect the standard
misalignment mechanism crucially depends on the potential’s variation with
temperature. If the CP-violating effect is sourced by an effective operator,
$\mathcal{O}_{\rm CPV}$, coupled to QCD, as in Eq. (7), then we expect
$K^{\prime}$ to be power-suppressed at $T\gtrsim T_{c}$ in a similar way as
for the topological susceptibility. However, if the CP-violating potential
does not depend on the temperature, as in the case of a UV source of PQ-
breaking exemplified by Eq. (12), then it may become competitive with the
topological susceptibility at temperatures relevant for misalignment.

In general, a CP-violating potential, parameterized by $K^{\prime}$ (see Eq.
(5)), dominates the misalignment dynamics if $K^{\prime}(T)\gtrsim K(T)$.
Today, this hierarchy is limited by the nEDM to be $K^{\prime}/K\lesssim
10^{-10}$. If $K^{\prime}$ is constant with temperature, then
$K^{\prime}\gtrsim K(T_{\rm osc})$ requires

$\displaystyle\frac{K(T_{\rm osc})}{K}\lesssim 10^{-10}~{}.$ (29)

From Eq. (28) we observe that the assumptions of an $\mathcal{O(1)}$
misalignment angle and $r=1$, i.e. that axions make up all of the dark matter,
implies that the CP-violating potential remains subdominant at $T_{\rm osc}$.
However, in the regime where the axion makes up only a sub-fraction of dark
matter, $K^{\prime}\gtrsim K(T_{\rm osc})$ can be achieved for $r\lesssim
5\times 10^{-4}$. This implies that CP-violating potentials significantly
impact misalignment for axions with

$\displaystyle m_{a}\gtrsim 5\times 10^{-3}\text{ eV}~{},$ (30)

where we still assumed $\delta_{\theta}=\mathcal{O}(1)$. We arrived at this
conclusion by starting with the assumption that the CP-preserving potential
dominates misalignment, so a more careful analysis is required in the regime
where the CP-violating potential is dominant.

The regime in which the CP-violating potential significantly modifies the
relic abundance was studied in Ref. 92. In their work, the authors considered
how the evolution generated by the temperature-dependent axion mass, given by
the fit in Eq. (25), is modified by the presence of a constant-temperature CP-
violating potential similar to Eq. (13). For such a potential, there are two
regimes with distinct phenomenology, which depend on the phase of the CP-
violating potential $\delta_{\theta}$ and the misalignment angle of the axion
$\delta$:

$\displaystyle\text{Smooth shift regime:}\qquad|\delta-\delta_{\theta}|$
$\displaystyle<\pi/n~{},$ $\displaystyle\text{Trapping
regime:}\qquad|\delta-\delta_{\theta}|$ $\displaystyle>\pi/n~{},$

where $n$ denotes the number of minima of the axion potential, see Eq. (13).
In either case, the axion initially rolls towards the nearest minimum of the
CP-violating potential before the time when it would have started oscillations
around the CP-preserving minimum. Eventually, the axion will transition to the
CP-preserving minimum as this becomes dominant. The crucial distinction is
whether the transition between these minima is adiabatic (smooth shift) or if
it takes place abruptly (trapping). The smooth shift will take place if the
axion initially oscillates in a CP-violating minimum neighbouring the CP-
preserving one. In this case, oscillations start in the CP-violating minimum
and continue to decay according to the conventional WKB evolution throughout
the transition. The effect is then to decrease the relic abundance by
triggering earlier oscillations and thus increasing the redshift.

Alternatively, if the axion initially starts oscillating in a CP-violating
minima that does not neighbour the CP-preserving one, then there will be a
potential barrier preventing the transition from taking place in an adiabatic
way. Instead, at the transition the axion will be suddenly released from the
higher CP-violating minimum and new oscillations will be triggered with an
amplitude set by the misalignment between the false and true minima. This
mechanism, known in the literature as trapped misalignment, has been discussed
in different contexts, see e.g. 93, 94, 95, 96. Interestingly, in this limit,
the axion abundance becomes approximately independent of $m_{a}$ (up to
changes in $g_{*s}$), and the present-day relic is controlled entirely by the
position of the trapped minimum and the amplitude of the CP-violating
potential. Thus, the dark matter abundance can either be increased or
decreased by the CP-violating potential. However, as argued above, if there is
no tuning then the CP-violating potential can only become significant in the
regime in which misalignment under-produces axion dark matter, cf. Eq. (30).
The most optimistic scenario is thus that the CP-violating potential could
enhance the fraction of dark matter that the axion can make up in the
$m_{a}\gtrsim 10^{-3}$ eV regime. Since the axion abundance is approximately
independent of $m_{a}$, the net effect is that the misalignment relic produced
at larger $m_{a}$ would be similar to the one at the transition point.

The reason why the CP-violating potential cannot generically modify the axion
dark matter relic density is because of the nEDM constraint. However, if the
phase of the CP-violating potential $\delta_{\theta}$ is tuned towards zero,
then the nEDM constraint can be relaxed circumventing the limitations above.
Ref. 92 finds that with sufficient tuning, the CP-violating potential in Eq.
(13) can reproduce the observed dark matter from misalignment for any $m_{a}$
compatible with observational constraints. For $m_{a}\gg 10^{-5}$ eV, which
conventionally under-produces axion dark matter, the axion abundance can be
enhanced through trapping. For $m_{a}\ll 10^{-5}$ eV, which conventionally
over-produces axion dark matter for $\delta\sim\mathcal{O}(1)$ misalignment
angles, the axion relic can be diluted if the axion starts in the smooth shift
regime. In the latter scenario, tuning in $\delta_{\theta}$ replaces tuning in
$\delta$.

##  3 CP-violating ALPs

The ALP, hereafter denoted by $\phi$, is a popular generalization of the QCD
axion discussed in Sect. 2, with the most prominent difference being that its
mass ($m_{\phi}$) and couplings to SM particles are arbitrary parameters to be
determined or bounded by experiments. Here, we only require that
$m_{\phi}\ll\Lambda$, where $\Lambda$ is a UV scale related to the origin of
the ALP field. In the following, we report on some recent results from Refs.
97, 98 regarding the construction of the most general CP-violating ALP
interactions and related contributions to different types of EDMs. The
treatment differs depending on whether the ALP mass is above the GeV scale 97,
as detailed in Sect. 3.1, or below it 98, as further discussed in Sect. 3.2.

###  3.1 CP-violating ALP Lagrangian: $m_{\phi}\gtrsim 1$ GeV

Let us start by considering the case $m_{\phi}\gtrsim\rm few~{}$GeV, so that
QCD can be treated perturbatively. Moreover, although we take $\Lambda\gtrsim
1$ TeV, we focus only on electromagnetic and strong interactions, since weak
interactions play a subleading role in our analysis.

The standard ALP effective Lagrangian 29 can be generalised to include also
CP-violating interactions. The most general $SU(3)_{c}\times U(1)_{\rm em}$
invariant $d=5$ effective Lagrangian below the electroweak scale, accounting
for CP-violating ALP interactions with photons, gluons and SM fermions, reads
97

$\displaystyle\mathcal{L}_{\phi}$ $\displaystyle\supset
e^{2}\frac{\tilde{C}_{\gamma}}{\Lambda}\phi
F\tilde{F}+g^{2}_{s}\frac{\tilde{C}_{g}}{\Lambda}\phi
G\tilde{G}+i\frac{v}{\Lambda}y^{ij}_{P}\phi\overline{f}_{i}\gamma_{5}f_{j}$
$\displaystyle+e^{2}\frac{C_{\gamma}}{\Lambda}\phi
FF+g^{2}_{s}\frac{C_{g}}{\Lambda}\phi
GG+\frac{v}{\Lambda}y^{ij}_{S}\phi\overline{f}_{i}f_{j}~{},$ (31)

where $f\in(e,u,d)$ denotes SM fermions in the mass basis, $i,j$ are flavor
indices, and $y_{S}$ and $y_{P}$ are hermitian matrices. $F$ and $G$ stand for
the QED and QCD field strength tensors, respectively, while $\tilde{F}$ and
$\tilde{G}$ are their duals. Contractions in Lorentz space are implicitly
assumed. Note that the interactions in the first line of Eq. (3.1) can be
understood to be invariant under the $\phi$ shift symmetry, as long as non-
perturbative effects related to boundary terms can be neglected in the case of
QCD. In fact, both $F\tilde{F}$ and $G\tilde{G}$ are total derivatives, while
pseudo-scalar interactions can be understood as the linear expansion of an
exponential parametrization of the ALP field. The latter interaction can be
rewritten in a shift-symmetric way through the $d=5$ operator
$\frac{\partial_{\mu}\phi}{\Lambda}\overline{f}_{i}\gamma^{\mu}\gamma_{5}f_{j}$,
upon an anomalous ALP-dependent field redefinition on the fermion fields,
which also redefines the couplings $\tilde{C}_{\gamma}$ and
$\tilde{C}_{g}$.444Note that the identification between the pseudo-scalar
operator in Eq. (3.1) and the derivative ALP interaction with the axial
current breaks down for processes involving the exchange of more than one ALP
field. In the following, we will focus exclusively on processes that entail
the exchange of a single ALP field. Therefore, the parameterization provided
in Eq. (3.1) suffices for our purposes. Instead, the interactions in the
second line of Eq. (3.1) break explicitly the ALP shift symmetry. In
particular, scalar interactions can be written, in the unbroken SM phase, in
terms of the $d=5$ operator $\phi H\overline{f}_{L(R)}f_{R(L)}$ thus
justifying the normalization factor $v/\Lambda$, with $v=246$ GeV. Notice that
$(\tilde{C}_{\gamma},\tilde{C}_{g},y_{P})$ and $(C_{\gamma},C_{g},y_{S})$
could have very different sizes as they stem from the shift-symmetry invariant
and breaking sectors, respectively. For a detailed account of the shift-
breaking orders in the ALP effective Lagrangian, see Ref. 99. Possible UV
completion of the CP-violating ALP effective Lagrangian in Eq. (3.1) will be
discussed in Sect. 3.3.

Since both the operators $XX$ and $X\tilde{X}$ ($X=F,G$), and scalar and
pseudo-scalar operators have opposite CP transformation properties,
$\mathcal{L}_{\phi}$ necessarily violates CP irrespectively of the CP-even or
CP-odd nature of $\phi$. CP-violating phenomena are commonly described by
means of Jarlskog invariants,555See Ref. 99 for a classification of the
Jarlskog invariants at the SM EFT level. i.e. rephasing-invariant parameters
which provide a measure of CP violation 57. In our ALP scenario, the full set
of Jarlskog invariants is given by

$\displaystyle C_{a}\tilde{C}_{b}~{},\quad
y^{ii}_{S}\,{\tilde{C}_{a}}~{},\quad y^{ii}_{P}\,C_{a}\,,\quad
y^{ii}_{S}\,y^{jj}_{P}\,,\quad y^{ik}_{S}\,y^{kk}_{\rm SM}\,y^{ki}_{P}\,,$
(32)

where $a,b=\gamma,g$ and $y^{kk}_{\rm SM}$ denotes a SM Yukawa coupling in the
diagonal basis. Notice that only the last invariant of Eq. (32) is sensitive
to flavor-violating effects.

####  3.1.1 ALP contribution to EDMs

Figure 2: Leading contributions to the semi-leptonic nucleon-electron
operators. The combination of light-blue and purple blobs refer to CP-
violating effective interaction vertices. Figure from Ref. 97.

The effective Lagrangian $\mathcal{L}_{\phi}$ defined in Eq. (3.1) at the
scale $\Lambda$ is renormalized at lower energies by QED and QCD
interactions.666The complete one-loop anomalous dimension matrix assuming the
full SM gauge group has been recently presented in Ref. 100. In the leading
logarithmic approximation, the solution of the renormalization-group equations
for the leptonic (pseudo)scalar couplings $y^{\ell\ell}_{S,P}$ gives 97

$\displaystyle y^{\ell\ell}_{S}$ $\displaystyle\simeq
y^{\ell\ell}_{S}(\Lambda)+\frac{6\alpha}{\pi}\frac{m_{\ell}}{v}e^{2}C_{\gamma}\log\frac{\Lambda}{\mu}~{},$
(33) $\displaystyle y^{\ell\ell}_{P}$ $\displaystyle\simeq
y^{\ell\ell}_{P}(\Lambda)-\frac{6\alpha}{\pi}\frac{m_{\ell}}{v}e^{2}{\tilde{C}}_{\gamma}\log\frac{\Lambda}{\mu}~{},$
(34)

where $\mu$ is the renormalization scale. Instead, in the quark sector, we
obtain 97

$\displaystyle\\!\\!y^{qq}_{S}\simeq$
$\displaystyle~{}~{}y^{qq}_{S}(\Lambda)\\!+\\!\frac{m_{q}}{v}\\!\left(\frac{6\alpha}{\pi}Q^{2}_{q}e^{2}C_{\gamma}\\!+\\!\frac{8\alpha_{s}}{\pi}g_{s}^{2}C_{g}\right)\\!\log\frac{\Lambda}{\mu}~{},$
(35) $\displaystyle\\!y^{qq}_{P}\simeq$
$\displaystyle~{}~{}y^{qq}_{P}(\Lambda)\\!-\\!\frac{m_{q}}{v}\\!\left(\frac{6\alpha}{\pi}Q^{2}_{q}e^{2}{\tilde{C}}_{\gamma}\\!+\\!\frac{8\alpha_{s}}{\pi}g_{s}^{2}{\tilde{C}}_{g}\right)\\!\log\frac{\Lambda}{\mu}~{}.$
(36)

Since in Eq. (3.1) we factored out the gauge couplings $e^{2}$ and
$g^{2}_{s}$, the coefficients $C_{\gamma,g}$ and $\tilde{C}_{\gamma,g}$ are
scale-invariant at one-loop order. Top and bottom contributions are taken into
account by the QCD trace anomaly in the gluon-gluon-ALP vertex after they have
been integrated out (see Fig. 2), yielding

$\displaystyle g_{s}^{2}C_{g}$ $\displaystyle\simeq
g_{s}^{2}C_{g}(\Lambda)+\frac{\alpha_{s}}{12\pi}\sum_{q=t,b}\frac{vy^{qq}_{S}}{m_{q}}~{},$
(37) $\displaystyle g_{s}^{2}\tilde{C}_{g}$ $\displaystyle\simeq
g_{s}^{2}\tilde{C}_{g}(\Lambda)-\frac{\alpha_{s}}{8\pi}\sum_{q=t,b}\frac{vy^{qq}_{P}}{m_{q}}~{},$
(38)

valid in the limit $m_{q}\gg m_{\phi}$, in agreement with Higgs low-energy
theorems 101.

Among the most stringent constraints on the CP-violating ALP interactions
stemming from Eq. (3.1), EDMs emerge as the most powerful probes. From the
experimental side, there is an outstanding program to improve the current
limits on permanent EDMs of molecules, atoms, nuclei and nucleons by several
orders of magnitude (see e.g. 102). It is therefore mandatory to provide a
general framework to systematically account for CP-violating ALP contributions
to EDMs, which was missing until recently (see however 103, 104, 105).

The leading low-energy CP-violating Lagrangian relevant for EDMs of molecules,
atoms, nuclei and nucleons reads 106

$\displaystyle\mathcal{L}_{\rm CPV}$
$\displaystyle=\sum_{i,j=u,d,e}\\!\\!\\!C_{ij}(\overline{f}_{i}f_{i})(\overline{f}_{j}i\gamma_{5}f_{j})+\alpha_{s}C_{Ge}\,GG\,\overline{e}i\gamma_{5}e+\alpha_{s}C_{\tilde{G}e}\,G\tilde{G}\,\overline{e}e$
(39)
$\displaystyle-\frac{i}{2}\\!\sum_{i=u,d,e}d_{i}\overline{f}_{i}(F\\!\cdot\\!\sigma)\gamma_{5}f_{i}-\frac{i}{2}\\!\sum_{i=u,d}g_{s}d^{C}_{i}\overline{f}_{i}(G\\!\cdot\\!\sigma)\gamma_{5}f_{i}+\frac{d_{G}}{3}f^{abc}G^{a}\tilde{G}^{b}G^{c}~{},$

where we omitted color-octet 4-quark operators (as they emerge only at one-
loop level in the ALP framework) and the dim-4 $G\tilde{G}$ operator. The
latter is assumed to be absent thanks to a UV mechanism addressing the strong
CP problem.

Within our EFT approach, $C_{ij}$, $C_{Ge}$ and $C_{\tilde{G}e}$ are generated
by the Feynman diagrams of Fig. 2 and read

$\displaystyle
C_{ij}\simeq\frac{v^{2}}{\Lambda^{2}}\frac{y^{ii}_{S}y^{jj}_{P}}{m^{2}_{\phi}}~{},\qquad
C_{Ge}=\frac{4\pi}{m^{2}_{\phi}}\frac{v}{\Lambda^{2}}C_{g}y^{ee}_{P}~{},\qquad
C_{\tilde{G}e}=\frac{4\pi}{m^{2}_{\phi}}\frac{v}{\Lambda^{2}}\tilde{C}_{g}y^{ee}_{S}~{}.$
(40)

The last term of Eq. (39) represents the Weinberg operator which is generated
by the representative diagrams shown in Fig. 3. The related Wilson coefficient
$d_{G}$ reads 97

$\displaystyle d_{G}$
$\displaystyle\simeq\frac{g_{s}\alpha_{s}}{(4\pi)^{3}}\sum_{i=t,b}\\!\frac{v^{2}}{\Lambda^{2}}\frac{y^{ii}_{S}y^{ii}_{P}}{4m^{2}_{i}}+\frac{3g_{s}}{\pi^{2}}\frac{g_{s}^{4}C_{g}\tilde{C}_{g}}{\Lambda^{2}}\log\frac{\Lambda}{m_{\phi}}~{},$
(41)

where the first term refers to the two-loop diagram and holds for $m_{q}\gg
m_{\phi}$. Instead, the second term of Eq. (41) arises from the one-loop
diagrams of Fig. 3.

Figure 3: Leading contributions to the Weinberg operator. Figure from Ref. 97.
Figure 4: Leading contributions to the fermionic (C)EDMs. Figure from Ref. 97.

Finally, we analyse the fermionic (C)EDMs induced by ALP interactions. The
leading contributions stem from the Feynman diagrams reported in Fig. 4 and
read 103, 97

$\displaystyle\frac{d_{i}}{e}\simeq$
$\displaystyle-\sum_{k}\frac{Q_{k}}{16\pi^{2}}\frac{m_{k}}{m^{2}_{\phi}}\frac{v^{2}}{\Lambda^{2}}\mbox{Re}\,(y^{ik}_{S}y^{ki}_{P})\,\ell(x_{k})$
$\displaystyle-\sum_{k}\frac{N_{c}\,\alpha\,Q_{i}\,Q^{2}_{k}}{8\pi^{3}\,m_{k}}\frac{v^{2}}{\Lambda^{2}}\left(y^{ii}_{P}y^{kk}_{S}f(x_{k})+y^{ii}_{S}y^{kk}_{P}g(x_{k})\right)$
$\displaystyle-\frac{Q_{i}}{2\pi^{2}}\frac{v}{\Lambda^{2}}e^{2}(y^{ii}_{S}{\tilde{C}}_{\gamma}-C_{\gamma}y^{ii}_{P})\log\frac{\Lambda}{m_{\phi}}$
$\displaystyle-\frac{3\,\alpha\,Q^{3}_{i}}{\pi^{3}}\frac{m_{i}}{\Lambda^{2}}e^{4}C_{\gamma}{\tilde{C}}_{\gamma}\log^{2}\frac{\Lambda}{m_{\phi}}$
$\displaystyle-\delta_{qi}\frac{2\,\alpha_{s}\,Q_{i}}{\pi^{3}}\frac{m_{i}}{\Lambda^{2}}e^{2}g^{2}_{s}(C_{\gamma}{\tilde{C}}_{g}+C_{g}{\tilde{C}}_{\gamma})\log^{2}\frac{\Lambda}{m_{\phi}}\,,$
(42)

in the EDMs case (where $i=e,u,d$, $q=u,d$, and $N_{c}=3$) and

$\displaystyle d^{C}_{i}\simeq$
$\displaystyle-\sum_{k}\frac{1}{16\pi^{2}}\frac{m_{k}}{m^{2}_{\phi}}\frac{v^{2}}{\Lambda^{2}}\mbox{Re}\,(y^{ik}_{S}y^{ki}_{P})\,\ell(x_{k})$
$\displaystyle-\sum_{k}\frac{\alpha_{s}}{16\pi^{3}\,m_{k}}\frac{v^{2}}{\Lambda^{2}}\left(y^{ii}_{P}y^{kk}_{S}f(x_{k})+y^{ii}_{S}y^{kk}_{P}g(x_{k})\right)$
$\displaystyle-\frac{1}{2\pi^{2}}\frac{v}{\Lambda^{2}}g_{s}^{2}(y^{ii}_{S}{\tilde{C}}_{g}-C_{g}y^{ii}_{P})\log\frac{\Lambda}{m_{\phi}}$
$\displaystyle-\frac{4\,\alpha_{s}}{\pi^{3}}\frac{m_{i}}{\Lambda^{2}}g_{s}^{4}C_{g}{\tilde{C}}_{g}\log^{2}\frac{\Lambda}{m_{\phi}}$
$\displaystyle-\frac{3\,\alpha\,Q^{2}_{i}}{2\pi^{3}}\frac{m_{i}}{\Lambda^{2}}e^{2}g_{s}^{2}(C_{\gamma}{\tilde{C}}_{g}+C_{g}{\tilde{C}}_{\gamma})\log^{2}\frac{\Lambda}{m_{\phi}}\,,$
(43)

for the CEDMs (where $i=u,d$). The loop functions are
$\ell(x)=(3\\!-\\!4x\\!+\\!x^{2}\\!+\\!2\log x)/(1-x)^{3}$ where
$x_{k}=m^{2}_{k}/m^{2}_{\phi}$ and, in the limit $x\gg 1$, $f(x)\approx(6\log
x+13)/18$ and $g(x)\approx(\log x+2)/2$. The first diagram of Fig. 4 accounts
for flavour-violating effects (for flavour-diagonal effects, see 107) while
the second diagram refers to Barr-Zee two-loop contributions.

The above expressions have been obtained employing a hard cutoff as a UV
regulator and assuming no significant cancellations with finite terms.
Although Eqs. (33)-(3.1.1) include only leading-order short-distance effects,
the bounds of Table 1 have been derived accounting also for one-loop QCD
running effects (improved with a two-loop running of $\alpha_{s}$ and quark
masses) from $\Lambda=1$ TeV to $m_{\phi}=5\,$GeV and the running of
$\mathcal{L}_{\rm CPV}$ from $m_{\phi}$ to the hadronic scale $\mu_{\rm
had}=1$ GeV 108.

Constraints on $d_{e}$, $C_{ij}$ and $C_{Ge}$ in Eq. (39), are obtained by
using the polar molecule ThO. Indeed, the electron spin-precession frequency
$\omega_{\text{ThO}}$ is affected by both $d_{e}$ and CP-odd electron–nucleon
($N$) interactions 109

$\displaystyle\omega_{\text{ThO}}=1.2\,\mathrm{mrad}/\mathrm{s}\left(\frac{d_{e}}{10^{-29}\,e\,\mathrm{cm}}\right)+1.8\,\mathrm{mrad}/\mathrm{s}\left(\frac{C_{S}}{10^{-9}}\right)~{},$

where the theoretical error is within a few percent while the experimental
limit $\omega_{\text{ThO}}<1.3\,\mathrm{mrad}/\mathrm{s}$ ($90\%$ C.L.) 110.
The coefficient $C_{S}$ stemming from the interaction
$\mathcal{L}\supset-\frac{G_{F}}{\sqrt{2}}C_{S}\overline{N}N\overline{e}i\gamma_{5}e$
109 is related to $C_{ij}$ and $C_{Ge}$ as
$C_{S}/v^{2}\simeq-17(C_{ue}+C_{de})+4.7\,\text{GeV}\,C_{Ge}$. The neutron EDM
receives contributions from quark (C)EDMs, the Weinberg operator and 4-quark
operators 111, 112, 82

$\displaystyle d_{n}$
$\displaystyle=0.784(28)\,d_{u}-0.204(11)\,d_{d}-0.55(28)\,e\,d^{C}_{u}-1.10(55)\,e\,d^{C}_{d}$
$\displaystyle+50(40)\,{\rm MeV}\,e\,d_{G}+30(20)\,{\rm
MeV}\,e\,(C_{ud}-C_{du})~{},$ (44)

and the current experimental bound is $d_{n}<1.8\cdot 10^{-26}\,e~{}$cm
($90\%$ C.L.) 50, 51. Finally, the EDM of the diamagnetic atom 199Hg is
generated by both nuclear and leptonic CP-odd interactions 109, 112

$\displaystyle d_{\rm Hg}\simeq 4.0\cdot
10^{-4}d_{n}-[2.8\,C_{S}-2.1\,C_{P}]\,10^{-22}\,e\,{\rm cm}~{},$ (45)

with $C_{P}\simeq C^{(0)}_{P}-C^{(1)}_{P}$ defined through the Lagrangian
$\mathcal{L}\supset-\frac{G_{F}}{\sqrt{2}}\overline{N}(C^{(0)}_{P}+\tau_{3}C^{(1)}_{P})i\gamma_{5}N\overline{e}e$.
Our bounds are set by employing
$C_{P}/v^{2}=350(C_{eu}+C_{ed})+1.1\,\text{GeV}\,C_{\tilde{G}e}$ and the
experimental limit $d_{\rm Hg}<6\cdot 10^{-30}e~{}$cm ($90\%$ C.L.) 113.

CP-violating invariant | Bound | Observable
---|---|---
$|C_{\gamma}{\tilde{C}}_{\gamma}|$ | $6.2\times 10^{-3}$ | $\omega_{\rm ThO}(d_{e})$
$|C_{g}{\tilde{C}}_{g}|$ | $1.4\times 10^{-6}$ | $d_{n}(d_{G})$
$|C_{\gamma}{\tilde{C}}_{g}|$ | $0.40$ | $d_{\rm Hg}(C_{P},C_{S})$
$|C_{g}\tilde{C}_{\gamma}|$ | $2.3\times 10^{-3}$ | $\omega_{\rm ThO}(C_{S})$
$|y^{ee}_{S}{\tilde{C}}_{\gamma}-y^{ee}_{P}C_{\gamma}|$ | $6.9\times 10^{-11}$ | $\omega_{\rm ThO}(d_{e})$
$|y^{uu}_{S}{\tilde{C}}_{g}-y^{uu}_{P}C_{g}|$ | $8.1\times 10^{-9}$ | $d_{n}(d^{C}_{u})$
$|y^{dd}_{S}{\tilde{C}}_{g}-y^{dd}_{P}C_{g}|$ | $6.5\times 10^{-9}$ | $d_{n}(d^{C}_{d})$
$|y^{ee}_{P}C_{g}|$ | $2.1\times 10^{-11}$ | $\omega_{\rm ThO}(C_{S})$
$|y^{ee}_{S}\tilde{C}_{g}|$ | $7.3\times 10^{-9}$ | $d_{\rm Hg}(C_{P})$
$|y^{uu}_{S}y^{dd}_{P}-y^{dd}_{S}y^{uu}_{P}|$ | $5.6\times 10^{-9}$ | $d_{\rm Hg}(C_{ud}-C_{du})$
$|y^{ee}_{S}y^{uu}_{P}|$, $|y^{ee}_{S}y^{dd}_{P}|$ | $4.2\times 10^{-13}$ | $d_{\rm Hg}(C_{P})$
$|y^{uu}_{S}y^{ee}_{P}|$, $|y^{dd}_{S}y^{ee}_{P}|$ | $2.1\times 10^{-13}$ | $\omega_{\rm ThO}(C_{S})$
$|y^{tt}_{S}y^{ee}_{P}|$ | $6.8\times 10^{-9}$ | $\omega_{\rm ThO}(C_{S})$
$|y^{bb}_{S}y^{ee}_{P}|$ | $1.7\times 10^{-10}$ | $\omega_{\rm ThO}(C_{S})$
$|y^{ee}_{S}y^{tt}_{P}|$ | $8.8\times 10^{-9}$ | $\omega_{\rm ThO}(d_{e})$
$|y^{ee}_{S}y^{bb}_{P}|$ | $3.9\times 10^{-9}$ | $d_{\rm Hg}(C_{P})$
$|y^{tt}_{S}y^{tt}_{P}|$ | $0.10$ | $d_{n}(d_{G})$
$|y^{bb}_{S}y^{bb}_{P}|$ | $5.9\times 10^{-5}$ | $d_{n}(d_{G})$
$|y^{ee}_{S}\,y^{ee}_{P}|$ | $1.0\times 10^{-10}$ | $\omega_{\rm ThO}(d_{e})$
$|y^{e\mu}_{S}\,y^{\mu e}_{P}|$ | $2.2\times 10^{-12}$ | $\omega_{\rm ThO}(d_{e})$
$|y^{e\tau}_{S}\,y^{\tau e}_{P}|$ | $3.1\times 10^{-12}$ | $\omega_{\rm ThO}(d_{e})$
$|y^{uu}_{S}\,y^{uu}_{P}|$ | $3.9\times 10^{-8}$ | $d_{n}(d^{C}_{u})$
$|y^{uc}_{S}\,y^{cu}_{P}|$ | $3.2\times 10^{-9}$ | $d_{n}(d^{C}_{u})$
$|y^{ut}_{S}\,y^{tu}_{P}|$ | $3.2\times 10^{-7}$ | $d_{n}(d^{C}_{u})$
$|y^{dd}_{S}\,y^{dd}_{P}|$ | $4.6\times 10^{-8}$ | $d_{n}(d^{C}_{d})$
$|y^{ds}_{S}\,y^{sd}_{P}|$ | $6.4\times 10^{-9}$ | $d_{n}(d^{C}_{d})$
$|y^{db}_{S}\,y^{bd}_{P}|$ | $1.7\times 10^{-8}$ | $d_{n}(d^{C}_{d})$
|  |

Table 1: Bounds on CP-violating invariants for $\Lambda=1$ TeV and
$m_{\phi}=5$ GeV. In the 3rd column we specify the observable and the leading
operator setting the bound (in brackets).

The EDM’s sensitivity to the CP-violating invariants defined in Eq. (32) are
reported in Table 1 for $m_{\phi}=5$ GeV. As we can see, 4-fermion operators
set the strongest bounds on several CP invariants. Moreover, the Weinberg
operator is particularly effective in constraining ALP couplings to top and
bottom quarks as well as to gluons. Interestingly, EDMs put severe bounds also
to flavor-violating sources of CP violation that are competitive with those
stemming from flavour-changing neutral current processes. Even if the relative
impact of the above contributions on the EDMs will depend on the specific ALP
model, it is instructive to consider a realistic example where
$y^{ii}_{S,P}\propto\frac{m_{i}}{v}$, $C_{\gamma(g)}$ and
$\tilde{C}_{\gamma(g)}\propto\frac{1}{16\pi^{2}}$. In this case, 4-fermion
operators are the best probes of CP violation followed by the electron EDM and
the Weinberg operator which show similar sensitivities. Let us also remark
that the electron EDM bounds are always stronger than the limits from the
anomalous magnetic moment of the electron, unless CP-violating phases are
smaller than about $10^{-4}$ 114. Future experimental projections are
particularly promising. The experimental bound on the neutron EDM measurement
should reach the level of $d_{n}\lesssim 10^{-28}e~{}$cm 102. There are also
plans to measure the proton and deuteron EDMs in electromagnetic storage rings
at the level of $d_{p,D}\lesssim 10^{-29}e~{}$cm 102. Furthermore, we expect
also one order of magnitude improvement on the current measurement of
molecular systems, e.g. the polar molecule ThO, which currently sets the most
powerful bounds on the electron EDM and electron-nucleon couplings. If the
above forecasts are respected, the limits on $d_{G}$ and quark (C)EDMs will
improve by roughly three orders of magnitude while the constraints on electron
EDM and 4-fermion contributions will be strengthened by one order of
magnitude.

###  3.2 CP-violating ALP Lagrangian: $m_{\phi}\lesssim 1$ GeV

In this section, we are going to extend the previous analysis to ALP masses in
the sub-GeV region, where non-perturbative methods, such as chiral
perturbation theory ($\chi$pt), have to be employed. The construction of the
effective chiral Lagrangian for an ALP interacting with photons and light
pseudoscalar mesons has been already discussed at length in the literature for
the case of a CP-odd ALP (see Refs. 29, 115, 116, 117, as well as 118, 119,
120, 121, 122 for analyses beyond leading order in $\chi\text{pt}$). The
development of the chiral Lagrangian for CP-even scalar interactions was
elaborated instead in Ref. 123. While we refer to Ref. 98 for the construction
of the most general chiral Lagrangian at $\mathcal{O}(p^{2})$ describing CP-
violating interactions of a light ALP with mesons and baryons, we report here
the main results needed to estimate the ALP contribution to the EDMs.

To match the notation of Ref. 98 we start from a slightly different version of
the ALP effective Lagrangian, compared to the one in Eq. (3.1), featuring a
derivative ALP coupling to the quark axial current, namely

$\displaystyle\mathcal{L}_{\phi}^{\text{QCD}}$
$\displaystyle=e^{2}\frac{\tilde{C}_{\gamma}}{\Lambda}\phi
F\tilde{F}+g_{s}^{2}\frac{\tilde{C}_{g}}{\Lambda}\phi
G\tilde{G}+\frac{\partial_{\mu}\phi}{\Lambda}\overline{q}\,\gamma^{\mu}Y_{P}\gamma_{5}\,q$
$\displaystyle+e^{2}\frac{C_{\gamma}}{\Lambda}\phi
FF+g_{s}^{2}\frac{C_{g}}{\Lambda}\phi
GG+\frac{v}{\Lambda}\,\phi\,\overline{q}\,y_{S}\,q~{},$ (46)

that is valid above the GeV scale. Here, $q^{T}=(u,d)$, while $Y_{P}$ and
$y_{S}$ are hermitian matrices.

Upon taking into account hadronization effects, following a chiral
representation of the ALP interactions with quarks and gluons, Ref. 98
obtained a low-energy effective Lagrangian describing the interaction of the
ALP with light particles: photons, leptons, pions and nucleons. The ALP
effective Lagrangian can be cast in two contributions displaying opposite CP
transformation properties, given respectively by (in the case of two active
flavours, $N_{f}=2$) 98

$\displaystyle\mathcal{L}_{\phi,\text{\,CP-even}}^{\chi\text{pt}}$
$\displaystyle=e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,FF+\frac{v}{\Lambda}y_{\ell,S}^{ij}\,\phi\,\overline{\ell}_{i}\ell_{j}+\kappa\,\frac{\phi}{\Lambda}\,\left[\partial^{\mu}\pi_{0}\partial_{\mu}\pi_{0}+2\partial^{\mu}\pi^{+}\partial_{\mu}\pi^{-}\right]$
$\displaystyle+\frac{\phi}{\Lambda}\,(\pi_{0}^{2}+2\pi^{+}\pi^{-})m_{\pi}^{2}\left[\frac{v}{2}\frac{\mathcal{Z}_{u}+\mathcal{Z}_{d}}{m_{u}+m_{d}}-2\,\kappa\right]+\frac{\phi}{\Lambda}C_{\phi\text{NN}}\overline{N}N$
(47)

and

$\displaystyle\mathcal{L}_{\phi,\text{\,CP-odd}}^{\chi\text{pt}}$
$\displaystyle=e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F\tilde{F}+i\frac{v}{\Lambda}y_{\ell,P}^{ij}\,\phi\,\overline{\ell}_{i}\gamma_{5}\ell_{j}+\frac{\phi}{\Lambda}\,\pi_{0}(\pi_{0}^{2}+2\pi^{+}\pi^{-})\left[\frac{m_{\pi}^{2}m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta^{A}_{ud}}{6f_{\pi}\Lambda}\right]$
$\displaystyle+\frac{2}{3}\frac{\Delta^{A}_{ud}}{f_{\pi}\Lambda}\frac{m_{\pi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\,\partial^{\mu}\phi\,(\pi_{0}\pi^{-}\partial_{\mu}\pi^{+}+\pi_{0}\pi^{+}\partial_{\mu}\pi^{-}-2\pi^{+}\pi^{-}\partial_{\mu}\pi^{0})$
$\displaystyle+\frac{\partial_{\mu}\phi}{\Lambda}\overline{N}\tilde{C}_{\phi\text{N}}\gamma^{\mu}\gamma_{5}N~{},$
(48)

where explicit expressions for the parameters appearing in the above
Lagrangians in terms of the short-distance parameters of Eq. (3.2) are
reported below in Eq. (3.2). In particular, the physical ALP mass after
diagonalization of the ALP-pion system reads 98

$m^{2}_{\phi}=M_{\phi}^{2}+m_{\pi}^{2}(\lambda_{g})^{2}\frac{m_{u}m_{d}}{(m_{u}+m_{d})^{2}}\frac{f_{\pi}^{2}}{\Lambda^{2}}+m_{\pi}^{2}\frac{M_{\phi}^{4}}{(m_{\pi}^{2}-M_{\phi}^{2})^{2}}(\Delta^{A}_{ud})^{2}\frac{f_{\pi}^{2}}{\Lambda^{2}}~{},$
(49)

where $M_{\phi}$ is the bare ALP mass and
$\lambda_{g}=32\pi^{2}\tilde{C}_{g}$.

| $c_{\gamma}$ | $y_{\ell,S}$ | $\kappa$ | $\mathcal{Z}$ | $C_{\phi\text{NN}}$
---|---|---|---|---|---
$\tilde{c}_{\gamma}$ | $\tilde{c}_{\gamma}\,c_{\gamma}$ | $\tilde{c}_{\gamma}\,y_{\ell,S}$ | $\tilde{c}_{\gamma}\,\kappa$ | $\tilde{c}_{\gamma}\,\mathcal{Z}$ | $\tilde{c}_{\gamma}\,C_{\phi\text{NN}}$
$y_{\ell,P}$ | $y_{\ell,P}\,c_{\gamma}$ | $y_{\ell,P}\,y_{\ell,S}$ | $y_{\ell,P}\,\kappa$ | $y_{\ell,P}\,\mathcal{Z}$ | $y_{\ell,P}\,C_{\phi\text{NN}}$
$\Delta_{ud}^{A}$ | $\Delta_{ud}^{A}\,c_{\gamma}$ | $\Delta_{ud}^{A}\,y_{\ell,S}$ | $\Delta_{ud}^{A}\,\kappa$ | $\Delta_{ud}^{A}\,\mathcal{Z}$ | $\Delta_{ud}^{A}\,C_{\phi\text{NN}}$
$\tilde{C}_{\phi\text{N}}$ | $\tilde{C}_{\phi\text{N}}\,c_{\gamma}$ | $\tilde{C}_{\phi\text{N}}\,y_{\ell,S}$ | $\tilde{C}_{\phi\text{N}}\,\kappa$ | $\tilde{C}_{\phi\text{N}}\,\mathcal{Z}$ | $\tilde{C}_{\phi\text{N}}\,C_{\phi\text{NN}}$
Table 2: Jarlskog invariants emerging from the interactions in Eq. (3.2) and
(3.2).

Note that, independently of the scalar or pseudo-scalar nature of the ALP
field, CP violation is unavoidable as long as at least one CP-even and one CP-
odd couplings from Eq. (3.2) and (3.2) are simultaneously non-vanishing. CP-
violating effects can be described in terms of low-energy Jarlsokg invariants,
which are reported in Table 2. In particular, the coefficients appearing in
Table 2 are related to the microscopic parameters of Eq. (3.2) via the
following relations 98

$\displaystyle c_{\gamma}$
$\displaystyle=C_{\gamma}-\frac{\beta^{0}_{\text{QED}}}{\beta^{0}_{\text{QCD}}}C_{g}~{},\qquad\tilde{c}_{\gamma}=\,\tilde{C}_{\gamma}-4N_{c}\text{
tr
}(Q_{A}Q_{q}^{2})\tilde{C}_{g}~{},\qquad\kappa=\frac{8\pi}{\alpha_{s}}\frac{g_{s}^{2}C_{g}}{\beta^{0}_{\text{QCD}}}~{},$
$\displaystyle\mathcal{Z}$
$\displaystyle=y_{S}+\frac{g_{s}^{2}C_{g}}{\beta^{0}_{\text{QCD}}}\frac{8\pi}{\alpha_{s}}\frac{M_{q}}{v}~{},\qquad\Delta^{A}_{ud}=Y_{P}^{u}-Y_{P}^{d}-16\pi^{2}\tilde{C}_{g}\frac{m_{d}-m_{u}}{m_{d}+m_{u}}~{},$
$\displaystyle\tilde{C}_{\phi\text{p}}$
$\displaystyle=Y_{P}^{u}\Delta_{u}+Y_{P}^{d}\Delta_{d}-16\pi^{2}\tilde{C}_{g}\left[\frac{m_{u}\Delta_{d}}{m_{u}+m_{d}}+\frac{m_{d}\Delta_{u}}{m_{u}+m_{d}}-\frac{m_{\phi}^{2}\Delta^{A}_{ud}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta_{u}-\Delta_{d}}{32\pi^{2}\tilde{C}_{g}}\right]~{},$
$\displaystyle\tilde{C}_{\phi\text{n}}$
$\displaystyle=Y_{P}^{u}\Delta_{d}-Y_{P}^{d}\Delta_{u}-16\pi^{2}\tilde{C}_{g}\left[\frac{m_{u}\Delta_{u}}{m_{u}+m_{d}}+\frac{m_{d}\Delta_{d}}{m_{u}+m_{d}}+\frac{m_{\phi}^{2}\Delta^{A}_{ud}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta_{u}-\Delta_{d}}{32\pi^{2}\tilde{C}_{g}}\right]~{},$
$\displaystyle C_{\phi\text{pp}}$
$\displaystyle=\frac{y_{q,S}^{u}\,v}{m_{u}}\,\sigma_{u}+\frac{y_{q,S}^{d}\,v}{m_{d}}\,\sigma_{d}+\frac{y_{q,S}^{s}\,v}{m_{s}}\,\sigma_{s}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})~{},$
$\displaystyle C_{\phi\text{nn}}$
$\displaystyle=\frac{y_{q,S}^{u}\,v}{m_{d}}\,\sigma_{d}+\frac{y_{q,S}^{d}\,v}{m_{u}}\,\sigma_{u}+\frac{y_{q,S}^{s}\,v}{m_{s}}\,\sigma_{s}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})~{},$
(50)

with the low-energy QCD matrix elements given by $\Delta_{u}=0.858(22)$ and
$\Delta_{d}=-0.418(22)$ 124, 125, while the values of $\sigma_{u}$,
$\sigma_{d}$ and $\sigma_{s}$ can be found e.g. in Ref. 126.

####  3.2.1 ALP contribution to EDMs

The effective Lagrangian provided in Eqs. (3.2)–(3.2) readily allows one to
compute the contributions of a light ALP to the proton EDM, with the relevant
topologies displayed in Fig. 5. Following the standard definition of the EDM
of a fermion given in Eq. (39), one finds 98

$\displaystyle d_{p}$
$\displaystyle\simeq-\frac{e\,Q_{p}}{4\pi^{2}\Lambda^{2}}\left[C_{\phi\text{pp}}\tilde{C}_{\phi\text{p}}+e^{2}m_{p}c_{\gamma}\tilde{C}_{\phi\text{p}}\left(6+2\ln\frac{\Lambda_{\text{ren}}^{2}}{m_{p}^{2}}\right)+e^{2}\tilde{c}_{\gamma}C_{\phi\text{pp}}\left(2+\ln\frac{\Lambda_{\text{ren}}^{2}}{m_{p}^{2}}\right)\right.$
$\displaystyle\left.+3\frac{Q_{p}^{2}}{\pi^{2}}m_{p}e^{6}c_{\gamma}\tilde{c}_{\gamma}\ln^{2}\frac{\Lambda_{\text{ren}}}{m_{\phi}}\right]~{},$
(51)

in the limit $m_{\phi}\ll m_{p}$ and setting the renormalization scale
$\Lambda_{\rm ren}\simeq m_{p}$.

Figure 5: Feynman diagrams contributing to the ALP-mediated effects to the
proton EDM, $d_{p}$. Figure from Ref. 98.

Due to their lack of a minimal coupling to the electromagnetic field-strength
tensor, neutrons cannot develop an EDM at leading order in $\chi$pt. However,
they do display non-minimal couplings to photons at next-to-leading order in
$\chi$pt 127, 128 which can eventually induce a non-zero value for $d_{n}$. In
the $N_{f}=2$ case, the relevant relativistic Lagrangian reads

$\mathcal{L}_{\gamma\text{N}}^{\text{NLO}}=\frac{F_{\mu\nu}}{4m_{N}}\left[\,\mathcal{C}_{p}\,\overline{p}\sigma^{\mu\nu}p+\mathcal{C}_{n}\,\overline{n}\sigma^{\mu\nu}n\,\right]~{},$
(52)

where $\mathcal{C}_{p}\simeq 1.79$ 125 and $\mathcal{C}_{n}\simeq 1.91$ 125
are measured low-energy constants. Given these interactions, one can then
compute the corresponding contribution to $d_{n}$, whose Feynman diagrams are
the same as for the proton case, finding the following order of magnitude
estimate 98

$d_{n}\sim-\frac{e\,\mathcal{C}_{n}}{4\pi^{2}\Lambda^{2}}\left[-\frac{3}{8}C_{\phi\text{nn}}\tilde{C}_{\phi\text{n}}\frac{\Lambda_{\text{ren}}^{2}}{m_{n}^{2}}+e^{2}(2m_{n}c_{\gamma}\tilde{C}_{\phi\text{n}}+3\tilde{c}_{\gamma}C_{\phi\text{nn}})\ln\frac{\Lambda_{\text{ren}}^{2}}{m_{n}^{2}}\right]~{}.$
(53)

Note that $d_{p}$ and $d_{n}$ have a comparable size while their current
experimental bounds read $d_{p}<2.1\times 10^{-25}\,e$ cm 129 and
$d_{n}<1.8\times 10^{-26}\,e$ cm (90% C.L.) 50, 51.

Constraints on the low-energy Jarlskog invariants are obtained by imposing the
existing bounds on $d_{p}$ and $d_{n}$. Turning on just two couplings at a
time, one finds the limits displayed in Table 3.

| $y_{S}^{u}$ | $y_{S}^{d}$ | $C_{\gamma}$ | $C_{g}$
---|---|---|---|---
$y_{P}^{u}$ | $2.8\times 10^{-12}$ | $1.8\times 10^{-12}$ | $8.0\times 10^{-8}$ | $1.1\times 10^{-10}$
$y_{P}^{d}$ | $9.9\times 10^{-12}$ | $6.4\times 10^{-12}$ | $1.1\times 10^{-7}$ | $3.8\times 10^{-10}$
$\tilde{C}_{\gamma}$ | $1.2\times 10^{-6}$ | $1.8\times 10^{-6}$ | $4.8\times 10^{-3}$ | $7.1\times 10^{-5}$
$\tilde{C}_{g}$ | $1.1\times 10^{-9}$ | $7.4\times 10^{-10}$ | $2.4\times 10^{-5}$ | $4.4\times 10^{-8}$
Table 3: Upper limits on Jarlskog invariants obtained from the bounds on the
neutron and proton EDMs assuming $\Lambda=1$ TeV 98. To make contact with the
notation in Eq. (3.1) here we have defined
$y_{P}^{q}=-2\frac{m_{q}}{v}Y_{P}^{q}$.

###  3.3 Towards a UV completion for the CP-violating ALP

It is interesting to speculate on the possible origin of the CP-violating ALP
effective Lagrangian in Eq. (3.1). The underlying UV dynamics can be naturally
conceived in the framework of strongly-coupled theories. A paradigmatic
example is given by the SM itself, when considering the effective interactions
of the neutral pion $\pi^{0}$ below the GeV scale (see e.g. 130). In this
setup, the role of the ALP is played by $\pi^{0}$, quark masses induce the
breaking of the shift symmetry while the QCD $\theta$ term provides the
necessary CP-violating source. Finally, electromagnetic interactions act as
mediators from the strong sector to the $\pi^{0}$. CP-odd pion interactions
include the terms

$A_{1}\frac{\pi^{0}}{f_{\pi}}F\tilde{F}+A_{2}\frac{\partial_{\mu}\pi^{0}}{f_{\pi}}\overline{e}\gamma^{\mu}\gamma_{5}e~{},\\\
$ (54)

where $A_{1}=\frac{\alpha}{4\pi}$ is the Wess-Zumino-Witten term and $A_{2}$
is radiatively induced by $A_{1}$ via electromagnetic interactions, so that
$A_{2}\sim(\frac{\alpha}{4\pi})^{2}$. CP-even pion interactions

$C_{1}\frac{\pi^{0}}{f_{\pi}}FF+C_{2}m_{e}\frac{\pi^{0}}{f_{\pi}}\overline{e}e~{},$
(55)

are instead sourced by the QCD $\theta$ term. In particular, $C_{1}$ emerges
from integrating out heavy charged-baryon loops. The CP-even Yukawa-like
couplings between pions and baryons appearing in such triangle loops naturally
emerge from the baryonic Lagrangian upon including the $\theta$ term in the
phase of the quark mass matrix. $C_{2}$ is in turn generated radiatively from
$C_{1}$ via electromagnetic interactions and thus scales as
$C_{2}\sim\theta\frac{\alpha}{4\pi}$. Strongly coupled new physics scenarios
emerging at the scale $\Lambda\gtrsim 1$ TeV, that mimic the pion dynamics of
the SM, can hence be conceived in analogy.

Other interesting scenarios motivating the study of a CP-violating ALP are
provided by relaxion models 131, aiming at solving the hierarchy problem
through an ALP field, the relaxion, which scans the Higgs boson mass in the
early Universe from a very high-energy scale $\mu\gg 1$ TeV down to the final
value of $\mathcal{O}(v)$. The presence of both the relaxion-Higgs mixing and
the relaxion-photon/gluon couplings does violate CP. As an explicit example
let us consider the following relaxion Lagrangian

$\displaystyle\mathcal{L}_{\text{Relax}}$
$\displaystyle=\frac{1}{2}\partial^{\mu}\Phi\partial_{\mu}\Phi-\frac{\Phi}{\Lambda}\left(\tilde{\mathcal{C}}_{g}\,g_{s}^{2}\,G\tilde{G}+\tilde{\mathcal{C}}_{W}\,g^{2}\,W\tilde{W}+\tilde{\mathcal{C}}_{B}\,g^{\prime
2}\,B\tilde{B}\right)$
$\displaystyle+\frac{\partial^{\mu}\Phi}{\Lambda}\sum_{f}\overline{\psi}_{f}\gamma_{\mu}\tilde{X}_{f}\psi_{f}-V(H,\Phi)~{},$
(56)

where $\Phi$ is the relaxion field and
$f=\left\\{\ell_{L},q_{L},u_{R},d_{R},e_{R}\right\\}$ are the chiral SM
fermions, and the relaxion-Higgs potential is given by (see e.g. 131, 132,
133, 134)

$\displaystyle V(H,\Phi)$
$\displaystyle=\left[-\mu^{2}+\tilde{g}\mu\Phi\right]|H^{\dagger}H|-M^{2}|H^{\dagger}H|\cos\left(\frac{\Phi}{\Lambda}\right)$
$\displaystyle+\lambda\,|H^{\dagger}H|^{2}+r\tilde{g}\mu^{3}\Phi+(m_{\phi}^{0})^{2}\Phi^{2}+\dots~{}.$
(57)

Of these terms, the linear one in $\Phi$ prompts the cosmological evolution of
the relaxion field, which allows for scanning of the Higgs VEV via the
relaxion-dependent Higgs mass parameter. The cosine term acts instead as a
backreaction, producing the potential barrier that eventually halts the
evolution of the relaxion field around the Higgs VEV. Expanding both the Higgs
and relaxion fields around their VEV,

$\Phi=\phi_{0}+\phi^{\prime}\qquad\text{and}\qquad
H=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ v+h^{\prime}\end{pmatrix}~{},$ (58)

generates in turn a mixing between the Higgs and the relaxion fields, with the
mass eigenstates, $\phi=c_{\theta}\phi^{\prime}+s_{\theta}h^{\prime}$ and
$h=c_{\theta}h^{\prime}-s_{\theta}\phi^{\prime}$, given in terms of the mixing
angle
$s_{\theta}\simeq\frac{1}{2\lambda}\frac{M^{2}}{v^{2}}\frac{v}{\Lambda}\sin\frac{\phi_{0}}{\Lambda}\ll
1$. The resulting interaction Lagrangian in the mass eigenbasis at
$\mathcal{O}(1/\Lambda)$ is given by

$\displaystyle\mathcal{L}_{\text{Relax}}$ $\displaystyle\supset
c_{\theta}\frac{\partial_{\mu}\phi}{\Lambda}\sum_{f}\overline{f}\,\gamma^{\mu}(X_{f}^{V}+X_{f}^{A}\gamma_{5})f-s_{\theta}\frac{\phi}{v}\sum_{f}\frac{m_{f}}{v}\overline{f}f$
$\displaystyle-
c_{\theta}\frac{\phi}{\Lambda}\Bigg{[}\tilde{\mathcal{C}}_{g}\,g_{s}^{2}\,G\tilde{G}+e^{2}\Big{(}\tilde{\mathcal{C}}_{\gamma}F\tilde{F}+2\frac{\tilde{\mathcal{C}}_{\gamma
Z}}{s_{w}c_{w}}F\tilde{Z}+\frac{\tilde{\mathcal{C}}_{Z}}{s_{w}^{2}c_{w}^{2}}Z\tilde{Z}\Big{)}$
$\displaystyle+\,\tilde{\mathcal{C}}_{W}\,g^{2}\,\left(W^{+\mu\nu}\tilde{W}^{-}_{\mu\nu}-2ig(s_{w}A_{\nu}+c_{w}Z_{\nu})W_{\mu}^{+}\tilde{W}^{-\mu\nu}\right.$
$\displaystyle\left.+2igW_{\mu}^{+}W_{\nu}^{-}(c_{w}\tilde{F}^{\mu\nu}+s_{w}\tilde{Z}^{\mu\nu})+\text{h.c.}\right)\Bigg{]}+s_{\theta}\frac{\phi}{v}\left(\frac{2}{3}\mathcal{C}_{g}\,g_{s}^{2}\,GG+2\mathcal{C}_{\gamma}e^{2}FF\right)$
$\displaystyle-\left(\varphi^{+}\varphi^{-}+\frac{\varphi_{Z}^{2}}{2}\right)\left[c_{\theta}\frac{\phi}{\Lambda}M^{2}\sin\frac{\phi_{0}}{\Lambda}+2s_{\theta}\lambda\phi
v+2c_{\theta}s_{\theta}\phi h\right]$
$\displaystyle+2s_{\theta}\frac{\phi}{v}\left(\\!m_{W}^{2}W^{+}_{\mu}W^{-\mu}\\!+\frac{m_{Z}^{2}}{2}Z_{\mu}Z^{\mu}\right)\\!+2s_{\theta}c_{\theta}\frac{\phi
h}{v^{2}}\left(\\!m_{W}^{2}W_{\mu}^{+}W^{-\mu}\\!+\frac{m_{Z}^{2}}{2}Z_{\mu}Z^{\mu}\right)$
$\displaystyle-3\lambda c_{\theta}^{2}s_{\theta}h^{2}\phi-\lambda
c_{\theta}^{3}s_{\theta}h^{3}\phi-c_{\theta}^{3}\frac{\phi
h^{2}}{\Lambda}M^{2}\sin\frac{\phi_{0}}{\Lambda}+\mathcal{L}_{\text{Higgs}}^{\text{kin}}(c_{\theta}h\rightarrow
s_{\theta}\phi)~{},$ (59)

where $\mathcal{L}_{\text{Higgs}}^{\text{kin}}$ is the Lagrangian piece
containing all the interactions between the Higgs boson, the electroweak gauge
bosons and the respective Goldstones that stem from the Higgs kinetic term.
Moreover, we have defined
$2X_{V/A}^{f}=(K_{f}^{\dagger}\tilde{X}_{R}^{f}K_{f}\,\pm\,U_{f}^{\dagger}\tilde{X}_{L}^{f}U_{f})$,
where $K_{f}$ and $U_{f}$ are the right-handed and left-handed fermion
rotation matrices needed to go from the interaction basis to the mass basis.

The Lagrangian in Eq. (3.3) has then to be evolved down to the QCD scale. This
procedure includes both integrating out heavy particle states and the running
of Wilson coefficients 100. Finally, the resulting Lagrangian can be matched
onto the effective one in Eq. (3.2), yielding

$\displaystyle C_{\gamma}$
$\displaystyle=2\,s_{\theta}\,\mathcal{C}_{\gamma}\frac{\Lambda}{v}~{},\qquad
C_{g}=\frac{2}{3}\,s_{\theta}\,\mathcal{C}_{g}\frac{\Lambda}{v}~{},$ (60)
$\displaystyle\tilde{C}_{\gamma}$
$\displaystyle=-2\,c_{\theta}\,\tilde{\mathcal{C}}_{\gamma}~{},\qquad\,\tilde{C}_{g}=-\,c_{\theta}\,\tilde{\mathcal{C}}_{g}~{},$
(61) $\displaystyle y_{s}$
$\displaystyle=-s_{\theta}\,\frac{M_{q}}{v}\frac{\Lambda}{v}~{},\quad\,Y_{P}=c_{\theta}\,\frac{K_{f}^{\dagger}\tilde{X}_{R}^{f}K_{f}-U_{f}^{\dagger}\tilde{X}_{L}^{f}U_{f}}{2}~{}.$
(62)

##  4 Conclusions

In this review, we explored the theoretical landscape of CP-violating axions,
delving into both the well-established framework of QCD axions and the broader
setup of ALPs. Our discussion centred around two main themes: the implications
of CP violation for axion-mediated forces in the context of QCD axions and the
effective interactions of CP-violating ALPs, emphasizing their contributions
to EDMs.

The structural sensitivity of the QCD axion to high-energy sources of CP
violation, as shown by the estimate in Eq. (11), suggests rethinking the role
of the axion, to be considered not just as a “laundry soap” of CP violation in
strong interactions, but rather as a low-energy portal to high-energy sources
of CP violation, thus turning the “strong CP problem” into the “strong CP
opportunity”. In particular, axion-mediated forces, which are enhanced by the
presence of a CP-violating axion coupling to nucleons, provide an extra
experimental handle for probing UV sources of CP violation in the quark
sector, with projected sensitivities that are stronger (in a given range of
axion masses, cf. Fig. 1) than current EDM limits. Moreover, if the origin of
CP-violating axion couplings does not stem from QCD-coupled physics, like e.g.
in the presence of PQ-breaking operators, the CP-violating axion potential is
unsuppressed above the temperatures of the QCD crossover, and the relative
enhancement compared to the CP-preserving QCD potential might lead to sizeable
deviations in the axion relic abundance from the misalignment mechanism.
However, as discussed in Sect. 2.4, the relative enhancement is only
sufficient to achieve such deviation in the regime in which the axion provides
a sub-fraction of dark matter, unless tuning is invoked to relax the nEDM
constraint.

In the second part, we reviewed the contribution of CP-violating ALP
interactions to EDMs of nucleons, nuclei, atoms and molecules, following two
recent works 97, 98, respectively for the case of ALP masses above and below
the GeV scale. In the latter case, we employed a chiral Lagrangian approach to
parametrize the general CP-violating interactions of the ALP with pions and
nucleons.

The analysis of CP-violating ALPs could be expanded in several directions. For
instance, it could be worth to construct explicit UV completions of the CP-
violating ALP effective Lagrangian, which were briefly sketched in Sect. 3.3.
Another interesting path would be to investigate whether a successful
baryogenesis mechanism can be driven by an ALP field 135, 136, 137, 138, 139,
thus providing a rationale for exploring CP-violating ALP interactions.

## Acknowledgments

The work of LDL and PP is supported by the European Union – Next Generation EU
and by the Italian Ministry of University and Research (MUR) via the PRIN 2022
project n. 2022K4B58X – AxionOrigins. The work of LDL, HG and PS is funded by
the European Union – Next Generation EU and by the University of Padua under
the 2021 STARS Grants@Unipd programme (Acronym and title of the project: CPV-
Axion – Discovering the CP-violating axion).

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